ASTRO 101: Black Holes Curtis Brown, Jeanette Gladstone, Ross Lockwood, & Sharon Morsink Updated September 20, 2021 Contents 1 Introduction to Black Holes 9 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Black Holes in Pop Culture . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.1 Star Trek: The Future Begins . . . . . . . . . . . . . . . . . . . . . . 11 1.2.2 Interstellar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.3 Disney’s The Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Connecting Gravity and Light . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.1 Gravity and Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.2 Melanoheliophobia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Getting on the Same Wavelength . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.1 Colours and Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.2 Light Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4.3 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4.4 Waves or Particles? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5 Working with Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5.1 Making Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5.2 Measuring Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5.3 Using Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.6 Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6.1 Shifting with sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6.2 Shifting with light . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6.3 astro-shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.7 Newtonian Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.7.1 Who is Newton? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.7.2 Universal Law of Gravitation . . . . . . . . . . . . . . . . . . . . . . 24 1.7.3 Earth Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.7.4 Newton’s Laws of Motion . . . . . . . . . . . . . . . . . . . . . . . . 26 1.7.5 Acceleration Due to Gravity . . . . . . . . . . . . . . . . . . . . . . 27 1.8 Escape Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.8.1 Interlude on weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.8.2 Weightlessness and Freefall . . . . . . . . . . . . . . . . . . . . . . . . 27 1 1.8.3 Escape Velocity in Modern Spaceflight . . . . . . . . . . . . . . . . . 28 1.8.4 Escape Velocity Formula . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.9 Dark Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.9.1 John Michell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.9.2 Trapping Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.9.3 Dark Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.10 What is a Black Hole? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.10.1 Dark Star versus Black Hole . . . . . . . . . . . . . . . . . . . . . . . 32 1.10.2 What isn’t a Black Hole? . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.10.3 Are Black Holes Dangerous? . . . . . . . . . . . . . . . . . . . . . . . 34 1.10.4 Black Holes in Manga? (Interview with Dr. Mimi Okabe) . . . . . . . 35 1.11 Black Hole Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2 Life and Death of a Star 38 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 The Stellar Nursery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.1 What is a star? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.2 Forming Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2.3 Where are the Sun’s siblings? (Interview with Dr. Erik Rosolowsky) . 40 2.3 Now That’s a Stellar Sequence! . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.1 The Main Sequence of Stars . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.2 Where Does Star “Colour” Come From? . . . . . . . . . . . . . . . . 41 2.3.3 Day and Night . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4 Energy Production in Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4.1 Building and Smashing Atoms . . . . . . . . . . . . . . . . . . . . . . 44 2.4.2 Chemical and Nuclear Reactions . . . . . . . . . . . . . . . . . . . . . 45 2.4.3 Nuclear Fusion in the Sun . . . . . . . . . . . . . . . . . . . . . . . . 46 2.4.4 The Iron Catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.5 Energy Loss in Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.5.1 The Hundred Thousand Year Journey . . . . . . . . . . . . . . . . . 47 2.6 The Sun’s Light and Life on Earth . . . . . . . . . . . . . . . . . . . . . . . 49 2.6.1 Safety Owl’s Message . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.6.2 Goldilocks and the Three Stars . . . . . . . . . . . . . . . . . . . . . 51 2.6.3 What is a Super-Earth? (Interview with Dr. Kelsey Hoffman) . . . . 52 2.7 End of a Star’s Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.7.1 The Main Sequence isn’t forever . . . . . . . . . . . . . . . . . . . . . 52 2.7.2 Moving on up... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.7.3 Becoming Giant...er . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.7.4 Burning Fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.7.5 Nurseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.8 Life After the Death of Low Mass Stars . . . . . . . . . . . . . . . . . . . . . 55 2.8.1 Stars Like Our Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.8.2 Our Sun’s Inevitable End . . . . . . . . . . . . . . . . . . . . . . . . 56 2.8.3 What are White Dwarfs? . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.8.4 Is a White Dwarf the only option? . . . . . . . . . . . . . . . . . . . 57 2 2.8.5 White Dwarfs in Binary Systems . . . . . . . . . . . . . . . . . . . . 58 2.9 Life After the Death of High Mass Stars . . . . . . . . . . . . . . . . . . . . 58 2.9.1 Explosions! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.9.2 Discovery of the first Neutron Star . . . . . . . . . . . . . . . . . . . 59 2.9.3 Pictures of Explosions! . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.9.4 Fool’s Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.9.5 Creating Compact Objects . . . . . . . . . . . . . . . . . . . . . . . . 60 2.9.6 Failed Supernova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.10 Summary: The Circle of Life . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3 The Structure of Spacetime 62 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2 Fishing in Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2.1 Why Did the Fish Cross the Waterfall? . . . . . . . . . . . . . . . . . 62 3.2.2 Crossing the Point of No Return . . . . . . . . . . . . . . . . . . . . . 63 3.2.3 Breaking Down the Analogy . . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Introducing Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3.1 Light Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3.2 A Pair of Powerful Postulates . . . . . . . . . . . . . . . . . . . . . . 65 3.3.3 Galileo to Lorentz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4.1 Introduction to Spacetime . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4.2 Spacetime diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.3 A Cheesy Story . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4.4 Could there be extra dimensions? (Interview with Dr. Gingrich) . . . 68 3.5 Effects of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.5.1 Special Events in Spacetime . . . . . . . . . . . . . . . . . . . . . . . 69 3.5.2 The Relativity of Simultaneity . . . . . . . . . . . . . . . . . . . . . . 70 3.5.3 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.5.4 Length Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.5.5 The Gamma Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.5.6 The Twin Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.6 The Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.6.1 Elevators in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.6.2 How do black holes fall? (Interview with Dr. Jeremy Heyl) . . . . . . 76 3.7 Curved Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.7.1 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.7.2 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.7.3 GPS Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.7.4 Gravitational Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.7.5 A brief history of relativity (Interview with Dr. Robert Smith) . . . . 80 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3 4 Sizing Up Black Holes 82 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2 May the Schwarz(schild) Be With You . . . . . . . . . . . . . . . . . . . . . 82 4.2.1 The Schwartz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2.2 Combing Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2.3 Static and Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2.4 Measuring Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3 Dancing with the Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.1 Pairs of Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.2 Centre of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.3 Playground Break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3.4 Kepler’s Laws of Planetary Motion . . . . . . . . . . . . . . . . . . . 86 4.4 Black Hole Weigh-In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4.1 Weight Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4.2 Black Hole Lightweights . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4.3 Black Hole Middleweights . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4.4 Black Hole Heavyweights . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4.5 The Black Hole Family . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4.6 Is there a limit to the size of a black hole? (Interview with Dr. Gregory Sivakoff) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.5 Stellar Mass Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.5.1 Birth in Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.5.2 Finding Stellar Mass Black Holes . . . . . . . . . . . . . . . . . . . . 90 4.5.3 Masses of Stellar Mass Black Holes . . . . . . . . . . . . . . . . . . . 90 4.5.4 Colliding Stellar Mass Black Holes . . . . . . . . . . . . . . . . . . . 91 4.6 Supermassive Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.6.1 The Supermassive Black Holes Zoo . . . . . . . . . . . . . . . . . . . 92 4.6.2 How do we know the masses? . . . . . . . . . . . . . . . . . . . . . . 93 4.6.3 Relationship to Dark Matter? . . . . . . . . . . . . . . . . . . . . . . 94 4.6.4 How do supermassive black holes form? (Interview with Dr. Daryl Haggard) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.7 Intermediate Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.7.1 Mind the Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.7.2 Black Hole Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.7.3 Higher Masses? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.7.4 Direct Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.7.5 Runaway Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.8 SUPERtiny Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.9 Summary: Preparing to Explore . . . . . . . . . . . . . . . . . . . . . . . . . 99 5 Approaching a Black Hole 100 5.1 Journey to a Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2 Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2.1 Jets at a Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2.2 Jet Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4 5.2.3 Why are black hole jets important? (Interview with Dr. Bryan Gaensler)102 5.3 Black Hole Companions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3.1 Types of Companion Stars . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3.2 Types of X-ray Binaries . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.4 Sipping on Star Soup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.4.1 The Roche Lobe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.4.2 Is this just for stars? . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.4.3 Mass transfer in black hole binaries . . . . . . . . . . . . . . . . . . . 105 5.4.4 Other Options for Mass Transfer . . . . . . . . . . . . . . . . . . . . 105 5.4.5 How can a star be like a vampire? (Interview with Dr. Craig Heinke) 106 5.5 Have a Corona! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.6 What is Accretion? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.6.1 Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.6.2 Formation of a Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.6.3 Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.6.4 How do black holes grow? (Interview with Dr. Robert Thacker) . . . 109 5.7 Spinning Through the Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.7.1 Moving Through the Disc . . . . . . . . . . . . . . . . . . . . . . . . 110 5.7.2 Note About Time Dilation and Redshift . . . . . . . . . . . . . . . . 111 5.7.3 Tidal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.7.4 Tidal Forces Around Black Holes . . . . . . . . . . . . . . . . . . . . 112 5.8 Innermost Stable Circular Orbit . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.8.1 Kepler’s Third Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.8.2 Breakdown of Kepler’s Law . . . . . . . . . . . . . . . . . . . . . . . 113 5.9 Summary: Teetering on the Edge . . . . . . . . . . . . . . . . . . . . . . . . 114 6 Crossing the Event Horizon 115 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 The Event Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2.1 The Boundary of a Black Hole . . . . . . . . . . . . . . . . . . . . . . 115 6.2.2 Curvature of Particle Paths . . . . . . . . . . . . . . . . . . . . . . . 116 6.2.3 The Schwarzschild Radius . . . . . . . . . . . . . . . . . . . . . . . . 116 6.2.4 The Ring of Fire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2.5 Shining a Light on Black Holes . . . . . . . . . . . . . . . . . . . . . 118 6.3 The Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.3.1 Entering the Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.3.2 Space and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.3.3 Representing a Black Hole . . . . . . . . . . . . . . . . . . . . . . . . 119 6.3.4 The Penrose Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.3.5 Time to Reach the Centre . . . . . . . . . . . . . . . . . . . . . . . . 120 6.3.6 When Maths and Physics Collide . . . . . . . . . . . . . . . . . . . . 121 6.3.7 Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.3.8 Cosmic Censorship Hypothesis . . . . . . . . . . . . . . . . . . . . . . 122 6.3.9 What happens at the singularity? (Interview with Dr. Valeri Frolov) 123 6.4 Spinning Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5 6.4.1 Effect of Spin on the ISCO . . . . . . . . . . . . . . . . . . . . . . . . 123 6.4.2 The Ergosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.4.3 Gravity Probe B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.4.4 Ring Singularities and the Inner Horizon . . . . . . . . . . . . . . . . 126 6.4.5 How can you determine if a black hole is spinning? (Interview with Dr. Fiona Harrison) . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.5 Wormholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.5.1 Wormholes in Science Fiction . . . . . . . . . . . . . . . . . . . . . . 127 6.5.2 Understanding Einstein-Rosen Bridges . . . . . . . . . . . . . . . . . 128 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7 Inside a Black Hole 131 7.1 Black Holes: The Final Frontier . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.1.1 The Quantum Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.2 Introduction to Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . 131 7.2.1 The Ultraviolet Catastrophe . . . . . . . . . . . . . . . . . . . . . . . 131 7.2.2 The Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.2.3 Matter Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.2.4 The Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . 134 7.3 Hawking Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.3.1 General Overview of Hawking Radiation . . . . . . . . . . . . . . . . 135 7.3.2 The Quantum Foam . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.3.3 Virtual Particles Near the Horizon . . . . . . . . . . . . . . . . . . . 137 7.3.4 What is Hawking radiation? (Interview with Dr. Don Page) . . . . . 137 7.4 Information in a Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.4.1 Information and Quantum Mechanics . . . . . . . . . . . . . . . . . . 138 7.4.2 The Information Preservation Station? . . . . . . . . . . . . . . . . . 139 7.4.3 What is quantum information? (Interview with Dr. Lindsay LeBlanc) 140 7.5 Black Hole Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.5.1 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.5.2 Introduction to Thermodynamics . . . . . . . . . . . . . . . . . . . . 141 7.5.3 The Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . 142 7.5.4 Entropy and Temperature of a Black Hole . . . . . . . . . . . . . . . 143 7.5.5 Black Hole Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 143 7.5.6 How do you cool a gas with lasers? (Interview with Dr. Lindsay LeBlanc)144 7.6 Lifespan of a Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.6.1 Temperature of a Black Hole . . . . . . . . . . . . . . . . . . . . . . . 145 7.6.2 Evaporation of Black Holes . . . . . . . . . . . . . . . . . . . . . . . 146 7.6.3 Black Holes and the Large Hadron Collider . . . . . . . . . . . . . . . 147 7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6 8 Hunting for Black Holes 150 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.2 Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.2.1 Observing the Sky . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.2.2 Radio Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 8.2.3 X-Ray Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.2.4 Why are X-ray telescopes useful for studying black holes? (Interview with Dr. Daryl Haggard) . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.3 Chopping up Rainbows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.3.1 Making Rainbows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.3.2 Kirchhoff and the Continuum . . . . . . . . . . . . . . . . . . . . . . 154 8.3.3 Absorption and Emission . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.4 Advanced Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8.4.1 Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8.4.2 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.4.3 Scattered Outta Compton . . . . . . . . . . . . . . . . . . . . . . . . 158 8.5 Black Hole Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.5.1 Modeling the Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.5.2 Multi-coloured Disc Model . . . . . . . . . . . . . . . . . . . . . . . . 160 8.5.3 Across the Mass Scale . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8.6 Staring into the Hot Mess . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8.6.1 Looking at the Corona . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8.6.2 Stepping up the Mass Scale . . . . . . . . . . . . . . . . . . . . . . . 162 8.6.3 Modeling the Mess . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 8.7 Beam Me Up! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.7.1 Emission from the jet . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.7.2 the Jet Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.7.3 Types of Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.7.4 Orientation of the System . . . . . . . . . . . . . . . . . . . . . . . . 164 8.7.5 What types of jets could a black hole have? (Interview with Dr. Gre- gory Sivakoff) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9 Our Eyes in the Skies 166 9.1 Introduction: Turn to face the strange . . . . . . . . . . . . . . . . . . . . . 166 9.2 To Feed Or Not To Feed? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.2.1 To Feed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.2.2 Not To Feed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 9.3 Companion Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 9.3.1 Black Hole X-ray Binaries . . . . . . . . . . . . . . . . . . . . . . . . 167 9.3.2 Classifying the Companion Star . . . . . . . . . . . . . . . . . . . . . 168 9.3.3 Examples of Low Mass X-ray Binaries . . . . . . . . . . . . . . . . . 168 9.3.4 Examples of High Mass X-ray Binaries . . . . . . . . . . . . . . . . . 169 9.3.5 What is a black hole outburst? (Interview with Dr. Aarran Shaw) . . 169 9.4 The Alternative Diets of Supermassive Black Holes . . . . . . . . . . . . . . 170 7 9.4.1 Snacking on Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 9.4.2 Slurping on Soup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.4.3 Black Hole Burps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.5 The Special Case of Sgr A* . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.5.1 Where Do We Live? . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.5.2 What’s So Special? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 9.5.3 Why is Sag A* important? (Interview with Dr. Fiona Shaw) . . . . . 172 9.6 The Hermits of the Black Hole Family . . . . . . . . . . . . . . . . . . . . . 173 9.6.1 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.6.2 Wine Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 9.6.3 Einstein Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 9.6.4 Gravitational Micro-Lensing . . . . . . . . . . . . . . . . . . . . . . . 175 9.6.5 What does microlensing have in common with exoplanet searches? (Interview with Dr. Kelsey Hoffman) . . . . . . . . . . . . . . . . . . 175 9.7 It’s on... Now what? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 9.7.1 Varying X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 9.7.2 High and Low? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 9.7.3 Very High . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 9.7.4 Feeding Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 9.8 Impact of Black Holes on Galaxies . . . . . . . . . . . . . . . . . . . . . . . . 178 9.8.1 Relative Scales of Supermassive Black Holes . . . . . . . . . . . . . . 178 9.8.2 Black Hole Feedback on the Host Galaxy . . . . . . . . . . . . . . . . 179 9.8.3 Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 9.8.4 What is the impact of a supermassive black hole on a galaxy? (Inter- view with Dr. Sarah Gallagher) . . . . . . . . . . . . . . . . . . . . . 181 9.9 Seeking Out The Elusive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 9.9.1 Intermediate Mass Black Holes . . . . . . . . . . . . . . . . . . . . . 182 9.9.2 HLX-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 9.9.3 M82 X-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 9.9.4 XJ 1417+52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 9.9.5 Can intermediate mass black holes be found in globular clusters? (In- terview with Dr. Craig Heinke) . . . . . . . . . . . . . . . . . . . . . 184 9.10 Summary: The End Of The Rainbow . . . . . . . . . . . . . . . . . . . . . . 185 10 Gravitational Telescopes 186 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 10.2 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 10.2.1 Gravitational Lensing by Neutron Stars . . . . . . . . . . . . . . . . . 186 10.2.2 Gravitational Lensing by Black Holes . . . . . . . . . . . . . . . . . . 187 10.2.3 The Photon Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 10.2.4 The Event Horizon Telescope . . . . . . . . . . . . . . . . . . . . . . 189 10.2.5 The Black Hole in M87 . . . . . . . . . . . . . . . . . . . . . . . . . . 189 10.3 Gravitational Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 10.3.1 Gravity at the Speed of Light . . . . . . . . . . . . . . . . . . . . . . 190 10.3.2 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 191 8 10.3.3 Surfing Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 10.4 Binaries and Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . 193 10.4.1 Binary Inspiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 10.4.2 Following Kepler’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . 193 10.4.3 Loss of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 10.4.4 How do binary pairs of black holes form? (Interview with Dr. Tyrone Woods) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 10.4.5 Inspiral and merger of Neutron Stars . . . . . . . . . . . . . . . . . . 195 10.4.6 Gravitational Observatories . . . . . . . . . . . . . . . . . . . . . . . 195 10.4.7 Merger Remnants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 10.4.8 Merger of a Neutron Star and a Black Hole . . . . . . . . . . . . . . . 196 10.4.9 Where does the gold in your ring come from? (Interview with Dr. Rodrigo Fernandez) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 10.4.10 How does a neutron star’s vibrations affect binary inspiral? (Interview with Dr. Jocelyn Read) . . . . . . . . . . . . . . . . . . . . . . . . . 198 10.4.11 Inspiral and Merger of Two Black Holes . . . . . . . . . . . . . . . . 199 10.4.12 Masses in the Stellar Graveyard . . . . . . . . . . . . . . . . . . . . . 199 10.4.13 Colliding Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 10.5 Gravitational Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 10.5.1 The Long Arm of Gravity . . . . . . . . . . . . . . . . . . . . . . . . 201 10.5.2 Two LIGOs are Better Than One . . . . . . . . . . . . . . . . . . . . 202 10.5.3 Lasers in Space! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 10.6 Pulsar Timing Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 10.6.1 Pulsar Positioning System . . . . . . . . . . . . . . . . . . . . . . . . 203 10.6.2 Squishing Signals in Spacetime . . . . . . . . . . . . . . . . . . . . . 204 10.6.3 What is a pulsar timing array? (Interview with Dr. Ingrid Stairs) . . 204 10.7 The Final Countdown! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 10.7.1 What We Know We Know . . . . . . . . . . . . . . . . . . . . . . . . 205 10.7.2 Black Hole Formation . . . . . . . . . . . . . . . . . . . . . . . . . . 205 10.7.3 Where Science Fiction Meets Science Fact . . . . . . . . . . . . . . . 206 10.7.4 Observations of the Black Hole Environment . . . . . . . . . . . . . . 207 10.7.5 What We Know We Don’t Know . . . . . . . . . . . . . . . . . . . . 208 1 Introduction to Black Holes Hello and welcome to the first module of Astro 101! In this module, you will become familiar with the basic structure of a black hole, learn the terminology used to describe them, and explore the history of black hole physics. 1.1 Introduction Welcome to Astro101. My name is Sharon Morsink, and I will be your professor. With me is Jeanette Gladstone, Ross Lockwood, and Curtis Brown who will be your guides throughout our voyage to a black hole. I did my PhD research on the insides of black holes when I 9 worked on my PhD in theoretical physics. Over the years, I have also studied gravitational radiation and X-ray emission from black holes and neutron stars, which are dense, compact objects. I’m fascinated by the force of gravity. How does gravity’s strength depend on an object’s mass and size? How does gravity cause light to travel on curved paths? Can we use the effects of gravity to observe black holes through gravitational lensing? What exactly are the gravitational waves that scientists are so excited about? Hi there, I’m Curtis. Some things you ought to know about me, when I’m not on my side hustle slinging plywood, I’m learning about black holes and studying engineering. I’ve been keen on learning about astrophysics since the very first time I watched my favorite science guy, Neil deGrasse Tyson. In my spare time, I make pipe cleaner mustaches, and sit above the streets of Toronto in my palace of textbooks. My interest in black holes started in my undergraduate courses in astrophysics. We learned that black holes are the endpoint of stellar evolution for high mass stars. What I’m curious to learn is how the mass of a star can create a gravitationally collapsed object, a black hole. I know it has something to do with the four-dimensional structure of spacetime, but the thing that I’m really inter- ested in, is learning why massive black holes are much safer for astronauts and spaceships to visit than very small black holes. I can’t wait to get started. I’ll see you in the stellar nursery. Hi, I’m Ross. I earned a PhD in Condensed Matter Physics, relating to quantum mechan- ics. As a result, I’m interested in the theories about microscopic black holes, and the effects caused by quantum mechanics like Hawking radiation. I’m excited about learning what happens inside of a black hole, what would happen to an astronaut and their equipment if they cross the event horizon, and what you would see if you look towards a black hole’s singularity? What would happen to an astronaut and the information they were collecting if they cross into a black hole, and what about the possibility of them escaping from the black hole? But I’m most excited to be one one your teachers throughout the course. I’ll see you in a little bit, when we begin talking about Newtonian gravity. Hi there, my name is Jeanette Gladstone, and I’m looking forward to helping you learn more about black holes while sharing some of my sci-fi favorites. I also have a PhD, but mine is in black holes, or rather, in observing them. Yes, I spent three years of my life in the UK testing theories and ideas that had been put forward to explain a strangely bright class of black hole. I know that sounds a bit of an oxymoron, but it’s true. At the end of those three years, I got to call myself Dr. Jeanette, and then moved across the pond to continue researching black holes here at the University of Alberta. My background means that my favorite sections lie towards the end of the course. So, I’m hoping you’ll stick around to modules eight and nine to find out how it’s actually possible to look at and study black holes. I hope to help you find the answers to questions such as how do astronomers look at a black hole, how do they study them, and what do they look like, and more. Thank you for coming on this journey with us through space and time, and I look forward to chatting with you again soon. In this course, we will be exploring these questions using animations, demonstrations, interactive learning objects, food analogies, calculations, readings, discussions, and other activities. In order to learn about black holes, we should first take a look at how black holes are portrayed in popular culture. 10 1.2 Black Holes in Pop Culture 1.2.1 Star Trek: The Future Begins Let’s start our journey into the realm of pop culture with a franchise that has been attracting fans for the last 50 years. A generation of Star Trek enthusiasts grew up watching Star Trek The Original Series, which first aired in 1966. A lot has changed in the time since Captain Kirk fought a Gorn by hand in the original series. Star Trek isn’t merely science fiction but is also known to challenge inequalities in society. But Star Trek is known for singlehand- edly inspiring a generation of scientists and engineers whose work saw human exploration of the moon and permanent laboratories in space. My favorite Star Trek movie reboot of the franchise in 2009 with an epic space adventure with the tagline, the future begins. The story begins following James Kirk’s exploits as a cadet in Starfleet Academy but an attack on the Vulcan homeworld forces the cadets to become crew of the newly commissioned USS Enterprise. Without spoiling too much of the movie, the writers of the film employed a strange form of fictional matter called red matter, which appears to create black holes that can consume entire planets. In the movie, the Vulcan homeworld has been attacked, and the planet’s surface collapses inwards. Although visually impressive, Star Trek gets a lot of the black hole physics wrong. For one, the audience is meant to understand that red matter black holes are traversable. Unfortunately, traversable wormholes are merely a theory and would require a super advanced civilization and a number of notable scientific discoveries to permit travel. In fact, travel through a wormhole, if possible, would likely expose travelers to a serious dose of radiation, making the journey fatal to squishy humans like me. On the other hand, Star Trek explores some very interesting physics of black holes, specifically how a spacecraft might escape if it becomes trapped within the gravitational pull of a black hole. In one scene, the enterprise is at maximum thrust and is still being accelerated towards the black hole. In order to save the crew, the chief engineer Scotty suggests that the last option is to eject the warp core and ride the shockwave from the explosion to safety. At best, it explores interesting science fiction concepts. At its worst, it makes some small, maybe forgivable errors, in scientific judgment for the sake of entertainment. 1.2.2 Interstellar I enjoy watching Star Trek and Doctor Who, but often the treatment of science in these shows doesn’t satisfy my sense of scientific consistency. One movie that did satisfy me was the 2014 movie Interstellar, which presents the science of black holes as accurately as possible. There is still black hole physics that is unknown, like what happens when you cross an event horizon? So, the movie makers do engage in some speculation about what happens inside a black hole. The black hole in Interstellar is a supermassive black hole named Gargantua. Two planets are in orbit around the black hole at a safe distance, which is a reasonable possibility even though in the TV show Doctor Who, they claim that planets orbiting black holes are impossible. It is important to the plot of Interstellar that Gargantua is a supermassive black hole, since the strength of the tidal forces that could spaghettify an astronaut is weaker for more 11 massive black holes. Although Gargantua causes enormous tides on Miller’s planet, the planet that orbits closest to Gargantua, the tidal forces aren’t strong enough to destroy the planet or the characters on the surface. Another cool bit of science that Interstellar gets right are the extreme time differences between Miller’s planet and the spaceship that orbits farther away from the black hole. When one hour passes on the planet, seven years pass for the astronauts far from the black hole. For me, the most exciting part of the movie is the view of the black hole and the disk of material flowing around it. This view of the black hole incorporates most of the science of how light travels on curved paths around regions with gargantuan gravity. The black hole Gargantua is extremely far away from us, but luckily the writers included a wormhole that allows a quick shortcut through space between Saturn and Gargantua. Black holes exist but wormholes probably don’t so this part is science fiction. In the movie, they speculate that the wormhole was constructed by aliens. I don’t know how to construct a wormhole but then I’m not an alien. I won’t spoil the ending of the movie for you. However, I’ll just let you know that there is a trip into the black hole. What happens when they enter the event horizon is pure spec- ulation, but who knows? Maybe the screenwriters are correct. 1.2.3 Disney’s The Black Hole In 1977, George Lucas presented his creation Star Wars: A New Hope to the world. It was packed with brand new special effects like fast-paced space battles and spectacular lightsaber duels. Sadly, Star Wars didn’t include anything that we would really consider a black hole, unless you include the insatiably hungry Sarlacc. So, we don’t have much material from the Star Wars universe. Fortunately though, Disney was aware of the success of Star Wars and in 1979 decided to fund their own space opera called The Black Hole. The Black Hole was the biggest production in Disney’s history with 20 million spent on the budget, allowing them to create incredible visual effects. In the movie, the crew of the spacecraft Palomino approaches a black hole on a scientific mission, only to discover another ship in orbit around it, the Cygnus, a long lost vessel. The crew of the Palomino must assess the dangers around the black hole and decide whether or not to mount a rescue mission. Disney’s The Black Hole was one of the first works of modern film making that attempted to recreate the environment around a black hole. Since we still don’t have pictures of black holes, the filmmakers relied on scientists of the day to tell them what it might look like. Have a look at this scene from the bridge of the Cygnus, a stark starry background is punctuated with a tear in the fabric of spacetime. At the end of the 1970’s, this was the picture of a black hole. In order to make such a convincing picture, the filmmakers relied on 20 years of work from scientists, like Roy Kerr, Roger Penrose, Werner Israel and Stephen Hawking, whose names come up a lot in this course. They had been working on models of how black holes behave and just what they might look like. In addition, the nearby black hole Cygnus X-1, which had then only been recently confirmed, became the namesake of this ship in the movie The Black Hole did a lot of things correctly that other movies failed to do, like an 12 accurate representation of an accretion disk for the time, and accounting for the time dilation effects around the black hole. However, in the film, the crew of the Palomino descend past the event horizon, where their interpretation of the interior is almost certainly false. Instead of space-bending physics, as the Cygnus descends into the black hole, it becomes a fiery realm like a portrayal of Dante’s Inferno. In reality, crossing the event horizon has dozens of dangers that would likely harm human explorers. A lot has changed in the 40 years since Disney released The Black Hole, especially related to black hole physics. We now know the strength of gravitational waves produced by merging black holes, and scientists are still developing theories to explain what the interiors of black holes might look like. This film introduces some of the interesting ideas about falling into a black hole, but there is one thing that I can tell you that none of these movies can. No one can tell you for certain what happens when you cross a black hole’s event horizon. 1.3 Connecting Gravity and Light 1.3.1 Gravity and Light The relationship between light and gravity is fundamental to our understanding of black holes. In order to tie these ideas together, we will have to relearn the concepts of gravity, space, time, and light that we’ve been presented in our favorite movies, TV shows and other great works of art. Popular culture doesn’t always get the science behind these concepts right. But it’s surprising how scientific accuracy informs spectacular effects in movies like in Interstellar. The scientific foundation for black holes has been built by hundreds of scientists contributing ideas throughout history. But one unmistakable scientist was responsible for the biggest developments, Einstein. Einstein’s theory of gravity, describes the gravitational attraction between massive ob- jects, such as a star or a black hole, by associating the force of gravity with an equivalent curvature of space, sort of like this curved sheet. All physical objects including rocketships and individual particles of light called photons, travel on curved spacetimes similar to the curved path that these marbles follow. In this simple demonstration, we have an up and down direction defined by the earth’s gravity, which gives us the effect of the marbles orbit- ing around the central depression. In outer space, near a star or a black hole, we don’t have earth’s gravity holding the marbles down on the curved sheet. Instead, we recognize gravity as curvature of space and time. This curved sheet provides valuable insight into how we can imagine the warping of spacetime. A black hole is a region of space where the force of gravity becomes so strong that the curvature of spacetime prevents light from escaping. In this curved sheet model, imagine that a marble represents light. If the marble is aimed away from the center and tossed with a high enough speed, the marble is able to escape from the depression in the center. The region outside of a black hole or a normal star is like this. Light that is emitted from objects in the neighborhood of a black hole are still able to escape. But there is a special spherical boundary surrounding the black hole, which scientists call the event horizon. Once a marble falls within the event horizon, it must continue falling inwards. You could try to aim a marble outwards, but you’d never be able to make the marble go fast enough that it can escape. 13 1.3.2 Melanoheliophobia To some people, the concept of a black hole is terrifying. A black hole represents an in- escapable vortex, whose strength can stretch astronauts into spaghetti, and even eat whole stars. Falling into a black hole is irreversible, and once inside, there is no opportunity to escape ever. To characterize this fear of black holes, people use the word melanoheliophobia, the fear of black holes. Much of that fear is misguided, as you will see, in black holes come in various flavors that are not equally dangerous. That’s good news if you are among those who suffer from melanoheliophobia, so sit tight. There is another misconception about black holes: that they suck things towards them. All astrophysical objects like the sun, the earth, and black holes have gravity, and gravity is an attractive force. This means that just as marbles are attracted by gravity to the earth, so too are they influenced by the gravity of other objects. If we throw a marble directly towards the earth, or the sun, or a black hole, then just like the marble on this curved sheet, the throne marble traveled directly towards the central depression. The strength of the gravitational depression depends on the mass of the object causing it. A star and a black hole with equal mass, even though they are different sizes, produce the same gravitational attraction far from their centers. If this depression represents a star, then the stars surface is big and the marble enters the star and is burned up. Traveling directly towards a star is a recipe for crispy bacon. Now, let’s pretend this depression represents a black hole. The gravity is exactly the same, but the black hole is much smaller than the star. So marbles can travel deeper into the depression before we lose track of it. When a marble enters the black holes event horizon, the curvature due to gravity prevents it from escaping and it becomes lost to us forever. So, even though directly traveling towards a star is a recipe for crispy bacon, traveling directly towards a black hole is a recipe for no bacon at all. Instead of traveling directly towards a star or a black hole, we can instead travel safely around it in orbital paths similar to how this marble can be made to circle or orbit around the depression. Since the curve sheet in our demonstration has friction, the marbles eventually lose energy and fall in towards the center. However, in space, there is virtually no friction, so it is possible for orbits to be stable for extremely long periods of time. For instance, the earth has been orbiting the sun for close to five billion years, and will continue to orbit it for several billion more years. The idea that gravity is really just the curvature of spacetime is a tough concept, which requires some knowledge of Einstein’s theory of gravity called general relativity, which we will be introduced too later on in this course. However, we can still get a taste of black hole physics by making use of an older less accurate description of gravity created by Sir Isaac Newton. In fact, several 18th century physicists were able to deduce the idea of a star, whose light can’t be seen using just their knowledge of Newton’s theory of gravity along with the value for the speed of light. Since the concepts of light and gravity are so important for understanding black holes, we will review the basics of light, Newtonian gravity, and some elementary physics principles as our starting point. 14 1.4 Getting on the Same Wavelength 1.4.1 Colours and Pitch It’s impossible to discuss the concept of a black hole without first delving into the details about light, or electromagnetic radiation. In the absence of light, we get darkness or black- ness, which is how black gets into the name black holes. Black holes do not allow light to escape from their interior, so they appear completely dark. Our intuition regarding the properties of light can sometimes be misguided, so let’s discuss some of light’s fundamental properties. When a beam of white light is passed through a prism, the white light that enters is separated into a rainbow of colours spreading out the other side of the prism. Each of the colours in the rainbow corresponds to a property of light called its wavelength. Wavelength is also related to the frequency of light, which is itself is related to the energy of a photon. Another way to think about light is by way of an analogy to sound. Sound waves also come in different frequencies, which we call the pitch of a sound. So colour is to light, as pitch is to sound. For instance, I can make sounds that have low pitch [la la la with an exaggeratedly low voice] or with a high pitch [la la la with a high falsetto voice], just as I can make low frequency light, like red, or high frequency light, such as blue. When a large tuning fork is struck, a low pitch sound wave is created. The ends of the fork vibrate back and forth slowly, at a low frequency. This vibration creates the pitch we hear, as it is transmitted through the air. When he strikes a small fork, a high pitch sound is created and the ends of the fork vibrate rapidly. Each of the fork’s tines are vibrating periodically, pushing on the air molecules back-and- forth, which results in the creation of sound waves. Sound waves repeat at set intervals, separated by a distance, which is the sound waves’ wavelength. Low pitch sounds have long wavelengths, while high pitch sounds have short wavelengths. Additionally, sound travels at some finite speed, which we call the speed of sound, which is much slower than the speed of light. This is why you will sometimes see the source of a distant sound, before you hear the sound itself. One of the most well known example of this is the thunder that follows the lightning. Here, we see Curtis and Ross are sending waves along this spring which travel using the same principles that sound waves do. As the wave in the spring passes by, the coils are compressed and stretched periodically, resulting in the motion of a wavefront along the length of the spring. This type of wave is called a longitudinal wave. This is similar to how the air molecules are pushed back and forth by a sound wave. Human hearing is limited to a range of pitches from about 20 hertz to 20,000 hertz. Which means that it is possible to create sound waves that have a pitch that is either too high or too low for humans to hear. A sound wave that is too low for us to hear is called infrasonic, and some animals, like Elephants, use infrasound to communicate. Although a human couldn’t hear it, it is possible from them to feel infrasound as a vibration. Sound waves that are too high pitched for us to hear are called ultrasonic, and are also useful for navigation in animals like bats. 15 1.4.2 Light Waves Just like sound waves, light waves are characterized by their properties of wavelength, fre- quency, and propagation speed. The speed of sound depends on things like the temperature and density of air, and even the pitch of the sound. This is contrasted by light, which has a fixed speed in vacuum, and very little wavelength dependence travelling through a medium like air. However, light waves travel at a incredibly high speeds compared to sound waves. The speed of light is just a whisper shy of 300,000 kilometers per second! The symbol “c” is used to denote the speed of light, which has a precise value of 299 million, 792 thousand, 458 meters per second, or 2.9979x108 metres per second. While sound waves are longitudinal compressions, light waves are actually a type of transverse electrical and magnetic wave. Scientists use the term electromagnetic radiation to describe light. Show the rainbow, and label the white space above and below the coloured regions as infrared and ultraviolet respectively, as they are mentioned ] The longest wavelengths light that our eyes are capable of detecting are approximately 700 nanometers long, which corresponds to a deep red. The shortest wavelength light that we can see is approximately 400 nanometers, a deep violet. Therefore, the range of light between 700 and 400 nanometers is often called “visible light”. This can be seen as the rainbow of colours we can see. In the order of longest to shortest wavelength the colours are red, orange, yellow, green, blue, indigo, and violet. Just as there are infrasonic and ultrasonic sounds that human ears can’t hear, there are colours that humans eyes can’t see. The visible spectrum is only a narrow slice of the entire of electromagnetic radiation. In order of longest wavelength waves to shortest wavelength waves, the bands in the electro- magnetic spectrum range from radio waves to infrared light. This is followed by the familiar visible spectrum, then ultraviolet light. As we approach shorter wavelengths we have X-rays, and the shortest wavelength light, gamma-rays. Going from long wavelength electromagnetic light to shorter wavelengths is the same as going from low frequency light to higher frequen- cies! Even though there are all of these types of electromagnetic waves, they all travel at the speed of light. 1.4.3 Photons We call individual particles of light photons. This is the same as saying that light is quantized, each photon is a small, discrete packet of energy. The energy of a photon is related to its wavelength and frequency, and therefore its colour. Within a beam of light, are trillions of photons, each with their own energy. The energy that a photon carries is proportional to its frequency, and inversely propor- tional to its wavelength. This means that X-ray photons, which have fast frequencies of oscillation, carry large amounts of energy. As a result, X-ray photons have very short wave- lengths, and can easily pass through human tissues, with the exception of bones. This allows doctors to take X-ray images of your skeleton. On the opposite end of the energy scale, radio waves have low oscillation frequencies, 16 and therefore long wavelengths, because they carry tiny amounts of energy. While exposure to x-rays can cause tissue damage, radio waves transport such small amounts of energy per photon, that they are considered safe for use in modern telecommunications. 1.4.4 Waves or Particles? We have talked about light in terms of waves, discussing features like wavelength and speed. But we have also talked about light in terms of particles, or photons. So which is it, are photons waves, or are they particles? Well, the physical theories that describe light tell us that light behaves BOTH as a particle and as a wave. It can take a little bit to get your head around the idea of light being both a wave and particle, and it took scientist a long time to work this out, with arguments debate over the nature of light and matter from the 1600s through till the early 1900s. This idea is now known as wave-particle duality, the duality being that light can act as either a wave or a particle depending on the situation you are considering. Now that we have covered the basics of light, let’s take a look at how we can measure and use light to gain a greater understanding of black holes and the universe. 1.5 Working with Light 1.5.1 Making Light For us to even know anything about light, we need ways of making it, measuring it, and using it. In order to do all these things, we, creatures made of matter, need methods of interacting with light. We know that matter is somehow responsible for the creation of photons, but until now we really haven’t considered how. Light production falls into two major categories: incandescence and luminescence. Incandescence is the production of light by any body that contains heat energy, the energy of vibration. Incandescence is how filament light bulbs produce light: by warming a metal filament inside of a lightbulb using electricity, the metal grows hotter and hotter, emitting more and more light as the temperatures increase! The scientific principle of Blackbody radiation explains how photons are created by the intense vibrations of atoms and electrons at high temperatures. Blackbody radiation is another name for incandencence. Blackbody radiation applies to any object above absolute zero, even those that are very cold, since even small atomic vibrations exist above absolute zero, the coldest possible temperature. The theory of blackbody radiation describes how the oscillations of atoms in objects creates light waves. Atoms and electrons sloshing back and forth due to thermal vibrations, or heat, act as tiny emitters, creating oscillating electromagnetic fields. This wiggle of the electromagnetic field can be wrapped up into a neat little bundle we call a photon. Luminescence is the production of light through atomic transitions, which are sometimes called “cold-body” radiation. In the planetary model of the atom, electron orbits move in orbits around the nucleus. When an electron jumps from one orbit to another, it emits (or absorbs) a photon of a specific energy to do this. There are many subcategories of luminescent processes; fluorescence 17 converts UV photons which we can’t see into visible light, phosphorescence releases energy stored in glow-in-the-dark objects, and triboluminescence produces light when you chew on hard candies like lifesavers. Now that we’ve got some light to work with, let’s make sure we can measure its properties. 1.5.2 Measuring Light When we want to take a measurement of light, what are we measuring? We know that the speed of light is denoted by the letter c, and is constant, at just under 3 times ten to the 8 meters per second. What is left is either a measure of the wavelength, the frequency or the energy that the photon packets carry. A typical red laser emits a beam of photons with a wavelength of 650 nanometers. It could equally be advertised as having photons oscillating with a frequency of 460 Terahertz, or even in terms of the photon energy, 1.9 electron volts. Electron volts may sound like a strange unit, one of many we will come across in this course. It is defined as the amount of energy that is gained, or lost, by the charge of a single electron moving across an electric potential difference of one volt. It is a miniscule amount of energy, equivalent to about 1.6 times ten to the minus 19 Joules. The naming and labelling of light can be confusing. The thing to remember is that a pho- ton’s energy, wavelength, and frequency can all be considered as equivalent ways to describe light. While a radio frequency photon emitted by a radio station is usually characterized by its frequency, say 102.9 megahertz, and an X-ray photon is usually characterized by its energy, say a kilo-electron-volt, visible light photons are often described by their wavelength between 400 and 700 nanometers. If you are given one of energy, frequency, or wavelength, there are some very simple mathematical relationships that allow you to determine the other quantities. The equation relating frequency and wavelength of a photon to the speed of light is: λf = c (1) The wavelength, lambda, times the frequency, f, is equal to the speed of light, c. Let’s double check the values we had for our red laser. We’ve been given its wavelength as 650 nanometers, so to determine its frequency we simply divide the speed of light, c, by the wavelength, lambda. Remember, in order to calculate this properly, we need to express both the speed of light and the wavelength using the common unit of meters: f650nm = 2.99× 108m/s 6.5× 10−7m = 4.6× 10 14Hz (2) So a red photon with a wavelength of 650 nm, has an oscillation frequency of 460 Tera- hertz, exactly what we calculated before! How much energy is our 650 nanometer photon carrying? Let’s use our last answer of 460 Terahertz to calculate the photon’s energy. This time, we’ll need another simple equation that relates the frequency of a photon with the energy it carries. The photon energy is given by the equation, E equals H times f E = hf (3) 18 E stands in for energy, and f stands for the frequency of a photon, but h is a new value in this equation. This h represents Planck’s constant, named after Max Planck, and relates the frequency of a photon with its energy. h has a value of 6.626 times 10 to the minus 34 in units of Joule-seconds. So now, our 650 nm wavelength photon, which oscillates with a frequency of 460 Tera- hertz, carries with it an energy equal to: E = 6.626× 10−34Js× 4.6× 1014Hz = 3.0× 10−19J (4) 3 times 10 to the minus 19 Joules?? That’s a tiny amount of energy! Each and every 650 nm wavelength photon carries an incredibly small amount of energy, but bright light sources, like lasers, produce tremendous numbers of photons, which is why they pack enough punch to damage sensitive tissues like the retinas in our eyes. Hence the laser safety label, and the common phrase in laser-labs, “Do Not Look Into Laser With Remaining Eye”. The relationships we have discussed here can be summed up in just three equations. The first two we used, that we just ran through, and a combination of them: E = hf, fλ = c, E = hc/λ . (5) These simple relationships are modern discoveries, relatively speaking, since they were developed in the early 20th century during the quantum revolution. With these three equa- tions, a technological revolution occurred that permitted the development of advanced optics and telescopes, capable of measuring light from even the far reaches of the cosmos. 1.5.3 Using Light The speed of light is known to incredible precision. Since the speed of light is well-known, it has become common for astronomers to measure the enormous distances in space in terms of the time it takes light to travel in a given amount of time. For example, the distance that light can travel in one second is known as a light-second. light-second = 299, 792, 458 m = 299, 792.458 km (6) One light-second is equal to two-hundred and ninety-nine million, seven hundred and ninety thousand metres, or two-hundred and ninety-nine thousand, seven hundred and ninety kilometers. A number this large can be hard to wrap your head around, so let’s compare this with the distance we can imagine, the distance between the Earth and the Moon. The distance between the Earth and Moon is 384,400 km, which is just a bit larger than one light-second. We could use the the unit kilometers, which is getting a bit crazy at this point, or we could use the unit of light-second. In this new unit, the distance between the Earth and the Moon is 1.3 light-seconds! In other words, it takes a photon of light 1.3 seconds to travel from the Earth to the Moon. If we now step up the size scale, and consider the Earth and our Sun, it takes light 8.3 minutes to travel from the Sun to the Earth, so we say that the distance is 8.3 light-minutes. Since this distance from the Earth to the Sun is so important in astronomy, astronomers also introduced a new distance, called an Astronomical Unit abbreviated to AU. An astro- nomical unit is the average distance between the Earth and the Sun and is equal to 1 AU = 8.3 light-minutes = 149.6 million km (7) 19 Yes, the measure of distance in kilometers is starting to get a bit crazy and messy. If we return to the speed of light measuring stick, and continue to shift the size scale further, we have had light-seconds, light-minutes and then a light-year is the distance that light travels in one year, 1 light-year = 9.5× 1012 km (8) A light-year sounds like a big distance but the distance between the Sun and the next closest star, Proxima Centauri, is larger than this, 4.2 light-years. The distance to the centre of the Milky Way galaxy is close to twenty-five thousand light-years. The final unit of distance we want to mention here is called a parsec. This distance is equivalent to 3.26 light-years, and was redefined in 2015 to be equal to 648 thousand divided by pi astronomical units. This unit of measurement, developed in the early 1900’s, may sound familiar to some of you, from a beloved character in Star Wars - A New Hope. Han Solo, owner of the Millennium Falcon, brags to Luke Skywalker and Ben Kenobi that, “It’s the ship that made the Kessel run in less than twelve parsecs!”. On first hearing this, you may think that a parsec is a measurement of time, but this is obviously not the case. According to fan theory, the Kessel Run is a heavily used smuggling route that normally takes 18 parsecs to navigate. However, Han claims he had shortened the journey by skirting a nearby black hole cluster called The Maw. This path took him closer to the black holes than others were willing to go, and shortened their route by 6 parsecs. However, in the original script written by George Lucas, Han’s line is to be delivered in such a way to denote he is “obviously lying”, in order to boast in front of Luke and Ben. Needless to say, the Millennium Falcon did prove to be a fast ship, and the way speed is portrayed in these films depends not only on visual effects, but also sound effects. Yes, you heard that right. . . the Doppler shift. 1.6 Doppler Shift 1.6.1 Shifting with sound Next up on our tour of the properties of light . . . and sometimes sound, is an effect first explained by an Austrian mathematician and physicist, Christian Doppler in 1842. In a paper entitled “On the coloured light of the binary stars and some other stars of the heavens”, Doppler presented his theory that the observed frequency of a wave depends on both the emitted light, and on the relative speed of the source and the observer. But what does this mean? Well, if we switch back to sound waves for a moment, we can use a real world analogy to explore this. I am sure many of you have been present as an emergency services vehicle or a train pass by. When this happens, what do you hear? Well, yes, you hear the siren, but what happens to the siren as an ambulance drives by? Well, if you listen closely you can hear that the pitch of the siren changes over time. As the ambulance moves towards you, you hear the siren at one pitch, but as it passes by, it seems to drop in pitch. Let’s listen to the sounds of a passing train, to compare the sound of the effect from a different source. 20 There is a clear drop in pitch here also, but what is the reason for this? Let’s take take the example of trains and look at this in more detail to explore the physics of the Doppler effect. If we look and listen to a train that is honking its horn while sitting at rest. We can think of the sound waves as circles moving away from the source. As the train sets off, we begin to see things change. The sound waves in front of the train begin to bunch up, or be pushed together. When the wavefronts of the sound waves are pushed closer together, the wavelength that a stationary person will hear will be smaller. Smaller wavelength corresponds to higher frequency and higher pitch. When a train or ambulance approaches, you hear a higher pitched sound. Now if we look at the waves behind the train, we can see that, as the vehicle moves forward, the waves appear to be more spread out. This apparent stretch results in longer wavelengths or lower frequencies, which results in a lower pitch. So as a train moves past us, the shift raises the pitch as it moves towards us, has no effect when it is next to us, and lowers the pitch as it moves away from us. Curtis has an electronic sound device that emits a sound wave with just one pitch. As he twirls the sound emitter in circles we can hear the sound waves change pitch to higher and lower pitches as the emitter moves towards and away from us. 1.6.2 Shifting with light This raising and lowering of pitch is known as the Doppler Effect, and as we hinted at earlier, it also applies to light. As light is emitted from a source, the waves being emitted can be squashed or stretch along the direction of motion. So if a star is moving towards us, the lightwaves moving ahead of the star can appear to increase in frequency or decrease in wavelength. This translates as a shift towards the bluer end of the electromagnetic spectrum. Conversely, as a star moves away from us, the waves will appear to be stretched, resulting in a longer wavelength and a redder look to the star. For light, the shifts caused by the Doppler effect are known as blueshift and redshift, although we should not that this is not confined to only the visible band of the electromag- netic spectrum. Both X-rays and radio waves can also be blue or red shifted, the colours just speak to the direction if the shift. 1.6.3 astro-shifts While stars do not normally race around in cars, they are moving around in space. Many stars that are observed in the night sky are actually in binary star systems. In such systems, we see stars in orbit around one another. If we view the pair of stars from the side, it will appear that one star is moving towards us while the other star moves away. This relative motion is detected as blueshifts and redshifts of the light that we detect from these stars. In this binary star system we see two yellow stars moving in circles. An astronomer is watching the stars orbit from a location far away to the left. The upper star is moving away 21 from the observer at a speed “v”. The observer will measure a longer wavelength for the light emitted by this star. The lower star is moving towards the observer at speed “v”. The observer will measure a shorter wavelength for the light emitted by this star. How much does the colour of the light change? We can use the Doppler Shift Formula to calculate the change in wavelength. ∆λ = λ× v c (9) The wavelength of the emitted light is represented by the Greek letter lambda. The change in the wavelength is represented by the Greek symbols “Delta lambda”. The symbol “v” represents the speed of the star, while “c” represents the speed of light. Stars emit all the colours of the rainbow, which results in a slightly off-white colour for all stars. But as a simple example, let’s look at a pair of stars that only emit yellow light with a wavelength equal to 600 nm. In this example, the stars are moving at a speed v = 3×106 m/s that is one hundred times smaller than the speed of light, so v/c = 3×106/(3×108) = 1/100. If we put these numbers into the Doppler Shift formula we find that the change in the wavelength that the astronomer measures is ∆(λ) = λ× v/c = 600nmx1/100 = 6nm. The observer measures a wavelength for the upper star that is 6 nm longer than the emitted light, lambdaobs = λ + ∆λ = 600nm + 6nm = 606nm, and we say that the light is redshifted. The observer measures a wavelength for the lower star that is 6 nm shorter than the emitted light, lambdaobs = λ −∆λ = 600nm − 6nm = 594nm, and we say that the light is blueshifted. We say that the light is either redshifted or blueshifted. But if we look at the colour scale we see that 594 nm and 606 nm are still in the range that we call yellow. Most stars move at slow speeds, so a human would have trouble seeing the change in colour. We need sensitive instruments called spectrographs to detect the changes in wavelength due to a star’s motion. The same is true of galaxies. Spiral galaxies are found spinning and spiralling in space. In fact our own galaxy, the Milky Way takes about 240 million years to complete one full rotation. When we look out at other galaxies, we can measure the light emitted at various points across the disc to deduce the galaxy’s speed of rotation. When measuring light from stars, we see a shift in the light towards shorter wavelengths from the side of the galaxy approaching us, it is blue shifted. If we look at the side of the disc moving away from us, we can see a shift towards longer wavelengths, so it is redshifted. It is interesting to note, that while one of our closest neighbours, M31 or the Andromeda Galaxy, is rotating, we also see an overall blue shift of the whole galaxy, implying that M31 is moving towards us! While there are many other examples of the use of redshift and blueshift in astronomy, the most well known Doppler shift in astronomy triggered the idea of the expanding universe. In the early 1900’s, the prevailing theory was that the Milky Way, our own galaxy, was pretty much the extent of the Universe. In the early 1920’s, an astronomer named Edwin Hubble was working at the Mount Wilson Observatory, in the USA. He was making measurements of the distances of various nebulae, only to find that some, including what was then known as the Andromeda nebula, were far too distant to be part of our galaxy. Instead these objects must be galaxies in their own right. This meant that Andromeda 22 nebula became known as the Andromeda galaxy, the one I just mentioned a few moments ago. Although the idea of multiple galaxies had been proposed as early as the mid 1700’s, It’s strange to think that the concept of galaxies is so new, this idea was only conclusively proven about one hundred years ago! In 1929 Hubble continued these observations, and found a relationship between the dis- tance and redshift of galaxies. Hubble found that galaxies that are outside of our local group of galaxies have light that is redshifted. Not only is the light redshifted, he also saw that the further away a galaxy is from us, the larger the change in the wavelength, or the larger the redshift. If the change in wavelength is interpreted as a Doppler shift due to motion, then Hubble’s observations suggest that galaxies appear to move away from us. Galaxies that are farther away also move away from us at a faster speed. The modern interpretation of Hubble’s observations is that the Universe is expanding which makes it look like the galaxies are moving away from us. The description of the expansion of the Universe is called The Big Bang Theory. The Big Bang explains the evolution of the Universe after its beginning. 13.82 billion years later, we find ourselves here on the Earth . . . learning about black holes. Speaking of which, if we are wanting to understand black holes, we need to understand something about gravity. For that, lets start with the simplest version, Newtonian gravity. 1.7 Newtonian Gravity 1.7.1 Who is Newton? Gravity is the force that keeps us standing on Earth’s surface. It’s the reason that a ball thrown upwards falls back towards the ground. It was Newton who realized that this force, gravity, doesn’t just affect physical objects here on Earth, but it is also responsible for the motion of the stars and planets. Gravity keeps the Earth moving in orbit around the Sun, and the Sun in orbit around the supermassive black hole at the center of the Milky Way Galaxy. Gravity is a central principle in black hole physics, because it is gravity that gives them their extreme properties. Until the year 1687, the year that Sir Isaac Newton put forth his vision of gravity, no one had a clear understanding of what causes the attraction of objects towards the ground. A similarly mysterious force was also keeping the Earth moving around the Sun. Even in antiquity humans understood that something held objects in place, but lacked the mathematical description. It was Newton who provided the first empirical description of how gravity works. Although Newton was the first to explain gravity mathematically, almost exactly 100 years earlier, in 1589, Galileo Galilei was busy investigating gravity, and his observations greatly advanced our understanding of the interactions between objects and their masses. Galileo theorized that falling objects of different masses would fall at the same rate, contrary to the Aristotelian belief that heavy objects fall faster than light objects. It is famously claimed that to prove this idea, Galileo climbed the Leaning Tower of Pisa and dropped two cannonballs with different masses, one heavier, and one lighter. He observed 23 that if both cannonballs were dropped simultaneously, they hit the ground at precisely the same time, independent of their weights. Galileo made the mistake of assuming that the gravitational force was a constant between two objects , with no relationship to the distance between them. Historians disagree whether this experiment really took place, because it is first mentioned almost 65 years after it supposedly took place, in a biography of Galileo by Vincenzo Viviani. One experiment done during Apollo 15’s mission to the Moon, demonstrates the principle that Galileo addressed. At the end of the last moon walk, on the Apollo 15 mission, astronaut David Scott performed the same demonstration with a hammer and feather in the vacuum of space. The result of course, is visible in this famous video: Shortly after Galileo’s death, mathematician and astronomer Johannes Kepler observed that planets trace ellipses through the solar system as they orbit the Sun. Kepler famously described the motion of the planets mathematically, laying the groundwork for the second last piece of the gravity puzzle, which was solved by Christiaan Huygens, who in the 1660’s, described the Law of Centrifugal Force. Together with the help of Edmund Halley, Christopher Wren, and Robert Hooke, Isaac Newton finally had all the clues he needed to piece together a mathematical description of gravity. In 1687, Newton’s book “Philosophiae Naturalis Principia Mathematica”, which translates to the “Mathematical Principles of Natural Philosophy”, Newton lays the mathe- matical foundations to explain all gravity related phenomena, including apples falling from trees and planets in orbit around stars. 1.7.2 Universal Law of Gravitation Gravity is an attractive force between two objects that have mass. Any object that we talk about in this course, with the exception of light, has mass. The Earth has mass, I have mass, and you have mass! There is therefore a gravitational attraction between the Earth and me, the Earth and you, but also between you and I, at any given time! The mathematical description of the force of gravity needs to take into account the mass of both objects, and also the distance in between them. In order to get useful information out of any equation, we also need a Universal Gravitational constant to tell us how strong the force will be, given the masses and distance. Let’s call the mass of the larger object capital-M [M], and the mass of the smaller object little-m [m]. The distance between the objects will be measured by little-r [r], and the Universal Gravitational constant will be denoted G. The force of attraction between these two objects will be directly proportional to their masses, but inversely proportional to the square of the distance separating them. Direct proportionality means that the force, F, will be equal to the Universe Gravitational constant, G, times capital-M times little-m. Finally, because of the inverse-square relationship, we divide the whole right-hand-side of the equation by r2. F = GMm r2 (10) This equation is called Newton’s Universal Law of Gravitation, and calculates the force be- tween two objects, no matter what their masses. In order to use this equation, we need 24 to consider the units of each term. G, the Universal Gravitation constant, has a value of 6.67x10−11 in units of Newton Meters-squared per kilogram-squared, whoo, that’s a mouth- ful. To make these units cancel out, you can see that capital-M and little-m cancel out the kilogram-squared terms, and that the distance squared cancels out the meters-squared term, leaving behind Newtons, which are a measurement of force. Notice how tiny the gravitational constant is. If we ask ourselves, “How much attractive force is felt between two objects, each weighing one kilogram, separated by one meter?” The answer, of course, is G times 1 kilogram, times one kilogram, divided by one meter squared. . . So 6.67x10−11 Newtons, or 66.7 picoNewtons. For comparison, 66 picoNewtons is about how hard you have to pull on two ends of a DNA molecule in order for it to unravel, but gravity acts on much larger scales, and is therefore comparatively weak. 1.7.3 Earth Gravity Let’s compare that to the force of gravity that I feel due to the Earth. Earth weighs 5.97x1024 kilograms, and I weigh about 75 kilograms. In order to calculate the force of gravitational attraction, we replace capital-M with Earth’s mass, and little-m with my mass. We also need to know how far apart the centre of Earth is from the center of me! Let’s take the radius of Earth’s surface to be ‘r’, and replace it with 6,378.1 kilometers - which we have to convert into meters - so 6,378,100 meters, which we square. And finally we replace the Universal Gravitation constant G with its value of 6.67x10−11 and its units Newton-meters-squared divided by kilograms-squared. Together, the units of meters cancel each other out, as do the units of kilograms, leaving Netwons in the result. I’ll get my calculator out and plug in the math... and I get a result of 735 Newtons. So I’m being pulled towards the center of the Earth with a force of 735 Newtons. The unit of force, Newtons, is sometimes difficult to put into context. Its related clas- sically with the acceleration of a mass by Newton’s second law, F = ma, which relates the force on a mass to how quickly the mass accelerates. Since I feel the force of gravity as 734.9 Newtons, I can calculate my acceleration due to gravity by dividing by my mass, 75 kilograms - which results in an acceleration of 9.798 meters per second-squared. You might recognize the coincidence. The acceleration I feel is very close to the value of Earth’s acceleration due to gravity, which is often denoted as a little-g, and has an average value of 9.807 meters per second squared. The reason these two numbers are different is because the strength Earth’s gravity varies over its surface. For example, you weigh about half a percent heavier when you are at Earth’s poles than you do when you are at Earth’s equator. In fact, Earth’s gravity varies A LOT over its surface because of the different densities of rocks, and the geography of different regions. Earth’s gravity diminishes by about one-fifth of one-percent from the surface to an altitude of 5 kilometers, so your height above or below sea level is a factor, but geology can account for another one-one hundredth of a percent difference in gravity. This map of the globe represents the difference in Earth’s gravity from the average value. Red indicates stronger gravity, and blue indicates weaker. The data was collected by a pair of satellites called GRACE, the Gravity Recovery and Climate Experiment. GRACE uses changes in Earth’s gravity to measure changes to HUGE masses of ice in polar regions. If 25 gravity decreases, scientists can determine how much glacial ice is melting in those regions, and this data can even tell us where vast underground reservoirs of water are filling up. 1.7.4 Newton’s Laws of Motion If Newton had accomplished nothing but the mathematical formulation for the law of Gravity, he would still go down as one of history’s greatest physicists, but he contributed much more to our understanding of the Universe. He revolutionized our understanding of motion, forces, and mechanics, with his Three Laws of Motion. Newton’s three laws can be stated in the following way: Newton’s First Law: “An object at rest will stay at rest, unless a force acts upon it. An object in motion (specifically uniform motion) will stay in motion, unless a force acts upon it.” It is interesting that we distinguish between and object at rest, and an object moving with a uniform velocity. As we get deeper into this course, you’ll understand that these two examples, and object at rest, and an object in uniform motion, are themselves within an inertial frame of reference. We could also think about a rocket moving in outer space at a constant speed. Unless the rocket were to fire its thrusters to exert a force in the opposite direction, the rocket will continue moving at a constant speed forever. Newton’s first law is also called the law of **inertia**. Inertia is the resistance an object has to changing its state of motion. Newton’s Second Law: “An object acted upon by a force will experience an acceleration in proportion to its mass.” This is the famous formulation which is described by the equation: F=ma, that we used earlier. Any force acting on an object will produce an acceleration in proportion to the mass of the object. So, for any given force, a small mass will accelerate quickly, but a big mass will accelerate slowly. Think about using a small motor on both a small boat and huge ship, the motor delivers the same force, but the ships accelerate at much different rates! Newton’s Third Law: “For every action, there is an equal and opposite reaction.” Newton’s third law is a little hard to wrap your head around, but it basically means this: any force which is imparted on one object, must also be imparted equally upon another. In other words, for all the force the Earth’s gravity is pulling upon you, you are also exerting a force, pushing down upon the Earth! This point confused me for some time as a student. Why is it that we say Earth’s gravity has a value of 9.81 meters per second squared? That’s a measure of acceleration! When I’m standing still on Earth’s surface I’m not actually accelerating anywhere, my acceleration is zero! Well, the truth of the matter is that the strength you exert to stand is the force pushing back on the Earth! The net force between you and the Earth is zero! Well what about if you aren’t standing on Earth’s surface, but you’ve gone skydiving and you are falling freely through the air. In this case you are accelerating at 9.81 m/s2, but you should also consider that Earth is accelerating towards you! The force is the same for you and Earth, but the acceleration of the two is different because you and Earth have vastly different masses! In this case, Earth would accelerate towards you at a TINY rate of about 1.23x10−22 meters per second squared! 26 1.7.5 Acceleration Due to Gravity Newton’s 2nd law of motion means that when we apply a force to an object, it will accelerate. Therefore when you apply a gravitational force to an object, it will accelerate. If we take Netwon’s Law of Universal Gravitation, F = GMm/R2, and set the force equal to the F in Newton’s second law, F = ma, the little-m masses on both sides of the equation cancel each other out: GMm r2 = F, F = ma (11) GM r2 = a (12) This is a simple way of calculating the acceleration due to gravity! When I stand on the surface of a planet that has a radius r, and a mass capital-M, then the acceleration due to gravity at the surface is given by ag = GM r2 (13) with a value of Newton’s Universal Gravitational constant G times the object’s mass M, divided by the square of the radius of the planet. If the planet is Earth, we use the symbol “g” to represent the acceleration due to gravity. We say that a body has a gravitational field when it has the potential to accelerate nearby objects toward it. Newton’s equations are robust enough to send Rockets to other planetary bodies. In order to do so, we need to further tie the concept of gravitational potential energy, the energy required to climb through a gravitational field, in order to calculate. . . escape velocity. 1.8 Escape Velocity 1.8.1 Interlude on weight Before we go into orbit, let’s discuss an important difference in physics, the difference between weight and mass. Mass is a property of an object that we can describe as the ability of an object to resist acceleration. Weight on the other hand depends on the local gravitational field. Mass always stays the same. If my mass is 75 kg, I will be 75 kilograms whether I’m on Earth, on the Moon, or in deep space. Weight is actually a measurement of the force felt by an object within a gravitational field, which means that weight can change in different gravities. It is the product of the mass and the local gravitational acceleration Weight = m× g in multiples of 1 Earth Gravity. On the Moon, where the gravity is roughly one-sixth of Earth’s gravity my mass is still 75 kilograms, but my weight is reduced by a factor of 6. That means that on the Moon, my weight would be 12.5 kilograms, even though my mass is still 75 kilograms! 1.8.2 Weightlessness and Freefall What does it mean to be weightless, if weight depends on the local gravity? Well imagine a region of space so far from stars and planets, that the local gravitational field is very 27 close to zero. What would someone with a weight of 75 kilograms feel when there is zero gravitational force on them? They would feel a weight of ZERO KILOGRAMS. So, when an astronaut is floating freely in space, are they weightless? NO! It is a common misconception that astronauts experience weightlessness when they are above Earth’s atmosphere where gravity is weak. In fact, there is still enough gravity in the environment around Earth that they still have a measurable weight. However, this is different from experiencing freefall, acceleration in a gravitational field that is not restricted by any other forces. Astronauts feel weightless because both the spacecraft and the astronauts are in a state of freefall, even though they still have weight. A body is in freefall whenever gravity is the only force acting on it. If I throw a ball downwards towards the ground, either here or up high in space, the only force acting on the ball while it is moving is gravity. While it’s moving, it is in a state of freefall and it experiences weightlessness. Since the force of gravity acts on it, the ball accelerates and moves towards the Earth until it hits the Earth. What do you think will happen when the ball is thrown horizontally? Newton was the first to imagine what would happen if you climbed a tall mountain in order to fire a cannonball horizontally. Newton reasoned that the cannon ball would curve towards the Earth, due to gravity. If the cannonball was fired at a fast speed, it would go a longer distance. Eventually, if the cannonball could be fired fast enough, it would fall towards the ground on a curved trajectory that matches the curvature of the Earth! This was the first reckoning of orbital motion! This is very similar to how flying is described in Douglas Adam’s Hitchiker’s Guide to the Galaxy series where it is stated: ”There is an art to flying, or rather a knack. The knack lies in learning how to throw yourself at the ground and miss.” When an astronaut orbits the Earth in the International Space Station, the only force acting on the astronaut is gravity. The astronaut is travelling in a stable orbit around the Earth, so although gravity is pulling on the astronaut towards the Earth, the circular motion makes it possible for the astronaut to “miss” the Earth. One way to experience weightlessness without being in orbit, or at a vast distance from the Earth, is to fly in an airplane on a parabolic trajectory. Special aircraft that can withstand many times the force of gravity navigate to a high-altitude before climbing into an inverted parabolic flight path. During the arc of the parabola, the airplane, and the occupants within it, only experience the force of gravity, and therefore feel weightless! These moments feel like zero gravity, but only last about 20 seconds. . . the airplane can’t stay in freefall very long, for obvious reasons! 1.8.3 Escape Velocity in Modern Spaceflight Rockets, like the Saturn V that carried the crew of Apollo 11 to the moon, must expend energy to climb through Earth’s gravitational field. The speed of a spacecraft dictates how high it will go in a given scenario. So, just how much energy is required for a rocket to escape from a planet entirely? Let’s consider an example of a rocket escaping from Earth. Kinetic energy is the energy associated with the speed of an object, which is supplied to the rocket by burning fuel and expelling it from the rocket’s nozzles. The energy required to break the 28 gravitational grasp of a planet like Earth depends on the mass of the planet, as well as its size. When a speed is associated with the kinetic energy of a departing rocket, we call it the escape velocity. Earth has an escape velocity of velocity is 11.2 kilometers per second, more than 40,000 kilometers per hour. Let’s not get too carried away though, getting to space is much more complicated than merely getting a vehicle to the right speeds. This calculation considers the pure physics involved in climbing out of a gravitational well, so we ignore otherwise important factors like air resistance. 11.2 kilometers per second is the instantaneous velocity you’d need, travelling upwards from the Earth’s surface, in order to escape Earth’s gravitational well. At sea level, 11.2 kilometers per second is equivalent to Mach 33, which is fast enough to make the air around the spaceship into a boiling plasma, so instead, rockets accelerate out of the atmosphere starting at a standstill. Although we used Apollo 11 to introduce you to the concept of Escape Velocity, it’s worth pointing out that, in order to reach the moon, the astronauts never exceeded Earth’s escape velocity. Since the Moon is gravitationally bound to Earth, a voyage there hasn’t escaped from Earth’s gravitational force! The Moon itself is also trapped within Earth’s gravitational well. Out of all the spacecraft launched by humanity, only a few have reached Earth’ escape velocity (those spacecraft which travelled to other planets in the solar system). But a small subset of spacecraft have voyaged well beyond Earth’s grasp, and escaped from the gravitational pull of the entire solar system. One such spacecraft, Voyager 2, launched in 1977, is now considered an interstellar traveller. The red line in this graph represents the changes in speed experienced by Voyager 2, from 1977 to 1989 on its journey past the outer planets. In order for Voyager 2 to achieve escape velocity from our solar system, it needed a gravity assist from the planet Jupiter. A gravity assist is a way for a space probe to boost its kinetic energy by stealing from the orbital energy of a heavy body, like Jupiter. Over the course of Voyager 2’s transit through the solar system, it was repeatedly boosted by encounters with planets: Saturn, Uranus, and Neptune chipped in too! At present , Voyager 2 is travelling at 15.4 kilometers per second, on its way to the outermost edge of the solar system. By contrast, the fastest humans have ever travelled was accomplished by the crew of Apollo 10 in 1969, achieving a top speed of 11.08 kilometers per second. However, their speed record was on their way back through Earth’s atmosphere! But even faster than the Voyager spacecraft, the current speed record held by a human object, is the New Horizons probe, which took pictures of Pluto in a flyby in 2015. New Horizons accelerated away from Earth, achieving a whopping 16.26 kilometers per second, making it the fastest spacecraft ever launched. 1.8.4 Escape Velocity Formula Deriving the formula for escape velocity is relatively straightforward. It involves setting the gravitational potential energy equation of an object on the surface of a body, G M m over R equal to its kinetic energy, one-half m ve squared: GMm r = 1 2 mv2e (14) 29 Since the little-m mass, which represents the mass of the object you want to move, appears on both sides of the equation, we can eliminate those from the equation. This means that the escape velocity of an object does not depend on its mass. Finally we can rearrange the terms of this equation, solving for ve: ve = √ 2GM r (15) So escape velocity, ve, can be calculated by multiplying two times the Universal Gravitational constant, G, the mass of the body, M, and dividing by the radius of its surface, r. . . all taken under a square root! So, increasing the mass of an object will increase escape velocity, and decreasing the radius of an object will also increase escape velocity. In order for spacecraft to escape from Earth, Escape velocity is required to ensure the gravitational potential well can be climbed. In fact, if you’d like to explore these ideas further, we’ve created an Escape Velocity Calculator that you can use to plan a mission across the Solar System. What could escape velocity possibly have to do with black holes? Chew on this for a minute: what if we calculated the density of an object, for which the speed of light is equal to the object’s escape velocity. . . The answer is something sinister. . . something. . . dark. 1.9 Dark Stars 1.9.1 John Michell If I asked you who first proposed the idea of a black hole, who would be your first guess? Perhaps Albert Einstein, Stephen Hawking, or Karl Schwarzschild? While these scientists have had a huge impact on black hole astrophysics, the idea of strong gravitational fields altering light was first described by an often-overlooked clergyman named John Michell. John Michell was the first to describe an object whose escape velocity exceeded the speed of light, or so called “Dark Stars”. The year was 1783, falling very close to the midpoint between Newton’s theory of Univer- sal Gravitation, and Einstein’s theory of Special Relativity. John Michell, a retired professor of Geology at Cambridge, was working as the Rector of Thornhill in England, and he used his spare time to fuel his scientific curiosity. In particular, working with the theories of light and gravity. John supposed that light consisted of particles (a topic of hot debate at the time), and that gravity acted upon the particles of light in the same way that gravity acts on objects. At the time there was no experimental evidence to think otherwise, and Newton’s gravity was considered a “Universal Law”. Rector Michell reasoned that objects within a gravity well require a certain amount of speed to reach infinity, the speed which we called Escape Velocity, and that for particular small and dense objects, Escape Velocity might exceed the speed of light! The French mathematician Pierre-Simon Laplace came up with the same idea in 1796, which he referred to as an invisible body. Although Laplace first wrote about invisible bodies in 1796, more than 10 years after Michell, this idea was probably developed independently 30 since there was very little scientific communication between France and England at that time. 1.9.2 Trapping Light Let’s have a look at the escape velocity equation again, but this time let’s do something silly. Instead of solving for the velocity, ve, let’s solve for the radius of an object with mass M, whose escape velocity is the speed of light, just a John Michell did! ve = √ 2GM r (16) Let’s denote the speed of light as the letter ‘c’, and use it to replace ve. c = √ 2GM r (17) In order to solve for the radius, ‘r’, we need to first square both sides of the equation so that c becomes c-squared, and the square-root sign on the right hand side goes away: c2 = 2GM r (18) And now we can multiply both sides by a factor of ‘r’ divided by c-squared, leaving us with the solution in terms of the radius. A body of mass M has an escape velocity equal to the speed of light when its radius is, r equals two Gee Em divided by c-squared: r = 2GM c2 (19) So what this means, is that given an object of mass M, we can calculate how small it would need to be in order to have an escape velocity equal to the speed of light! Let’s try Earth’s mass for fun. Inserting M equals 5.972×1024 kg into the equation yields a radius of a puny 8.87 millimeters, like this tiny ball less than one centimeter on a side! So if this ball weighed the same as the entire Earth, it would have an escape velocity equal to the speed of light from its surface! 1.9.3 Dark Stars Our Sun’s escape velocity is 617.7 kilometers per second, given a solar mass and the average solar radius. In order for the Sun’s escape velocity to increase to the speed of light, or 300,000 kilometers per second, its radius would have to be reduced from 695 700 kilometers to a radius smaller than 2.953 kilometers. What would the Sun looks like if it were compressed to 2.953 kilometers? With its escape velocity equal to the speed of light, light would no longer escape from it, so it would appear dark. Also, any light falling towards the Sun would disappear completely the moment it crossed the Sun’s dark surface. In fact, nothing could escape from the object’s surface, because we know that the speed of light is an upper limit in the universe. 31 Using only classical physics, Michell was the first to describe “Dark Stars”, by trying to determine a method for measuring the distance and brightness of stars. Instead, he invented the first description of a black hole: an object massive enough to prevent light from escaping it. Additionally, Michell also predicted one of the most interesting results in black hole physics. You see, the equation that we naively replaced Escape Velocity for the speed of light? Well that equation comes up again once we encounter General Relativity, as the solution for the event horizon of the simplest kinds of black holes, Schwarzschild black holes. r = 2GM c2 (20) 1.10 What is a Black Hole? 1.10.1 Dark Star versus Black Hole The basic idea behind a dark star only requires the knowledge of 18th century physics. If a star is dense enough, its escape velocity will be the speed of light, making it impossible for light emitted by the star to escape the star’s gravity! The idea of a dark star as proposed by John Michell is not correct, but it is still important since it introduces some ideas that apply to black holes even in modern theories. The problem is that the concept of a dark star uses Newton’s older theory of gravity, instead of Einstein’s newer theory. That being said, Newton’s theory of gravity is pretty good approximation to Einstein’s theory of gravity when gravity is weak. It was good enough for us to plan a mission to the Moon! What I mean by weak is this: calculate the escape velocity from a planet or star, and compare the value of its escape velocity to the speed of light. If the escape velocity is tiny compared to the speed of light, then we say that gravity is weak, and Newton’s theory of gravity is a good enough approximation. For example, the escape velocity from the Earth is 11.2 km/s, but the speed of light is approximately c = 27, 000ve (21) Twenty-seven thousand times larger. Since the escape velocity from Earth is so small com- pared to the speed of light, Newton’s gravity is good enough for most calculations near the surface of the Earth. But if the escape velocity is larger, say 10% the speed of light, or larger, that means that Newton’s theory of gravity is no longer sufficient to calculate the strength of gravity. Since Albert Einstein’s equations correctly describe relativistic effects at high speeds, they improved on Newton’s theory of gravity. This means we can predict what happens in sit- uations with strong gravity. Einstein’s theory of gravity is called the Theory of General Relativity. In general relativity, mass, energy, and angular momentum are all responsible for creating curvature in spacetime. The curvature of spacetime then causes planets, stars, and light to travel on curved paths. To create a dark star we might start with a large star and compress it inwards to make it smaller and denser while keeping the amount of mass unchanged. As the star shrinks in 32 size, the escape velocity from the surface becomes faster and faster until it becomes equal to the speed of light. At this point, Newton’s theory of gravity just predicts that light won’t be able to escape from the star and it will appear dark. However, the predictions from Einstein’s theory of gravity demonstrate a so-called “dark star” would exert a much stronger force due to gravity than predicted by Newton. This additional inwards gravitational force makes it impossible for a star to have a stable size. In order for stars to exist, there is a delicate balance between its gas molecules, which exert a net outwards pressure, that is exactly balanced by the attraction of gravity, allowing stars to stay the same size over time. When a star gets so small that its escape velocity is the speed of light, then the required outward gas pressure is infinite! There is no way to create infinite gas pressure, so the star is unstable and begins collapsing inwards. A black hole is what remains after a star is unable to resist gravity and collapses inwards. A black hole does not have a surface, but there is a special boundary that surrounds a black hole called an event horizon. In the case of the simplest black hole, the event horizon is a sphere with a radius called the Schwarzschild radius with the value REH = 2GM c2 (22) Event horizon radius equals 2 times G times the mass of the black hole, divided by the speed of light squared. The amazing thing about the formula for the event horizon radius is that it is exactly the same equation that Michell derived for the radius of a dark star! The event horizon radius is a boundary for light rays. If an astronaut shines a flashlight outside of the event horizon, the light rays can escape and be seen by astronomers far away from the black hole. But if the flashlight is at or inside of the event horizon all light emitted will be trapped inside of the black hole. And it’s not just light! Massive objects like cakes, or rockets, or astronauts can escape as long as they are outside of the event horizon radius and their rocket is good enough. But if a cake-eating astronaut crosses the boundary defined by a black hole’s event horizon, no escape is possible. The name “black hole” didn’t enter common usage until 1967, where it was popularized by John Wheeler. Before then, astronomers used the name “totally gravitationally collapsed objects” to describe black holes. This is an accurate phrase, but difficult to say, so it’s not surprising that the name “black hole” caught on so quickly with scientists and science fiction writers alike! The distinguishing difference between Michell’s dark stars, and black holes as they are described in General Relativity, is whether or not the star within the dark boundary main- tains a surface. Michell didn’t consider what would happen to the surface of a star when its escape velocity reaches the speed of light. Scientists now believe that the creation of an event horizon causes all the material hidden behind it to continue collapsing inwards, with no chance of a stable surface. 1.10.2 What isn’t a Black Hole? There are other dark objects that astronomers make reference to, but they aren’t black holes. For instance, there are dark nebula, which consist of clouds of cool molecules and dust that 33 block out passing light. These types nebula can be observed if they lie between us and a bright source of light since we will see that some light is blocked out by the nebula. One famous example is the Horsehead Nebula. The horsehead is a dense, cool, cloud that blocks out the red light that is emitted behind it, allowing us to see it. In addition, dust emits infrared light, so we can detect dust clouds if we use an infrared telescope! Dark matter is a hypothesized type of matter that was introduced to explain the motions of stars and gas in galaxies. Dark matter is a type of matter that doesn’t emit light, which means it can’t be observed directly. However, dark matter does have mass, so there is a mutual gravitational attraction between dark matter and the stars and gas in a galaxy. The gravitational attraction of the dark matter affects how the stars in a galaxy move, allowing scientists to infer the existence of dark matter by their observations with theoretical models. In the 1970’s Vera Rubin observed spiral galaxies and measured the speeds of their stars. She showed that the fast speeds of these stars implies the existence of dark matter. A tiny amount of the dark matter could be black holes, but most of the dark matter is a type of particle called a WIMP, which means Weakly Interacting Massive Particle. Physicists are trying to detect WIMPs using the Large Hadron Collider in Geneva Switzerland and SNOLAB in Sudbury Canada as well as other laboratories. So far the dark matter WIMPs have not been detected. The main thing that dark matter and black holes have in common is that they are both detected by observing their gravitational interactions with luminous objects! Dark energy is the name for another mysterious force which appears to act in opposition to the force of gravity. When astronomers observe galaxies far, far away, they measure that more distant galaxies appear to move away from us more quickly than galaxies that are close. This is one of the pieces of evidence that our Universe is expanding from a moment in our history, referred to as the “Big Bang”. Since all these galaxies have mass they are gravitationally attracted to each other, we might expect that the rate of the Universe’s expansion should slow down over time. Instead, there is evidence that the expansion is speeding up, as if there were a repulsive force, like a very large-scale kind of “anti-gravity”. This force, called Dark Energy, has nothing to do with black holes. However, there are some theorists who have considered types of stars that have some dark energy in them to help combat gravitational collapse. 1.10.3 Are Black Holes Dangerous? Black holes may give some people melanoheliophobia, but in most ways they are no more dangerous than any other star in the sky. For instance, entering into a black hole is dangerous, since once you pass through the event horizon, you can’t get out. But if you enter into a star, the hot gas would burn you up too. I would say they are both equally dangerous. There are safe ways to visit a star or a black hole. Instead of travelling directly towards a black hole, you could orbit the black hole, just as you can orbit around a star. For instance, the Earth orbits around the Sun in a safe, stable orbit. Similarly, the Earth could orbit a black hole with the same mass as the Sun, and at the same distance, making the orbit just as safe and stable as it is now. Unfortunately it would be very cold around a black hole since the sunlight that warmed us would no longer be able to reach us! There is nothing about the black hole’s gravity that would suck in the Earth. 34 Black holes *can* become dangerous if they are surrounded by an orbiting disc of hot gas, that looks similar to the rings of Saturn. The disc of gas could emit high energy X-rays, so if you were to approach the black hole’s disc you could receive an unhealthy dose of radiation. For this reason, in the movie Interstellar, the script writers decided to make the disc of gas orbiting their black hole be relatively cool so that it only emits visible light and no harmful x-rays. The tidal force, which is a difference in the strength of gravity at different locations, can become very strong around a black hole. In fact, when it comes to the tidal force, the smaller a black hole is, the more dangerous it becomes. An astronaut venturing too close to a small black hole would be stretched by gravity into long thin spaghetti-like strands. Out of all the types of black holes, the most dangerous are thought to be isolated black holes. In isolation, a black hole does not have a companion star or an orbiting disc of gas, making them extremely difficult to see. Due to their difficulty of detection, it’s possible that you could accidentally stumble across one and inadvertently cross into its event horizon while you are exploring the Universe. Gravitational lensing by the black hole’s mass will distort the images of background stars, so the presence of an isolated black hole could still be deduced, if you are careful! 1.10.4 Black Holes in Manga? (Interview with Dr. Mimi Okabe) Q: What is your favourite depiction of a black hole in manga? Dr. Okabe: This is actually from a Japanese manga series called Doraemon and it features a robotic cat from the 22nd century who helps this human boy throughout his childhood years until adulthood. In this particular story, it begins with the boy, his name is Nobita, who leaves the dinner table because he’s unable to eat which makes his parents really worry about him. The robotic cat finds a solution and he pulls out this futuristic device from his magical pocket called the mini black hole. And the black hole is kind of explained in this little panel here as a graveyard of outer space. It swallows up everything in sight and not even light can escape it. What Doraemon, the little the robotic cat, does it gives Nobita this little speck of the black hole which he consumes and it enables him to finish his meal. And then the following day, Nobita is talking to his friend about how many bowls of rice he can eat and this leads to the friends having an eating contest. In order to win, Nobita goes back home and eats more of the black hole which eventually does enable him to win the contest but then he falls asleep after eating such a big meal. Every single time he inhales and exhales, because he’s snoring, so when he inhales everything that’s within his gravitational pull gets sucked in. The story ends with his mother really frustrated at the fact that all this junk has accumulated at their doorstep. I think this is another very playful and interesting way that mediates our fears of the black hole. Q: How are black holes depicted in Japanese popular culture? Dr. Okabe: I think in Japanese popular culture the black hole is made manifest in the form of like a superpower. And I’m only going to talk about some of them but everyone’s familiar with Pokemon. Yes, okay I don’t know very many people who don’t know what Pokemon is. But you see here this is a Pokemon named Gardevoir. That’s a genderless Pokemon and it has this power to generate a black hole as a kind of a last resort to use it to protect its Pokemon master if he or she is faced in mortal danger. 35 The other one is this guy. So this is an action figure from Japanese manga and anime series called Kinnikuman or muscle man. He’s the villain of that series in particular. He’s basically a personification of the black hole as you can see BH, by black hole. He has a superpower to generate several black holes at once in which he can kind of travel through them. So the idea of the worm hole and the black hole is slightly confused it seems. But his ultimate power is that his head basically functions like a vacuum in which he sucks up his opponents and throws them into this alternate universe and supposedly the black hole. Of course the science is never really explained. But the black hole also tends to be an attribute of villains. But not always the case. I guess my last example. I show this one because it’s also a very popular anime series in Canada. So this guy is featured in a Japanese anime as well as manga called Inuyasha which was broadcast in Canada sometime ago. I grew up watching it here. But he has his power or rather the male line in his family is cursed with this black hole on his palm. It’s never really called the black hole in the anime series but it functions in the same way and that every single time he opens up his palm, whatever’s within his gravitational pull gets sucked in and overuse of this power risks him being sucked in to his own hand which I guess scientifically it doesn’t make any sense. Yeah but this guy’s name is Miroku from Inuyasha actually. So, those are just some I guess fun ways, entertaining ways in which the the black hole is represented in Japanese popular culture and anime as a kind of superpower. But it’s not only in Japan actually, in North America you have Silver Surfer. He’s a Marvel hero. I think he’s able to manipulate cosmic power and generate black holes as the superpower too. It’s very cool. Dr. Okabe’s webpage: https://www.ualberta.ca/modern-languages-and-cultural-studies/ people/contract-academic-staff/mimi-okabe.html 1.11 Black Hole Basics In science fiction, plots often involve travelling astronomical distances. This is often achieved by using a type of “warp drive” which circumvents the limit imposed by the speed of light, allowing faster-than-light travel. Warp drive is central to space travel in shows like Star Trek, putting it squarely in the realm of fiction. But there are some physicists who have proposed ideas for warp-drive-like phenomena. One such idea is the Alcubierre drive, named after theoretical physicist Miguel Alcubierre, which would warp spacetime with exotic forms of matter. Alternatively many novels, movies and TV shows look for “short-cuts” through space called wormholes. Traversable wormholes have been used in many science fiction stories, such as Carl Sagan’s Contact, and Star Trek’s Deep Space Nine. These traversable wormhole are better than black holes for long-distance space travel, but unfortunately they require large amounts of an exotic type of matter undiscovered to science. Unlike wormholes, there is strong evidence that black holes exist in our universe. Black holes fall into several categories. Some black holes are formed by natural methods, such as the collapse of a high mass star. Black holes with masses ranging from five times the Sun’s mass to about 60 times the Sun’s mass, called stellar mass black holes, have been observed in our own galaxy and in nearby galaxies. We also see evidence of supermassive 36 black holes with masses that are about a million to 10 billion times more massive than the Sun at the centres of galaxies. It’s also possible to form mini black holes artificially in experiments involving colliding proton beams at the Large Hadron Collider. To start us off in our journey to a black hole, we need to learn more about the destination. Specifically, let’s begin our understanding of the lifecycle of black holes by examining how they are formed in the first place. . . Through the birth, life, and death of massive stars: the progenitors of black holes. 37 2 Life and Death of a Star 2.1 Introduction Just as the natural cycle of life ends with death, so too does a star follow a familiar timeline: birth, life, and eventually... yes. . . death. Let me reassure you right now, our Sun is in no danger of dying anytime soon, unless you consider 5 BILLION YEARS soon. In the distant future the Sun will die, but that’s a good thing: the death of a star provides fertile material and fresh elements for the formation of new stars and star systems. In fact, our own solar system is thought to contain elements created by the death of several earlier star systems. All the elements heavier than hydrogen, helium, and lithium are produced in the fusion reactions of stars. In order to understand how stars come to die, we need to first understand how they live. Most stars, like our Sun, live long and uneventful lives. Some astronomers might even call stars like our Sun “boring”! These stable, long-lived stars make great neighbourhoods for planetary formation; and assuming a planet is at just the right distance from its host star, we might expect life to spring forth from a primordial ocean. Some of the most interesting objects in the universe are also produced in the final stages of a star’s life. Strange remnants persist after the death of a star: stars like our Sun produce white dwarf stars and some high mass stars result in the formation of neutron stars. Instead of a gentle death, high mass stars use up their fuel very quickly, and die in MASSIVE supernova explosions. If the star is massive enough, its death results in and even more exotic object. . . You guessed it, a BLACK HOLE! 2.2 The Stellar Nursery 2.2.1 What is a star? In order to fully understand the story of black holes, it’s important that we start at the beginning. To know why black holes are formed, we must first understand why the objects that form black holes are formed. As stated in the introduction to this module, stellar mass black holes are one of the two possible products of violent explosions of high mass stars which occur at the end of the star’s life. These explosions, called Type-II or core-collapse supernovae, occur in stars at least eight times as massive as our Sun. When a massive star experiences a supernova event, the amount of energy released is on the order of ten to the forty-six joules. That’s enough energy to last the Sun, at its present rate of energy output, 825 billion years. For reference, our Solar System, along with our Sun, has existed for just 5 billion years. The Universe has existed for a mere 13.8 billion years. Clearly that is a huge amount of energy! If you’re anything like me, right now you’d have a ton of questions, starting with: How is it that some stars meet such violent ends? How do stars even form in the first place? Tech- nically speaking, what is a star? Let’s begin by answering the simplest of those questions, “What is a star?”. Simply put, a star is a big ball of gas. 38 A ball of gas which is gravitationally bound, and dense and hot enough to sustain a nuclear fusion reaction at its core. Our Sun is one such object. It, like all other main sequence stars, produces energy by fusing hydrogen into helium in its core. Most stars are spherical, or, if they happen to rotate quite quickly, we call them oblate spheroidal, because they’re slightly squished. 2.2.2 Forming Stars So now that we have a working definition of what a star is, let’s move on to the next question: “How do stars form?”. Stars form in clouds of gas and dust which are particularly cold and dense, at least by interstellar standards. These regions are known as molecular clouds, because their temperatures are low enough to allow molecules to form. Molecular clouds are just one component of all the gas and dust in the space between stars, known as the interstellar medium, or ISM. What distinguishes molecular clouds from other gas and dust in the ISM is the effect gravity has on them. Molecular clouds are cold (between 10 and 30 Kelvin) and dense (several hundred molecules per cubic centimetre), meaning there are plenty of particles in close proximity to each other, and at the same time, relatively little gas pressure. These two conditions are each very important for star formation, as they allow the inward force of gravity to overpower the outward force of the gas pressure, and initiate the collapse of the cloud. As the cloud contracts, it releases gravitational potential energy. This energy is converted into thermal energy which in turn increases the pressure within the gas. Without some way of removing thermal energy, the gas pressure would build and eventually stop contraction of the cloud altogether, prior to the formation of a star. What is needed, then, is a way to get energy out. A way to get energy out so that gravity still has the advantage and contraction can continue. Thermal energy manifests itself in the random motions and frequent collisions of molecules. Collisions between molecules in the gas cloud can excite the molecules, allowing them to pro- duce light that can escape the cloud. And so without a buildup of thermal energy and gas pressure, the cloud is free to continue contracting. However, as contraction continues the central region within the cloud eventually becomes so dense that light emitted by molecules and by dust grains has a hard time escaping. More particles present in a given volume of the cloud means an increased likelihood of absorption of the light by other molecules and subsequent conversion of that energy back into thermal energy. Over time, the cloud’s increasing density will result in nearly all of the radiation being trapped within the central region of the cloud. When this radiation trapping occurs, pressure in the central region increases to a level which slows the rate of contraction: this is the formation of a protostar. When observed through telescopes, protostars look much the same as regular stars in that they have similar luminosities and surface temperatures. The difference lies underneath, as protostars are not yet hot enough to sustain fusion reactions. In order to become hot enough to sustain fusion, protostars must gather more material and squish it. Material surrounding the protostar feeds down onto it, and at the same time gravity continues to slowly squish this protostellar material into smaller and smaller regions. 39 As the protostar contracts and heats the fusion rate increases, and the heat generated by these nuclear reactions provides a pressure force that slows the contraction caused by gravity. When the core temperature of the protostar reaches about a million Kelvin, the winds generated at the protostar’s surface blow the surrounding gas and dust away, ending the accretion phase. Now without its source of additional material, the protostar continues to slowly contract and heat until the core temperature reaches ten million Kelvin, at which point fusion becomes stable ...and we have a star! Fusion rates become stable because the forces in the interior of the star become balanced. The nuclear reaction rate is now high enough that it produces the necessary heat and pressure to prevent the star from collapsing further due to gravity. When the gravity and gas pressure forces are in balance, we call this state hydrostatic equilibrium. The net force on material within the star is zero. The star can remain stable in this state for billions of years. Our Sun is currently about 5 billion years old and in a state of hydrostatic equilibrium. It will remain in this stable state for another 5 billion years. It’s time again for us to confess about a lie of omission we’ve been telling. Until this point we’ve been considering a scenario of star formation which isn’t perfectly realistic. We’ve been considering a single cloud in isolation, when in reality, individual sites of star formation are often influenced by other nearby sites of formation, and by nearby newborn stars. In truth, large molecular clouds fragment as they contract into several smaller cloud cores and from these one or more stars form. Often what we have is several neighbouring sites potentially each producing several stars. And then there’s the matter of those additional dynamical aspects we also “forgot” to mention. More than just two forces are present in molecular clouds as they contract. In addition to gravity and gas pressure, magnetic fields affect molecular clouds by slowing their contraction. Magnetic fields cause particles in the cloud to move in such a way that they exert a friction on each other, hindering motion within the gas and helping to “prop” up the cloud against gravity. Turbulence also plays an important role. Gas clumps moving relative to each other at large speeds act to shear the cloud apart rather than facilitate the cloud’s contraction. In the later stages of star formation, material surrounding a protostar will coalesce into a disk and the protostar itself will eject material from the system via large jets. So, sufficed to say, star formation is very complex. But, star formation is also incredibly commonplace. Several stars finish forming in our galaxy every year. And in total, our galaxy contains roughly a hundred billion stars. And the key to our final question, “How do some stars meet such violent ends?”, lies in the variety of stars which result from the formation scenario. 2.2.3 Where are the Sun’s siblings? (Interview with Dr. Erik Rosolowsky) Q: Where are the Sun’s siblings? Dr. Rosolowsky: One of the outcomes of our thinking about the sun growing up or being born in a neighborhood and then growing up with a bunch of other stars is that we actually have stellar siblings. Those stellar siblings have left the nest, this cluster in which we formed and have gone out, and we think that right now we have stellar siblings that are spread throughout the galaxy. The stars that are formed in clusters ultimately dissolve into 40 the background of stars within the galaxy. But if we look out, we can actually see individual stars out there that have basically the same pattern of chemical enrichment as the sun, and we think that those were the stars that we were formed with. We can never know because we’d have to unwind billions of years of going around the center of the galaxy and map it back and make sure we’re at the same place, and that’s something we really can’t understand from the dynamics of the stars around us. But the signature does point to every now and then we can look out and we pass one of the siblings stars near which we were born. Dr. Rosolowsky’s webpage: https://sites.ualberta.ca/~erosolow/ 2.3 Now That’s a Stellar Sequence! 2.3.1 The Main Sequence of Stars A Hertzsprung-Russell Diagram (or HR Diagram) is a tool common in astrophysics, used for the purpose of analyzing the properties of populations of stars. It’s a simple two axis plot with luminosity increasing as you go up on the vertical axis and temperature increasing as you move leftward on the horizontal axis (the temperature axis is flipped). By observing a large number stars and plotting each one as point on one of these diagrams we can begin to notice several patterns, perhaps the most striking of which is what we call the Main Sequence. The Main Sequence of stars represented on the HR diagram is a roughly diagonal swath of points stretching from the low luminosity, low temperature region of the diagram to the high luminosity, high temperature region of the diagram. These are the stars which originated from the formation scenario we described in the previous section. Fusion rates have stabilized in their cores and they are living out their “adulthood” in a state of hydrostatic equilibrium. The Main Sequence phase of a star’s life, when considered relative to formation and “retirement”, meaning prior to the death of the star, is the longest. As a note we consider “death” of a star to be any end state, such as the formation of a White Dwarf, Neutron Star, or Black Hole. We’ll see more about these in the coming sections. During the Main Sequence phase a stable source of fuel is present in the form of hydrogen, which the star consumes, converting it through fusion processes into helium. “Elderly” stars which have left the Main Sequence source their energy from not only hydrogen, but from other elements as well. For Main Sequence stars there is a strong relationship between mass and luminosity. The more massive the star, the brighter it is. The intense gravity of a massive star means its core will be denser, and as a result, hotter. This is important, because fusion rates, meaning the rate at which energy is produced, is highly dependent on core temperature. So the more massive a star, the more energy per unit time it produces, and as this energy leaves the core and eventually reaches the surface, we observe a greater luminosity, meaning a brighter star! 2.3.2 Where Does Star “Colour” Come From? You’ll notice in taking this course that we often refer to stars by their “colour”. So when we say colour, what do we mean? Well, as we learned in Module 01, electromagnetic radiation, or light, is a spectrum, visible light being just one portion of it. 41 What we call “blue” is just an even smaller portion of this spectrum. Instead of simply calling it blue, we could define it in numbers, because, as we know, light is characterized by its wavelength. Blue light has a wavelength of about 450 nanometres. So when we refer to a “blue” star, what we are saying is that much of its radiation is coming from this portion of the spectrum. When we say that a star is “bluer” than another star what we mean is that the bulk of the bluer star’s radiation is coming from even shorter wavelength light. The same can be said of red and redder stars, as a redder star will have the bulk of its radiation in longer wavelength light. Just like the filament of a lightbulb, a star’s light is produced by incandescence, or formally blackbody radiation. Blackbody radiation is temperature dependent. The hotter a blackbody radiator is, the brighter it is. As the temperature of a blackbody emitter increases or decreases it also “changes colour”. This is why colder stars appear dim and red, and hotter stars are brighter, and bluer. The hottest stars in the sky, blue hypergiants, are upwards of 40,000 degrees Centigrade, and can shine about 5 million times brighter than our own Sun! We say that the hotter stars are blue, and the colder stars are red, but the reality with blackbody radiation is that every star produces at least a little bit of every colour of the spectrum! When we say a star is blue, we are saying that the majority of its radiation is being produced in this portion of the spectrum. This majority exists because each blackbody emitter has a spectrum that is peaked, meaning there is wavelength at which the star produces more light than at any other wave- length. These peak wavelengths are directly related to the surface temperature of a star. The relationship is described by Wien’s Law which takes the form of this equation: λPeak = 0.0029 m K T (23) The peak wavelength, lambda peak, is equal to point zero zero two nine metres Kelvin divided the temperature, T, of the blackbody emitter. As we mentioned earlier, more massive stars are brighter because they’re more luminous as a result of having higher fusion rates. And despite having more material to burn, more massive stars live shorter lives. The more massive a main sequence star is, the quicker it exhausts the fuel in its core. A hot blue star might live on the Main Sequence for ten million years, whereas a dim red star could live as long or longer than a trillion years. That’s a hundred thousand times longer! This is why we like to personify stars. We like to think of blue stars as rock stars that “live fast and die young”. Red stars live long and much more uneventful lives. 2.3.3 Day and Night At midday there is one star visible to our eyes: the Sun! At this point you might be wondering where the Sun lies on an HR Diagram, meaning how does it compare to other stars? Well, the Sun happens to be pretty average. It’s not terribly hot or particularly cold. The Sun’s spectrum peaks at a wavelength of about 500 nanometres, which is greenish. It appears whitish or yellowish to our eyes because it is emitting a lot of light across the entire visible spectrum. 42 In terms of mass, the Sun lies fairly close to the lower end of what’s possible for a Main Sequence star. At the high end, massive blue stars can be a couple hundred times as massive as our Sun. At the low end, a red Main Sequence star can be as massive as a tenth of our Sun. The most massive star that is currently known lives at the edge of our galaxy. Scientists have given it the name R136a1, but we will call it Rob. Rob has been measured at a whopping 256 Solar masses. However, Rob has lost a great deal of material through fusion and winds, it’s thought that 20% of its mass has been ejected already. This would mean that at birth, Rob was about 320 solar masses! In terms of luminosity, the Sun is fairly average. Blue stars can be several million times as luminous as our Sun, and red stars can be much dimmer, such that their luminosity is only a ten thousandth that of our Sun. One of the difficulties associated with constructing an HR Diagram of a population of stars is figuring out how bright the stars actually are versus how bright they look to us in the night sky. Luminosity is a measure of how bright a star actually is. Apparent brightness is how bright a star will look to us. Apparent brightness is affected by distance, because the further away a star is, the dimmer it will appear to be. And we’re not interested strictly in appearances. We want to be able to infer characteristics of the stars themselves. What we need to do then, is measure the brightness of a star knowing already how far it is from us. That way, we can correct for its “appearance”, meaning its distance, and compare it fairly to other stars. Astronomers have spent hundreds of years constructing catalogues of objects with known properties (distances and luminosities), so that when we discover new objects we can know much more about them! Being able to measure apparent brightness and distance accurately is very important. It means we can construct an accurate HR Diagram. And that’s crucial, because an HR Diagram is a tool which allows us to learn even more, specifically characteristics about an entire population of stars. For instance, we can use an HR Diagram to learn about the age of a population of stars. We do this by determining the Main Sequence Turnoff Point. As a star exhausts the hydrogen fuel supply in its core, its time on the Main Sequence ends. This is because the surface of the star cools as its core runs out of fuel, and so it moves rightward on the HR Diagram, away from that diagonal swath of stars. Imagine we have a large population of stars with a large variety of masses, all born roughly at the same time. The hot blue stars, if you remember, “live fast and die young”. In other words, the stars in the high luminosity, high temperature region of the HR Diagram will exhaust the fuel in their cores first. They will move off the Main Sequence and continue with the later stages of their lives. Next to move off the Main Sequence will be the stars which are slightly less massive than those first stars, because, as you remember, the lower the mass of the star, the longer its Main Sequence lifetime. Over time the Main Sequence will slowly be eaten away as less and less massive stars begin to move away. The point along the Main Sequence where stars are “departing” is called the Main Sequence Turnoff Point. We can measure the properties of the stars at this point, and because we know how long every type of Main Sequence star typically lives, we can learn the age of the entire population of stars. 43 2.4 Energy Production in Stars 2.4.1 Building and Smashing Atoms Stars are powered by a NUCLEAR REACTION at their core. Fusion is the source of a star’s power, which keeps them hot, and allows them to produce light. Fusion is also the force that keeps a star from collapsing! Since the atoms at the core of the Sun are heated to incredible temperatures, their motion and collisions create a gas pressure that pushes material outward, counteracting the gravitational forces that pull material inward. This balance of outward gas pressure and inward gravitational forces keep a star in hydrostatic equilibrium. Chemical reactions are just one of the ways that energy can be released as heat. For millennia, humans have harnessed the energy of chemical reactions with the power of the campfire! Fire is a reaction that breaks the chemical bonds in materials like wood, releasing the excess energy as light and heat. However, the Sun requires a much more powerful source of energy to continuously burn for its 10 billion year lifetime. If the Sun were made of wood, and burned by conventional combustion, it would only last a few thousand years! Nuclear reactions are about a million times more energetic than chemical reactions, so they are a much better source of energy for stars to use. In fact, researchers here on Earth are trying to replicate the conditions at the centre of the Sun, so that humanity can enjoy the abundant energy of nuclear fusion. In order to understand the difference between chemical and nuclear reactions, we need to understand the structure of an atom, its nucleus, and some subatomic particles like protons and neutrons. All atoms consist of a small, dense nucleus, and a cloud of electrons bound by electromag- netic forces. Within the nucleus, there are two major components: protons and neutrons. Both protons and neutrons are made up of quarks, protons in such a way that they end up with a positive charge, and neutrons, which are neutrally charged. Both protons and neutrons weigh about the same, but the neutron is a tiny bit heavier. Since the proton has a positive charge, and like-charges repel, all the protons in the nucleus will repel one another. . . so why don’t nucleuses explode due to this repulsion?? Gravity and electromagnetism are only two of the four forces that exist in Nature. The strong nuclear force is the third, and is responsible for tightly binding Protons and Neutrons together! The strong nuclear force only works over very short distances, too short for us to expe- rience in everyday life, but it is so strong that it can overcome the electrostatic repulsion between protons within the nucleus. The fourth force in Nature is called the weak nu- clear force, and allows protons and neutrons to transform into one another. These types of transformations are the evidence we have that protons and neutrons aren’t themselves fundamental particles; they are composed of even smaller particles called quarks and gluons! On the other hand, electrons are fundamental particles, scientists don’t think we can take electrons apart into smaller particles. With a mass that is two-thousand times smaller than a proton, electrons are zippy particles that have negative electric charge. Additionally, all particles in Nature have an anti-particle, kinda like an evil counterpart. Anti-particles share the same mass as their normal-particle partners, but they have the opposite charge. For example, the antiparticle version of an electron is called a positron. When electrons and positrons come close to each other, they are attracted together by their 44 opposite charges, and they destroy each other in an explosion of pure energy! Finally, the tiniest particles involved in nuclear reactions are neutrinos, a name that means “little neutral ones”. Neutrons have a very very tiny mass, so small that it is difficult to measure! We call neutrinos “weakly-interacting” particles since they do not have electric charge, nor do they feel the strong nuclear force. The only forces that affect neutrinos is gravity (like all particles), and the weak nuclear force. This makes them very hard to detect, since they emit no light, and can pass through many thousands of kilometres of lead without colliding with any other particles! That’s our particle-physics recap. Now let’s look at some practical examples: the simplest atom, Hydrogen. Most Hydrogen atoms contain only one proton in the nucleus, with a single electron orbiting far from the atom’s nucleus. In cartoon pictures like this one, the orbitals are shown as circular planetary-like orbits, but that is not at all a correct picture of the atom! On small scales, the behaviour of atoms are governed by quantum physics, so a better picture of the hydrogen atom would be smeared out into probability clouds! A scale model of hydrogen wouldn’t look like this though. . . The distance between the electron and the proton is about 100,000 times wider than the radius of the proton. If you wanted to make a scale model of Hydrogen, the distance between the electron and the proton should be 100,000 times larger than the radius of the proton! This is often why you hear the claim that atoms are mostly empty space. For example, if this marble were the size of the proton, the electron would be more than a kilometer away! The proton, neutron, and electron are elementary particles. The strong and weak nuclear forces govern their behaviour at very high energies, but in regular everyday life, we interact with matter through the electromagnetic force, which governs chemistry 2.4.2 Chemical and Nuclear Reactions A typical chemical reaction, like hydrogen and oxygen reacting to form water, is a process that breaks and forms chemical bonds between atoms. These chemical bonds are a compli- cated function of how the electrons are shared by different types of atoms, different elements. For example, if two hydrogen atoms come together with an oxygen atom, they can form a molecule of water, H2O, by sharing electrons in a covalent bond. The production of water is an example of an exothermic reaction, which means that it releases heat. Chemical reactions interact through the electromagnetic force. What kind of reactions are nuclear reactions then? Nuclear reactions only take place between the protons and neutrons within the nuclei of the atoms. Since protons are all positively charged, they repel one another, but are held together by the strong nuclear force between the nucleons. By adding or subtracting protons and neutrons, new atomic nuclei can be created, but this takes a TREMENDOUS amount of energy. In a nuclear reactions, protons and neutrons can also be converted into one and another, and new atomic nuclei can be created. There are two types of nuclear reactions: fusion and fission reactions. For very large atoms, like Uranium-238, the proton-to-proton repulsion is so strong, that across the width of the atom, there is enough force to overcome the strong nuclear binding energy! Uranium-238 nuclei split on a timescale of 4.4 billion years, and when they do, they produce a Thorium atom through the emission of an alpha particle! Alpha particles are just naked helium nuclei, with no electrons to cover them up! When large atoms split into 45 smaller ones, we call the process nuclear fission. I remember that fission breaks nuclei apart, but the phrase “Fish ‘n Chips” or “Fission Chips”. When I eat FISSION chips, I break them into smaller pieces! Current nuclear reactor technologies here on Earth use fission reactions to release energy. NASA is even investigating nuclear fission for future space engines. You can also combine nuclei together, the reverse of fission, in a process called nuclear fusion. The word fusion means “the process of joining two or more things together to form a single entity”. In a nuclear fusion reaction, two or more small nuclei are joined together to form a bigger nucleus. Just like Jazz Fusion is a musical combination of Jazz, Funk, Rock, and Blues music so too can Protons and Neutrons JAZZ FUSION together to form bigger nuclei! Both fusion and fission reactions can release energy, but it depends on the details of the reaction. In the Sun, fusion reactions that combine the nuclei of four Hydrogen nuclei together to produce one Helium nucleus, plus energy, are what produces most of the Sun’s energy. 2.4.3 Nuclear Fusion in the Sun The most important nuclear reaction taking place in the core of the Sun is the fusion of Hydrogen into Helium. This reaction releases nuclear energy, the energy which powers the Sun. Rather than a cartoon, let’s approach hydrogen fusion with more scientific notation. . . 4 H → He + 2 e+ + 2 ν + γ (24) The reaction takes place in a few steps. Four hydrogen atoms produce one Helium nucleus plus two positrons plus two neutrinos plus one gamma ray photon. A positron is denoted as “e plus” and is the anti-matter partner of the electron. A neutrino is denoted with the greek letter “nu”. The greek letter “gamma” is used to denote light. The Hydrogen fusion reaction is called sometimes called the “p-p” chain since it involves protons (in other words “p”). We can add up the mass of the four original Hydrogen nuclei and compare it with the mass of the Helium nucleus that is produced. We find that the Helium weighs less than the sum of the 4 Hydrogen atoms weights! Some mass is lost during this fusion reaction! However, the mass is not really lost, it has transformed into thermal energy, as described by Einstein’s famous equation: E equals M C-squared. E = mc2 (25) In this equation, m is the mass lost in the equation, c is the speed of light, and E is the energy that is released by the reaction which now heats up the Sun’s core! The term c-squared is such a large number, that even the tiny mass that is consumed in the reaction is amplified to become a large energy. The difference in mass, before and after a nuclear reaction takes place, is called the mass defect. The larger the mass defect, the larger the amount of energy that will be released in the reaction. Another related quantity is the binding energy of a nucleus. Any nucleus is made up of N-numbers neutrons and Z-numbers of protons where N and Z depend on the 46 element. The binding energy is defined by adding the mass of all the protons and all the neutrons and subtracting the mass of the nucleus and multiplying by c-squared: Binding Energy = (Zmp +Nmn −mnucleus)c2 (26) This binding energy is the amount of energy that you can get out of a reaction if you bind all the protons and neutrons into a nucleus. Alternatively, if you want to rip apart a nucleus, the binding energy is the amount of energy that you have to apply. In order for the fusion of Hydrogen into Helium to take place, the positively charged protons have to come close to each other before they can fuse. But positive particles repel each other! We need to give the protons some extra energy so they can get close enough so the strong nuclear force can glue them together. This requires that the conditions in the Sun’s core be very hot and dense. Only the inner 25% of the Sun is hot and dense enough for nuclear fusion to take place. The centre of the Sun is about 25 million Kelvin. The outer parts of the Sun are too cool for nuclear reactions to take place. Since Hydrogen is slowly being transformed into Helium in the core of a star, this means that the star is slowly using up its fuel. Eventually the core of a star will be depleted of Hydrogen. The end of Hydrogen fusion in the core of a star signals the end of the Main Sequence stage of a star’s life. 2.4.4 The Iron Catastrophe Nuclear Fusion of Helium into Carbon also releases energy, so this and other nuclear reactions that build up higher mass elements can take place in stars. But the heaviest element that can be formed by the nuclear fusion at the core of stars is iron. Iron is a special element. In any nucleus, there is an interplay between the strong nuclear force which has a small distance range that glues together protons, and the electrostatic force, which is long range that wants to keep protons apart. Since the strong force is only strong at small distances, there is a special element, iron, which has the most tightly bound nucleus. This graph shows the binding energy of the nucleus of different elements. Energy is released in reactions that transform light elements into heavier elements, corresponding to downwards on this graph. Nuclear fusion releases energy when elements with masses as large as iron are formed. Similarly, nuclear fission can release energy as high mass elements are split to smaller ones, until they are as small as iron. But we cannot gain energy from iron by either breaking it apart, or by smashing two iron nuclei together! As no energy can be gained, a star that has accumulated iron in the process of nuclear fusion has nothing left for the star to feed on. Iron can’t be used as fuel, and the star must die! This is sometimes called the Iron Catastrophe, and leads to the death of a star. 2.5 Energy Loss in Stars 2.5.1 The Hundred Thousand Year Journey At the core of all stars there is a glowing nuclear furnace. We can be sure of this because we can measure how many neutrinos are streaming out of the Sun with experiments like 47 the Sudbury Neutrino Observatory. This property of neutrinos, that they can pass through the core of a star virtually unimpeded, is because neutrinos are weakly interacting particles, which is to say that they do not interact with electromagnetic forces. Light particles, or photons, DO interact electromagnetically, and thus they have a very difficult time leaving the core of a star. Let’s carve into our own Sun and examine how the energy produced at the core is radiated away as the light that we see here on Earth. The innermost region of the Sun, extending from the center to about one-quarter the radius of the Sun, is the region where nuclear fusion takes place. This is aptly named, the thermonuclear energy core. In this innermost region, the temperature, pressure, and density are extremely high – suitable for elements to combine in the process of fusion. The temperature has been estimated to be 1.55 times 10 to the power of 7 Kelvin, roughly 15 million degrees! The density of material is nearly 14 times the density of lead, and the pressure is almost a billion times atmospheric pressure on Earth. Needless to say, you wouldn’t want to go there, but if you did find yourself there you’d want to escape pretty fast! In order for energy to escape from this region, it needs to be carried away by one of three processes: conduction, convection, and radiation. Conduction, the propagation of heat through a solid, is inefficient in the Sun, because, well. . . the Sun isn’t a solid. In this cen- termost region, extending to about three-quarters the diameter of the Sun, energy is carried by radiative processes, the motion of photons! Extending outward from the thermonuclear energy core, the radiative zone is a region of the Sun where photons dominate the energy flow towards the surface of the Sun. You’d think that light, travelling just over a billion kilometers per hour, wouldn’t take long to escape from the core of the Sun. Indeed, if there was nothing impeding the progress of photons inside the Sun, they would take about 2.3 seconds to cross the Sun’s nearly 700 thousand kilometer radius! But photons are impeded by the materials the Sun is composed of. Instead of 2.3 seconds, photons take an average of one-hundred and seventy THOUSAND YEARS to escape into space!! That’s right! The energy that we observe in the form of sunlight has been working its way to the surface of the Sun for hundreds of thousands of years! Within the first three- quarters of the diameter of the Sun, photons bounce around on very short paths, randomly “walking” their way towards the surface. Each step taken by a photon amounts to about one centimeter, and the energy must be carried nearly seven hundred thousand kilometers! Just like the little balls in this toy version of a rain stick, photons jostle through the material of the sun, colliding and careening their way to the surface. Not every step will be directed towards the surface of the Sun, but photons will preferentially migrate towards the surface due to the temperature gradient. If you consider that this rain stick has a gravity gradient, instead of a temperature, it’s not a bad analogy for the motion of photons within the Sun. Have a look at the math: if you took a one centimeter step every second, it would take you over 2,000 years to walk 700,000 kilometers. But, that’s assuming you walk in a straight line! If you stumbled about randomly, say after getting off a dizzying rollercoaster, it would take you quite a bit longer to get where you want to go! Eventually, the energy from the core of the Sun stumbles its way to the surface! From the centermost thermonuclear energy core, encompassing the first one-quarter of 48 the Sun’s radius, through the radiative zone, from one-quarter to about seven-tenths of the Sun’s, photons finally encounter the convective zone, the outermost layer of the Sun’s surface. As you move outward the average temperature of the Sun drops. At the boundary of the convective zone, it is cool enough for electrons and protons to join in hydrogen atoms, which happen to be very good at absorbing photons. Radiative energy transfer is no longer the dominant process, and convection takes over. Atoms heated at the bottom of the convective zone become buoyant, and the mass of hot atoms and plasma rise to the surface, like bubbles in a lava lamp. Cooler gas at the surface of the Sun sinks down to replace the hot gas that is rising, and the process continues. Once the hot gas reaches the surface of the Sun, energy stored in the heat of the gas is emitted as blackbody photons, which can then travel across the vast distances of space (virtually) unimpeded! Now that there are finally some photons escaping the Sun, let’s put on our safe, solar- filtering glasses, and have a look! 2.6 The Sun’s Light and Life on Earth 2.6.1 Safety Owl’s Message ALERT! ALERT! DO NOT LOOK DIRECTLY AT THE SUN The Sun is bright! And our squishy, liquid-filled, human eyes have evolved for use in an average solar-irradiance environment. That’s a mouthful, but the lesson is simple: DO NOT LOOK DIRECTLY AT THE SUN, even using sunglasses!, doing so can permanently damage your eyes, and your ability to see! In extreme cases, you may become permanently blind! There are safe ways to look at the Sun, either by reducing the total light that we observe with our eyes, OR by narrowing the spectrum of interest into a short range of colours. Common light reducing tools are Number 14 Welder’s glass, or an astronomy-specific pair of Solar Sunglasses like these. Alright, now that we’re finally equipped to look at the Sun, let’s have a look. . . Wow. The Sun’s surface, which is called the photosphere, is a roiling inferno of activity! Bright patches mottle the surface, separated into small cells by darker boundaries. Once in awhile you encounter a very dark patch, a Sun Spot! If you look closely, you can also see hot gas above the surface following the Sun’s magnetic field lines. Every detail we see on the surface of the Sun is the result of the thermonuclear reaction at the core. The photons that eventually escape from the surface of the Sun are not the same ones that began the journey in the nuclear furnace at the stellar interior. Although the energy was produced by fusion, that energy went through several stages on its 100,000 year journey. The most notable was the journey through the convective zone, where our photon’s energy was locked away in the vibrations of the hydrogen and helium gases as they floated to the surface. Once exposed to the vast vacuum of space, the heat energy contained within those vibrations can now escape freely as photons. 49 Since these photons originated from hot dense gas, they are mostly the result of blackbody radiation. Some sprinklings of atomic hydrogen emission and absorption are present in the Sun’s spectrum, but the dominant component of the spectrum is the result of the Sun’s surface temperature. Since we can measure what the peak wavelength of the Sun’s blackbody emission is, we can use Wien’s law to calculate the average surface temperature of the Sun too! Since the Sun’s peak wavelength is around 500 nanometers, which is to say, a yellow- greenish colour, we’ll plug that number into Wien’s law in order to calculate the surface temperature. TK = (2.898× 10−3 mK) λpeak = 2.898× 10−3 mK × 5.00× 10−9 m (27) The Sun’s surface temperature is equal to Wein’s constant, two point eight nine eight times ten to the negative three, divided by the peak wavelength, 500 nanometers... Which results in a temperature of 5796 Kelvin! The fact that the Sun does not look green, when in fact it has a peak emission wavelength at green, is due to the fact that the Sun also emits lots of red and blue light, and when you combine red, green and blue light you get white light! But what about the strange pattern on the surface of the sun? What causes this pattern to emerge? These are called Solar Granules, and are the result of convection in the photosphere. Hot gases rising from the stellar interior are visible as bright patches of yellow, but what happens to them once they are at the surface? At the surface these gasses emit light, and in doing so, cool down. The cooler gases are now more dense, and therefore less buoyant, and they begin descend- ing in zones at the boundaries of the hot spots. These cooler gases are visible as the grain-like boundaries on the Sun’s surface, and this is where the cool gases begin their descent. The Sun’s photosphere is a layer of gas that becomes cooler in the outer layers. This image of the Sun’s visible light shows all the colours of the rainbow, and corresponds to blackbody emission from the lowest region of the photosphere. As the light travels outwards through the photosphere, some of the light with special colours is absorbed by the cooler Hydrogen and other elements. When the light is absorbed at these colours, we see a black line instead of the colour. We call this an absorption spectrum. The Sun doesn’t just produce good old-fashioned visible light, Wien’s law tells us that it also produces high-energy UV radiation and X-rays too! These images come from NASA’s Solar and Heliospheric Observatory, called SOHO for short, and show the Sun at wavelengths that our eyes cannot see. This image, for example, was taken with a peak wavelength in the ultraviolet, or 19.5 nanometers, revealing even more of the stellar atmosphere that was visible to the naked eye. Not only that, but in the UV, there is much more contrast, so activity in and above the photosphere is much more apparent. The outermost regions of the Sun’s atmosphere are called the Chromosphere and the Corona. The corona is much much hotter than the Sun’s surface, which scientists think is a result of the tremendous energy contained within the magnetic fields generated by the Sun. 50 2.6.2 Goldilocks and the Three Stars The energy produced at the core of the Sun is the same energy that (nearly) all of life depends on. From the water cycle, to the web of life, the majority of the energy required to sustain all plants and animal life on Earth comes from the Sun. The energy that we release when we burn hydrocarbon-based fuels can be traced back through geological time, to the moment when an ancient plant absorbed light from the sun, which eventually contributed to coal seams and oil wells where we derive the fuel for modern life. This is contrasted by modern renewable fuel sources like solar energy, that convert direct sunlight into electricity. In a sense, almost all energy sources can be considered “Solar Energy”. Aside: Explanations of the habitable zone typically exclude geothermal energy, and the extremophiles that live off the energy expelled by hydrothermal vents, since that energy comes from Earth’s formation and our molten rock core. One other non-solar energy sources is nuclear fission, which can occur naturally, but hasn’t been shown to support any known life forms. We happen to be in a relatively stable energy environment around the Sun. If Earth were closer to the Sun, like Mercury and Venus, it would be much too hot for life to exist as we know it. Similarly, if Earth were farther away from the Sun, like Mars’ orbital radius, it would be much too cold for most life forms to survive. Asking where a planet can have liquid water, or more importantly the conditions required to sustain life, is a fundamental question that many people have asked throughout human history. Astrobiologists, people who study both astronomy and biology, call this region the Habitable Zone. Generally speaking, the Habitable Zone is a region surrounding a star where a planetary body like Earth would be able to support liquid water. There is some flexibility in this definition, since it would include frigid planets where the maximum temperature is just above the freezing point of water, and intensely hot planets whose minimum temperature dips just below the boiling point of water. Let’s look at our own solar system, and see where we would draw the boundaries of the habitable zone. Scientists believe that our own solar system’s habitable zone extends a close to the Sun as Venus, at just under three quarters of an astronomical unit, to as far as twice Mars’ orbit, or twice 1.5 A.U. Arguments vary regarding what we consider “normal life”, since we have examples here on Earth of extremophiles that can live at extreme temperatures, so we’ll just have to leave the boundaries of the habitable zone to what they are: estimates. What if Earth were in orbit around a different type of star though, instead of the G-type star we currently orbit? An M-type star, like our nearest stellar neighbour Proxima Centauri, are the cooler Red-dwarf cousins of our Sun. Since they have a lower surface temperature, the habitable zone around it to contract to smaller distances. If Earth stayed the same distance from an M-type star like this as it is to our Sun, it would quickly turn into an icy snowball at the edge of the new habitable zone! On the other hand, if the Sun were suddenly replaced with a more massive, and therefore higher temperature star, like a blue-blazing B-type star like Bellatrix, the habitable zone would move outwards in the solar system, leaving Earth a burnt crisp! Similar effects can happen as stars age as well. As stars move off the Main Sequence 51 they enter a series of life stages where luminosity and radius tend to increase. We’ll see more about in the next section. 2.6.3 What is a Super-Earth? (Interview with Dr. Kelsey Hoffman) Q: What is a Super-Earth? Dr. Hoffman: In the field of exoplanet studies, something that has made me very interested and excited is the discovery of all these different planets that we have no analogs in our own solar system. So these are the planets that are more massive than the Earth and less massive than Neptune. So the so-called Super-Earths are Sub-Neptunes. The reason they’re so interesting is we don’t know what the composition is or the structure of these planets are. So people will talk about maybe the Super-Earths are habitable, but we don’t even know if we can have a rocky, hard surface for people to live on, on a Super-Earth. So I’m very interested in studying the structure and the composition of these planets. Q: What is the SETI Institute? Dr. Hoffman: So SETI Institute stands for the Search for Extra-Terrestrial Intelligence. It has two main parts; the SETI part where they have radio telescopes where they listen to see if they can hear any signals that could maybe become from alien sources. Another part of the SETI Institute is The Carl Sagan Institute, where people are doing work on trying to understand planets and habitability and asteroids. There’s people who work on Mars missions. Then a lot of people work on the Kepler project of looking for planets and so my work is to help look for those planets and then with doing the microlensing search using the Kepler field, then we are also getting an understanding of the difference occurrence rates of compact objects and just understanding stellar evolution and the systems that you might be finding planets in. 2.7 End of a Star’s Life 2.7.1 The Main Sequence isn’t forever Within this module, we have looked at the birth and life of stars. We have explored the stellar nurseries and discovered that stars, just like humans, walk different paths. Yes, we all eat, sleep, and explore life, or . . . in a star’s case burn fuel, and shine bright. However, just as there are different paces to life for us humans, so too can stars live and die in many different ways. Now that we know the basics, let’s explore the life of rockstars, and that of an average Joes. . . in the stellar sense that is. Let’s see how different stars move towards the end of their lives. Here, we will discover that the life and subsequent death of a star are determined at birth, by the star’s mass. How can this be the case? The story of a star’s death begins as the point at which the star leaves the safety and security of the main sequence. The main sequence is the long main track observed in the Hertzsprung-Russell diagram. We have learned that stars seen in the upper left of this track are hot, blue and massive, weighing 10 to 100 times the mass of our Sun, or possibly more. Stars at the lower right end of the track are cool (by stellar standards). They are red and may only contain a tenth 52 of the mass of our Sun. We have also learned that blue stars tend to be called high mass, while stars like our Sun, or smaller, are often called low mass stars. When a star leaves the main sequence of the Hertzsprung-Russell diagram, the star is seen to move towards the right of this plot. This movement is the result of the star becoming redder. But what does change in colour represent? What changes in the star’s interior are powering this shift? And what happens in the time between the departure from the main sequence and the stars demise? That is what we are about to explore. 2.7.2 Moving on up... When a star’s core runs out of hydrogen fuel, the core can no longer sustain the outward radiative force that balanced the force of gravity, which pulls everything inwards. Therefore, the star will no longer be in hydrostatic equilibrium. This loss of radiation pressure results in the star’s core collapsing. The collapse of the core in turn causes the temperature at the interior of the star to increase. When a star burning primarily hydrogen suffers from such a collapse, it will contract until the core reaches about 100 million degrees. At that temperature, the star can begin to burn helium in its core. Helium will become the main source of energy for the star at this point in its life, as it fuses to create carbon and other trace elements. However, if we look just outside the helium core, we can see that the core is surrounded by a shell of hydrogen that is also burning. The hydrogen burning shell is slowly consuming more material as it moves outwards through the star. As the helium burns hotter than its predecessor, the hydrogen core, it burns more rapidly. Therefore, this phase of a star’s life is shorter. Once the helium core is exhausted, the core will, once again collapse. As it collapses, the temperature will, once again increase. If the collapse leads to a temperature of 600 million degrees, the star is able to burn carbon. Such a core would be surrounded by both a helium and hydrogen shell. Once the carbon burning is complete, the cycle can again repeat, leading to neon, oxygen or even silicon burning, with the core becoming heavier and heavier. As each layers is hotter than the last, the star burns through its core more rapidly. For example, a star could take billions of years to burn through its hydrogen, whilst it may only take hundreds of years to feed on a carbon core. By the time we reach silicon, it’s possible for a star to consume its core in about a day! The multiple layers of the star seen here have led to these stars sometimes gaining the nickname of onion stars, as onions have layers . . . just like ogres . . . or cakes . . . or parfaits . . . everybody likes parfaits! Hmmmm . . . where was I . . . back to stars . . . yes they have layers. OK, well this all stops with iron. When learning about fusion and fission, we discovered that the most tightly bound atom is iron, and that no more energy can be gained by either breaking iron apart, or by smashing iron nuclei together. As a result, there is no fuel left for the star to burn through. Fusion in its core must stop, and the star will die. We will explore the death of stars in more detail in the next two videos. Meanwhile ... 53 2.7.3 Becoming Giant...er So far, we have been looking at what is going on inside the star, but what effect does this have on the outside of the star, or rather what effect does this have on what we can see? When stars run out of the current fuel in their core, we discovered that they cores collapse. This continues until the star is able to burn a new type of fuel. So for example when a star initially runs out of hydrogen, it will collapse until it the helium fires begin to burn. What effect does this have on the envelope of the star. The envelope is outer hydrogen rich region of the star that is not involved in nuclear fusion. It is the outer layer of the star. Well, as the core collapses, the energy generated by this collapse drives the diffuse enve- lope outwards. And so the star expands. As it becomes less dense, and cools, it reddens. This means that we would see the star appear to grow in size, while becoming redder. If we return to the Hertzsprung-Russell diagram, we will see that this means the star leaves the main sequence and becomes either a red giant or a supergiant. This expansion and reddening happens each time the core runs out of fuel, and searches for a new source of food. As a result’ stars can move great distances from the main sequence over time. However’ as a star lives most of its life on the main sequence, it is the loss of hydrogen in the star’s core that is easiest to mark. A star’s departure from the main sequence is known as the stars turnoff. What kind of size difference would we expect to see? Well, in the case of our Sun, we would expect that the surface of the Sun would reach out and swallow up the Earth, possibly even extending out to Mars. Don’t worry though, we have about 5 billion years before that is going to happen . . . so we have a little more time to crack the problem of space travel and venture out to explore the universe before our Sun dies. You never know, we might come back to watch the death of our Sun on the day 5 point 5 slash apple slash 26, as they did on Platform 1 in the Doctor Who episode, “The End of the World” that aired in 2005. 2.7.4 Burning Fuel So what about stars other than our Sun? How will their mass effect their lifespan? How will their life, and life span be affected by the fuel they burn? If we start by considering the life of an average Joe, that is to say a low mass star, they can spend 10 to 100 billion years on the main sequence, slowly burning through their hydrogen cores. At the end of this time, they will depart the main sequence, obtaining their middle age spread, puffing up to become Red Giants. In this phase of a low mass stars life, our average joe will carry on, plodding along as it slowly consumes at helium core. At this point, the core will collapse, but will never reach a temperature that is hot enough to ignite its carbon fires. The star will instead end its life quietly, missed only by its family of planets and its close friends. But what about the rockstars, the high mass blue stars? What path do these stars take? High mass stars, like many well known rock stars, live fast and die young . . . in a huge explosion! Explosions so massive that they can be witnessed in other galaxies. The rockstars of the stellar nursery, may be more massive, and so contain more hydrogen, 54 but they are also much hotter. High mass star burn through their hydrogen cores at a much faster rate, departing the main sequence after only maybe a million years. At this point the high mass star will switch to burning helium, then carbon, and so on, building up multiple layers. The number of layers the onion star builds up, depends on its initial mass. The more massive the star was at its birth, the closer it can get to an iron core. During this time, our rockstar will go through a giant, and possibly even a supergiant phase, as the outer envelope swells, before our rockstar dies. The explosive nature of our rockstars ending is explored in more detail later in this module. 2.7.5 Nurseries Our average joes and our rock stars were all born in the same stellar nursery. These stars were all born around the same time, and yet they live very different lives, at very different paces. By taking measurements of stars to obtain their colour and brightness, we are able to learn more about where this star is in its life cycle. However, we can also use this information to find out about all the stars that came from the same nursery. If we take measurements of lots of stars from any given cluster, and when I say cluster, I mean all our nursery graduates that are still around, we can create a Hertzsprung-Russell diagram for that cluster. This will provide us with a clear view of the current turnoff from the main sequence within that cluster. If we check what type of stars that are currently turning off, we be able to determine how long there would have been living on their diet of hydrogen in their cores. This will in turn tell us the age of the cluster. By taking the measurements of multiple stars,we can tell us if this was the graduating class of 100 million, 1 billion or even 10 billion years ago! If the cluster is still very blue, then it must be young, while redder cluster will be old. Clusters of stars change colour with age, just as stars do. The hotter bluer stars die out first, then the average stars, and finally the red dwarfs. Eventually, all stars die off, resulting in the production of elements that can feed into the next generation of stars. This stellar death is the next avenue to be explored in our journey through the life and death of a star. 2.8 Life After the Death of Low Mass Stars 2.8.1 Stars Like Our Sun The Sun is a fairly ordinary star. There are stars that are much smaller than the Sun, and also stars that are much much larger than the Sun! Stars with larger masses than the Sun are also hotter and brighter than the Sun. Brighter stars produce energy at a faster rate than the Sun, which means that they run out of fuel faster. Brighter stars also have a more spectacular death than our Sun.. Yes, I said that our Sun will eventually die. . . don’t worry though, it won’t happen for approximately five billion years, and it will be a rather slow, boring death. In case you’re still worried about the death of the Sun, let’s put five billion years into perspective. We often think about the Pyramids of Giza as old, but they were built only 55 about five thousand years ago, within the realm of written history. Earlier still, the oldest archeological artifacts from North America’s first aboriginal peoples date from fourteen thou- sand years ago, which, from an astronomical perspective is still very recent history compared to the first modern humans, homo sapiens sapiens who lived about two hundred thousand years ago. To put that into perspective, that’s about 800 generations of humans, or a time when your great-great-great-great... ...great-great-great-great-grandfather or grandmother lived! And that’s only 200,000 years! We’ll need another 25,000 of those to get to 5 billion! Let’s jump back one thousand times further: if we multiply two hundred thousand years by one thousand we get two hundred million years ago, which is the beginning of the Jurassic age when dinosaurs roamed the Earth and Plesiosaurs, like this one, swam in the oceans. This is also close to the time it takes the Sun to make one full orbit around the centre of the Milky Way galaxy. That means the Sun celebrates its galactic birthday every 200 million years! If we multiply two hundred million years by ten we get to two billion years ago, which is when the Earth’s atmosphere first became rich in Oxygen, a milestone in the evolution of plants and animals on Earth. The Sun is still older than 2 billion years though, since the Sun and the Earth were formed close to five billion years ago. This means that the Sun isn’t even halfway through its lifecycle yet, and it still has about 5 billion years to go! 2.8.2 Our Sun’s Inevitable End It is convenient to classify the main sequence stars into two groups: high mass stars with a mass eight times the Sun’s mass or larger that die in a supernova explosion and low mass stars with less than eight solar masses that have gentle deaths. When a star runs out of Hydrogen in its core, it transforms into a red giant star which is stable for a period of time that is about one tenth as long as the time period that it is a main sequence star. For instance, the Sun is estimated to have a total lifetime of ten billion years as a main sequence star, and another one billion years as a red giant star. During the red giant stage of a star’s life it swells out to a larger size that could be ten or a hundred times larger than it was during the Main Sequence phase. When it swells, its surface cools off, so from Wien’s law, its colour becomes redder. Hence the name, “Red Giant”! The star will expand during the red giant phase. During this phase its mass stays ap- proximately constant, but its radius become larger. The acceleration due to gravity depends on a constant divided by the square of the star’s radius. This means that as a star expands, the acceleration due to gravity at the surface of the star becomes smaller. If an astronaut visits the star now, and floats near the surface of the star, the astronaut will feel a weaker gravitational force towards the star when the star expands! This makes it easier for the astronaut to escape from the stars gravitational pull as they leave. What is true for the astronaut is also true for the gas in the outer layers of a red giant star. As the star expands, the outer layers aren’t very strongly attracted to the rest of the star. Any small disturbance can end up pushing the outer layers outwards. Eventually low mass stars end up as red giants that shed their outer layers into a beautiful cloud of gas confusingly called a “Planetary Nebula”. This image is a particular planetary 56 nebula that is commonly called the Ring Nebula. The radius of the nebula is approximately one light year across, which is much larger than the size of the red giant whose outer layers dispersed to create the nebula. At the centre of the nebula you can see a bright star. That star is the leftover core of what was once a red giant star. That leftover core is called a white dwarf star. 2.8.3 What are White Dwarfs? A white dwarf star is a rather peculiar type of star. The strangest thing about a white dwarf is that it can keep its size constant in time without any nuclear fusion keeping it hot. A typical white dwarf has a mass that is about the same as the Sun, but a size that is closer to the size of the Earth! The composition is mainly Carbon that has solidified into a crystal structure. The electrons zoom around wildly creating an outward pressure that balances gravity. The electrons travel around rapidly due to a quantum mechanical effect called degeneracy pressure. Degeneracy pressure is due to the Pauli Exclusion Principle, that states that particles such as electrons or neutrons are not allowed to exist in exactly the same state. If we were to try to make the particles have exactly the same location with zero speed, they would be identical. In order to keep their identities unique, the particles move quickly with different speeds. As a result they move around and bump into each other creating a gas pressure. This effect does not depend on temperature. So if a white dwarf cools down the degeneracy pressure will still keep the star a constant size. This means that a white dwarf star could potentially live forever! When we look at planetary nebulae, we always see a white dwarf in the centre. Ultraviolet photons emitted by the white dwarf star ionize the gas in the nebula causing it to glow with beautiful colours. Over the course of thousands of years, the gas in the nebula slowly starts to disperse and mix with the other gas between the stars, and the white dwarf cools. After a few tens of thousands of years, the white dwarf will be isolated, and will slowly cool down and fade away. 2.8.4 Is a White Dwarf the only option? In the early 1900’s, astronomers thought that all stars ended up as white dwarfs. That changed in 1930 when a young Indian astrophysicist named Chandrasekhar started thinking about the electrons zooming around in a white dwarf star, keeping it from gravitationally collapsing. Chandrasekhar realized that in order to keep a high mass white dwarf from collapsing, the electrons would have to travel at faster and faster speeds. If you extrapolate, the prediction is that at some critical stellar mass, the electrons would have to move at speeds faster than light! Chandrasekhar knew that it is impossible for electrons to travel faster than the speed of light. This led him to calculate the largest mass that a white dwarf star can have. This mass is now called the Chandrasekhar mass and is MCH = 1.4MSun (28) 57 One point four times the mass of the Sun. What happens if you try to add extra mass to a stable white dwarf star that has a mass equal to the Chandrasekhar mass? Well, electron degeneracy pressure will not be capable of combatting gravity and the star will collapse! 2.8.5 White Dwarfs in Binary Systems Approximately half of all stars have a binary companion. We will learn later on in Module 5 that if the two stars are very close to each other, mass can flow from one star to the other. There are many binary systems where a white dwarf star gains mass from another star. If the added mass makes the white dwarf’s mass go over the Chandrasekhar limit then the white dwarf will implode. This implosion heats the star, allowing the Carbon to explosively fuse into heavy elements such as Uranium. The explosion destroys the star and sends heavy elements outwards. This type of explosion is called a Type Ia Supernova. After a supernova explosion takes place there is a hot glowing cloud of gas left over called a “Supernova Remnant”. In the year 1572, the Danish astronomer Tycho Brahe observed a bright new star appear in the sky and then fade away. The new star was the supernova explosion. In modern times, when we point a telescope in the part of the sky where Tycho Brahe observed the supernova we see this glowing and expanding supernova remnant. We now call this supernova remnant “Tycho” after Tycho Brahe. This picture might look similar to the picture of the Ring nebula that we showed earlier. But the Ring nebula is smooth while Tycho is very lumpy. This is evidence that the process that formed the Ring nebula was gentle, while the formation of Tycho was violent. In addition, Tycho is emitting X-rays, while the Ring mainly emits visible and ultraviolet light. The point sources of light are foreground stars that are located between us and Tycho. There does not appear to be any stellar core left over after this supernova. It is probably true that all Type Ia Supernovae do not leave behind any remnant star. Could our Sun die in a supernova explosion? The Sun will eventually transform into a white dwarf star. But since the Sun does not have a companion star, it will not be in any danger of gaining so much weight that it explodes. 2.9 Life After the Death of High Mass Stars 2.9.1 Explosions! High mass stars, larger than eight times the Sun’s mass, live through multiple red giant stages allowing them to transform into giant layered stars with the heaviest elements at the centre and layers of lighter elements on top. Eventually a core of iron and nickel forms. Since heat-releasing nuclear reactions are not possible in iron, the star develops an energy crisis. No more nuclear reactions can take place, so the core will cool down causing the atoms to move slower, and the gas pressure to drop. If the core gas pressure drops, it can’t balance gravity and the star collapses! We don’t quite know all the details but it appears that as long as the original main sequence star’s mass was less than 30 times the Sun’s mass, then as the star collapses and becomes denser, the collapse halts suddenly. The sudden braking of the collapse takes place 58 if the conditions allow the formation of an ultradense type of star called a neutron star. A neutron star has a hard surface, so when the neutron star forms, collapsing gas from the outer layers of the star will smash into the surface and “bounce” outwards ploughing into more infalling gas. The result is an explosion. This is the trigger for a “Core Collapse Supernova” which is also called a Type II supernova. Normally neutrons don’t appear by themselves outside of the nucleus of an atom. This is because they are unstable when they are isolated and decay into a proton and an electron. However, if protons and electrons are forced to come too close together, it is possible for them to combine and transform into a neutron. During the collapse of the core of a high mass star, the elements are squashed into such a small volume that lots of protons and electrons combine to form neutrons. This leads to a neutron-rich gas that continues to become very dense. Neutrons are particles that obey the Pauli Exclusion Principle that also governs the electrons in a white dwarf star. The Pauli Exclusion Principle means that the neutrons try to keep their own unique identities as they are forced to occupy smaller regions of space. The result is that the neutrons zoom around and create a degeneracy pressure that pushes outwards and balances gravity. A neutron star maintains hydrostatic equilibrium through this process that is called neutron degeneracy pressure. 2.9.2 Discovery of the first Neutron Star The concept of a neutron star was proposed in the 1930’s soon after the discovery of the neutron. However, many astronomers doubted the existence of neutron stars and black holes. Most astronomers thought that all stars end up as white dwarf stars when they die. In 1967, this erroneous belief changed when an astronomy PhD student named Jocelyn Bell observed pulses of radio waves. Jocelyn Bell’s goal for her PhD thesis research was to observe quasars using a radio telescope. Today we understand that quasars are supermassive black holes at the centres of galaxies, but in the 1960’s these were mysterious, unexplained objects. During her search for quasars, she found something totally different, the pulsed radio emission with a very regular pulsation period of 1.337s. She suspected that this might be a new class of astronomical object, so she searched the sky in other directions and found a few more similar types of pulsed radio sources. These sources of pulsed radio emission were named pulsars. Soon after the discovery of pulsars it was understood that they are rotating neutron stars. The discovery that neutron stars are a possible endpoint of stellar evolution opened up the possibility that even more exotic objects could exist: black holes! 2.9.3 Pictures of Explosions! This is the Cassiopeia A supernova remnant which is the gas leftover from a Type II super- nova. The gas is millions of degrees and glows in the X-ray part of the spectrum. The small inset box shows a small point of light which is the hot, newly formed neutron star found at the centre of the supernova remnant. An artist has used their imagination to draw a picture of what the neutron star might look like, since no telescope has ever imaged a neutron star’s surface with more detail that the point of light in this picture. 59 Neutron stars are tiny stars. A typical neutron star has radius that is about 10 km, about the size of a city. (Remember that white dwarf stars are close to the size of the Earth, so neutron stars are much tinier!) The only object smaller than a neutron star but with the same mass is a black hole! For instance, a black hole with the same mass as a neutron star would be about 3 times smaller in radius. This is perhaps one of the least visually interesting pictures taken by the Hubble Space Telescope. It is an image of the closest neutron star which is 400 light years away. Most neutron stars are thousands of light years away. It is not possible with today’s technology to resolve features on something that small and far away. 2.9.4 Fool’s Black Holes Just like white dwarf stars, neutron stars also have an upper mass limit. However, unlike white dwarf stars, the value of this upper mass limit is not known exactly. The maximum allowed mass is larger than two times the mass of the Sun, and less than three times the mass of the Sun, but we don’t really know the value more accurately than this. If a neutron star gains mass above this maximum mass, it will start to collapse, probably forming a black hole! The maximum allowed mass for a neutron star is very important for identifying black holes. Often we observe X-ray emitting binary star systems. The X-ray emitting properties of neutron stars and black holes are very similar, so it’s easy to confuse one for the other. One way to tell them apart is to measure the mass of the X-ray emitting object, as we’ll learn how to do in Module 4. If the mass is larger than 3 times the Sun’s mass, then it’s a black hole! If the mass is less than 3 solar masses, then it could be either a neutron star or a black hole. In all cases where the mass is smaller than 3 solar masses astronomers have found some other evidence, such as pulsed emission from the surface which allows for the identification of the object as a neutron star. 2.9.5 Creating Compact Objects The highest mass main sequence stars are thought to form black holes when they run out of fuel. However, there are still many open questions about how the black holes form. For instance, how massive must a star be in order for it to form a black hole? The standard limit that is usually quoted is that the mass when it was a main sequence star should be larger than 30 solar masses. However, this limit is not really known that accurately. It could be a bit smaller or larger. One way to produce a black hole, suggested in the 2009 Star Trek movie, is to inject a planet with something called “Red Matter”. We have no idea what red matter might be, so this definitely in the realm of science FICTION! Some core collapse supernovae might produce black holes instead of neutron stars. As- tronomers carefully examine supernova remnants to see if any evidence of a black hole (in- stead of a neutron star) can be found. So far there hasn’t been any discovery of a black hole found inside of a supernova remnant. However, since black holes are difficult to detect, this doesn’t mean that there aren’t any. 60 2.9.6 Failed Supernova Another idea is something called a “Failed Supernova”. The left image from 2007 shows the red supergiant star N6946-BH1, which has a mass that is about 25 times larger than the Sun. In 2015 an image of the same star-field shows no star! Astronomers did see the star get a little bit brighter, but there was no supernova explosion before it disappeared. Astronomers are now carefully watching this region to look for signs for the formation of an accretion disk. It’s possible that we have actually caught a star in the act of collapsing to form a black hole! Alternatively, in some cases it might be possible for the birth of a black hole to produce a burst of gamma-rays! Gamma-ray bursts are short-lived bright bursts of gamma-rays, generally seen in far away galaxies. This movie shows the whole sky (mapped onto a sphere) as viewed by gamma-ray telescopes. On April 27th, 2013, the gamma-ray burst 130427A occurred as shown in this movie. Over the next few hours astronomers observed this region using telescopes sensitive to X-rays, visible and radio waves. In some cases, when astronomers observe the place where the burst occurred over the next few days, fainter light with the same spectrum as a core-collapse supernova appears. In other words, it appears that some Gamma-ray bursts are ultra-bright supernovae! These ultra-bright supernovae are sometimes called “hypernovae”. It is very likely that some of these hypernovae explosions produce black holes. 2.10 Summary: The Circle of Life All stars are born from the accretion of interstellar gas and dust. When enough hydrogen and helium gas accretes by the forces of gravity, stable nuclear fusion sets in at the core of the new star. Small stars burn for many billions of years, and enjoy a long retirement before their gentle end. We are lucky that our own Sun is one such middle aged star! Humanity, and all life on Earth will continue to enjoy the Sun for billions of years to come. On the other hand, large stars, ones with more than 8 times our Sun’s mass, live life ON THE EDGE. They burn through vast amounts of hydrogen and helium, and continue to brighten as they fuse heavier and heavier elements. Just like a rock star who lives life on the edge, the fast pace is enough to wear out anyone - and massive stars ROCK and ROLL into early deaths, and violent ones at that: supernova explosions. Not all stars that explode at the end of their lives end up as black holes. Only stars which are much more massive than our Sun go on to produce black holes. Hydrostatic equilibrium is the principle that explains why stars don’t collapse at different stages of their life. Each of the remnants: white dwarf stars and neutron stars, depend on different types of pressure to maintain their surfaces. However, black holes don’t have a surface, per se, because there is no force strong enough to overcome the force of gravity around a black hole. . . Black holes generate such a strong gravitational field that they permanently warp space and time, creating a hole in the fabric of the Universe. 61 3 The Structure of Spacetime 3.1 Introduction In order to understand more about black holes, we need to understand the concept of space- time. Most people are familiar with the concepts of space and time and think of them as separate quantities. But Einstein’s theories of Special and General Relativity show us that different observers do not agree on measurements of distance and intervals of time. What everyone can agree on, is the mixture of these two concepts, in the framework called spacetime. Spacetime is used to explain all of the strange effects we encounter in the theories of Special and General Relativity. For example, why is the speed of light the ultimate limit in the universe? Why do moving clocks run slow compared to stationary clocks? An interesting way of understanding the basic properties of black holes is to consider a simple example involving sound waves instead of light. In 1972, black hole physicist William Unruh devised a thought experiment consisting of a fish, calling out to its friend as it falls over a waterfall. At a certain point in the waterfall, the speed of the water exceeds the speed of sound, and the falling fish can no longer be heard by its friend above. This is an example of a “sonic black hole”, an analogy that uses sound waves instead of light waves to help us understand black hole physics. Of course, this example makes LOTS of assumptions, for one, I don’t know how a fish would yell underwater, but that’s besides the point. Let’s “dive right in” and see how a fish experiences water as spacetime, and what it can tell us about the nature of black holes. 3.2 Fishing in Spacetime 3.2.1 Why Did the Fish Cross the Waterfall? Suppose we have two fish, swimming in a gentle stream, above a waterfall. Let’s assume that since the fish are immersed in water, they don’t really have a concept of what water is. They consider water to be a natural, immutable part of their environment. As you well know, water is not the framework of the universe, but to a fish, it might as well be. So, our two happy fish are living their fishy lives, exchanging fishy details about fish-stuff by communicating through sound waves in the water. We’ll assume now that the fish don’t communicate visually, and that all of their communication is limited to sound. Maybe it’s night in the stream, or maybe the fish are unable to see because of mud in the water. Consider the speed of the stream. Since it’s flowing slowly, the fish can stay relatively stationary with just a few swishes of their tails. The motion of the water is similar to the effects that we perceive as spacetime. Keep in mind, both the fish and the water can have independent speeds, and that there is no universal limit for how fast they can go. So it is possible for the fish, and the water itself, to exceed the speed of sound in water. Our fish can’t swim that fast. . . but perhaps one of our fish is researching a faster-than-sound jetpack... At one end of the stream, the current is drawn into a rushing waterfall. The water pouring over the top of the waterfall gets faster as it falls, travelling faster and faster towards the bottom. The waterfall can be divided into two regions with a short transition between them. 62 The first region is the top, where the stream’s water begins to accelerate, but the water flows slower than the speed of sound. Below this first region, there is a transition where the speed of the water is equal to the speed of sound in water. In the lower region, the water is flowing faster than the speed of sound. When both fish are above the waterfall, they can carry out a fishy conversation without any difficulty. When they speak, the speed of the stream has very little effect on how the sound travels between them. But what do you think will happen if one of the fish is carried over the top of the waterfall? Let’s examine what the two fish experience as the travelling fish descends through the three regions of the waterfall. 3.2.2 Crossing the Point of No Return In the first region of the waterfall, where the water is gently accelerating over the edge, our adventurous fish is heard yelling for the help of his friend. “Help me! I’m being swept over the waterfall!” [Ross says in a higher-than-normal voice.] The sound waves emitted by the yelling fish propagate back up the stream, towards the stationary fish upstream. Since the velocity of the water in the first region is lower than the speed of sound, the fish can still be heard by its companion. However, since the fish is yelling in flowing water, the sound waves travel at the speed of sound minus the speed of the water in waterfall. The stationary fish will hear the falling fish’s voice as being deeper the faster the water in the waterfall flows. This is due to the Doppler effect. “Help me! I continue to be swept further down this waterfall!” [Ross says in a deep voice]. At the transition between regions, the speed of the water is equal to the speed of sound. At this point in the waterfall, any sounds emitted by the falling fish would appear to be completely stationary! This corresponds to an infinite doppler shift, and the sound waves would be stretched by the motion of the water. The sound emitted by the falling fish would be so low, it would be inaudible to the fish upstream. So, what exactly happens when the falling fish is carried beyond this point? At the point in the waterfall where the speed of the water is equal to the speed of sound, the infalling fish’s calls can no longer be heard by the fish upstream. This region in the waterfall is similar to the event horizon of a black hole. Recall that the speed of light is the escape velocity from an event horizon, so similarly, the part of the waterfall where the stream flows at the speed of sound is just like an event horizon. Since the information carried by sound is trapped, we can call this a “sonic event horizon”. 3.2.3 Breaking Down the Analogy What does the infalling fish experience? Well, beyond the awareness that it’s going over a waterfall, it’s experience is almost indistinguishable from the stationary fish! Since the infalling fish is accelerating at the same rate as the water around it, it feels like it’s in a perfectly still environment, oblivious to the peril that it’s in! Not only that, but the infalling fish would continue to hear its companion upstream! Why? Because the upstream fish’s sound waves are being carried along, and accelerated 63 with the flow of the water, instead of against it! So communication into the “sonic event horizon” is possible, just like it’s possible to send light rays into a black hole. This analogy illustrates a couple of important points about black holes, but it does have some limitations that we need to consider. First and foremost, a black hole has a singularity at its core. Unlike a waterfall, which has a bottom and a region for water to flow outwards, there is no escaping from a fall into a black hole. We might try to illustrate this by adding sharp rocks at the bottom of the waterfall, which obliterate anything, including the fish that encounter it. However, we also said that the speed of light is the ultimate limit. Not the speed of sound. In fact, this gives our adventurous little fish an opportunity to escape! Perhaps the motivation for the falling fish to go over in the first place is because it’s an inventor fish, who has discovered the secret to underwater rocket technology! Perhaps, this inventor fish accidentally dropped a rocket pack over the waterfall, and was on a mission to retrieve it! If the fish reaches its rocket pack before hitting the rocks at the bottom of the waterfall, it can accelerate to a speed faster than that of the water’s flow, and return to the safety of the stream, where its companion anxiously awaits. This analogy does demonstrate some key physics with respect to the behaviour of sound waves in the region around an event horizon, similar to the behaviour of light around black holes. Some scientists have created “bathtub drain black holes” in the laboratory in order to gain a better understanding of the environment around black holes. These drain holes probe the behaviour of event horizons in much the same way that our fish encountered a sonic event horizon. So long and thanks for all the fish. 3.3 Introducing Special Relativity 3.3.1 Light Puzzle Just as sound waves propagate in water, in was once believed that light waves propagate in a medium called Aether. This presented a an opportunity for experimental physicist to measure the motion of the Earth with respect to the aether! In 1887, two scientists named Michelson and Morley conducted an experiment to do just that. Interferometers leverage the wave nature of light to measure the difference in lengths between different beam paths. When coherent light, light with that has the same phase, such as laser light, is split between two different paths, when the light is recombined its brightness will depend on how different the two paths were. The original Michelson-Morley interferometer was developed to determine if the “flow of Aether” caused a delay in one of the device’s two arms, instead of a difference in the arms lengths. Like a boat travelling against the current of a river, the theory of light back then predicted that moving Aether would cause a delay in the “upstream” arm. The opposite was discovered, that there was never any delay, no matter what orientation of the device with respect to its motion through the supposed Aether. This proved that light waves didn’t require a medium like Aether to travel in, and as a consequence, the speed of light is a constant! This forced them to conclude that the aether did not exist, light would always be 64 seen travelling at the same speed. That puzzled scientists worldwide. . . well, except for one: Einstein. Einstein imagined what it would be like to see the universe from the perspective of a beam of light. He asked questions like, “How would a photon perceive the passage of time?” and “Will distances shrink and stretch depending on the motion of an observer?” Of relevance to him was the fact that experiments had proven that light waves were special compared to sound waves, or water waves, in that they didn’t require a medium through which to propagate. In helping us to understand these new revelations, Einstein had to tackle problems which few could even consider. 3.3.2 A Pair of Powerful Postulates In one of Einstein’s most famous papers, entitled “On the Electrodynamics of Moving Bod- ies”, he introduces two very important ideas. Ideas which are among the foundational postulates of Modern Physics. They are: One: The Laws of Physics are the same in all inertial frames of reference. Two: Light moves at the same speed relative to all observers. That first postulate seems reasonable. The Laws of Physics are the same for me here, as they are for you sitting there. The Laws of Physics are the same on the Moon as they are on Earth. In Physics this principle is essential, as we use it all the time in order to learn about places which are far from us. An inertial frame of reference is either an experiment at rest, or one moving with a constant velocity. Inertial frames are not accelerating. For example, someone standing in a high-speed train would experience the same laws as someone stationary on the ground, so long as neither is accelerating. The first postulate is intuitive to human beings, which makes the second one impossibly hard to believe at first glance. Einstein’s second postulate was that light moves at the same speed relative to all observers. So, if we were to measure the speed of light from a fast moving astronaut, we do not add the astronaut’s speed to the speed of light! Weird right? The speed of light always comes out to the same value, no matter how fast the astronaut is travelling. Einstein realized that if the laws of physics are the same for all observers, then all observers must agree on the value of the speed of light. 3.3.3 Galileo to Lorentz You’ve probably experienced some of the strange effects of changing references frames before. Have you ever been parked in a car, when an adjacent car starts moving? In some cases, your brain tricks you into thinking you’re moving, instead of the other car! Since motion is relative, we can always choose a stationary reference frame, even if there is relative motion to something else. Let me explain. Suppose you’re riding in the passenger seat of a self-driving car. Some fast cars are passing you in the left lane, and you are passing slow cars in the right lane. If all the cars are moving at constant, but different speeds, each car is their own inertial reference frame. If you observe the cars from the ground, they all appear to be moving. If however, you choose 65 a reference frame of the car in the middle lane, the cars on the left appear to be moving forward, and the cars in the right lane appear to be moving backwards. Without the road in the background, we can’t figure out how fast the cars are travelling. [ROAD DISAPPEARS] The only thing we can tell is how fast they are travelling with respect to one another. Try this the next time you are a passenger in a moving vehicle: without looking at your speedometer OR the ground (maybe close your eyes) try to figure out how fast you are going! It’s not an easy task. Now, with your eyes open, try to figure out how fast other cars are travelling relative to you! If you get good at this kind of reference frame transform, eventually you can imagine that you are never moving, even if you are walking! The universe is moving around you! Is there a “correct” or universal frame of reference? Is there a special place in the universe that we can call “stationary”? No! The special theory of relativity states that all motion is relative. Thinking back to the car examples, you could easily have chosen to occupy one of the slow cars in the right lane, and observed the other cars moving forward. Changing your frame of reference by adding or subtracting speeds is called boosting, or computing a Galilean Velocity Transformation. I think boosting sounds cooler though. Until the discovery of Special Relativity by Einstein, these sorts of transformations were thought to hold true in all situations. We can go from frame to frame through the simple addition of velocities and consider how objects move in different frames through the same, linear process. However, this becomes problematic when light is introduced. Instead of driving during the day, what if our cars are driving at night. They’ll need to turn their headlights on! If the centre car turns on their headlights, they will see photons leaving at ‘c’, the speed of light. Let’s boost into the reference frame of the speeding cars, say by adding 10 kilometers per hour of speed to our reference frames. Does the speed of the light coming from the middle car’s headlights appear faster, or slower? In fact, it’s neither! Regardless of what reference frame you choose, the photons coming out of the headlights always appear to move at the speed of light! The speed of light from the middle car is not ‘c’ minus 10 km/h due to the motion of the cars. It’s always ‘c’. Weird, huh? Einstein thought so too. Clearly, if Einstein’s second postulate, that all observers measure the speed of light as a constant, is to hold true, we need some other kind of transformation group so that every observer can see the light beams moving relative to themselves at ‘c’. In order to do such a thing, Einstein realized that our intuitions about space and time must be incorrect, and that a new theory is required to describe how all observers, moving at different speeds, measure the speed of light to be constant. 3.4 Spacetime 3.4.1 Introduction to Spacetime Last section ended with the idea that the speed of light is the same in all inertial frames. Now, let’s connect this to the idea of spacetime and events. Two astronauts moving at constant speeds relative to one another will measure the speed of light from all sources of light to have the exact same value, which we write as a lower case ‘c’. A constant speed is just the distance travelled divided by the time that has elapsed. 66 Einstein realized that one way to explain why two astronauts measure the same speed for light is because the two astronauts do not agree on the definitions of space and time. Instead of thinking about space and time as separate concepts, Einstein realized that he needed to consider the combination of both space and time, a concept he called, spacetime. The Universe consists of four dimensions. Three of those dimensions are spatial: moving in the up/down dimension, the left/right dimension, and the forward/back dimension. The last dimension is time: the past/future dimension, although we don’t say something moves in time, just that the flow of time itself carries us towards the future and away from the past at a speed of one second. . . per second. 3.4.2 Spacetime diagrams Let’s exercise our imagination to better understand how humans perceive time. Our senses collect information from the space around us, but we exist only at a single *moment* in time. None of our senses can perceive the passage of time directly, or changes in time for that matter. The best humans can do is to create memories of the past, which allows us to effect change in their future environment. In that sense, we only experience a narrow slice of time. So how do we effectively imagine what a 4-dimensional spacetime looks like? Well, let’s start by considering an easier picture, with fewer dimensions altogether. Suppose my fingers are limited to motion in only one spatial dimension, like walking along my arm. I can move them forward along my arm or backwards along my arm, but it’s too narrow for left and right, and my fingers are too weak for up and down. In essence, I’ve compressed 3 dimensions of space into a single dimension, so now, my fingers’ position can be characterized by a single point, along a line. If the fingers walk along my arm, from elbow to hand, it takes some amount of time. Since we’ve got my fingers’ distance on the horizontal axis, let’s now plot time along the vertical axis. This diagram is called a spacetime diagram. Since we humans only see a narrow slice of the spatial dimensions, we need to reveal how my fingers’ position changes with time. This curve, for example, shows that my fingers walk back and forth across my arm, and also move forward in the time dimension. If we mask everything but a narrow slice, we get back to the original representation of my fingers’ position in space. This path is my fingers’ worldline. Since photons must travel at the speed of light, a plot of position of a photon on this spacetime diagram will zoom outwards in a perfectly straight line. Normally this line is drawn so that light rays make a 45 degree angle from the space and time axes. This line has a special name, it’s called the light cone. Since nothing can travel faster than light, all objects, even objects like you and me, can never escape from our individual light cones! It’s hard to convey what spacetime actually is. The best I can do is to imagine a three- dimensional object, and then try to extend its dimensionality across time. So even though I’m human shaped in 3 dimensions, I’m actually a long tube in a 4 dimensional spacetime! In essence, if you slice this 4-D tube in the time dimension, a 3-D version of me at that particular point is the “slice”! 67 3.4.3 A Cheesy Story The four dimensional spacetime we live in is similar to a block of cheese, if we reduce space down to only two dimensions. We can imagine the ends of the cheese as the two “space” dimensions, and the length of the cheese as the “time” dimension. So a hole in swiss cheese becomes a cheese-being, living inside of a block of cheese-time, or space-cheese, or. . . Space- Cheese-Time. You get the idea. A cheese-being at rest thinks of time running along the long axis of the cheese, and the two-dimensional space that it lives in is the flat end of the cheese. Humans are all-powerful higher-dimensional beings with a cheese knife, and we can slice up the block of cheese into thin slices. Each “slice” of cheese represents a single moment of time as experienced by the cheese-being. At one end of the cheese, there are no cheese-beings, but as we slice through their time dimension, we discover the birth of a cheese-being, growing to larger sizes, and eventually disappearing! Just like our 4-dimensional human tube, from our perspective, a cheese being has been born, lived a fruitful life, and died a cheesy death. Here’s where things get interesting though. . . A different cheese-being that moves at a constant speed through the cheese will slice up the cheese in a different direction. The moving cheese-being will see different size slices and think the times and sizes are different, as if the space-cheese-time itself were somehow warped! However, if we were to take the same cheese block and slice it up in yet another direction, we can do it such that both observers agree on where the bubbles are in spacetime, but they can’t agree on how it was sliced! We can only speculate whether Einstein used cheese to describe spacetime, but if he did, surely we can all agree that it must have been tasty! 3.4.4 Could there be extra dimensions? (Interview with Dr. Gingrich) Q: Could there be extra dimensions? Dr. Gingrich: We all know four dimensions or three dimensions, and the idea is that if we do have extra dimensional space, where is it? So it has to be, first of all, small, otherwise will be detected. If it was infinite like our other three dimensions then, of course, we would see it. So they are very small and they are finite in the sense that they don’t extend to infinity. They’re very small and so we think of extra dimensions as curled up so that if you traveled in them, if you could, you would come around after a while to where you started from. If you make them small enough, you don’t see them, yet they’re still allowed in the sense that they don’t change the observable physics enough, that you could still have them yet they will be undetectable till now or hopefully in the near future. Q: How would the extra dimensions affect gravity? Dr. Gingrich: Well, the small dimensions allow gravity to go into those dimensions but not other particles. So if you imagine the electrons and the older particles that interact by electromagnetic and strong nuclear forces, they can’t go into the extra dimensions otherwise we would see matter disappearing or come into the extra dimensions and we’d see it appear- ing. So the idea is that, matter as we know it exists in the four dimensions, but gravity can go into the extra dimensions. This is beneficial because it explains that gravity is very weak on the sub-atomic scales which it is. It is a very weak force compared to all the other forces 68 at very small scales or distances, and so that allows us to also justify why gravity or explain why gravity is so weak at such small distances. Dr. Gingrich’s webpage: https://apps.ualberta.ca/directory/person/gingrich 3.5 Effects of Special Relativity 3.5.1 Special Events in Spacetime If you decide to hold a party, you need to tell your guests where and when they should arrive. It’s not enough to simply tell your friends where the party is, without telling them when it occurs, and vice versa. Therefore, when we use the word “event”, we are using it to describe where AND when something happens. Events can describe things like: your arrival at a party, or that time when you spilled your drink, even something as simple as snapping your fingers [SNAP] can be characterized as an event. I could say that I snapped my fingers 5 seconds ago at this location, right here. [POINTS DOWN to emphasize here]. So an event must include details of both a position and a time. Our Universe has three spatial dimensions, and so a defined event would look something (x,y,z,t), where x, y, and z define a position, and t defines a time. A great example of this occurs on the popular TV Show Big Bang Theory. In an episode entitled, “The Cushion Saturation” Dr. Sheldon Cooper explains the first time he sits in his favourite spot on the couch as follows, “In an ever-changing world it is a single point of consistency. If my life were expressed as a function on a four-dimensional Cartesian coordinate system, that spot, at the moment I first sat on it, would be (0,0,0,0).” Sheldon feels most at comfortable and at home at the x, y, and z coordinates of his spot, or (0,0,0). Although he can return to that spot on the couch many times, and he does, he can never return to the exact moment when he first sat on the couch, to experience the event again. The reason for this is that the four coordinate of the event is time, t, is always increasing as time passes. Suppose he originally sat down to enjoy a 40-minute episode of Star Trek, we can describe the end of the show as an event that happened at (0,0,0,40 minutes). The only way to return to the original event would be use a time machine such as the one Leonard Hofstadter took a ride in the Big Bang Theory episode “The Nevada Annihilation”. If we were to return to Sheldon’s first time on the couch, and saw Penny riding a skate- board past the scene, Sheldon would see her in his reference frame. Would Penny see the first moment that he sits in his spot, followed by a 40 minute episode of Star Trek, in the same order Sheldon experiences it? Considering our previous discussion about slicing-up spacetime, do you think all observers see these two events in the same order, with the same time interval between events? Let’s this further using one of Sheldon’s favourite objects: trains. 69 3.5.2 The Relativity of Simultaneity Sheldon is preparing a trip on the Napa Valley Wine Train, in his favourite 1915 Pullman Standard Lounge Car. Ever the physicist, Sheldon would like to conduct an experiment to test the consequences of a constant speed of light. To do this, he sets up a light source in the centre train car, and has two of his friends standing in as light detectors in the caboose and the engine, at either ends of the train. While the train is in the station, we say it is at rest. Here, in the station, Sheldon conducts his first experiment by flashing the light and recording the arrival times that his friends measure at each end of the train. Since the distance to the light source is the same for both friends, they each detect the light at the same time. We call this a simultaneous detection. The train leaves then leaves the station and begins its journey, travelling at a constant speed through the mountains. Sheldon prepares the experiment again, this time with the train in motion. Since the train is travelling at a constant speed, once again the flash of light arrives at each of his friends at precisely the same time. In both the stationary case, and the one with the moving train, the observers are at rest with respect to Sheldon and the light source. As a result they observe the pulses arriving simultaneously on each occasion. Sheldon wants to know what his experiment would look like if he were stationary with respect to a moving train, so he gets off at the next stop. He gets ahead of the train and gets set up to redo his experiment as the train passes by at a constant velocity. This time, the light source flashes the moment that it passes Sheldon. Since the train is moving from left to right, Sheldon observes the light pulse arrive in the Caboose of the train first, followed by the arrival of the pulse in the Engine of the train second! Sheldon is puzzled! He observed the light pulse arrive at the back of train first, while his friends onboard report that the light pulses arrived simultaneously! In Sheldon’s frame of reference, the light pulses do not arrive simultaneously, even though in his friends’ frame of reference, they do! The disagreement between observers is a result of light travelling a constant speed, no matter how quickly the source of light moves. This is called the Relativity of Simultaneity and it describes that stationary and moving observers will report the order of events differently, depending on their proper motion with respect to one another. As strange as it seems, both Sheldon and his friends on the train are right! In some cases, the order of events depend on the motion of an observer! Einstein explained this through the concepts of length contraction and time dilation. 3.5.3 Time Dilation When a moving car shines its headlights, the emitted light always travels at the same speed. It doesn’t matter how fast the car moves. But the car can’t travel faster than light. What would you see if you could travel at the speed of light? Suppose that you could fly away from the Sun at the speed of light. Then, light emitted by the Sun wouldn’t be able to catch up to you! One of the weird results due to the invariance of the speed of light is that people in 70 different inertial frames don’t agree on the order of events. However, they all agree that the events actually took place! So, why don’t observers in different reference frames agree on the order of events? In the cheesey spacetime analogy, a moving observer slices spacetime at an angle with respect to the stationary observer. Not only will the observers see events occurring in a different order, but they’ll also measure length and time differently. This spacetime diagram shows lots of events taking place at different times and locations, as experienced by a stationary observer. The stationary observer sees that all of the events on a horizontal line take place at the same time, but different locations. Another observer is moving with respect to the stationary observer in the x direction. They measure time and space using this spacetime diagram where both time and space are rotated. The moving observer sees that all of the events on a slanted line take place at the same time. Both the stationary and moving observers will see the same events, but since their co- ordinate systems are slanted, they do not agree on the times and locations of the events! However, the two observers can agree on the path that light takes. In both cases, a beam of light follows the same path! One way to measure time is with a light clock. Laser light is bounced off a mirror, and the clock ticks when the laser light returns. For example, we could shine a laser beam at the Moon. The Apollo astronauts left a mirror on the Moon, so if we aim carefully, the light will bounce off the mirror and return to the Earth, taking 2.6 seconds. If the light keeps bouncing back and forth, we have a clock that measures out time in units of 2.6 seconds. Sheldon has built a much smaller light clock that fits into his train. The laser is located on the floor and the mirror is on the ceiling, so that the light bounces up and down. The distance between the floor and the ceiling is d. In one full tick of the light clock the light has to travel a distance that is two times d, taking a time that we’ll call t zero. c = 2d t0 (29) Speed is defined as the distance travelled divided by the time taken to travel. Since light travels at the speed of light, c equals 2 times d divided by t-zero. We can rearrange the equation to solve for t-zero, the time between ticks t0 = 2d c (30) Now Sheldon leaves the train, and watches the train move to the right at speed v. His friends are on the train watching over the light clock apparatus. Since the train is moving at a constant velocity, the friends on the train see the light clock tick exactly the same as when the train was at rest. When Sheldon watches the train go by, the light is emitted from the floor when the train is at one location. By the time that the light hits the ceiling, the train has moved to the right. And the train has moved even further to the right when the light returns to the floor. The time for the light to travel from the laser source, to the mirror and back again is represented by the symbol “t”. The vertical distance between the floor and the ceiling is still “d”, but during the time “t”, the laser source has travelled through a horizontal distance of 71 “v” times “t”. The light ray has to travel through a slanted distance that is 2 times “z”. Z is the hypotenuse of the triangle joining the source of light and the mirror. We can see that z is longer than the vertical distance between the floor and ceiling. Since the light is now travelling through a longer distance, but travelling at the same speed, the time has to be longer too! t = 2z c (31) Similar to the stationary case, the time for one tick of the clock is 2 times z divided by c. Sheldon thinks the moving clock’s time interval, t, is longer than the time interval, t-zero measured by the friends. We call the lengthening of the time intervals for moving clocks time dilation. If we do the math we find that t = t0√ 1− (v c )2 . (32) During the train ride, less time has passed for the friends, so they haven’t aged as much as Sheldon. We say that the moving clocks run slow with respect to a stationary clock. Time dilation was tested experimentally in an airplane in 1971. The European Space Agency is planning to launch a new experiment called “ACES” (short for Atomic Clock Ensemble in Space) which will be added to the International Space Station. This experiment will test the time dilation due to the spacecraft’s motion around the Earth! 3.5.4 Length Contraction Another weird prediction of Einstein is called Length Contraction, which makes moving objects look smaller in the direction of their motion. The proper length of an object is its length when measured at rest. Sheldon can measure the length of the train by bouncing light down its length and measuring the time that it takes the light to travel back and forth across the train. This length we call L-zero. When he leaves the train and watches it move to the right, his friends can shine the laser beam, allowing him to measure the length of the train, which is denoted L. When the light moves from the back of the train to the front, the train has moved forwards, so the light travels a little bit farther than the true length of the train. When the light travels from the front to the back, it travels through a slightly shorter distance than the true length of the train. L = L0 √ 1− (v c )2 (33) The final result is that the measured length of a moving object is smaller than the length measured when it’s at rest. This effect is called Lorentz Contraction, and only affects the length of the object in the direction of motion. The dimensions of a moving object perpendicular to the direction of motion are not affected by the motion. 72 3.5.5 The Gamma Factor This function with the square root is called the Lorentz factor, or the gamma factor: γ(v) = 1√ 1− (v c )2 (34) The Lorentz factor is a convenient tool when discussing length contraction and time dilation, because it converts these equations: L = L0 √ 1− (v c )2 (35) t = t0√ 1− (v c )2 (36) into these, vastly simpler forms: L = L0 γ(v) (37) t = γ(v)t0 (38) Here’s a table of some relativistic speeds, given in fractions of the speed of light, c, and their associated Lorentz factor. So, someone travelling at 10% the speed of light, or 0.1c, sees a 0.5% shortening in the length of stationary objects, and 0.5% increase in the flow of time. That’s not that much! And even at half the speed of light, it’s only a 15% change. You need to be travelling at almost the speed of light to see significant changes. At 90% the speed of light, the Lorentz Factor is more than two! Here’s a graph illustrating how quickly the Lorentz factor changes at very high speeds. Now it’s time to think about the issue of relatives in special relativity. Since a moving observer appears to be stationary in their own frame of reference, who can we trust when we say that lengths have been contracted and clocks have been slowed? To illustrate the problem, we need to introduce you to the Twins Paradox. 3.5.6 The Twin Paradox Let’s imagine that twins, named Leia and Luke, are preparing for an interstellar voyage. Leia will stay behind on Earth to monitor the journey, while Luke, the adventurous one, takes off on a spaceship capable of reaching nearly the speed of light. Since Leia stays here on Earth as Luke speeds away, Luke’s clock appears to slow down the faster his ship zooms off. But to Luke, Leia appears to be speeding away, so to Luke . . . Leia’s clocks have slowed down. Both observers see the other’s clock as being slow, while their own clock ticks at regular speed. How can that be? Surely, both observers can’t be right! Welcome to the Twin Paradox, proposed by Einstein, not as a paradox, but as a pecu- liarity of special relativity. In Einstein’s 1905 paper, he reasoned that if two clocks were synchronized, and one of them were to go on a lengthy journey, the travelling clock would return to the original 73 location with its time lagging behind the stationary clock. However, since relativity says that either clock could view the other as the one in motion (that is, the travelling clock could consider itself at rest, and the stationary clock would therefore be the moving one), couldn’t the stationary clock instead be the one lagging behind when the other returns again? Most explanations of the twin paradox talk about the acceleration of the travelling twin. It’s the travelling twin that experiences the effects of time dilation, returning to Earth much younger than the twin that stayed behind. However, the twin paradox can be explained without accelerations at all! Let’s watch as Luke flies to a nearby star system 6 light years away, while Leia stays behind on Earth. Since Luke can travel a significant fraction of the speed of light, say 0.6c. When we say that something is travelling at 1c, it is the same as saying it goes 1 light year per year. So if Luke travels at 0.6c, he will travel 0.6 of a light year for each year of travel. As such, Leia will say that Luke’s journey takes 10 years. Let’s carefully assess what Luke observes on his journey. Luke sets his spaceship to travel at speed 0.6c. In doing so, the distance to his destination changes. At 0.6c the length between Earth and the destination is shortened by 20%. So, instead of 6 light years, the star appears to be only 4.8 light years away from Luke’s perspective. At 0.6c, Luke times his arrival as only 8 years after his departure from Earth! Meanwhile, Luke has turned his ship around, and begins the journey back to Earth. Again, at 0.6c. So the same length contraction applies. Instead of flying across 6 light years of space, Luke only flies the length-contracted 4.8 light years, and he is back from the distant star system after another 8 year journey. For Leia, Luke has been gone 20 years, but from Luke’s perspective, he has only been gone for 16 years. Luke is now 4 years younger than Leia! Hopefully you can now see why Einstein said this is a peculiarity and not a paradox. Luke, the moving observer, sees space itself as foreshortened, but in Leia’s reference frame, the distances haven’t changed, instead the shape of Luke’s ship. We should note that Einstein developed Special Relativity to describe physics for ob- servers who are not experiencing a strong gravitational force. But what happens near an object with a strong gravitational field, such as a black hole? Einstein realized that he had to modify his theory of special relativity and make it more general to allow for gravity. Einstein called his relativistic theory of gravity, “General Relativity”. In order to understand the theory of General Relativity, we’ll begin with Einstein’s first ponderings on the subject . . . something called the Equivalence Principle... 3.6 The Equivalence Principle 3.6.1 Elevators in Space If humans decide to explore other solar systems, we will need to get there using Generation Ships. Ships like these can carry a self-sustaining human colony, which can survive for many generations. This is the premise of a television series called “Ascension”. The characters in the show walk around as if they are in Earth gravity, which is generated by the acceleration of their ship. But without windows to see outside, can the characters tell the difference between 74 uniform acceleration of their spaceship and simply being stationary on Earth’s surface? Suppose you awaken in just such a situation. You are trapped in a small room, unable remember how you got there, and with no way of seeing what’s outside. Could you tell the difference between the room being here on Earth, or in an accelerating rocketship? Einstein realized that these two scenarios are indistinguishable, so long as the room is small enough. An experiment that you could try, in your small room, is to drop a ball. There are two possible outcomes, depending on where the room is. If it’s here on Earth, the ball will appear to accelerate downwards, falling to the ground, due to gravity. In the second scenario, where you are on an accelerating rocketship, far from any sources of gravity, the ball still appears to fall, only this time, it’s because the rocket is accelerating upwards! If the acceleration of the rocket is precisely 9.81 meters per second squared, the motion of the ball will be indistinguishable from the effects of gravity on Earth. Nice try, but dropping the ball won’t tell you which of the two scenarios you’re in. You might say that you “dropped the ball” by dropping the ball. Not knowing whether it’s the rocketship accelerating, or the ball accelerating due to gravity, is called the Equivalence Principle. In fact, the formal definition of the equivalence principle given by Einstein states, “We assume the complete physical equivalence of a grav- itational field and a corresponding acceleration of the reference system”. What Einstein discovered is that gravity is not a force, as it was believed to be by Newton and classical physicists. In small regions of space, gravity is indistinguishable from acceleration. Now, we should note that Einstein’s statement of the equivalence principle requires that your choice of a reference frame is small. Earth’s gravity, for example, doesn’t change much between your toes and your head. However, if you chose a tall reference frame, or an extremely wide reference frame, it would become pretty obvious that you were on Earth, just by measuring changes in gravity as you moved around. Gravity would become weaker the higher you went from Earth’s surface. The force of gravity on the surface always points to the center of the Earth, so if you have a large horizontal frame, the direction balls will fall will be different. The converse of our ball analogy is also true. When an astronaut is orbiting the Earth, they appear weightless, because they are in “free-fall”. In free-fall, the force due to gravity is exactly matched by the centripetal acceleration towards the Earth. This is similar to riding your favourite “Drop-of-Doom” amusement park rides, where you feel temporarily weightless. Astronauts are in a perpetual Drop-of-Doom! This would be indistinguishable from an astronaut who has run out of fuel far from any sources of gravity! Many science fiction shows, including Star Trek portray “artificial” gravity, but fail to explain the technology necessary in order to produce these fields! However, some science fiction gets it right! In the movies ‘2001: A Space Odyssey’ and ‘Interstellar’, gravity is produced by rotating the spacecraft to introduce centripetal acceleration, which mimics gravity. An example of the equivalence principle at work. If a ball thrown on Earth travels on a curved path due to gravity, we’d expect the same curvature to occur on an accelerating rocketship. Einstein’s explanation of gravity introduces the idea of the thrown ball’s trajectory being the shortest possible path through a curved spacetime. Gravity itself, Einstein believed, was the result of spacetime being curved by mass and energy. 75 3.6.2 How do black holes fall? (Interview with Dr. Jeremy Heyl) Q: How do black holes fall? Dr. Heyl: One fun thing that I did recently with this was to ask the question whether black holes, like if you think of Galileo, the first astronomer, there’s this story of him drop- ping things off towers and whether they fall faster or slower depending on what they’re made of. You can do the same experiment with black holes, because there’s black holes in galaxies, and these galaxies are falling through the universe. You can ask, does the black hole fall any differently than the rest of the galaxy, for example? Amazingly, I mean, maybe not surprisingly, they do! But what’s maybe more surprising is that you can actually pose the question that, ”Hey, how does a black hole fall? Does it fall like normal stuff or not?” I mean, we’re not very precise, they fall like normal stuff to within 50 percent, maybe about as precise as Galileo was, 400 years ago. But I mean, we can get better. Dr. Heyl’s webpage: https://jeremy.coolpulsars.org/ 3.7 Curved Spacetime 3.7.1 Invariants One of the most important realizations Einstein made while developing Special Relativity is that there is no such thing as a universal time, or distance. Instead, Special Relativity introduced the notion of an invariant spacetime interval. When astronauts travel at different speeds, and experience differences in the duration of time intervals, they are effectively trading distance for time or vice-versa. They are experiencing a distortion, or warping of space and time together. This is required in order for all observers to agree on the speed of light. Einstein realized that he could explain the effects of gravity by combining the equivalence principle with the concept of an invariant spacetime interval used in special relativity. When Einstein developed the General Theory of Relativity, he came to the realization that Gravity IS the warping of spacetime. So stars like our Sun, which have strong gravi- tational fields due to their large mass, actually bend and stretch the fabric of the universe itself! The warping of spacetime causes planets and light to travel on curved paths near massive objects. An early test of General Relativity depended on the bent spacetime around the Sun. In 1919, the astronomer Sir Arthur Eddington led an expedition to an island off the west coast of Africa in order to measure how much the gravity of the Sun warped spacetime. They did it by observing a total eclipse of the Sun. During the eclipse a star could be seen next to the eclipsed Sun. (Aside: Those of you who have observed a total solar eclipse know that the Sun looks eerily like a black hole in the sky, surrounded by white hair!) The locations of all the stars in our neighbourhood of the galaxy were mapped more than 100 years ago, so it was known that this star should really be located directly behind the Sun when viewed from the Earth on the day of the eclipse! So how did Eddington and his team see the star? The light from the star travelled on a curved path around the Sun to the astronomers’ telescopes! This effect was predicted by Einstein’s theory of general relativity! 76 The measurement they made confirmed, and has since been reconfirmed, that the theory of General Relativity is accurate, and that spacetime itself is bent by matter. This changes our notion of what a straight line is, of course, because if spacetime itself is bent, how can we possible know that we are going in a straight line?? Instead of calling them “straight lines” in General Relativity, we call them geodesics, which represent the “straight” path of an object in a curved spacetime. 3.7.2 Geodesics Even though geodesics represent straight lines in curved spacetimes, they wouldn’t be con- sidered straight by our standards. Just as the Sun’s mass bends the spacetime around it, any light crossing bent spacetime will appear to have its path bent. Since we are talking about curving spacetime, let’s consider the surface of this Chalk Ball as a section of a curved two-dimensional space. This works equally well if you imagine the Chalk Ball to be Earth. If I asked you to draw a straight line between two points on opposite sides of the ball, the same as asking the flight path between two cities on Earth, you might be tempted to draw along an equatorial line to join them together. Even though I asked you to draw a straight line, it’s curved! Instead, the distance between two points on a curved surface is considered “straight” if it’s also the shortest line joining the two points together. These curves are called geodesics. If you look at the flight path of an airplane from Toronto to London, the airplane crosses the ocean near Greenland! The shortest path joining the two cities is a curved path! The same is true for any object travelling through curved spacetime. And now, where do you think the MOST convoluted curvatures of spacetime in the universe exist? That’s right. . . BLACK HOLES. Not only do black holes warp spacetime, they warp it to the point that even light will travel on highly curved paths. Photons, by definition, travel on geodesic paths in spacetime. Close to the black hole, the curvature becomes so high, that light is bent into paths that all terminate at the black holes’ singularity. General relativity interprets gravity as the warping of spacetime. When we view a picture of the gravitational field around a massive object, it’s usually represented as a depression in space, however, we also need to understand that gravity also warps the passage of time. It’s strangely difficult for the human mind to grasp the concept of warping spacetime. We understand what it means to bend, or warp, a material like plastic. But what does it mean when the actual space and time that we live in are bent and twisted? In a sense, warped spacetime means that the paths we choose to cross space and time will be shorter or longer, both in distance between two points, and in the duration it takes to travel between them, depending on what the gravitational fields are along the path. Let’s focus specifically on how gravitational fields warp the time component, in an effect named Gravitational Time Dilation. Let’s start with an example by considering two astronauts exploring an unstudied planet around a distant star. . . Perhaps planet “E” in the nearby TRAPPIST-1 system, which we’ll shorten to TRAPP-E. One astronaut needs to stay with the ship in orbit around the parent star, while the other astronaut descends to TRAPP-E’s surface. Since we are talking about time, both astronauts will need to carry clocks, which they synchronize before they separate. Far from the surface of the planet, both clocks tick in perfect synchronicity. 77 One astronaut now descends to the surface of TRAPP-E, which is deep within the grav- itational well of its parent star, ready to collect data! Since the planet itself is deeper in the star’s gravitational well, and therefore experiences a greater gravitational force, the space- time in the vicinity of the planet will also be warped. The effect that the warping has on the astronaut’s clock causes it to tick more slowly than the one in orbit. For every tick of the clock on the surface, the orbital clock ticks more rapidly. On the surface of TRAPP-E, the astronaut doesn’t experience the change in the passage of time, because all biological processes are likewise slowed down by the warping of gravity. Just like the ticks of the clock, a distant observer would see the heartbeat of an astronaut on the surface to beat more slowly. Once the surface mission is complete, the two astronauts rejoin one another in orbit around TRAPP-E. The astronaut who stayed in orbit will be dismayed. . . they experienced a longer time than the astronaut who was on the surface! Depending on the duration of the stay, and the strength of the gravitational field, the astronaut who went down to the planet’s surface will experience fewer ticks of the clock, and therefore be several seconds younger than the one that stayed in orbit! To calculate how time is warped in a strong gravitational field, the following equation is employed: ∆tplanet = ∆torbit √ 1− 2GM Rc2 (39) ∆tplanet, the elapsed time on the surface of the planet, is equal to ∆torbit, the elapsed time on the orbiting spaceship times the square root of one minus 2 times G times Mass divided by R times c-squared. In this formula, the mass and radius refer to the mass and radius of the planet. But if instead of a planet you were a distance R from a star or black hole with mass M, you could use the same formula. The important thing in this formula is that the quantity inside the square root sign is smaller than one, so the amount of time that passes when you’re in a gravitational well is smaller than if you’re out in space far from a gravitating object! Note that this formula doesn’t make sense if the ratio of the mass to the radius gets too large! This formula only makes sense if R > 2GM/c2 (40) R is larger than 2 times G times mass divided by c2. 3.7.3 GPS Satellites You might think that your everyday life is not affected by time dilation due to special or general relativity. However, you may be surprised to learn that almost everyone carries a piece of technology that would be useless without both theories: GPS. The Global Positioning System that you use every time you navigate with a map on your smartphone, depends on Einstein’s theories of relativity to function correctly. Handheld GPS works because the device inside your smartphone is capable of measuring and comparing the signals from multiple satellites in orbit around Earth. These satellites are placed in well-known orbits and carry very precise clocks. By broadcasting a timing signal 78 that can be picked up on a GPS receiver, the difference in timing signals from different satellites can be used to triangulate your position. Since GPS satellites travel at about 14,000 km/h, they experience a very slight time dilation due to special relativity. Each day a satellite’s clock would appear to slow down by about 7 microseconds. That doesn’t sound like much, but if you neglected this drift, your GPS would accumulate an error of about 2 kilometers every day. General Relativity predicts that the clocks aboard a GPS satellite, travelling at an alti- tude of 20,000 kilometers, would appear to tick faster than clocks on Earth. Every day, a satellite’s clock would appear to speed up by 45 microseconds, compared to clocks on Earth’s surface. If this error wasn’t corrected, the GPS would accumulate an error over 13 kilometers a day. Since special relativity works to slow down the apparent rates of the clocks on a GPS satellite, and general relativity works to speed up their apparent rates, the combined effects add up to a 38 microsecond per day error. Without accounting for relativity, our GPS devices would drift by over 10 kilometers every day, roughly the same as 12 centimeters a second! Luckily, we know about the effects of relativity, so we can correct for this drift. GPS devices are some of the most robust tests we have for Einstein’s theories of relativity. In the movie Interstellar, the main character Cooper is sent to retrieve a fellow explorer, Mann, from the surface of Miller’s planet orbiting the nearby black hole, Gargantua. On the surface of Miller’s planet while an observer experiences one hour of time, people on earth will experience 7 years of time. This is an example of the correct use of an effect called Gravitational Time Dilation. 3.7.4 Gravitational Redshift Since Gravitational Time Dilation slows down the passage of time in intensely strong grav- itational fields, it equally affects physical processes that are time dependent. That means that someone observing an astronaut in orbit around a black hole would see their clocks ticking slow AND their hearts beating slower, AND everything about them slowed down. So what happens to a beam of light when it’s generated deep in the gravity well near a black hole? The beam of light experiences Gravitational Redshift. Recall the doppler effect that we discussed earlier. When a moving object, like a rocket ship, is emitting light, the light can be blueshifted or redshifted depending on the ship’s motion towards or away from the observer. If a ship were to accelerate away from you, you would see the light from its engines become redder and redder as it accelerated to ever increasing speeds. Light emitted from deep within a gravitational well has to work against gravity in order to leave a planet or a star or the region near a black hole. When light travels away from a planet, the photon has to convert kinetic energy into gravitational potential energy. If we remember that red photons have less energy than blue photons, we can predict that photons emitted from the surface of a star will appear redder to an observer far from the star. This effect is called Gravitational Redshift. The gravitational redshift effect is very small, but it has been measured in the light emitted by a white dwarf star, and it agrees with the predictions of general relativity! 79 3.7.5 A brief history of relativity (Interview with Dr. Robert Smith) Q: What is the history behind Einstein’s prediction of the bending of light rays? Dr. Smith: Einstein develops special relativity in 1905, the famous publication. By 1907 though, he’s beginning to realize that he would like to generalize special relativity. No longer talking about reference frames which are moving uniformly with respect to one another. But what about reference frames that are moving in an accelerated fashion with respect to one another? What about rotating reference frames? So, by 1911, he’s got his first, what’s called the Prague theory of general relativity and that will change. By 1913, he’s got a version which he thinks is superior. Then, the final version of general relativity that Einstein would develop, he’s got by the end of 1915 start of 1916. But the famous light deflection that he predicts in 1911, comes out of an earlier version of general relativity. So, he calculates how big should the deflection of light be, as light, from a distant star just passes by the sun. How much will it be affected? The calculation he makes turns out to be an answer of less than one second of arc. An astronomer at the Berlin Royal Observatory, a man called Erwin Finlay-Freundlich, is really interested in Einstein’s theory, is one of the very few astronomers who’s actually interested in trying to test the results of general relativity. So, Freundlich tries to look at past photographs of eclipses. Can he look at these earlier photographs and see star images, and compare the positions of those stars with the Sun there at the eclipse? And also when the sun has moved away from the star, because then we should see the deflection. But the images were just not good enough for a range of different reasons, so he decides in 1913 that he would like to engage in an expedition. Go to the Crimea in August 1914, take photographs of the sun during eclipse in order to get the star positions during the eclipse. Take photographs at a different time when the sun is moved away, compare the star positions between the two sets of photographs, and then you’ve got the measurement of the deflection of light. What was the effect of World War I on the history of relatvity? Dr. Smith: Freundlich leads an expedition and they go to Russia, and the eclipse will occur at the end of August 1914. But this is not a good time to be a German traveling in Russia, because World War I breaks out and so Freundlich and his companions and their instruments are taken into custody, they’re thought to be spies and they’re held in Odessa for several weeks before, in fact, they’re released but the instruments are confiscated. But what is very interesting about this expedition is if they had in fact gone ahead, made the measurements of the deflection of light, they would have come up with an answer which would have been roughly twice the answer that Einstein had calculated, based on one of his earlier versions of general relativity. So, the actual shift that Einstein predicted later would be 1.75 seconds of arc. At this point, the prediction that Einstein makes is 0.85 seconds of arc. The first measurements made at an eclipse of the deflection of light that are successful anyway, the result is 1.64 seconds of arc. So, what that means is, if Freundlich had gone ahead, made the observations in 1914, the answer would have been about double Einstein’s prediction. Then Einstein would have come along later and redone his calculations, ”Oh look, my answer is now double what I pretty much got the amount that the light is deflected by.” Now, would that have aided the acceptance of general relativity? I suspect not because coming in after the fact, after the measurements have been made and say, ”Oh, I goofed by a factor of two” is not really a very convincing way to verify your theory I think. 80 Dr. Smith’s webpage: https://apps.ualberta.ca/directory/person/rwsmith Dr. Smith’s book on the Hubble Space Telescope: https://www.amazon.com/Hubble-Imaging-Space-National-Geographic/dp/B00B9ZE8TY 3.8 Summary Earlier in this module, we came to understand how event horizons work by drawing an analogy to a fish falling over a waterfall. At a certain point in its descent, the falling fish can no longer communicate with its friend above, because the water is flowing downwards at a greater speed than the speed of sound. Luckily in this story, we rescued our fishy friend using a rocket pack. This was possible because the speed of sound is not a universal limit. The speed of light, on the other hand, is. Were we to drop our fishy friend into a real black hole, a rocket pack would be of little use, because once you’re beyond the event horizon of a real black hole, escape is NO LONGER POSSIBLE. But don’t worry, no fish were harmed in the making of this course! We also learned about a revolution in physics which began in the early 1900s. Einstein, in a single year and with four extraordinary papers, turned physics on its head. One of the results of these papers was that space and time were no longer absolutes, and in their place was this amalgamation called spacetime. In this framework, relative motion is what matters, and many of our traditional notions about how the universe works no longer apply. Some truly weird things start happening, like length contraction, time dilation, and relative simultaneity. When Einstein went about generalizing his Theory of Relativity, he realized that there ought to be equivalence between gravitational fields and the acceleration of a reference system. Meaning, if we stuff you in a windowless rocket you can’t tell the difference between a rocket sitting at rest on the surface of the Earth or a rocket that is accelerating uniformly upwards at a rate equal to the acceleration due to gravity on Earth’s surface. Under these conditions, the two are indistinguishable. In addition, General Relativity comes with its own quirks, one of which being that mass deforms spacetime. This is what causes the effect known as gravitational lensing, something not exclusive to black holes but something they are nonetheless known for: that is, their ability to bend light rays. In the next module, we’ll explore black holes in more detail and learn how to weigh a black hole! 81 4 Sizing Up Black Holes 4.1 Introduction Black holes represent spacetime that is so warped, light itself is unable to climb beyond the event horizon. The properties of a black hole depends on its mass and its spin. Observations of black holes support a categorization system that divides black holes by mass: stellar mass, intermediate mass, and supermassive black holes. The black hole categories reflect the environments where the black holes live, and the ways these black holes form. Stellar mass black holes form from the collapse of a star. Less is known about the formation of intermediate, and supermassive black holes. The stellar mass black holes may act as the seeds for larger black holes that grow as other stars and black holes collide and merge. Let’s explore the characteristics of each category of black hole. 4.2 May the Schwarz(schild) Be With You 4.2.1 The Schwartz In the 1987 movie, “Spaceballs”, which is a parody of the Star Wars franchise, a mysterious force called the “Schwartz” allows Dark Helmet to battle Lone Starr. Upon engaging in battle, Dark Helmet says to Lone Starr, “I see your Schwartz is as big as mine”. Strictly speaking, the Schwartz has nothing to do with black holes. But this scene is an easy way to remember that the Schwarzschild radius is a measurement of the size of a black hole. In order to “size up” a black hole, you need to be able to deduce how massive it is! The first major tool for discovering the properties of black holes was developed by Ein- stein, who first published his field-equation formalism for General Relativity in 1915. Einstein himself wasn’t confident that his equations allowed exact solutions, I mean, just look at this mess. . . Rµν − 1 2 Rgµν + Λgµν = 8piG c4 Tµν (41) Well luckily, there are some people out there that don’t listen to Einstein, and amazingly, a solution to Einstein’s equations was developed by German astronomer Karl Schwarzschild the same year that Einstein introduced General Relativity. Schwarzschild’s solution to Ein- stein’s field equations gave us the first glimpse at the nature of black holes. The result, of course, is the expression for the Schwarzschild radius: rs = 2GM c2 (42) Which relates the mass of a black hole with the radius of its event horizon. The best part, Karl’s solution to Einstein’s equations were exact, making Einstein ex- tremely happy! At first, it wasn’t recognized the Schwarzschild solution described black holes, in those early days they called them totally gravitationally collapsed objects. Only much later on, in 1967, were they formally called “black holes”. Schwarzschild provided a convenient equation for deducing the radius of a black hole, knowing only its mass. But that 82 still leaves us with a problem: we can’t travel to a black hole to measure the size of its event horizon, so how are we supposed to deduce the mass of a black hole in the first place? Karl Schwarzschild was serving in the German army when he wrote the solution in a letter to Einstein in December 1915. Unfortunately, Schwarzschild was suffering from a skin condition, and succumbed to it only six months later. He died on May 11th, 1916, without knowing the impact his work would have in the era of Modern Physics. 4.2.2 Combing Black Holes Why then, is it so hard to measure the properties of black holes? Think about it like this: humans have 5 senses that we can use to touch, taste, smell, see, and feel objects in our environment. We’re really good at it: we can distinguish millions of colours, thousands of smells, hundreds of tastes, excetera. There’s a reason for the english idiom, “A picture is worth a thousand words”. But black holes have none of these things. They can’t be tasted, smelled, they have no colour, make no sound, but above all, they emit no light. Back in 1967 an astrophysicist named Werner Israel, a professor here at the University of Alberta, was the first to demonstrate why black holes were so elusive. He showed that if any large feature existed on the black hole’s surface, like “mountains” or “hair”, they would tend to smooth out. Scientists now call this the “No Hair Theorem”. The name is used to illustrate that black holes have no other independent characteristics than their Mass, Charge, and Spin. There are no black holes with ponytails, mullets, or mohawks, because if they indeed had any kind of identifying feature on their event horizon... well, it would get sucked into the black hole! A black hole’s event horizon is a featureless boundary. Just like the scene in Spaceballs where Dark Helmet is searching the desert on the Moon of Vega. . . If you try to comb a black hole, you aren’t going to find anything! 4.2.3 Static and Spin When you rub a balloon against your head, you create a difference in charges between your head and the balloon: your hair, charged with static electricity, will stand up straight, while the balloon can now stick to a nearby wall. A law of physics, called the Law of Conservation of Charge tells us that charge can’t be created or destroyed, so even when charges are separated, they are always balanced between positive and negative charges. Since black holes don’t have hair, you can’t rub a balloon against them. . . But a black hole can have charge. This might happen if say, a statically charged balloon is carried across the event horizon, but not the head of hair with the balancing charge. However, there is a force of Electrostatic Attraction between the (+) and the (-), which, like gravity, will pull these two objects together. This means that any Charged Black Hole will attract the opposite charges, and eventually it will become neutral! Most black holes are theorized to be neutrally charged. The opposite is true for Black Hole Spin. We mostly talk about black holes that have no spin, but what would happen if say, I throw in a frisbee that is rotating? Any of the angular momentum carried by the frisbee will have to be transferred to the black hole once it crosses the event horizon. 83 A Schwarzschild black hole is a non-spinning black hole, and these types of black holes aren’t very realistic. Out in space, even a spinning frisbee is enough to impart spin on a black hole! 4.2.4 Measuring Mass We’ve spent an awful lot of this lesson telling you what you can’t measure about black holes. So now we need to ask what can you measure about black holes. Well, a black hole all by itself isn’t going to tell you everything t you want to know. A second object is required, something that is big enough, and even bright enough for us to see, which will interact with the black hole, and therefore reveal its secrets! I’m leaving you in good hands, but before I go. . . May the Schwarzschild be with you! 4.3 Dancing with the Stars 4.3.1 Pairs of Stars Without hair, black holes are incredibly difficult objects to observe. In order to truly measure their mass, a second object, hopefully one with “hair”, needs to be in the vicinity of the black hole. Fortunately, most stellar systems contain two or more massive bodies. Our home star, the Sun, is an isolated star. The closest star to us, named Proxima Centauri is 4.2 light years away. But many stars, including Proxima Centauri live in groups of two, three, or even four stars. In fact, systems consisting of two stars, called binaries, are just as common as single stars. Binary systems, of two stars, can consist of any combination of stars, neutron stars, and black holes. They are scientifically important since they allow us to measure the masses of the components in the system, and in some cases determine their sizes. Since the most accurate method for determining a black hole’s mass is to observe its orbital dance with a companion star, we need to examine how stars move when they have a gravitational dance partner. When a compact object, either a black hole or a neutron star, are in a dance with a companion star, it is sometimes hard to determine what the identity of the compact object is. This is because neutron stars can look strikingly similar to black holes, leading to cases of mistaken identity! The weight of the compact object, or really, its mass, is the key difference that distinguishes between these two types of compact objects. Neutron stars cannot be heavier than three times the mass of our Sun, or else they would be dense enough to become a black hole. If we see a compact object hiding in a binary system that might be a black hole, we call it a Black Hole Candidate. If we can measure the mass of the candidate and it is larger than 3 times the mass of the Sun, then we are confident that the object is not a neutron star, and we call it a black hole. 4.3.2 Centre of Mass When two stars are in a binary system, they orbit on either circular or elliptical paths around a point in space called the Centre of Mass. 84 Let’s start off with circular orbits with both stars moving in a circle around the centre of mass. The centre of mass is a balance point that always falls on a line connecting the two stars. If the two stars have the same mass, then the system is perfectly balanced. Notice in this animation that the two stars have the same mass and move on the same circular path around the Centre of Mass. The Centre of Mass is always directly in between the two stars. We call the time that it takes for a star to travel one time around the centre of mass the orbital period. Notice that both stars take the exact same amount of time to make one full orbit. If the two stars in a binary system have different masses, the balance point between them will no longer be equal distance between them. Instead, the higher mass star will be closer to the centre of mass, while the lower mass star will orbit further away. In this picture, the high mass blue star has two times more mass than the little red star. The little red star orbits in a circle with a radius that is two times larger than the radius of the circle that the big blue star orbits on. 4.3.3 Playground Break The situation is similar to when my little sister and I played together when we were children. My sister is four years younger than me, so when I was eight years old and she was four years old, I weighed two times more than she did. Sometimes we played on a teeter-totter. If we both sat in the seats at equal distances from the pivot point, then the teeter-totter was unbalanced and I could stay low and keep my sister up high. This led to her crying which wasn’t very nice. The only way to stop her crying was if I moved closer to the pivot point. Once I found the correct balance point we could then swing up and down and have fun. The kids playing on a teeter-totter are similar to two stars in a binary system. The pivot point is like the centre of mass. The kids are like the stars, except that the stars move in circles while the kids move up and down. The larger mass kid, or star, have to be located closer to the Centre of Mass, while the smaller mass kid, or star, are far from the Centre of Mass. We can quantify the relative distances that the stars have to be from the centre of mass in order to balance the binary. The two stars have masses M1 and M2. The first star is at a distance a1 from the centre of mass, while the second star is located at a distance a2 from the centre of mass. The equation relating the star’s masses and locations is M1a1 = M2a2 (43) M1 times a1 equals M2 times a2. In order to balance the binary, we have to balance the equation so that the larger mass star has the smaller distance from the centre of mass. In this picture, we can see by eye that a1 is two times larger than a2. This tells us that M2 is two times larger than M1. In a circular binary, the total distance between the stars is constant in time as the stars move in their circular orbits. If we define the total distance between the star to be “a”, then you can see from the picture that a = a1 + a2, (44) 85 a equals a1 plus a2. Sometimes a binary star system consists of a bright, easy to see star, and an invisible star. Even though we can’t see the invisible star, we can deduce its presence when we see the bright star moving in circles around a point in the sky. The empty point in the sky is the centre of mass, and the invisible star is also orbiting the same point. The invisible star might be a dim star like a neutron star, or it could be something that isn’t really a star, like a planet, or a black hole! Often we can’t see the orbital motion as circles in the sky because the stars are too far away. But what we can do is detect doppler shifts in the light emitted by the stars. When a star is moving towards us, its light is blueshifted, and when it is moving away from us the light is redshifted. When we see light from a star periodically doppler shifted from blue to red to blue, we can deduce that we are observing a binary star system. This is actually the most common way that binaries are detected. 4.3.4 Kepler’s Laws of Planetary Motion The motion of the planets around the Sun is similar to a binary system. Suppose that we ignored all the planets except for Jupiter, since Jupiter is the biggest planet in our solar system. Then the Sun and Jupiter are like a binary system. Since the mass of the Sun is one thousand times larger than Jupiter, we shouldn’t expect that the center of mass is exactly at the centre of the Sun! Jupiter is massive enough to pull the centre of gravity away from the centre of the Sun, somewhere closer to the Sun’s surface. Johannes Kepler studied the motions of the planets around the Sun and came up with a set of three “laws” of planetary motion. The first law is the law of orbits that states: Planets orbit the Sun in an ellipse where the Sun is located at one focus. An ellipse is a shape that is sort of like a squashed circle with a short dimension and a long dimension. It has two special points inside that are called the focus points. A circle is a special type of ellipse where both dimensions are the same size, and the two focus points converge to a point at the centre of the circle. Although the planets travel on elliptical orbits, these ellipses are almost circular for most of the planets. The Earth’s orbit is just a tiny bit elliptical, and the Earth is closest to the Sun in January, and furthest from the Sun in July. In a binary star system, Kepler’s first law tells us that the stars orbit on ellipses with the centre of mass located at one of the focus points. Kepler’s second law is the law of equal areas. Imagine a line joining the Sun and the planet. As the planet moves, the line sweeps out an area. Kepler’s second law is that the line sweeps out equal areas in equal times. This is easier to think about in terms of an elliptical chocolate cake. Suppose that we want to cut slices of cake from the focus to the edges of the cake, but everyone should the same amount of cake. If we cut a wedge near the focus, then we need to cut a wide wedge since the length of the cuts are short. If we cut a wedge at the opposite side of the cake, the cuts will be long, so the wedge needs to be skinny. If we cut the cake like this, everyone gets the same amount of cake and everyone is happy! 86 Based on the equal areas law, a planet travels faster when it is near the Sun, and slower when it is further from the Sun. Here on Earth, that means that in January, when the Earth is closest to the Sun the Earth travels faster and in July, when the Earth is farther from the Sun it travels slower. As a result, the Earth spends more time in the outer part of its orbit. Kepler’s third law originally only applied to planets, but Newton improved it so that it could be used to describe the orbits of stars in a binary system. Kepler’s third law is an equation that relates the masses of the stars to the orbital period and the total distance between the stars. The equation is M1 +M2 = a3 P 2 (45) Mass-1 plus Mass-2 equals the cube of the total distance between the stars, divided by the square of the orbital period of the stars. In this equation, a represents the total distance between the stars measured in Astronomical Units, which is the distance between the Earth and the Sun. The time for the stars to orbit around the centre of mass, the orbital period, is represented by the letter P. The period is measured in units of years. The sum of the masses of the two stars is in units of the Sun’s mass. We had better do an example! Suppose that we observe two stars in a binary. We can easily time how long the orbits are by watching for a while, and find that the stars take 2 years to make one full orbit. Measuring the distance between the two stars is harder to do, but we learned that the distance is 4 astronomical units. Using Kepler’s 3rd law, we can calculate the sum of the masses in this star system to be M1 +M2 = 43 22 = 4× 4× 4 2× 2 = 16MSun (46) M1 plus M2 equals four cubed divided by two squared. You can probably do this math in your head, or use a calculator to find that the sum of the two masses is 16 times larger than the Sun’s mass. Unfortunately, this only gives us the value of M1 plus M2, so we’ll need some more information to find out the masses of each star as individuals. Perhaps we can measure the properties of the light from one of the stars and it is identical to our Sun! Then it would be logical to assume that its mass is the same as the Sun, so M1 = MSun (47) M1 equals MSun. Then it is simple to solve for the mass of the other star M2 = (16− 1)MSun = 15MSun (48) M2 equals sixteen minus one equals fifteen solar masses! This is a simplified example, but this is how astronomers measure the masses of stars and black holes! Now that we can finally deduce the mass of a black hole in a binary pair, we’ve can now distinguish between different masses of black holes. Just like sports are often divided into “weight classes”, so too are black holes lumped into major categories. 87 4.4 Black Hole Weigh-In 4.4.1 Weight Training **DING DING DING** ON THIS SIDE OF THE BINARY SYSTEM WE HAVE A COM- PACT OBJECT! LET’S WELCOOOME NEUTROOON STAAAAAAAAR! AAAAAAND IN THIS CORNER WE HAVE A BLACK HOOOOOLE — STELLAR MAAAAAAAAAASSSS!!! AND THREE! TWO! ONE! ORBIT! When a black hole is in a binary system, we are able to deduce how massive it is. But that alone isn’t enough to really say what the black hole is like. We need to explore the way that black holes are classified. Any black hole astrophysicist will tell you about the three basic weight classifications of black holes: stellar mass, intermediate mass, and supermassive black holes. There are other ways of classifying black holes as well, like whether they have charge or spin. But there might be some black holes that don’t fit into those categories at all! 4.4.2 Black Hole Lightweights Stellar mass black holes are the lightweights of the black hole family. They are produced at the end stages of a star’s life, and range between about 3 solar masses up to about 100 . This is due to the 20 to 30 solar mass minimum mass for a dying star scientists theorize is required in order to create a black hole. Also, main sequence stars only get as big as about 100 solar masses. Since stellar mass black holes require the collapse of a star, they are sometimes called collapsars. During the collapse, lots of gas will be expelled before the black hole forms. The resulting black hole will have a much lower mass than the progenitor star. The collapse of a supergiant star might produce a three solar mass black hole. The Schwarzschild radius equation tells us that a 3 solar mass black hole would have an event horizon roughly 9 kilometers in radius. That’s an easy ratio to remember: for every solar mass, the radius of the event horizon gets roughly 3 kilometers larger. A 9 kilometer radius black hole would have a footprint similar to the size of Manhattan. The diameter is double the radius, so a three solar mass black hole would have a diameter of 18 kilometers. That would make its diameter roughly twice the height of Earth’s highest mountain, Mount Everest! If instead you measure 18 kilometers across the ground, you could walk across it in a little more than 3 hours... As long as you could survive the extreme gravity! 4.4.3 Black Hole Middleweights Intermediate mass black holes are the middleweights of the black hole family. These black holes aren’t the result of a stellar collapse, but of an existing stellar mass black hole growing by consuming gas, dust, stars, and other black holes to become more massive. Intermediate mass black holes are classified in a range between 100 and 100,000 solar masses. At 100 solar masses, the lightest of the intermediate mass black holes would be roughly 300 kilometers in radius, which means they would span the orbital height of the International Space Station, which orbits Earth 400 kilometers above the surface. 88 4.4.4 Black Hole Heavyweights Supermassive black holes, the heavyweights of the black hole family, occupy everything above 100,000 solar masses. These are the black holes that reside at the center of galaxies, and are some of the oldest objects in the Universe. Weighing 100,000 solar masses, the diameter of the smallest supermassive black hole’s event horizon would be around 600,000 kilometers. That’s about 40% as big as the Sun! A black hole eclipse would sure look strange! The biggest black holes that astrophysicists theorize are limited to no heavier than 50 BILLION Solar Masses! That would make the largest mass black holes about 150 BILLION Kilometers across! At that size, the largest black holes would still be. . . smaller than our own solar system, when you consider that it goes well beyond Pluto to the edges of the Oort Cloud, which is over 5 QUADRILLION Kilometers from the Sun. A photon, travelling at the speed of light, would still need 58 days to go that distance! 4.4.5 The Black Hole Family We only calculated the smallest black hole in each of the categories in order to illustrate where the boundaries are in the range of black hole sizes. In reality, these boundaries are more like scientific guidelines than they are rules. The black holes that we have observed and measured fall all across this scale. The first black hole merger, detected by LIGO, created a stellar mass black hole weighing 60 solar masses. We suspect that intermediate mass black holes could be found at the center of smaller, dwarf galaxies, which can be found in orbit around our own Milky Way Galaxy. They fall into the intermediate category, but below 10,000 solar masses they are just below our guidelines for supermassive black holes. The black hole at the centre of our galaxy is Sagittarius A*. It weighs just over 4 MILLION solar masses, our galaxy’s heavyweight contender. 4.4.6 Is there a limit to the size of a black hole? (Interview with Dr. Gregory Sivakoff) Q: Is there a limit to the size of a black hole? Dr. Sivakoff: In theory, there is no upper limit that I know of for how massive a black hole can grow. It needs to have nearby material that it can accrete from eventually will fall onto the black hole. So in some ways you might say, well, a black hole can only grow as big as the galaxies in. A matter of fact, most black holes don’t grow bigger than about one 10th of a percent of the galaxy they’re in. Because not all of the galaxies material is by the black hole. The biggest black holes that I know of are probably about 10 billion times the mass of the sun, or in that rage. It’s unlikely that we’re going to get much bigger black holes because we don’t have much bigger galaxies than these. So perhaps in time, if a lots of galaxies were to merge, you might make even bigger black holes, but I doubt you’re going to see something significantly bigger than that. Dr. Sivakoff’s webpage: https://sites.ualberta.ca/~sivakoff/ 89 4.5 Stellar Mass Black Holes 4.5.1 Birth in Isolation Stellar mass black holes are born during the violent deaths of massive stars. If the star in question wandered through space as an isolated star, like our Sun, then a black hole formed as a result of the star’s death would also be an isolated black hole. Since isolated black holes wander through space without a companion, they have little to feast upon. As a result, there is little material for them to consume, and so, they can virtually undetectable. Such isolated black holes can also be formed by a different route. Stars are more generally birthed in multiples, so as binaries, triplets, quadruplets and so on. If one of the stars is a high mass star, it will burn through its fuel more rapidly than others in its family. At the end of its life it will explode in a violent supernova explosion. This explosion can provide a ‘kick’ that, if strong enough, would break up the family, throwing the black hole out into space by itself. The ejected black hole may even be on a trajectory that throws it out of the disc of its host galaxy! If, however, the kick is small, the binary will remain intact, providing the black hole with a handy food source to munch on for many years to come. 4.5.2 Finding Stellar Mass Black Holes Since stellar mass black holes are the endpoint of a star’s life, they can occur anywhere you would find stars. This means that stellar mass black holes are found scattered throughout galaxies. They can be seen around the core of a galaxy, or at its outerlimits. Almost all of the known stellar mass black holes have been found in binary systems. But this doesn’t mean that black holes in binaries are more common than isolated black holes. Instead, this is most likely due to selection bias. In science, whenever we choose a sample based on how easy it is to find instead of how common it truly is, we introduce a bias, called selection bias. Actively feeding black holes in binaries are much easier to spot than their fasting coun- terparts. That means that we are much more likely to detect stellar mass black holes in binaries when we look out at the night sky. 4.5.3 Masses of Stellar Mass Black Holes When stellar mass black holes live in a binary system with a companion star, we can deter- mine the black hole’s mass using Kepler’s laws of motion. If the companion star is a regular star such as a main sequence star or a giant star that orbits close to the black hole, gas from the companion star can flow towards the black hole, and be captured in an accretion disk. This hot disk of gas will emit X-rays, which we can detect using an X-ray telescope. Sometimes we can see the motion of the companion star. This allows us to measure the orbital period and the distance between the black hole and the star, and then calculate the black hole’s mass using Kepler’s 3rd law. Black holes whose existence and mass has been determined through observations of the light emitted by their companion star are shown in this diagram. Many black hole binary 90 systems discovered through their X-ray emission have been detected in our own galaxy, and a few have been discovered in other nearby galaxies. The masses of stellar mass black holes detected this way range from about 3 times the Sun’s mass up to about 30 times the Sun’s mass. The question marks show that a couple of these measurements are rather uncertain. Whenever we try to identify stellar mass black holes, we have to make sure that we aren’t fooled into mis-identifying a neutron star as a black hole. On this diagram, the known neutron star masses are shown in orange, and the black holes observed with X-ray telescopes are shown in purple. Neutron stars can’t have a mass larger than three times the Sun’s mass, so a dark object larger than this limit is assumed to be a black hole. This dividing line is not accurately known, by the way. The limiting mass for a neutron star could be smaller, for instance around 2.5 times the Sun’s mass, in which case slightly smaller mass black holes could potentially exist. The purple circle with a mass near 3 solar mass marked with a question mark could actually be either a black hole or a neutron star. At the moment we don’t have enough information to know for sure, although I personally think that it’s a black hole. 4.5.4 Colliding Stellar Mass Black Holes A different type of binary system is one where the black hole’s companion is either a neutron star or another black hole. In these binary systems, there is no gas flowing from the compan- ion star to the black hole, and no accretion disk, so we don’t see X-rays. However, a totally different type of radiation is emitted from the system, called gravitational radiation. We will learn more about gravitational radiation in Module 10. Gravitational radiation is detected by laser-based detectors that stretch out a few kilometers named LIGO and VIRGO. When two black holes are in a binary system, they orbit their common centre of mass, but lose energy through gravitational radiation, causing the black holes to spiral inwards towards each other, eventually merging into one larger black hole! Similarly, a neutron star and a black hole can spiral inwards and merge, also forming a black hole. LIGO and VIRGO have detected many of these merging black hole systems, all of which are located in other galaxies. Most of these have masses that range up to 100 times the Sun’s mass, and are classified as stellar mass black holes. This diagram includes the black hole masses measured both before the merger and after the merger. The merger events have been rearranged to make it clearer which black holes were formed from the mergers of smaller black holes. Arrows join two black holes which then join together to make a larger black hole. The two smaller mass black holes don’t exist anymore, and today only the larger black hole exists, as far as we know. The diagram also shows two pairs of neutron stars in orange that merged to form a higher mass object that might be either a neutron star or a black hole, and a neutron star and a black hole that merged to form another stellar mass black hole. The mergers of black holes and neutron stars are another way that stellar mass black holes can form! 91 4.6 Supermassive Black Holes 4.6.1 The Supermassive Black Holes Zoo Supermassive black holes are the largest mass black holes, and are found at the centres of galaxies. Galaxies are large collections of stars, ranging from small galaxies consisting of hundreds of millions of stars up to the largest galaxies, which can have upwards of one trillion stars. Galaxies fall into several categories, developed by astronomers, based on their shapes, like spiral galaxies, and elliptical galaxies. There are also galaxies that don’t fit into the standard classifications! Our classification system has really become a zoo of galaxy types! Among the unusual creatures in the galactic zoo, Active Galaxies are those which are thought to host a supermassive black hole at their centre. Let’s visit the zoo! The galaxy NGC 7742 is an example of a Seyfert galaxy. Seyfert galaxies are spiral galaxies with unusually bright central regions. The properties of the light emitted from the bright cores is not typical of regular starlight. Normally the light from a galaxy is mainly blackbody emission from the stars, as we learned about earlier. But the central regions of Seyfert galaxies also have a bright emission spectrum which indicates extremely hot gas. We will learn what an emission spectrum is in a later module. The first quasar discovered, in 1963, is called 3C 273. In the first pictures taken of the quasar, its features could not be resolved and it looked much like a star. However, the properties of this star seemed very peculiar, so it was called a “Quasi-Stellar Object”. This name was later shortened to “QSO”, or “quasar”. Modern images show that the quasar isn’t a star, but the ultra-bright nucleus of a galaxy. In left picture, the quasar is the very bright point of light in the middle of the box. (The spikes that are visible are artifacts of how the light is collected by the telescope.) You might be able to see faint light from the galaxy surrounding the bright quasar, and there is a jet of gas poking down and to the right. In order to take picture on the right, the Hubble Space Telescope placed a shield over the bright light which allowed it to take a picture of the galaxy. In 1929 astronomers observed a star-like object that they called BL Lacertae (or BL Lac for short). This thing also looked like a star through a telescope, but its brightness varied so wildly that it could sometimes be 15 times brighter than it was during the previous month! At the time, it was classified as a variable star, but when better telescopes observed BL Lac, evidence for a galaxy could be seen surrounding the bright point of light. Other similar galaxies have since been discovered, and they have been called “BL Lac Objects” since they seem very similar to the original, BL Lacertae. More recently astronomers have been calling these galaxies Blazars. When blazars are imaged, we find them at the centre of elliptical galaxies. Galaxies are made up of lots of stars, so we expect the light from a galaxy to look like the light from stars. Stars emit most of their light in the ultraviolet, visible, and infrared part of the electromagnetic spectrum. One thing that stars do not emit significant amounts of are radio waves. That means that normally, we would not detect much radiowave emission from galaxies. However, a small fraction of galaxies emit lots of radio waves! We call these galaxies “radio galaxies”. Most radio galaxies are also giant elliptical galaxies. 92 Active galaxies that are part of this “zoo” share many similar properties. Typically they have a bright central object, called an Active Galactic Nucleus (AGN), which is surrounded by the fainter stars that make up a spiral or elliptical galaxy. Have a look at this diagram which represents a unified model for all of these active galaxies. The unified model for an Active Galactic Nucleus has a supermassive black hole at its centre, surrounded by a disc of gas that orbits the black hole and emits lots of energy. In some cases there is a jet of gas shooting out from the central object. We will learn more about discs and jets in a later module. The model predicts that when you look directly down the jet, you see a blazar. If you look at the jet from an angle so you can also see into the accretion disc, then you see a radio-loud quasar. If you are looking at the edge of the disk so that the inner part of the accretion disc is blocked you will see a radio galaxy. If there is no jet, then there will be very little radio emission and you’ll see a Seyfert galaxy, which also might be called a radio-quiet quasar. So the difference in the names you call a supermassive black hole really just depends on what your point of view is! One thing in common with all active galaxies is that the supermassive black hole at the centre are consuming lots of gas, enough to power their energy output. The active galaxies are feeding their black holes, which allows the black holes to grow. The supermassive black holes may have started out as large stellar mass or intermediate black holes many billions of years ago and grew over time. On the other hand, sometimes there is evidence for supermassive black holes which aren’t being fed tremendous amounts of gas! Like the supermassive black hole at the core of our own Milky Way Galaxy, these black holes are quiet, and much harder to detect. 4.6.2 How do we know the masses? So, just how massive are these supermassive black holes? Although we can’t put a black hole on a scale to figure out how much it weighs, we can still measure its mass! Kepler’s laws of orbital motion apply to planets, stars, and black holes. If we can find a star orbiting a black hole at the centre of a galaxy, we can determine its mass! The best example is the black hole known as Sagittarius A*, (or Sag A*) located at the centre of our galaxy. It is possible for astronomers to observe stars orbiting in the region within a few light years of the centre of our galaxy. Astronomers have tracked the orbits of these stars for more than twenty years, which can be seen in this video. In this video the centre of our galaxy is marked with a star symbol, and the coloured circles are mark the position of false-coloured stars. Over the years, the stars’ paths trace out ellipses. Watch the star labelled “SO-2”. This star (and the others) moves faster when it is close to the black hole and slower when it is further out, just as required by Kepler’s equal areas law. The stars in this video orbit the invisible point at the centre of galaxy in a manner similar to how the planets orbit our Sun! We can use Kepler’s 3rd law of motion to compute the mass of the invisible object located at the star symbol. The astronomers who measured the mass had to take into account the three dimensional nature of the orbits of the stars. The star SO-2 takes 15.9 years to make one full orbit. When a star travels on an elliptical orbit, the distance “a” that appears in Kepler’s third law is equal to half of the length of the long axis of the ellipse. In the case of SO-2, the value of a is one thousand astronomical 93 units. Kepler’s third law for SGR A* is: MSGRA∗ +MSO−2 = a3 P 2 = (1000)3 (15.9)2 = 4× 106MSun (49) Mass of SGR A* plus Mass of SO-2 equals a-cubed divided by P 2 equals one thousand cubed divided by fifteen point nine squared equals four million solar masses! The mass of SO-2 is tiny compared to the mass of SGR A*, so we can approximate this as MSGRA∗ = 4× 106MSun (50) Mass of SGR A* equals 4 million solar masses. If you compare the distance between SGR A* and the orbiting star to our own solar system, 1000 AU would be well beyond the orbit of the planets, extending into the region where comets orbit the Sun. The closest star to us, Proxima Centauri, is two hundred thousand astronomical units distant. So this is a gigantic mass packed into a tiny volume of space! Whatever is located at the centre of our galaxy can’t be stars since we would be able to see them if they were there. A black hole is the only plausible way to get such a large mass into such a tiny volume. Our galaxy’s supermassive black hole is actually considered a small fry in the heavyweight division, compared to other galaxies’ central supermassive black holes. The giant elliptical galaxy M87, has a black hole at its centre that is approximately seven billion solar masses. 4.6.3 Relationship to Dark Matter? You may have heard that galaxies have dark matter, and may be wondering whether the supermassive black holes could be the mysterious dark matter. There are many reasons why supermassive black holes are not galactic dark matter. The most important thing to know is that the mass of a supermassive black hole is just a tiny fraction of the mass of the galaxy that it lives inside. In addition, supermassive black holes are found at the centre of a galaxy. The dark matter in galaxies has a mass that is larger than the mass of all the stars in the galaxy, and is spread out throughout the galaxy in a giant sphere that surrounds the galaxy. 4.6.4 How do supermassive black holes form? (Interview with Dr. Daryl Hag- gard) Q: How do supermassive black holes form? Dr. Haggard: Supermassive black holes are the kind that have a million to a billion times the mass of our sun. So, these are the really massive galaxies. We think there’s one in the center of pretty much every big galaxy, like the Milky Way and bigger ones and smaller ones. So, supermassive black holes, when they have a lot of stuff when they’re growing, when they’re actively feeding, and have a lot of stuff flowing in through an accretion disk onto the black hole, and that stuff, as it falls towards the black hole, gets hot, super hot to the point where it emits that really high energies of X-rays and gamma rays. These hot accretion disks are so bright. So, just to say that again, this is mass outside the event horizon so it’s still able to let photons out. So, it’s bright. It’s not stuck inside where the photons can’t 94 get out. But it’s all this mass that’s swirling around the outside of the event horizon of the black hole. It gets so hot and so bright that we can see these objects all the way out to the edges of the observable universe. So, we can see them out at redshifts of seven or eight. This is way back when the universe was really young. We study that radiation from the hot, the heat, and the photons coming off of this material as it tries to flow in towards the black hole. We can see that the mass at the center is already enormous, it’s already millions or billions of times the mass of the sun. So, the mystery for supermassive black hole scientists is how did you get something that massive so early on in the life of the universe. It means, you had to get all that mass down into a black hole at very early times. So, one way you can make black holes smaller kinds of you explode stars and the stars collapse back down, and can give you a black hole. But those black holes are only maybe 10 or 30 times the mass of the Sun. So, if you want to build up take building blocks of 30 solar mass things, and build up to a million solar masses or a billion solar masses, you’d need a billion or a million of those objects to crash together in LIGO-like mergers. We don’t think that’s very likely, that would be a whole lot of stars exploding. We would see all these starbursts all over the edges of the universe. Or you could just force mass in there really fast. But it’s basically... It’s hard we don’t have a good physical model for how you can get so much mass into a black hole so early in the life of the universe. So, that’s a mystery I would really love to work on more, to think about the formation of black holes in the very early universe, and then how they evolve as we move toward the present day. We’re only now just getting a new generation of space telescopes that can allow us to probe black hole growth, way at those very early times. It’s hard work. It’s very finicky, the galaxies they live in are small and sometimes this bright hot gas around the black hole, just sort of out shines everything else. It’s really hard to see the environment, and to understand what the building blocks were that gave you that supermassive thing just really fast. They basically grew really fast, and we don’t know how they could possibly grow that fast. Dr. Haggard’s webpage: https://www.dhaggard.physics.mcgill.ca/ 4.7 Intermediate Black Holes 4.7.1 Mind the Gap So far we have looked at the extremes of the astrophysical black hole scale: stellar mass black holes and supermassive black holes. There is a good reason for this. These extremes are the most well known cases. If we consider all the measured masses of black holes to date, so as of late 2017, we see a cluster of objects in the range of about 5 to 20 solar masses, with some reaching as high as possibly 70-80 solar masses. These are the stellar mass black holes. There are also a large number of objects towards the right of this plot, at the highest mass end. These are the supermassive black holes that are thought to reside in the centres of most galaxies. Masses have also been obtained for many other low mass compact objects. These low mass stellar remnants populate the far left of this plot and are classified as either neutron stars or white dwarfs. 95 And in the middle of this plot there is a great deal of empty space. Should there not be something lying in the middle? This is one of the many questions perplexing astronomers today. If we were to find black holes lying in the centre of this plot they would be known as intermediate mass black holes, and although the search for these sources is ongoing, intermediate mass black holes have proven themselves to be very elusive. Intermediate mass black holes are just that, they have masses that lie between the heaviest stellar mass black holes, and the lightest supermassive black holes, making them intermediate on the mass scale of astrophysical black holes. Hence the name, but what are these objects, how are they made and why should we care about them? Intermediate mass black holes weigh-in at more than 100 times the mass of our Sun, reaching up to one hundred thousand times our Sun’s mass. They are thought to be too big to form due to the death of stars that exist in our universe today. If this is the case, how are they made? 4.7.2 Black Hole Mergers The first intermediate mass black hole was discovered in 2019, by two gravitational wave telescopes, LIGO, located in the United States, and VIRGO, located in Italy. Stellar mass black holes were first observed in the 1970’s, while the first confirmations of the existence of supermassive black holes were made in the 1990’s. We had to wait almost 30 years to observe an intermediate mass black hole! This long wait is for many reasons. For instance, they are probably much more rare than stellar mass or supermassive black holes. But the most important reason is that we needed a new type of telescope! The confirmation of stellar mass black holes only required regular ground-based telescopes to observe and measure the orbits of the companion stars. Supermassive black holes are much farther away, and the launch of the Hubble Space Telescope allowed the 1994 observation of gas orbiting the centre of the galaxy M87 to confirm the existence of a supermassive black hole in that galaxy. Gravitational Wave Observatories can detect the motion of black holes, even when there is no light emitted! The first gravitational wave observatory, named LIGO, first detected stellar mass black holes in 2015. In 2019, a collaboration between LIGO and VIRGO detected two black holes with masses of 66 and 85 solar masses merging into a larger black hole with a mass of 142 solar masses, large enough to be classified as intermediate mass! This observation shows us how intermediate mass black holes can be built by merging together smaller mass black holes. In fact, since 66 and 85 solar masses are already fairly high, it is hypothesized that these two black holes are the result of mergers of a previous generation of lower mass black holes! As newer gravitational wave observatories in Japan, India, and Australia start operating, it is likely that more intermediate mass black holes will be detected, which will give us a better understanding of the population of black holes that grow through the merger of smaller black holes. 96 4.7.3 Higher Masses? Now that we’ve found evidence for intermediate mass black holes with a mass close to the lower boundary of 100 solar masses, what about higher masses, such as 1,000 or 10,000 times the Sun’s mass? When we look at far away objects like quasars, we’re seeing the objects as they looked at the time that the light was emitted, which was at a time when the Universe was young. For instance, with some of the farthest away quasars, we’re seeing the quasars as they appeared when the Universe was only about 800 million years old. These quasars are galaxies with supermassive black holes that are a few million times more massive than the intermediate mass black hole that LIGO and VIRGO discovered. Current calculations suggest that there isn’t enough time to construct these supermassive black holes through the mergers of millions of 100 solar mass black holes. Instead, theories predict that an earlier population of intermediate mass black holes with masses around 1,000 to 10,000 solar masses were formed in the early years of the universe, and these were the seeds of the supermassive black holes. If this idea is correct, there should be some left-over intermediate mass black holes. There are two known methods for forming larger intermediate mass black holes: direct collapse and runaway formation. 4.7.4 Direct Formation One theory suggests that these behemoths were formed earlier in the universe, when it was a much simpler place . . . chemically speaking that is. The first stars formed, when the Universe was only about 100 million years old. At this time the universe contained only the simplest elements, so there was only really hydrogen and helium. This early in the Universe, stars could become much larger than they are today, sometimes containing upwards of hundreds of solar masses! We have already learned that massive stars burn hotter, brighter and quicker than their low mass counterparts. This was true for those first stars too, such huge stars would have very short lives indeed. We have also seen that massive stars can lose mass through winds. What we have not yet mentioned though, is that the power of the wind is a function of the star’s chemistry. Astronomers have found the the metallicity of a star, or the amount of metals it contains, affects the strength the star’s wind. And here is where I should point out a quirk of astronomy. Forget your high school chemistry class for a moment. According to astronomers, the universe is made up of hydrogen, helium and metals. Anything that contains more than two protons is a metal . . . strange but true in astronomical circles . . . anyway back to the stellar winds. Therefore the metallicity of a star, or region give you an indication of how much of these “metals” are present. When the metallicity is essentially zero, we find that the star loses little to no mass via winds, irrespective of its size. This means that these first stars would have lost very little mass by the ends of their lives. And at the end of that short life? Well, here is another place where the life of a star changes. 97 Given the huge mass contained in these first stars, astronomers think that they would NOT have ended in a huge explosion, as massive stars do today. Instead, it is thought that, once the star ran out of fuel, the force of gravity would be so strong that all of the star would collapse directly down to the black hole. The outer envelope of the star would not be blown off as it is today, it would be dragged down into the black hole to join the core of the star. This stellar death is called direct collapse. This means the the first stars in the universe may have collapsed to form black hole weighing hundreds of solar masses. They would have created intermediate mass black holes 4.7.5 Runaway Formation Some theoretical astronomers have suggested that intermediate mass black holes could also form by a process known as runaway formation. Runaway formation can only occur in dense regions. Dense regions are areas in space where many stars are clumped closely together, as they are in some stellar clusters. Within the central regions of this cluster, you can think of the stars as dancers in a club. They are each moving around each as they travel under the influence of gravity. If two of these stars get too close together, they can start orbiting as binaries do, or they can spiral in towards each other and merge. This new star will have more gravity and attract other nearby stars. As they spiral in and merge, the object at the centre will have even more gravity and the cycle will continue, allowing this object to grow and grow, until the gravity of this object is so strong the supermassive star is forced to collapse to make an intermediate mass black hole. 4.8 SUPERtiny Black Holes So that’s it, there are only stellar mass, intermediate, and supermassive black holes. . . OR ARE THERE SUPER tiny black holes too? Well the smallest black holes we’ve observed are all stellar mass black holes and bigger. But. . . the General Theory of Relativity doesn’t put a lower limit on how small black holes can be, only on how big stars must be to form them. If there is a different mechanism to form black holes, other than a stellar collapse, it might be possible for tiny little black holes to exist! Let’s get the Schwarzschild equation out, but this time instead of putting in HUGE masses, like stars, let’s see what happens when we try smaller masses, like Earth’s! Earth weighs the equivalent to 3 MICRO-solar masses, and it has a corresponding Schwarzschild radius of 9 MICRO-kilometers, or 9 millimeters! That’s about the size of a small ball. So imagine that. In order to create a black hole with Earth itself, you would need to devise a way to compact the ENTIRE PLANET into a tiny ball about this size. I can’t think of any gentle ways of doing that! What if, instead of a calculating the Schwarzschild radius for a massive black hole, we use our own weight in the formula? I weigh about 75 kilograms, so if we calculate the Schwarzschild radius for a mass of 75 kilograms. . . My Schwarzschild radius would be one times 10 to the minus 25 in meters! That’s 10 MILLION times smaller than the classical size of a proton in the nucleus of an atom! 98 Tiny black holes aren’t anything to worry about. First of all, we haven’t ever actually seen a tiny black hole, and we don’t know for sure if humans can create them. It may have been possible for small black holes called primordial black holes to have been created in the moments after the Big Bang, but so far there is no conclusive evidence of their existence. Tiny black holes, because of their diminutive size, have a hard time eating! Imagine, you are a tiny, 10 to the power of minus twenty five meter sized black hole, and you need to try and eat a proton-hamburger that, to you would be the SIZE OF THE SUN!!! But, there’s an even more mysterious reason not to fear small black holes. . . They evaporate! Don’t worry, we’ll explain more when we bring in Quantum Mechanics. 4.9 Summary: Preparing to Explore Black holes can range in size, from mini, to MASSIVE, but so far astrophysicists have only collected observations from black holes in the stellar mass and supermassive categories. These are the black holes thought to form from stellar collapse, which then grow by feeding on materials in their environment. To determine a black hole’s size, we can use the Schwarzschild equation. But to do that, we must determine a black hole’s mass, which requires careful observations of black hole companions. Those companions are the food that black holes feed on, which allows them to grow, progressing to larger and larger masses. That needn’t scare you. As we’ll soon discover, it is much nicer in the vicinity of a supermassive black hole than it would be near a stellar mass black hole. And that’s important to remember, because the next stage of our journey will take us into the environment around a black hole! 99 5 Approaching a Black Hole 5.1 Journey to a Black Hole Black holes are a common element in many science fiction movies and TV shows. In the 2014 movie Interstellar the fictional crew of the spacecraft Endurance visits a supermassive black hole called Gargantua. Similarly, in Disney’s 1979 film ‘The Black Hole’, the crew of the spaceship Cygnus plunges into a black hole. In this module we will approach the nearby (only 6 thousand light years away!) stellar- mass black hole known as Cygnus X-1. From a safe distance we’ll make our way towards its event horizon. Along the way we will make observations, examining some of the major structures that are common in the environment around black holes. Our journey will finish at the inner edge of the black hole’s accretion disc, teetering in the closest stable orbit around Cygnus X-1. In Modules 6 and 7, we will plunge past the inner edge of the black hole’s accretion disc to gaze upon Cygnus X-1’s event horizon, exploring the latest theories, and their predictions about what a black hole’s interior might be like. Is Cygnus X-1 hiding a wormhole, or is there a destructive singularity lurking at its core? From our vantage point here on Earth, Cygnus X-1 appears in the constellation of, well. . . Cygnus the swan, one of my favourites - because it’s visible on summer nights in the Northern hemisphere. With a small telescope, the companion star to Cygnus X-1’s black hole is readily visible. Cygnus X-1 was the first confirmed black hole. Canadian astronomer, Tom Bolton, stated “it is inevitable that we should also speculate this might be a black hole” in his 1972 paper, while its status as a black hole was confirmed in 1974. Through a telescope, the stellar companion to Cygnus X-1 is a faint source of light along the neck of the swan. But a closer X-ray examination of Cygnus X-1 reveals a bright X-ray source originating from the system. What is creating all of this X-ray light? Closer to the black hole is a pancake-like feature, called the accretion disc. This squashed doughnut ring of material is the food that is feeding the black hole, so I hope you aren’t too hungry. I know Ross is. As we continue our approach, we will encounter first-hand some of the strange effects produced by the extreme gravity of Cygnus X-1. Friction, tidal forces, and gravitational effects make this journey extremely dangerous. What dangers can you identify, and how close to a black hole do you think it would be safe for an astronaut to approach? 5.2 Jets 5.2.1 Jets at a Distance From a distance, you might expect that there isn’t much to see when looking at a black hole like Cygnus X-1. However, black holes can be some of the brightest objects in the night sky. This is not due to the black hole itself emitting light, but is an indirect result of the effects the black hole has on the space around it. The bright lights that we see emanating from the regions around black holes are due to the material the black hole is “feeding” on, and from any material escaping from its “mouth”. If the black hole’s gravity is so strong, how can some of this material escape? 100 One of the first things we detect from Cygnus X-1 are the radio waves and X-rays being produced by cone-like structures that seem to originate from its rotational axis. These structures are called astrophysical jets, and are one of the few structures associated with black holes that have been imaged directly. These cone-like structures can funnel material away from the black hole, and in the case of Cygnus X-1, release the power of more than a thousand Suns. Astrophysical jets are thought to be powered by the material falling into a black hole (or another compact object), but the formation of jets is not yet fully understood. Imagine water cascading over a waterfall: the turbulence at the bottom and the resulting mist are analogous to the way that jets are formed. We still don’t know what jets are composed of, but leading theories suggest that they are either electrically neutral combinations of electrons, atomic nuclei, and positrons OR a positron-electron plasma. This material is responsible for creating the light that we detect when we see jets. 5.2.2 Jet Structure Near the inner edge of the accretion disc, hot material interacts with the the magnetic field generated within the disc. Due to these extreme conditions, not only is the material hot enough to be in a state of matter called plasma, but this plasma interacts strongly with magnetic fields to produce light in a process called synchrotron radiation. Although the precise mechanism of interaction is still unknown, there is no doubt that huge amounts of energy escape from the infalling material in the form of a jet. In fact, there is so much energy within the escaping material, that it can reach 10 percent of the speed of light. When this occurs, we say that the jet is “relativistic”. Relativistic jets can be seen emanating from compact objects like neutron stars and black holes. Jets originate at the magnetic poles of the compact object and are, in general, aligned with the spin-axis of the compact object. This is not always the case, however, as jets can be slightly offset from the spin-axis. The magnetic poles of the Earth, for example, are offset from the rotational poles. This slight offset in jets can be observed as a “wobble”, or as a lighthouse effect. The lighthouse effect occurs as the light from a jet sweeps across our field of view, just like the light from a lighthouse sweeps past you as it rotates. While this effect can be seen with any type of compact object, including Cygnus X-1, the lighthouse effect is more commonly associated with a class of neutron stars known as pulsars. The jets of Cygnus X-1 are 100 times longer than the distance between the Earth and our Sun when imaged in X-rays. On the other hand, if we look at the radio emission of Cygnus X-1, there is evidence that the jets extend even farther, possibly 600,000 times the distance between Earth and our Sun. If a jet is not pointed towards us, the wobble can be seen through other effects. A good example can be seen in the relativistic jet originating from the elliptical galaxy M87. M87 contains one of the largest known supermassive black holes, which powers itself by devouring material at a rate equal to one solar mass every 10 years! M87’s jet is a jaw-dropping 5,000 light-years in length, spanning more than 4% of its host galaxy. These NASA images of M87 show clear bends and kinks that come from both the wobble of the jets and from their interactions with material in the surrounding space. Since the 101 material within a jet is travelling so quickly, we’d expect it to have features related to its motion, and indeed, we do! Material from the jets is subject to the same doppler shifts that we discussed in Module 1, which is evidenced by the fact that material travelling away from us appears dimmer and “redder”, versus the brighter and “bluer” jets that are directed towards us. A good example of this was a recent survey by the NASA Chandra observatory of Pictor A, a galaxy containing a supermassive black hole. NASA calls this black hole the “Death Star” because of the powerful beams of energy it generates. This is one of the best images we have of a complete system around a supermassive black hole. In this image the central black hole of Pictor A is obscured by an intense X-ray source, depicted by the colour blue, which is also the source of the jets. The blue jet that is visible on the right-hand side of this image is pointed roughly towards us, but the counter jet on the other side is pointed away. Due to doppler shifting, we aren’t able to see the counter jet. The red clouds labelled in this picture are also an important part of the environment around black holes. These are called Radio Lobes because they produce significant amounts of radio frequency radiation. We mentioned earlier that the size of the jet in M87 is more than 5,000 light-years long. . . The jets emanating from Pictor A have been estimated to be 800,000 light years in length, more than 8 times the span of the entire Milky Way Galaxy. With some jets from black holes exceeding the size of most galaxies, it’s safe to say that some jets generated from black holes are among the biggest structures in the Universe. 5.2.3 Why are black hole jets important? (Interview with Dr. Bryan Gaensler) Q: What is important about a black hole’s jets? Dr. Gaensler: Everyone thinks the black holes are sucking things in. But it turns out that one of the most important things about black holes is that they also shoot stuff out. Now the stuff that comes out is not material that’s fallen into the black hole because that can never escape, but material that just misses and then gets squirted out in these two high- speed jets. I’d like to understand how the universe became magnetic. That looks like one of the leading candidates for this, are black holes. As the material gets squirted out from the black hole at high speed, what keeps these jets long and straight is actually magnetism. So effectively, its magnetic fields that have been blasted into space at high-speed by the black holes. So it could turn out that the origin of all the magnetism in the universe is black holes, and like some high-speed water jet, they have actually sprayed magnetic fields all over the universe. Dr. Gaensler’s webpage: https://www.dunlap.utoronto.ca/~bgaensler/ 5.3 Black Hole Companions 5.3.1 Types of Companion Stars Black holes can have low mass or high mass companion stars. The type of companion determines how mass is transferred, meaning how the black hole is fed. High-mass stars 102 generate significant stellar winds, which can transfer large amounts of mass into the black hole. On the other hand, low-mass stars have much weaker winds, so in order for them to feed the black hole, they need to be in perilously close proximity. Our black hole binary, Cygnus X-1, contains a high-mass star. This companion to Cygnus X-1 is a blue supergiant variable star called “HDE 226868” or as I like to call it, “Lunch”. I say this because “Lunch” orbits the central black hole at a measly 0.2 astronomical units, which is half the distance from the Sun to Mercury, and this is a SUPERGIANT star! 5.3.2 Types of X-ray Binaries If the black hole is actively feeding on the companion star, we will be able to see this clearly in the X-ray portion of the spectrum. In fact, because these systems are so bright in X-rays, they are often referred to as X-ray binaries. We should note, however, that this term refers to a system containing a star and a compact object. As we’re aware, compact objects can be either neutron stars or black holes. As such, it is important that when we observe these systems we try to find the mass of the compact object. If the mass of the compact object is more than 3 solar masses it must be a black hole, if it’s lighter than that, it could be a neutron star. In some cases though, it can be very hard to tell the mass of the compact object. When astronomers are unsure of the characteristics of systems like this they list them as black hole candidates. There are two types of X-ray binaries: high mass X-ray binaries (HMXBs) and low mass X-ray binaries (LMXBs), but this classification is not based on the mass of the compact object. It may seem strange to you at first, but X-ray binaries are classified by the mass of the companion star, not the compact object. The companion stars in Low Mass X-ray Binaries have masses that are the mass of the Sun or smaller. High Mass X-ray Binaries have companion stars that are at least ten times more massive than the Sun. Anytime astronomers come up with a classification like this, you’ll find some objects that don’t quite fit, so we also have an in between group that are sometimes called “Intermediate Mass X-ray Binaries” but their properties are usually pretty similar to the Low Mass X-ray Binary group. Why would astronomers choose to classify a binary system based on the type of compan- ion star rather than the type of compact object? The reason for this classification is that the properties of the system depend more on the type of donor star than on the type of compact object. What this means is that the observations of these systems vary more dramatically if you compare a high mass and low mass X-ray binaries, than if you were to compare a stellar mass black hole and neutron star that were both feeding on ....say, a low mass star. Typically low mass companions are small in size, while the high mass companions are large. Small stars can orbit closer to a black hole than large stars. Kepler’s laws of motion tell us that stars with small orbital separations orbit with faster speeds and take shorter amounts of time to orbit. Low Mass X-ray binaries typically have short orbital periods that can range from less than an hour to many hours. Meanwhile, the larger companions in High Mass X-ray binaries orbit further away from the centre of mass of the system and can take a few days to complete an orbit! 103 This means that the feeding, or mass transfer mechanism, and so the rate of mass transfer, along with the orbital period can be greatly impacted by the type of companion star. 5.4 Sipping on Star Soup 5.4.1 The Roche Lobe After examining Cygnus X-1’s blue supergiant companion up close, we now shift our view to the black hole and companion star system as a whole. We know that these companions can either be high mass blue stars like HD 226868, or low mass stars, like our Sun. How exactly does a black hole “eat” from a companion star? If the black hole is feeding on its companion star, then material from the star must be transferred to the black hole by some mechanism. The answer to this question is explained by the work of a French astronomer, named Edouard Roche. He developed a model for the transfer of material from a star to a black hole (or other stars), known as the Roche lobe. To understand the Roche lobe, let’s consider two scenarios: a single star and a system of two stars. In either scenario, the force of gravity will have some say in whether material will be drawn into the star. In the case of the single star, if we were to draw lines or contours of constant gravitational potential, we would need to create a series of circles originating from the star. These drawings are is similar to topographical maps of a mountains. How does this change when another star is nearby? When two stars are in a binary system, the gravity of the two will interact. These two stars would be in orbit around one another, or rather around their common centre of mass. However, we must also consider in addition to the gravitational force the force due to the relative motion of the stars: the centrifugal force! Think about a child’s roundabout in a play park: once you kick off and begin spinning, you can feel a force pushing you outwards. This is called the centrifugal force. As a result of the rotation of our star system, we have gravity pulling inwards, and a centrifugal force pushing outwards. It is the combination of these two forces that are represented by the lines of constant potential in binary systems. If we now build up lines of equal potential around two stars, we will initially see circles around each of these stars. However, as these rings get larger and closer together, their shape begins to change. They are slowly stretched out in the direction of the opposing star. The circles begin to morph into teardrops. This stretch, or distortion, increases until they connect, forming a figure of eight around the two objects. Each of these teardrops, or lobes is called a Roche lobe. It is the Roche lobe for the star it contains. Any material that is inside the lobe is gravitationally bound to that star. You can think about gravitational lobes like two lakes occupying adjacent valleys, separated by mountains. The lake’s watersheds don’t share any water unless they fill to a mutual height, which we typically call a “watershed divide”. Similarly, material within a Roche Lobe is bound unless there is a point where the potential is equal between the two stars. The point where these teardrops meet is known as a Lagrange point. This first Lagrange point is commonly labelled L1. If you were an astronaut situated at L1 you would feel an equal gravitational pull towards each of these stars. 104 But there are other points where we could also feel an equal pull between the two stars. If we continue to map the lines of equal potential, we can find other Lagrange points. As you can see, there are four other points that surround the binary system. 5.4.2 Is this just for stars? Although we’ve used an example of two stars, you can draw similar lines of constant potential around any other pair of massive bodies, including the Sun and Earth, but more importantly, between a black hole and its stellar lunch! Lagrange points are special regions in space where the gravitational potential is relatively flat, making it easy for spacecraft to hover there. Considering the Earth-Sun system, the first Lagrange point lies on their connecting line. This region, known as L1, is used extensively as a “parking spot” for telescopes, since they can hover at L1 using small amounts of fuel. As such, this location has been used as a prime spot for astrophysical observations, which we will discuss further in Module 9. 5.4.3 Mass transfer in black hole binaries Black hole binaries are a type of system which contain two massive bodies, and so these systems also have Roche lobes and Lagrange points. But how does this help us understand the transfer of matter? The transfer of material from a star to a black hole is a gravitational effect. If material from the star moves outside its Roche lobe, gravity can pull it towards the black hole instead. But if material inside the Roche lobe is gravitationally bound to the star, then how can it move outside this boundary? The easiest way for this to happen, is if the star begins to fill its Roche lobe. As stars like our Sun get older, they will swell up to become red giants. At this time, the star could grow to fill its Roche lobe. At that point material can spill over, across the boundary at L1. This star stuff will then start falling in towards the black hole. Stars can also fill their Roche lobe if the Roche lobe shrinks. How can this happen? The Roche lobe would get smaller if the bodies in the system move closer to each other, or as astronomers call it, the binary becomes more compact. There are many ways that a binary system can become compact. One method involving gravitational radiation will be discussed in Module 10. This image shows 16 black hole binaries that all live in our galaxy. At the top of the image you can see our Sun, and the distance between it and the planet Mercury. Mercury orbits the Sun at just over a third of the distance between the Sun and Earth. Yet most of these systems are much smaller, or much more compact. 5.4.4 Other Options for Mass Transfer So far we have looked at material being stripped away from the surface of the star as it overflows it Roche lobe. This material then crosses the first Lagrange point, L1, and starts falling in towards the black hole. This process is known as *Roche Lobe Overflow*. It can provide a fairly stable way of feeding a black hole for quite some time. But is it the only way to transfer mass from the star to the black hole? 105 No, there is another option: many stars have winds. Our Sun does, but its winds are considered puny by stellar standards. Massive blue stars can blow away up to 100 million times more mass than our own Sun does. Such strong winds can be captured by the gravity of the black hole and fall in towards the event horizon. This type of accretion is known as *wind fed*. While wind fed mass transfer is only really an option for high mass stars, Roche lobe overflow can occur with any type of star, as long as the companion star fills its Roche lobe. Our good friend Cygnus X-1, is one such system, with a high-mass companion that is BOTH overflowing its Roche lobe, AND feeding the black hole with high velocity stellar winds. 5.4.5 How can a star be like a vampire? (Interview with Dr. Craig Heinke) Q: How can a star be like a vampire? Dr. Heinke: These stellar vampires would be a star that is kind of dead by itself, but is sucking the life from another star. If you have a binary star which consists initially of a high-mass star, and then another star, it could be either high or low mass, and the high-mass star reaches the end of its life, explodes, and becomes either a black hole or a neutron star now. In certain circumstances, these two stars we brought close enough together that the black hole or the neutron star or even a white dwarf can tear material off of the edge of the other star. This happens when they’re so close together that the gravitational force from the compact object, on the nearest edge of the other star is as strong as the gravitational force from that star on its own material. The material kind of leaks over, its kind of pulled over. This material then funnels down through what we call an accretion disk down towards the black hole or neutron star. Of course, if it’s a black hole, then we don’t know anything that happens once it passes the event horizon. But as it funnels down towards it, it’s brought up to very high velocities and very high energies by the gravitational force of this black hole for instance and then as it as it funnels down, it releases a huge amount of that energy as visible radiation and X-ray radiation that we can see from far away. Dr. Heinke’s webpage: https://sites.ualberta.ca/~heinke/Research.html 5.5 Have a Corona! Now that material is streaming off the black hole companion, we can monitor material as it progresses closer and closer to the black hole. This can be clearly seen in a simulation created by researchers at North Carolina State University, which models material transferring between two stars. We clearly see a stream of material being stripped away from the star and spiralling in towards the star. Although this is a simulation of the interaction between two stars, the same physics applies to our black hole binaries. Looking more closely at the material spiralling in, we see a dim, cloud-like feature around the black hole and inner disc: this is the black hole’s corona. Astronomers have detected evidence that the corona is a cloud of fast moving electrons that hang out near the black hole, and they think that coronas come in two flavours. The first flavour, called the “lamp- post model”, which does not sound appealing at all, explains the corona as a source of light that sits close to the rotational poles of the black hole. The second model, deliciously called 106 the “sandwich model” explains the corona as a larger cloud that envelops the central disc. Think of it like this: the corona is like the two halves of a bagel surrounding a accretion disc of . . . (yuck) ...Nutella? Not my cup of tea, but Ross loves it! In this artist’s visualization, the corona is this diffuse purple light enveloping the black hole. The corona is thought to be powered by the intense interactions between the chaotic magnetic fields generated by the material falling into the black hole, and the resulting hot plasma. The black hole in this artistic rendering is located in the centre of the galaxy Markarian 335, in the direction of the constellation Pegasus. Observations of the supermas- sive black hole have revealed powerful coronal outbursts that accelerate material to nearly 20% the speed of light. However, we would like to emphasize that scientists are still in the very early stages of understanding the environment near the black hole, and that research into the corona of black holes is ongoing. 5.6 What is Accretion? 5.6.1 Accretion We’ve encountered the word, “Accretion” several times now in this module, but what exactly does it mean? In astrophysics, accretion is the word used to describe the process of gas and dust being collected together by gravitational forces. Accretion can happen on many different scales. For example, in the early history of our own solar system, gravity caused the accretion of smaller particles into larger and larger ones, eventually growing from tiny grains of dust, planetesimals, and finally into the planets that we know today. Accretion is also responsible for gathering clouds of hydrogen into stars, and stars into galactic discs. But most importantly for us, accretion is the process by which black holes are fed. When a star, or a cloud of material is present near a black hole, it experiences the gravitational effects of the black hole, and the force of attraction accelerates material towards the black hole. Since particles in the cloud are free to move around, they experience viscosity, an effect of the friction during collisions with neighbouring particles. Viscosity does two things within accretion discs: 1) it slows the particles’ orbits, allowing them to fall further into the gravitation well, and 2) it heats up the particles, which in turn causes them to glow red-hot, a process that creates light called blackbody radiation. Much of this energy is derived from the gravitational potential energy of the materials, which we talked about in Module 1, when we discussed escape velocity. 5.6.2 Formation of a Disc Accretion is not as simple as it seems. . . Why are moons, planets, and stars - which were all created by accretion processes - spherical, instead of disc shaped, like the rings of Saturn, or the disc of a galaxy? The answer has to do with angular momentum. You are probably already familiar with linear momentum, which is a product of an object’s mass and its velocity. Momentum, in this sense, is a conserved quantity. Another way of describing it is through Newton’s First Law, “An object in motion stays in motion.” 107 precisely because it has momentum! Momentum = p = m× v (51) What if an object isn’t moving along a path, but it’s spinning in place? In that case, we now have another conserved quantity, called angular momentum. Angular momentum is a product of mass, velocity, AND the distance from the origin of the spin. Usually, physicists tidy up the angular momentum equation by considering the sum total of all the masses being considered and their distances from the origin, into a neat quantity called Moment of Inertia. That way, angular momentum, L, can be expressed in a similar way to linear momentum: as the product of Moment of Inertia, I, multiplied by the Angular velocity, omega: L = I × ω (52) What’s important here isn’t the form of the equation, but rather the statement that Angular Momentum is a Conserved Quantity, or, in plain language, “An object which is spinning will continue to spin”. Here’s what I mean. . . If I have these extra weights in my hands, and I extend my arms, I will have a large moment of inertia. If I pull inwards, my moment of inertia will decrease. So if I’m spinning slowly with my arms out, I will have a good amount of angular momentum. What will happen if I pull my arms inward? Let’s see: Did you see that? As I pulled my arms in, I started to rotate faster! I was conserving angular momentum! As you’ll see shortly, the exact same thing happens to material in an accretion disc when it goes into smaller and smaller orbits. But weren’t we supposed to be talking about discs versus spheres though? What does angular momentum have to do with that? Well, if a structure is accreting out of a rotating cloud, the angular momentum will dictate in what direction, and how fast, the final object will spin. Suppose we started with a nice big nebula, and we want to condense a star out of it. From a distance, it probably doesn’t look like the nebula has much angular momentum. . . but it does: because all of these particles are far from the center, they have HUGE moments of inertia. . . As those particles collapse due to gravity, they must rotate faster and faster to conserve angular momentum. Since the material in the nebula began with some amount of angular momentum, the new, smaller structure must preserve the original amount by rotating faster. Why is Earth spherical instead of flat, then? For structures like Earth, which are solid objects held together by their own gravity, the centrifugal force does flatten it a little bit, which is why we call Earth an “oblate-spheroid” instead of a sphere. If Earth were to rotate faster and faster, eventually the centrifugal forces would exceed the internal stress, and Earth would torn apart. Angular momentum causes rotating objects to flatten into discs. 5.6.3 Mass Transfer Earlier, we said that accretion was the process by which black holes are “fed”. When physicists say “feed a black hole”, we are describing the transfer of material and energy towards the black hole’s event horizon, in the same way that when you’re eating, you are transferring material, and energy, into your own body’s mouth horizon. 108 When we were discussing Newtonian Mechanics, in Module 1 we used two equations to describe two important types of energy: gravitational potential energy, and kinetic energy. Any matter in a gravitational field has potential energy, which it can give up as it descends into the gravitational well, by converting the potential energy into kinetic energy. Let’s have a look at some matter, falling into a black hole. Recall, for a black hole with mass M and radius R, and a test particle with a mass of little-m, the gravitational potential energy for the infalling particle is: Epotential = GMm R (53) If the particle starts with zero velocity (somewhere infinitely far from the black hole), much of that gravitational potential energy will turn into kinetic energy, in which case, we set kinetic energy, one-half em vee squared, equal to the gravitational potential energy: 1 2 mv2 = GMm R (54) Since the mass M of a black hole can be LARGE, and the radius can be MINISCULE, even the tiniest particles can be accelerated to INCREDIBLE speeds, however, it’s worth noting that these equations are classical. All of the energy of the infalling particle has to go somewhere, and indeed, much of that energy becomes heat, which is then radiated away in the form of light. In the centermost regions of accretion discs around black holes, the disc material can become so incredibly hot that it produces enough light to push back against the infalling material. Here, we use a specific name for when the force of gravity pulling material inward is equal to the radiation pressure pushing material outward: it’s called the Eddington limit, named after Sir Arthur Eddington. The Eddington limit describes a natural limit to how much material can be captured from the accretion disc around a black hole, based on the power output, or luminosity of the infalling material. This limit is expressed in equation form by: LEdd = 4piGMmpc σT (55) If the luminosity of the disc exceeds the Eddington limit, material in the disc will be pushed outwards from the interior of the disc, and if the luminosity is below the limit, gravity will pull more material in. This equation is powerful, because it allows scientists to estimate the minimum mass of a black hole, simply by measuring the system’s luminosity. 5.6.4 How do black holes grow? (Interview with Dr. Robert Thacker) Q: How do black holes grow? Dr. Thacker: The thing that really puzzles me about black holes is really the question of how they grow, and this to a large extent is one of the great unsolved problems in physics. We understand the general process, but the way that the material gets down from very large scales down to like scales of the solar system literally in some cases, is not really well under- stood. So the process by which material goes into what we call the accretion disk around 109 the black hole, which gets hot just for the same reason, rub your hands on your thigh, on your jeans, you go hand’s going to get help from friction, same kind of processes are going on. But all of the detailed physics of how that material goes through that disk and then into the event horizon of the black hole, like the very edge of the black hole, you can’t see it beyond that point. It’s not really that well understood. You’ve to get rid of something called angular momentum, that spinning motion. You’ve to get rid of that somehow, pass it out to other material, and all of those details are really difficult to understand. You then got to connect that up to really big scales of a galaxy too. So these two things are naturally linked in the real world, but for a physicist who’s trying to understand this, they’re really hard to put the two things together in a way that bonds them together in a concrete manner. So working on that is a really hard thing. That small-scale process really impacts the big scales, and so that’s what I really want to know. We’re trying to get more information about it but it’s really hard to do. Dr. Thacker’s webpage: http://www.ap.smu.ca/~thacker/index.html 5.7 Spinning Through the Disc 5.7.1 Moving Through the Disc Now, let’s consider what it would be like to be a little piece of dust in the accretion disc around a black hole. Since our little piece of dust is in orbit, along with all the other little pieces of dust, we say that it has both angular momentum (which prevents it from falling further inwards), and gravitational potential (which is trying to pull the dust further inwards). How can this particle migrate through the disc? Well, on it’s own, it can’t! What do we know so far? Well, for one, we know that, due to Kepler’s laws, that the orbital speed of our piece of dust is related to its distance from a central object. When our dust particle is far away from the central object, its orbital speed is low. When it’s close to the central object, the orbital speed is high! We need a second little piece of dust to really understand how dust migrates through the disc. Let’s say our first piece of dust, Dust A, is in the accretion disc at a distance little-a from the central black hole. Now consider Dust A’s buddy, Dust B, a little further away (but not too far!) from the central black hole. Since Dust A is closer, it will be moving a little faster, and since Dust B is further, Dust B will be travelling a little slower. Every once in awhile, Dust A will catch up with, and bump into Dust B. This will cause Dust A to slow down, and Dust B to speed up! The result is that the slower Dust A will fall further into the disc (gaining kinetic energy), and Dust B will use its speed boost to find a stable orbit further out (gaining potential energy). In an ideal case, we aren’t losing any energy, but reality is far from ideal. . . which means that both Dust A and Dust B will be slightly hotter than they were before their collision. And that heat can be carried out of the disc by thermal radiation (which is how we see them)! As energy is being carried away from the system, material in the disc will be pulled further inward. In a real disc, there are innumerable dust particles participating in these collisions, so instead of looking at individual collisions, scientists often consider the physics of a large 110 number of interactions. Material moves inwards through the accretion disc by losing energy through a viscous force. This process turns the gravitational potential energy of the disc into heat, which is carried away by radiation, allowing more material to “feed” into the black hole. 5.7.2 Note About Time Dilation and Redshift Until now, we haven’t really considered what effects the black hole has on the accretion disc other than gravity, obviously, but we should briefly discuss how Gravitational Time Dilation, Gravitational Redshift, and the Doppler Effect have on the properties of the disc. Our friendly little dust particles that were exchanging energies were also aging at different rates. If each of Dust A and Dust B were carrying tiny dust clocks with them, they would tick at different speeds. Clock A, being close to the black hole, would appear to a distant observer to tick more slowly compared to Clock B, which is further out in the disc. Practically, what this means is that if you wanted to time travel into the future all you would need to do is get very close to a black hole for a small amount of time. In that way, the clocks in the rest frame of the universe would appear to tick faster, and you would re-emerge from near the black hole into a time-shifted future! Astrophysicists need to account for Gravitational Time Dilation when they are accounting for the rates that particles emit radiation. A hot particle, for example, emitting radiation from near a black hole, would appear to be radiating at a much slower rate. On top of that, due to Gravitational Redshift, the light being emitted from the accretion disc close to a black hole will lose energy as it climbs out of the black hole’s gravitational well, thereby becoming redshifted from its original emission wavelength. But wait, there’s more! Due to the rotation of the accretion disc, observers will also measure a difference in the intensity and wavelength of light, depending on whether material in the accretion disc is approaching, or moving away from the observer. For a spinning disc, the material moving away from the observer will be dimmer, and redder, whereas the material moving towards the observer will be brighter and bluer! In order to truly understand black holes, each of these effects, along with many more, need to be taken into account. Talk about a fun puzzle for theoretical physicists! 5.7.3 Tidal Forces Now that we have a good grasp on what we would see in the environment around a black hole, we should now ask what an observer would feel in the environment around a black hole. Normally when we think of spacecraft in orbit around Earth we imagine an astronaut experiencing the sensation of weightlessness, with gravity serving only to keep them from being flung out into deep space. But there is another effect that becomes significant around black holes that will be noticeable to an astronaut nearby: tidal forces. Tidal forces are named after the tides here on Earth, which we experience as the rise and fall of sea levels that occurs periodically. Humans have speculated about the cause of tides for millennia. Today we know cause of tides is the combined gravitational forces of the Moon and Sun gently pulling on water in the ocean or rather, tidal forces act on everything on Earth, but we only notice it when it’s creating ocean tides. 111 For a simple spherical object, gravity acts to pull objects towards the center of mass. However, when a second gravitational body is introduced the forces are now the sum of the gravitational forces due to both bodies. This presents an interesting dilemma, since the gravity from the second body changes strength with distance, and thus the tidal forces will have different values and directions over the surface of the first. These tiny differences in gravitational forces are all that are needed for an object to experience tidal forces. On Earth, we observe this tiny change in force as a major change in the height of sea water. In some cases, like the Bay of Fundy in Canada, the sea level can vary by as much as 16.3 meters, tall enough to swamp an entire 5 story building! If small forces like these can create big effects here on Earth, what do you imagine the tidal forces near a black hole might be like? 5.7.4 Tidal Forces Around Black Holes In our daily lives, we experience one Earth-gravity worth of acceleration, it’s the force that keeps us stuck to the ground. However, there is a very slight difference between the forces that pull on our feet compared to the forces pulling on our head, unless of course you are laying perfectly level. Let’s calculate the difference in acceleration by a rearranged version of Newton’s formula for universal gravitation: a = 2GMh r3 (56) Here, ‘a’ will be the difference in acceleration between two points separated by height ‘h’, above a body of mass M and radius r. Let’s see what the difference is for a person on the surface of Earth by inputting Earth’s mass of 5.97×1024 kg, and radius of 6.37×106 m. For someone my height, about 1.8 meters tall, they will experience a difference in acceleration of a miniscule 5.5× 10−6m/s2. Compared to one Earth-G, of 9.8m/s2, that’s less than 2 parts per MILLION. Definitely not something we can sense. Let’s do the same thing again, but this time taking the mass and radius of the nearby black hole Cygnus X-1. It has a mass of approximately 15 solar masses, or 3× 1031 kilograms. For simplicity, let’s say this is a Schwarzschild black hole with a radius of 44 kilometers Plugging in the numbers gives us a difference in acceleration between my head and my feet of, wow, hold on a minute here... EIGHT MILLION TIMES THE FORCE OF GRAVITY ON EARTH Of course, we wouldn’t survive such incredible differences in forces, and scientists have named this effect “Spaghettification”, which isn’t so much delicious as it is horrifying. Es- sentially, as you approach a stellar mass black hole, you will eventually be pulled into a thin strand that once called itself human. Not a pretty way to go. But that was for a small, stellar mass black hole. Let’s see how it would affect someone around a supermassive black hole. Sagittarius A* is the name of the black hole at the center of our galaxy, the Milky Way. Weighing in at just over 4 MILLION solar masses, SGR A* has a corresponding Schwarzschild radius of 12 MILLION kilometers, more than 8 times the diameter of our Sun. Putting those values into our equation yields a difference in acceleration of. . . a measly one-ten-thousandth of gravity on Earth? What’s going on here??? 112 As it turns out, the larger the black hole, the more gradual the changes in the gravitational field as one makes their approach. In the movie Interstellar, the writers chose to use a fictional supermassive black hole called Gargantua, which is why the crew of Endurance were able to get so close to the event horizon without feeling tidal forces. However, massive tidal forces were apparent on Miller’s planet when gigantic waves circulated around the planet. For supermassive black holes, the tidal forces at the event horizon are gentler than the forces around a smaller black hole. In fact, if you were a space traveller, you would need to be extremely careful in the area around a supermassive black hole, because it’s possible you could cross the event horizon and not even realize it! 5.8 Innermost Stable Circular Orbit 5.8.1 Kepler’s Third Law After a thorough investigation of the accretion disc around the black hole, we will end this module teetering at the edge of stability. . . No, we’re not discussing the black hole’s event horizon just yet, but we can stay indefinitely in a stable orbit around a black hole, as long as we don’t cross a boundary called the Innermost Stable Circular Orbit. If Cygnus X-1 were not rotating, it would permit stable orbits at 3 times the distance of the event horizon. (We’ll learn in Module 6 what happens when a black hole rotates). This would allow stable orbits as close as 130 km from the centre of Cygnus X-1, or 90 km from its event horizon. The Innermost stable circular orbit is the boundary that distinguishes between stable orbits, which don’t require energy for an object to stay in orbit there, and unstable orbits, which will pull you in towards the black hole’s event horizon, unless you have very powerful engines like the Enterprise in Star Trek. The Innermost Stable Circular Orbit, which we will henceforth call the I.S.C.O., or ISCO, defines the inner edge of the accretion disc, beyond which, material will fall freely and become captured by the black hole. 5.8.2 Breakdown of Kepler’s Law Somewhere between the accretion disc and the event horizon, Newtonian gravity stops being a good approximation, and the gravitational field becomes stronger than Newtonian gravity predicts. This increased strength of the gravitational field is due to corrections by Einstein’s theory of General Relativity. The result is that the gravitational becomes much stronger within the region bounded by the ISCO, and stable orbits predicted by Newtonian gravity are no longer stable. Since the gravitational potential around a black hole is represented by the Schwarzschild potential, there are 5 kinds of orbits that we can find. There are stable and unstable circular orbits, bound precessing orbits, scattering orbits, and PLUNGING orbits. For a Schwarzschild black hole, the solution to the equations of general relativity tell us that the peak of the potential occurs at: R = 6GM c2 , which should be looking pretty familiar right now. Recall that we’ve already encountered the radius of a Schwarzschild black hole when we were experimenting with the escape velocity formula. The Schwarzschild radius, which describes the event horizon of a black hole occurs at: R = 2GM c2 . Quite the coincidence, huh? The only difference between the Innermost Stable Circular Orbit, and the 113 Event Horizon is a multiple of three. Actually, this makes life quite a bit easier for us, since we can simply state: The innermost stable circular orbit of a non-rotating black hole is three times farther from the black hole than its event horizon. Key here: non-rotating. We haven’t discussed much about rotating black holes yet, and we will get more into in Module 6, however, I can’t help myself here, because the ISCO will actually change when we are considering rotating black holes. This was apparent in the movie Interstellar, when the crew of the Endurance visit Miller’s planet very close to Gargantua’s event horizon. They were able to do it safely, because a rotating black hole can actually have an ISCO that is much closer to the event horizon. In an extreme case, a rotating black hole could have an ISCO coincident with its event horizon, but we will see that in an upcoming lesson. One final note about innermost stable circular orbits: we’ve only covered matter inter- acting with the gravitational field of the black hole. If instead we looked at light we would have found a different result. The ISCO for light orbiting a nonrotating black hole can be a factor of two closer to the event horizon. Photons can get trapped in circular orbits at the light-ISCO radius, and orbit for many times before escaping. We could then detect these photons which would look like they came from a sphere of light surrounding the black hole sometimes called the photon sphere. 5.9 Summary: Teetering on the Edge On our approach to a black hole, using our example of Cygnus X-1, we identified a number of structures that are visible from the environment around the black hole. We now find ourselves teetering at the edge of stability, on the precipice of the Innermost Stable Circular orbit. Looking away from the black hole, we can see the bright material within the accretion disc, being fed by a nearby companion star within its Roche lobe. Looking above the disc around the black hole, a faint glow is evidence of the corona, and stretching brightly off the poles of the black hole we see towering jets, columns of accelerating plasma that can stretch thousands and millions of lightyears in length. Our next step takes us within the ISCO radius, where without a powerful rocket engine to escape, we will eventually fall across the black hole’s. . . EVENT HORIZON!! 114 6 Crossing the Event Horizon 6.1 Introduction Welcome to the black hole! Now that we have arrived in orbit near a black hole’s event horizon, it’s time to dip down into the interior. The event horizon a strange one-way street in the universe, preventing light and material that have crossed from returning. But the event horizon is thought to hide something even stranger in the interior of the black hole, the singularity. We’ll also put a positive spin on things and find out what happens when a black hole rotates. Finally, we’ll take a look at the black hole’s weird cousin, the wormhole, and see if it’s possible to travel to distant regions of the universe. 6.2 The Event Horizon 6.2.1 The Boundary of a Black Hole Black holes have an “inside” and an “outside” separated by a boundary called the event horizon. What is an event horizon? What does it look like, if it looks like anything at all? Let’s explore this concept by revisiting what we mean when we talk about the surface of an object like the Sun. We often think of the Sun (or any other star) as a big ball of gas which has a surface. But it’s an oversimplification to say that all of a star’s gas lies inside this surface. Some of of the sun’s material is continually escaping from the hot and energetic “surface”. The solar wind pushes a small amount of the Sun’s gas all the way to the outer edges of the solar system (this is the material the generates auroras here on Earth, after all). For stars, we generally define the “surface”, known as the photosphere, to be the outermost layer of the Sun. The photosphere is what we see when we look at the Sun in visible light. If we try to look deeper inside the Sun, the hot gas blocks light, so we can’t actually see deeper than the photosphere. Beyond the photosphere there are additional regions of the Sun where gas interacts, such as the chromosphere and corona, but those outer layers are very faint and difficult to see. As you can imagine, saying exactly where the Sun’s surface is located, is a matter of scientific definition! Although black holes do not have a “surface”, scientists have defined a boundary to separate the “interior” of the black hole from its “exterior”. A black hole’s event horizon is a boundary that separates the black hole’s interior, which we are unable to see, from the outer region. However, unlike stars, the black hole’s event horizon is much easier to define, because it’s impossible for gas or light to escape from the event horizon! Therefore, we say that the event horizon is the “surface”, or boundary of a black hole, not as a rigid body, but as the point-of-no-return for material that has fallen in. Why isn’t it possible to have a ball of gas inside of the event horizon? It all comes down to the concept of hydrostatic equilibrium. In Module 2, we discovered that hydrostatic equilibrium is the balance between gravity and gas pressure in the interior of stars. Grav- itational attraction tries to bring all the gas in a star towards the star’s centre. But gas 115 pressure creates an outward force that prevents further gravitational collapse. When they are in balance, the star is stable and can stay the same size for a long time (like our Sun). Suppose we take a star and compress it into a smaller volume, overpowering the gas pres- sure at the interior. The matter in the star will be squashed and feel a stronger gravitational pull towards the centre, which requires a larger gas pressure to push outwards in order to balance the star. Is it possible to continue compressing the star into smaller and smaller regions? NO! If you compress the gas within the star’s Schwarzschild radius, the gas pressure required to balance gravity becomes infinite! It’s not possible to create infinite gas pressure, so gravity wins the battle and the star’s gas will have to continue falling inwards! It is impossible for any matter to be at rest inside of the event horizon of a black hole. 6.2.2 Curvature of Particle Paths The event horizon can be understood by observing how light rays are bent by the gravita- tional field of a massive object. We know that if there is no gravity, then light travels in straight lines, just like this ball rolling on a flat surface travels in a straight line. Just as the sheet is deformed by the presence of the weight, space time is deformed by massive objects, like a star or a black hole! When we roll a ball on the curved sheet, it doesn’t travel in a straight line. Instead its path is curved towards the central mass. The closer the ball’s starting point is to the mass, the more the path of the ball becomes curved. Light is deflected in an analogous way by the mass of a star or a black hole. If a star and a black hole have the same mass, the deflection angle is the same for photons travelling on paths that are the same distance from the object (assuming the light path stays outside of the object). 6.2.3 The Schwarzschild Radius We must remember that a black hole with the same mass as a star is much more dense. If you recall that the Sun’s radius is seven-hundred-thousand kilometers, but a black hole with the same mass as the Sun has an event horizon radius that is only 3 kilometers. This means that it’s possible to get much closer to the centre of a black hole than a star! The radius of a non-rotating black hole is sometimes called the Schwarzschild radius, named after Karl Schwarzschild, the first person to solve Einstein’s equations for strong gravitational fields. Einstein’s equations were thought to be so difficult, that Albert Einstein himself, said that nobody would ever be able to solve them! However, only a year after Einstein published the equations, Karl Schwarzschild found the first solution which happened to describe a non-rotating black hole. What a coincidence that Schwarzschild, whose name means “Black Shield” in German, was the first to describe the concept of a black hole! In Module 4 we explored the equation for the Schwarzschild radius, which is two times the mass times G over c squared. RS = 2GM c2 (57) In this equation RS is the distance corresponding to the Schwarzschild radius, M is the black hole’s mass, G is Newton’s gravitational constant, and c is the speed of light. For 116 black holes that don’t rotate, the event horizon is a sphere with a radius that is simply proportional to the black hole’s mass. So if you double the mass of a black hole, the radius of the sphere doubles too. If you put the numbers in for the Sun, you’ll find that the Schwarzschild radius is 3 kilometers. Physicists often simplify key equations by folding terms that occur repeatedly. In this case, the equation for the Schwarzschild radius is simplified so that it is equal to 3 kilometers times the black hole’s mass divided by the Sun’s mass. RS = 3km( M MSun ) (58) We now have the event horizon radius scaling with the ratio of masses, or in other words, it’s dependent on how much more massive the black hole is than the Sun! This is helpful as it makes the numbers a little easier. If we were to visit Cygnus X-1 which has a mass that is 15 times larger than the Sun’s mass, the radius of the black hole would be 3 kilometres times 15 which equals 45 kilometres. Still pretty small! The supermassive black hole at the centre of the Milky Way has a mass that is 4 million times larger than our Sun. This means that its event horizon is 12 million kilometers. That might sound like a big distance, but the largest black hole in our galaxy is smaller than the distance between our Sun and Mercury! A black hole would need to have a mass that is 50 million times larger than the Sun before the event horizon would be as large as the distance between the Sun and the Earth. Since the event horizons of supermassive black holes are further away from the black holes’ centre, the tidal forces at the event horizon are *smaller* for black holes with larger masses! As we learned earlier, tidal forces can be pretty hazardous to an astronaut’s health! This means that if you get to choose which black hole to visit, you should choose a larger mass black hole. It is estimated that a black hole should be at least one thousand solar masses in order for it to be safe to visit. Since the radius of a black hole is proportional to its mass, if matter falls into a black hole, the event horizon grows larger. In most cases, this is an incredibly small change. However, if two black holes collide, they can merge into one significantly larger black hole. In case you’re wondering, Stephen Hawking proved that it’s impossible for a black hole to split into multiple black holes! We’ll talk more about this in Module 7. 6.2.4 The Ring of Fire If we shine a flashlight in the direction of a black hole, the closer the light is aimed towards the black hole, the more curved the light beam will become. As we aim the flashlight closer and closer, we discover that there is a special distance at which light from our flashlight begins travelling in circles around the black hole! This is an area called the photon sphere, which corresponds to a radius that is 1.5 times larger than the radius of the event horizon. The photon sphere is similar to the innermost stable circular orbit for particles, except that the circular photon orbits are unstable. If a photon becomes trapped within the photon sphere only a small “nudge” is enough to kick the photons away from black hole, OR to send them spiralling inward. Scientists call this collection of circular photon orbits “The Ring of Fire”. In this com- puter simulation, a black hole is surrounded by the purple and red accretion disk. Some of 117 the light emitted by the accretion disk travels close to the black hole, and becomes trapped in circular orbits for a while, before escaping. An observer would see these escaping photons forming a bright ring around the black hole. Current telescope technology is on the verge of capturing images of a photon sphere. New telescopes, like the Event Horizon Telescope, and others in development should be able to capture images of the ring of fire. If we aim our flashlight closer to the black hole than the circular photon orbit, photons emitted from the flashlight will move on plunging orbits and will end up crossing the event horizon. Any light entering the event horizon is unable to escape. If we could directly image the region outside the event horizon of a black hole, we would see a region with no light emission that is sometimes called “the black hole’s shadow”. 6.2.5 Shining a Light on Black Holes What will happen to an astronaut that is far from the black hole? Let’s consider a situation in which both an astronaut and a distant observer are equipped with flashlights capable of emitting a pulse of light once per second. If they shine this pulsing flashlights at one another while the astronaut falls towards the event horizon, what observations would we expect them to see? We already learned that gravitational time dilation will stretch out the time intervals that the far away observer sees. As the astronaut falls into the event horizon, the time interval between the light pulses received by the observer stretch to infinite amounts of time, even though the astronaut may have only spent a few hours falling into the black hole. Since the event horizon is a one-way street in spacetime, the astronaut falling in towards the black hole will continue receiving the signals from a distant observer at exactly the same rate of one pulse per second. The astronaut even continues to receive the signals after crossing into the black hole’s event horizon! Remember, the event horizon is asymmetric, just like a one-way street: light can enter the black hole, but it can’t escape. The in-falling light pulses from the distant observer don’t change as they pass through the event horizon to be observed by the astronaut. Just as there was no wall of gas left over from us compressing a star; there is nothing special marking the location of the event horizon. This makes a trip to a black hole extremely dangerous. If you manage to survive the tidal forces near a black hole, it is easy to accidentally cross over the event horizon since it seems like an unremarkable location in space when you’re travelling through it. So if you do travel to a black hole be sure to calculate exactly where the event horizon is before you approach! We know that anything that enters a black hole’s event horizon cannot escape, not even light. This means that you can’t sneak a look at what’s inside and let people outside know what’s happening. While you obviously wouldn’t risk putting your head into a black hole, if you are sitting comfortably inside your rocket, orbiting just outside of the event horizon, you could lower a camera past the event horizon. However, in order to take a picture of the inside of a black hole, the electrons in the camera’s circuitry would need to travel faster than the speed of light to send any information back up to your spaceship!. We expect that the structure holding the camera would be ripped apart and the camera would fall inwards before any photos could even be taken. Even though we can’t in theory pass information about the interior of black hole past the 118 event horizon, we can deduce some of the properties of a black hole’s interior. One object, theorized to exist by Sir Roger Penrose, is called the singularity: an object so foreign to the laws of physics, that our understanding of them is incomplete. Singularities are thought to be such dreadfully ugly objects, that we think event horizons themselves are there to shield us from seeing it. This (yet unproven) conjecture is called the Cosmic Censorship hypothesis, and we will go into more gory detail in the next section. 6.3 The Singularity 6.3.1 Entering the Black Hole Now let’s do something unwise and travel past the event horizon of a black hole. Knowing that we’d like to get there without being spaghettified, let’s choose a supermassive mass black hole as our destination, so that tidal forces don’t rip us apart on our approach. Once we pass through the event horizon we will be in the strange world of a black hole’s interior. Although it is impossible to send information about the inside of the black hole to the universe beyond the event horizon, there are no laws of physics that would prevent us, observers within the event horizon, from making scientific discoveries. The first thing that we would notice looking away from the black hole, is all of the light emitted by the stars and galaxies outside of the black hole. It definitely isn’t black inside a black hole! If we shine a flashlight, we find that no matter what direction we try to aim the light, the rays always end up pointing inward, to smaller values of the black hole’s radius. 6.3.2 Space and Time Before we examine the peculiarities inside of the event horizon, it’s worth pointing out just how strange our universe actually is. We have 3 spatial dimensions that allow us to move about front and back, left and right, as well as up and down. We also consider time a dimension, even though we can only move forward it in. Something very peculiar happens to the dimensions of space and time at the event horizon of a black hole. Within the event horizon the radial coordinate, which measures how far you are from the black hole’s singularity switches meaning with the dimension of time. Think about it like this: when we are out and about in the universe, we cannot go backwards in time, but in the interior of a black hole we can no longer go backwards in space! This may give you a headache, but moving to smaller values of radius is really the same thing as moving towards a time in the future. Escaping the black hole would require that you move to larger values of radius, which is equivalent to going backwards in time. Since you can’t go backwards in time, you have no choice but to continue to future times (just as we have to) which is the same thing as moving towards smaller values of radius towards the centre of the black hole. 6.3.3 Representing a Black Hole This might not make much sense if you are thinking of a black hole as a sphere surrounding the point r = 0. This is a good enough picture for the region outside of the event horizon, but it is NOT a good representation of the inside of a black hole. 119 We can make a better picture of a black hole by first thinking about how to represent a star that has the same size for all time. In this diagram since the star is a sphere, all we show is the size of the star’s radius. Time runs upwards in this diagram, and to the right we plot distance from the centre of the star. Since the star has the same size for all time, the surface of the star is just a straight vertical line. On this diagram light rays travel on 45 degree angle lines. Now let’s draw a picture of a star that is a sphere that is collapsing to become smaller in size. We are using the same coordinates on this graph, so the surface of the star is a curve instead of a straight line. As time increases upwards on this graph, the distance between the surface of the star and the centre of the star decreases with time. 6.3.4 The Penrose Diagram Now let’s take the collapsing star and allow it to form a black hole. In this picture we have the same surface of the star that gets smaller as time increases. But at one special moment of time, the surface of the star is at the same location as the Schwarzschild Radius RS. At this moment of time the event horizon forms, and is represented by a straight line drawn at a 45 degree angle. The region “below” the event horizon is the region outside of the black hole. The region “above” the event horizon is the inside of the black hole. The jagged line corresponding to what we thought was a “point” is actually a time in the future! This is a simplified version of a Penrose Diagram, which is a tool scientists use to un- derstand the interiors of black holes. More advanced versions of Penrose Diagrams further “compactify” the dimension of space to a finite region. 6.3.5 Time to Reach the Centre Since the radial coordinate “r” takes on the characteristic of a time coordinate, smaller values of distance from the centre correspond to later times. There is no way to avoid the flow of time, so any object that is dropped into the event horizon ends up falling to the centre at r=0. In this diagram, light rays travel on upward paths at 45 degrees. Light emitted in the region outside of the event horizon can go in two directions: the right, which means escaping from the black hole, or to the left, which means falling into the event horizon. Light that is emitted inside of the event horizon still travels on upward-directed 45 degree angle lines. Light that sent in either left or right travelling directions hits the jagged r = 0 line. Having a powerful rocket engine won’t help you escape either. All this can do is slow down the inevitable, since your rocket can’t travel faster than light. The amount of your own personal proper time that it takes to fall from the event horizon to the centre of the black hole depends on the mass of the black hole. A higher mass black hole is larger in size, and the fall takes more time. The time it takes to reach the centre is characterized by this tidy equation: Tfall = 15 · 10−6s( M MSun ) (59) Time to fall equals Fifteen microseconds times the black hole’s mass divided by the Sun’s mass. (I’ve done the hard work of deriving this from a fall to the centre of a solar mass black hole, and got 15 microseconds. The scaling comes in the form of the mass fraction.) 120 So for the black hole Cygnus X-1, with a mass that is 15 times larger than our Sun’s, the equation dictates that you’d have about 0.2 milliseconds to relish in the experience of touring of the black hole’s interior. This amount of time is about how quickly your eyelids take to blink, so you wouldn’t even be able to think about snapping a photo of the scene using your camera! We now have another great reason to visit high mass black holes: we can have more time to enjoy the sights! For example, if we were to choose a supermassive black hole that is one billion times the mass of the Sun, we’d have around 4 hours of proper time to fall from the event horizon to the singularity at the centre. The centre of the black hole, at zero radius, is the location of the singularity. Since this location corresponds to a time in the future, you won’t see or experience it until the precise moment of time when you reach it. How would objects behave when they arrive at the singularity? Well, since observers aren’t able to report back what happens there, we examine what theoretical models of the interior tell us. No matter how massive the black hole, the equations suggest that all of the mass that has fallen into the black hole accumulates at the centre, and is squashed into zero volume! They also predict that an observer would feel infinitely strong gravitational and tidal forces, from which no known object would survive destruction! 6.3.6 When Maths and Physics Collide At this point you might be a bit confused about the terminology, so let’s unpack the word singularity a bit more. To start, I’ll state that physics and mathematics have an unequal relationship. In order for physicists to make predictions about physical processes, we need the mathematical equa- tions in order to describe likely outcomes. However, it’s possible to write down all sorts of mathematical expressions that don’t seem to have any connection to physics at all. Many times in the history of science mathematicians have come up with equations that don’t seem to have anything to do with physics, and only many years later does some physicist discover that the equations actually describe some physical phenomena! Mathematical equations that describe physical processes are limited to certain circum- stances. Outside of those limits, the equations begin to fail, giving nonphysical answers. For an example, consider Newton’s equation for the attractive gravitational force between two objects with mass M1 and M2 separated by a distance r: F = GM1M2 r2 (60) What happens if we allow the distance ‘r’ between the two objects to come infinitely close together, allowing the masses to occupy the same spot in space? In that case we would set r in the equation to zero, which would mean that we would be dividing by zero in this equation! Dividing by zero is undefined in mathematics, and is normally something that you should avoid doing. In order to make sense of this situation, we provide some physical context. What we should remember is that mass takes up space, so it is physically impossible for the centres of two masses to have zero separation. Since Newton’s equation of gravity fails when r is 121 zero, we should treat them only as a good description of nature if the distance between the objects is bigger than zero. The result when r=0 is called a singularity. What this tells us is that at r=0 our equation just doesn’t make any sense. This type of situation is one that prompts us as scientists to look for a new explanation, and more specifically, a new equation! We already learned that Newton’s equation of gravity is an approximation to those of Einstein. Does that mean Einstein’s description of gravity could help us remove the pesky singularity at the centre of black holes? Unfortunately, the answer is no. In fact, Einstein’s equations predict a divergence of the gravitational fields, meaning the problem of the sin- gularity gets even more troublesome than what we would otherwise predict using Newton’s equation. 6.3.7 Quantum Gravity One shortcoming of Einstein’s equations for gravity is that they do not include our modern knowledge of quantum mechanics. Quantum mechanics distinguishes itself by introducing the concept of wavefunctions to describe the positions of particles. Quantum mechanics governs the behaviour of particles at scales where Einstein’s equations fail. Einstein’s equations for gravity assume that we know the locations and speeds of particles exactly. However, the Heisenberg uncertainty principle, a foundational concept of quantum me- chanics, tells us that there is a limit to how precisely we can determine the location and speed of particles. In order for physicists to understand the behaviour of the singularity, we need to combine quantum theory with general relativity, which remains a mystery at present. If we knew how to create just such a theory we would call it “Quantum Gravity”. One proposed method is called “String Theory”, and is a promising set of equations that might describe quantum gravity. However, scientists have not yet managed to solve these equations or make any useful predictions with them. You, dear listeners, could take this up as a challenge, the prize for successfully deriving a theory of quantum gravity could win you a Nobel Prize! Many physicists think that as you fall in towards the black hole’s singularity, the standard equations of Einstein’s gravity describe what happens to you for most of the trip, until the distance (or really time) between you and the singularity becomes much smaller than the size of an atomic nucleus. The region of spacetime that is close to the singularity requires quantum gravity for accurate predictions. Since we don’t understand quantum gravity we can only speculate: perhaps quantum gravity removes the concept of a singularity and could potentially describe the existence of a nice remnant! We really have no way of knowing, since the singularity occurs at a time in the future, it can’t affect us as we fall in. Ultimately, our lack of understanding of quantum gravity doesn’t affect our journey to the centre of a black hole until the moment when we’re about to reach the singularity. 6.3.8 Cosmic Censorship Hypothesis As we have alluded to before, the singularity inside of a Schwarzschild black hole is a bit of a pesky mathematical object. The singularity simultaneously exists everywhere at the interior of a black hole, but only at a specific moment in time. Given that singularities in our universe are not visible to us (as far as we can tell, they are all hidden behind event 122 horizons), scientists have conjectured the existence of a principle to hide singularities from view, called the Cosmic Censorship Hypothesis. Einstein’s gravitational equations predict many different types of singularities. We’ll encounter a new type of black hole singularity shortly: a ring singularity. These other singularities are more like a wall that has a fixed location in space and lasts for a long time. Suppose you run towards a wall. You can see it in front of you as you approach it, and if you don’t stop, you’ll smash into it. A singularity that is like a wall is called a “naked singularity”. A naked singularity is problematic since we don’t have any way to predict what it might emit. The Cosmic Censorship Conjecture states that in realistic astrophysical situations, naked singularities can’t form. In other words, the laws of physics keep singularities cloaked by event horizons. Possibly only singularities like the Schwarzschild version can exist. The cosmic censorship conjecture is yet unproven, but at present we do not see any evidence for naked singularities existing in nature. (Aside: well, with one exception: the big bang itself is quite possibly a naked singularity). It’s likely that in nature all singularities are surrounded by event horizons, which is why I’m excited about direct observational evidence for the event horizon of a black hole! 6.3.9 What happens at the singularity? (Interview with Dr. Valeri Frolov) What happens at the singularity of a black hole? Dr. Frolov: When you fall down to the center this squeezing and stretching forces increase infinitely. They destroy all the elements, all the baryons, all the elementary parti- cles and when you come to the center physically, theoretically this forces grow to infinity. Physically, it means that you cannot believe in the prediction of general activity in this domain. So everything will be destroying there is no clocks, no rulers, you cannot measure time space and question is, is it final state? So will space-time disappear or it will be some continuation? This is an open question. Dr. Frolov’s webpage: https://sites.ualberta.ca/~vfrolov/index.html 6.4 Spinning Black Holes 6.4.1 Effect of Spin on the ISCO There have been only a few occasions where we have discussed the properties of a rotating black hole. Back in Module 05, for example, we discovered that the rotation of a black hole changes the location of the Innermost Stable Circular Orbit. For a non-rotating black hole, the ISCO was 3 times further from the centre of the black hole than its Schwarzschild event horizon, BUT for a rotating black hole, the ISCO can shrink until it exactly matches up with the black hole’s event horizon. It should be noted that as the black hole spins faster and faster, it also pulls the event horizon inwards. Both the ISCO and the event horizon can be as small as half a Schwarzschild radius for a maximally rotating black hole. In fact, now would be a good time for us to fess up about a white lie we’ve been telling you: any realistic black hole will have angular momentum and will therefore be spinning. We 123 know that black holes must be spinning, because infalling particles carry angular momentum, and we have a pesky Law of Conservation of Angular Momentum which tells us that particles will contribute their angular momentum to the black hole. While it’s convenient for scientists to learn about the properties of black holes from non-rotating solutions, the reality is that “perfect” Schwarzschild black holes are unlikely to exist! We should also note something very strange about black hole rotation. . . there is a limit to how quickly they can spin! We will talk a little more about the mathematics behind this concept shortly, but the basic idea is that a rotating black hole drags spacetime along with it. Similar to the way water spirals down a drain, spacetime rotates around a rotating black hole. This is a process called frame-dragging. The rotation of a black hole depends on its original spin, and the cumulative effects brought about by all the material that has fallen into it. A problem arises when we begin to talk about angular momentum. Before, we had taken an object’s mass and multiply it by the distance from the point of rotation to calculate its moment of inertia. See the problem here? If all the mass of a black hole is trapped within a zero volume singularity at its centre, how is it possible for a black hole to have a moment of inertia? Well, in 1963 from a scientist named Roy Kerr developed a solution to Einstein’s Field equations which precisely describe the properties of a rotating black hole. In the original research, Kerr describes how he characterizes the angular momentum of a black hole, but mentions how the moments of inertia cannot be characterized, except to say that they are very small! The rotation of black holes is usually characterized by a number between zero and one, called a, which is calculated by the equation a = Jc M2G (61) J is the angular momentum of the black hole, M is its mass, and G is Newton’s gravitational constant. (If you haven’t figured it out already, lowercase-C is always used to describe the speed of light). If a black hole is not rotating at all, unlikely I know, ‘a’ takes on the value of zero. If a black hole is spinning “maximally” , meaning it has reached that upper limit we mentioned before, a takes on the value of one. As the black hole spins faster, the event horizon and the ISCO are pulled inwards. The ro- tating black hole’s event horizon shrinks to about half the size of a non-rotating Schwarzschild black hole with the same mass, and the ISCO decreases from 3 times the Schwarzschild ra- dius, to coincide with the event horizon at half a Schwarzschild radius for maximal rotation. With that said, I now feel comfortable showing you how the radius of the ISCO changes from 3 times Schwarzschild radius, down to one half, as the rotation of the black hole changes from a = 0 to a = 1. In case you were wondering, the maximum allowed spin frequency at the event horizon of a Solar mass black hole is 16,000 Hertz. The event horizon of a black hole with the same mass as the Sun can spin as fast as sixteen thousand times per second! For other masses, we can simply calculate the maximum spin rate as 16,000 Hertz times the mass of the Sun divided by the black hole’s mass: FrequencyMax = 16, 000Hz · (MSUN/M) (62) 124 Therefore higher mass black holes must spin at slower rates. For a black hole about 36 times the mass of the sun, you would find that the maximum rotation corresponds with 440 hertz, or for you music aficionados, the same as the Concert-A note. The underlying mathematics of Kerr’s solution would take an entire course to discuss, but it’s impact within the scientific community is characterized best by Nobel Prize winning physicist Chandrasekhar, who said: ”In my entire scientific life, extending over forty-five years, the most shattering experience has been the realization that an exact solution of Einstein’s equations of general relativity, discovered by the New Zealand mathematician Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the universe.” 6.4.2 The Ergosphere As a rotating black hole twists the spacetime around it, like the ripples in a whirlpool, the twisting and warping of spacetime itself begins influencing the particles and objects within it. Far from the black hole, these forces gently swirl objects around the black hole. The closer you approach the event horizon of a rotating black hole, the more extreme this interaction becomes, eventually pulling anything falling past the event horizon into complete lockstep with the black hole. Let’s have a top down look at a non-rotating Schwarzschild black hole. In this diagram, there is a central, non-rotating black hole, and each point here represents some source of light. When a light source is far from the black hole, the light propagates outwards in all directions. Now, lets impart some spin on this black hole, changing it from a Schwarzschild black hole into a rotating Kerr black hole. Since the Kerr black hole drags spacetime, light sources far from the black hole begin to see a shift in the direction their light “spheres” propagate. But, there is an even more interesting change when a black hole is rotating. . . Not only does the event horizon shrink from the non-rotating Schwarzschild radius down to about half its normal size for maximal rotation, but particles falling directly inward begin spiralling around the black hole, even though there are no forces acting on them! There is a special distance from a rotating black hole that defines the a region called the Ergosphere. The outer boundary of the ergosphere is called the stationary limit, a region within which, no matter how fast and powerful of a spacecraft you have, the spiralling of spacetime cannot be defeated. Outside of the stationary limit, a spacecraft can “park” with respect to the black hole. Within the stationary limit, no spaceship can ever appear at rest to a distant observer. Even spacecraft entering the ergosphere orbiting opposite to the rotation of the black hole eventually will eventually be pulled by the spiralling spacetime into a co-rotating trajectory. Although the word “sphere” is part of “ergosphere”, the ergosphere is not actually spheri- cal, but rather an ellipsoid. While the event horizon is still spherical, the ergosphere envelops the event horizon, only touching at the spin axis of the event horizon. It’s good to remind ourselves that the ergosphere and the event horizon are “boundaries” not “objects” so they don’t interact with each other in the same way that particles interact within them. I’ll emphasize now, that a clever spaceship captain can still escape from the ergosphere, and can in fact steal rotational energy from the black hole! The word Ergosphere comes from the Greek root, “Ergon”, which means “Work”. The Ergosphere is so named, because 125 it is theoretically possible to extract the energy from the black hole’s rotation with some clever tricks. For example, from within the Ergosphere, you could throw the ship’s garbage against the rotation of the black hole, accelerating the ship forward, and in the spiralled spacetime, end up with more kinetic energy than you started with! In a case like this, you are “stealing” energy from the black hole’s rotation. Roger Penrose first described this process of stealing energy from a rotating black hole in 1971, which is why we call it the “Penrose Process”. Without going into detail, within the ergosphere it’s possible for the energy of a particle to become “negative”, a consequence of the changing coordinate system at the stationary limit. Ultimately what this means is a super-advanced civilization could survive around a rotating black hole, extracting a surplus of energy from the Penrose process, until the black hole’s rotational energy has been sapped. They could also do the reverse, storing energy as the angular momentum of the black hole, and extracting it at a later time. 6.4.3 Gravity Probe B It might seem far fetched to you to be talking about “spiralling spacetime” and “frame- dragging”, but it is possible to measure the gravitational effects of a rotating body without black holes at all. In fact, a space probe, aptly called ‘Gravity Probe B’ was launched back in 2004 to investigate just how strong the frame-dragging effects are here on Earth. Gravity Probe B carried 4 incredibly precise gyroscopes in order to measure these effects. At the time of their construction, these gyroscopes were the most spherical objects ever made, differing from a perfectly round by no more than 40 atoms for a sphere roughly the size of a ping pong ball. Since the effects are quite a bit weaker around a planet like Earth compared to black holes, it took 4 years of operations before NASA reported agreement with Einstein’s theory of General Relativity. 6.4.4 Ring Singularities and the Inner Horizon We’ve been hiding a few details of Kerr black holes “behind the veil” as it were. The event horizon of a Kerr black hole should really be called its “Outer Horizon”, because the mathematics tell us that there must be another “Inner Horizon” hidden inside. The outer horizon is basically the same as the event horizon, its the boundary from which nothing can escape. Even if you’ve fallen through the outer horizon, it’s still possible to receive information from beyond the event horizon. . . right up until you fall through the inner horizon, often called the “Cauchy horizon”. The Cauchy horizon marks a boundary within a black hole where information from the entire history of the universe is compressed. An observer approaching the Cauchy horizon would see more and more of the history of the universe, essentially being battered by the extreme energies that are compressed within that region. Crossing the Cauchy horizon would be perilous enough simply due to the incredible energy densities one would need to survive, but there is yet another mathematical danger lurking within the Cauchy horizon: the Kerr black hole’s Ring Singularity. Unlike the point-like singularities we’ve been discussing for Schwarzschild black holes, the singularity of a rotating Kerr black hole is a ring, instead of a point. The Cauchy horizon may be 126 the universe’s last stand at preventing observers from violating Cosmic Censorship, and glimpsing the singularity. What happens if the black hole’s spin increases? The distance between the outer and inner horizons becomes smaller and the two horizons coincide if the black hole has maximal rotation. If the black hole spin faster than the maximal rotation rate then the equations predict that the horizons will disappear and the singularity will be visible to the whole Universe! Since there will be no event horizons, the resulting thing will not be a black hole. Instead we call it a naked singularity. Cosmic Censorship predicts that it is impossible to spin a black hole faster than the maximal amount. One final note about rotating black holes: IF it were possible to pass through the Cauchy horizon, and IF it were possible to survive the cosmic censor, it might be possible that an extremely talented astronaut could pilot their ship past the ring singularity, and emerge into another universe. What that universe might look like, or what you would find there is still unknown. While it may be tempting to plunge into a Kerr black hole, hoping to survive the journey to a new universe, there may be less dangerous possibility, which we will talk about next: wormholes. 6.4.5 How can you determine if a black hole is spinning? (Interview with Dr. Fiona Harrison) How can you determine if a black hole is spinning? Dr. Harrison: NuStar is able to tell whether black holes are spinning or not. Black holes astrophysically are pretty simple, they only have two parameters, mass and spin. And when matter falls onto a black hole, attracted by its gravity, it organizes itself into what’s called an accretion disk. And friction in this disk heats the material up so that when you get very close to the black hole, if it’s super massive black hole, it’s emitting in the optical or ultraviolet. And then you have regions where particles get accelerated very close to the speed of light, and these emit high energy X-rays. These high energy X-rays can act like a light bulb, shining down on this accretion disk and reflecting X-ray light off of it. Now, by then analyzing that reflected X-ray light, we can tell the geometry of the accretion disk, and in particular, most importantly, how close does it come to the black hole. If the black hole is not spinning, the closest it can come is six gravitational radii, all right? Don’t worry too much if you don’t know what a gravitational radius is, it’s six, okay? If the black hole is spinning maximally, then the accretion disk can come down to one gravitational radius. And we can tell this by analyzing that reflected X-ray spectrum off of the accretion disk. And therefore tell how rapidly the black hole is spinning. Dr. Harrison’s webpage: https://pma.caltech.edu/people/fiona-a-harrison 6.5 Wormholes 6.5.1 Wormholes in Science Fiction I grew up in the orchards of the Okanagan Valley, in Canada, and apples have been an important part of my life. Apples also played an important part in the work of Newton, as 127 he reputedly described the moment he began to wonder about gravity as being the result of seeing an apple fall out of a tree. Often, an apple is used as an analogy to describe another important concept in gravitational physics. I am talking about a specific solution to Einstein’s field equations: the Schwarzschild wormhole. Of course, the name wormhole comes from the idea that worms or caterpillars who feast on the flesh of the apple could tunnel through the interior of the apple, in order to create a shorter path between two points on the surface of the apple. In this analogy, the skin of the apple is our regular 4-dimensional spacetime, and the flesh of the apple is some higher dimension in a hyperspace. Carl Sagan famously said about wormholes: “It is just possible that you might emerge in another part of space-time – somewhere else in space, some when else in time.” So what exactly is a wormhole? The short answer is that a wormhole looks like a black hole, but instead of a trash-compacting singularity at the centre, a wormhole opens back up into a distant region of spacetime. In 1916, Ludwig Flamm was studying the Schwarzschild black hole solution to Einstein’s field equations, when he discovered that a second solution was possible. This second solution described a white hole: a region in space that ejects matter from its event horizon. Flamm then lined up and joined the necks of both the black hole and white hole, and BOOM: the concept of a spacetime bridge was born. Today, we call this an Einstein-Rosen bridge, after it was rediscovered by Einstein and Rosen in 1935. It wasn’t until 1957 that the word “wormhole” was first used to describe a connection between two points in spacetime by scientists Charles Misner and John Wheeler. Let’s be very clear right off the bat here: white holes and wormholes are purely hy- pothetical, and, unlike black holes, there is no observational evidence of their existence. Mathematically speaking, wormholes can exist. Not only can they tunnel through space, but it’s also possible for them to tunnel through time. Throughout the 60s, 70s, and 80s, wormholes entered popular culture through novels like Madeleine L’Engle’s “A Wrinkle in Time”, Joe Haldeman’s “The Forever War”, and Carl Sagan’s “Contact”. Even modern video games like Portal and Portal 2 employ wormholes as their central game mechanic. In fact, the mathematical development of wormhole theories seems to be heavily influenced by science fiction. 6.5.2 Understanding Einstein-Rosen Bridges When Carl Sagan was writing his science fiction novel “Contact” he approached the famous black hole physicist Kip Thorne and asked him how it would be possible for a human being to travel vast distances across the galaxy using a rotating black hole. Thorne suggested instead that Carl consider travelling through wormholes. When Thorne put pen to paper to start figuring out the mathematics, they discovered that wormholes are inherently unstable. Not only would the neck of a Schwarzschild wormhole be too narrow to permit the passage of a human being, but the wormhole itself would close up extremely quickly, making it possible only to squeeze a bit of information through before the wormhole is destroyed. However, Thorne realized that the neck of a wormhole could be held open by some kind of material that would repel the wormhole’s “walls” gravitationally. In reality, we have no idea what kind of material this would be, all “regular” material in our universe acts through gravitational attraction. If regular matter won’t do the job, Thorne posited that a spherical wormhole could be kept open by using a form of material with a negative energy 128 density. (In fact, when University of Alberta physicist Don Page was approached by Thorne, Page demonstrated that ANY shape of wormhole requires negative energy density to be held open, in much more elegant mathematics). Physicists call this “exotic material”, and although there are no examples that we know of, there is nothing written in the laws of physics that prevents it from existing in our universe. The distinguishing feature between unstable and traversable wormholes is therefore the presence of this exotic material. One such traversable wormhole was theorized by a scientist named Homer Ellis, who demonstrated a solution to the Einstein field equations that permits safe passage through the wormhole in either direction. Named after him, the Ellis Wormhole was used as a template for the wormhole in Interstellar, which carries the crew of the Endurance from orbit around Saturn to Gargantua, in a distant region of the universe. We don’t touch much on the science of time travel, but one of my favourite movies on the subject has to be ‘Back to the Future’. In it, Doc Brown accidentally sends Marty McFly travelling backwards in time from the year 1985 to 1955, in his time travelling DeLorean. One of my favourite fan theories for how the DeLorean works is that the Flux Capacitor stores enough energy, 1.21 GIGA-watts (not Jija, as doc brown says), to create the negative energy density required to amplify a tiny wormhole, just big enough, and just long enough for the DeLorean and its passenger to squeeze through before the wormhole closes behind them. 6.6 Summary Black holes that occur in nature are most likely to be rotating Kerr black holes. If you tried to travel in a straight line towards the centre of such a rotating black hole, the frame- dragging effect will instead cause your path to spiral in towards the black hole, like a boat caught in the current of a whirlpool. With that being said, it would be possible to cross the event horizon of a supermassive black holes safely. These are the black holes found at the centres of galaxies, which have massive accretion discs and energetic jets. It is nearly impossible to represent the view near a black hole with 100 percent scientific accuracy. In Interstellar, for example, the Doppler-shifted light from the accretion disc was reduced to avoid confusing the audience. However, there are some simulations that are much more accurate, scientifically speaking. This simulation shows the approach to a realistic black hole with an accretion disc and jets. The set of circles at the bottom left are a map showing us where we are in relation to the black hole. The green represents the region outside of the innermost stable circular orbit. The yellow region is the area outside of the photon sphere where photons can orbit the black hole. The outer and inner red circles are the outer and inner event horizons. The clock at the bottom right shows the time on a clock falling in, as read by an observer far from the black hole. As we approach the black hole we see the two orange coloured jets streaming outwards horizontally and the rotating accretion disc. As a result of all the dust and gas orbiting the black hole, it is difficult to see the location of the event horizon. As we fall closer to the innermost stable circular orbit, our friends far from the black hole see our clock appear to tick very slowly. As we cross through the Event Horizon we continue to see the glowing gas outside of the black hole. Light from all the stars and galaxies in the Universe enters the 129 black hole and is concentrated at the inner horizon, leading to a blinding flash of light at the instant we smash into the singularity. A trip into a supermassive black hole would take us about 4 hours, but to an observer far away, our trip across the event horizon appears to take an infinite amount of time! They would watch our clocks slow down the closer we approached the horizon, and would quickly get bored. University of Alberta scientists Eric Poisson and Werner Israel were the first physicists to show that the inner, or Cauchy horizon of a black hole is not traversable. For my own PhD work, I demonstrated that for any realistic black holes , once you enter a black hole’s event horizon, you cannot avoid hitting the singularity. So, if you are interested in travelling to distant parts of the Universe, entering a black hole is not a good idea. The singularity at the core of a black hole is an unavoidable obstacle. So, is it possible for spacetime to warp in such a way that does permit long distance travel in the universe? In theory, yes... Wormholes are hypothetical spacetime structures that could act as bridges to different parts of the Universe. But wormholes come with their own set of problems and challenges - and aren’t much more likely to help you travel throughout the universe! 130 7 Inside a Black Hole 7.1 Black Holes: The Final Frontier 7.1.1 The Quantum Zone You unlock this door with the key of the singularity. Beyond it is a commymy, bination of dimensions - dimensions of space, of time, of. . . imagination. You’re falling into a reality of both order and chaos, of theories and information. You’ve just crossed over...into the Quantum Zone. 7.1.2 Introduction Just like the 1960s cult classic, the Twilight Zone, the interior of a black hole is governed by mysterious forces and strange effects, which sometimes behave in unexpected ways. Why are bigger black holes colder than small ones? How can tiny black holes evaporate out of existence? The answers will require us to leave the comfortable world of gravitation, and examine the effects caused by the random and unruly laws of. . . Quantum Mechanics. Until now, we’ve assumed that anything falling beyond the event horizon of a black hole meets an untimely demise on the journey to the singularity. But one principle in quantum mechanics tells us a conflicting story: that information can never be destroyed. What is left of all the star-light, the radio transmissions, the books and teapots that happen to fall into a black hole? Can we know what has fallen into a black hole? The answer to these mysteries are central to the search for a theory of quantum gravity. Now it’s time to explore the intersection of quantum physics with black holes. 7.2 Introduction to Quantum Mechanics 7.2.1 The Ultraviolet Catastrophe The story of Quantum Mechanics really begins at the end of the 1800s, a time when scientists thought that the properties of light were completely understood. Their understanding, one which is still fully relevant today, was that visible light is a manifestation of electromagnetic waves. Light had been shown in experiments to be a wave corresponding to changing electric and magnetic fields, and Maxwell had introduced his equations describing the behaviour of light waves. There was one problem that remained with their wave description of light, and it really troubled physicists! The standard description of light waves did not correctly describe the light emitted by hot objects, an emission known as blackbody radiation. We know that if we turn on a stove, the element will become hot and glow red. But when we consider light using the physical laws as laid out in the 1800s, we would instead predict that our stove will emit MORE blue light than red, and EVEN MORE ultraviolet light than blue. The theory of light created in the 1800s predicted the wrong spectrum of colours for blackbody emitters. Theory predicted that as the frequency of light increases the light becomes more and more intense, contributing more and more energy to the spectrum! This 131 is clearly wrong, as it results in the stove emitting an infinite amount of energy! When you turn on your stove, it becomes hot, and the energy emitted, in turn heats your food. However, the energy emitted by your stove isn’t infinite, otherwise turning it on would destroy the planet! This problem of supposedly infinite energy being emitted by blackbodies came to be known as the ultraviolet catastrophe. The ultraviolet catastrophe was a huge problem for theoreticians of the era, and its resolution became a turning point in the history of physics. 7.2.2 The Photon In 1899, Max Planck was commissioned to study the efficiency of incandescent light bulbs, which shine due to their heat. That same year he proposed a theory that resolved the problem of the “infinite” energy emitted by blackbodies, by introducing a new idea: that electromagnetic waves can transport only a special amount of energy called a “quantum”, instead of any arbitrary amount of energy. This was the key to explaining why the energy emitted by a lightbulb, or any other hot object, is not infinite. The new equations Planck developed were the mathematical foundation to resolving the ultraviolet catastrophe. They correctly described the colour and energy emission properties of a blackbody! This new concept of a “quantized” packet of light energy was called the photon. To accompany this concept, Planck also introduced a new physical constant, which is now named after him, called “Planck’s constant”. h = 6.6× 10−34Js (63) The letter h is often used for Planck’s constant, and that is equal to 6.6 times 10 to the minus thirty-four Joule seconds. The “minus 34” tells you that Planck’s constant is a very, very small number. It is hard to wrap your head around how small this number is, or what it means, but we’ll investigate its implications in more detail soon. Planck’s introduction of the tiny constant h was mainly a trick to make the maths work properly for blackbody radiation. He didn’t suggest that it had any true physical meaning. However, Albert Einstein realized that the constant Planck created does imply a new physical reality: It implies that light can be thought of as a particle. In a brilliant paper published in 1905, Einstein showed that particles of light, or photons, come in units that are called quanta, where a specific amount of energy is related to each colour of light. If we have light with a certain colour, we know its wavelength and its frequency. Einstein introduced a simple equation for the energy of the photon of that colour; E = hf (64) The energy of the photon is equal to h, which is Planck’s constant, times frequency. This radical idea led to Einstein receiving the Nobel Prize for his work in Physics in 1922. So why was the concept of a photon such a radical idea? It’s radical because it claims that there is a smallest, indivisible amount of energy for light of any particular colour. Think about our modern understanding of matter. We know that matter is made up of fundamental particles that are indivisible. For instance, an electron is a particle that has a specific amount of electric charge and a specific amount of mass. It is possible to have zero 132 electrons, or one electron, or two, or any whole number of electrons. However, you cannot have half an electron; matter comes in clumps! The concept of a photon is similar. When we see light of some colour, the light is arriving as a group of photons. There might be one, or two, or five million photons, but never half a photon. Light comes in clumps! Light clumps are clumps of energy, while matter clumps are clumps of mass. Since the value of h is so tiny, the amount of energy carried by most photons of light is also tiny. Normally when we see light, or when we capture it on the back of our eyes using our retinas, the number of photons is enormous, and so we don’t notice the clumpy nature of the light. For instance, if you are a couple metres away from a standard incandescent light bulb, about one trillion photons of yellow light will enter your eye every second! The particle nature of light is more obvious when we do experiments with very short wavelength radiation, like X-rays or gamma-rays. When radiation has a long wavelength, such as radio waves, the wave nature of the light is more apparent, since an individual photon in the radio spectrum has a wavelength much larger than a human! The quantized nature of light is a fundamental principle in modern physics. Lasers, for instance, can only be understood if we acknowledge the existence of photons! The interesting thing about light is that it can be described using the properties of both waves AND particles. Our being able to describe light in these two ways is called the wave- particle duality of light. In fact, we are able to not just describe photons, but all fundamental particles, as both particles and waves. Wave-particle duality is one of the foundational ideas of quantum mechanics. 7.2.3 Matter Waves Einstein came up with the radical idea that photons can have both wave and particle prop- erties. So what about matter? How can matter ever behave like a wave? In 1923, a graduate student named Louis de Broglie asked this question and hypothesized that a particle such as an electron could have some “wave-like” features. One of the most fundamental fea- tures of a wave is that it has a distance over which its properties repeat. This distance is what we call the wavelength, and we use the Greek letter “λ” to represent wavelength. De Broglie’s hypothesis was this: if a particle has a known mass and velocity, then it ALSO has a wavelength! λ = h mv (65) Lambda equals Planck’s constant divided by mass multiplied by velocity. We now call this the de Broglie wavelength. Since the mass and velocity are in the denominator of this equation, the wavelength becomes large if either the particle’s mass or velocity are small. So, since an electron has a smaller mass than a proton, if they are both travelling with the same speed, the electron will have the larger wavelength. By the way, setting the speed to zero in this equation makes no sense! Since anything that has a temperature above absolute zero has some motion, we don’t ever actually have particles with zero speed. 133 One of the important properties of waves is that it doesn’t make sense to talk about where a wave is located. Say I asked you to point to the location of a water wave travelling across the ocean, could you point to a specific location that defines where the wave is? It can be difficult to define. When a wave is forced to travel through a small region, like this hole in the wall, it spreads out. The resulting circular wavefronts can be seen in this photo of ocean waves passing through a hole in a rock wall. We call the effect of spreading waves diffraction. But can matter really act like a wave? When you are considering distances that are smaller than the wavelength for our clump of matter, as is often the case in quantum me- chanics, some strange patterns begin to emerge. In this animation we see a stream of particles travelling towards a wall with two holes. In this picture, the de Broglie “De Broye” wavelength of the particles is tiny compared to the distance between the slits. Since the wavelength is very tiny, the particles move the way you would expect particles to move and travel through one slit or the other and land in two tidy stripes on the far wall. Now we have the same experimental set-up but we choose particles with a large De Broglie wavelength compared to the slit separation. We could do this by either choosing slower speeds, or smaller mass particles. When the particles pass through the slits, they spread out in the same way that water or light waves would spread and they interfere with each other. The resulting pattern when they hit the wall has a series of stripes called an interference pattern. The interference pattern is not obvious at first, but over time as more and more particles arrive and interfere with each other a series of bright and dark stripes is seen. An example of this would be in the use of an electron microscope. An electron microscope is similar to an optical microscope, but instead of photons to see the detail in an image, an electron microscope shines a beam of electrons, by using their wave nature. Actually, since the electron’s matter-wavelength is so small compared to the wavelengths of photons, an electron microscope has more powerful magnification than regular light-based microscopes. 7.2.4 The Uncertainty Principle The fact that matter can also be described using the properties of waves tells us that it wouldn’t make sense for us to know the exact location of a matter particle. Instead, we say that the particle is likely inside of an envelope bounded by its de Broglie wavelength. But the de Broglie wavelength depends on how fast the particle is moving! If the particle is moving fast, then its wavelength is small and if the speed is slow, then its wavelength is large. This idea led Werner Heisenberg to introduce a concept called the Uncertainty Principle. It is difficult to say exactly where a wave is or what exactly its momentum is, in fact there is a limit to exactly how well we can figure out these quantities. In other words, the position and momentum of any wave has uncertainty. Heisenberg bridged the idea of uncertainty to particles using his formula for matter-waves, which resulted in the creation of the Heisenberg Uncertainty Principle. Here’s an example: Let’s say we are studying a beam of protons, and we want to know where they are and how fast they are going. The Heisenberg Uncertainty Principle sets a limit to how well we can know each quantity, as they relate to each other. This is a strange 134 aspect of quantum mechanics! The more you know about a particle’s position in space, the less you know about it’s momentum through space! For instance, if we could pin down the proton’s location to a small distance interval, called δx, we could only determine the proton’s momentum to within a small interval of momentums according to the Heisenberg Uncertainty Relation: δxδp ≥ h¯/2 (66) Where δx times δp is greater than or equal to hbar over 2. Here, the little squiggly D’s are the greek letter δ, which we use to denote small values in mass and momentum. In this equation we also use symbol h-bar. This is not a typing error. This is the symbol that represents the reduced Planck’s Constant. h¯ = h/(2pi) (67) “Hbar equals h divided by 2 pi” since some physicists think that writing hbar is easier than dividing by “2 pi”. The important thing that the Uncertainty Principle tells us is that you can’t know exactly where a particle is located and what its speed is. If you are very certain about where a particle is located, then you can’t accurately know the particle’s speed, and vice versa. There is also an energy and time version of the Uncertainty Principle. The equivalent energy-time version of Heisenberg’s Uncertainty Principle is: δEδt ≥ h¯/2 (68) Here, we see that delta E times delta t is greater than or equal to hbar divided by two. We will explore what this means for black holes in the following section! 7.3 Hawking Radiation 7.3.1 General Overview of Hawking Radiation So far, quantum mechanics seems pretty weird, eh? Niels Bohr, responsible for the planetary model of the hydrogen atom once said, “Anyone who is not shocked by quantum theory has not understood it.” While that may be the case, another famous physicist, Richard Feynman, once said, “I think I can safely say that nobody understands quantum mechanics.” Now, Feynman didn’t mean that quantum mechanics is a useless theory, but that the quantum world behaves so strangely compared to our macroscopic world, that our human intuitions cannot be relied upon to predict what will happen. Quantum tunnelling is just one such example of how strange quantum mechanics can be. How can tiny particles vanish on one side of a wall, only to appear on the other side? Well, if we recall from our last lesson, we don’t know exactly where particles are as a result of the Uncertainty Principle. If we don’t know exactly where something is, how could a wall know either? That’s the basic principle behind quantum tunnelling, that indeterminacy, or uncertainty, can lead to a whole host of outcomes! But quantum tunnelling is related to a process within black holes called Hawking Radiation, so we need to understand that in order to continue! 135 As we saw previously, one of the fundamental relations of quantum mechanics is the Heisenberg Uncertainty Principle. In a practical sense, it states that no matter how accurate our instrumentation, the experiments we conduct will always be limited in the accuracy of their results by at least hbar divided by two. But it’s not just our instruments, the entire universe obeys this uncertainty principle. The Uncertainty Principle has an important implication when we think about the vac- uum. Normally, you think of the vacuum as a void. Space that is completely devoid of material, matter, and energy of any kind. In the context of quantum mechanics that simply cannot be true, because if there is absolutely nothing, that would imply certainty. Certainty about the fact that there is no energy or mass in the vacuum. If you are certain that there is no energy in the vacuum, then you are saying that the uncertainty in the energy, delta-E, is zero. However, the energy-time version of the Heisenberg Uncertainty Principle states that δEδt ≥ h¯/2 (69) δE times δt is greater than or equal to hbar divided by two. In essence you can never be totally certain that you have a true vacuum, since δE has to be larger than zero! The uncertainty principle tells us that what we think is a vacuum, actually has, for brief moments of time delta-t, particles appearing and disappearing. This quantum view of the vacuum is sometimes called the quantum foam. 7.3.2 The Quantum Foam In general, the uncertainty principle is meaningless in everyday life. We never measure things so precisely that we would run into such a limit. But what if instead, we zoomed down into the quantum world, what do you suspect we’d see? Let’s shrink down in spatial dimensions, to about the size of an atom, and in the time dimension, so that we are living through nanoseconds as if they were seconds. . . Already you can see that spacetime itself is strange at the quantum scale. Scientists call this. . . mess. . . the Quantum Foam. This is an artist’s illustration of the quantum foam, and as you can see the foam looks to be FROTHING with activity? What’s going on down here? Since we’ve shrunk ourselves down in space and in time, we’re living in the cold real- ity of what the Uncertainty Principle enforces on the universe. . . for lack of a better term, uncertainty. Down here, the foam can be essentially anything. A proton-antiproton pair here, an electron-muon-neutrino entanglement there. The quantum foam might even generate tiny wormholes! As long as these species last for only a short time they can have lots of energy. δEδt ≥ h¯/2 (70) Physicists call particles born of the quantum foam, virtual particles, because for all intents and purposes, they exist for such a short period of time, and we in the classical world barely measure their existence.. Essentially we can “borrow” energy from the universe, as long as we pay it back very quickly! 136 7.3.3 Virtual Particles Near the Horizon So now a question we might ask is: What happens if we put a black hole nearby, and zoom in on the quantum world at the boundary of the event horizon? Do virtual particles get pulled across the event horizon? Classically, we’d say no. For one, the quantum foam is not a result of classical theories like general relativity, but given its existence anyway, we generally think that virtual particles don’t last long enough to fall in. However, we also have to consider where a particle is, because in this universe we obey the laws quantum mechanics! So, here we are, among virtual particles in the quantum foam, and this black region is the event horizon of our black hole. Let’s see if we can pin down an electron and its antiparticle, the positron, when they emerge from the quantum foam. This particle pair is called Positronium, because it exists, albeit temporarily, as a quasi- atom, until the universe decides its time is up, and it vanishes back into the quantum foam. But the diagram here doesn’t show a complete picture, because the electron and positron in this image have definite positions. If we were truly to draw this properly, each would have a much larger probability density, which is to say, regions within which the particles are likely to exist. Probability densities, characterized by something called the wavefunction, are places where there are probably particles. So for our example positronium, there’s a 100% chance that we’d find them inside this boundary. But look, the way we’ve drawn this, there’s a good chance that we could find the blue particle, not just on the boundary of the event horizon, but within it entirely. That’s a bit goofy, you might think. The particle-antiparticle pair have come into exis- tence, and before they could repay their energy to the universe, ONE OF THEM VANISHED INTO THE BLACK HOLE! So, now we’re in a pickle. We needed that blue particle to an- nihilate the red one. . . but instead, now we just have a red one left, with no companion! And it does not want to stick around! Essentially what you’ve just witnessed is Hawking Radiation, the process by which par- ticles can kind of *wave hands in a questioning gesture* escape from a black hole. When a particle-antiparticle pair pop out of the quantum foam right on the boundary of a black hole, the outgoing particle actually steals energy from the black hole system. It doesn’t have to pay the Universe back for the energy, instead the black hole had to pay for that part of that equation! Not only does that mean that particles appear to come from the event horizon, but also that Quantum Mechanics allows black holes to evaporate slowly! So it’s pretty obvious that quantum mechanics can do some strange stuff. They can whittle away at a massive black hole, until nothing remains. In the next lessons we’ll develop a further understanding of this concept by examining a black hole’s temperature, its entropy, and eventually, we’ll determine how long a black hole has to live in the face of quantum mechanics... 7.3.4 What is Hawking radiation? (Interview with Dr. Don Page) What is Hawking radiation? Dr. Page: There are different ways to think about Hawking radiation. But one way is 137 to think that in quantum mechanics, because of the uncertainty principle, what you might think of empty space is not really an empty. It’s seething with particle, anti-particle pairs that form and then annihilate. In some sense, they borrow energy, but then, they have to return it back and so, they can exist for very short time. So you can have, for example, an electron positron pair that’s produced, but it annihilates really quickly in the vacuum. But what happens is if you have a black hole, you have this black hole horizon and if one of the two particles in the pair falls behind the horizon, then it’s no longer left outside to annihilate with what’s outside, so that the other particle of the pair that’s outside the black hole, can escape. So this escaping particle then is the hawking radiation. So that’s one way to view it. There’s other ways to talk about just evolving Quantum Field Theory in curved space-time, that the mathematics gets rather difficult. A longer interview with Dr. Page where he talks about Stephen Hawking: https://www.ualberta.ca/science/news/2018/march/stephen-hawking-don-page.html 7.4 Information in a Black Hole 7.4.1 Information and Quantum Mechanics Why are we so concerned about quantum mechanics as it relates to black holes? Well, the existence of Hawking Radiation presents an interesting problem to black hole physicists. . . Classical black holes can be characterized by three numbers: their mass, angular momentum, and charge. This is a concept called the “No-hair theorem”, which is a way of saying that information (which is what they mean by hair, in this instance), can’t escape from a black hole. But Quantum Mechanics tells a completely different story, which has lead to one of the biggest unsolved problems of our time: the black hole information paradox. In addition to quantum mechanics enabling processes like Hawking Radiation, it also throws a bit of a wrench into theories that try to combine Quantum Mechanics with General Relativity. In particular, quantum mechanics requires information to be preserved. So, in order for astrophysicists to have a complete description of black holes, they will have to find a way to explain the so-called black hole information paradox. Information can be defined as something which answers a question. When physicists talk about information, they are generally asking questions about the characteristics and state of something. So when we ask about the mass of a star, or we measure what its spectral output is, we are generating information. Now imagine that you are the commander of a scientific research vessel that is in orbit around a black hole, collecting measurements about the properties of the black hole. You can very easily deduce the mass of the black hole by observing, measuring, and timing how long it takes to orbit the black hole at a given distance. You can also measure the spin of the black hole by comparing the size of the Innermost Stable Circular to the size of the Event Horizon. And, by dropping a couple of test charges and observing their behaviour, you could tell the charge of the black hole (scientists think that most black holes have zero charge anyway). So, you’ve collected all this data, but OH NO, YOU ACCIDENTALLY CROSSED THE EVENT HORIZON! Luckily you and your crew have survived, but the future is bleak. . . 138 Given the fuel you have in reserve, you can only postpone your collision with the singularity by a few hours, maybe a few days at most. Is there any way that you could send what you learned back to a distant observer, or does the event horizon prevent information from escaping? Classical physics, that is to say General Relativity, has some very bad news for you. There is no way for you to transmit your data across the event horizon. But being the good spaceship captain that you are, you’ve brushed up on your quantum mechanics, and you know about two important principles in quantum mechanics: quantum determinism and the principle of reversibility. Let’s start by unpacking the easy one: reversibility. Reversibility is the idea that physical laws can be used to predict the future or the past of a particular system. If we look at a comet shooting by, not only can we predict where it will be decades from now, but also where it was decades in the past. In some sense, reversibility also tells us that we can go backwards from one state just as well as we can go forwards. In reality, we know that reversibility is more complex than that: shattered mugs don’t suddenly unshatter themselves, but we’ll address that when we discuss entropy. On the other hand, quantum determinism is a much stranger idea. Determinism on its own is the idea that we can predict the outcome of any interaction if we have sufficient information. If I throw a ball, and you know its direction and how fast it’s going, where it lands is predetermined. But, quantum mechanics basically deconstructs the notion of classical determinism because of the Heisenberg Uncertainty Principle. Since we have limits on what we can measure, we also have limits on what we can predict. Quantum determinism is then telling us a slightly different version of the story: we can predict with certainty what the probabilities of outcomes in quantum mechanics will be! Since the principles of quantum determinism and reversibility are fundamental to quan- tum mechanics, we have run into a problem: these two principles together mean that IN- FORMATION MUST ALWAYS BE PRESERVED. BUT, the no-hair theorem for black holes says that information falling into a black hole disappears when it crosses the event horizon. Many scientists postulate that the information is destroyed somewhere at the interior of the black hole. In the next lesson we’ll delve further into this investigation, examining specific theories which attempt to explain the black hole information paradox. 7.4.2 The Information Preservation Station? So, which is it? Is information preserved by quantum mechanics? Or is it destroyed by general relativity? There are many possibilities: the information could be destroyed, the information could leak out of the black hole gradually, or maybe information could escape in a powerful explosion, perhaps the black hole itself saves the information like a giant hard drive. The sad truth, is that we just don’t know. Although there are no known laws that unify gravitation with quantum mechanics, many researchers, including Stephen Hawking, now believe that information is preserved, somehow, and deeply linked to the process of Hawking Radiation. Some researchers have had other. . . interesting ideas... One such interesting idea is the concept of a new boundary within the event horizon, the information firewall. In 2012 a group of physicists, Almheiri, Marolf, Polchinski, and 139 Sully, (shorted to AMPS), introduced the concept of a boundary of high energy quanta that destroy all incoming information - the AMPS firewall. The AMPS firewall cleans up some of the inconsistencies of quantum mechanics by providing a mechanism to destroy, or scramble, the incoming information. But some scientists think the firewall creates more problems than it solves... Present day research done by Don Page at the University of Alberta suggests that the Information Firewall described by the AMPS group could produce a Naked Singularity. If indeed there is a firewall within a black hole, Page and his collaborators demonstrate the the firewall can migrate to a region outside of the event horizon, allowing the singularity to become visible to distant observers! We already know how much physicists abhor a naked singularity! One of Page’s research collaborators, Misao Sasaki has said, “If a firewall exists, not only would an falling object be destroyed by it, but the destruction could be visible, even from the outside.” This is a very complex idea, but it brings up an interesting motivation for physicists. If the firewall idea is right, it means that there is new physics for us to consider. If the firewall idea is wrong it means that we’ve uncovered some potential flaws in older physical theories. Although there is no consensus about how the Black Hole Information Paradox will be resolved, there are several leading theories about the fate of information that has fallen into a black hole. One such theory, put forth by Stephen Hawking when he originally described Hawking Radiation in 1976, predicts that the outflow particles from the black hole would have unpredictable properties. So then. . . what does Hawking Radiation look like? 7.4.3 What is quantum information? (Interview with Dr. Lindsay LeBlanc) Q: What is quantum information? Dr. LeBlanc: On a quantum level, information is what we call qubit. So, it’s storing information, not classically like a one or a zero that you would have in a regular computer, but a quantum system can exist in what’s called a superposition. It can hold information as some probability of being in say left, and some probability of being in right. Those labels are completely random, it doesn’t matter what they are. But at the same time, you hold two possibilities at once and it is this superposition or this probabilistic holding of information that makes quantum information so much different and actually so much more powerful, because you can actually store a lot more information in one physical thing as opposed to a bit. Q: What experiment would you like to do with a black hole? Dr. LeBlanc: I don’t know a lot about black holes, but I would be interested in looking at how quantum phenomena actually interact in the system. So, we often think of black holes as being a phenomenon of gravity, and gravity and quantum mechanics are two sepa- rate things in physics that we think should be connected, but we actually don’t have a very good idea of how they are connected. So, black holes is very interesting object that perhaps we could start thinking about what happens to quantum things as they interact in a black hole. So, if you can imagine having a quantum system where you have maybe two particles that are entangled, which means that if you do something to one particle, automatically something happens to the other and there’s this intrinsic connection between them that you can’t break and so, if you could send one particle into the black hole, you still have the other 140 one and how it behaves should tell you something about what’s happening to the one inside this black hole. But black hole, you obviously can’t get any information about that. So, yeah, how we would ever do an experiment about that? I don’t know but I think it’s an interesting problem to think about. Dr. LeBlanc’s webpage: https://sites.google.com/ualberta.ca/ultracold 7.5 Black Hole Thermodynamics 7.5.1 Entropy Since we seem to have found ourselves in the business of throwing stuff into black holes and waiting to see what pops out, a natural question to ask might be: how does the black hole change when something falls in (or comes back out!)? Of course, we do our best to respect the “No-hair theorem”, but we can ask more about the black hole than just what happens to the mass, spin, and charge. After all, if all you’ve given me are those three values for any particular black hole, I can’t tell you anything about its history compared to a black hole with the exact same three characteristics. Somehow, part of the black hole’s history is hidden from observation, but astrophysicists have yet another trick up their sleeve: entropy! Entropy is a property that all matter has, and is usually explained as a measurement of the disorder in a system. That’s a pretty good explanation, but it’s quite a bit easier to explain by an example: your bedroom. Entropy is one of the main reasons why we don’t experience the reversibility of the universe. Just like your bedroom, there is only one way that a mug is unshattered, but an uncountable number of ways that it can be smashed. What this means is that we rarely see mugs spontaneously going from shattered to unshattered, or dirty rooms spontaneously becoming clean, because there are MANY ways for those systems to evolve (most of them messier than before). For this reason, entropy is not considered to be time reversible. 7.5.2 Introduction to Thermodynamics The behaviour of all matter as it relates to entropy, temperature, and energy is codified in a field of physics called thermodynamics. The word thermodynamics comes from the Greek word therme, meaning heat, and dynamis, which means power. Thermodynamics is there- fore the study of converting heat into power, and historically it was motivated throughout the 1800’s by the invention of steam engines and later combustion engines. However, ther- modynamics has become an essential tool for astrophysicists, since stars are basically the equivalent of nuclear engines! There are four basic laws in thermodynamics, and you’ll find them surprisingly simple: ...the Zeroth Law is a statement that the temperatures of two things are the same if they are in thermal equilibrium. This is a great law, because if it weren’t true, we couldn’t trust thermometers! . . . the First Law is a statement that energy is conserved in isolated systems. This is the principle that Thermoses operate on: they attempt to keep your food hot by isolating the interior of the thermos from the outside world. Isolation means (ideally) no energy loss to the outside environment, and so your food stays hot! 141 ...the Second Law is a statement that heat flows from hotter objects to colder objects, and NOT the other way around. This might make some intuitive sense. When we warm ourselves by the campfire, we don’t think about cold flowing out of our bodies, we think of hot flowing in. ...the Third Law is a statement that as we cool things down to absolute zero, all processes stop. This is an extension of our experience, since humans can’t survive for long below about -40 Celsius, but using a fridge is exactly the third law in practice: cold food lasts longer because decay processes slow down at low temperatures. 7.5.3 The Second Law of Thermodynamics The Second Law seems pretty obvious, eh? Heat flows from hot things to cold things. And this usually progresses until both objects are the same temperature, right? But how would you go about measuring how far along this equalization is? The answer is entropy! Just like the example with your messy bedroom, entropy measures how “messy” a system is. As it turns out, this is directly related to an object’s temperature! The formulaic definition of entropy is this: S = k · log(W ) (71) where S is the entropy of a system, Kay is Boltzmann’s constant, and W is the number of equiprobable states that the system can be in. Ludwig Boltzmann was so proud of this equation that it was engraved on his gravestone. It is easiest to think back to your messy bedroom for this one: there is only ONE equiprobable state for a clean room, so entropy is LOW. There are many MILLIONS of ways your room can be dirty, so entropy of a messy room is HIGH. Here’s the formula to calculate the entropy of a black hole, based on the area of its event horizon, S-blackhole is equal to: SBH = kA 4l2p (72) Where S is the total entropy, equal to k (Boltzmann’s constant) times the area of the event horizon, divided by four times the square of lp, the Planck length. The Planck length is defined by: lp = √ h¯G c3 = 1.6× 10−35m (73) The Planck length is a very small distance which is commonly thought to be the smallest distance that can be described by relativity without quantum physics. In some sense, the equation for the black hole’s entropy says that each little bit of entropy that falls into a black hole is encoded as a small patch on the event horizon one planck length across. Before we try and apply entropy to a black hole, take a moment to think. Does a black hole represent a “clean” state of all the matter that has fallen in, or is it “messy”? 142 7.5.4 Entropy and Temperature of a Black Hole Black holes are strange objects within the study of thermodynamics, a fact recognized by scientist Jacob Bekenstein. In the early 1970s Bekenstein asked himself a question very similar to the one I just asked you. At the beginning of the 1970s, it was still believed that nothing could escape from a black hole, so a question like, “How hot is a black hole?” made no sense! Since nothing comes out of a black hole, the temperature should be zero! Bekenstein recognized that, if black holes didn’t have a well-defined temperature, maybe they had a well-defined entropy. He went on to prove that a black hole’s entropy is propor- tional to the area of the black hole’s event horizon. But since entropy was well characterized, shouldn’t temperature follow? We can’t just stick a thermometer into a black hole in order to see if it’s running a fever, instead we’ll have to consider what is emitted from the black hole as Hawking Radiation causes it to slowly evaporate. Indeed, this was the motivating question that led Stephen Hawking to derive the frame- work for Hawking Radiation. Since quantum mechanics permits black holes to “evaporate”, then they certainly don’t have zero temperature! Hawking’s student at the time, Don Page, took the concept of Hawking Radiation further, deriving what temperature we could expect a black hole to radiate. So what do you think? Are black holes hot or cold? If you said cold, you’re mostly right! The temperatures of black holes are given by the simple equation: Tee equals Kappa divided by 2-Pi: T = κ 2pi (74) Where T is temperature, 2 and pi are just constants, and this Kappa is just the surface gravity of the black hole. The surface gravity is a way of expressing the acceleration due to gravity at the event horizon, but in weird units. Let’s skip the lengthy derivation for a Schwarzschild black hole and get right to the result: The surface gravity of a black hole is proportional to one-over-four-Em: κ = 1 4M (75) You recognize this right? The surface gravity is inversely proportional to the mass of a black hole. . . So small black holes have big surface gravities, and big black holes have small surface gravities. . . This will become important shortly. 7.5.5 Black Hole Thermodynamics Since we can now calculate the temperature and entropy of a black hole, we can reconsider the meaning of the thermodynamical laws that got us here! In black hole physics the laws of thermodynamics can be restated as they apply to black holes like this: ...the Zeroth Law states that a black hole’s gravity is the same across its surface! While gravity and temperature are different things, they are closely related in black hole physics! ...the First Law states that changes to a black hole depend on the mass and energy consumed by the black hole. This is a statement of energy conservation. 143 ...the Second Law states that the area of a black hole always increases as matter falls in. The event horizon can shrink when Hawking radiation is emitted. However, there is a generalized second law that states that the sum of the area of a black hole and the entropy of the emitted Hawking radiation always increases. [There is also a 3rd law, based on spinning black holes that is a bit more complicated and isn’t covered in this course.] So, now that we have a theory of black holes compatible with the laws of thermodynamics, what can we say about them that we didn’t know before? Simply this: black holes have well defined temperature and entropy. So a black hole has temperature, so what? Did you notice how the mass of a black hole relates to its temperature. Yeah, this is where things get strange. The more MASSIVE a black hole is, the COLDER it becomes! The opposite is true too! The smaller a black hole is, the HOTTER it becomes. Now that’s strange! To calculate the temperature of a Schwarzschild black hole, this is the full equation: T = h¯c3 8piGMkB (76) Using this equation with a one-solar mass black hole yields a temperature of a chilling 60 nano-Kelvin! That’s closer to absolute zero than most scientists have been able to achieve in the lab. The lowest recorded temperature of anything on Earth was a molecular gas created here at the University of Alberta by professor Lindsay LeBlanc, at 40 nano-Kelvin, that’s a mere 20 nano-Kelvin colder than a solar mass sized black hole! How small do you think a black hole would need to be for it to be room temperature? And what would happen as a black hole “cools off” through Hawking Radiation?? 7.5.6 How do you cool a gas with lasers? (Interview with Dr. Lindsay LeBlanc) How cold is your research? Dr. LeBlanc: My research involves looking at really cold gases of atoms. We make them really cold by shining lasers on them and we can slow the gases down to very slow speeds, which is equivalent to being at low temperatures. We’re interested in these because they display very quantum mechanical behaviors at these low temperatures. So we take this gas of quantum stuff and do experiments studying the basic properties of quantum mechanics with these gases. What do you mean by ”cold”? Dr. LeBlanc: In this context, cold temperatures means that the particles are moving very slowly. So temperature is really just a measure of how fast things are going on average. So what we’re doing is slowing things down to slow enough speed that they’re actually getting close to almost stopping, which should be close to absolute 0. How does this compare to interstellar space? Dr. LeBlanc: The temperature for interstellar space is around three Kelvin, we cool our gases to about 40 nanokelvin, so that is about a billion times colder, maybe not quite, 100 million times colder. What it means is that they’re just moving about maybe 10,000 times slower. So our atoms move at about two millimeters per second, at room temperatures, these 144 atoms would move it a few hundred meters per second, and in interstellar space it would maybe a few tens of meters per second. How do you cool a gas with lasers? Dr. LeBlanc: Laser cooling is an interesting principle. It works because lasers are in some sense perfect. Lasers have a single color of light, the light moves all in one direction, and it’s what we call coherent. So it waves altogether all the time. You see this when you see a laser pointer. That’s why a laser pointer is different than a light bulb, a laser pointer will shine and you get a spot in one spot, a light bulb, the light just spreads out everywhere. So the spreading out everywhere is associated with some randomness, whereas the laser having one point is a perfection. So we’re taking that perfection and transferring it to our gas of atoms when we’re doing laser cooling. So what we do is we take lasers and we shine them from all directions at a gas of atoms and in three-dimensions, that means that we have six beams coming in all three directions. Now, those atoms that sit at the center or atoms that wander into that intersection region of the six lasers, they see that laser light and we’ve tuned that laser light to be very close to what we call a resonance frequency. So it’s the color of light that interacts most strongly with this gas of atoms. We can set it up so that those atoms only see one of those beams at a time. It depends on how they’re moving and it’s related to something called the Doppler effect, which you might know from moving firetrucks, you hear the sound change as it moves by you. The atoms actually see the color of the light change just a little bit as they move towards or away from these beams of light. So as the atom moves towards one of the beams, it will actually absorb that light more strongly. So if it’s moving this way, it’s going to absorb the light from this beam, which means it gets pushed back towards the center. But if it starts moving towards this beam, it’s going to get pushed by that beam. So we call this an optical molasses actually, because the atoms stuck at the center of these six beams are a little like a molasses, they’re a thick, viscous, almost like a fluid. Whichever way they try to move, they’re opposed by these laser beams and it’s actually a very strong effect. So we can just turn these six laser beams on, even in just a room temperature gas of atoms and all of a sudden we’ll just collect a sample of very cold atoms at this intersection of all of these six laser beams. 7.6 Lifespan of a Black Hole 7.6.1 Temperature of a Black Hole Quantum Mechanics tells us that we can’t know exactly where a particle is located. In particular, it could be located at any position within its de Broglie wavelength. If the de Broglie wavelength of a particle is larger than the size of the event horizon, then there is no reason to think that it is actually inside of the black hole! Quantum mechanics tells us that due to the wave nature of particles, mass that has fallen into the black hole can potentially get out. This concept is called “Quantum Tunnelling” and leads to the process of Hawking Radiation. Stephen Hawking calculated the probabilities for particles escaping from a black hole. Hawking showed that the number of particles with various energies that escape from a black hole corresponds to exactly the same emission as blackbody radiation. So Hawking proved that black holes are actually blackbody emitters! Very cold blackbody emitters, but still 145 this was a rather surprising result! So, if an object is hot and emits as a blackbody, this means that it has a temperature. We learned earlier in this course that a hot object’s temperature is inversely related to the peak wavelength through Wien’s law. The peak wavelength is the wavelength where the most particles (or photons) are emitted. Wien’s law is T = W λpeak (77) where T is the temperature of a hot object and λ is the peak wavelength.” We know if the peak wavelength is red, corresponding to 700 nanometers, the temperature of the object is 4100 Kelvin. If the wavelength is longer (in the infrared or radio parts of the electromagnetic spectrum), then the temperature will be cooler. So what would be the temperature of a black hole? As we already know, the event horizon radius of a one solar mass black hole is three kilometers. A hot object whose peak wavelength is this large has a very small temperature, which we saw in the last section is 60 nanoKelvin. A lower mass black hole will have a smaller event horizon, so the emitted Hawking radiation will have a smaller peak wavelength. Smaller peak wavelength means higher tem- perature. This leads to the inverse relationship between black hole mass and temperature. Where does the energy for this radiation come from? The energy is coming from the mass of the black hole. The process of Hawking radiation slowly converts the mass of the black hole into energy, by allowing the mass to leak out of the black hole. The Hawking radiation that the black hole emits is blackbody radiation, which has no dependence on the composition of the material that originally fell into the black hole. The emitted Hawking radiation depends only on the black hole’s mass and angular momentum, so it preserves the “No Hair” property of black holes. This is another aspect of the black hole information paradox! The emitted radiation doesn’t convey any of the information that had previously been captured by the black hole. 7.6.2 Evaporation of Black Holes Hawking radiation is a quantum mechanical process that allows mass to slowly leak out of the black hole’s event horizon into the region outside. If a black hole loses mass, we know that its event horizon radius must shrink. Shrinking the event horizon radius means that the temperature will increase. Since higher temperature hot objects are also brighter, this means that the rate that mass is lost will increase. So as black holes lose mass, and become smaller, they also radiate faster! Uh oh, this could get messy! The result that Hawking radiation causes a black hole to lose mass is called Black hole evaporation. If you were to extrapolate to zero mass, the equations would predict infinite brightness and temperature! This extrapolation is most likely not correct, and requires a quantum theory of gravity to predict the outcome correctly. One possibility is that all the information that was lost inside the black hole is finally released in the moments before the Hawking radiation process is finished! The process of black hole evaporation takes a long time for the black holes that we see in nature, but is rapid for black holes with tiny mass. For instance, a black hole like Cygnus 146 X-1 would take 1068 years. TEN TO THE POWER OF 68. . . YEARS. That’s an ENORMOUS amount of time, longer than the present age of the universe by a factor of an OCTODECILLION, a one followed by 57 zeros! Since Cygnus X-1 won’t evaporate anytime soon, the Hawking radiation plays no important role in its lifespan. In addition, Cygnus X-1 is accreting mass from its companion, so it is gaining mass at a much, much faster rate than the black hole evaporation rate. The X-rays emitted by the accretion disc are enormously bright in comparison to the Hawking radiation, making it impossible to detect the faint signal from the black hole’s evaporation. At present, Hawking radiation has never been detected from any black hole. This is due to the very long wavelength associated with it, and the very low temperature and energy associated with astrophysical black holes. In order to detect Hawking radiation we need to hunt for a special set of circumstances. First of all, it would be best to look for an isolated black hole that is not accreting matter from a companion star. Secondly, the mass should be very small so that the temperature is high enough that we could detect it. How small should the mass be? If we could find a black hole with a mass the same as our Moon, then the Hawking temperature would be 1.6 Kelvin, which would be difficult, but possible, to measure. Any black hole with a mass smaller than our Moon would be hotter and easier to detect. If we want to see the whole evaporation process, it would be convenient if the time for the evaporation to take place would be less than the present age of the Universe, 13.8 billion years. A black hole with a much smaller mass, such as 1012 kg (a trillion kilograms) or smaller would take less than 13.8 billion years to evaporate. That might sound like a large mass, but a trillion kilograms is approximately the mass in a large mountain here on Earth. That may still sound like a large mass, but to astrophysicists, that’s tiny! Unfortunately, we don’t know about any mechanisms for creating mountain-mass black holes from astrophysical objects like stars or planets or asteroids. One possibility is that very small black holes, called primordial black holes could have been created in the big bang. However, no evidence for Hawking radiation from these conjectured primordial black holes has ever been seen. If you are really interested in detecting Hawking radiation, you’ll have to artificially create your own black hole. . . by smashing protons and anti-protons together at incredible speed! 7.6.3 Black Holes and the Large Hadron Collider Making your own black hole sounds kind of silly. Wouldn’t it be dangerous to create a black hole on Earth? Well, we just learned that small black holes evaporate quickly by emitting Hawking Radiation, so if it’s a really tiny one, it won’t last long enough to capture matter from Earth! It is theoretically possible to create one using a particle accelerator such as the Large Hadron Collider, also known as the LHC in Geneva, Switzerland. The LHC accelerates protons and antiprotons in opposite directions along a circular path. When the particles are moving fast enough, the protons and antiprotons beams are crossed and allowed to smash into each other. If the proton and antiproton get closer together than the Schwarzschild radius for two protons, they could form a tiny black hole before they get 147 a chance to annihilate each other! The tiny black holes formed in the LHC would evaporate very quickly (less than a second) and we could then observe the energy released. Scientists are looking for the signature of the energy released from Hawking radiation, but so far no evidence for black holes created in the LHC has been found. We should remember that the Earth’s atmosphere is bombarded by natural high energy particles called Cosmic Rays that possibly originate from the jets of supermassive black holes in far away galaxies. Some of these cosmic rays could interact with atoms in the Earth’s atmosphere and create a tiny black hole. No evidence for this process has ever been seen. This tells us that either tiny black holes are difficult to create, or when they are created, the radiation that they emit is difficult to detect. Some people expressed concern about the possible harm to the Earth if a tiny black hole were created in the LHC. However, the very short lifetime due to Hawking radiation would make these tiny black holes harmless. In addition, in the five billion years of the Earth’s history, no cosmic ray collision, which could be much more energetic than the LHC, has ever created a black hole that has harmed the Earth. We also have not seen any other planets or stars harmed by interactions with tiny black holes, so there is no evidence of risk. Physicists who study these tiny black holes are confident that their existence does not endanger life on Earth. 7.7 Summary Over the last few modules we have journeyed to the black hole Cygnus X-1 to explore the black hole binary system, and see the wondrous sights outside the event horizon. We then dove inside to explore the mysteries hidden behind the event horizon. We learned that the boundary between the inside and outside of a black hole is actually rather fuzzy once we include the laws of quantum physics. In fact, quantum mechanics suggests that black holes aren’t actually black and they can actually leak out their contents, and eventually disappear! Scientists today have different levels of understanding of the phenomena associated with black holes. First of all, there is excellent evidence that black holes do exist. The fundamental characteristics of a black hole, such as the location and properties of the event horizon, innermost stable circular orbit, spatial curvature and time dilation are well-understood and are no longer controversial. We see excellent evidence of jets, accretion discs, and other features of accretion in many black hole systems. However, we do not yet have a full understanding of all these processes. Thankfully, we have lots of telescopes observing these systems and we are building a very good understanding of these structures, which we will begin to explore more in the next few modules. Crossing through the event horizon, we enter a region where it is predicted to be impos- sible for light (or anything else) to escape, so we have no observational confirmation of the mathematical theories describing the interior. However, as long as we don’t include quan- tum physics, it is possible to write out all the equations describing the inside of a black hole including the singularity. Quantum mechanics, the science of the very small, and how it applies to black holes is still a big open question. Since quantum mechanics is normally only important for describing 148 small objects, the introduction of the ideas of quantum physics mainly only affects tiny black holes. For the smallest black holes, we cannot be certain whether particles are inside or outside of the event horizon. This leads to the prediction of Hawking radiation which culminates in the evaporation of a black hole. A related problem is that of information: is it truly destroyed when it enters a black hole? This is a question many astronomical theorists are still pondering. However, we should remember that quantum mechanics is probably also important if we want to understand the singularity of all black holes, whether or not they are small. Addressing questions such as the properties of the singularity, information loss, and the final result of black hole evaporation will require physics that has not yet been uncovered: a quantum theory of gravity. There are only hints of what a quantum theory of gravity might look like, but the full details are beyond our present understanding. Now that we have explored some really big unknowns, it’s time to look more closely at the really cool phenomena that we can see when we point our telescopes at black holes! 149 8 Hunting for Black Holes 8.1 Introduction In order to find black holes, scientists need to choose the appropriate tools. Of course, we can’t yet travel to a black hole - but what we can do is collect light from the structures around a black hole. In order to choose the right tools, we need a better understanding of the light that is emitted from a black hole, and from the structures in its neighbourhood. The structures near a black hole can emit light in parts of the electromagnetic spectrum (or the extended rainbow) ranging from the lowest energy radio waves, to visible light, to high energy X-rays! We need to choose the right type of telescopes that will allow us to view the light emitted from black holes. Let’s begin by discussing optical telescopes. 8.2 Telescopes 8.2.1 Observing the Sky So, you want to find a black hole? Let’s get started by evaluating some of the tools available to us. Obviously, when astrophysicists study objects in the sky, they typically use telescopes! Telescopes allow us to collect light and produce images of the features near a black hole. But there are many different types of telescopes so it is important to choose the best one in order to be certain that what you are looking at is, in fact, a black hole! You may have seen a telescope like this one that we have at the University of Alberta’s observatory. This is a fairly standard type of telescope that gathers and focuses light using curved mirrors, called a “reflecting telescope”. Historically, it was much easier to make lenses, rather than mirrors, so early telescopes, like the one Galileo used to discover the moons of Jupiter, are called “refracting telescopes”. In modern times it is much easier to build large mirrors than it is to build large lenses, so the ENORMOUS telescopes used in astrophysical research are reflecting telescopes. When we talk about ground-based reflecting telescopes, the majority collect light in the optical spectrum, that is, light which humans can see with their eyes. For example, this Schmidt-Cassegrain telescope has two mirrors, a big, primary mirror at the back, and a secondary mirror at the front (it also has a corrective lens, but this is quite an expensive telescope!). The purpose of a telescope’s primary mirror is to collect as much light as possible from faint objects. The larger the diameter of the mirror, the more light is collected (think of it like a big bucket), and it becomes easier to see faint celestial bodies. But a larger mirror also means a larger, more expensive telescope, which is the tradeoff we have to pay in order to better resolve the features of distant objects. For example, if something looks like a blurry smudge through a small telescope, a bigger telescope will produce a clearer image, without even changing the magnification! Larger telescopes may also allow more magnification of the image, but magnification should be second to the mirror diameter. Dim and distant astrophysical objects, like nebulae and galaxies, are easily resolved by telescopes with low magnification, but require lots of collected light in order to be visible. 150 My family invested in a 4-inch Newtonian reflector when I was young (it was a present for my dad). This telescope encouraged my interest in astronomy, and it taught me an important lesson: if you are purchasing a telescope, do not purchase one that advertises its magnification! This is the sign of a poor quality telescope. Instead, a good quality telescope is described by the diameter of the primary mirror (or, for a refracting telescope, the diameter of its primary lens). For instance, our telescope at the University of Alberta has a HUGE 14-inch diameter mirror (sorry about the Imperial units, they are still common among telescope manufacturers). The 14-inch mirror can collect much more light than smaller telescopes, revealing dim structures in the night sky like the Ring Nebula in the constellation Lyra. Large ground-based research telescopes, like the two Gemini telescopes in Chile and Hawaii have mirrors that are 8 metres in diameter. But that’s a drop in the bucket compared to some on-going construction projects like the “Extremely Large Telescope”, or ELT, which will have the largest compound-mirror of any telescope in history! The ELT’s compound mirror will have an effective diameter of 39 metres, itself made up of a collection of smaller mirrors that can be aimed independently. Scientists call this independent motion “adaptive optics”, because the mirrors need to move in order to cancel out turbulence in Earth’s atmosphere. Often, they measure these disturbances using powerful lasers! Here’s the Subaru Telescope calibrating its optics: The Earth’s atmosphere is turbulent and the rapid motion of air pockets in the atmo- sphere smears out the light from stars and makes them appear blurry. What would be the best place to construct these massive telescopes be? Of course, you’d want to build them near the top of mountains! This is not to get them closer to the stars! By situating a tele- scope on the top of a mountain we decrease the amount of atmosphere between the telescope and the stars, which improves the seeing. “Seeing” is actually a technical term used by astronomers. If the atmosphere is calm and the images seen through the telescope are crisp and steady, we say “The seeing is good tonight!”. There is an obvious way to avoid the blurring effects of the Earth’s atmosphere: launch a telescope into space! Of course, you’re probably already familiar with the Hubble Space Telescope, but did you know about its successor, the James Webb Space Telescope? The telescopes that we’ve just looked at are all visible light telescopes, also called optical telescopes. These are telescopes that can detect light visible to our eyes (along with some neighbouring wavelengths of infrared and ultraviolet light, which are just beyond our eyeballs’ range of vision). This type of telescope is capable of viewing the stellar companions in black hole binary systems. However, since most of the energy emitted by a black hole is in parts of the electromagnetic spectrum that our eyes cannot detect, we will need to investigate other types of telescopes capable of detecting light that is invisible to our biological eyes. 8.2.2 Radio Telescopes Black hole jets emit radio waves, part of the spectrum emitted by hot plasma within the jet, so a radio telescope is an important tool for the black hole astronomers. Remember that radio waves are the very lowest energy and thus the longest wavelength part of the electromagnetic spectrum. Since radio waves are electromagnetic waves, or photons, they still travel from the black holes towards us at the speed of light. Radio waves have long 151 wavelengths that range from millimetres to metres in length, which is the reason why radio antennae have to be very long. You might be familiar with radio waves that you receive when listening to a radio station. If, for example, you were listening to a station at 102.9 on the dial, meaning you were capturing photons with frequencies of 102.9 megahertz, they would have a wavelength of approximately 2.92 meters. . . just longer than the span of my arms. Recall, green lasers have tiny wavelengths, measured around 532 nanometers. Radio telescopes, like the ones that make up the Very Large Array, which was featured in the movie Contact, are usually large dishes instead of antennae. Light waves from neigh- bouring radio telescopes in an array can be combined using a technique called interferometry which allows the whole group of telescopes to act as one large one. The effective size of a radio array is similar in size to the distance between the dishes! The largest single dish telescope, called FAST, the Five hundred meter Aperture Spherical Telescope, is 500 metres in diameter and located in China. If you’ve seen the James Bond movie Goldeneye, you might recognize the Arecibo radio telescope in Puerto Rico, where Bond defeats Trevelyan. Radio telescopes located at different parts of the Earth, as shown on this map, are being used as one Earth-sized radio telescope called “The Event Horizon Telescope”. It is observing Sagittarius A*, the supermassive black hole at the centre of our galaxy. We will discuss the Event Horizon Telescope’s observations in Module 10! 8.2.3 X-Ray Telescopes On the other end of the electromagnetic spectrum, at high energies, black hole accretion discs produce X-rays. So an X-ray telescope would be the best tool in our hunt for black holes. But there’s one problem. . . The Earth’s atmosphere absorbs X-rays! Actually, that’s a really good feature of our atmosphere. If X-rays could make it through the atmosphere to the ground, we would be constantly irradiated. X-rays have even more energy than ultraviolet light, the light that causes sunburns, so tanning under the X-ray light from a black hole would burn you to a crisp! Don’t worry though, the X-rays used in doctors’ and dentists’ offices are produced in safe quantities. Radiation therapy used to treat cancers are a good example of the damage x-rays can do, to the cancers, of course! Since the Earth’s atmosphere protects us from cosmic X-rays, an X-ray telescope needs to be launched above the atmosphere and into space. This diagram shows how much the Earth’s atmosphere blocks light with different wavelengths. Visible light and radio waves can penetrate through the Earth’s atmosphere. However, gamma-rays, X-rays, ultraviolet and infrared radiation are blocked by the atmosphere, so telescopes that can observe light at these wavelengths usually orbit the Earth. The Chandra X-ray telescope is an important black hole detecting telescope. It allows astronomers view X-ray images and spectra of black holes. Chandra is named after the Indian physicist Chandrasekhar who is famous for his theoretical work on black holes, neutron stars, and white dwarf stars. He also prefered to be called “Chandra”. Another orbiting X-ray telescope is called XMM-Newton. Although Chandra is a better telescope for creating detailed X-ray images, XMM-Newton is a better telescope for determining the wavelengths of the X-rays. A new telescope called Athena, with a planned launch date in 2028, will combine the best features of both of these two telescopes. 152 NuSTAR is another X-ray telescope that orbits the Earth, but the long length of the telescope allows astronomers to view much higher energy X-ray photons than the Chandra observatory. This allows NuSTAR to detect processes taking place very close to the black hole’s event horizon. A nice X-ray telescope called NICER was mounted on the International Space Station and is orbiting the Earth on the ISS! NICER is designed to accurately give the time of every X-ray photon that strikes it. This allows NICER to detect rapid changes in X-rays that are emitted by black holes and neutron stars. Astronomy that makes use of accurate photon timing is sometimes called “Time Domain Astronomy”. This is just a small selection of some of the telescopes that are used to study black holes! A completely different way to detect black holes is through gravitational radiation. We will learn more about gravitational radiation in Module 10. 8.2.4 Why are X-ray telescopes useful for studying black holes? (Interview with Dr. Daryl Haggard) Q: Why are X-ray telescopes useful for studying black holes? Dr. Haggard: And I was sort of a training junior astronomer at the time that the Chandra X-ray Observatory launched, but that was in 1999 and started to kind of giving us data as a scientific community in 2000. And so my first introduction to doing observational astronomy was through the X-ray lens of the Chandra X-ray Observatory, which still is a major workhorse in a major part of the research I do today. So the beautiful thing about X-rays is that they’re so high energy that they penetrate through a lot of dust and gas. So think about what you’re trying to do when you want to study the black hole at the center of the Milky Way galaxy. You’re in the Earth or sitting on the surface of the Earth, and you’re in this big spiral galaxy and you’re looking through the galaxy to try to see this object that’s at the center of the galaxy. So you’ve got to look through all of that junk, like all those stars, all that gas, all that dust. So the beautiful thing about X-rays is that they’re really high energy, they just penetrate right through all of that stuff not all of it, but most of it in the same way that they penetrate through your own flesh and they’re only stopped by your bones, right? So you know X-rays are really good at penetrating stuff. So X-rays are great for studying accretion disks of objects partly because they penetrate all this stuff, and also because those accretion disks I keep saying they’re really, really, really, really hot. So the temperatures that you need are millions to billions of Kelvin. That’s a really hot. I forget what the conversion is to centigrade, but anyway 10 to the 8 Kelvin is the temperature on average of hot X-ray emitting gas. And so that’s a great way to study these extremely hot environments near the supermassive black hole, plus it has this added advantage that doesn’t get absorbed easily by the material along your line of sight. 8.3 Chopping up Rainbows 8.3.1 Making Rainbows Have you ever wondered how we can determine so much about objects in the Universe? Like the Sun, how do we know that it’s made up of 74 percent hydrogen? Or that its surface temperature is almost 6000 Kelvin? Or that the Sun rotates once every 24.5 days? 153 The answers to all of these questions can, in some way, be tied to a field of science called spectroscopy, which is the study of light and its interactions with matter. When a scientist observes and records light, they produce a spectrum, or many spectra, which tell them how much of each colour, or wavelength of light, is produced by the objects they are studying. When an astronomer produces a spectrum they are spreading incoming light into a band of colours, just like you see when light from the Sun is spread into a rainbow. This is exactly Newton’s famous prism experiment, where he demonstrated that white light is actually a combination of every colour of the rainbow. An astronomer collecting spectral data is doing much the same thing (the act of separating light into its component colours), only modern equipment uses special instruments called diffraction gratings. Just like Newton’s prism, a diffraction grating separates light into its component colours. However, diffraction gratings use the physics of waves to spread light out, instead of refraction. An example of this type of interference is a compact disc - the rainbows produced are from the interference of light rays reflected by the lines etched on the CD’s surface. Spectrometers can be combined with telescope technology for use in astrophysics. Tele- scopes equipped with spectrometers can do what’s called imaging spectroscopy. By collecting a spectrum for each pixel of an image, scientists can distinguish what colours are produced in different locations on the same object. A spectral image is a type of image that allows us to do very special things. Even two regions which look the same to our eyes, if we examine their spectra overlaid on one another, there can be noticeable and important differences! From a spectral image, we can extract a wealth of information about different parts of astrophysical objects. Since each pixel corresponds to a different location in the sky, we can examine all the different parts of what we call extended sources. An “extended source” is something that doesn’t appear point-like to us, they extend over a region of the sky. Extended sources appear to be large either because they are big, or because they are very close to us. Some examples of extended sources are the Sun, a nebula, or even a galaxy. Spectroscopy is one of the most frequently used tools in an astronomer’s toolbox. Without spectroscopy, we would struggle to define important characteristics of the objects we view: speeds, temperatures, and compositions, to name a few. 8.3.2 Kirchhoff and the Continuum When matter in outer space interacts with light, we see the result as a spectrum. In astro- physics, we discuss three primary types of spectra, summarized by Kirchoff’s Three Spectral Laws. The first of Kirchoff’s laws describes the conditions for blackbody emission, while the other two laws deal with atomic emission and atomic absorption, which we’ll cover in the next lesson. Kirchhoff’s first law states that, “A luminous solid, liquid or gas emits light of all wave- lengths”. This law is a description of the type of light given off by blackbody emitters. You are probably familiar with at least one type of blackbody emitter: your kitchen stove! If it has electric coil burners like mine does, it will emit a deep red light when you’re cooking! This might be confusing, since Kirchhoff’s law says it should emit all wavelengths of light. Well, blackbodies do emit all wavelengths of light, just at vastly different intensities. For example, there are so few X-rays being emitted at this temperature, that we can totally 154 ignore that part of the spectrum. On the other end, the burner feels hot because, along with a lot of red light, the burner is also emitting plenty of infrared light. The spectra of thermal or blackbody emitters are continuous (like rainbows). You may remember from Module 02 that perfect blackbodies absorb all incoming light across all wavelengths and are completely non-reflective. The kind of light they emit will depend solely on their temperature. This temperature effect can be seen when examining a fire poker as it gets hotter and hotter. First there is “red hot”, then “orange” and “yellow hot”, and finally “white hot”. The surfaces of stars, which are close approximations to blackbodies, exhibit the same properties, though they can get even hotter, getting to “blue hot” for certain types of stars. As a thermal emitter gets hotter, its peak emission is shifted into more energetic portions of the electromagnetic spectrum. Black hole accretion disks can be so hot that their peak emission isn’t “blue hot”, it isn’t “ultraviolet hot”, it’s “x-ray hot”! When plotted, blackbody spectra are smooth, continuous curves that look like the hill of a rollercoaster. Let’s see how changing the temperature of a blackbody emitter changes its spectrum. Two laws govern the shape of a blackbody spectrum. The first, Stefan-Boltzmann Law, states that a hotter object emits more light at every wavelength. What does this mean for our plot? Well, higher temperature, larger curve! Additionally, the second law, Wien’s Law, states that hotter objects emit light with greater average energy. What does this mean for our plot? Well, a change in temperature will skew the peak of this graph. Hotter objects move the peak towards shorter wavelengths (higher energies) and lower temperatures will move the peak towards longer wavelengths (lower energies). Recall that frequency and wavelength are inversely related, so the same graph plotted for frequency instead of wavelength will look reversed! A higher temperature means that the peak emission of a hotter object is shifted “right”, towards higher frequency (which is more energetic). 8.3.3 Absorption and Emission Kirchhoff’s second and third laws are concerned with how emission and absorption spectra are produced. Producing these types of spectra relies on a process which affects individual atoms and molecules, called luminescence. Luminescence is the process that produces light when electrons drop from higher energy states within an atom or molecule, to lower energy states. Kirchoff’s second law states, “A low density, hot gas seen against a cooler background emits a bright line or emission line spectrum.” As we mentioned before, luminescence is responsible for this: when electrons transition from higher energy states to lower energy states, they emit light based on how far (or how many states) they drop. The reverse process can also happen, which is described by Kirchhoff’s third law, “A low density, cool gas in front of a hotter source of a continuous spectrum creates a dark line or absorption line spectrum.” So, if the right energy of light is shining through a low density gas like a nebula, electrons can steal energy from passing photons in order to climb the to higher energy states. By absorbing light, the low density, cool gas takes away portions of the continuous spectrum of the background emitter, leaving behind dark absorption lines. Energy level transitions are an effect of quantum mechanics. Electrons surrounding the 155 nucleus of an atom are able only to accept quanta, or in other words specific amounts of energy. When a passing photon or a collision between atoms in a gas has the right amount of energy, the electron will transition to a higher energy state. Transitioning to higher energy is much like climbing a ladder: you can only exist on the top of each rung. You can attempt to place your foot in between the rungs, but all that will result in is a banged up shin! Shortly after an electron transitions to higher energy, it spontaneously transitions back down to a lower energy state. The energy the electron had has to go somewhere, and what happens is the atom produces a photon which has the same energy as the difference between the higher energy state and the lower energy state. A large number of these downward transitions will produce a bright emission line in the spectrum, and a large number of upward transitions will produce a dark absorption line. It’s important to note now, how we can determine chemical composition using emission and absorption spectra. Each type of atom, or molecule, has a unique set of energy levels which produce a unique set of emission lines, meaning each type of atom or molecule can be characterized by a unique spectrum. This is a lot like the fingerprints on your hand. Every person has a different set of fingerprints, and every type of atom produces a different emission and absorption spectrum. A lot of work has been done by scientists to study the spectra of atoms and molecules in the lab, so we know very well what they look like. When we look at an object in the sky we can match parts of its spectrum to specific atoms and molecules to determine what it’s composed of! Here’s a great diagram that helps us explain Kirchhoff’s three laws. If a light source, like a star, is shining through empty space, we see the blackbody spectrum it produces as a continuous rainbow of colours, like in the leftmost image. If the star has a cooler outer atmosphere, or photosphere, then some of the cold atoms will absorb photons with specific colours, causing electrons to move to higher-energy states. This removes light from the continuous spectrum of the background source, creating an absorption spectrum, as shown in the centre image. If instead, light from a star strikes a nebula from the side, electrons will be excited and will subsequently fall to lower energy states. In doing so they will emit light with specific frequencies in all directions, including ours!.This produces an emission-like spectrum, like in the rightmost image. Keep in mind that we are generalizing the emission and absorption processes, and that both happen simultaneously, and are mediated by the temperature of an object. Both emission and absorption spectra are important in determining the chemical abun- dances in objects in space. We examine how bright emission lines are or how dark absorption features are, and how each is associated with a different element (or molecule) to determine how much and what is prevalent in all sorts of astrophysical objects. This is the first step in characterizing a black hole system, but advanced techniques can tell us even more... 8.4 Advanced Illumination 8.4.1 Blackbody Radiation To recap: we know that hot objects emit light by producing a continuous spectrum known as a blackbody spectrum. We know that low-density clouds of hot gas emit light at specific frequencies. And we know that low-density clouds of cold gas in between us and hot sources 156 absorb light at specific frequencies. Let’s have a look again at a black hole system with an accretion disk. Can you guess what kind of light we will see from the hot, dense, accretion disc? If you said that accretion discs emit blackbody radiation, you would be correct! However, since the accretion disc can become incredibly hot, that is, millions of kelvin, the peak wavelength, according to Wien’s law, is not in the blue part of the spectrum, it’s not even in the ultraviolet. . . Accretion discs are so hot, that many of them have a peak wavelength in the X-ray part of the spectrum! We also previously learned the the black hole itself emits a type of radiation called Hawking Radiation. Unfortunately, scientists have not yet measured any form of Hawking radiation, because the blackbody radiation from the accretion disc of black holes is PHE- NOMENALLY brighter than the light emitted through the Hawking process. In addition, the peak wavelength for Hawking radiation emitted by a solar mass black hole would be a couple hundred kilometers long, which means that no radio telescope on Earth could detect it. 8.4.2 Synchrotron Radiation We’ve previously mentioned the magnetic fields around black holes, but we haven’t delved to much into the physics of their interactions with the matter in the black hole neighbourhood. The reason is that the magnetic fields around black holes are poorly understood: how they form, how they are powered, and the effects they have on light generated in the black hole environment. We do suspect that magnetic fields are responsible for boosting the energy of particles in the vicinity of the black hole by a process known as synchrotron radiation. A magnetic field exerts a force on electrons and magnetic materials like iron filings. We can easily see the effect of a bar magnet’s magnetic field on iron filings by bringing iron filings close to a bar magnet. The magnetic field forces the iron filings to move, and they form a pattern that shows curves that stretch from the North and South magnetic poles. We call the lines that we see in this pattern “magnetic lines of force”. An electron that moves across a magnetic field line feels a force that pushes the electron into a circular path around the magnetic field line. An electron that has some amount of momentum also in the direction of the magnetic field line will experience the same effect, but appear to be moving on a spiral path circulating around a magnetic field line. Synchrotron radiation was first seen in laboratories that accelerate electrons. These laboratory accelerators are called synchrotrons, so when the radiation was first observed in 1947, it was called “synchrotron radiation”. One modern synchrotron is the Canadian Light Source located in Saskatoon, Saskatchewan, capable of producing some of the brightest light on Earth. Synchrotron radiation is produced when electrons travel in curved paths, and photons are emitted in the direction that the electron is travelling, like the headlights of a car going around a curve. I prefer to think about screams produced on rollercoasters, they are the loudest during the tightest curves! Since spiralling electrons are moving so quickly, they will naturally emit high energy photons due to the path’s curvature. When radiation is emitted in one direction we say that the radiation is “beamed”. This is very different from blackbody radiation, which is equally bright in all directions. Radiation 157 that is equally bright in all directions is called “isotropic”. Synchrotron radiation is generally associated with “beamed” emission in the black hole jets, which we’ll discuss shortly. The brightness of the light emitted at the different wavelengths depends on the strength of the magnetic field and the energy of the electrons. One beautiful example of synchrotron radiation can be seen in this true-colour visible light image of the Crab Nebula. The Crab nebula is a supernova remnant, which has, at its centre, a type of neutron star called a pulsar. The pulsar is one of the bright white sources near the centre of the image. The faint blue light in this image is created by synchrotron emission from electrons that have been accelerated by the neutron star’s strong magnetic field. Depending on how fast the electrons are accelerated by the pulsar’s magnetic field, synchrotron radiation can also produce light in the form of radio waves, X-rays, and even high energy gamma-rays. But because this image is only in visible light, we can’t see those forms of radiation. The red light in the image is an emission-line spectrum coming from excited hydrogen gas. This image of the radio galaxy Cygnus A shows a bright point of light and two bright regions stretching out from the point of light to about one hundred thousand light years in opposite directions. The red colour represents radio emission, showing that the central point of light is the location of a supermassive black hole, and the two bright regions are glowing due to electrons emitting synchrotron radiation. 8.4.3 Scattered Outta Compton Another process which modifies the electromagnetic radiation present in the environment surrounding a black hole is called Compton Scattering. When high-energy photons, like X- rays, scatter off of electrons in a low-density gas, the photons can lose energy in the collision. Just like billiard balls bouncing off of one another on a pool table, in Compton scattering, photons collide with electrons. After the collision, the electron and the photon move off in different directions, with the outgoing photon having lost energy to the electron. The photon which has lost energy travels away redshifted: with a longer wavelength than it started out with. Scattering of photons off of electrons also takes place for other types of light, not just X-ray photons like we’ve see in our description of Compton Scattering. For lower energy photons (like the light visible to the human eye) the wavelength of the photons are much larger and quantum effects are less important. For visible-light photons, the collision results in the light changing direction but it doesn’t result in a change of colour. A similar process to Compton Scattering, called Inverse Compton scattering is, well, exactly the inverse of Compton Scattering. Instead of a photon losing energy in a collision with an electron, Inverse Compton scattering describes a process which results in an increase in photon energy. When a photon collides with a high-energy electron, the electron gives some of its energy to the photon. This increases the energy of the photon, resulting in its wavelength being blueshifted! In order for Inverse Compton scattering to take place, a source of electrons which are moving close to the speed of light is required. Regions near a black hole such as the corona, are places where Inverse Compton scattering is likely to take place. There are many more methods by which light can be emitted, absorbed, and shifted. Some processes like, “Synchrotron Self-Compton emission” are a combination of the ones we 158 just covered: if electrons are spiralling around magnetic field lines at relativistic speeds and emitting synchrotron radiation, the photons emitted can then scatter off of the high-energy electrons and gain even more energy! We know our audience loves learning about changes to light, but we must now move on to determine where in a black hole system these sorts of processes dominate. 8.5 Black Hole Discs 8.5.1 Modeling the Disc So far in this module, we have explored HOW to look at black holes. Now we get to the fun stuff . . . now we get to actually see what astronomers see! Let’s examine the information received from black holes with various telescopes, and find out how astronomers use this information to learn more. Active black hole binaries, like our friend Cygnus X-1, are impressively bright in the X-rays. Black hole binaries can be about ten-to-the-power-of-ten times brighter than our Sun in X-rays . . . that is ten-billion times brighter! This means that in the X-ray part of the spectrum, black hole binaries really stand out! Stellar mass black holes, have two main features in the X-ray band. The first we associate with the hot accretion disc. The material in the discs around stellar mass black hole is travelling so quickly that it can reach temperatures of millions of degrees kelvin, which means that the peak of the discs emission is in the X-rays. Earlier in this module, we mentioned that the emission from a disc would be blackbody- like, but what does a disc look like in X-rays, and what can it tell us about the black hole? In this segment, we will only need the disc component (dotted line). Can we make it solid and in colour. In the next two segments/lessons we will add on the other two components. Don’t worry about the points, we don’t really need them at present, we are just focussing on the lines. The horizontal axis of this plot is photon frequency, which you may recall is inversely related to the wavelength of light. So as we move along this x-axis from left to right, the photon frequency will increase. This increase in frequency means that the wavelength of the light is getting shorter. The vertical, or y-axis is labeled relative brightness. The units of this axis are a little strange, and so we have left them off. We don’t really need to worry them at this point though. All we need to remember is that that things get brighter, the further up the scale you go. If we now look at the spectrum itself, we can see the feature, or shape that astronomers have associated with the disc. It looks kind of like a hill. We have a steady slope building from lower photon frequencies or longer wavelengths, up to a peak at higher photon frequencies or shorter wavelengths. This steady slope is known as the tail of the disc. After the peak, there is then a sharp drop off - or turnover, towards higher photon energies. Astronomers call this feature, a component of the spectrum, in the same way that the disc is only one part, or component, of the black hole binary system. What would cause this shape? Well, the emission from a disc is thought to be powered by blackbody radiation. 159 However, if we now plot our disc spectrum next to a blackbody spectrum, we can see that they don’t quite match up. How can this be? 8.5.2 Multi-coloured Disc Model The easiest way to think about this is to imagine that the disc is made up of a series of narrow rings. Each of these rings is emitting radiation at a different temperature. As we move inward through the disc, from ring to ring, each subsequent ring gets smaller and hotter. The hotter the ring, the more the peak of its spectrum become shifted towards higher photon frequencies. Now do you see why it’s not one blackbody spectrum? If we plot a spectrum for each of the rings and stack them all together. . . we get an overall shape that matches the observed spectrum of a disc. Astronomers call this the multi-coloured disc model, since each ring has a peak that is a different wavelength or colour from the next. The multi-coloured disc model produces a close fit to observational data from accretion discs. While this model is simple, it is widely accepted by the astronomical community. It has been accepted because the model provides a reasonable description of the accretion disc emission and can provide key information relating to the black hole. The peak temperature of the multi-coloured disc model tells you the peak temperature of the inner ring of the disc. If this last ring is at the innermost stable circular orbit around the black hole, then this temperature can give us information relating to the mass of the black hole. However, we should note that, although astronomers have used the multi-coloured disc model for decades, and probably will continue to do so for a while to come, this simple model has some issues. Why do we have these issues? Well, it is at this point I should remind you that astronomy is different from many other areas of science. Most scientists come up with theories and then try and test them, by looking at the objects the are investigating, by picking them up and turning them around, by exploring things from every angle. Quite often scientists can poke and prod the things they are interested in, maybe even pull them apart, and . . . hopefully . . . put them back together again. This is not the case in astronomy. All the objects astronomers investigate are out there in space. They are too far away for us to visit . . . yet, and so we only have the small amount of light they send in our direction to help us piece together their inner workings. So what are the problems with the multicoloured disc model? The first issue is that, in order for the temperature of the innermost ring to provide information on the mass of the black hole, we have to work on the assumption that that a black hole’s accretion disc extends all the way down to the ISCO. But is that a safe assumption? Even the nearest active black holes are so far away that we cannot directly image the inner disc. As such, we don’t know how safe it is to assume that the accretion disc of a black hole extends to the ISCO . . . . But we suspect that it does, when the disc spectrum dominates ! What other problems could there be with the multi coloured disc model? Secondly, this model does not take into account the spin of the black hole. As we saw in Module 6, if the 160 black hole is spinning, the ISCO can shrink from three times the Schwarzschild radius down to about half the Schwarzschild radius. So, for a given temperature or radius, the spin of a black hole can change our estimate of the black holes mass by up to factor of 6. In order to overcome this issue astronomers would need to also know the spin of the black hole. If the spin of the black hole is known, then astronomers will fold this into the mass calculations to improve our estimate of the black holes mass. In most cases, however, we don’t know how fast these objects are spinning. When we don’t know the spin, astronomers tend to assume zero spin, knowing that this will add an additional error onto the estimate of the black holes mass. The final issue with the multi-coloured disc model is that it is a very simple model. This model does not take into account all of the physics of the accretion disc. The material in the disc is an incredibly hot material that is thought to be some kind of plasma. Plasmas have been found to act like fluids. It is also thought that there are magnetic fields threaded throughout the disc. However, to account for these factors, you must perform very complex magnetohydrodynamical calculations. But these calculations usually require the a few spare days, weeks or even months, even with the help of a supercomputer. So observers tend to leave these calculations to the theorists, and continue to use their simple toy multi-coloured disc blackbody model when playing with data. 8.5.3 Across the Mass Scale As we mentioned earlier, black hole binaries, like our good friend Cygnus X-1, are bright in the X-ray portion of the electromagnetic spectrum. This is why astronomers often call these systems “X-ray binaries”. Their accretion disc spectra peak in the X-ray band of the electromagnetic spectrum, with the tail extending through the ultraviolet waveband, and even into visible wavelengths. How would this change if we were to move up the mass scale for black holes, if we were to consider intermediate or even supermassive black holes? More massive black holes are larger. Both the event horizon and the ISCO are found at greater distances from the black holes singularity. This would mean that the innermost ring of the accretion disc would be at a greater distance from the centre of the black hole, and so will be cooler. As cooler discs emit at lower photon frequencies, the peak of the disc spectrum for a supermassive black hole would be in the ultraviolet part of the spectrum. Given that the disc of supermassive black holes peaks in the ultraviolet, you might be surprised to learn that there are few observations of supermassive black holes made with ultraviolet telescopes. Most supermassive black holes are in galaxies that are moving away from us as the universe expands. Therefore, the ultraviolet photons emitted from their discs are significantly redshifted, and can be observed in the optical or even near-infrared parts of the spectrum. 8.6 Staring into the Hot Mess 8.6.1 Looking at the Corona In exploring the X-ray spectrum of black hole or X-ray binaries, such as Cygnus X-1, we noted that there are two major spectral components. The first of these two features, is related to 161 the disc spectrum of the X-ray binaries. Here we will explore the second major component. Astronomers have found that the best explanation for this component is a diffuse region of gas that emits via inverse Compton scattering. This region is often labeled the corona, but is also sometimes called the hot inner flow. Also as paragraph develops add in small side figure of black hole binary graphic from module 5, show system interacting, highlighting corona] The component of the spectrum stretches out over a large fraction of the X-ray band, slowly increasing in brightness as it extends to higher photon frequencies, or shorter and shorter wavelengths, before turning over and dropping off. Astronomers think that the emission we observe from this region is fed by the disc. Coronal photons are thought to originate from the accretion disc. This means that when photons escapes the disc in the direction of the corona, they can interact with the electrons in the corona via inverse Compton scattering, and gain energy while moving through the corona. This increase in photon energy can either be small, or large. This means that the photons escaping the corona can have a wide range of energies, which results in the long, slowly increasing slope observed in this spectrum. We should also note that electrons in the corona are slowed down slightly as they give energy up to passing photons. 8.6.2 Stepping up the Mass Scale The coronal spectrum component of a black hole spectrum is clearly visible in the X-ray band for X-ray binaries. Would this still be the case if the black hole was larger? We saw earlier in this module that the disc component of the spectrum is shifted to ultraviolet wavelengths for supermassive black holes. Is the same true for the coronal component? The photons from the inner disc feed the corona, where they gain energy via inverse Compton scattering. Therefore, the temperature or wavelength of the inner disc photons will impact the temperature and wavelength we observe in the coronal component of the spectrum. Just as the temperature of the disc can be tied to the mass of the black hole, the corona can also be impacted by the mass of the black hole. For supermassive black holes, the disc emission is seen to peak in the ultraviolet. As such, the coronal component is shifted to longer X-ray wavelengths. However, if the supermassive black hole resides in a galaxy that is moving away from us, the coronal emission will be redshifted into the optical and ultraviolet wavebands 8.6.3 Modeling the Mess Now that we know what the emission from the corona looks like, what can this tell us about the structures around the black hole? There are a couple of different theories for the location of the corona. The first of these is the the “lamppost model”, which suggests that the corona is a cloud of gas that sits at a certain height above, and below, the accretion disc, as it it were suspended on a pole or lamppost. The second is more of a sandwich model, in which the corona, and hot inner flow extends out above and below the accretion disc, as well as in towards the black hole. In this case, the disc and corona are in contact with one another. 162 Given the current data that is available from black holes, we know the interactions that are happening within the corona, but do we have lamppost hanging out above black hole accretion discs or are they wrapped up and bagels? As yet, this is still unclear, astronomers are working to answer this question. 8.7 Beam Me Up! 8.7.1 Emission from the jet Although structures like jets emit in the X-ray region of the spectrum, the X-ray band is dominated by the disc and corona, and so X-ray jet emission can be hard to detect. Jet emission more commonly associated with radio wavelengths. In this lesson, we will take a look at the spectrum of a black hole to see why this may be the case, and explore the mechanism that creates this emission, discovering what this can tell us about black hole systems. If we return to the plot we have been examining during this module, we can see the addition of this new radio component. From right to left, the new component begins in the same region of the plot as the spectrum from the disc and corona. The overlap in this region of the spectrum is due to all three components emitting a portion of their energy in the X-ray band. The new radio component then extends to longer wavelengths, or lower photon frequencies, peaking then tailing off into radio frequencies. 8.7.2 the Jet Spectrum The jet of the black hole is responsible for this new component of the spectrum. Jets are powered by synchrotron radiation, energizing photons through interactions with electrons that are trapped in circular orbits around the magnetic fields within the jet. While we don’t fully understand the mechanism that is used to launch the jet, astronomers suggest that the magnetic fields within the jet can be thought of as a tangled mess of spaghetti that has been stretched out in one direction. The stretched spaghetti causes the jet to transfer energy and angular momentum into the surrounding area. Particles energized within jets can extend out to incredibly large distances. Similar to the multi-colour disc model, accurate descriptions of jets require us to consider smaller slices in order to account for the different energies supplied by synchrotron emission. If the jet is cut up into narrow discs along its length (like slicing a banana into small circular pieces), we can plot the spectrum of synchrotron emission from each of these discs. As we move away from the central black hole, the number of particles decreases, along with the strength of the magnetic field. By adding the contributions from each disc of the jet, we can recreate the spectrum of the jet. 8.7.3 Types of Jets One mystery astronomers are trying to solve is why there appears to be two types of jets around black holes. The first type of jet is a continuous jet, and just like water spouting from the nozzle of a hose, the continuous jet is a continuous stream of particles constantly flowing outward along the path of the jet. 163 The second type of jet has multiple names, but is seen as clumps of particles being emitted out of the jet. Scientists call these clumps “burps, bullets or ejecta”. Just like the continuous jets, jet ejecta provide a route for the spread of energy and angular momentum into the area surrounding the black hole. As a result, jets can sometimes be a mechanism to feedback energy into the region surrounding the black hole system. 8.7.4 Orientation of the System We haven’t yet considered how the orientation of a black hole system affects our observations. Since jets tend to align with the spin axis of black holes, the direction of their spin determines which way the jet points, and how much light escapes. If we are viewing a black hole system with its accretion disc edge on, the jets appear to extend across the disc. If, on the other hand, we see the accretion disc from a top down perspective, the jet is pointed directly at us, and appear to be much brighter than jets viewed edge on. This is because the emission is beamed towards us, increasing the energy of the photons we receive. If the system is at some intermediate angle, we will see a blueshift in the jet angled towards us, while the jet angled away from us will be redshifted. We will also see a that the jet angled in our direction is brighter than its counterpart. We should note that the difference in brightness and colour is due to the beaming effect, it does not mean that one jet is actually more powerful than the other, or that they are emitting at different wavelengths. If the jet is offset from the spin axis of the black hole system, then we may also be able to detect a wobble in the jet. This has recently been seen in observations of a black hole binary known as V404 Cyg. A research team, including professors here at the University of Alberta, have been looking at V404 Cyg to investigate its jet in more detail. 8.7.5 What types of jets could a black hole have? (Interview with Dr. Gregory Sivakoff) Q: What types of jets could a black hole have? Dr. Sivakoff: So the material comes into the black hole, falls onto the black hole. If you have your accretion disk, roughly speaking, there is a jet perpendicular to the accretion disk. During the initial stages of the outburst, there is a jet which is like one continuous long burp. It’s just a jet coming out. Sort of maybe imagine your water hose, you turn your water hose on and your water is coming out, except this is probably more like a cone of material coming out. It’s a very, very narrow cone of material coming out in both directions, perpendicular to disc. So that’s what we call a compact steady jet. We call it a compact jet because it turns out that with all of our best instruments, we can’t really see the structure of this too well as we go down the jet. We can tell some of it but we can’t tell in detail. Play video starting at ::55 and follow transcript0:55 Now it turns out that black holes have another type of jet behavior. And they undergo a switch where all of a sudden they go from this, sort of nice little compact steady jet to a sudden burst where instead of the compact steady jet, that compact steady jet seems to be gone. And you get these blobs of material that go out with time. So these blobs of material are jet ejecta. In the Canadian press, we call these things black hole burps. The Americans call them black hole bullets. 164 Q: Why do we have burps and bullets? Dr. Sivakoff: One of the reasons why they’re called bullets, perhaps, in the American press, is that we talk about the trajectory that these things have. And so these objects go in a straight line. And that trajectory is called a ballistic trajectory. And so, it is not a small stretch to go from a ballistic trajectory to a black hole bullet. The particular press release officer at the University of Alberta was not a fan of that particular description and went with burps. I think it’s a wonderful analogy because we can talk about the feasting that the black hole is doing and relating it to their burps. Although they have different types of burps. Some are sort of your short staccato burps, and some of them are these long continuous burps. 8.8 Summary In this module we learned how astronomers observe black holes using the electromagnetic spectrum. To explore the nature of black hole systems, we require telescopes that allow us to image a black hole’s features in a large range of wavelengths, from radio to X-ray and beyond. Astronomers use the technique of spectroscopy to spread the light into an extended “rainbow” so that we can understand the radiative processes like synchrotron radiation and Compton scattering that take place near the black hole. Spectroscopy allows us to see the fantastic sights such as the jets, corona, and accretion disc around a black hole! Now that we know the components of a black hole spectrum, why do astronomers continue to look at black holes? The reason we keep looking, is that the spectra of black holes, like the black hole systems themselves, change in time. In the next module we will take this a step further to examine what changes we observe over time. We will explore black holes that are on a “strict diet” as well as those that are “extreme eaters” and learn how their meals change how they appear to us. 165 9 Our Eyes in the Skies 9.1 Introduction: Turn to face the strange Astronomers have been looking at black holes since 1963. We have learned a lot, but if we know all of this, why do astronomers keep asking for more money to build better telescopes? Why keep looking at black holes? What more can we learn about these systems, that we don’t already know? After more than 50 years, why haven’t astronomers cracked the mysteries of black holes? Black hole binaries have helped us navigate our way through much of this course. We have picked then apart and put them back together again, discovering what they contain and how they work in the process. The black holes that we have explored have a companion star that is sending mass towards the black hole. The material that is stripped from the star passes through a disc and corona to get to the black hole, unless it is thrown out via a jet. Looking at black hole binaries with visible light alone can be quite limiting, so we ex- panded our view to include radio, infrared, ultraviolet, and X-ray telescopes. These obser- vations help astronomers learn about the underlying physics of each of the components of the binary system. The reason for continued observations and studies is that a black hole’s properties change over time. The brightness of a black hole depends on what the black hole is eating and whether it is actively feeding at all. If the black hole is eating, is it a leisurely afternoon tea, or a crazy pie-eating competition? The rate at which black holes consume food can dramatically affect what we see through our multi-wavelength spectacles. 9.2 To Feed Or Not To Feed? 9.2.1 To Feed When is a black hole eating? And when is it taking a rest? The black hole candidates that we observe in the sky aren’t always dark, and they don’t always have bright accretion disks either. Black holes change in brightness and emission wavelengths, depending on how they are eating! A better question to ask is: how does a black hole actually reach its food? We know that if anything strays too close, the black hole will gobble it up. Almost by definition, black holes are like that! If a star, cloud of gas, salmon, spacecraft or astronaut venture too close to a black hole, they will be pulled inwards, into the accretion disc. The emissions from the accretion of material onto the black hole causes the black hole’s accretion disk to become visible across the electromagnetic spectrum, from long radio wavelengths to short X-rays and gamma rays. It’s only when a black hole is feeding, that astronomers are able to investigate the type of food that it’s feasting on. It could be sipping on a star, nibbling on a nebula, digesting dust, or even slurping up spaghettified space travellers. An actively accreting black hole can provide astronomers with the opportunity to test and gain a greater understanding of the underlying physics governing the processes which 166 feed black holes, including opportunities to put general relativity to the test in the strongest gravitational environments known to science. One of the effects astronomers measure is the doppler shifts that occur due to the rotation of the accretion disc. If we look at the disc up close, we can see that one side is moving towards us, while the other is moving away. As the disc spins, the light emitted from the side moving towards us is blue shifted, because the light’s wavelength is compressed, while the photons we receive from the side moving away from us are redshifted, their wavelengths have elongated. Observing accreting black holes is a major test in determining the veracity of competing scientific models. The corona, for example, is thought to be described by the lamppost model, or the sandwich model. My stomach already likes the sound of the sandwich model better! A black hole’s jet also comes in two flavours, and observations can help us understand the relationship between them. But first, we need to learn about black holes that aren’t eating! 9.2.2 Not To Feed A hungry black hole drifts through space, with nothing to eat. Since it isn’t emitting light, black holes that don’t have enough food are nearly impossible to detect, but scientists are building better tools all the time. However, the reason these drifting dark spheres are interesting is because there are a large number of them. Mathematically speaking, there are many more black holes out there than the ones we do see, simply because they can’t be seen! On the British television series, Red Dwarf, the computer Holly says, “The thing about a black hole, its main distinguishing feature is its black! And the thing about space, your basic space colour is its black! So how are you supposed to see them?” Well, it’s hard, but not impossible to find these isolated black holes. Since black holes have a strong gravitational field, they create large curvature in the spacetime around them. Curved regions in spacetime can act as lenses, which can reveal a black hole due to the warped background images of distant stars and galaxies. In fact, a hungry black hole gives astronomers the best evidence for the foods they dine on! For example, if a black hole is in a wide binary system, far from its companion star, we would distinguish between the star’s light and the light produced by the accretion disc. A wide binary system like this can tell us a lot about the black hole: its mass, and therefore its size, just to give you an example. But if the black hole is close enough to its companion, the material that it draws inward can get so hot and bright, that they become brighter than the parent star itself! The light being emitted from the star becomes difficult to distinguish from the light from the disc. In a sense, a binary system like this might look to astronomers the way a firefly dancing above a campfire might look to you from across a dark field! 9.3 Companion Stars 9.3.1 Black Hole X-ray Binaries Stellar mass black holes are most easily identified when they are accompanied by a companion star. The gravitational effect of the black hole on the companion star can help give us the location where the black hole might be hiding. 167 When the companion is a low mass star, such as in low mass X-ray binaries, the gas from the companion star flows to the black hole via the Roche Lobe overflow that we studied in an earlier module. Also recall that high mass stars tend to have larger outflows of material in the form of powerful “stellar winds”. In high mass X-ray binaries, the mass lost through winds ends up being accreted onto the black hole. We called this “wind fed accretion”. However, we should note that high mass stars can also feed black holes via Roche lobe overflow. 9.3.2 Classifying the Companion Star Stars are usually classified by observing their colour in visible light since this is the portion of the spectrum where they are usually brightest. We already learned that the accretion discs around black holes will also emit some amount of visible light. This means that if we want to view the companion star of a black hole we’ll have to wait until the black hole has finished “eating” a major meal so that its disc isn’t emitting light which would otherwise pollute our image. When astronomers want to classify a star, they look at it using different coloured filters to determine the star’s properties. Low mass stars with masses less than the Sun’s mass, are dim and have colours that range from yellow to orange to red. High mass stars are bright and are blue in colour. Since low mass stars are dim, they can be difficult to detect, so sometimes we have trouble detecting the companion in a Low Mass X-ray Binary, and the binary is classified based on its X-ray emission instead. The companion stars in High Mass X-ray Binary systems are usually easier to see since they are so bright, meaning that in many cases we can also obtain a detailed spectrum of the star. So it isn’t at all surprising that the first confirmed black hole, Cygnus X-1, has a bright blue, high mass companion star. However, accretion discs can also look very blue, bluer in fact than hot blue stars. This means that when the disc is bright it can be incredibly hard to work out what kind of star is feeding the compact object. 9.3.3 Examples of Low Mass X-ray Binaries X-ray images of black holes are not quite as impressive to look at as some of the other types of images we have seen in this course. They can be fairly featureless, with just a series of dots scattered a black section of the sky, except of course, when they suddenly change! The left image shows an X-ray image of the sky near our old friend Cygnus X-1. In the left image (taken before June 2015) we see four bright X-ray point sources. Cyg X-1 is the brightest X-ray source in Cygnus and a High Mass X-ray Binary. Cyg X-3 was the 3rd X-ray source discovered in Cygnus, and is a Low Mass X-ray Binary. At this moment it is unknown whether there is a neutron star or a black hole in Cyg X-3. 3A 1954+319 is also a Low Mass X-ray Binary, most likely harbouring a neutron star. Cyg A is a supermassive black hole, but it looks dim because it is in a galaxy far, far away ...while the other sources are in our own galaxy. A small “x” marks a spot where the Low Mass X-ray binary V404 Cyg suddenly became as bright as Cyg X-1 and Cyg X-2 in June 2015. V404 Cyg is close to eight thousand light years away from us. The companion star is a Type K star, which means that it’s orange in colour, and has a mass that is just forty 168 percent of the Sun’s mass 0.4MSun. The black hole has a mass that is seven times the Sun’s mass, so there is no danger that this is a neutron star masquerading as a black hole! In this movie, the black hole and its accretion disc are the blue-ish-white light in the centre of the image. The accretion disc suddenly erupted on June 26, 2015, emitting X-rays in all directions. These X-rays form a spherical wave front that expands and collides with dust clouds far away from the black hole. The red rings are X-rays that are reflected off of dust that lies between the black hole and the Earth. Although the wave front is a sphere, we see circles since the dust clouds are a series of surfaces between the black hole and the Earth. Another example of a Low Mass X-ray Binary is the X-ray source X9 in the globular star cluster named 47 Tuc. A globular cluster is a dense star cluster that can have as many as a million stars! Since the stars are closer to each other than in the part of the galaxy where we live, the stars can easily hook-up with other stars to form binary systems, through dynamical formation. So if you were to randomly choose a globular cluster to look at with an X-ray telescope, you’d have a good chance of finding an X-ray binary. The X-ray binary X9 is still classified as a “candidate” black hole since its mass is not yet measured. But the orbital period is very small, only 25 minutes and the companion star is most likely a white dwarf star. 9.3.4 Examples of High Mass X-ray Binaries The companion star to the black hole Cygnus X-1 is easily seen as the bright star in the very centre of this visible light image of the constellation Cygnus. Since this is a visible light image, we can’t see the accretion disc of the black hole. The red light is coming from glowing Hydrogen gas in a nearby star-forming region. Cygnus X-1’s companion star is named HDE 226868, but for obvious reasons we normally call it “Cygnus X-1’s companion star”. The companion is a Type O Supergiant Star that has a larger mass than the black hole! The companion star’s mass is 19 times larger than the Sun, while the black hole is 15 times the Sun’s mass. The two objects orbit their common centre of mass (which is closer to the companion) once every 5.6 days. Both high mass and low mass X-ray binaries are spotted scattered throughout galaxies. They are relatively easy to spot because the black holes have their dinner sitting right next to then in the binary system. What happens when we switch to other size scales? What are the alternative diets for supermassive black holes? 9.3.5 What is a black hole outburst? (Interview with Dr. Aarran Shaw) Q: What is a black hole outburst? Dr. Shaw: An outburst is a sudden increase in luminosity of a source. A source will go from bubbling around in what we call this low quiescent state, where you either don’t see it because it’s just not giving off enough x-rays, or optical light, or it’s existing at a lowish luminosity that we can’t see. Then, it will increase its luminosity or its brightness by a factor of 1000, and that’s what we call an outburst. Then it will start to decay back to this level that is known to be normal for this source. Q: Do you have any favourite black holes? 169 There are a few black holes that jump out or you are being extremely cool, I guess is the word. So two years ago or 2.5 years ago now, there was a very bright outburst of an X-ray binary called V404 Cyg, and this was the first time this source had been seen in outburst for 26 years. So outburst in 1989, is one of the most famous black holes in the galaxy. The great thing about outbursting in the modern era of astronomy is that, we have a huge array of facilities our disposal. So back in 1989, we looked at it with optical telescopes, and maybe an x-ray telescope. There weren’t many around, but now we have all sorts of facilities that we can coordinate via the power of the Internet essentially to look at this thing at the same time. So we have optical telescopes looking at it at the same time as radio telescopes, and all the x-ray observatories that are currently in orbit around the Earth. From the simultaneous astronomy, we can actually find out how the different wavelengths of light interact with each other, what the interplay is there. So that was fascinating. Somewhere that I was involved with, we found that we could measure the base of the jet. So there’s a jet that is coming out of the top of the black hole, and coming out of the poles of the black hole, and we can use this simultaneous x-ray and optical view to look at what the interplay is between the x-ray light and the optical light. We can see that the x-ray light, and the optical light are delayed by around 0.1 seconds. This gives us an idea of the size of the base of this jet. I thought that was absolutely fascinating, so that’s really cool stuff that happened very recently. Dr. Shaw’s webpage: https://greatbasinobservatory.org/dr-aarran-shaw 9.4 The Alternative Diets of Supermassive Black Holes 9.4.1 Snacking on Stars Much of our discussion around the feeding of black holes has revolved around stars, the snack of choice stellar mass black holes. Gourmet! Since stellar mass black holes make up the majority of binary pairs we associate with stellar companions, what do you think a supermassive black hole likes to snack on? Well, if you were thinking, “Surely they also eat stars. . . ” you’d be correct! But they are also voracious consumers of the gas and dust that happen to fall in too close. Some of the most spectacular phenomenon theorized to occur in the environment around supermassive black holes are called Tidal Disruption Events. These feeding frenzies are so violent, that the stars being consumed are completely torn apart! During a tidal disruption event, stars passing close to a supermassive black hole are disrupted by the strong tidal forces of the black hole’s gravitational field. The tidal forces deform the star into a long string of hot glowing gas, just like this artist’s impression of what the disrupted star might look like. Astronomers can detect these tidal disruption events as a sudden increase in the brightness of light. However, direct imaging of these disruptions is not yet possible, which is why we only show an artist’s’ drawing here. The gas from the destroyed star then accretes into the disc of the supermassive black hole, feeding its insatiable appetite. This effect was used in the Doctor Who episode “The Impossible Planet”, when the Scarlett system, which was home to the Peluchi, was drawn out into a red cloud before being accreted onto the black hole. 170 9.4.2 Slurping on Soup Supermassive black holes can feed on stars, but the gas that black holes devour doesn’t have to come from a disrupted star. Are there other options? What about clouds of dust and gas? The centres of galaxies can be messy places after all. Astronomers are puzzled by the sheer size of supermassive black holes. The mass at their interiors came from somewhere! Until recently, scientists thought that supermassive black holes feed on a steady diet of hot ionized gas from the halo of the galaxy. Similarly, supermassive black holes can feed on cold molecular gas clouds, sometimes the remnants of ancient supernova explosions. In this case, the food is more like a soup, with some lumpy noodles and vegetables in it! In this image the supermassive black hole is hidden away in the centre of the galaxy in the cluster Abell 2597, which is shown as blue light in this image. The red blobs show the locations of cold clouds of molecular carbon monoxide gas. When a lumpy carbon monoxide chunk is consumed by a black hole, it temporarily blocks the light emitted from the accretion disc and jets, like the black blob in the inset of this image. All of this food talk is giving me indigestion. . . 9.4.3 Black Hole Burps Supermassive black holes are not the most eloquent of dinner guests either. Not only do they DEMAND MORE, they don’t care whether you serve them a delicious apple pie, made from scratch, or a pile of rocks! While they do clean up after themselves, we can all agree that they need to work on their manners. No one is going to sit at a table with a supermassive black hole! We’ve also learned that the jets of a black hole are hot gases being accelerated into interstellar space. This is kinda like a black hole. . . you know. . . burping. While gas flows inwards towards the black hole, the material and energy can be thrown out in the form of powerful relativistic jets. On closer inspection, astronomers have also found that supermassive black holes can also eject gas from the region even closer to the event horizon. Here, we see the Whirlpool galaxy, a beautiful example of two colliding galaxies. The smaller galaxy, visible in the inset, has a bright X-ray source, visible in blue, indicating the presence of a supermassive black hole. The blue arcs are bands of hot, glowing gas, being ejected from the region around the black hole. This is evidence of black holes burping! Excuse me! We have a supermassive black hole at the centre of our galaxy, Sagittarius A*. I wonder what its favourite foods are? 9.5 The Special Case of Sgr A* 9.5.1 Where Do We Live? Our Sun resides in a little corner of the universe known as the Milky Way, so named because of the milky path it leaves across our night sky. . . The milky path we see at night is made up approximately a hundred billion stars, each in a lazy orbit around the galaxy that takes around two hundred million years to orbit the centre of our galaxy. 171 An observer on Earth looking towards the center of the galaxy would see the constellations of Scorpius, the scorpion, and Sagittarius, the archer, (I think it looks more like a teapot). Sagittariuis where our central supermassive black hole gets its name, Sagittarius A-star, shortened to SGR A* as a nickname. This constellation is easiest to observe in July and August, when it’s visible in the evening sky. Looking at the constellation of Sagittarius, we see that the bulge of the Milky Way is located at the westernmost end of the teapot. Since Earth hangs out in one of the outermost arms of the Milky Way, tilted with respect to the plane of the galaxy, it’s actually easier for observers in the southern hemisphere to enjoy the view of the galactic bulge. Notice that the more we zoom into the center of the galaxy, the more crowded the stellar environment. 9.5.2 What’s So Special? Even though we call Sagittarius A-star a supermassive black hole, it’s a lightweight contender in the supermassive category. One of the reasons SGR A* isn’t bigger, is that it isn’t presently eating very much at all! It sits in a region with many stars, but none have strayed within gravitational reach for it to grab. So what’s special about SGR A*? Well, it’s probably the only supermassive black hole within about 2.5 million light years (the next nearest known supermassive black hole is at the core of the Andromeda galaxy). Due to the proximity to Earth, SGR A* is the most studied supermassive black hole, and we are fortunate that it isn’t currently eating. This gives us an excellent opportunity to see the environment around it, without being blinded by the glare from the accretion disc. If SGR A* isn’t currently feeding, how do we even know it’s there? Well, scientists using ESO’s Very Large Telescope have actually imaged the core of the galaxy, and revealed the motion of stars and dusts clouds around the central supermassive black hole. From 2000 to 2011, this video shows the stars orbiting SGR A*, which reveal to astronomers just how massive the black hole is. In fact, one of the objects, called G2, isn’t a star at all, but a cloud of molecular dust. Researchers predicted that G2 would be captured by the gravity of the supermassive black hole, with a collision around 2014. Scientists predicted that it would be consumed by SGR A*, causing it to light in the X-ray spectrum. However, when G2 passed the central object, not much happened! It’s still a mystery why G2 wasn’t eaten by the black hole. 9.5.3 Why is Sag A* important? (Interview with Dr. Fiona Shaw) Q:Why is Sagittarius A* important? Dr. Harrison: Sagittarius A-star is the very heart of the Milky Way. And at the very heart of the Milky Way, there’s a supermassive black hole that’s 4 million times the mass of the sun. But what’s mysterious about this black hole is unlike many others that are quite active, there’s a lot of dust and gas around it, but it’s very quiet. It doesn’t emit much radiation. You can see it in the radio. And about once a day, you can see it flare in the X-ray. And what these flares can teach us is what’s going on in the region very close to this supermassive black hole. And eventually, we’re going to have a huge array of radio telescopes that may even be able to image the event horizon of this supermassive black hole. It will 172 be our first real view of the event horizon. We could tell, for example, if this black hole is spinning or not, which would tell us something about how it formed. And so by studying Sagittarius A star across the entire spectrum, because it’s so close, we can learn what’s going on with these dormant black holes. What is the center? It’s the closest supermassive black hole that we have to study. 9.6 The Hermits of the Black Hole Family 9.6.1 Gravitational Lensing Until now, we have only discussed black holes that are either in a binary star system, or located at the centre of a galaxy. Surely those aren’t the only ways to find black holes! Indeed, scientists think that there are isolated black holes, lurking within our galaxy. So, how do we detect them? We need to remember the presence of gas between the stars, called the Interstellar Medium. If an isolated black hole travels through a gas cloud, we would expect some of the gas to accrete onto the black hole. The accreting gas should emit X-rays, which could potentially be detected! Although astronomers detect lots of X-rays emitted from gas clouds, conclusive evidence for isolated black holes using this method hasn’t been detected. If there isn’t light being generated by an isolated black hole, is it still possible to detect them? Of course! Black holes change the gravitational field in their local environment, so light passing by is influenced by the gravitational field. Since black holes can strongly warp spacetime, they cause light to travel on curved paths. We see the light from behind the black hole, as being warped by gravity! This is called Gravitational Lensing. Light can’t escape if it enters the black hole’s event horizon. So if a black hole were to get in the way of our view of Orion, we might see something like this computer-generated image, with a dark circular region where we see no stars. The black circle corresponds to the event horizon of the black hole. The black hole is blocking our view, in the same way a cat blocks your view when it decides it’s time for attention! But if we look carefully at the area around the dark region, it looks as though many more stars have appeared around its outer edge. The strange appearance of these additional stars is an optical illusion. What we are actually seeing is multiple images of the stars that reside in the background . For instance, you should be able to see two images of the group of three “belt” stars on either side of the black hole. Similarly, you can see two images of the very bright star called Sirius. What we would see is more complicated than just a black circle in the sky! Instead light from the far away stars curves around the black hole and arrives in our eyes as though the star is located at many locations! The black hole’s gravity distorts the image of the background stars, giving away its presence! In reality there is no black hole close enough to give us a view like this distorted picture of Orion, but we can use the concept of gravitational lensing to identify some isolated black holes. 173 9.6.2 Wine Glass If this lensing effect is difficult to to wrap your head around, you probably feel like a photon that’s been bent in the spacetime around a black hole. Let’s have a look at a real world example of image distortion by your dining room glasses. Stemmed glassware are simple vessels that can be filled with a liquid. Since an empty glass has curved edges, it bends and warps light, just like a lens. When the glass passes in front of a picture, like this star map, a distorted view of the background becomes visible. Since the curvature of the glass allows multiple images of the same object, we see multiple images of some stars. Depending on where the stars are, they can also be distorted into rings. 9.6.3 Einstein Ring When gravity is weak, light travels along paths that we consider straight. Since gravity can distort space time, photons are forced to travel along geodesics, which are curved paths that bend around the black hole. To us, the resulting images have multiple views of the same object, along with great arcs of refracted light, similar to what we saw in the glasses. For this reason, when the gravity of a massive object curves the path of light, we call the massive object a gravitational lens. A gravitational lens can be a star like our Sun, a galaxy, or a black hole - the more massive, the better. When the faraway object, the nearby mass, and the Earth have perfect alignment, the image that we see is called an Einstein Ring. This diagram demonstrates the perfect align- ment between the Earth, a nearby galaxy, and a faraway star. Light from the star can be emitted on paths that go around the galaxy and reach the Earth in many different ways. The light can go over, or under, or beside the galaxy. The result is that the light in our telescope from the distant star looks like it comes from a ring in the sky that surrounds the nearby galaxy! Here is an example of an Einstein Ring captured in a photo taken by the Hubble Space Telescope. The fuzzy orange blob in the centre of the picture is a nearby galaxy. The blue circular halo is a distant galaxy lying behind the orange galaxy. The mass of the nearby orange galaxy warps spacetime, and the light from the faraway blue galaxy appears like a ring due to the gravitational lensing effect. The ring is not a perfect circle due to the fact that the orange galaxy’s mass is not located at one point. If the source of light and the lens mass do not line up perfectly, we see multiple copies of the same faraway galaxy. In this picture we see four lights that are all images of the same far away quasar that is behind the nearby galaxy. The nearby galaxy is the fuzzy light in the centre of the 4 quasars. This is called an Einstein cross since it resembles a cross. When astronomers view images of gravitational lensing they can compute the mass con- tained in the nearby galaxy. In many cases, the mass calculated from gravitational lensing is larger than the mass inferred from looking at the bright stars. This is one method used to show the existence of dark matter in galaxies! Although black holes could be a type of dark matter, most dark matter is not made of black holes. 174 9.6.4 Gravitational Micro-Lensing The size of an Einstein ring is related to the mass of the nearby object. For the images that we’ve shown, the nearby mass is a galaxy, and galaxies have gigantic masses, larger than a hundred billion Suns. The large mass gives a large deflection and a bigger-looking Einstein ring. If the lens mass is small, where small means similar to the Sun’s mass, and the distance is far from us, then the size of ring will be too small for a telescope to resolve. Instead of seeing a ring, the light from the far away star will appear brighter. This situation is called gravitational microlensing. The mass causing the lensing could be a dim star like a brown dwarf, or a black hole. Astronomers have been monitoring many stars in a nearby galaxy to look for the mi- crolensing brightening effect due to isolated black holes in our galaxy. If the black hole is travelling between the Earth and the far away stars, then they will appear brighter while the black hole is in front of them. This gives us a view of a star that appears brighter for a short time period. Microlensing by black holes is very rare, but it is seen occasionally. This image shows a gravitational lensing event that occurred in 1996. The upper left panel shows a dim star on April 28, and the lower left panel shows the same star on November 15. The image on November 15 is brighter, and a further analysis showed that the star appears brighter because a black hole with a mass around ten times larger than the Sun passed by. Only a handful of black holes have been found this way, since this is a really rare event. 9.6.5 What does microlensing have in common with exoplanet searches? (In- terview with Dr. Kelsey Hoffman) Q: What does microlensing have in common with exoplanet searches? Dr. Hoffman: So there is two different main aspects of it. One is looking for objects, whether they be planets or compact objects. The other part is understanding the structure inside these objects. So first of all, the looking for them is through the transit method, where you have your source, like your sun-like star and a planet that orbits in between you and the sun, or the star. When a planet does this though, you’ll see the light from the source decrease. Now when it comes to compact objects, and you may be replaced like an Earth-like planet with a white dwarf which has the same radius. When that is going in front because of the mass of that white dwarf, it actually acts as a lens and magnifies what you see. Then in this case, you get something instead of the light going down, it increases in what we call microlensing. Q: What is gravitational lensing? Dr. Hoffman: So in the gravitational lensing, the mass of the object allows the light will now bend around the object instead of being blocked by it. In these cases you get an amplification because the light bending will create different images of this and so then it increases the light that you see. 175 9.7 It’s on... Now what? 9.7.1 Varying X-rays Black holes will eat whenever there is something for them to grasp in their gravitational clutches. The rate they eat depends on how much material can be captured, which means that, just like people, black holes will eat at different rates, at different times. Sometimes, you are having an off day and you don’t really want anything to eat, or you may be fasting. Yet on highdays and holidays you may overindulge, eating way more than normal. Although black holes don’t really choose their food in the same way that we do, they can go through cycles of feast or famine . . . or anything in between. This change in food intake can result in a change in the strength of the different components of the spectrum. So what are these changes and what do they look like? If we take another look at our spectrum of an accreting stellar mass black hole, we can see the different components that we have learned about. We have an accretion disc, a corona and a jet. If we zoom in on this plot, to take a look at just the X-ray band we can see the two X-ray components more clearly. Here, we see that the disc is dominating the emission from the black hole. While the corona slopes gently upwards as we move to shorter wavelengths. Now, let’s look at some real data taken from our old friend Cygnus X-1 and see how this compares. If you recall, Cygnus X-1 is a black hole binary system that contains a stellar mass black hole, weighing in at about 15 times the mass of our Sun, and a hot blue companion star. If we look as the X-ray spectrum of this source, we can see the large bump at lower X-ray energies, which is explained by thermal emission from the accretion disc. The bump’s long tail, at it is sometimes referred to, extends to higher photon frequencies, or shorter wavelengths. The spectrum looks pretty similar to what we were expecting to see. So what was seen during a different observation of Cygnus X-1? Well, something quite different. The blue line has a steeper incline, increasing towards higher energies, that seems to peak in the same area of the plot that we would expect to see the corona. What is happening here? The truth is, scientists haven’t collected enough information to know for sure. This is the type of change that drives astronomers to continue investigating these sources, and for them to ask the questions about what could be causing these changes. 9.7.2 High and Low? Our view of black holes changes over time. Sometimes we can get more emission from the disc, while at other times we can get receive more photons from the corona. What physical mechanism could be driving this? When we look back at the image we built up during module 8, we saw the disc extending in towards the black hole with a corona that could be explained as either the lamppost model or the sandwich model with light coming from both the corona and the disc as material moves in towards the black hole. If our portion sizes our changing, how does that affect our view of the system 176 . . . assuming we could go for a visit? Well, let’s simplify things to start off with. We are going to stick with the sandwich model for the rest of this video . . . given our love of food. If we start off with a sandwich that is overly stuffed ... Here the filling, or our disc takes up a lot of our view, so you could say it’s dominating the picture. In fact it is stretching all the way down to the innermost stable circular orbit or ISCO. This is when the disc is at its brightest. It can be so bright in fact, that the emission from the accretion disc can be the brightest component in the optical band of the spectrum. When this is the case, we would not be able to see what kind of star the black hole is consuming. The bread of our sandwich is almost missing. The corona is so thin and wispy that we don’t really receive too many photons from it The sketch we see here of our spectrum matches quite closely with the spectrum of Cygnus X-1, shown in red. Astronomers call this the high state. The name is historical as it comes from the early days of X-ray astronomy. It’s known as the high state because it had a higher luminosity . . . it was the brighter option. Have we mentioned that astronomers like simple names . . . Following that line of thought, the next state I would like to mention is the low state . . . so named because it’s the fainter one. When black holes are in the low state, the sandwich is switched ... where the disc may feel thin, sort of stretched, like too little butter scraped over too much bread. In the low state the innermost part of the disc is not at the ISCO, it is found some distance away. We learned earlier that we can think of a disc as being made up of many rings. As we progress inwards through the disc, the temperature of each ring increases. This means that the highest temperature we detect from the disc, also known as its peak temperature, would come from the innermost ring of the disc. In the low state, the inner disc is further from the black hole. This means that the disc spectrum is cooler and so shifted to the left of the plot. We also find that the disc is fainter. So faint, in fact, that this is a great time to check out their companion stars. Here we a lot more bread for our sandwich, with the corona dominating the innermost regions around the black hole. With this increase in corona, we receive more photons, as it begins to dominate the X-ray spectrum. This is more akin to the spectrum of Cygnus X-1 shown here in blue. These are two striking different views of the same source, built with the same components. But what about the jet? It turns out that, although the disc and corona appear to be permanent features, the jet is not. The jet is strongly associated with the low state. Although astronomers don’t fully understand how jets are launched, they have found strong ties between the emission seen in X-rays and the radio emission. By combining information relating to the brightness of these electromagnetic bands, you can obtain estimates on the mass. 9.7.3 Very High In order to investigate the high and low states that had been seen in the emission from stellar mass black holes in binary systems, astronomers continues to make observations of these sources. Over time, they found that black holes could get even brighter, which seemed 177 to correlate with a change in the shape of the spectrum. This called for a new accretion state, known as the very high state. If we break this model apart to build a picture of this we find that there is both a lot of disc, possibly extending down to the ISCO, and a lot of corona. In this case we seem to have a more balanced sandwich, with both a good amount of filling and bread . . . tasty! During the very high state it’s possible to see a jet, although they are not always present. 9.7.4 Feeding Cycles We have explored three black hole brightness states. In addition, some astronomers are lobbying for a fourth intermediate state that seems to live somewhere between the high and low states we have already discussed. But how do these states relate to one another? By looking many times at multiple sources, astronomers have seen cycles emerging within many systems. Cycles start with black holes that are either off, or in the low state. When a feeding frenzy occurs, the black hole will rapidly brighten. This can take as little as hours to occur, and given the rapid rise, many times this can be missed by observers. After the low state, the black hole transitions to the high state, and possibly even the very high state . . . or beyond!!! Black holes can hang out in the high state for a while, depending on their food source, with some sources, like GRS 1915+105 seeming to stay this in state for decades. Towards the end of their dinner sitting, they slowly return back to the low state before fading away. These cycles are called outbursts. These outbursting cycles can also take place in super- massive black holes, but over much longer timescales, with outburst lasting centuries . . . to long for an astronomer to observe in their lifetime 9.8 Impact of Black Holes on Galaxies 9.8.1 Relative Scales of Supermassive Black Holes Supermassive black holes, with masses larger than one million solar masses MBH ∼ 106M (78) are found at the centres of most galaxies, but what effect do these black holes have on the galaxies themselves? Do the properties of the host galaxy affect its central black hole? These questions are important areas of active study in the field black hole feedback, a forefront area of black hole physics today. Let’s look again at SGR A*, the black hole at the centre of our own galaxy, the Milky Way. SGR A* has a mass of four million times the mass of the Sun. Although this seems impressively large, it is small when you compare it to the total mass of the entire Milky Way galaxy. SGR A* weighs 4 million times the mass of the Sun 4× 106M, but the Milky Way weighs in at close to 1 trillion solar masses ∼ 1012M . Therefore, our galaxy is one million times heavier than the black hole at the centre! SGR A* contributes only a tiny amount to the total mass of the galaxy. Similarly, the size of the black hole SGR A* is also tiny compared to the size of our galaxy. The radius of SGR A*’s event horizon extends about twelve million km, which is 178 itself almost thirty times larger than the distance between the Earth and the Moon . . . a reasonable size. However, the event horizon is much smaller than the distance between the Earth and the Sun, which is about 150 million km. The black hole’s accretion disc is larger, but it is still less than a light year across. In contrast, the Milky Way galaxy is huge, with a diameter of more than one hundred thousand light years! To give you a sense of this astronomical size scale ... if SGR A*’s accretion disc were shrunk to the size this penny, our galaxy would still be as big as the Earth! Supermassive black holes living at the centres of other galaxies also have relatively tiny masses and sizes compared to their host galaxies. For instance, the galaxy M87 has a supermassive black hole with a mass of three billion solar masses. The mass of its host galaxy, M87 is more than two trillion solar masses, which is more than a thousand times larger than its central black hole’s mass. This suggests that the overall impact of a supermassive black hole on its host galaxy should be quite small... The funny thing is . . . this doesn’t seem to be the case. In the 1990’s, astronomers measured the masses of many supermassive black holes, along with the mass of their host galaxies. They found a strong correlation between the black hole’s mass and the galaxy’s mass. Essentially, they found that larger mass galaxies have larger mass black holes at their centres. This suggests that galaxies and their central black holes have a symbiotic relationship, meaning that as one grows, so does the other. This opened up an interesting new area of study, that is still under investigation today. How can they impact each other so much? 9.8.2 Black Hole Feedback on the Host Galaxy The exchange of mass and energy between galaxies and their central black holes is called feedback. The galaxy supplies some of the gas and dust that accretes onto the central black hole, causing the black hole to slowly grow. As the gas accretes, thermal heating causes some of this gas to be ejected from the region near the black hole, in the form of high speed jets. These jets can extend thousands of light years, which heats up the gas and dust in the central parts of the galaxy. So, although the galaxy is feeding the black hole, the black hole throws some of the gas back at high speeds, that feed back into the galaxy! The outflowing gas can affect the galaxy in two ways. Firstly, the outflowing gas can push the interstellar gas outwards, clearing large regions of gas and dust. By clearing out areas of gas and dust, there is a lack of the material required to form new stars, and so this can slow down the birth rate for stars in this region. This may happen in the brightest active galactic nuclei, helping regulate the rate of star formation. The second effect of outflowing gas can, strangely, have the opposite effect. If the black hole’s jets interact with large gas clouds, they can compress the clouds, triggering the for- mation of new stars. When this happens, the black holes actually help galaxies form new stars. So the central supermassive black holes of galaxies can suppress or promote the formation of stars that could potentially have planets, and maybe even life! In other words, black holes aren’t just doom and gloom or death through spaghettification! 179 9.8.3 Cosmic Rays We think of astronomy as a science where cosmic objects like stars or galaxies are studied by observing the light that they emit. But there’s more to the Universe than just light! The Universe consists of uncountable elementary particles like electrons, protons, and neutrons, that make up all the elements, such as hydrogen, helium, and so on. These in turn make up all the molecules and matter! Amazingly, here on Earth we often detect particles that originate from cosmic sources, particles that come from distant parts of the Universe! These particles are called Cosmic Rays, and can carry tremendous amounts of energy! The highest energy cosmic ray ever measured, affectionately named the ‘OMG Particle’, carried 48 joules of energy. To put that into context, that’s about the same as a baseball pitched at a speed of 28 meters per second, but since all that energy was carried in an atomic nucleus, it was travelling only a whisper slower than the speed of light! When astronauts travel beyond the protection of Earth’s atmosphere, they report strange flashes of light visible, even when their eyes are closed. Although this phenomenon hasn’t been studied in detail, scientists think that the astronauts are observing flashes of light when a cosmic ray travels through their eyes! The high-energy particles emit Cherenkov radiation, as they pass through. This phenomenon is known as the Cosmic Ray Visual Phenomena. In order to understand the nature of Cosmic Rays, which can range from single electrons to the nuclei of heavy atoms, we must identify an energy source capable of accelerating them. Lower energy cosmic rays, like the ones we detect from the Sun, are accelerated by the Sun’s rapidly changing energetic magnetic fields. Higher energy cosmic rays originate from outside of our Solar System, and are a common occurrence. Some are accelerated during supernova explosions taking place in our galaxy, but where else can they originate? Most cosmic rays are charged particles like protons, or occasionally, a heavy atomic nucleus, like an atom of iron. But we also know about even smaller fundamental particles, like the neutrino. We’ve already seen that neutrinos, which are uncharged particles with tiny masses, can travel incredible distances without being stopped by interactions with other types of matter. This makes neutrinos almost impossible to detect, since it’s extremely rare for them to participate in particle interactions. This is why neutrino observatories are some of the most specialized detectors on the planet. Neutrino detectors regularly see neutrinos created in the Sun’s core. In 1987, the Kamiokande II detector in Japan detected neutrinos from the supernova explosion called SN 1987A. This detection led to the award of the Nobel Prize in Physics to Masatoshi Koshiba, the leader of the Kamiokande experiment, in 2002. More recently the Ice Cube detector, located in Antarctica detected some very high energy neutrinos that are unlikely to be emitted by a supernova. The source of Ice Cube’s highest energy neutrinos is still a mystery, but one possibility is that the neutrinos may have come from the jets of a supermassive black hole! Occasionally, detectors such as the Pierre Auger Cosmic Ray Observatory in Argentina, detect ultra-high energy particles like the OMG particle. The term Ultra-high energy refers to particles whose energies are millions of times more energetic than anything humans can create. What we mean by this is that the highest energy particles that humans have created are 40 million times less energetic than the highest ultra-high cosmic rays ever seen. Ultra-high energy cosmic rays can’t originate from supernova explosions, since scien- 180 tists have determined through simulations that a supernova can’t accelerate a particle to ultra-high energies. The origin of ultra-high energy cosmic rays is still deep mystery in astrophysics. We do know that the highest energy cosmic rays almost certainly originate from sources vast distances beyond our galaxy. Again, one plausible explanation is that the jets of supermassive black holes at the centres of distant galaxies are capable of accelerating particles to incredibly high speeds. If this is true, then our detectors here on Earth we are occasionally registering matter which escaped from the neighbourhood of a black hole! 9.8.4 What is the impact of a supermassive black hole on a galaxy? (Interview with Dr. Sarah Gallagher) Q: What is the impact of a supermassive black hole on a galaxy? Dr. Gallagher: When supermassive black holes are growing and they have the accre- tion disk, you have material that’s swirling in and all that light basically can drive a wind, and the way that works is the actual individual photons. The individual light particles have momentum that carry a punch, and so when they encounter gas they can blow these really energetic winds. The way they might impact their host galaxy is that depending on how much mass is in the wind so the mass could easily be a few times the mass of our sun, which is comparable to the amount of gas that’s falling into the black hole per year. So few times the mass of the Sun per year could be coming out as winds and a few times the mass of the Sun could be falling into the black hole causing the mass to grow. So, if you have that much gas that’s being blown out from the central part of the black hole system in the center of a galaxy and its going out at tens of thousands of kilometers per second that gas has a lot of energy and what it can do is it can impact the other gas in the galaxy and it can do. We’re not quite sure what it could do, but what it could do there’s sort of two things that it could do. One is that you have gas that’s coming out at very high velocity from the center of the galaxy near the black hole and it can basically plow into other gas in the galaxy. If it has enough energy it can actually plow it out of the entire galaxy, and what happens is if it blows all the gas out of the galaxy, it can completely shut off the star formation because that gas is the fuel of star formation. What it could also do if it’s not going as fast as it could plow into the gas in the galaxy and it can compress it, and if it can compress it it might actually trigger star formation and cause star formation to occur. So, either of those two is our plausible explanations, but what we really need to do is we need to learn more about the properties of the actual winds. What are the velocities? What are the geometries? How much mass is in them? Once we have a better handle of that then we’ll have a better idea of how it actually could ease impacting the galaxies that it lives in. But those are sort of the most extreme scenarios that could enhance star formation by causing the gas clouds to compress or it could just blow out all the gas and shut off star formation, and that’s why the event of being a quasar which is something that all massive galaxies go through, it’s kind of like being a teenager. It’s just like this phase of life. It’s kind of dramatic. It’s potentially unpleasant and you’re definitely different afterwards, but it’s not clear what the change ac- tually will be. So that’s something that I think is really fascinating and one reason why if you really want to understand how galaxies evolve, you have to understand how their black holes grow because this quasar phase could be really fundamentally important to shaping them over time. 181 Dr. Gallagher’s webpage: https://physics.uwo.ca/~sgalla4/index.html 9.9 Seeking Out The Elusive 9.9.1 Intermediate Mass Black Holes Stellar mass black holes have been studied for approximately the last 50 years. Supermassive black holes have been observed for almost as long. However, the identification and study of intermediate mass black holes is still in its infancy. The reason for the lack of observations of intermediate mass black holes is not due to lack of interest. On the contrary, intermediate mass black holes are thought to be the seeds of supermassive black holes, and so investigations into their nature could help unlock the mysteries of the formation of supermassive black holes. The study of intermediate mass black holes is such a new field, because members of this class of black holes have been incredibly elusive. In the rare instances when intermediate mass black hole were identified, their classification was contentious, and later many were disproved. At present there are only a handful of strong candidates for the intermediate mass black hole class. 9.9.2 HLX-1 SO 243-49 HLX-1 is the first candidate intermediate mass black hole we will discuss. This source first hit the headlines in 2009, when its discovery was announced in the journal Nature. This black hole resides in a star cluster that is in orbit around the galaxy ESO 243-49, a galaxy that is about 320 million light years away from us. The designation HLX means “Hyper-Luminous X-ray Source”, while the 1 follows the same convention as other black holes we have discussed in the course. This is the brightest X-ray object in its host galaxy, outside the galaxy’s core. Astronomers were quick to follow up on this source, making observations in multiple wavebands, from radio all the way through to X-rays and gamma rays. The temperature of an accretion disc around a black hole can give us an indication of the black hole’s mass. The more massive the black hole, the cooler the disc appears to be. We have also found that more massive black holes tend to be brighter, or more luminous, than their lower mass cousins when they are actively feeding. ESO 243-49 HLX-1 is approximately a thousand times brighter than of a stellar mass black hole at the same distance would be. The main difference observed between ESO 243-49 HLX-1, and say . . . our old friend Cygnus X-1 is that it appears brighter and its spectrum has been shifted towards the redder end of the spectrum. As astronomers watched over the next few years, we saw repeated brightening of this source. ESO 243-49 HLX-1 appeared to increase in brightness in a period of just over a year. The spectrum of this source was seen to change shape with brightness. The time-dependent emission of ESO 243-49 HLX-1 is very similar to the behaviour of X-ray binaries, leading astronomers to the belief that it is also in a binary system. This means that the black hole ESO 243-49 HLX-1 may be actively consuming a star. However 182 the cyclical nature of the outburst lead some to suggest that this black hole may not be slowly sipping on the surface of the star, but may instead be ripping off larger chunks before gobbling them up in feeding frenzies. This would trigger the outbursting behaviour observed that mirrors that of outbursts of X-ray binaries. What could cause these periods of glut and fasting? The periodic feeding frenzies can be explained if the binary system’s orbit is elliptical instead of circular. This means that the distance between the star and the black hole changes periodically during the orbit. When the star is at its furthest point from the black hole, no mass will be transferred from the star to the black hole. However, when the star is close to the black hole, the gravitational pull of the black hole will increase, allowing the black hole to attract large amounts of material from the star to the accretion disc. This famine and glut cycle has been suggested as a way to create the regular outbursts witnessed from ESO 243-49 HLX-1! Further studies have indicated that this black hole weighs in at about the thousand times the mass of our Sun, and that is resides in the centre of a dwarf galaxy that has been stripped and squashed during its interactions with ESO 243-49. These interactions may have triggered the formation of the star now feeding this black hole, and may have pushed it into the elliptical orbit that now feeds the beast. 9.9.3 M82 X-1 The cigar-shaped galaxy M82 is the home of another candidate intermediate mass black hole. M82 is a starburst galaxy where stars form at a rate much higher than in our galaxy. The X-ray inset shows a bright X-ray source named X-1 which is possibly an intermediate mass black hole. M82 X-1 lies outside of the galaxy’s core. It was first detected in 1978, when it piqued the interest of astronomers due to its strange brightness. The reason that this source was thought to be strangely bright, is that it emits light at a much greater rate than we would expect to see if the source was a stellar mass black hole. The luminosity of M82 X-1 could have been explained by a supermassive black hole, and yet supermassive black holes are found in galaxy centres, while M82 X-1 does not lie in the centre of its host galaxy. This left astronomers with two options. The first option is that this black hole lies in the intermediate mass black hole range. If this was the case, it could explain the luminosity of M82 X-1. However, an alternative theory emerged, the idea that this may be a stellar mass black hole undergoing a feeding frenzy. If it is a stellar mass black hole then it is accreting at previously unobserved gigantic rates. As more observations have been made of this source, the mass estimate for M82 X-1 has bounced between stellar mass and intermediate mass over the last 40 years. Today, evidence seems to be mounting in support of an intermediate mass black hole, with a mass estimated in the range range of 400 to 100,000 solar masses. It is interesting to note, however, that M82 X-1’s close neighbour M82 X-2 has recently shown just how extreme accretors can get! 183 In 2014, a team working with data from the NuSTAR, Chandra and Swift satellites discovered that the previously observed X-ray source M82 X-2 is in fact a class of neutrons star known as a pulsar. A pulsar is a rotating neutron star, that shoots jets of emission from its poles. These jets sweep across our field of view as the star rotates, like a lighthouse beam that sweeps over the rocks and out to sea. The striking thing about this neutron star, is that it is 100 times brighter than theory predicts that it could be. This means that M82 X-2 is one of the most extreme objects in our galaxy, and it could be used to help us understand how supermassive black holes could grow so quickly in the early universe. Enough about neutron stars though . . . let’s get back to intermediate mass black holes. 9.9.4 XJ 1417+52 The next candidate intermediate mass black hole is XJ 1417+52. This black hole has a mass that lies at the upper boundary for intermediate mass black holes, approaching the range of the supermassive black holes. Omega Centauri is a globular cluster found in the constellation of Centaurus. It is located at almost 16,000 light years away, and is the largest globular cluster in the Milky Way. This star cluster may be the home of our final example of a candidate intermediate mass black hole. Unlike the other candidate intermediate mass black holes, this black hole was not identified through its X-ray emission! Most likely this black hole is not actively feeding since there is no substantial food source nearby. The candidate intermediate mass black hole in Omega Centauri was inferred by observing the motions of stars using optical telescopes. The stars in a globular cluster move around randomly like a swarm of bees. The speeds of the stars are related to the total mass of the star cluster. From the observations of the stars’ motions the mass at the centre of the cluster can be inferred. Different observations suggest that a black hole with a mass in the range of 10 to 15 thousand solar masses lies at the centre of Omega Centauri. While there are a few other candidate intermediate mass black holes, their numbers are limited, and many are still contentious. Many candidate intermediate mass black holes have been ruled out upon further study. The search for the elusive intermediate mass black hole is an active area of research, a field that is definitely ongoing! 9.9.5 Can intermediate mass black holes be found in globular clusters? (Inter- view with Dr. Craig Heinke) Q: Can intermediate mass black holes be found in globular clusters? Dr. Heinke: If you smash several black holes together in the center of a globular cluster, they end up with an unusually high mass black hole. So when I say unusually, so black holes are thought to be born initially from stars at masses somewhere between three and a few dozens of solar masses, say 3-50 solar masses, give it a random number. Well, if you smash a bunch of these together, you might end up with something that’s hundreds or thousands of solar masses, and so that we call an intermediate mass black holes because they’re also the giant black holes at the centers of galaxies, the supermassive ones. So these intermediate 184 mass black holes, people have long thought that there might be some of these at the centers a globular clusters. But we haven’t found convincing evidence for them yet. There’s some very suggestive evidence in some globular clusters, but so far none of it has been confirmed by multiple lines of evidence. So that’s an area that future very large telescopes, such as the 30-meter telescope or the European Extremely Large Telescope. These might be able to give us stronger evidence as to whether these large black holes lurk at the centers. globular clusters. 9.10 Summary: The End Of The Rainbow Black holes don’t emit light, but that doesn’t mean that we can’t detect them! Just one of the ways is through their interactions with background light, gravitational lensing. When a black hole is in a binary system, the companion star can be seen. In some cases mass flows from the companion star into an accretion disk surrounding the black hole, which then emits light. The electromagnetic radiation emitted by the disk, jet, and corona varies in time as the rate of infalling gas changes. Supermassive Black holes in the centres of galaxies also give away their presence through the energy released by accretion. Jets of gas, energized by gravitational potential energy interact with the black hole’s host galaxy, affecting the formation of nearby stars in the galaxy. Supermassive black hole jets are thought to accelerate particles, such as protons and neutrinos! Perhaps this is the origin of cosmic rays? Isolated black holes most certainly exist in our galaxy, but are hard to detect. They can give away their location when they travel between a distant star and the Earth. The gravitational micro-lensing effect briefly causes the light from the distant start to appear brighter! Scientists have learned all these things through the observation of light, or electromagnetic radiation. But this isn’t the only tool they have at their disposal. . . Recent projects like LIGO and Virgo are built to be incredibly sensitive gravitational wave observatories. What are they looking for? They are looking for evidence of merging black holes which generate gravitational radiation. That’s right. . . waves in the fabric of spacetime. 185 10 Gravitational Telescopes 10.1 Introduction Our view of black holes has been limited to the electromagnetic radiation that the material falling into a black hole gives off. But this emitted light will have to travel on curved paths in order to reach our telescopes. As a result, our views of structures near black holes will be warped. New observations by the Event Horizon Telescope will be able to see the distortions of space caused by a black holes strong gravity! Visible light is not the only form of electromagnetic radiation, and in order to understand black holes we have had to observe them using X-ray, ultraviolet, and radio telescopes. Similarly, electromagnetic radiation is not the only type of radiation: when masses such as black holes move rapidly, gravitational radiation is emitted! Modern technological advances allow scientists to see with a new kind of eyes, ones that don’t see light, but see the stretching and squeezing of spacetime itself! By using gravity to probe into the heart of black holes, we can determine details that would otherwise be hidden behind the dust and gas of an accretion disk. Gravitational waves, predicted by Einstein, have recently become the hot new tool to generate scientific data on the behaviours of black holes and compact objects in extreme orbits. In order to see a gravitational wave, there are a couple of options: we can measure the influence of a passing wave on laser beams, or we could precisely time how the pulses of fast rotating pulsars change when a gravitational wave passes by. Let’s start with the simplest way that a black hole’s gravity affects our observations: gravitational lensing. 10.2 Gravitational Lensing 10.2.1 Gravitational Lensing by Neutron Stars When light enters glass, the direction that the light travels changes, or bends, in a process called refraction. If the surface of the glass is curved appropriately, we produce a lens, which can magnify, diffuse,or distort the light rays from objects in the background. Since gravity causes light to travel in curved spacetime, any object with a gravitational field appears to bend light as a gravitational lens. The examples that we looked at earlier, like this image of the Cheshire Cat galaxy group, were cases where the light was emitted far away from the closer lensing mass. In this image the arcs making the smile and outline of the face are images of galaxies lying far behind the eyes and nose galaxies whose mass is warping spacetime. However, if we consider ultra-dense stars with GARGANTUAN gravitational fields the light that they emit will also travel on curved paths. Neutron stars are examples of stars with strong enough gravity that they exhibit gravitational lensing. Their emitted light travels part of the way around the star before it can escape to be observed by telescopes. By observing how light is bent around neutron stars, we can understand the properties of these stars better, which will help us keep from confusing neutron stars with black holes in the future. If you look at my head, you can only see the front of my head, and you have no way to know what the back of my head looks like, unless I turn around. This is because in situations 186 with weak gravity like here on Earth, light travels on straight paths from my head to the camera. This animation shows the effect of a neutron star’s gravity on the light that it emits. In these cartoon animations, we show a dim star with one bright spot on it. The left animation shows an image of the star as though its gravity has no effect on the light that it emits. In this case we see the hot spot for only half of the rotation period and it is eclipsed when it is on the back side of the stars. Since the hotspot appears brighter than the surrounding star, a plot of the overall brightness would increase as the hotspot rotates into view, and decreases as it becomes hidden again. The right animation shows the same star, but now it includes the gravitational effects. The star’s strong gravity causes the light emitted by the star to travel on curved paths, so that parts of the star that would normally be hidden are distorted into view. The strong gravity allows us to see the bright spot all of the time, even when it is on the back side of the star. The star’s gravitational field distorts the circular spot’s image so that it looks like a thin curve, and it allows us to see around the spin poles. Since the hotspot is now visible throughout the rotation of the star, there is much less variation in the overall brightness with time, as shown on the graph below the animation. The star is rotating so fast that the equator is moving at thirty percent of the speed of light. When the spot is moving towards us, the light it emits is blueshifted, and when the spot is moving away from us it is redshifted by the Doppler effect. When light is Doppler shifted, there is another relativistic effect called “Doppler Boosting” which makes the blueshifted light appear brighter, and the redshifted light appear dimmer. If you pay attention to the brightness scale on the right, you’ll notice that the spot is brightest when it is moving towards us, and it’s dimmest when it is moving away from us. 10.2.2 Gravitational Lensing by Black Holes Black holes do not have a surface, so we can’t observe hot spots on rotating black holes in the same way we observe them on neutron stars. But we do know that there are light-emitting structures like accretion discs that orbit black holes. Images like this one, drawn by artists, represent accretion discs around black holes. In most pictures, the artist has drawn the accretion disk as though the black holes’ gravity does not warp spacetime or the paths that light rays follow, so we see the accretion disc as flat. But if a neutron star’s gravity can distort the paths of light, then a black hole’s can too! This vinyl record represents a crude model of an accretion disk, and the hole at the centre represents a black hole. If we ignore the black hole’s strong gravity then when you look at this disc, the light rays travel from the disc to the camera on straight lines, and the disk looks flat. A black hole with an event horizon with the same size as this hole, has a Schwarzschild radius of 3.6 mm, which corresponds to a mass that is about half of the planet Venus squished into this hole. The strong gravity of the black hole will distort your image of the back of the disc. You will still see the front of the disc since the light rays don’t have to pass by the black hole to get to the camera. The light emitted from the front of the disc will be slightly curved leading to some difficult to detect distortions. Light emitted by the back of the disc has to travel close to the black hole in order to get 187 to the camera. Light from the back is bent by gravity to travel up and over the black hole to the camera. As a result, this back curve will appear to look like an arch over the black hole. But the disc also has a bottom that emits light! Light emitted by the bottom can travel down and below the black hole to the camera. The bottom of the disc will look like a second arch below the black hole. A more accurate drawing of an accretion disc around a black hole will show the disc arching over and below the black hole. The movie Interstellar features a view of a supermassive black hole with an accretion disc. The director, Christopher Nolan, wanted to have a fairly realistic view of the disc that includes gravitational lensing, so he consulted physicist Kip Thorne, who recently shared the 2017 Nobel prize in physics. The resulting image produced for the movie is shown here. In this image we can see the front of the disc , the top of the back of the disc, and the bottom of the back of the disc. However, the black hole created for the movie required some simplifications. First, they wanted a black hole that would be safe for the astronauts to visit, so they did not include a jet. This suggests that Gargantua is not accreting enormous amounts of matter. They also chose a colder-than-usual accretion disk, which is only a few thousand degrees Kelvin so that it emits ultraviolet, visible, and infrared light, but practically no harmful X-rays. When we looked at the light from rotating neutron stars, we saw that the Doppler- boosting effect makes the blue-shifted side of the star appear brighter than the red-shifted side. The accretion disc orbits the black hole at high speeds. This side coming towards you is blueshifted and should appear brighter than the side moving away from you. The director was worried that people watching the movie might get confused if the Doppler-boosting effect were included, so they left it out from the rendering. This computer animation shows the results of a magnetohydrodynamic computer sim- ulation of an accretion disc around a black hole. The researchers are simulating realistic patterns in the disc which makes it easier to see the motion of the gas. At the start of the animation we are looking down on the disc, and then we move downwards so that we are viewing the black hole from just above the plane of the disc. The warping effect of gravitational lensing becomes more apparent as we look into the equatorial plane. When we are viewing the disc from the side, we can see that the side that is spinning towards us is brighter than the side spinning away from us. 10.2.3 The Photon Sphere Light near a rotating black hole can travel on curves that maintain a constant distance from the black hole, but trace out a spherical shell, like the photon in this animation. These paths are unstable, so a photon travelling on this path can easily be pushed outwards or inwards. If they are pushed inwards the photons can cross the event horizon and become lost inside the black hole. If the photons are pushed outwards, they can escape to be seen by a telescope. Recall the black hole’s innermost circular orbit for photons, or photon sphere. Photons within the photon sphere travel in spherical orbits, sometimes called the “ring of fire”. Within the ring of fire, there is a black region called the “black hole’s shadow”, a region where light can no longer escape to outside observers. 188 10.2.4 The Event Horizon Telescope The Event Horizon Telescope is a collection of radio telescopes scattered across many loca- tions over the Earth. The Event Horizon Telescope is observing Sagittarius A*, the black hole at the centre of our galaxy, and eventually it will become sensitive enough to detect the black hole’s shadow. Computer simulations suggest that SGR A* should look something like this video. This computer simulation shows an accretion disc orbiting around a black hole, and shows a ring of fire around a central, dark feature. However, the image that the Event Horizon Telescope is creating will not look as sharp, since the picture will be averaged over time, and there will be blurring due to light scattering off of interstellar gas and dust. This observation will be the most detailed image of the region directly outside of a black hole’s event horizon! 10.2.5 The Black Hole in M87 Our home galaxy, The Milky Way is located in a small group of about 30 galaxies known as “The Local Group” for lack of a better name. Groups of galaxies with more than 100s or 1000s of galaxies are known as clusters. The Virgo cluster of galaxies, located 65 million light years away from us, has over 2000 member galaxies. Deep in the heart of the Virgo Cluster of galaxies, an enormous black hole hides inside the largest galaxy in this cluster. This giant elliptical galaxy is named M87, and has a mass that is about 10 times larger than the Milky Way. Images of the galaxy M87 show a long jet that is thousands of light years long, reach- ing out from the central region of the galaxy. This jet led astronomers to suspect that a supermassive black hole lies at the centre of the galaxy. In 1994, the new Hubble Space Telescope, was pointed at the nucleus of M87. At the core of the galaxy, the telescope was able to resolve a disk of gas, suggesting orbital motion around the galaxy’s centre. Hubble’s spectrograph was able to detect Doppler-shifted emission lines on opposite sides of the disk at distances of 60 light-years from the galaxy’s centre. The redshifted and blueshifted spectral lines allowed the measurement of the gas’ velocity which was then used with Kepler’s laws to find the mass at the centre. The resulting mass of 2.5 billion solar masses is large enough that the only sensible conclusion is that a supermassive black hole lies at the centre of this galaxy. Back at this time, I was a student studying black holes, so it was really exciting to hear about this observational confirmation of the existence of supermassive black holes! The gigantic mass of the black hole, and the relatively close distance, on astronomical scales, led the Event Horizon Telescope team of scientists to choose the black hole in M87 as their first target. In April 2017 they pointed radio telescopes located on multiple locations on the Earth at M87. The Black hole at the centre of the galaxy M87 is sometimes called M87-star. More recently, it has been renamed Po¯wehi “Poh-veh-hee” which is a Hawai’ian name meaning “the adorned fathomless dark creation.” After 2 years of data analysis, they announced their results and images in April 2019. The orange “doughnut” image swept the internet by storm. But what is this image actually showing us? 189 The Event Horizon Telescope observes radio waves, so the orange colour is a false-colour image that maps the brightest radio waves to yellow and dimmer radio waves to dark orange. The astronomers could have shown this picture in black and white, but the use of colour does help bring out details to our eyes. This set of images shows how the image of the black hole changes day by day. The general shape of a Photon Ring that is bright on one side and dim on the other, surrounding a black region known as the black hole’s shadow stays constant. But you can see that the ring develops little bumps that move around a bit on timescales of days. The ring shape comes about as photons travel around the black hole on curved paths. A bright region on the far side of the black hole will emit light that travels around the black hole to the Earth. Since the light can travel over, under, or beside the black hole we see a ring of light surrounding a dark region. The outer edge of the black hole’s shadow has a radius Rshadow = √ 27 GM c2 ' 2.6× 2GM c2 (79) which is about 2.6 times the event horizon radius of a Schwarzschild blackhole. Measuring the size of the image allows an independent measurement of the black hole’s mass! The Doppler boost effect is responsible for the bright and dim regions seen in the image. Regions of gas that are moving towards us are blueshifted and appear brighter, while regions that are moving away are redshifted and appear dimmer to us. The astronomers computed thousands of model accretion discs, traced the light rays around the black hole, and produced images similar to this one. Then the blurring effect of the gas between the black hole and the Earth was added to the simulated image. The simulated blurred image is then compared to the observed image in order to validate the physical interpretation of the image. This is a really remarkable observation and analysis! The next black hole that the Event Horizon Telescope Collaboration plans to observe is Sagittarius A*, the supermassive black hole at the centre of the Milky Way. You might think that SGR A* would be easier to observe than M87*. The centre of the Milky Way is 1000 times closer to the Earth than the galaxy M87, so the black hole’s shadow should look larger. However, the mass of SGR A* is about 1000 times smaller than M87*’s mass. This makes the event horizon and the size of the shadow smaller. The overall effect is that the shadow will appear to have a similar size! The real problem is that SGR A* is known to vary rapidly on time scales shorter than a day. This will make the image jump around and blur it. So the observation of our own black hole’s shadow will be even more difficult than the observation of M87*’s. 10.3 Gravitational Radiation 10.3.1 Gravity at the Speed of Light One of the foundational principles of special relativity is that light always travels at the same constant speed. The speed of light speed is fast, but it still takes 8.3 minutes to travel the enormous distance from the Sun to the Earth. This means that if the Sun were to suddenly change brightness, there would be a delay of 8.3 minutes before we would see the change here on Earth. 190 Suppose that powerful aliens with advanced technology managed to change the Sun’s gravity, maybe by removing a big blob of hot gas. Removing the gas also causes the Sun to dim at the same time. If we calculate the gravitational attraction between the Sun and the Earth, Newton’s equations have no time dependence. If the Sun’s mass suddenly changes, as it would if aliens removed a large portion of gas, Newton would have predicted that Earth would feel a different force due to gravity instantaneously, with no delay. If gravity changes instantaneously, as Newton predicted, it would still take photons emitted from the Sun 8.3 minutes to reach Earth, revealing the alien actions as the Sun becomes dimmer. This means that we could receive information about changes to the Sun at a speed faster than light by using gravity! This instantaneous transmission of information is sometimes called ”Action At A Distance”. Einstein realized that Newton’s theory of gravity was wrong, because it implies that Action at a Distance takes place, and that gravity could transmit information faster than the speed of light. Although there were no experiments Einstein could conduct that showed that Action at a Distance is incorrect, Einstein knew that it would imply that there could be ways to transmit information at speeds faster than light, which would contradict the principles of special relativity. Einstein theorized that gravity could be made compatible with special relativity if changes in gravitational fields are transmitted by “gravitational waves” that obey the speed of light limit in the Universe. Gravitational waves, also called “gravitational radiation”, are an important part of Einstein’s “general theory of relativity”. Let’s return to the aliens who are changing the Sun’s mass and brightness. Einstein’s theory predicts that it takes 8.3 minutes for the gravitational force felt by the Earth to change. So changes in gravity travel at the same speed as light! 10.3.2 Electromagnetic Waves The relationship between the force of gravity and gravitational radiation is similar to the relation between electrostatic forces and the emission of electromagnetic radiation, or light!. An electron, or a proton are just two examples of charged particles, which create an electro- static field. Electrostatic fields can be attractive, if the force is between opposite charges, or repulsive, if the force is between similar charges. By rubbing a balloon against your head you can cause negative charges (electrons) from your hair to migrate to the balloon, leaving your head positively charged. If you then hold the negatively charged balloon near your positively charged head, your hairs will stand up, following electrostatic field lines. The word “static” means that the system doesn’t change with time. A positive charge (like some hair) will feel an attractive electrostatic force that will feel a force towards the negatively charged balloon. The electric field of the balloon influences nearby charged objects, but in a static system there is no electromagnetic radiation or light emitted in this situation. If we want to produce electromagnetic radiation, we need to create periodic changes in the static charges. This can be accomplished by changing the position of a charged balloon, say by waving it back and forth rapidly. Positively charged hair will be attracted first in one direction, then the other, and so on. This is a situation which produces a time changing electric (and magnetic!) field that creates a disturbance called electromagnetic radiation, also known as light. 191 For example, if the balloon is moved back and forth once per second, it generates a changing electric field that moves away from the source at the speed of light. One complete cycle of the balloon corresponds to one cycle of the light wave. So if the balloon moves at one hertz, it produces a photon with a wavelength of equal to the distance light travels in one second! That’s 299,792 kilometers, a photon at the very long end of the wavelength spectrum, in the radio frequencies! This electromagnetic wave travels at the speed of light, perpendicular to the direction that the balloon is being waved, and information about the changing balloon position reaches your hair after a time equal time = d/c (80) to the distance divided by the speed of light. When the wave passes by a charged particle, the particle will feel a time-changing force that will cause it to oscillate back and forth in a direction perpendicular to the direction the wave travels in. Normally we don’t use balloons to create electromagnetic waves. Instead we might have an alternating current travel through a dipole antenna to create radio waves, a long wave- length form of light. The important thing to know is that light is only emitted when there is are time-changing electric charges or magnets. 10.3.3 Surfing Spacetime A star or planet has a gravitational field that doesn’t change with time, and unlike electro- static forces, gravity is only attractive. A small mass placed near a heavy planet will feel an attractive gravitational force that will pull it towards the surface. If there is no motion of the masses, there will be no gravitational radiation emitted. Just like the electrostatic example, in order to excite gravitational waves, we are going to need a gravitational field that changes with time! When gravitational waves are created the move away from their source at the speed of light, and as the wave moves it distorts spacetime by stretching and compressing spatial dimensions periodically in the two directions perpendicular to the direction that the wave travels. These waves are called, transverse waves, and can be understood in a simplified sense by the analogy of a transverse wave on this rope that Curtis and Ross are playing with. In this video, we see Ross wave the rope up and down, which is sort of like a gravitational wave source. The wave travels along the rope horizontally from left to right and then from right to left. At each point between Ross and Curtis, the rope is forced to move up and down at the same frequency as Ross’ hand. In a transverse wave, the movement of the medium is in a different direction than the direction that the wave travels in. For example, if a gravity wave passes through me from back to front, I would first appear to get stretched vertically, and then horizontally as the gravity wave passed me by! We’ve exaggerated the effects of a passing gravity wave by a lot, but the principle is correct. Suppose that a gravitational wave travels from the ceiling to the floor through my body. Focus on my arms. The effect of the gravitational wave will be to stretch one arm outwards and compress the other arm inwards. Then the second part of the wave’s periodic motion 192 will cause the opposite changes in each arm. Periodic motions of my arms will then occur when the wave passes through my body! If a gravitational wave were to pass by your ear, it would cause your eardrum to start vibrating so you could in principle hear a gravitational wave if you could had sensitive enough ears! For this reason, scientists often convert gravitational waves into equivalent sound waves, which is how we got the CHIRP sound of two merging black holes. In reality the waves are very weak and the change in arm lengths are microscopic. In the Star Trek Next Generation episode called “Hero Worship”, the Enterprise is violently rocked by gravitational waves. This is rather difficult to understand since in reality the waves are very weak, and the Enterprise would have to be right next to a ridiculously strong explosive event in order to get such a strong rocking event. Scientists on Earth have been working hard for decades to build detectors sensitive enough to detect gravitational waves and directly detected them for the first time in 2015. 10.4 Binaries and Gravitational Waves 10.4.1 Binary Inspiral When two stars are in orbit around one another in a binary system, their positions change periodically in time. Since the gravitational force between an astronomer and the stars depends on the distance and direction to the stars, an observer will feel a time-changing gravitational force as the two stars orbit one another. As we watch the stars in orbit, we will see the brightness of the system change as they pass in front of one another, as they continue their dance through the universe. To preserve causality, there has to be a delay in the change in gravitational force that is synchronized with the brightness changes as the locations of the stars change. In other words, the two star’s changing positions cause gravitational waves to be emitted by the binary system! These gravitational waves will distort spacetime and cause objects far away to be squeezed and stretched periodically. The energy for these distortions is carried away from the binary system by a wave. This means that gravitational waves carry energy away from the binary, and the binary loses orbital energy. 10.4.2 Following Kepler’s Laws The stars in a binary system are moving in ellipses, in accordance with Kepler’s laws. One component of the total energy is their kinetic energy, which is kept in balance with their gravitational potential energy. By summing both the kinetic and gravitational potential, we obtain the total energy for the system. This total energy is large when the stars are far from one another, and smaller when the stars are closer to each other. This effect is caused by the emission of gravitational radiation: outgoing waves carry energy away from the binary, causing the stars to fall inwards, orbiting closer and closer to each other, shrinking the orbit of the system, and, instead forming an inwards spiral. The process of gravitational wave emission, and the inward spiral, is incredibly slow. Therefore, at any moment of time we can rearrange Kepler’s third law to relate the distance between the stars, “a”, to the time it takes the stars to make one full circle, “P”. 193 P = √ a3 M1 +M2 (81) Here we see that P, the orbital period is equal to the square root of a-cubed divided by the sum of the masses of the two stars. As the stars slowly spiral inwards, their masses do not change. Only the distance between the star change. In Kepler’s third law, when “a” gets smaller, so does the orbital period, P. The stars take less time to orbit when they are close to one another. 10.4.3 Loss of Energy The result of the loss of energy through gravitational radiation causes binary star systems to slowly spiral inwards. As this process continues, the stars’ orbits speed up as they come closer together, appearing to spin more rapidly around one another. This slow death spiral will eventually cause the two stars to collide, merging into one big star. This sounds dangerous, and how common would such a phenomena be? This death spiral only affects binary stars that are both tiny and extremely close together. In order for two stars in a binary to spiral in and merge within the present age of the Universe, so about thirteen billion years, the distance between the stars has to be no bigger than the diameter of our Sun! That’s pretty close. . . Typical main sequence stars that are found in binary pairs in our Galaxy, orbit much farther away from each other than one astronomical unit, and if you recall, one astronomical unit is the distance between the Sun and the Earth. So, these typical stellar binaries are in no danger of undergoing inward death spirals. As an example, the brightest star that we can see Earth’s night sky is called Sirius, the dog star. It is found in the constellation Canis Major, which is named after the mythical dog that guards Europa from an abduction by Jupiter. Sirius is actually a binary system composed of two stars that orbit each other every 50 years. The distance between the two stars is similar to the distance between our Sun and the planet Neptune. Gravitational radiation emitted by the Sirius binary system is so weak that, currently, it is impossible for us Earthlings to detect! The inspiral is so mind-bogglingly slow that it would take 10 Zettayears, or 10 to the power of 22 years before the two stars merge! This would be long after both stars burn through their nuclear fuel anyway. In fact, the inspiral if the Sirius binary system takes such a long time that we can ignore the effect of gravitational radiation on the system, and all other binaries of main sequence stars. The only stars that are small enough to allow orbits to be close enough together to emit gravitational waves at an observable rates are white dwarf stars, neutron stars, and black holes! 10.4.4 How do binary pairs of black holes form? (Interview with Dr. Tyrone Woods) How do binary pairs of black holes form? Dr. Woods: So I think something that is fascinating everybody in the Black Hole com- munity right now is the recent discovery by LIGO, demonstrating the existence of binary 194 pairs of black holes. How these objects actually form still remains a mystery, although we have a lot of ideas. Generally, one major picture is that a pair of stars born together even- tually die together, one forming a black hole and then the other. Another idea is that to get around some of the complicating factors in terms of trying to actually make a binary pair of black holes together starting from scratch is to form these objects initially born separately in a globular cluster, a very dense system of stars. And then many of which orbit our own Milky Way. These are sort of the building blocks of galaxies. These systems move around within their den system and interact far more often than they would in a lower density stellar environment like our Milky Way. And they can swap partners and maybe in this way bring together two black holes in a close enough orbit that they eventually merge but we don’t know which is the right answer. Dr. Woods’ webpage: https://www.tewoods-astro.com/ 10.4.5 Inspiral and merger of Neutron Stars The first detected binary star system demonstrating the emission of gravitational waves, was a binary composed of two inspiraling neutron stars. This system was first observed by Russell Hulse and Joseph Taylor, using the Arecibo Radio Telescope in 1974. It contains a pulsar and a neutron star that orbit one another once every 8 hours. Therefore they are separated by a mere 3 light-seconds, which is similar to the diameter of the Sun. Together, the two neutron stars slowly spiralling in towards each other, allowing Hulse and Taylor to detect the change by measuring the orbital period of the pair over time. They found that the orbit of the two stars slows by approximately one tenth of a millisecond every year. This tiny change in the orbital period agrees with predictions of gravitational radiation coming from Einstein’s theory of general relativity! Although indirect, this was the first evidence for gravitational radiation, demonstrating that gravitational waves do carry energy away from binary systems. This lead Hulse and Taylor to be awarded the Nobel Prize in Physics in 1993. The two neutron stars in the Hulse-Taylor binary system are radiating gravitational waves so slowly, that the stars won’t collide for another several billion years. If we are wanting to catch a neutron star collision, direct measurements of the source with a radio telescope just doesn’t cut it! So, in the years since 1974 scientists around the world have been working on the development of sensitive gravitational observatories that allow us to measure the stretch and squeezing of spacetime as a gravity wave passes by. 10.4.6 Gravitational Observatories Modern gravitational observatories use lasers to detect gravitational waves. These machines can identify the spiralling of neutron stars at a much later stage. These neutron star binaries have got so close together that they are able to orbit one another more than 25 times per second! Once a pair of neutron stars orbit this quickly, they radiate energetic gravitational waves, the energy of the emitted gravitational waves increases more and more, until the two stars spiral close enough that they collide and finally merge. A merger like this can happen on the order of minutes! 195 In August of 2017, such an inspiral of a neutron star binary was detected. In this instance, astronomers also managed to observe the same part of the sky with gamma-ray, X-ray, visible light, and radio telescopes. Several seconds after the gravitational wave observatories detected the merger of the neutron star binary, astronomers saw a bright gamma-ray burst in the same direction. Gamma-ray bursts come in two types: short bursts that last for less than a second, and long bursts that last for closer to a minute. Long gamma-ray bursts may be associated with collapsars or hypernovae which might form some black holes. However, in August 2017, the astronomers witnessed a short gamma-ray burst right after the merger. This confirmed a suspicion that astronomers had that short gamma-ray bursts are the result of neutron stars smashing into one another. The smashing together of neutron stars initiates nuclear reactions that allow the forma- tion of more massive than those formed in stars, that is elements that are heavier than iron. This includes elements such as gold, platinum, and heavy radioactive elements! So the next time you wonder where the gold for jewelry comes from, some large fraction of your ring or necklace is debris from the collision of two neutron stars. These heavy elements can be blown off into space and recycled into the next generation of stars or planets. But what is made at the core of the merger? 10.4.7 Merger Remnants When two neutron stars merge there are two possible outcomes for the object left behind. The remnant of the merger could either be a more massive neutron star or it could become a newly formed black hole. The outcome for the merger depends on the mass that remains. Neutron stars have a maximum allowed mass. This maximum mass is not exactly known, but it’s somewhere between 2 and 3 times the mass of the Sun. Astronomers commonly state that the maximum mass is 3 solar masses or ease of definition, but the largest mass is probably a bit smaller than this. If we knew this mass more accurately, this would be helpful when we try to distinguish between neutron stars and black holes. When the two neutron stars merge, if the total merged mass is smaller than the maximum neutron star mass, then we end up with a neutron star. But if the mass is too high, the neutron star will be unstable and will collapse to form a black hole. The collapse of a neutron star into a black hole should also create gravitational waves, but these would be more difficult to detect. There was no evidence for waves like these from the 2017 neutron star merger. Unfortunately this means that we just don’t know the whether the collision resulted in a larger neutron star, or a new black hole. This is an example of when no detection does not answer the question. The total mass of the merged object is 2.7 solar masses, and it is very likely that this is larger than the maximum allowed neutron star mass. However, we don’t yet have any evidence that tells us if a black hole formed. 10.4.8 Merger of a Neutron Star and a Black Hole We know that binary systems composed of two regular main sequence stars are common. When one of the stars is a high mass star it can evolve into either a black hole or a neutron 196 star, creating a binary system with a main sequence star companion. When the companion is close enough to the black hole or neutron star, or if the main sequence star evolves into a red giant star, it can overflow its Roche lobe, creating an accretion disk and an X-ray binary system. If the companion star has a high enough mass, the companion could transform into a neutron star or black hole too! For instance, the companion star in the Cygnus X-1 binary system has a high mass and will eventually evolve into either a neutron star or a black hole. This leads to the formation of binary systems composed of two black holes or two neutron stars. Both of these types of binaries have been observed by radio telescopes and gravitational wave detectors. But these binary systems could also evolve into a mixed binary with one neutron star and one black hole! Alternatively, a single black hole could encounter a binary system that has a neutron star or a black hole with a low mass companion if they are located in a dense cluster of stars. In the chaotic three-body encounter that results, the lower mass companion can be flung away, leaving behind a binary composed of two black holes or a black hole and a neutron star. Since a neutron star is very tiny, it doesn’t overflow its Roche lobe when it orbits in a system with a black hole. This means that these systems are unlikely to emit X-rays. So far, no binary system of this sort has been observed in X-rays, radio waves, or other type of electromagnetic radiation. The first detections of a neutron star and a black hole in a binary system occurred in 2020, when the LIGO and VIRGO gravitational wave observatories detected gravitational radiation emitted by two of these systems merging in far away galaxies. When a neutron star and a black hole collide, the neutron star will not survive. There are two types of outcomes when they merge. One possibility is that the neutron star will be distorted by the black hole’s tidal force and then swallowed whole by the black hole. In this case, we wouldn’t expect to see any light coming from the neutron star before it plunges into the black hole. Another possibility is that the black hole will tidally disrupt the neutron star before they meet up. In this case much of the neutron star’s gas will orbit the black hole. We would expect the neutron star’s remains to be extremely hot, and we might see the gas emit light before it enters the black hole, perhaps creating a short gamma-ray burst. In either case, the black hole will consume much of the neutron star’s material, and gain weight and grow in size! When news of the neutron star - black hole mergers came out, astronomers around the globe searched for remnant electromagnetic radiation, but none was found. This doesn’t mean that no light was emitted by the neutron star, though. These mergers occurred in far away galaxies, so the light may have been too dim to see. Also, the gravitational wave detectors couldn’t narrow down the exact galaxies, making the search difficult for the as- tronomers. In future years, the gravitational wave detectors in Japan and India will also be able to observe these types of mergers. This will give us a much better chance of observing X-rays or other electromagnetic radiation coming from these mergers. 197 10.4.9 Where does the gold in your ring come from? (Interview with Dr. Rodrigo Fernandez) Q: Where does the gold in your ring come from? Dr. Fernandez: This ring is made of white gold, and we think that the gold among those other heavy elements comes from collisions of the neutron stars, that astronomers called neutron star mergers. Very recently, a few months ago, it was announced that the first neutron star merger was observed in gravitational waves and in photons by the LIGO Collaboration and a number of other observers in the world. The signature of the formation of gold and other heavy elements called the r-process was detected in the signal. We have evidence that a significant fraction of that gold in the universe came from these types of events. Now, whether this gold in my ring comes from one close by merger or from many of them and then they all got mixed up somehow, we don’t know. But it could as well be just one that happened not so far in the past. Q: How are heavy elements produced in the R-process? Dr. Fernandez: R-process is one of many ways in which nature can create chemical elements, and we call those nucleosynthesis channels. We have, for instance, alpha capture, as I mentioned before, in which you take helium nuclei at the centers of stars and just stick it into a carbon nucleus and then you have oxygen, then you stick alpha particles helium again, and then you have a heavier nucleus. That’s one way of doing it. Then the other process is a special way of making heavy nuclei. The ”r” comes from rapid. It precisely means rapid neutron capture process. Rapid compared to Beta decay. As you might know, a neutron in isolation is not stable. You put a neutron in space and it will decay into a proton in about 15 minutes. Proton plus electron plus neutrinos to conserve lepton number, energy-momentum, etc. It turns out that the most efficient way that we know of to make things like gold and uranium is to take a seed nucleus such as iron, and then bombard it with neutrons and make it grow in mass. Now, why neutrons? Neutrons do not have electric charge. If you try to send a proton against an iron nucleus, the proton is going to be repelled by the electric forces but the neutron goes right in. Okay. So do you just pile up the neutrons and make this nucleus grow in mass up to a point at which it has so many neutrons that it’s unstable so it starts decaying some of those neutrons into protons and it becomes slowly a stable neutrons. That’s the r-process. That is the way we know how to make things as heavy as lead that has a mass number of 200 or more. As opposed to just using charged particle reactions. Dr. Fernandez’s webpage: https://sites.ualberta.ca/~rafernan/ 10.4.10 How does a neutron star’s vibrations affect binary inspiral? (Interview with Dr. Jocelyn Read) Q: How does a neutron star’s vibrations affect binary inspiral? Dr. Read: When we talk about a binary of black holes or a binary of neutron stars orbiting each other, the main signature of the gravitational waves comes at twice the fre- quency of the orbit basically. Each time one of the objects passes, say, in front of you, that’s stirs up another cycle of the gravitational wave signal which eventually reaches the observer. But just like you have, say, a wineglass has a particular resonant frequency and you see, say, an opera singer sings the right note and the wineglass can shatter. So one of 198 the research projects that I worked on, was understanding how the energy of the orbiting system, when the orbital frequency, the frequency of the orbits matches up with the natural frequency of an individual star that can transfer energy into the star’s vibration. So this can do some pretty dramatic things like similar to how a wineglass can shatter if you hit the right note, the solid crust of the neutron star might shatter if it sweeps through the right frequency in the orbit. So some collaborators and I worked through, that that might cause some high-energy flares that you could see in association with a gravitational wave signal or independently even. So these oscillations taking away energy from the orbits, if they’re strong enough and it has to be a huge amount of energy to affect the gravitational wave, but they can also drive the stars to merge earlier, so instead of spending the energy in orbiting, the energy goes into the vibration of the stars instead and that actually leads them to crash together. So they’re losing energy to gravitational waves and then they lose extra energy to this vibration. Dr. Read’s webpage: https://physics.fullerton.edu/~jread/ 10.4.11 Inspiral and Merger of Two Black Holes As the two black holes merge, for a brief period the black hole is a highly distorted mess. But the No Hair theorem tells us that the normal state for a rotating black hole has a smooth event horizon. During the final merger stage the distorted black hole emits gravitational radiation in a process called “Quasi-Normal Ringdown” where all the bumps and wiggles get smoothed out. The final resulting rotating black hole is a Kerr black hole of the sort that we looked at earlier in the course. The data from black hole and neutron star mergers helps astrophysicists understand the distribution of black hole and neutron star sizes in our galaxy, but also in the wider Universe. The merger of neutron stars can produce either a new neutron star, or a new black hole. When black holes merge, they form heavier black holes. 10.4.12 Masses in the Stellar Graveyard This diagram summarizes the known black hole food chain. This diagram shows all of the known neutron star and black hole masses, smaller than 200 solar masses. Astronomers have observed radio waves, visible light, X-rays, and gamma rays from a few thousand neutron stars. But only about 50 of these neutron stars are located in a binary star system, allowing their masses to be measured. We know the masses of around 30 black holes in X-ray binary systems, shown with purple circles. These black holes typically have masses between about 5 and 20 solar masses. You can see that the black holes and neutron stars are separate populations, when separated by mass. During LIGO’s first observing run, between September 2015 and January 2016, we were really excited when they announced 3 merger events, each event corresponding to the merger of two black holes into a larger black hole. After upgrading the LIGO detectors in the United States and adding the VIRGO de- tector in Italy, they added another 7 merger events and a neutron star merger in the next observation period. 199 After even more improvements to all of the detectors, the observation starting in April 2019 saw many more mergers! The full collection of black hole and neutron star mergers as known in 2021 are shown in this diagram. The first thing that you’ll notice is that LIGO and VIRGO have seen lots of black holes! In fact, whether you consider the black holes before or after merger, more black holes have been detected through their gravitational radiation than have been seen in X-rays! There is a population of black holes that participate in a merger that have masses very similar to the masses of black holes in X-ray binary systems. X-ray binary systems like Cygnus X-1 might be a typical progenitor of these merging systems. We also see a population of black holes with masses much larger than the black holes in X-ray binary systems, including the first black hole in the Intermediate mass range. Maybe some or all of these higher mass black holes are the result of an earlier generation of black holes that merged. There are some particularly cool events. There are the two cases where two neutron stars merged into a mystery object, that is either a high mass neutron star or a low mass black hole. One of these mergers resulted in a short gamma-ray burst, which was particularly exciting! I am personally hoping that one day they see two lower mass neutron stars smash together to create an object with a mass close to two times the Sun’s mass, since this would definitely be a new neutron star! We also have seen a neutron star and a black hole merging, which destroyed the neutron star and added mass to the black hole. Another interesting event was the merger of a mystery object with a mass of 2.6 solar masses with a higher mass black hole. This mystery object is either a high mass neutron star, or one of the lightest known black holes! A few interesting facts about the black holes discovered by LIGO and VIRGO have emerged. Many of their spins have been measured, and their spins don’t always point in the same direction, which is what one would expect if they formed in an X-ray binary. This instead suggests that some of these black holes formed in a dense star cluster. Most of these mergers took place in galaxies that are hundreds of millions of light years away from us, meaning that they took place in the far distant past when the Universe was younger. From this population of black hole mergers, we can infer that 8 billion years ago black holes merged about 3 times more frequently than they do today. LIGO, VIRGO, and the new Japanese detector, KAGRA, will start observing again in 2022, and we’re looking forward to seeing lots more black hole and neutron star collisions! 10.4.13 Colliding Galaxies If we now consider the other end of the black hole mass spectrum . . . By this I mean, let’s switch up to the size of galaxies. Observations of galaxies suggest that almost all galaxies have a supermassive black hole at their centres. Astronomers have been observing galaxies for a long time, and while we are used to seeing pretty pictures of spirals and swirls, astronomers have discovered that things can also get messy. This is because galaxies can also collide! 200 When galaxies collide their black holes can form a binary system which will emit gravi- tational waves, inspiral and merge. The merger of supermassive black holes hasn’t yet been observed, but hopefully they will be one day soon! Ground based gravitational wave observatories are responsible for the direct detection of merging black holes in the stellar mass range, but future orbiting space-based gravitational wave detectors are needed before we can detect the merger of supermassive black holes. We have discussed a lot about the detections of gravitational waves up to this point, but how do these detectors actually measure the effects of a passing gravitational wave? That is next on our agenda, so let’s find out now ... 10.5 Gravitational Telescopes 10.5.1 The Long Arm of Gravity Gravitational waves are extremely weak. These waves wouldn’t be felt by a human, for example, or any living creature for that matter. So, in order to detect gravitational waves, scientists must use the most sensitive instruments ever invented. And what devices are suitable for measuring incredibly small changes? Lasers, of course! And that means we’ll also need our good friend the Michelson-Morley interferometer. Remember, interferometers leverage the wave nature of light to measure the difference in lengths between different beam paths. When coherent light, light with that has the same phase, such as laser light, is split between two different paths, when the light is recombined its brightness will depend on how different the two paths were. The original Michelson-Morley interferometer was developed to determine if the “flow of Aether” caused a delay in one of the device’s two arms, instead of a difference in the arms lengths. Like a boat travelling against the current of a river, the theory of light back then predicted that moving Aether would cause a delay in the “upstream” arm. The opposite was discovered, that there was never any delay, no matter what orientation of the device with respect to its motion through the supposed Aether. This proved that light waves didn’t require a medium like Aether to travel in, and as a consequence, the speed of light is a constant! In order to leverage the sensitivity of interferometers, astronomers built the Laser Inter- ferometer Gravitational-Wave Observatory, whose acronym is L I G O, pronounced LYE-GO, or sometimes LEE-GO. These are two massively improved versions of the Michelson-Morley interferometer, built on either side of the continental United States of America. One of the detectors is in Hanford, Washington, while the other one is precisely 3,002 kilometers away in Livingston, Louisiana. They each have two arms, but instead of short, meter-long arms, like the original Michelson-Morley interferometer, each of the arms of the LIGO observatories is four kilometers long! In addition to the length, each arm bounces light from the laser source back and forth about 280 times, making each arm of LIGO equivalent to the length of 1120 kilometers! Why are LIGO’s arms so long? Well, it’s hunting for some of the weakest signals the universe has ever thrown at us! In fact, in order for LIGO to detect the strongest gravitational waves, it needs to be able to distinguish a change over the 4 kilometer length of its detector arms, a difference in length one-thousand times smaller than the radius of a proton!!! Think 201 about that! LIGO needs the precision to measure if the length of its arms changes by LESS THAN THE WIDTH of an atom! But one LIGO isn’t enough to catch a gravitational wave, we need at least two! 10.5.2 Two LIGOs are Better Than One There are several major Gravitational Wave Observatories in operation around the world. The two LIGO observatories were the first, followed by VIRGO, and a host of others in operation and under construction around the world. If one LIGO detector is sensitive enough to measure a change in its arm length down to a level below the width of an atom, why make more? Well, the first, and most important reason for a second LIGO is noise! Yes, sounds, footsteps, earthquakes, and cosmic gravity- quakes - they all register in LIGO as a change in the length of LIGOs arms, and in order to filter out footsteps from colliding black holes, you need a second detector. LIGOs second detector, built in Livingston, Louisiana, has a different set of sounds, footsteps, and earthquakes. . . but presumably the two LIGO detectors would both see the same gravitational wave coming from an intergalactic source. To determine whether a wiggle in one LIGO detector is the result of a gravitational wave, scientists compare the data from the second. If there is a wiggle at nearly the same time (remember, the detectors are separated by 3,002 kilometers), with nearly the same shape, scientists can be confident that they really saw a gravitational wave, and not some researcher sneezing in the control room! But there’s another important reason to have more than one gravitational observatory: direction! With only two LIGO detectors, an incoming gravitational wave will not have a well defined direction. Just like having two ears gives us stereo hearing, two LIGOs let us determine approximately where the source is, although with only two, there is still uncer- tainty about which direction it came from. In order to pin down the source of gravitational waves, a third gravitational observatory is necessary. In the case of the kilo-nova explosion resulting from the merger of two neutron stars in 2017, the gravitational wave signal was also detected by a third gravitational wave observatory: VIRGO, in Italy With all three gravity wave observatories up and running, most major astrophysical mergers will be detected. Over the next few decades, these types of observatories will get more and more sophisticated, detecting dozens of compact object collisions in the Universe. But things will get really interesting once we send these massive observatories into SPACE! 10.5.3 Lasers in Space! In order to really make the most sensitive gravitational wave observatories, scientists work hard to remove sources of error. Just like telescopes, which are better if they are on moun- taintops, but best if they are in space, a space-based gravitational observatory wouldn’t have to worry about earthquakes, or someone tripping on a banana peel near the detector. No, the next generation of gravitational observatories will also be built in space! The Laser Interferometer Space Observatory, whose acronym is L I S A, which is universally pronounced as LEE-SAH, but I would argue should equally be pronounced L-EYE-SAH in this case! LISA consists to three spacecraft, which will trail behind Earth in its orbit around the Sun, flying in a triangular formation. Each of LISA “arms” extend between the three 202 spacecraft, and instead of a puny 1,120 kilometer effective arm length that LIGO has, LISA will have three 2.5 million kilometer long arms! LISA will still be sensitive to small changes in the length between their arms, but will an incredibly sensitivity of 20 picometers over the 2.5 million kilometer long arms! As a result, LISA will be able to detect much smaller and quieter collisions that LIGO, but also begin probing into the processes by which compact objects are captured by, but not colliding with black holes. Beyond LISA, which won’t even launch until the early 2030s, future gravitational wave observatories will measure the rotation of compact objects, like pulsars. 10.6 Pulsar Timing Arrays 10.6.1 Pulsar Positioning System Since existing gravitational observatories can only detect the strongest waves created in the collisions of massive black holes and neutron stars, future detectors are being made with ever increasing sensitivities to find more subtle changes in the fabric of spacetime. One concept being developed is called a pulsar timing array, which would allow scientists to probe Einstein’s general theory of relativity, and the effects of gravitational waves, over thousands of lightyears. Since a pulsar is a rotating neutron star which emits jets of radiation, if the beam of jet points towards Earth, we detect a short radio burst. Those radio bursts arrive at regular intervals, sweeping across the Earth for every rotation of the pulsar. The fastest spinning pulsar, PSR J1748-2446ad, which lives within the globular cluster Terzan 5, 18,000 light years from Earth, rotates 716 times per second, which would sound like an F5 tone if the radio pulses were converted to sound waves. A spinning pulsar is of great interest, because some pulsars’ rotation rates are incredibly stable, so much so, they can rival the precision of atomic clocks! So, PSR J1748, the fastest rotating pulsar, has been measured to rotate exactly once every 0.001395952482 seconds, with an error of less than 600 femptoseconds. This incredibly precise timing is one of the most accurately measured observables in all of astrophysics! By the way, this pulsar was discovered by Dr. Jason Hessels who graduated with a BSc in Honours Physics from the University of Alberta! The precision of a pulsar’s rotation rate is very much like a clock, ticking at regular intervals. And, just like the effects of the gravitational doppler shift that redshift photons as they escape from a gravity well, gravitational waves alter the timing of pulses from pulsars! In order to actually do anything useful though, you need several pulsars in an array. . . and now you know why they are called pulsar timing arrays. It may be easier to imagine pulsar timing arrays as similar to the technology that un- derpins the Global Positioning System, or GPS. The GPS sensors in Smartphones and Nav- igation devices work by listening carefully for radio signals from GPS satellites high in orbit above Earth. By comparing the arrival times of the pulses from each GPS satellite, your device can triangulate your position on the surface of the Earth. NASA’s NICER/SEXTANT X-ray telescope on the International Space Station is ob- serving a collection of X-ray pulsars to test out the feasibility of using pulsars as future navigational aids. By listening to the regular pulses from several nearby pulsars, you could 203 triangulate your position anywhere in the interstellar space around the pulsars. This map, created by the Jet Propulsion Laboratory, was affixed to the Pioneer 10 spacecraft, which, after completing a survey of Jupiter, became the first satellite with sufficient escape velocity to leave the solar system. The image shows the relative positions of pulsars near Earth, with their particular timings encoded on the line that joins them. If this map were discovered, the position of Earth could be deduced. But our civilization hasn’t reached the point of navigating with pulsar timing arrays, instead, we are patiently listening to them for evidence of large scale gravitational waves passing in between the Earth and the pulsars. 10.6.2 Squishing Signals in Spacetime When a gravitational wave passes in between Earth, and a pulsar, it causes a distortion of spacetime that affects how signals propagate. Generally speaking, the signals from pulsars will either appear delayed or accelerated due to the influence of a gravitational wave. Just like a black hole creates a gravitational potential well, and the associated effects of gravitational redshift and time dilation, so too can a gravitational wave create a measurable effect as it passes by. Imagine, for example, that you usually drive to work or school on a flat road. The time it takes you to get from point A to point B along your route takes about the same amount of time. What would happen, if all of a sudden, along your route, a hill appeared? What about a depression in the road? In both cases, the time it takes to go from A to B will change! A gravitational wave in spacetime is just like this hill, only since it’s a wave, it will be a moving hill! If the timing of a gravitational wave is just right, it compresses the spacetime that the pulsar signals are travelling through, offsetting the arrival time by a small difference, which is called the timing residual. The timing residual is a measurement of the difference between the expected arrival time of a signal, and the observed arrival time. Since gravitational waves both stretch and squeeze spacetime, whether the signal is delayed or accelerated will depend on the geometry of the pulsar timing array and the incident gravitational waves. So far, this method of monitoring pulsar timing arrays has not resulted in any obser- vations of gravitational waves. However, techniques like these are a complement to the interferometer-based gravitational observatories, and will eventually contribute to the detec- tion of more massive binary collisions. 10.6.3 What is a pulsar timing array? (Interview with Dr. Ingrid Stairs) Q: What is a pulsar timing array? Dr. Stairs: For the pulsar timing array, there we’re looking at an array of pulsars over the sky, but basically all around the sky. So we really need an international collaboration to use telescopes all over the world to do this and the idea is to look for correlations. So commonalities in the pulse arrival times of these pulses that are around the sky. So since they’re acting like lighthouses though, these are very very regular spinners, stable rotators. So if they have pulses that come a little earlier, a little late there’s some reason for it. And with the pulsar timing of array, we’re looking for pulsars that are close together in the sky 204 to be early together or late together. Pulsars that are sort of 90 degrees apart to be out of sync, one comes early, the other comes late. And then pulsars that are a 180 degrees apart on the sky, opposite sides of the sky to be more in sync again, not completely, but mostly. And so if we can see that pattern in the timing from the pulsars all over the sky, we will have some confidence that we’ve seen a gravitational wave passing near the Earth. And probably we’re going to see gravitational waves from a whole collection of these supermassive black hole binaries, which are slowly spiraling into each other. We’re catching them, we’re aiming to catch them at orbital periods of a few years. So not right at the point where they’re about to merge, but getting in there. And we think these should be all over the sky, because galaxy mergers are happening all over the sky. And if every galaxy has a supermassive black hole at its center which we think they do, then those two black holes should naturally sink in together toward the center of the new galaxy and produce gravitational waves as they’re doing it. 10.7 The Final Countdown! 10.7.1 What We Know We Know Black holes in the media are portrayed as matter-thirsty objects with infinite hunger. They tear astronauts apart, cause incalculable damage, and facilitate impossible feats of time travel. These often misunderstood objects, have been discovered and studied only over the last half century. Black holes reveal to us the depth and breadth of the known universe, and the tremendous success of General Relativity - but they do hide their “dark sides”. As the brightest objects in the universe, quasars and blazars are powered by supermassive black holes accreting material into accretion discs. Recent black hole mergers, observed by LIGO reveal just how many black holes are out there. And observations from the Event Horizon Telescope give us our first glimpse at SGR A*, our galaxy’s very own supermassive black hole. But, physicists are well aware that we have much more to learn about black holes, and to pass that knowledge on to artists and filmmakers who have a passion for science. 10.7.2 Black Hole Formation In this course we learned that black hole formation, in the most basic sense, involves squash- ing the mass of an object down into a small volume with radius less than or equal to the Schwarzschild radius. It’s difficult to make a black hole, since as matter is squeezed into smaller volumes, other forces will tend to oppose this process and push outwards. For in- stance, in a star, nuclear reactions in the core heat up the star and provide an outward gas pressure that balances gravity. This is why we have stars that can live long stable lives of billions of years, potentially providing heat to life on orbiting planets! Black holes can form when very high mass stars can’t provide enough heat from nuclear reactions to sustain the gas pressure and keep the star in hydrostatic equilibrium. The loss of nuclear fuel leads to the implosion of the star, and in some cases, this creates a black hole. Another way of creating a black hole is to smash two neutron stars together. If the resulting mess of material has a high enough mass, it too can collapse into a black hole. I don’t know about you, but I’m feeling overwhelmed by all the information I learned 205 from the course! I really want to go back now to the scenes in Star Trek that we discussed at the beginning of the course, to compare what we saw then, to what we know now. As you probably recall, in the 2009 Star Trek movie reboot, “Red Matter” is employed as a quick method of creating a black hole. While there is still no scientific basis for Red Matter, or even the creation of a black hole through current human technologies, the destruction of the planet Vulcan due to a black hole is still a frightening thought. We know now, that even IF it were possible to create a black hole, say in a particle accelerator like the Large Hadron Collider, the black hole would be tiny! It would be so small, that its temperature would be MILLIONS of times hotter than the Sun, and quantum effects would quickly work on evaporating the black hole through Hawking Radiation. If the mass of the Red Matter black hole were similar to the mass of a proton, then the Hawking radiation would make the black hole unstable and it would disappear in a tiny fraction of a second after it was formed. This would create a small burst of energy, but not one anywhere near powerful enough to destroy a planet. That’s not where the scientific inaccuracies stop either. Let’s say that the Red Matter did create a miniature black hole that caused the collapse of the Planet’s surface. . . the planet wouldn’t vanish, as it did in the movie, instead, the infalling matter would accrete around the black hole, heating up to thousands and millions of degrees. Instead of watching the collapse, you would instead see blinding x-ray radiation (literally blinding, if you happen to be close!). Although scientists have never observed such a low mass black hole, it’s likely that all the models applied to the accretion discs of larger black holes would scale appropriately. I don’t bring up these issues because the scientific inaccuracies make Star Trek a poor movie, quite the opposite in fact. Without shortcuts around some of the difficult scientific principles, Star Trek would be more akin to just another documentary about black holes. It is interesting, however, when scientific principles are applied correctly, like in Interstellar, as the whole story gains significance. It might be a stretch to imagine oneself in the Star Trek universe, given it is well rooted in the realm of science-fiction. I think we can all more easily envision ourselves in a slightly more scientifically accurate future like that of the film Interstellar. 10.7.3 Where Science Fiction Meets Science Fact We can often find shortcuts taken by filmmakers and artists while portraying the astonishing environment around black holes. But the reality of black holes is much stranger than anything that has yet been captured on film. Science fact, as it were, still beats science fiction for the strangeness of black holes. Einstein’s insights into the structure of spacetime, insights that required a powerful imagi- nation, along with a knack for mathematics, were a giant leap from the Newtonian framework of gravitation. As a result, we know that in our universe there is a trade-off between how quickly you can travel and how quickly the clocks in your reference frame tick. Even thinking about it now is giving me a brain-cramp! Imagining spacetime is nothing compared to speculating about the interior of black holes. In both Disney’s “The Black Hole” and Christopher Nolan’s “Interstellar”, characters are portrayed crossing the event horizon. As astounding as the effects may be, the events portrayed in these movies taking place 206 within the event horizon of a black hole is pure speculation. Physicists make use of Penrose diagrams to try and explain the interior of black holes. Modern theories predict that anything that enters into a black hole’s event horizon, will have to collide with the singularity, which is likely fatal. We talked about the escape from a black hole as an impossibility. But we also know that the process of Hawking Radiation allows the escape of particles from a black hole when pairs of particles and anti-particles is created near the event horizon of a black hole. So what is it? Can particles escape from black holes, but not something big - like me? As quantum physicists are keen on saying, “Information Cannot Be Destroyed”, so what happens when I drop a memory device into a black hole? Can I read that information out of Hawking Radiation at a later time? Does the black hole somehow encode everything falling into it, as the Universe’s most compact hard drive? Until we can observe black holes in greater detail, we may not be able to learn the answers to these questions. It has been an honour teaching you. 10.7.4 Observations of the Black Hole Environment The idea of a black hole has been around for a long time, and more recently, a key component of many science fiction tales. Optical observations of black hole binaries have allowed us to look at the companions to black holes, watching them move and orbit around the binary’s common centre of mass. Using this information, we have been able to find out more about the types of stars that hang out with black holes, we have been able to learn how they can transfer material to the black hole, and how black hole can get their food! X-ray and radio observations of black hole binaries have given us to opportunity to learn more about the what’s going on close to the black hole, by giving us insight into accretion processes. These views allow us to test theories about the physics of matter in the presence of extreme gravity! While we are currently unable to visit a black hole ourselves, observations of them have taught us much more about the universe. The closest known black hole to us, is a black hole known as V616 Monocerotis. It lives about 3500 lightyears away from us. V616 Mon is a black hole a that lives in a binary system with orange coloured companion star. The black hole weighs in at about 7 solar masses. The farthest known black hole, ULAS J1342+0928 was discovered in 2017. This super- massive black hole has a mass of 800 million solar masses. Its light has taken 13.1 billion years to get to us, and was emitted only 690 million years after the big bang! The discovery of distant black holes allows us to learn more about the early universe, the formation of the first supermassive black hole and the formation of galaxies! The largest known black hole is S50014+81, an optically violent variable quasar. This black hole’s mass is 40 billion solar masses, and is also one of the most luminous black holes emitting radiation equivalent to 10 to the power 14 Suns. The faintest black hole is something harder to determine. One contender is Swift J1357.2- 0933, a stellar mass black hole in a binary system, located only 4,900 lightyears away, and emits light that is only 100 times brighter than the Sun in the X-ray band. 207 10.7.5 What We Know We Don’t Know I want to thank you for joining us on this learning expedition. While black holes appear to be mysterious, we have learned that the basic ideas and observations can be described using known scientific principles. Astrophysicists are well aware that a theory of quantum gravity is required to properly explain quantum phenomena associated with black holes on tiny scales. However, there are plenty more Unknown Unknowns that we can only just begin pondering about - like the nature of gravitational waves. The observations of gravitational radiation from merging black holes and neutron stars has opened up a new way to learn about black holes. We can’t know what will be discovered but we can guess at some possibilities. I’m hoping we will observe gravitational waves from merging supermassive black holes at the centres of galaxies! It also may be possible to see gravitational waves when stars are tidally disrupted by black holes. I hope that you have gained the tools to understand new discoveries about black holes, and to communicate those ideas clearly! Maybe one day I will read about your new discovery. Thank you! 208