Radiation Transport Around Kerr Black Holes Jeremy David ...

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6 Thank you to Paul Schechter, who took me under his wing my first day on the job; Fred Rasio, who introduced me to black holes and general relativity; Jack Wisdom, who trained me in the art of chaotic dynamics; Scott Hughes, whose office was always wide open for me to come in and bounce ideas off the white board; Jackie Hewitt, for offering an outsider’s point of view and fitting me into her crazy schedule. Arlyn Hertz for fitting me into Jackie’s schedule and helping make the defense run so smoothly. Martha Bezzat for answering all my random questions and requests for help with the mail, fax machine, and MIT bureaucratic regulations, always promptly and with a smile. And of course, many many thanks to my advisor, Ed Bertschinger, who provided the inspiration and direction for much of this work. His careful, critical eye kept me in line, and his encouragement this past year brought the final product to completion. Much of this work was supported by his NASA ATP grant NAG5-13306. Again, the ÒÐÞ¹¹ÖÛÜÒÝÖÒ¸Ð¹ÑÜØÕÒÑÒÝ biggest accolades are for my wife Nomi, for putting up with me and supporting me, especially these past few months. I couldn’t have done it without you. Ô¸ÜÜÞÐÕÔÔÔ¸ÒÔÒÒÐÝÒÔØÐÜÒÔÑ ¸ÓÜÓÜÐÖÓÝÜÓÝÜÐÖÜÒ¸ÐÝÒÓÔÖÐÝ¸ÝÐÐÔ ÞÛÜØ¹¹ÞÒÐÖÒ¸ÞÖÒÝÐÜÒÖÒÝÐÝÒÐÖ The heavens proclaim the glory of God; the sky declares His handiwork — Psalms 19:2 The wise man does not speak in the presence of one who is greater than he in wisdom; he does not interrupt the speech of his companion; he is not hasty to answer; he questions and answers properly, to the point; he speaks on the first point first, and on the last point last; regarding that which he has not learned he says: “I have not learned”; and he acknowledges the truth. — Ethics of the Fathers 5:9
Contents 1 Introduction and Outline 15 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Outline of Methods and Results . . . . . . . . . . . . . . . . . . . . . 22 1.3.1 Ray-tracing in the <strong>Kerr</strong> Metric . . . . . . . . . . . . . . . . . 22 1.3.2 The Hot Spot Model . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.3 Steady-state Disks . . . . . . . . . . . . . . . . . . . . . . . . 27 1.3.4 Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4 Alternative QPO Models . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 Ray-Tracing in the <strong>Kerr</strong> Metric 35 2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.1.1 Boyer-Lindquist Coordinates . . . . . . . . . . . . . . . . . . . 38 2.1.2 Doran Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 40 2.1.3 Analytic Methods . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2 Geodesic Ray-tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2.1 Tetrads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.2.2 The Radiative Transfer Equation . . . . . . . . . . . . . . . . 50 2.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.4 Broadened Emission Lines from Thin Disks . . . . . . . . . . . . . . . 59 2.4.1 Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.4.2 Observations of Iron Emission Lines . . . . . . . . . . . . . . . 63 3 The Geodesic Hot Spot Model 69 3.1 Hot Spot Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.1.1 Overbrightness and QPO Amplitudes . . . . . . . . . . . . . . 71 3.1.2 Harmonic Dependence on Inclination and Spin . . . . . . . . . 73 3.2 Non-circular Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3 Non-planar Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7
Page 1: Radiation Transport Around Kerr Bla
Page 5: 5 Acknowledgments Gravity can not b
Page 9 and 10: CONTENTS 9 A Formulae for Hamiltoni
Page 11 and 12: List of Figures 1-1 Sample of obser
Page 13 and 14: List of Tables 3.1 Black hole param
Page 15 and 16: Chapter 1 Introduction and Outline
Page 17 and 18: 1.2. HISTORICAL BACKGROUND 17 to th
Page 19 and 20: 1.2. HISTORICAL BACKGROUND 19 disk
Page 21 and 22: 1.2. HISTORICAL BACKGROUND 21 tral
Page 23 and 24: ¼ÐÀÓÐÒÖ× 1.3. OUTLINE OF ME
Page 25 and 26: 1.3. OUTLINE OF METHODS AND RESULTS
Page 31 and 32: 1.4. ALTERNATIVE QPO MODELS 31 isot
Page 33 and 34: 1.4. ALTERNATIVE QPO MODELS 33 jets
Page 35 and 36: Chapter 2 Ray-Tracing in the Kerr M
Page 37 and 38: 2.1. EQUATIONS OF MOTION 37 Killing
Page 39 and 40: 2.1. EQUATIONS OF MOTION 39 flat sp
Page 41 and 42: 2.1. EQUATIONS OF MOTION 41 conveni
Page 43 and 44: 2.1. EQUATIONS OF MOTION 43 in sepa
Page 45 and 46: 2.2. GEODESIC RAY-TRACING 45 basis
Page 47 and 48: 2.2. GEODESIC RAY-TRACING 47 throug
Page 49 and 50: 2.2. GEODESIC RAY-TRACING 49 with a
Page 51 and 52: 2.2. GEODESIC RAY-TRACING 51 since
Page 53 and 54: 2.2. GEODESIC RAY-TRACING 53 dx µ
Page 55 and 56: 2.2. GEODESIC RAY-TRACING 55 In the
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2.3. NUMERICAL METHODS 57 a long pa
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2.4. BROADENED EMISSION LINES FROM
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Chapter 3 The Geodesic Hot Spot Mod
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3.1. HOT SPOT EMISSION 71 Figure 3-
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3.2. NON-CIRCULAR ORBITS 77 Figure
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3.2. NON-CIRCULAR ORBITS 79 Figure
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3.2. NON-CIRCULAR ORBITS 81 paramet
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3.2. NON-CIRCULAR ORBITS 83 at (2/3
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3.2. NON-CIRCULAR ORBITS 85 form at
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3.3. NON-PLANAR ORBITS 87 3.3 Non-p
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3.3. NON-PLANAR ORBITS 89 Table 3.1
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3.4. SUMMARY 91 binaries in many wa
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Chapter 4 Features of the QPO Spect
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4.2. PARAMETERS FOR THE BASIC HOT S
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4.3. PEAK BROADENING FROM HOT SPOTS
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4.3. PEAK BROADENING FROM HOT SPOTS
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4.4. DISTRIBUTION OF COORDINATE FRE
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4.5. ELECTRON SCATTERING IN THE COR
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4.6. FITTING QPO DATA FROM XTE J155
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4.6. FITTING QPO DATA FROM XTE J155
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4.7. HIGHER ORDER STATISTICS 117 ob
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4.7. HIGHER ORDER STATISTICS 119 bi
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4.7. HIGHER ORDER STATISTICS 121 Fi
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4.7. HIGHER ORDER STATISTICS 123 12
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4.8. SUMMARY 125 els. Clearly the h
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Chapter 5 Steady-state α-disks The
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5.1. STEADY-STATE DISKS OUTSIDE THE
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5.2. GEODESIC PLUNGE INSIDE THE ISC
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5.2. GEODESIC PLUNGE INSIDE THE ISC
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5.3. NUMERICAL IMPLEMENTATION 143 F
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5.3. NUMERICAL IMPLEMENTATION 145 a
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5.3. NUMERICAL IMPLEMENTATION 147 (
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5.4. OBSERVED SPECTRUM OF THE DISK
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Chapter 6 Electron Scattering If we
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6.1. PHYSICS OF SCATTERING 159 Θ
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6.1. PHYSICS OF SCATTERING 161 the
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6.1. PHYSICS OF SCATTERING 163 and
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6.1. PHYSICS OF SCATTERING 165 colo
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6.2. EFFECT ON SPECTRA 167 Let g(z)
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6.2. EFFECT ON SPECTRA 169 (1979);
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6.2. EFFECT ON SPECTRA 171 Figure 6
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6.3. EFFECT ON LIGHT CURVES 173 Fig
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6.4. IMPLICATIONS FOR QPO MODELS 17
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Chapter 7 Conclusions and Future Wo
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7.1. SUMMARY OF RESULTS 183 the low
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7.2. CAVEATS 185 back to the coordi
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7.3. NEW APPLICATIONS OF CURRENT CO
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7.4. NEW FEATURES FOR CODE 189 grou
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7.5. DEVELOPMENT OF NEW MODELS 191
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Appendix A Formulae for Hamiltonian
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195 The relevant spatial derivative
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Appendix B Summing Periodic Functio
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199 Over a sample time T f ≫ T l
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Bibliography Abramowicz, M. A., Lan
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BIBLIOGRAPHY 203 Damour, T., & Espo
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BIBLIOGRAPHY 205 Hawler, J. F., & G
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BIBLIOGRAPHY 207 McClintock, J. E.,
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BIBLIOGRAPHY 209 Psaltis, D. 2001b,
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BIBLIOGRAPHY 211 Stern, B., et al.
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