Radiation Transport Around Kerr Black Holes Jeremy David ...

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102 CHAPTER 4. FEATURES OF THE QPO SPECTRUM Figure 4-2: Geodesic coordinate frequencies as a function of radius for a black hole with mass M = 10M ⊙ and spin a/M = 0.5. The radius of the inner-most stable circular orbit r ISCO is where ν r → 0. The commensurate radius r 0 is where the ratio of azimuthal to radial coordinate frequencies is 3:1.
4.4. DISTRIBUTION OF COORDINATE FREQUENCIES 103 nν φ ± ν r will be a Lorentzian of width ( ∆ν = ∆r n dν φ dr ± dν ) r . (4.10) dr r 0 Unlike in the previous section where the finite lifetimes gave a single width for every QPO peak, now each peak in the power spectrum will be broadened by a different but predictable amount. Note in particular how the peaks at the higher harmonics with n > 1 will be significantly broader (and thus lower in amplitude) than the fundamental. Another important feature evident from Figure 4-2 and equation (4.10) is that, due to the opposite-signed slopes of ν r (r) and ν φ (r) around r 0 , the beat mode at ν φ +ν r remains very narrow, while the peak at ν φ −ν r is quite broad. These features should play a key role in using the power spectrum as an observable in understanding the behavior of geodesic hot spots. As in Section 4.3, the first step in producing the theoretical power spectrum is to calculate the Fourier amplitude in each mode with the full three-dimensional raytracing calculation of emission from a single periodic hot spot at r 0 . Again, the linear properties of the problem allow us simply to sum a series of Lorentzians, each with a different amplitude, width, and location (peak frequency), to get the total power spectrum. The peak amplitudes A j are given by the ray-tracing calculations, the locations ν j from the geodesic coordinate frequencies and their harmonics, and the widths ∆ν j from equation (4.10). Since the QPO peak broadening is most likely caused by a combination of factors including the hot spots’ finite lifetimes as well as their finite radial distribution, the simulated power spectrum should incorporate both features in a single model. Now the computational convenience of Lorentzian peak profiles is clearly evident, since the net broadening is given by the convolution of both effects and the convolution of two Lorentzians is a Lorentzian: [L(ν 1 , ∆ν 1 ) ⋆ L(ν 2 , ∆ν 2 )](ν) = L(ν tot , ∆ν tot )(ν), (4.11) where the peak centers and widths simply add: ν tot = ν 1 +ν 2 and ∆ν tot = ∆ν 1 +∆ν 2 . In the case where one or both of the Lorentzians is not normalized, the amplitude of the convolved function is given as a function of the individual peak amplitudes and widths: A tot = π A 1A 2 ∆ν 1 ∆ν 2 ∆ν 1 + ∆ν 2 , (4.12) where A 1 and A 2 are the peak amplitudes of the respective Lorentzians [A j = 1/(π∆ν j ) corresponds to a normalized function.] Figure 4-3 shows the power spectrum for a collection of hot spots orbiting near
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Radiation Transport Around Kerr Bla
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5 Acknowledgments Gravity can not b
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Contents 1 Introduction and Outline
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CONTENTS 9 A Formulae for Hamiltoni
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List of Figures 1-1 Sample of obser
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List of Tables 3.1 Black hole param
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Chapter 1 Introduction and Outline
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1.2. HISTORICAL BACKGROUND 17 to th
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1.2. HISTORICAL BACKGROUND 19 disk
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1.2. HISTORICAL BACKGROUND 21 tral
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¼ÐÀÓÐÒÖ× 1.3. OUTLINE OF ME
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1.3. OUTLINE OF METHODS AND RESULTS
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1.4. ALTERNATIVE QPO MODELS 31 isot
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1.4. ALTERNATIVE QPO MODELS 33 jets
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Chapter 2 Ray-Tracing in the Kerr M
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2.1. EQUATIONS OF MOTION 37 Killing
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2.1. EQUATIONS OF MOTION 39 flat sp
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2.1. EQUATIONS OF MOTION 41 conveni
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2.1. EQUATIONS OF MOTION 43 in sepa
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2.2. GEODESIC RAY-TRACING 45 basis
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2.2. GEODESIC RAY-TRACING 47 throug
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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
Page 57 and 58: 2.3. NUMERICAL METHODS 57 a long pa
Page 59 and 60: 2.4. BROADENED EMISSION LINES FROM
Page 69 and 70: Chapter 3 The Geodesic Hot Spot Mod
Page 71 and 72: 3.1. HOT SPOT EMISSION 71 Figure 3-
Page 77 and 78: 3.2. NON-CIRCULAR ORBITS 77 Figure
Page 79 and 80: 3.2. NON-CIRCULAR ORBITS 79 Figure
Page 81 and 82: 3.2. NON-CIRCULAR ORBITS 81 paramet
Page 83 and 84: 3.2. NON-CIRCULAR ORBITS 83 at (2/3
Page 85 and 86: 3.2. NON-CIRCULAR ORBITS 85 form at
Page 87 and 88: 3.3. NON-PLANAR ORBITS 87 3.3 Non-p
Page 89 and 90: 3.3. NON-PLANAR ORBITS 89 Table 3.1
Page 91 and 92: 3.4. SUMMARY 91 binaries in many wa
Page 93 and 94: Chapter 4 Features of the QPO Spect
Page 95 and 96: 4.2. PARAMETERS FOR THE BASIC HOT S
Page 97 and 98: 4.3. PEAK BROADENING FROM HOT SPOTS
Page 99 and 100: 4.3. PEAK BROADENING FROM HOT SPOTS
Page 101: 4.4. DISTRIBUTION OF COORDINATE FRE
Page 105 and 106: 4.4. DISTRIBUTION OF COORDINATE FRE
Page 107 and 108: 4.5. ELECTRON SCATTERING IN THE COR
Page 113 and 114: 4.6. FITTING QPO DATA FROM XTE J155
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Page 117 and 118: 4.7. HIGHER ORDER STATISTICS 117 ob
Page 119 and 120: 4.7. HIGHER ORDER STATISTICS 119 bi
Page 121 and 122: 4.7. HIGHER ORDER STATISTICS 121 Fi
Page 123 and 124: 4.7. HIGHER ORDER STATISTICS 123 12
Page 125 and 126: 4.8. SUMMARY 125 els. Clearly the h
Page 127 and 128: Chapter 5 Steady-state α-disks The
Page 129 and 130: 5.1. STEADY-STATE DISKS OUTSIDE THE
Page 139 and 140: 5.2. GEODESIC PLUNGE INSIDE THE ISC
Page 141 and 142: 5.2. GEODESIC PLUNGE INSIDE THE ISC
Page 143 and 144: 5.3. NUMERICAL IMPLEMENTATION 143 F
Page 145 and 146: 5.3. NUMERICAL IMPLEMENTATION 145 a
Page 147 and 148: 5.3. NUMERICAL IMPLEMENTATION 147 (
Page 149 and 150: 5.4. OBSERVED SPECTRUM OF THE DISK
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5.4. OBSERVED SPECTRUM OF THE DISK
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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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