Radiation Transport Around Kerr Black Holes Jeremy David ...

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176 CHAPTER 6. ELECTRON SCATTERING scales are on the order of milliseconds. More promising is the application to hot spots from AGN sources or the X-ray flares around Sgr A ∗ (Baganoff et al., 2001). By integrating over broad energy bands such as those typically used in RXTE observations, we can increase our “signal” strength while sacrificing spectral resolution. For millisecond periods, there will still not be nearly enough photons to provide phase resolution, but these features may show up statistically in the power spectrum or bispectrum, as explained in Chapter 4. Figure 6-9 shows a set of integrated light curves for a variety of optical depths. The black hole and hot spot parameters are as in Figure 6-7, here assuming a thermal emission with hot spot temperature T hs = 1 keV. The photons are integrated over the energy range 0.5-30 keV. As the optical depth to electron scattering increases, the rms amplitude of each light curve decreases as the photons get smoothed out in time. Similarly, due to the average time delay added to each photon by the increased path length, the relative location of each peak is shifted later in time. These amplitudes and phase shifts are listed in Table 6.1 for inclinations of i = 45 ◦ and 75 ◦ . At even higher optical depths, the average light curve intensity begins to decrease, as more photons end up getting captured by the black hole and never reaching the observer. Table 6.1: Amplitudes and phase shifts of light curve peaks for hot spot inclinations of i = 45 ◦ and 75 ◦ for a range of optical depths τ es . The amplitude quoted is the standard deviation σ(I) normalized by the mean intensity µ(I). The phase shift is where the peak intensity is located in time, relative to that of the unscattered light curve. 45 ◦ 75 ◦ τ es σ/µ phase shift σ/µ phase shift (periods) (periods) 0 1.14 0 1.51 0 0.5 0.53 0.01 0.88 0.01 1.0 0.27 0.02 0.46 0.03 2.0 0.086 0.06 0.12 0.06 3.5 0.021 0.29 0.035 0.31 5.0 0.018 0.35 0.021 0.40 In all likelihood, the relative phase shifts would be nearly impossible to detect, regardless of the instrument sensitivity, since to do so would require measuring the light curve from a single coherent hot spot at two different optical depths. It is difficult to imagine a scenario where the coronal properties could change on such short time scales [yet it is possible that a fixed hot spot on the surface of an X-ray
6.3. EFFECT ON LIGHT CURVES 177 Figure 6-9: Energy-integrated light curves for a hot spot with orbital parameters as in Figure 6-7. The emitted spectrum is assumed to be thermal with a hot spot temperature T hs = 1 keV, integrated over 0.5 − 30 keV in the observer’s frame. With increasing optical depth to scattering, the rms amplitudes decrease significantly, and their peaks move slightly to the right, due to the time delay from repeated scattering events. pulsar might actually be used for this technique; see Ford (2000) and Gierlinsky, Done, & Barret (2002)]. However, the higher harmonic peaks of the different light curves may in fact be measurable with the next-generation X-ray timing mission, or under extremely favorable conditions, even with RXTE. In Figure 6-10 we show the damping of the Fourier modes A n /A 0 with increasing optical depth. Not only does the overall amplitude of modulation decrease with increased scattering, but also the relative amplitudes of the higher harmonics (n > 1) decreases relative to the fundamental (n = 1). While the absolute peak shifts for hot spot light curves at different optical depths would probably not be detectable, the relative shifts of simultaneous light curves in different energy bands may be observable, at least on a statistical level with a crosscorrelation analysis. Since the average scattering event boosts photons to higher energy bands and also causes a net time delay due to the added geometric path, the light curves in higher energy bands should be delayed with respect to the lower energy
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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
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2.2. GEODESIC RAY-TRACING 51 since
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2.2. GEODESIC RAY-TRACING 53 dx µ
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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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Page 183 and 184: 7.1. SUMMARY OF RESULTS 183 the low
Page 185 and 186: 7.2. CAVEATS 185 back to the coordi
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Page 189 and 190: 7.4. NEW FEATURES FOR CODE 189 grou
Page 191 and 192: 7.5. DEVELOPMENT OF NEW MODELS 191
Page 193 and 194: Appendix A Formulae for Hamiltonian
Page 195 and 196: 195 The relevant spatial derivative
Page 197 and 198: Appendix B Summing Periodic Functio
Page 199 and 200: 199 Over a sample time T f ≫ T l
Page 201 and 202: Bibliography Abramowicz, M. A., Lan
Page 203 and 204: BIBLIOGRAPHY 203 Damour, T., & Espo
Page 205 and 206: BIBLIOGRAPHY 205 Hawler, J. F., & G
Page 207 and 208: BIBLIOGRAPHY 207 McClintock, J. E.,
Page 209 and 210: BIBLIOGRAPHY 209 Psaltis, D. 2001b,
Page 211 and 212: BIBLIOGRAPHY 211 Stern, B., et al.
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