Dynamical and Radiative Modeling of Sagittarius A* A dissertation presented by Roman V. Shcherbakov to The Department of Astronomy in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Astronomy Harvard University Cambridge, Massachusetts May 2011 ⃝c 2011 { Roman V. Shcherbakov All rights reserved. Thesis advisor Author Ramesh Narayan Roman V. Shcherbakov Dynamical and Radiative Modeling of Sagittarius A* Abstract Sgr A* in our Galactic Center is the closest supermassive black hole (SMBH) with the largest event horizon angular size. Most other SMBHs are likely in the same dormant low-luminosity accretion state as Sgr A*. Thus, the important physical effects in lives of BHs can be best observed and studied in our Galactic Center. One of these effects is electron heat conduction. Conduction may be the main reason why Sgr A* is so dramatically underluminous: it transfers heat outwards from the inner flow and unbinds the outer flow, quenching the accretion. In Chapter 3 I build a realistic model of accretion with conduction, which incorporates feeding by stellar winds. In a model with accretion rate < 1% of the naive Bondi estimate I achieve agreement of the X-ray surface brightness profile and Faraday rotation measure to observations. An earlier model proposed in Chapter 2 with adiabatic accretion of turbulent magnetized medium cannot be tweaked to match the observations. Its accretion rate appears too large, so turbulent magnetic field cannot stop gas from falling in. Low accretion rate leads to a peculiar radiation pattern from near the BH: cyclo- synchrotron polarized radiation is observed in radio/sub-mm. Since it comes from several Schwarzschild radii, the BH spin can be determined, when we overcome all modeling challenges. I fit the average observed radiation spectrum with a theoretical spectrum, which is computed by radiative transfer over a simulation-based model. Relevant plasma effects responsible for the observed polarization state are accurately computed for thermal iii Abstract iv plasma in Chapter 4. The prescription of how to perform the correct general relativistic polarized radiative transfer is elaborated in Chapter 5. Application of this technique to three-dimensional general relativistic magneto hydrodynamic numerical simulations is reported in Chapter 6. The main results of analysis are that the spin inclination angle is ◦ ◦ estimated to lie within a narrow range θest = 50 − 59 , and most probable value of BH spin is a∗ = 0:9. I believe the researched topics will play a central role in future modeling of typical SMBH accretion and will lead to effective ways to determine the spins of these starving eaters. Computations of plasma effects reported here will also find applications when comparing models of jets to observations. Contents Title Page ........................................ i Abstract ......................................... iii Table of Contents .................................... v Citations to Previously Published Work ....................... ix Acknowledgments .................................... x 1 Introduction 1 1.1 Place of Sgr A* in the Cosmos ......................... 1 1.1.1 Formation Scenarios of Seed Black Holes ................ 1 1.1.2 Mergers and Accretion over Hubble Time ............... 2 1.1.3 Recent and Present-day Activity .................... 4 1.2 Summary of Sgr A* Observations ........................ 9 1.2.1 Sub-mm and Radio Observations .................... 9 1.2.2 Infrared Observations .......................... 13 1.2.3 X-ray Observations ............................ 13 1.3 Summary of Sgr A* Modeling .......................... 15 1.3.1 General Flow Structure ......................... 15 1.3.2 Collisionless Effects ........................... 18 1.3.3 Emissivity and Radiative Transfer ................... 19 1.4 Thesis Outline .................................. 20 v Contents vi 2 Spherically Symmetric Accretion Flows: Minimal Model with MHD Tur- bulence 23 2.1 Introduction .................................... 24 2.2 Spherical Model .................................. 27 2.2.1 Dynamics ................................. 28 2.2.2 Evolution of Turbulence ......................... 31 2.2.3 Correspondence to Numerical Simulations . 34 2.2.4 Magnetic Helicity ............................. 38 2.2.5 System of Equations with Source Terms . 40 2.3 Boundary Conditions and Parameters ..................... 43 2.3.1 Outer Medium Transition ........................ 44 2.3.2 Transition to Rotationally Supported Flow . 45 2.4 Results ....................................... 49 2.4.1 Maximum Rate Solution ......................... 49 2.4.2 Solution with Effective Angular Momentum Transport . 58 2.5 Discussion of the Model ............................. 64 2.5.1 Real Flow ................................. 65 2.5.2 Treatment of Magnetic Field ...................... 66 2.5.3 Radiative Cooling ............................ 70 2.5.4 Convection & Diffusion ......................... 71 2.5.5 Equation of State ............................. 72 2.6 Observations ................................... 73 2.7 Conclusions .................................... 75 2.8 Appendix: Analytical Tests ........................... 76 2.9 Appendix: Self-Similar Solution ......................... 80 2.10 Appendix: Convection .............................. 84 3 Inflow-Outflow Model with Conduction and Self-Consistent Feeding for Sgr A* 91 3.1 Introduction .................................... 92 Contents vii 3.2 Observations ................................... 93 3.3 Stellar Winds Feeding .............................. 98 3.4 Dynamical Equations ............................... 100 3.4.1 Energy Transport Mechanism . 100 3.4.2 System of Equations . 101 3.5 Solutions and Discussions ............................ 105 4 Propagation Effects in Magnetized Transrelativistic Plasmas 111 4.1 Introduction .................................... 112 4.2 Calculations .................................... 114 4.2.1 Geometry of the Problem . 114 4.2.2 Linear Plasma Response . 114 4.2.3 High Frequency Limit . 117 4.2.4 Components in High-Frequency Limit . 118 4.2.5 Fitting Formulas for Higher Temperatures . 120 4.2.6 Exact Plasma Response . 123 4.2.7 Eigenmodes ................................ 123 4.3 Applications .................................... 126 4.3.1 Dispersion Measure . 127 4.3.2 Magnetized Radiative Transfer . 127 4.4 Discussion & Conclusion ............................. 131 5 General Relativistic Polarized Radiative Transfer: Building a Dynamics- Observations Interface 133 5.1 Introduction .................................... 134 5.2 Newtonian Polarized Radiative Transfer . 136 5.3 Derivation of Response Tensor . 143 5.3.1 General Isotropic Particle Distribution . 144 5.3.2 Thermal Particle Distribution . 147 5.3.3 Rotation of Thermal Response Tensor . 149 Contents viii 5.4 Extension to General Relativity . 151 5.4.1 Transformation to Locally-flat Co-moving Frame . 153 5.5 Application to Compact Objects . 155 5.6 Discussion & Conclusions ............................ 159 6 Constraining the Accretion Flow in Sgr A* by General Relativistic Dy- namical and Polarized Radiative Modeling 162 6.1 Introduction .................................... 164 6.2 Observations ................................... 169 6.3 Dynamical Model: 3D GRMHD Simulations . 173 6.3.1 Governing Equations . 173 6.3.2 Physical Models ............................. 176 6.3.3 Numerical Methods . 178 6.3.4 Resolution and Spatial Convergence . 179 6.3.5 Ceiling Constraints ............................ 181 6.3.6 Temporal Convergence . 181 6.3.7 Evolved Disk Structure . 183 6.4 Averaged Dynamical Model . 185 6.4.1 Averaging ................................. 185 6.4.2 Extension to Large Radii . 188 6.4.3 Electron Temperature . 189 6.5 General Relativistic Polarized Radiative Transfer . 193 6.6 Statistical Analysis ................................ 197 6.7 Results ....................................... 201 6.8 Discussion and Conclusions . 217 6.9 Appendix: Radiative Transfer Convergence . 223 7 Discussion and Future Directions 227 Citations to Previously Published Work The results reported in Chapter 2 were published as "Spherically Symmetric Accretion Flows: Minimal Model with Magnetohydro- dynamic Turbulence," Roman V. Shcherbakov, 2008, ApJS, 177, 493 [arXiv:0803.3909 [astro-ph]]. Chapter 3 appears in its entirety in the paper ”Inflow-Outflow Model with Conduction and Self-consistent Feeding for Sgr A*," Roman V. Shcherbakov and Frederick K. Baganoff, 2010, ApJ, 716, 504 [arXiv:1004.0702 [astro-ph]]. The investigation of plasma effects described in Chapter 4 is published as "Propagation Effects in Magnetized Transrelativistic Plasmas," Roman V. Shcherbakov, 2008, ApJ, 688, 695 [arXiv:0809.0012 [astro-ph]]. The formalism of ray tracing through plasma near black hole elaborated in Chapter 5 is published as "General Relativistic Polarized Radiative Transfer: Building a Dynamics - Observations Interface," Roman V. Shcherbakov and Lei Huang, 2011, MNRAS, 410, 1052 [arXiv:1007.4831 [astro-ph]]. The culmination of the thesis, sub-millimeter modeling of Sgr A* written in Chapter 6, is currently under the second round of review in Astrophysical Journal "Constraining the Accretion Flow in Sgr A* by General Relativistic Dynamical and Polarized Radiative Modeling," Roman V. Shcherbakov, Robert F. Penna, and Jonathan C. McKinney arXiv:1007.4832 [astro-ph]. In the interest of brevity, I have omitted from my thesis a relevant paper published in 2009 that I was a