Theoretical Astrophysics

Matthias Bartelmann

PHYSICS TEXTBOOK

An Introduction

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Matthias Bartelmann

Theoretical Astrophysics

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Matthias Bartelmann

Theoretical Astrophysics

An Introduction

WILEY-VCH Verlag GmbH & Co. KGaA

The Author

Prof. Dr. Matthias BartelmannUniversitt HeidelbergInst. Theoretische AstrophysikAlbert-berle-Str. 269120 HeidelbergGermany

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V

Contents

Preface XI

Acknowledgements XV

Colour Plates XVII

1 Theoretical Foundations 11.1 Units 11.1.1 Lengths, Masses, Times, and Temperatures 11.1.2 Charges and Electromagnetic Fields 11.1.3 Natural Constants 21.2 Lorentz Invariance 21.2.1 The Special Lorentz Transform 41.2.2 Minkowski Space 61.2.3 Some Properties of the Minkowski World 91.2.4 Relativistic Dynamics 121.3 Electromagnetism 161.3.1 Field Tensor and Sources 161.3.2 Lorentz Transform of the Electromagnetic Field 181.3.3 Maxwells Equations 191.3.4 Energy-Momentum Conservation 211.3.5 LinardWiechert Potentials and the Larmor Formula 221.3.6 The Lorentz Force 261.4 Elementary Kinetic Theory 281.4.1 The BBGKY Hierarchy and the Boltzmann Equation 281.4.2 Collision Terms 311.4.3 Diffusion in Phase-Space: The FokkerPlanck Approximation 311.4.4 Diffusion in Absolute Momentum 341.4.5 Calculation of the Diffusion Coefficient D2 35

Further Reading 37

2 Radiation Processes 392.1 Thomson Scattering 392.2 Spectra 452.3 Synchrotron Radiation 50

VI Contents

2.3.1 Larmor Frequency and Relativistic Focussing 512.3.2 Synchrotron Power 532.3.3 Synchrotron Spectrum 532.4 Bremsstrahlung 582.4.1 Orbit of an Electron Scattering off an Ion 582.4.2 Fourier Transform of the Orbit 612.4.3 Integration over Impact Parameters 622.4.4 Average over Electron Velocities, Thermal Bremsstrahlung 632.5 Radiation Damping 652.5.1 Damping Force 652.5.2 Transfer of Energy from a Moving Charge to a Radiation Field 692.6 Compton Scattering 722.6.1 Energy Change in the Scattering Process 722.6.2 Net Energy Transfer 742.6.3 The Kompaneets Equation 772.7 Radiative Quantum Transitions 832.7.1 Transition Probability 832.7.2 Perturbing Hamiltonian 842.7.3 Decomposition of the Electromagnetic Field 872.7.4 Dipole Approximation 882.7.5 Cross Sections 902.7.6 Photoionisation Cross Section 922.8 Shapes of Spectral Lines 942.8.1 Natural Line Width 952.8.2 Collisional Broadening 972.8.3 Doppler Broadening of Spectral Lines 982.8.4 The Voigt Profile 992.8.5 Equivalent Widths and Curves-of-Growth 1002.9 Radiation Quantities 1032.9.1 Specific Intensity 1042.9.2 Moments of the Intensity 1052.9.3 Relativistic Invariance of I/3 1072.10 The Planck Spectrum and Einstein Coefficients 1092.10.1 The Planck Spectrum 1102.10.2 Transition Balance and the Einstein Coefficients 1152.11 Absorption and Emission 1172.11.1 Absorption Coefficients and Emissivity 1172.11.2 Radiation Transport in a Simple Case 1192.11.3 Emission and Absorption in the Continuum Case 1212.11.4 Energy Transport Through Absorbing Media 124

Further Reading 126

3 Hydrodynamics 1273.1 The Equations of Ideal Hydrodynamics 1273.1.1 Particle-Current Density and Energy-Momentum Tensor 127

Contents VII

3.1.2 Collisional Invariants and the Fluid Approximation 1303.1.3 The Equations of Ideal Hydrodynamics 1343.2 Relativistic Hydrodynamics 1393.2.1 Hydrodynamic Equations 1393.2.2 Hydrodynamics in a Weak Gravitational Field 1423.2.3 Gravitational Field Equation 1433.2.4 The Combined Set of Equations 1443.2.5 Perturbative Analysis 1453.3 Viscous Hydrodynamics 1483.3.1 Diffusion of Particles, Momentum and Internal Energy 1483.3.2 The Equations of Viscous Hydrodynamics 1523.3.3 Entropy 1543.3.4 Fluids in a Gravitational Field 1553.3.5 The Tensor Virial Theorem 1573.3.6 Transformation to Cylindrical or Spherical Coordinates 1603.4 Flows under Specific Circumstances 1623.4.1 Sound Waves 1633.4.2 Polytropic Equation of State 1643.4.3 Hydrostatic Equilibrium 1673.4.4 Vorticity and Kelvins Circulation Theorem 1703.4.5 Bernoullis Constant 1723.4.6 Bondi Accretion 1753.4.7 Bernoullis Law for Irrotational, Non-Stationary Flows 1783.4.8 Diffusion of Vorticity 1793.4.9 The Reynolds Number 1793.4.10 HagenPoiseulle Flow 1803.5 Shock Waves 1833.5.1 The Method of Characteristics 1833.5.2 Steepening of Sound Waves 1863.5.3 The RankineHugoniot Shock Jump Conditions 1873.5.4 Shock Velocity 1913.5.5 The Sedov Solution 1923.6 Instabilities 1953.6.1 Gravity Waves 1973.6.2 The RayleighTaylor Instability 1983.6.3 The KelvinHelmholtz Instability 1993.6.4 Thermal Instability 2023.6.5 Heat Conduction 2063.6.6 Convection 2093.6.7 Turbulence 210

Further Reading 213

4 Fundamentals of Plasma Physics and Magneto-Hydrodynamics 2154.1 Collision-Less Plasmas 2154.1.1 Shielding and the Debye Length 215

VIII Contents

4.1.2 The Plasma Frequency 2194.2 Electromagnetic Waves in Media 2194.2.1 Polarisation and Dielectric Displacement 2204.2.2 Structure of the Dielectric Tensor 2224.3 Dispersion Relations 2254.3.1 General Form of the Dispersion Relations 2254.3.2 Transversal and Longitudinal Waves 2274.3.3 Longitudinal and Transversal Dielectricities 2274.3.4 Landau Damping 2304.4 Electromagnetic Waves in Thermal Plasmas 2324.4.1 Longitudinal and Transversal Dielectricities 2334.4.2 Dispersion Measure and Damping 2364.5 The Magneto-Hydrodynamic Equations 2384.5.1 Assumptions 2384.5.2 The Induction Equation 2404.5.3 Eulers Equation 2414.5.4 Energy and Entropy 2434.5.5 Incompressible Flows 2454.5.6 Magnetic Advection and Diffusion 2454.6 Generation of Magnetic Fields 2464.7 Ambipolar Diffusion 2494.7.1 Velocity-Averaged Scattering Cross Section 2504.7.2 Friction Force and Diffusion Coefficient 2524.8 Waves in Magnetised Cold Plasmas 2544.8.1 The Dielectric Tensor 2544.8.2 Contribution by Ions 2574.8.3 Dispersion Relations in a Cold, Magnetised Plasma 2594.8.4 Longitudinal and Transverse Waves 2614.8.5 Faraday Rotation 2634.9 Hydromagnetic Waves 2664.9.1 Linearised Perturbation Equations 2664.9.2 Alfvn Waves 2694.9.3 Slow and Fast Hydro-Magnetic Waves 270

Further Reading 272

5 Stellar Dynamics 2735.1 The Jeans Equations and Jeans Theorem 2735.1.1 Collision-Less Motion in a Gravitational Field 2735.1.2 The Relaxation Time Scale 2755.1.3 The Jeans Equations 2775.1.4 Jeans Equations in Cylindrical and Spherical Coordinates 2805.1.5 Application to Spherical Systems 2815.1.6 The Tensor Virial Theorem in Stellar Dynamics 2865.1.7 Jeans Theorem 2885.2 Equilibrium and Stability 290

Contents IX

5.2.1 The Isothermal Sphere 2905.2.2 Equilibrium and Relaxation 2945.2.3 Linear Analysis and the Jeans Swindle 2955.2.4 Jeans Length and Jeans Mass 2975.2.5 Disk Potentials 2985.2.6 Fluid Equations for Two-Dimensional Systems 3015.2.7 Dispersion Relation 3025.2.8 Toomres Criterion 3045.3 Dynamical Friction 3055.3.1 Deflection of Point Masses 3065.3.2 Velocity Changes 3085.3.3 Chandrasekhars Formula 308

Further Reading 312

6 Brief Summary and Concluding Remarks 313

Index 315

XI

Preface

This book is not in any sense complete or exhaustive, and it is not meant to be. Itssubject, theoretical astrophysics, is vast and cannot possibly be comprehensivelycovered in a single volume.

This book has a rather different purpose. It is intended as a textbook for stude