Theoretical Astrophysics: An Introduction

Theoretical Astrophysics......Page 1
Contents......Page 7
Preface......Page 13
Acknowledgements......Page 17
Colour Plates......Page 19
1.1.2 Charges and Electromagnetic Fields......Page 23
1.2 Lorentz Invariance......Page 24
1.2.1 The Special Lorentz Transform......Page 26
1.2.2 Minkowski Space......Page 28
1.2.3 Some Properties of the Minkowski World......Page 31
1.2.4 Relativistic Dynamics......Page 34
1.3.1 Field Tensor and Sources......Page 38
1.3.2 Lorentz Transform of the Electromagnetic Field......Page 40
1.3.3 Maxwell's Equations......Page 41
1.3.4 Energy-Momentum Conservation......Page 43
1.3.5 Liénard–Wiechert Potentials and the Larmor Formula......Page 44
1.3.6 The Lorentz Force......Page 48
1.4.1 The BBGKY Hierarchy and the Boltzmann Equation......Page 50
1.4.3 Diffusion in Phase-Space: The Fokker–Planck Approximation......Page 53
1.4.4 Diffusion in Absolute Momentum......Page 56
1.4.5 Calculation of the Diffusion Coefficient D2......Page 57
Further Reading......Page 59
2.1 Thomson Scattering......Page 61
2.2 Spectra......Page 67
2.3 Synchrotron Radiation......Page 72
2.3.1 Larmor Frequency and Relativistic Focussing......Page 73
2.3.3 Synchrotron Spectrum......Page 75
2.4.1 Orbit of an Electron Scattering off an Ion......Page 80
2.4.2 Fourier Transform of the Orbit......Page 83
2.4.3 Integration over Impact Parameters......Page 84
2.4.4 Average over Electron Velocities, Thermal Bremsstrahlung......Page 85
2.5.1 Damping Force......Page 87
2.5.2 Transfer of Energy from a Moving Charge to a Radiation Field......Page 91
2.6.1 Energy Change in the Scattering Process......Page 94
2.6.2 Net Energy Transfer......Page 96
2.6.3 The Kompaneets Equation......Page 99
2.7.1 Transition Probability......Page 105
2.7.2 Perturbing Hamiltonian......Page 106
2.7.3 Decomposition of the Electromagnetic Field......Page 109
2.7.4 Dipole Approximation......Page 110
2.7.5 Cross Sections......Page 112
2.7.6 Photoionisation Cross Section......Page 114
2.8 Shapes of Spectral Lines......Page 116
2.8.1 Natural Line Width......Page 117
2.8.2 Collisional Broadening......Page 119
2.8.3 Doppler Broadening of Spectral Lines......Page 120
2.8.4 The Voigt Profile......Page 121
2.8.5 Equivalent Widths and Curves-of-Growth......Page 122
2.9 Radiation Quantities......Page 125
2.9.1 Specific Intensity......Page 126
2.9.2 Moments of the Intensity......Page 127
2.9.3 Relativistic Invariance of I/3......Page 129
2.10 The Planck Spectrum and Einstein Coefficients......Page 131
2.10.1 The Planck Spectrum......Page 132
2.10.2 Transition Balance and the Einstein Coefficients......Page 137
2.11.1 Absorption Coefficients and Emissivity......Page 139
2.11.2 Radiation Transport in a Simple Case......Page 141
2.11.3 Emission and Absorption in the Continuum Case......Page 143
2.11.4 Energy Transport Through Absorbing Media......Page 146
Further Reading......Page 148
3.1.1 Particle-Current Density and Energy-Momentum Tensor......Page 149
3.1.2 Collisional Invariants and the Fluid Approximation......Page 152
3.1.3 The Equations of Ideal Hydrodynamics......Page 156
3.2.1 Hydrodynamic Equations......Page 161
3.2.2 Hydrodynamics in a Weak Gravitational Field......Page 164
3.2.3 Gravitational Field Equation......Page 165
3.2.4 The Combined Set of Equations......Page 166
3.2.5 Perturbative Analysis......Page 167
3.3.1 Diffusion of Particles, Momentum and Internal Energy......Page 170
3.3.2 The Equations of Viscous Hydrodynamics......Page 174
3.3.3 Entropy......Page 176
3.3.4 Fluids in a Gravitational Field......Page 177
3.3.5 The Tensor Virial Theorem......Page 179
3.3.6 Transformation to Cylindrical or Spherical Coordinates......Page 182
3.4 Flows under Specific Circumstances......Page 184
3.4.1 Sound Waves......Page 185
3.4.2 Polytropic Equation of State......Page 186
3.4.3 Hydrostatic Equilibrium......Page 189
3.4.4 Vorticity and Kelvin's Circulation Theorem......Page 192
3.4.5 Bernoulli's Constant......Page 194
3.4.6 Bondi Accretion......Page 197
3.4.7 Bernoulli's Law for Irrotational, Non-Stationary Flows......Page 200
3.4.9 The Reynolds Number......Page 201
3.4.10 Hagen–Poiseulle Flow......Page 202
3.5.1 The Method of Characteristics......Page 205
3.5.2 Steepening of Sound Waves......Page 208
3.5.3 The Rankine–Hugoniot Shock Jump Conditions......Page 209
3.5.4 Shock Velocity......Page 213
3.5.5 The Sedov Solution......Page 214
3.6 Instabilities......Page 217
3.6.1 Gravity Waves......Page 219
3.6.2 The Rayleigh–Taylor Instability......Page 220
3.6.3 The Kelvin–Helmholtz Instability......Page 221
3.6.4 Thermal Instability......Page 224
3.6.5 Heat Conduction......Page 228
3.6.6 Convection......Page 231
3.6.7 Turbulence......Page 232
Further Reading......Page 235
4.1.1 Shielding and the Debye Length......Page 237
4.2 Electromagnetic Waves in Media......Page 241
4.2.1 Polarisation and Dielectric Displacement......Page 242
4.2.2 Structure of the Dielectric Tensor......Page 244
4.3.1 General Form of the Dispersion Relations......Page 247
4.3.3 Longitudinal and Transversal Dielectricities......Page 249
4.3.4 Landau Damping......Page 252
4.4 Electromagnetic Waves in Thermal Plasmas......Page 254
4.4.1 Longitudinal and Transversal Dielectricities......Page 255
4.4.2 Dispersion Measure and Damping......Page 258
4.5.1 Assumptions......Page 260
4.5.2 The Induction Equation......Page 262
4.5.3 Euler's Equation......Page 263
4.5.4 Energy and Entropy......Page 265
4.5.6 Magnetic Advection and Diffusion......Page 267
4.6 Generation of Magnetic Fields......Page 268
4.7 Ambipolar Diffusion......Page 271
4.7.1 Velocity-Averaged Scattering Cross Section......Page 272
4.7.2 Friction Force and Diffusion Coefficient......Page 274
4.8.1 The Dielectric Tensor......Page 276
4.8.2 Contribution by Ions......Page 279
4.8.3 Dispersion Relations in a Cold, Magnetised Plasma......Page 281
4.8.4 Longitudinal and Transverse Waves......Page 283
4.8.5 Faraday Rotation......Page 285
4.9.1 Linearised Perturbation Equations......Page 288
4.9.2 Alfvén Waves......Page 291
4.9.3 Slow and Fast Hydro-Magnetic Waves......Page 292
Further Reading......Page 294
5.1.1 Collision-Less Motion in a Gravitational Field......Page 295
5.1.2 The Relaxation Time Scale......Page 297
5.1.3 The Jeans Equations......Page 299
5.1.4 Jeans Equations in Cylindrical and Spherical Coordinates......Page 302
5.1.5 Application to Spherical Systems......Page 303
5.1.6 The Tensor Virial Theorem in Stellar Dynamics......Page 308
5.1.7 Jeans' Theorem......Page 310
5.2.1 The Isothermal Sphere......Page 312
5.2.2 Equilibrium and Relaxation......Page 316
5.2.3 Linear Analysis and the Jeans Swindle......Page 317
5.2.4 Jeans Length and Jeans Mass......Page 319
5.2.5 Disk Potentials......Page 320
5.2.6 Fluid Equations for Two-Dimensional Systems......Page 323
5.2.7 Dispersion Relation......Page 324
5.2.8 Toomre's Criterion......Page 326
5.3 Dynamical Friction......Page 327
5.3.1 Deflection of Point Masses......Page 328
5.3.3 Chandrasekhar's Formula......Page 330
Further Reading......Page 334
6 Brief Summary and Concluding Remarks......Page 335
Index......Page 337