Astrophysical Flows

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1 The basic fluid equations......Page 13
1.1.1 Mass conservation......Page 14
1.1.2 Momentum conservation......Page 15
1.2 The Lagrangian derivative......Page 16
1.3 Conservation of energy......Page 17
1.4 The equation of state and useful approximations......Page 18
1.4.2 Adiabatic flow......Page 19
1.5 The MHD approximation......Page 20
1.5.1 Notation and units......Page 22
1.6.2 Advection of vortex lines......Page 23
1.7 Conservation of energy......Page 24
1.7.2 Magnetic energy......Page 25
1.8 Further reading......Page 26
1.9 Problems......Page 27
2 Compressible media......Page 29
2.1.1 Small-amplitude sound waves......Page 30
2.1.2 Fourier transforms and the dispersion relation......Page 32
2.1.3 Waves in a magnetic medium......Page 33
2.1.3.1 Wavefronts parallel to the magnetic field......Page 35
2.1.3.2 Wavefronts perpendicular to the magnetic field......Page 37
2.2 Non-linear flow in one dimension......Page 38
2.2.1 Regions of influence......Page 41
2.2.2 Development of shocks......Page 43
2.2.3.1 Non-magnetic fluid......Page 45
2.2.3.2 Magnetic fluid......Page 48
2.2.4 Shock waves in general......Page 49
2.4 Problems......Page 50
3.1 Steady inflow/outflow......Page 56
3.1.1 Bondi accretion......Page 57
3.2 Explosion in a uniform medium......Page 62
3.2.1 Shock conditions......Page 63
3.2.2 Similarity variables......Page 64
3.2.3 The similarity (Taylor–Sedov) equations......Page 66
3.2.4 Solving the Taylor–Sedov equations......Page 68
3.4 Problems......Page 70
4.1 Models of stars......Page 72
4.2 Perturbing the models......Page 74
4.3 Eulerian and Lagrangian perturbations......Page 75
4.3.1 The perturbed velocity......Page 77
4.4 Adiabatic perturbations – a variational principle......Page 78
4.4.1 Implications......Page 81
4.4.2 Implication for stability......Page 82
4.4.2.2 Instability......Page 83
4.5 The Schwarzschild stability criterion......Page 85
4.6 Further reading......Page 86
4.7 Problems......Page 87
5 Stellar oscillations – waves in stratified media......Page 90
5.1 Waves in a plane-parallel atmosphere......Page 91
5.1.1 Local analysis......Page 94
5.2 Vertical waves in a polytropic atmosphere......Page 96
5.2.2 The governing equation......Page 97
5.2.3 Solution of the equation......Page 98
5.4 Problems......Page 99
6 Damping and excitation of stellar oscillations......Page 102
6.1 A simple set of oscillations......Page 103
6.2 Damping by conductivity......Page 104
6.2.1 An alternative derivation......Page 105
6.3 The effect of heating and cooling – the…......Page 107
6.4 The effect of opacity – the k-mechanism......Page 109
6.4.2 Perturbing the underlying oscillations:Q…......Page 111
6.5 Further reading......Page 113
7.1 Magnetic buoyancy......Page 114
7.2 The Parker instability......Page 118
7.2.1 Modes with ky = 0......Page 121
7.2.2 Modes with kx = 0, ky…......Page 122
7.4 Problems......Page 123
8 Thermal instabilities......Page 125
8.1 Linear perturbations and the Field criterion......Page 126
8.1.2 No net heating or cooling, but small conductivity......Page 128
8.1.3 Slow cooling......Page 129
8.2 Heating and cooling fronts......Page 130
8.4 Problems......Page 132
9.1 The Jeans instability......Page 135
9.2.1 Equilibrium configuration......Page 137
9.2.2 Stability analysis......Page 139
9.3 Stability of a thin slab......Page 140
9.4 Further reading......Page 142
9.5 Problems......Page 143
10 Linear shear flows......Page 146
10.1 Perturbation of a linear shear flow......Page 147
10.3 Rayleigh's inflexion point theorem......Page 148
10.4 Fjørtoft's theorem......Page 150
10.5 Physical interpretation......Page 151
10.6 Co-moving phase......Page 153
10.7 Stratified shear flow......Page 154
10.8 The Richardson criterion......Page 156
10.10 Problems......Page 157
11.1 Rotating fluid equilibria......Page 162
11.2 Making rotating stellar models......Page 163
11.3 Meridional circulation......Page 166
11.3.1 The basic snag with meridional circulation......Page 167
11.4 Rotation and magnetism......Page 168
11.6 Problems......Page 169
12.1 Incompressible shear flow in a rigid cylinder......Page 170
12.1.1 Axisymmetric perturbations: Rayleigh’s criterion......Page 173
12.2 Axisymmetric stability of a compressible rotating flow......Page 174
12.2.1 Equilibrium......Page 175
12.2.2 Stability......Page 176
12.2.3 The Solberg–Høiland criterion......Page 177
12.3 Circular shear flow with a magnetic field......Page 179
12.3.1 Local analysis......Page 182
12.3.2 The Balbus–Hawley instability......Page 183
12.4.1.1 Axisymmetric perturbations – the Toomre criterion......Page 184
12.4.1.2 Non-axisymmetric disturbances – spiral arms......Page 186
12.6 Problems......Page 188
13.1 The non-rotating `star’......Page 190
13.2 Uniform rotation......Page 193
13.2.1.1 The Chandrasekhar –Friedmann–Schutz instability......Page 194
13.2.2 The r-modes......Page 195
13.2.2.2 The CFS instability continued......Page 198
13.4 Problems......Page 199
14.1 Equilibrium configuration......Page 203
14.2 The perturbation equations......Page 205
14.3.1 Large m......Page 207
14.3.2 Thin cylindrical shell......Page 208
14.5 Problems......Page 209
References......Page 211
Index......Page 215