1 Introducing Asteroseismology ................................. 1
1.1 Introduction ............................................ 1
1.1.1 The Music of the Spheres ......................... 1
1.1.2 Seeing with Sound ................................ 2
1.1.3 Can we "Hear" the Stars? ......................... 6
1.1.4 Pressure Modes and Gravity Modes ................. 7
1.2 1-D Oscillations ........................................ 7
1.2.1 1-D Oscillations on a String ..................... 7
1.2.2 1-D Oscillations in an Organ Pipe ................ 8
1.3 2-D Oscillations in a Drum Head ......................... 8
1.4 3-D Oscillations in Stars .............................. 10
1.4.1 Radial Modes .................................... 11
1.4.2 Nonradial Modes ................................. 12
1.4.3 The Effect of Rotation .......................... 15
1.4.4 So how does Asteroseismology Work? .............. 17
1.4.5 p Modes and g Modes ............................. 17
1.5 An Asteroseismic HR Diagram for p-Mode Pulsators ....... 21
1.6 A Pulsation HR Diagram ................................. 23
1.6.1 How do Stars Pulsate: The Relevant Time
Scales .......................................... 23
1.6.2 Why do Stars Pulsate: Driving Mechanisms ........ 26
1.6.3 What Selects the Modes of Pulsation in Stars? ... 28
2 Stellar Oscillations across the Hertzsprung-Russell
Diagram ..................................................... 31
2.1 Stellar Evolution in a Nutshell ........................ 33
2.2 Variability Studies from Large-Scale Surveys ........... 38
2.2.1 Hipparcos ....................................... 38
2.2.2 Ground-Based Surveys ............................ 39
2.3 Oscillations Near the Main Sequence .................... 42
2.3.1 Solar-Like Oscillations in Solar-Like Stars ..... 42
2.3.2 γ Dor Stars ..................................... 45
2.3.3 δ Sct Stars ..................................... 49
2.3.4 SXPhe Stars ..................................... 53
2.3.5 Rapidly Oscillating Ap Stars .................... 56
2.3.6 Slowly Pulsating В Stars ........................ 60
2.3.7 β Сер Stars ..................................... 65
2.3.8 Pulsating Be Stars .............................. 68
2.4 Oscillations in Pre-Main-Sequence Stars ................ 72
2.5 Pulsations in Evolved Stars with M ≤ 9M ............... 75
2.5.1 RRLyrae Stars ................................... 75
2.5.2 Cepheids ........................................ 80
2.5.3 RV Tauri Stars .................................. 86
2.5.4 Mira and Semi-Regular Variables ................. 88
2.5.5 Solar-Like Oscillations in Red Giants ........... 92
2.6 Pulsations in Evolved Stars with M ≥ 9M ............... 95
2.6.1 Periodically Variable В and A Supergiants ....... 96
2.6.2 Wolf-Rayet Stars ............................... 101
2.6.3 The Role of Core g Modes in Supernova
Explosions ..................................... 103
2.7 Compact Oscillators ................................... 104
2.7.1 Variable Subdwarf В Stars ...................... 105
2.7.2 White Dwarf Stars .............................. 110
2.7.3 Neutron Stars .................................. 124
2.8 Pulsations in Binaries ................................ 126
2.8.1 Tidal Perturbations of Free Oscillations ....... 127
2.8.2 Tidally Induced Oscillations ................... 128
2.8.3 Are the SX Phe Stars all Blue Stragglers? ...... 129
2.8.4 Are all Dusty RV Tauri Stars Binaries? ......... 131
2.8.5 Hydrogen-Deficient Carbon Stars and Extreme
Helium Stars ................................... 131
2.8.6 Pulsating sdB Primaries ........................ 133
2.8.7 Pulsating Cataclysmic Variables ................ 134
2.8.8 X-Ray Burst Oscillations ....................... 136
2.9 Conclusions ........................................... 136
3 Theory of Stellar Oscillations ............................. 137
3.1 General Hydrodynamics ................................. 137
3.1.1 Equations of Hydrodynamics ..................... 137
3.1.2 The Adiabatic Approximation .................... 142
3.1.3 Equilibrium States and Perturbation Analysis ... 143
3.1.4 Simple Waves ................................... 147
3.2 Equilibrium Stellar Structure ......................... 154
3.2.1 Basic Properties of Stellar Evolution .......... 154
3.2.2 Microphysics of Stellar Interiors .............. 161
3.2.3 Standard Stellar Evolution ..................... 167
3.2.4 Complications .................................. 179
3.3 Equations of Linear Stellar Oscillations .............. 188
3.3.1 The Oscillation Equations ...................... 188
3.3.2 Linear, Adiabatic Oscillations ................. 195
3.3.3 The Dependence of the Frequencies on the
Equilibrium Structure .......................... 199
3.4 Asymptotic Theory of Stellar Oscillations ............. 202
3.4.1 The Cowling Approximation ...................... 203
3.4.2 Trapping of p and g Modes ...................... 203
3.4.3 Asymptotic Properties of Frequencies and
Eigenfunctions ................................. 212
3.5 Computed Properties of Modes of Oscillation ........... 222
3.5.1 Results for the Present Sun .................... 222
3.5.2 The Classification of Modes .................... 229
3.5.3 Results for the Models with Convective Cores ... 231
3.6 Variational Properties of Stellar Adiabatic
Oscillations .......................................... 234
3.6.1 The Oscillation Equations as Linear
Eigenvalue Problems in a Hilbert Space ......... 235
3.6.2 Effects on Frequencies of a Change in the
Model .......................................... 238
3.7 Driving Mechanisms .................................... 240
3.7.1 The Work Integral .............................. 241
3.7.2 The Condition for Instability .................. 246
3.7.3 Effects of Convection on Stellar Stability ..... 253
3.7.4 Excitation of High-Order g Modes ............... 256
3.7.5 Stochastic Excitation of Oscillations .......... 258
3.8 Effects of Rotation ................................... 265
3.8.1 A Simplified Description of the Effect of
Rotation ....................................... 265
3.8.2 The Effect of Large-Scale Velocities on the
Oscillation Frequencies ........................ 266
3.8.3 The Effect of Pure Rotation .................... 269
3.8.4 Properties of Rotational Splitting ............. 272
3.8.5 Effects of Rotation on Low-Frequency Modes ..... 276
3.8.6 Higher-Order Rotational Effects ................ 284
3.8.7 Effect of Rotation on the Excitation of
Oscillations ................................... 290
4 Observational Techniques ................................... 295
4.1 Duty Cycle ............................................ 297
4.2 Time .................................................. 298
4.3 Photometry ............................................ 301
4.3.1 Sources of Error in Photometry ................. 302
4.3.2 Differential Photometry ........................ 309
4.3.3 High-Speed Photometry (Non-differential
Photometry) .................................... 311
4.3.4 Filters ........................................ 313
4.4 Spectroscopy .......................................... 315
4.4.1 Wavelength Stability and Low Frequency Noise
in Spectroscopy ................................ 318
4.4.2 High-Resolution Spectroscopy and Line Profile
Variations ..................................... 319
4.4.3 Requirements of Spectroscopy for
Asteroseismology ............................... 322
4.4.4 Observational Line Diagnostics ................. 325
4.4.5 Increasing the Signal-to-Noise Ratio of the
Line Profile Variations ........................ 328
4.4.6 Increasing the Radial Velocity Precision in
the Context of Exoplanet Finding and Solar-
Like Oscillations .............................. 331
4.4.7 Disentangling Spectra to Interpret the
Oscillations of Double-Lined Binaries .......... 333
5 Frequency Analysis ......................................... 337
5.1 Harmonic Analysis by Least Squares .................... 338
5.1.1 Searching for a Single Frequency ............... 338
5.1.2 Searching for Multiple Frequencies ............. 341
5.2 Non-parametric Frequency Analysis Methods ............. 342
5.2.1 String Length Methods .......................... 342
5.2.2 Phase Dispersion Minimization .................. 348
5.3 Parametric Frequency Analysis Methods ................. 350
5.3.1 The Continuous Fourier Transform of an
Infinite Time Series ........................... 351
5.3.2 The Continuous Fourier Transform of a Finite
Time Series .................................... 352
5.3.3 Real Life: The Discrete Fourier Transform ...... 354
5.3.4 The Classical Periodogram ...................... 360
5.3.5 The Lomb-Scargle Periodogram ................... 361
5.4 Significance Criteria ................................. 363
5.5 Error Estimation of the Derived Frequencies ........... 366
5.5.1 Data without Alias Problems .................... 366
5.5.2 Data suffering from Aliasing ................... 368
5.6 The use of Weights in Merging different Data Sets
for Frequency Analysis ................................ 369
5.7 Damped Oscillations ................................... 372
5.8 Eliminating Aliases ................................... 374
5.9 Conclusions ........................................... 375
6 Mode Identification ........................................ 377
6.1 Mode Identification from Multicolour Photometry ....... 379
6.1.1 General Considerations ......................... 380
6.1.2 Detailed Description ........................... 383
6.1.3 Mode Identification Schemes .................... 393
6.2 Mode Identification from High-Resolution
Spectroscopy .......................................... 405
6.2.1 Calculation of Theoretical Line-Profile
Variations ..................................... 406
6.2.2 Line Profile Fitting ........................... 414
6.2.3 The Moment Method .............................. 415
6.2.4 The Pixel-by-Pixel Method ...................... 434
6.3 Mode Identification from Combined Photometry and
Spectroscopy .......................................... 440
6.4 Towards Mode Identification from Combined
Interferometry and Spectroscopy? ...................... 442
6.5 Towards Mode Identification from Eclipse Mapping? ..... 445
7 Applications of Asteroseismology ........................... 447
7.1 Helioseismology ....................................... 447
7.1.1 Introduction ................................... 447
7.1.2 Analysis of Solar-Oscillation Observations ..... 456
7.1.3 Observational Results on Solar Oscillations .... 461
7.1.4 Properties of Solar Oscillations ............... 470
7.1.5 Principles of Inverse Analysis ................. 488
7.1.6 Inversion for Solar Structure .................. 500
7.1.7 Results on Solar Structure ..................... 505
7.1.8 Results for Solar Rotation ..................... 522
7.1.9 Temporal Variations of the Solar Interior ...... 529
7.2 Solar-Like Pulsators .................................. 535
7.2.1 Observational Aspects .......................... 535
7.2.2 Asteroseismic Diagnostics ...................... 539
7.2.3 The Binary α Centauri A and В .................. 558
7.2.4 The Subgiant η Bootis .......................... 576
7.2.5 The Red Giant ε Ophiuchi ....................... 591
7.3 Heat Driven Main Sequence Stars ....................... 603
7.3.1 The β Сер Star V836 Centauri ................... 603
7.3.2 The β Сер Star ν Eridani ....................... 614
7.3.3 The β Сер Binary θ Ophiuchi .................... 625
7.3.4 HR 1217 among the roAp Stars ................... 634
7.4 Compact Pulsators ..................................... 646
7.4.1 The GW Vir Star PG1159-035, GW Vir itself ...... 646
7.4.2 The DB White Dwarf GD 358, V777Her ............. 651
7.4.3 The Subdwarf В Star PG 0014+067 ................ 656
7.4.4 The Subdwarf В Eclipsing Binary PG 1336-018,
NY Vir ......................................... 661
8 The Future ................................................. 669
8.1 Space Missions ........................................ 670
8.1.1 CoRoT .......................................... 671
8.1.2 The Kepler Mission ............................. 672
8.1.3 BRITE .......................................... 673
8.1.4 PLATO .......................................... 673
8.1.5 Solar Missions ................................. 674
8.2 Ground-Based Networks and Antarctica .................. 675
8.2.1 SONG: A Ground-Based Radial Velocity Network ... 675
8.2.2 Antarctic Asteroseismology ..................... 676
A Summary of the Different Classes of Stellar Pulsators ...... 679
В Properties of Legendre Functions and Spherical Harmonics ... 683
B.l Properties of Legendre Functions ...................... 683
B.2 Properties of Spherical Harmonics ..................... 685
С Mathematical Preliminaries ................................. 689
C.l Formulation of Oscillation Equations in Complex
Form .................................................. 689
C.2 Vector Operators in Spherical Polar Coordinates ....... 690
D Adiabatic Oscillations in an Isothermal Atmosphere ......... 693
D.l Equilibrium Structure ................................. 693
D.2 Oscillation Properties ................................ 694
D.3 Boundary Conditions in a Stellar Atmosphere ........... 696
E Asymptotic Theory of Stellar Oscillations .................. 699
E.l A General Asymptotic Expression ....................... 699
E.2 JWKB Analysis ......................................... 701
E.3 The Duvall Law for p-Mode Frequencies ................. 705
E.3.1 Frequencies in Polytropic Envelopes ............ 707
E.3.2 Frequencies of Low-Degree Modes ................ 709
E.4 Asymptotic Properties of Eigenfunctions ............... 710
E.4.1 Asymptotic Properties of the p-Mode
Eigenfunctions ................................. 712
E.4.2 Asymptotic Properties of the g-Mode
Eigenfunctions ................................. 715
Bibliography .................................................. 717
Subject Index ................................................. 845
Object Index .................................................. 861
Acronym Definition Index ...................................... 865
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