Fundamentals of Astrophysical Fluid Dynamics: Hydrodynamics, Magnetohydrodynamics, and Radiation Hydrodynamics

634 Pages·2020·10.001 MB·English

by  Shoji Kato, Jun Fukue

Astronomy and Astrophysics Library
Shoji Kato · Jun Fukue
Fundamentals
of Astrophysical
Fluid Dynamics
Hydrodynamics, Magnetohydrodynamics,
and Radiation Hydrodynamics
SeriesEditors: MartinA.Barstow,UniversityofLeicester,Leicester,UK
AndreasBurkert,UniversityObservatoryMunich,Munich,
Germany
AthenaCoustenis,Paris-MeudonObservatory,Meudon,
France
RobertoGilmozzi,EuropeanSouthernObservatory(ESO),
Garching,Germany
GeorgesMeynet,GenevaObservatory,Versoix,Switzerland
ShinMineshige,DepartmentofAstronomy,Kyoto,Japan
IanRobson,TheUKAstronomyTechnologyCentre,
Edinburgh,UK
PeterSchneider,Argelander-InstitutfürAstronomie,Bonn,
Germany
StevenN.Shore,UniversitàdiPisa,PISA,Italy
VirginiaTrimble,UniversityofCalifornia,Irvine,USA
DerekWard-Thompson,UniversityofCentralLancashire,
Preston,UK
Moreinformationaboutthisseriesathttp://www.springer.com/series/848
Shoji Kato (cid:129) Jun Fukue
Fundamentals
of Astrophysical Fluid
Dynamics
Hydrodynamics, Magnetohydrodynamics,
and Radiation Hydrodynamics
ShojiKato JunFukue
KyotoUniversity DivisionofScienceEducation
Kyoto,Japan OsakaKyoikuUniversity
Kashiwara,Osaka,Japan
ISSN0941-7834 ISSN2196-9698 (electronic)
AstronomyandAstrophysicsLibrary
ISBN978-981-15-4173-5 ISBN978-981-15-4174-2 (eBook)
https://doi.org/10.1007/978-981-15-4174-2
©SpringerNatureSingaporePteLtd.2020
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Preface
Astrophysical fluid dynamics is one of the important branches of astrophysics.
In astrophysics, we apply all kinds of physics to investigate and understand the
structure and evolution of the Universe. Because information from the Universe
were primarily brought to us through optical observations, physics of radiative
transfer was first developed in astrophysics. Since then, various branches of
physicsincludingnuclearphysics,highenergyphysics,andgeneralrelativityhave
been applied and developedin astrophysics. The application of fluid dynamics to
astrophysicsis,however,relativelylate,comparedwiththeabove-mentionedfields
ofphysics,sincefull-blownstudiesondynamicsofastrophysicalobjectsbeganafter
elucidationoftheirstructure.
Inthepresentadvancedstagesofastrophysics,however,studiesonastrophysical
objects from the dynamical viewpoints are important. Since many astrophysical
objects are gaseoussystems subject to magnetic and radiation fields, their studies
from the viewpoints of magnetized fluids and radiative fluids are important. The
purpose of this textbook is to provide the basics of astrophysicalhydrodynamics,
magnetohydrodynamics,andradiationhydrodynamicstoadvancedundergraduates
andabove.Atthesametime,wehopethatthematerialspresentedinthistextbook
are helpful for researchers to organize their knowledge on astrophysical fluid
dynamics.
Thepresentbookconsistsofthreeparts.InPartI,wedescribethefundamental
and important processes in astrophysical fluid dynamics, and, in Part II, some
important magnetohydrodynamical processes are presented. In Part III, starting
fromthebasicradiativetransfer,wedescriberadiativefluiddynamics.
Finally,weappreciatemanycolleagueswemetinourresearchactivities.Without
collaborations and arguments with them, the present book could not be written.
We thank the Springer Nature for publishing this textbook of astrophysical fluid
dynamics.
Kyoto,Japan ShojiKato
Osaka,Japan JunFukue
2February2020
v
Contents
1 Introduction................................................................. 1
1.1 ObjectsTreatedinAstrophysics..................................... 1
1.2 Gaseous States of Astrophysical Objects
onTemperature–DensityDiagram................................... 2
1.2.1 ConditionforEnsembleofFreeParticles ................. 2
1.2.2 ConditionofDegenerateGas .............................. 6
1.2.3 RelativisticGases........................................... 8
1.2.4 GyrationofElectronsaroundMagneticFields............ 9
1.3 CharacteristicsofPhysicalStatesofAstrophysicalObjects ....... 9
1.3.1 GravitationalForceandSelf-Gravity...................... 10
1.3.2 MagneticFields............................................. 11
1.3.3 RadiationFields ............................................ 12
PartI HydrodynamicalPhenomenainAstrophysicalObjects
2 BasicEquationsforHydrodynamics..................................... 17
2.1 ConditionforFluidApproximations................................ 17
2.2 EulerianDescriptionofHydrodynamicalEquations ............... 18
2.3 CommentsonEnergyConservation................................. 19
2.3.1 MechanicalEnergyandItsConservation.................. 20
2.3.2 ConservationofTotalEnergy.............................. 20
2.3.3 ConservationofThermalEnergy .......................... 22
2.4 BernoulliTheoremandVorticityConservation..................... 22
2.4.1 BernoulliTheorem.......................................... 22
2.4.2 HelmholtzTheoremandHelicityConservation........... 23
2.5 LagrangianDescriptionofDisplacementVectorξ ................. 25
2.5.1 Equation Describing Small-Amplitude
PerturbationsinTermsofDisplacementVectorξ ........ 27
2.5.2 HermitianCharacterofOperatorL(ξ) .................... 28
vii
viii Contents
2.5.3 WaveEnergyandItsConservation ........................ 29
2.5.4 OrthogonalityofNormalModes........................... 30
Reference..................................................................... 31
3 AstrophysicalFluidFlows................................................. 33
3.1 ParkerWindandBondiAccretion................................... 33
3.1.1 BasicEquationsforSteadySphericalFlow............... 34
3.1.2 SoundSpeedandCriticalPoints........................... 35
3.1.3 SolutionBehaviorNeartoCenteroratInfinity........... 36
3.1.4 MachNumberandSingularPointAnalysis............... 37
3.1.5 AGeneralArgumentonTopologyaround
CriticalPoint................................................ 39
3.2 AccretionDisks ...................................................... 43
3.2.1 StandardAccretionDiskModel ........................... 43
3.3 SlimDisksandAdvection-DominatedAccretionFlows........... 48
3.3.1 SlimDisks .................................................. 48
3.3.2 Advection-DominatedAccretionFlows(ADAFs)........ 50
3.3.3 ThermalEquilibriumCurves............................... 50
3.4 MeridionalCirculationinStars...................................... 52
3.5 Self-SimilarSphericalAccretion.................................... 53
3.5.1 Self-SimilarTreatments.................................... 54
3.5.2 TypicalSolutions ........................................... 56
References.................................................................... 56
4 WavePhenomenainAstrophysicalObjects ............................. 59
4.1 WavesinGravitationallyStratifiedAtmosphere.................... 59
4.1.1 Brunt–VäisäläFrequency................................... 60
4.1.2 AcousticandGravityWaves............................... 61
4.2 RadialandNonradialOscillationsofStars.......................... 64
4.2.1 BasicEquationsDescribingNonradialOscillations
ofStars...................................................... 64
4.3 WavesinRotatingSphericalShell................................... 68
4.3.1 Rossby–HaurwitzWaves................................... 68
4.4 WavesinNonself-GravitatingDisks ................................ 72
4.4.1 UnperturbedState .......................................... 73
4.4.2 SmallAmplitudeAdiabaticPerturbations................. 74
4.4.3 EigenvaluesandEigenfunctions........................... 75
4.4.4 ClassificationofOscillationModesandRossby
WaveMode ................................................. 77
4.5 Self-GravitatingDisksandToomre’sQParameter ................ 79
4.5.1 DerivationofDispersionRelation ......................... 80
4.5.2 Collisionless Stellar Disks and Comparison
withGaseousDisks......................................... 84
4.6 ShockWavesandShockConditions ................................ 88
4.6.1 Growthof Finite-AmplitudeAcoustic Waves
toShockWaves............................................. 89
Contents ix
4.6.2 ConditionofDiscontinuityatShockFront................ 92
4.6.3 StrengthofShockinCaseofIdealGases ................. 94
4.6.4 EntropyIncreasebyShockDiscontinuity................. 95
4.7 PropagationofBlastWavesandSedov–TaylorSimilarity
Solution............................................................... 96
References.................................................................... 98
5 ConvectionandRelatedTopics............................................ 101
5.1 ConditionofConvectiveInstability ................................. 101
5.1.1 Boussinesq Approximation and Criterion
ofConvectiveInstability ................................... 104
5.2 MixingLengthTheoryandBeyond................................. 107
5.3 GenerationofOscillationsbyTurbulence........................... 109
5.3.1 Lighthill Theory of Excitation of Acoustic
OscillationsbyTurbulence................................. 110
5.3.2 StochasticExcitationofOscillationsbyGoldreich
andKeeley .................................................. 115
References.................................................................... 117
6 DynamicalInstabilityandDynamicalExcitationofOscillations ..... 119
6.1 IntroductoryRemarks................................................ 119
6.2 Rayleigh–TaylorInstabilityandRayleighCriterion................ 120
6.2.1 Rayleigh–TaylorInstability................................ 120
6.2.2 RayleighCriterion.......................................... 122
6.3 Solberg-HøilandCriterion ........................................... 122
6.3.1 DerivationofDispersionRelation ......................... 123
6.4 GravitationalInstability.............................................. 126
6.5 Kelvin–HelmholtzInstability:ResonantInstabilityI............... 129
6.5.1 WaveEnergyandResonantInteraction ................... 131
6.6 CorotationResonance:ResonantInstabilityII...................... 134
6.6.1 SpiralDensityWavesinSelf-gravitatingDisks........... 135
6.6.2 P-ModeOscillationsinNonself-GravitatingDisks....... 143
6.6.3 RossbyWaveInstability.................................... 148
6.7 Wave-Wave ResonanceInstabilityin DeformedDisks:
ResonantInstabilityIII............................................... 152
6.7.1 Quasi-Nonlinear Coupling between Two
OscillationsthroughTidalWave........................... 154
6.7.2 WaveEnergyandItsSign.................................. 157
6.7.3 ConditionsofResonantGrowth ........................... 159
6.7.4 AnApplication:SuperhumpsofDwarfNovae............ 160
References.................................................................... 162
7 InstabilitiesDuetoDissipativeProcessesI(SecularInstability)...... 165
7.1 ThermalInstabilityinOpticallyThinMedium ..................... 165
7.1.1 QualitativeArgumentonStabilityCriterion .............. 166
7.1.2 AnalyticalDerivationofInstabilityCriterion............. 168
x Contents
7.2 Flash,Flicker,andShellBurninginStars........................... 170
7.2.1 HeliumandCarbonFlash.................................. 171
7.2.2 FlickerandShellBurninginBinaryStars................. 172
7.3 ThermalandSecularInstabilitiesinDisks.......................... 173
7.3.1 ThermalInstabilityinStandardDisks..................... 174
7.3.2 Secular(Viscous)InstabilityinStandardDisks........... 176
7.3.3 StabilityAnalysesonEquilibriumSequence.............. 179
7.4 Goldreich–Schubert–FrickeCriterion............................... 181
7.5 GravothermalInstability............................................. 185
7.5.1 DerivationofInstabilityCriterion ......................... 186
7.5.2 Relation Between Stability Criterion
andEquilibriumSequence ................................. 190
7.5.3 PhysicalProcessesofGravothermalInstability........... 193
References.................................................................... 194
8 OverstabilityDuetoDissipativeProcessesII: Excitationof
Oscillations.................................................................. 195
8.1 ExcitationofStellarOscillationsbyThermalProcesses........... 195
8.2 ExcitationofDiskOscillationsbyViscousProcesses.............. 200
8.3 DoubleDiffusiveInstability ......................................... 203
8.3.1 OverstableConvectioninUniformlyRotatingFluid ..... 204
8.3.2 Overstable Convection in Chemically
InhomogeneousMedia..................................... 208
8.3.3 Goldreich–Schubert–FrickeCriterion-II .................. 210
References.................................................................... 212
9 GeneralRelativisticHydrodynamics .................................... 213
9.1 RelativisticHydrodynamicEquations............................... 213
9.1.1 RelativisticEquationofState .............................. 215
9.2 GeneralRelativisticSphericalStars................................. 218
9.2.1 StaticLineElementwithSphericalSymmetry............ 218
9.2.2 Tolman–Oppenheimer–VolkoffEquationand
CollapseofStars............................................ 219
9.3 SchwarzschildMetricandKerrMetric.............................. 221
9.4 ParticleMotionsinSchwarzschildandKerrMetrics............... 224
9.4.1 AngularVelocityofCircularMotions..................... 224
9.4.2 Stability of Circular Motions and Epicyclic
Frequency................................................... 226
9.4.3 EpicyclicFrequency........................................ 229
9.5 RelativisticTori....................................................... 231
9.5.1 DerivationofEq.(9.85) .................................... 236
References.................................................................... 237