Theoretical Astrophysics: An Introduction

342 Pages·2013·5.57 MB·English

by  Matthias Bartelmann

PHYSICS TEXTBOOK
Matthias Bartelmann
Theoretical
Astrophysics
An Introduction

MatthiasBartelmann
TheoreticalAstrophysics
RelatedTitles
Stepanov,A.V.,Zaitsev,V.V.,Nakariakov, Morison,I.
V.M.
Introduction toAstronomyand
Coronal Seismology Cosmology
WavesandOscillationsinStellarCoronae
2008
2012
ISBN:978-0-470-03333-3
ISBN:978-3-527-40994-5
Stahler,S.W.,Palla,F.
Creighton,J.D.E.,Anderson,W.G.
The Formationof Stars
Gravitational-Wave Physics
and Astronomy 2004
ISBN:978-3-527-40559-6
AnIntroductiontoTheory,Experiment
andDataAnalysis
2011 Shore,S.N.
ISBN:978-3-527-40886-3
The Tapestryof Modern
Astrophysics
Kopeikin,S.,Efroimsky,M.,Kaplan,G.
2003
RelativisticCelestial
ISBN:978-0-471-16816-4
Mechanics of the Solar System
2011 Rybicki,G.B.,Lightman,A.P.
ISBN:978-3-527-40856-6
Radiative Processesin
Astrophysics
1979
ISBN:978-0-471-82759-7
Matthias Bartelmann
Theoretical Astrophysics
An Introduction
WILEY-VCH Verlag GmbH & Co. KGaA
TheAuthor AllbookspublishedbyWiley-VCHarecarefully
produced.Nevertheless,authors,editors,and
Prof.Dr.MatthiasBartelmann publisherdonotwarranttheinformation
UniversitätHeidelberg containedinthesebooks,includingthisbook,to
Inst.TheoretischeAstrophysik befreeoferrors.Readersareadvisedtokeepin
Albert-Überle-Str.2 mindthatstatements,data,illustrations,
69120Heidelberg proceduraldetailsorotheritemsmay
Germany inadvertentlybeinaccurate.
LibraryofCongressCardNo.:
appliedfor
BritishLibraryCataloguing-in-PublicationData:
Acataloguerecordforthisbookisavailable
fromtheBritishLibrary.
Bibliographicinformationpublishedbythe
DeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhis
publicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableonthe
Internetathttp://dnb.d-nb.de.
©2013WILEY-VCHVerlagGmbH&Co.KGaA,
Boschstr.12,69469Weinheim,Germany
Allrightsreserved(includingthoseoftranslation
intootherlanguages).Nopartofthisbookmay
bereproducedinanyform–byphotoprinting,
microfilm,oranyothermeans–nortransmitted
ortranslatedintoamachinelanguagewithout
writtenpermissionfromthepublishers.
Registerednames,trademarks,etc.usedinthis
book,evenwhennotspecificallymarkedas
such,arenottobeconsideredunprotectedby
law.
PrintISBN 978-3-527-41004-0
ePDFISBN 978-3-527-66978-3
ePubISBN 978-3-527-66977-6
mobiISBN 978-3-527-66976-9
CoverDesign Adam-Design,Weinheim
Typesetting le-texpublishingservicesGmbH,
Leipzig
PrintingandBinding MarkonoPrintMedia
PteLtd,Singapore
PrintedinSingapore
Printedonacid-freepaper
V
Contents
Preface XI
Acknowledgements XV
ColourPlates XVII
1 TheoreticalFoundations 1
1.1 Units 1
1.1.1 Lengths,Masses,Times,andTemperatures 1
1.1.2 ChargesandElectromagneticFields 1
1.1.3 NaturalConstants 2
1.2 LorentzInvariance 2
1.2.1 TheSpecialLorentzTransform 4
1.2.2 MinkowskiSpace 6
1.2.3 SomePropertiesoftheMinkowskiWorld 9
1.2.4 RelativisticDynamics 12
1.3 Electromagnetism 16
1.3.1 FieldTensorandSources 16
1.3.2 LorentzTransformoftheElectromagneticField 18
1.3.3 Maxwell’sEquations 19
1.3.4 Energy-MomentumConservation 21
1.3.5 Liénard–WiechertPotentialsandtheLarmorFormula 22
1.3.6 TheLorentzForce 26
1.4 ElementaryKineticTheory 28
1.4.1 TheBBGKYHierarchyandtheBoltzmannEquation 28
1.4.2 CollisionTerms 31
1.4.3 DiffusioninPhase-Space:TheFokker–PlanckApproximation 31
1.4.4 DiffusioninAbsoluteMomentum 34
1.4.5 CalculationoftheDiffusionCoefficientD 35
2
FurtherReading 37
2 RadiationProcesses 39
2.1 ThomsonScattering 39
2.2 Spectra 45
2.3 SynchrotronRadiation 50
VI Contents
2.3.1 LarmorFrequencyandRelativisticFocussing 51
2.3.2 SynchrotronPower 53
2.3.3 SynchrotronSpectrum 53
2.4 Bremsstrahlung 58
2.4.1 OrbitofanElectronScatteringoffanIon 58
2.4.2 FourierTransformoftheOrbit 61
2.4.3 IntegrationoverImpactParameters 62
2.4.4 AverageoverElectronVelocities,ThermalBremsstrahlung 63
2.5 RadiationDamping 65
2.5.1 DampingForce 65
2.5.2 TransferofEnergyfromaMovingChargetoaRadiationField 69
2.6 ComptonScattering 72
2.6.1 EnergyChangeintheScatteringProcess 72
2.6.2 NetEnergyTransfer 74
2.6.3 TheKompaneetsEquation 77
2.7 RadiativeQuantumTransitions 83
2.7.1 TransitionProbability 83
2.7.2 PerturbingHamiltonian 84
2.7.3 DecompositionoftheElectromagneticField 87
2.7.4 DipoleApproximation 88
2.7.5 CrossSections 90
2.7.6 PhotoionisationCrossSection 92
2.8 ShapesofSpectralLines 94
2.8.1 NaturalLineWidth 95
2.8.2 CollisionalBroadening 97
2.8.3 DopplerBroadeningofSpectralLines 98
2.8.4 TheVoigtProfile 99
2.8.5 EquivalentWidthsandCurves-of-Growth 100
2.9 RadiationQuantities 103
2.9.1 SpecificIntensity 104
2.9.2 MomentsoftheIntensity 105
2.9.3 RelativisticInvarianceof Iω/ω3 107
2.10 ThePlanckSpectrumandEinsteinCoefficients 109
2.10.1 ThePlanckSpectrum 110
2.10.2 TransitionBalanceandtheEinsteinCoefficients 115
2.11 AbsorptionandEmission 117
2.11.1 AbsorptionCoefficientsandEmissivity 117
2.11.2 RadiationTransportinaSimpleCase 119
2.11.3 EmissionandAbsorptionintheContinuumCase 121
2.11.4 EnergyTransportThroughAbsorbingMedia 124
FurtherReading 126
3 Hydrodynamics 127
3.1 TheEquationsofIdealHydrodynamics 127
3.1.1 Particle-CurrentDensityandEnergy-MomentumTensor 127
Contents VII
3.1.2 CollisionalInvariantsandtheFluidApproximation 130
3.1.3 TheEquationsofIdealHydrodynamics 134
3.2 RelativisticHydrodynamics 139
3.2.1 HydrodynamicEquations 139
3.2.2 HydrodynamicsinaWeakGravitationalField 142
3.2.3 GravitationalFieldEquation 143
3.2.4 TheCombinedSetofEquations 144
3.2.5 PerturbativeAnalysis 145
3.3 ViscousHydrodynamics 148
3.3.1 DiffusionofParticles,MomentumandInternalEnergy 148
3.3.2 TheEquationsofViscousHydrodynamics 152
3.3.3 Entropy 154
3.3.4 FluidsinaGravitationalField 155
3.3.5 TheTensorVirialTheorem 157
3.3.6 TransformationtoCylindricalorSphericalCoordinates 160
3.4 FlowsunderSpecificCircumstances 162
3.4.1 SoundWaves 163
3.4.2 PolytropicEquationofState 164
3.4.3 HydrostaticEquilibrium 167
3.4.4 VorticityandKelvin’sCirculationTheorem 170
3.4.5 Bernoulli’sConstant 172
3.4.6 BondiAccretion 175
3.4.7 Bernoulli’sLawforIrrotational,Non-StationaryFlows 178
3.4.8 DiffusionofVorticity 179
3.4.9 TheReynoldsNumber 179
3.4.10 Hagen–PoiseulleFlow 180
3.5 ShockWaves 183
3.5.1 TheMethodofCharacteristics 183
3.5.2 SteepeningofSoundWaves 186
3.5.3 TheRankine–HugoniotShockJumpConditions 187
3.5.4 ShockVelocity 191
3.5.5 TheSedovSolution 192
3.6 Instabilities 195
3.6.1 GravityWaves 197
3.6.2 TheRayleigh–TaylorInstability 198
3.6.3 TheKelvin–HelmholtzInstability 199
3.6.4 ThermalInstability 202
3.6.5 HeatConduction 206
3.6.6 Convection 209
3.6.7 Turbulence 210
FurtherReading 213
4 FundamentalsofPlasmaPhysicsandMagneto-Hydrodynamics 215
4.1 Collision-LessPlasmas 215
4.1.1 ShieldingandtheDebyeLength 215
VIII Contents
4.1.2 ThePlasmaFrequency 219
4.2 ElectromagneticWavesinMedia 219
4.2.1 PolarisationandDielectricDisplacement 220
4.2.2 StructureoftheDielectricTensor 222
4.3 DispersionRelations 225
4.3.1 GeneralFormoftheDispersionRelations 225
4.3.2 TransversalandLongitudinalWaves 227
4.3.3 LongitudinalandTransversalDielectricities 227
4.3.4 LandauDamping 230
4.4 ElectromagneticWavesinThermalPlasmas 232
4.4.1 LongitudinalandTransversalDielectricities 233
4.4.2 DispersionMeasureandDamping 236
4.5 TheMagneto-HydrodynamicEquations 238
4.5.1 Assumptions 238
4.5.2 TheInductionEquation 240
4.5.3 Euler’sEquation 241
4.5.4 EnergyandEntropy 243
4.5.5 IncompressibleFlows 245
4.5.6 MagneticAdvectionandDiffusion 245
4.6 GenerationofMagneticFields 246
4.7 AmbipolarDiffusion 249
4.7.1 Velocity-AveragedScatteringCrossSection 250
4.7.2 FrictionForceandDiffusionCoefficient 252
4.8 WavesinMagnetisedColdPlasmas 254
4.8.1 TheDielectricTensor 254
4.8.2 ContributionbyIons 257
4.8.3 DispersionRelationsinaCold,MagnetisedPlasma 259
4.8.4 LongitudinalandTransverseWaves 261
4.8.5 FaradayRotation 263
4.9 HydromagneticWaves 266
4.9.1 LinearisedPerturbationEquations 266
4.9.2 AlfvénWaves 269
4.9.3 SlowandFastHydro-MagneticWaves 270
FurtherReading 272
5 StellarDynamics 273
5.1 TheJeansEquationsandJeans’Theorem 273
5.1.1 Collision-LessMotioninaGravitationalField 273
5.1.2 TheRelaxationTimeScale 275
5.1.3 TheJeansEquations 277
5.1.4 JeansEquationsinCylindricalandSphericalCoordinates 280
5.1.5 ApplicationtoSphericalSystems 281
5.1.6 TheTensorVirialTheoreminStellarDynamics 286
5.1.7 Jeans’Theorem 288
5.2 EquilibriumandStability 290