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ASTROPHYSICS PROCESSES
Bridgingthegapbetweenphysicsandastronomytextbooks,thisbookprovidesphysical
explanationsoftwelvefundamentalastrophysicalprocessesunderlyingawiderangeof
phenomenainstellar,galactic,andextragalacticastronomy.Thebookhasbeenwritten
forupper-levelundergraduatesandgraduatestudents,anditsstrongpedagogyensures
solidmasteryofeachprocessandapplication.Itcontainstutorialfiguresandstep-by-
stepmathematicalandphysicaldevelopmentwithrealexamplesanddata.Topicscov-
eredincludetheKepler–Newtonproblem,stellarstructure,radiationprocesses,special
relativityinastronomy,radiopropagationintheinterstellarmedium,andgravitational
lensing.ApplicationspresentedincludeJeanslength,Eddingtonluminosity,thecooling
of the cosmic microwave background (CMB), the Sunyaev–Zeldovich effect, Doppler
boosting in jets, and determinations of the Hubble constant. This text is a stepping
stonetomorespecializedbooksandprimaryliterature.Reviewexercisesallowstudents
to monitor their progress. Password-protected solutions are available to instructors at
www.cambridge.org/9780521846561.
Hale Bradt isProfessorEmeritusofPhysicsattheMassachusettsInstituteofTech-
nology (MIT). During his 40 years on the faculty, he carried out research in cosmic
rayphysicsandx-rayastronomyandtaughtcoursesinphysicsandastrophysics.Bradt
founded the MIT sounding rocket program in x-ray astronomy and was a senior or
principalinvestigatoronthreemissionsforx-rayastronomy.HewasawardedtheNASA
ExceptionalScienceMedalforhiscontributionstoHEAO-1(High-EnergyAstronomical
Observatory)aswellasthe1990BuechnerTeachingPrizeoftheMITPhysicsDepart-
mentandsharedthe1999BrunoRossiprizeoftheAmericanAstronomicalSocietyfor
his contributions to the RXTE (Rossi X-ray Timing Explorer) program. His previous
book, Astronomy Methods: A Physical Approach to Astronomical Observations, was
publishedbyCambridgeUniversityPressin2004.
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Coverinformation
Viewsoftheentireskyatsixwavelengthsingalacticcoordinates;theequatoroftheMilky
Waysystemisthecentralhorizontalaxisandthegalacticcenterdirectionisatthecenter.
Exceptforthefarinfraredx-raysky,thecolorsrepresentintensitywiththegreatest
intensitieslyingalongtheequator.Inallcases,theradiationshowsanassociationwiththe
galacticequator,thegeneraldirectionofthegalacticcenter,orboth.Themapsarein
frequencysequenceaslistedhere:toptobottomonthebackcoverfollowedonthefront
coverbytopinset,backgroundmap,lowerinset.
Radioskyat408Hzexhibitingadiffuseglowofsynchrotronradiationfromtheentire
sky.High-energyelectronsspiralinginthemagneticfieldsoftheGalaxyemitthisradiation.
NotetheNorthPolarSpurprojectingabovetheequatortotheleftofcenter.Fromthree
observatories:JodrellBank,MPIfR,andParkes.[GlynHaslametal.,MPIfR,SkyView]
Radioemissionat1420MHz,thespin-flip(hyperfine)transitioninthegroundstateof
hydrogen,whichshowsthelocationsofcloudsofneutralhydrogengas.Thegasisheavily
concentratedinthegalacticplaneandmanifestspronouncedfilamentarystructureoffthe
plane.[J.Dickey(UMn),F.Lockman(NRAO),SkyView;ARAA28,235(1990)]
Farinfrared(60–240µm)skyfromtheCOBEsatelliteshowingprimarilyemissionfrom
smallgrainsofgraphiteandsilicates(“dust”)intheinterstellarmediumoftheGalaxy.The
faint,largeS-shapedcurve(onitsside)isemissionfromdustandrocksinthesolarsystem;
reflectionofsolarlightfromthismaterialcausesthezodaicallightatopticalwavelengths.
Colorcoding:60µm(blue),100µm(green),240µm(red).[E.L.Wright(UCLA),COBE,
DIRBE,NASA]
Opticalskyfromamosaicof51wide-anglephotographsshowingmostlystarsinour
MilkyWayGalaxywithsignificantextinctionbydustalongthegalacticplane.Galaxiesare
visibleathighergalacticlatitudes,themostprominentbeingthetwonearbyMagellanic
Clouds(lowerright).[(cid:2)CAxelMellinger]
X-rayskyat1–20keVfromtheA1experimentontheHEAO–1satelliteshowing842
discretesources.Thecirclesizerepresentsintensityofthesource,andthecolordenotesthe
typeofobject.Themostintensesourcesshown(green,largercircles)signifycompact
binarysystemscontainingwhitedwarfs,neutronstars,andblackholes.Otherobjectsare
supernovaremnants(blue),clustersofgalaxies(pink),activegalacticnuclei(orange),and
stellarcoronae(white).[KentWood,NRL;seeApJSuppl.56,507(1984)]
Gamma-rayskyabove100MeVfromtheEGRETexperimentontheCompton
Gamma-RayObservatory.Thediffuseglowfromthegalacticequatorisduetothe
collisionsofcosmic-rayprotonswiththeatomsofgasclouds;thenuclearreactionsproduce
thedetectedgammarays.Discretesourcesincludepulsarsandjetsfromdistantactive
galaxies(“blazars”).[TheEGRETteam,NASA,CGRO]
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ASTROPHYSICS PROCESSES
HALE BRADT
MassachusettsInstituteofTechnology
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cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,Sa˜oPaulo
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
(cid:2)C H.Bradt2008
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2008
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationData
Bradt,Hale,1930–
Astrophysicsprocesses/HaleBradt.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-521-84656-1(hardback)
1.Astrophysics. I.Title.
QB461.B67 2008
523.01–dc22 2007031649
ISBN 978-0-521-84656-1hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
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To mythreesisters,
Val,Abby,andDaleAnne
TheyaremyfansandItheirs
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vi
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Contents
Listoffigures pagexv
Listoftables xx
Preface xxi
Alsobytheauthor xxv
Acknowledgments xxvii
1 Kepler, Newton, and the mass function 1
1.1 Introduction 2
1.2 Binarystarsystems 2
(cid:2) (cid:2) (cid:2)
Celestiallaboratories Visualbinaries Eclipsingbinaries
Spectroscopicbinaries
1.3 KeplerandNewton 9
Kepler’slaws(M≫m) (cid:2)Ellipse (cid:2)TheNewtonianconnection (cid:2)
Earth-orbitingsatellites–Orbitchange–Launchinclination
1.4 NewtoniansolutionsM≫m 15
(cid:2) (cid:2)
Componentsoftheequationofmotion Angularmomentum(KeplerII)
Ellipticalmotion(KeplerI)–Trialsolutiontransformed–Radialequation
transformed–Solution (cid:2)Angularmomentumrestated (cid:2)Periodand
(cid:2)
semimajoraxis(KeplerIII) Totalenergy
1.5 Arbitrarymasses 22
Relativemotions–Relativecoordinates:reducedmass–Equationofmotion–
EquivalencetotheM ≫mproblem (cid:2)Solutions–Angularmomentum–
Ellipticalmotion–Periodandsemimajoraxis(KeplerIII)–Totalenergy
1.6 Massdeterminations 28
Massfunction (cid:2)Stellarmassesfromcircularorbits–Massivecentralobject–
Circularorbits–Spectroscopicbinary (cid:2)Stellarmassesfromellipticalorbits–
Orbitalelements–Visualbinary:relativeorbit–Visualbinary:twoorbits–
Spectroscopicbinary–MassofablackholeinCygnusX-1–Massesof
neutron-starpulsars
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viii Contents
1.7 Exoplanetsandthegalacticcenter 39
Exoplanets (cid:2)Galacticcenter–Stellarorbits–Distancetothegalacticcenter–
Massiveblackhole
2 Equilibrium in stars 49
2.1 Introduction 50
2.2 Jeanslength 50
(cid:2)
Collapsecriterion Criticalmass
2.3 Hydrostaticequilibrium 52
(cid:2)
Balancedforces Pressuregradient
2.4 Virialtheorem 54
Potentialandkineticenergies (cid:2)Derivation (cid:2)Stars (cid:2)Clustersofgalaxies–
Spatialdistribution–VirialMass
2.5 Timescales 59
Thermaltimescale (cid:2)Dynamicaltimescale (cid:2)Diffusiontimescale–
One-dimensionalrandomwalk–Three-dimensionalwalk–Meanfreepath–
Solarluminosity
2.6 Nuclearburning 65
Stableequilibrium–Coulombbarrier–Nuclearwarmer (cid:2)Proton-proton(pp)
chain–Nuclearinteractions–Baryon,lepton,andchargeconservation–Energy
conservation–pep,hep,andBereactions (cid:2)CNOcycle (cid:2)Energyproduction–
Yieldpercycle–Sunlifetime–Energy-generationfunction
2.7 Eddingtonluminosity 73
Forcesonchargedparticles–Radiativeforce–Balancedforces (cid:2)Maximum
starmass (cid:2)Massaccretionrate–Neutron-staraccretion–Accretion
luminosity–Massiveblackholes
2.8 Pulsations 78
Heatengine (cid:2)Conditionforpulsations (cid:2)Ionizationvalve–Transitionzone–
Variablesasdistanceindicators
3 Equations of state 87
3.1 Introduction 88
3.2 Maxwell–Boltzmanndistribution 89
One-dimensionalgas (cid:2)Three-dimensionalgas–Maxwell–Boltzmann
distribution–Momentumspace–Distributionofmomentummagnitude
3.3 Phase-spacedistributionfunction 92
(cid:2) (cid:2)
Maxwell–Boltzmannin6-Dphasespace Measurablequantities Specific
intensity–Particlenumber–Energyandphotons–Liouville’stheorem–
Conservationofspecificintensity–Relativityconnection
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Contents ix
3.4 Idealgas 97
Particlepressure–Momentumtransfer–Averagekineticenergy (cid:2)Equation
ofstate–Physicalform–Macroscopicform(idealgaslaw)
3.5 Photongas 101
3.6 Degenerateelectrongas 102
Fermionsandbosons–Spin–Pauliexclusionprinciple–Degeneracy–
Statisticsanddistributionfunctions (cid:2)One-dimensionaldegeneracy–Plotsof
2-Dphasespace–Fermimomentum–Compressionandcooling–Temperature
(cid:2)Three-dimensionaldegeneracy–Fermimomentum–Fermifunction–Fermi
energy–Pressuresofelectronsandprotons (cid:2)NonrelativisticEOS–Average
kineticenergy–Pressure (cid:2)RelativisticEOS (cid:2)SummaryofEOS
4 Stellar structure and evolution 117
4.1 Introduction 118
4.2 Equationsofstellarstructure 118
Fundamentalequations–Hydrostaticequilibrium–Massdistribution–
Luminositydistribution–Radiationtransport (cid:2)Convectivetransport–
Conditionforconvection–Adiabatictemperaturegradient (cid:2) Secondary
equations
4.3 Modelingandevolution 124
Approachtosolutions (cid:2)Sun (cid:2)Main-sequencestars–Spectraltypes–
Convectiveregions (cid:2)Hertzsprung–Russelldiagram–Color-magnitude
diagram–Effectivetemperatureandradius (cid:2)Giantsandsupergiants (cid:2)
Evolutionofsinglestars–Solarevolution–Massivestars–Gamma-ray
bursts–Globularclusters–Openclusters–Variablestars (cid:2)Scalinglaws–
Matterdensity–Pressure–Temperature–Luminosity–Massdependence–
H-Rdiagramcomparison–Homologytransformations
4.4 Compactstars 142
Whitedwarfs–Mass-radiusrelation–Stability–SiriusB–Chandrasekhar
masslimit (cid:2)Neutronstars–Radiusofaneutronstar–Equationsofstateand
structure–Evidenceforneutronstars–Maximummass (cid:2)Blackholes–Event
horizon(Schwarzschildradius)–Angularmomentum–Innermoststableorbit–
Broad,distortedironline–Plancklength–Particleacceleration–Evaporation–
Existenceofblackholes
4.5 Binaryevolution 157
Timescales (cid:2)Gravitationalradiation–Energylossrate–Finalchirp (cid:2)Tidal
interaction (cid:2)Magneticbreaking (cid:2)Effectiveequipotentials–Rochelobes–
Lagrangian-pointpositions (cid:2)Accretion–Starseparation–Periodchange–
Stellarwinds–Pulsarwindandx-rayirradiation (cid:2)Suddenmassloss–
Semimajoraxisandperiod–Eccentricity–Unbindingoftheorbit (cid:2)
Evolutionaryscenarios–High-massx-raybinaryandbinaryradiopulsar–
Pulsarevolution–Low-massx-raybinary (cid:2)Neutron-starspinup
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x Contents
5 Thermal bremsstrahlung radiation 181
5.1 Introduction 182
5.2 Hotplasma 183
5.3 Singleelectron-ioncollision 185
Radiationbasics–Radiatedelectricvector–Poyntingvector–Larmor’s
formula (cid:2)Energyradiatedpercollision (cid:2)Frequencyoftheemittedradiation
5.4 Thermalelectronsandasingleion 190
Single-speedelectronbeam–Powerfromtheannulus–Powerperunit
frequencyinterval (cid:2)Electronsofmanyspeeds
5.5 Spectrumofemittedphotons 193
Volumeemissivity–Multipleiontargets–Exponentialspectrum–Gaunt
factor–HIIregions,andclustersofgalaxies (cid:2)Integratedvolumeemissivity–
Totalpowerradiated–Whitedwarfaccretion
5.6 Measurablequantities 199
Luminosity (cid:2)Specificintensity(resolvedsources)–Emissionmeasure–
DeterminationofTandEM (cid:2)SpectralfluxdensityS(pointsources)–
Uniformvolumeemissivity–Specificintensityandfluxdensitycompared
6 Blackbody radiation 205
6.1 Introduction 205
6.2 Characteristicsoftheradiation 208
Specificintensity–Rayleigh–JeansandWienapproximations–Peak
frequency–Wavelengthunits (cid:2)Luminosityofaspherical“blackbody”–
Energyfluxdensitythroughafixedsurface–Effectivetemperature (cid:2)Radiation
densities–Energydensity–Spectralnumberdensity–Cellsinphasespace–
Totalnumberdensity–Averagephotonenergy (cid:2)Radiationpressure–Beamof
photons–Momentumtransfer–Photonpressure (cid:2)Summaryofcharacteristics
(cid:2)Limitsofintensity–Particlesadded–Surfaceoflastscatter–Temperature
limit–Blackandgraybodies
6.3 Cosmologicalexpansion 222
Adiabaticexpansion–Photons–Comparisonwithparticles (cid:2)Roomof
recedingmirrors–Hubbleexpansionandfundamentalobservers–Reflections
frommirrors–Wavelengthandroomsize (cid:2)Spectralevolution–Number
spectraldensity–Temperatureandintensity
6.4 Mathematicalnotes 230
(cid:2)
Riemannzetafunction Rootsofatranscendentalequation
7 Special theory of relativity in astronomy 233
7.1 Introduction 234
7.2 Postulatesofspecialrelativity 234
