10d4120090291whereistheperiodandthe nal029isthescatteraroundthemeanrelationTheothernumbersareestimatederrorsinthemeanrelationitselfThisformulahasbeentakenfromareviewby5Theslo ID: 147220
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RadialStellarPulsationsUptothispoint,youhavestudiedstarsinhydrostaticequilibrium.Inthenextfewlectures,wewillconsideroscillatingstars.Webeginwiththesimplestbutperhapsmostimportantcase:radialpulsators.Thesestarsremainsphericallysymmetric,thoughnothydrostatic.1TwoimportantclassesofradialpulsatorsareRRLyraeanddeltaCepheidvariables.(Theterm\variable"canbeappliedtoanystarwhosebrightnessvaries.Inadditiontovarioustypesofpulsators,theseincludeeclipsingbinaries,novae,andotherstarswhosebrightnessvariationshavenothingtodowithpulsation.)RRLyraesandCepheidsareimportantbecausetheyareusedasdistanceestimators.Thesestarsarebrightandcanberecognizedatgreatdistances.Furthermore,theirperiodsarecloselyrelatedtotheirluminosities.Hencebymeasuringtheperiodweknowtheabsolutemagnitudeofsuchastar,andbycomparingthiswiththeapparentmagnitude,wededucethedistance.DeltaCepheidsareevolvedstarswithmasses5 10M ,usuallyinthehelium-core-burningphase.TheybelongtoPopulationIandarefoundinthedisksofspiralgalaxies.(ThereareothersortsofCepheidvariables,forexamplebetaCepheids.ThedeltaCepheidsarethebrightestandmostuseful,however,sooneusuallycallsthemsimply\Cepheids,"orsometimes\classicalCepheids.")TheirabsolutevisualmagnitudesareMV= 1to 6(comparetheSun:MV; =+4:83).Theperiod-luminosityrelationforCepheidswasdiscoveredbyHenriettaLeavittin1906([4]),andithasplayedanimportantroleintheGalacticandextragalacticdistancescaleeversince.ArecentdeterminationisMV= 2:88(0:20)log 10d 4:12(0:09)[0:29];(1)whereistheperiod,andthe nal[0:29]isthescatteraroundthemeanrelation.Theothernumbersareestimatederrorsinthemeanrelationitself.Thisformulahasbeentakenfromareviewby[5].Theslope(thecoecientoflogabove)canbedeterminedfromanygroupofCepheidsatacommonbutunkowndistance(e.g.intheLMC)andisthereforerelativelyuncontroversial.Butthezero-point(givenas 4:12above)mustbedeterminedbycalibrationagainstCepheidswhosedistanceisknown,andisstillvigorouslydebated.CepheidsremaincrucialpartofthedistanceladderandhenceofthemeasurementoftheHubbleconstant.RRLyraestarsareoldhorizontal-branchstars(hencealsocoreheliumburning),lowmetallicity,andmasses.1M .TheybelongtoPopulationII;theyarefoundinglobularclustersandintheGalacticbulge.ThehorizontalbranchissonamedbecauseitformsasequenceofroughlyconstantluminosityintheHRdiagram.ThereforeRRLyraescoveramuchsmallerrangeofluminositiesthanCepheids;typically,MV+0:5!+0:2.TheyarefainterthantheCepheids,thoughstillmuchbrighterthantheSun.Withintheirnarrowrangeofmagnitudes,however,theRRLyraesalsosatisfyaperiod-luminosityrelation.ARoughExplanationofthePeriod-LuminosityRelationCepheids,RRLyraes,andmanyotherpulsatorsarefoundinanarrowandnearlyverticalstripintheHRdiagram.Thusthee ectivetemperaturesofthesestarsarecloselysimilarandonlyweaklydependentonluminosity(Te 61103K).Weshallseewhythisistrueshortly.SinceL=4R2T4e andTe constant,L/R2:(2)Theperiodofthefundamentalradialmodeofpulsation,inwhichtheentirestarexpandsandcontractstogether,isdeterminedbythemeandensity:(G) 1=2/M R3 1=2:(3)EliminatingthestellarradiusRbetweentheselasttwoequations,one nds/L3=4M 1=2:(4) 1Amoredetailedtreatmentofthissubjectisgiveninxx38-39ofthetextbook[3].1 SincetheclassicalCepheidsarepost-main-sequenceobjects,wecannotusethemain-sequencemass-luminosityrelationtoeliminateMinfavorofL.Nevertheless,sinceLisusuallyasensitivefunctionofmass,massisinsensitivetoluminosity.Thusequation(4)suggests/Lxwithx3=4sincedM=dL�0.However,Te isnotstrictlyconstantwithintheinstabilitystrip:itslowlydecreaseswithincreasingL.Theempiricalrelation(1)indicates/L0:870:06ifoneignoresvariationsinthebolometriccorrection.FrequenciesofAdiabaticRadialModesTheradialmodesofastarcanbelikenedtostandingsoundwavesinanorganpipe.Forourpurposes,anorganpipeisacylindercappedatoneendbutopenattheother.Thepressureandvelocityperturbationsinastandingacousticwaveare90outofphase.Thusattheclosedendofthepipe,thevelocityvanishesandthepressure\ructuationsaremaximal.Similarly,atthecenterofaradiallypulsatingstar,theradialvelocityhasanodeandthepressurehasanantinode.Attheopenendoftheorganpipeandatthesurfaceofastar,thesituationisreversed:thepressure\ructuationsvanishandthoseofthevelocityaremaximal.(Thelatterboundaryconditionisinaccuratefororganpipes,becauseitdoesn'tallowforradiationofsound,butisbetterforstars.)Inthisapproximation,thelengthofthepipe,a,isonequarterofthewavelengthofthefundamental.Thusthelowestnote|thefundamental|hasperiod0=4a=vs.Thesoundspeedofthestarvarieswithradius,andfurthermoregravitationalforcesmodifytheperiod,butneverthelesswemayexpectthat04R vs:(5)Theequationofhydrostaticequlibrium,dP dr= GMr r2;(6)canbeverycrudelyapproximatedbyP RGM R2;orP GM R;(7)wherePandrepresenttheaveragepressureandtheaveragedensityinthestar.ThecombinationP=isrelatedtothesoundspeed,vs:v2s@P @S 1P :(8)Sincetheadiabaticindex 1isalwaysoforderunity,itfollowsthatvsGM R1=2:(9)Puttingthisintoeq.(5),we ndthatthefundamentalfrequencyisoforder04GM R3 1=2;(10)inagreementwitheq.(3).Infact,forann=3=2polytropewith 1=5=3,theroughestimate(10)isremarkablyaccurate:theactualnumericalcoecientturnsouttobe3:815insteadof4.Forastarhavingtheequilibriumstructureofann=3polytropebutrespondingtoadiabaticperturbationswith 1=5=3(thisisacrudemodeloftheSun),thecoecientis2:065[[1],Table8.1].ByvariationalmethodsakintoRayleigh-Ritz,whichyoumayhaveencounteredinquantummechanics,itispossibletoestablishthefollowingboundsontheangularfrequency(!02=0)ofthefundamentalmode:(3 1 4)GM R3!20(3 1 4)jWj I;(11)2 Figure1:TheCarnotcycle.inwhichW ZGm rdm(12)isthegravitationalpotentialenergyofthestarand2 3I2 3Zr2dm(13)isthemomentofinertia.Foraderivation,seeChapter8of[1].Itisassumedherethatpressureanddensity\ructuationsareconnectedbytheadiabaticrelation(8)with 1constantthroughoutthestar.Thelefthandinequalityineq.(11)mustbereversedif 14=3.Inthiscasebothlimitsimplythat!20,sothatthefundamentalmodegrowsexponentially.LikeawhitedwarfabovetheChandrasekharmass,suchastarisunstabletogravitationalcollapse.Inanorganpipe,higherharmonicsexistinadditiontothefundamental,withwavelengthsn=4a=(2n+1)andperiodsn=n=vs.Becausethesoundspeedvarieswithradiusinagivenstar,itspulsationalharmonicsarenotsosimplyrelated.However,theperiodsdecreasemonotonicallywithincreasingnumbersofnodes.Thenthovertonehasnnodesinthevelocityordisplacement,notcountingthenodethatisalwayspresentatr=0.Theratio1=0oftheperiodofthe rstovertonetothatofthefundamentalmeasuresthedegreeofcentralconcentrationofthestar;thisratiotendstoincreaseasthecentraldensityincreaseswithrespecttothemeandensity.Forexample,1=0is0:465and0:738forthe(n; 1)=(3=2;5=3)and(3;5=3)polytropes,respectively([1]).Excitationofradialpulsations:TheKappaandEpsilonMechanismsIftheadiabaticapproximationwereexact,orinotherwordsiftheentropyofeverygaselementwereconstantthroughthepulsationcycle,thentheenergyinpulsationswouldbeconserved.Oscil-lationsofstarswith 1&#x-5.1;䝀4=3throughoutmostoftheirinteriorwouldneithergrownordecay.Onemightthinkthatnonadiabatice ects,suchasviscosityandradiativedi usion,wouldtendtodamptheoscillations.Thisisindeedusuallythecase.Undercertaincircumstances,however,nonadiabatice ectsmayactuallyexcitepulsations.Toseehowthiscanhappen,itisusefultoconvertourorganpipeoftheprevioussectionintoaheatengine.Filltheopenendwithamovablepistonandallowthepipetomakethermalcontactwithhotandcoldheatbaths.Justtoremindyou,theclassicalCarnotengineworksasfollows(refertoFigure1):3 A!B:Thepistoniscompressedadiabatically,thatis,atconstantentropy:SAB=0.PressureP,volumeV,andtemperatureVvaryasP/V 1andT/V1 1.ThetemperatureincreasesfromTAtoTB=TA(VA=VB) 1 1.B!C:Thegasisbroughtintocontactwiththehotterheatbath,Thot=TB,andallowedtoexpandatconstanttemperature.SincePV/TandTisconstant,P/V 1.Duringthisstep,theentropyincreasesbySBC=QBC=Thot,whereQBCistheheataddedtothegas.C!D:Thegasisdetachedfromtheheatbathandallowedtoexpandadiabaticallyuntilitstem-peraturefallsbacktoTA=Tcold.AtthispointthevolumeislargerthanVAbecauseoftheincreaseinentropyduringthepreviousstep.D!A:Finally,thegasisbroughtintocontactwiththecoldbathattemperatureTcoldandisothermallyrecompressedtoitsoriginalvolumeVA.Inthisstep,SDA=QDA=Tcold.SinceS(afunctionofstate)mustreturntoitsoriginalvalue,SDA= SBC.SoQBC+QDA=QBC(Thot Tcold)=Thot�0.Thusnetheatisabsorbedbytheengine,andenergyconservationrequiresthattheheatisconvertedtowork.ThenetworkdonebythegasonthepistonisW=IABCDAPdV;(14)whichistheareaenclosedbythecurveinthe(V;P)plane.Sincetheinternalenergy(U)returnstoitsoriginalvalueafteronecompletecycle,HdU=0.ThereforeusingtheFirstLaw,dU=TdS PdV,wecanrewritetheworkdoneasW=ITdS:(15)ClearlyW�0alonganyclockwiseclosedpath,notjustthosecomposedofadiabatsandisotherms.Henceifthegastendstogainentropywhenitiscompressed(athightemperature)andtoloseitwhenexpanded(atlowertemperature),thenthenetworkdoneispositive.IntheCarnotenginedescribedabove,heatandentropyaregainedorlostonlywhenthegasisincontactwithaheatbath.Atthecenterofastar,however,heatandentropyarecontinuallyproducedbynuclearreactions.Inequilibrium,heatproductioninthecoreisexactlybalancedbyheatlossfromthesurface,i.e.,bythestellarluminosity.Stellarpulsationdisturbsthisbalanceby(i)modulatingthenuclearreactionrateinthecore(epsilonmechanism);andmoreimportantly,(ii)bymodulatingtheradiativeluminosity(kappamechanism).E ect(i)isalwaysdestabilizingbecausethenuclearreactionrateincreaseswithincreasingtemperatureandpressure:thusthechangeinreactionratetendstoaddentropytothecorewhenitisathigherthanequilibriumtemperature.Exceptinverymassivestars(&100M ),theepsilonmechanismtendstobeweak,sothatasmallamountofdissipationisenoughtopreventgrowth.Atthelowertemperaturescharacteristicoftheouterenvelopesofstars,theopacitymayactuallyincreasewithtemperature(at xedentropy,forexample)becausethegasisincompletelyionized.Insuchacase,luminositydecreaseswhenthestarcontractsanditstemperaturerises,trappingheatintheinteriorandraisingtheentropy;thereverseistruewhenthestarexpands.Duringoneperiodofoscillation,mostmassshellsthenexecuteaclockwiseclosedpathinthe(S;T)and(V;P)planes,leadingtoanincreaseinmechanicalenergy.Thereasonfortheincreaseofopacitywithtemperatureisroughlyasfollows.Inthesolarphotosphere,almostallofthehydrogenatomsareintheirgroundstate(n=1).Toexcitesuchanatomintothehigherstatesrequiresphotonsenergiesof10:2eV,whereasthecharacteristicphotonenergyat5700KisonlykT=0:49eV.Henceneutralhydrogencontributeslittletotheopacity,whichisinfactdominatedbytheweaklyboundH ion.Atmodestlyhighertemperatures104K,asmallbutsigni cantproportionofthehydrogenliesinexcitedstates,wherethespacing4 betweenadjacentlevelsissmaller(/n 3),asistheionizationenergy(/n 2),sothattheopacityisdominatedbytheexcitedneutralhydrogen.Atthesetemperatures,@=@T�0.Atstillhighertemperatures,thefractionofthehydrogeninexcitedboundstatesactuallydecreaseswithincreasingtemperaturebecausemostofthehydrogenisalreadyionized;sincefreeelectronsaremuchlesse ectivethanboundonesatabsorbingphotons,thehydrogenopacitythendecreaseswithfurtherincreasesinT.However,atT4104K,heliumisintransitionbetweensinglyanddoublyionizedstatesandtendstocause@=@T�0.Theseopacitye ectsconstitutethekappamechanism(ii).Wecanunderstandthephysicsinabitmoredetailbyreturningtotheworkintegralintheform(15).ThegaswithintheCarnotengineisuniformintemperature,soSrepresentsthetotalentropyofthisgas.Inthecaseofastar,sincethetemperaturevarieswithradius,wemustevaluateTdSseparatelyforeachmassshell;henceS(m;t)becomestheentropyperunitmassatmassfractionm(thisnotationwillbeusedinplaceofMrinthislecture),andW=t0+Zt0dtMZ0dmTdS dt:(16)ReplaceT(m;t)!T(m)+T(m;t)andS(m;t)!S(m)+S(m;t),whereT(m)andS(m)arethetime-averagedtemperatureandentropyoftheshell,andTandShavezerotimeaverage.ThentheworkintegralbecomesW=IdtZdmTdS dt:(17)Thelimitsofintegrationarethesameasbeforebutwillnotbewrittenoutexplicitly.Noticethatthisissecondorderintheoscillatingquantities;thezeroth-ordertermsvanishbecausedS=dt=0,andthe rst-ordertermsvanishinthetimeaverage: TdS dt =T dS dt =0:Inthefundamentalradialmode,theentirestarexpandsandcontractstogether.HencethesignofTisindependentofmassfraction,andsincethepulsationisapproximatelyhomologous|i.e.,thefractionalchangeinradiusr=randvolumeV=Visapproximatelythesameformostmassshells|eventhemagnitudeofT Tisapproximatelyconstantwithmassfraction.Thusitisusefultopullthefractionaltemperatureperturbationoutofthemassintegral:WIdt T T mZdmTdS dt;(18)wheretheanglebracketsindicateanappropriateaverageovermassattimet.Finally,ifweignorechangesinthenuclearreactionrate,thento rstorder,TdS dt @L @m(m;t):(19)Themassintegralineq.(18)nowyieldsL(r;M) L(r;0),andthis=L(r;M)sinceL(r;0)=0evenunderperturbation.WritingL(t)L(M;t)forthechangeinthetotalstellarluminosity,wehaveatlastW Idt T T mL(t):(20)Theluminosity,opacity,andtemperaturearerelatedbyL(m;t)= 162r4c @Prad @m;(21)5 wherePrad=1 3aT4.Atsucientdepthbelowthephotosphere,wemayevaluateLfromthisequationusingtheadiabaticvaluesforT,,andr.Variationsinopacity,radius,andradiativegradientcontributeadditivelyto rstorder.Clearlyif@ @TS=@ @T+@ @T@ @TS�0;thentheopacitywillincreaseduringcontractionsThesecondtermontherighthandsidetendstobepositivesince(@=@)T0and(@=@T)S�0.The rstcanbepositiveatlowtemperatures:seeFigure17.5inthetextbook[3].ThiswilltendtomaketheproductLT0ineq.(20).Thedecreaseinradiusduringcontractionsreinforcesthistendencybecauseofther4factorineq.(21).Ontheotherhand,Prad&#x-5.1;䘘0incontractionsandthisimplies@Prad=@m0inahomologouscontraction:@ @mPrad=Prad Prad@Prad @m:Sothe rsttwoe ectsmustbestrongerthanthelastinordertoguaranteethatW�0,i.e.,inordertoincreasethemechanicalenergyofthepulsation.Intheargumentabove,weevaluatedthechangeinluminosityusingtheadiabaticvariationsinT,r,and.Thisisonlyanapproximation,sinceitispreciselythedeviationfromadiabaticbehaviorthatmakesgrowthpossible.Theapproximationisvalidifthelocalthermaltimescaleoftheionizationzone(i.e.theregionwhere@=@T�0)islongcomparedtothepulsationperiod.Thelocalthermaltimescalemaybede nedasthethermalenergyofthematterexteriortothebaseoftheionizationzone,dividedbytheluminosity:ttherm(M m)P L m=mioniz::(22)Ifttherm ,thentheadiabaticapproximationfailscompletelyintheionizationzone:theradiativegradientwillquicklycompensateforanychangeinopacitysothattheluminosityisnota ectedbythepulsationandW0.AsTe increases,theionizationzonesmoveclosertothesurfaceandtthermdecreases,andatsomepointtheinstabilityisquenched.Thisdeterminestheblue,i.e.high-Te ,edgeofthe\instabilitystrip"intheHRdiagramwhereCepheidsoccur.Ontheotherhand,iftheionizationzoneliestoofarbelowthesurface,thenttherm bothintheionizationzoneandintheregionsimmmediatelyaboveit;modulationofthelocalluminosityintheionizationzonethenhaslittlee ectonL(t)emergingfromthesurface,whichisdeterminedbylocalconditionsinsomeshallowerzonewherettherm.Furthermore,asTe decreasesandtheionizationzonesretreattogreaterdepths,theybecomeconvective,atwhichpointtheopacitynolongercontrolstheradiative\rux.Thesee ectsdeterminetherededgeoftheinstabilitystrip.Ourderivationofthekappamechanismisimpressionisticratherthanrigorous.Theproperwaytodeterminepulsationalstabilityistolinearizethefullnon-steadystellarstructureequationsandsolveforthecomplexeigenfrequenciesdirectly.Thisisdescribedin[1]andwillnotberepeatedhere.DirectDeterminationofDistance:TheBaade-WesselinkMethodWehavenotedthatonceaperiod-luminosityrelationhasbeenestablished,pulsatingstarscanserveasdistanceindicators.Remarkably,however,thedistancetoapulsatingstarcanbedetermineddirectly,inprinciple,withoutpriorknowledgeofthePLrelation.The\ruxreceivedfromastarofradiusR,distanceD,ande ectivetemperatureTe isF=R2T4e D2:(23)The\ruxande ectivetemperatureofapulsatingstarvarywithtime,butthesearedirectlymea-surableifwehavephotometryinmultiple lters,orbetteryetseriesofspectra.Takingthelogofeq.(23)anddi erentiating,onehasd dtlnF=4d dtlnTe +2 RdR dt:(24)6 Figure2:LightcurvesofafewoftheLMCCepheidsfoundbytheOGLEIIproject[7].Noticethatthedistancehasdroppedoutbecauseitisconstant.Onecansolvetheaboveforthestellarradiusintermsofmeasurablequantities:R=2dR dtd dtlnF 4d dtlnTe 1:(25)HeredR=dtismeasuredbythedopplershiftofstellarabsorptionoremissionlines.OnceRisknown,eq.(23)canbesolvedforD.Anobviousconsistencycheckisthatthederiveddistanceshouldnotvaryoverthepulsationcycle.Thiselegantmethodsu erssomepracticaldiculties.First,determiningTe fromcolorsorspectraisnottrivial.Thestardoesnotradiateasablackbody|ifitdid,therewouldbenospectrallineswithwhichtomeasuretheradialvelocity.Second,thespectrallinesusedforthevelocityandthecontinuumrepresentedbyTe arenotformedatquitethesameradius.Thismeansofcoursethatthelinesandthecontinuummayhavedi erentvelocities.Bothdicultiescanbeconqueredinprinciplebydetailedphysicalmodelingofthemovingstellaratmosphere,butthisiscomputationallychallenging.The eldisstillunderdevelopment.Incidentally,eq.(25)doesnotrequirethatthepulsationbeperiodic.ThesametechniquehasbeenusedtomeasuredistancestoextragalacticTypeIIsupernovae,butthisapplicationiscalledbyitspractitioners\TheExpandingPhotosphereMethod"[[6],[2]].7 Figure3:I-bandperiod-luminosityrelationofthefundamental-modeCepheidsintheLMC(upperpoints)andSMC(pointsshifteddownby0:8mag)[8].CalculationoflinearadiabaticradialpulsationsTodeterminethepulsationaleigenmodesofagivenstellarmodel,onestartswiththefullradialforceequationintheform@2r @t2m= 4r2@P @m+Gm 4r4:(26)Thereisanequilibriumstateinwhichr(m;t)=r0(m)andbothsidesoftheequationvanish.De ner(m;t)r(m;t) r0(m),andsimilarlyP(m;t),(m;t),etc.Presumingthattheseperturbationsaresmall,expand(26)to rstorder,obtainingr= 4r2@P @m 4Gm 4r5r:Thefollowingequationsareslightlytidierifr0ratherthanmisusedasindependentvariable.Tosavewriting,thesubscriptwillbeomittedfromr0,butitshouldberememberedthatunlesspre xedby,rwillrefertotheequilibriumradiusofagivenmassshell.Thenthelastequationbecomesr= 1 @P @r+4GMr r3r:(27)Wehaveputm!Mrtoremindourselvesthatitreferstothemasscontainedwithintheequilibriumradiusr.Thedotsrefertoderivativeswithrespecttotatconstantmorr0.Asyet,wehavenotusedtheassumptionofadiabaticity,butweareabouttodoso:P=@P @S= 1P ;where 1@lnP @lnS:(28)Inagas-pressure-dominatedstar, 15=3everywhere|inradiativeaswellasconvectiveregions|because 1describesnottheequilibrium(inwhichentropymayvarywithradius)buttheresponseofagivenmasselementtoadiabaticperturbations.Conservationofmassimplies = 1 r2@ @r r2r:(29)8 Bycombining(28)and(29),oneconverts(27)toanequationinvolvingralone:r=1 @ @r 1P r2@ @r r2r+4GMr r3r:(30)Foraneigenmode,allmassshellsshouldvibrateatthesamefrequency:thatis,r(m;t)=(r)e i!t;sothat !2=1 d dr 1P r2d dr r2+4GMr r3:(31)Fornumericalwork,itisoftenconvenienttounpack(31)asapairofordinarydi erentialequationsbyintroducingr 2d(r2)=dr,whichisessentiallyjust =,sothatd dr= 2 r;1 d dr( 1P)= !2 4GMr r3:(32)Theboundaryconditionatr=0isobviously=r=0.Theboundaryconditionatr=R(theequilibriumouterradius)isnotsoobvious,butitcanbeobtainedfromtherequirementthatshouldberegularasr!R:thatis,(r)andatleastits rstfewderivativeswithrespecttorshouldexistatr=R.[Infact,therequirementofregularityasr!0givesthecentralb.c.,(0)=0,ascanbeseenfromthe rstofequations(32).]Letusrewritethesecondofequations(32)intheform 1P d dr 1GMr r2= !2+4GMr r3:(33)Thecoecientofd=drinthisequationisv2s,whichisproportionaltotemperatureforanidealgasandthereforeverysmallatthesurfacecomparedtoitsvalueatthecenter.Infact,forthepurposeofcomputingmodesofoscillation,oneoftenapproximatesthehydrostaticequilibriumstructurebymodelsinwhichv2s!0asr!R.Ifthesolutionistoberegular,thend=drwillbe niteasr!R;infactd=dr=R2there.Sothe rsttermonthelefthandsideof(33)essentiallyvanishesatr=R,andtheboundaryconditionsbecome= 1 1RR3!2 GM+4atr=R(34)=0atr=0:(35)Thesurfaceboundarycondition(34)isonlyapproximateandwouldbreakdownforhigh-frequencymodeswhosewavelengthisshorterthanthepressurescaleheightatthephotosphere;intheSun,thisoccursfor.3min,whereas0; 1hr.Wemaysolveequations(32),(34)&(35)byshootingtoa ttingpoint,i.e.integratingoutwardfromr=0andinwardfromr=Rtoanintermediateradiusr tButwehavethreeunknownstoguess:(0),(R),and!itself.Fortunately,becausethedi erentialequationsarelinearandhomogeneous,wedonothavetosolveforallthreesimultaneously.Wecanfreelyrescalethedependentvariablesbyacommonfactor|andwemaychoosethesescalingfactorsindependentlyfortheinwardandoutwardintegrationstominimizethemismatchatr t.Thisfreedomreducesthenumberofmatchingconditionsfromtwotoone: outward= inwardatr=r t:(36)Sincebothsidesofthisequationareindependentofthescalingfactors,wemaychoose(0)and(R)arbitrarily|Igenerallyjustsetbothtounity.Thenwehaveaone-dimensionalsearchover!2toperforminordertosatisfy(36).Thelowestrootfor!2isthefundamentalmode.After nding!,wemayeasilyrescale(0)and(R)soastoobtaincontinuoussolutionsforand.9 References[1]J.P.Cox.TheoryofStellarPulsation.PrincetonUniversityPress,1980.[2]R.G.Eastman,B.P.Schmidt,andR.Kirshner.TheAtmospheresofTypeIISupernovaeandtheExpandingPhotosphereMethod.ApJ,466:911{937,August1996.[3]R.KippenhahnandA.Weigert.StellarStructureandEvolution.Springer,1990.[4]H.Leavitt.1777VariablesintheMagellanicClouds.Ann.HarvardColl.Obs.,60(4),1908.[5]B.F.MadoreandW.L.Freedman.TheCepheiddistancescale.PASP,103:933{957,September1991.[6]B.P.Schmidt,R.P.Kirshner,andR.G.Eastman.ExpandingphotospheresoftypeIIsupernovaeandtheextragalacticdistancescale.ApJ,395:366{386,August1992.[7]A.Udalski,I.Soszynski,M.Szymanski,M.Kubiak,G.Pietrzynski,P.Wozniak,andK.Zebrun.TheOpticalGravitationalLensingExperiment.CepheidsintheMagellanicClouds.IV.CatalogofCepheidsfromtheLargeMagellanicCloud.ActaAstronomica,49:223{317,September1999.[8]A.Udalski,M.Szymanski,M.Kubiak,G.Pietrzynski,I.Soszynski,P.Wozniak,andK.Zebrun.TheOpticalGravitationalLensingExperiment.CepheidsintheMagellanicClouds.III.Period-Luminosity-ColorandPeriod-LuminosityRelationsofClassicalCepheids.ActaAstronomica,49:201{221,September1999.10