1988Apj. . .329. .803A the Astrophysical Journal, 329:803-807,1988 June 15 © 1988. the American Astronomical Society. All Right

1988ApJ. . .329. .803A 2 between observedclusterH-Rdiagramsandcomputedevolu- qualitative featuresofstellarevolution.Theagreement The AstrophysicalJournal,329:803-807,1988June15 formation isproperlytreatedinthenumericalcodes,and tionary tracksisquitegood(seeIbenandRenzini1984fora © 1988.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. recent review).Thisagreementimpliesthatthephysicsofgiant main sequence,areforcedtotheredinH-Rdiagram,and description ofwhatcausesastartobecomeredgiantexists. However, nocompellingqualitativeorsemiquantitative this sensetheproblemofgiantformationhasbeensolved. forces thestartoHayashitrack,andanswerissimple. perature oftheHayashitrack.Theimportantquestioniswhat 47rRo-T£, where7^isthenearlyconstanteffectivetem- the Hayashitrackitsexpansiontoenormoussizeinresponse tion. Inparticular,Idescribewhystars,oncetheyleavethe The envelopeofasolar-typestarnearcorehydrogenexhaus- to itslargeluminosityisexplainedbytherelationL= track iswellunderstood(Hayashi1966),andonceastaron of redgiantstarsinthisarticle.ThephysicstheHayashi ultimately intersecttheHayashitrack.Idonotpresentmodels My purposeinwritingthispaperistosuggestsuchadescrip- in thesestarsisunimportantforargumentsandcanbe tion ispredominantlyradiative(thethinouterconvectionzone neglected), andtheopacityisgivenbyKramerslaw to increased flowofenergy.AsIshowin§II,theenvelopeadjusts the envelopemustadjustitsstructuretoaccommodate the good approximation.Astheluminosityofstarincreases star theignitionofahydrogen burningshellsourcecausesthe increases theexpansionismodest,butforlarge(in a by decreasingitsdensityandexpanding.Forsmallluminosity which Kramersopacitylawholds. IftheKramersopacitylaw can betransportedbyaradiativeenvelopeofgivenmass in giant branch. star ultimatelyintersectstheHayashitrackandascends red the starisforcedsharplytoredinH-Rdiagram. The sense tobemadedefinitein§II)theexpansionisdramatic and diverges asthemaximumluminosity isapproached.Inareal is usedforalltemperaturesand densitiestheradiusofstar 1 The formationofredgiantsisonethemostdramatic In §II,Ishowthatthereisamaximumluminositywhich AlfredP.SloanFoundationFellow. luminosity exceedingthecriticalvalue.IfKramerslawisusedforalltemperaturesanddensities, larger thanacriticalvalue,andIarguethatthetransitiontoredgiantstructureistriggeredbystar’s convection zone. in excessofthecriticalluminositycanbeaccommodatedbyforcingmostmassenvelopeinto radius ofthestardivergesascriticalluminosityisapproached.Inrealstarsradiativeenvelopeexpands as theluminosityincreasesuntilstarintersectsHayashitrack.Onceontrack,luminosities Subject headings:stars:evolution—interiors © American Astronomical Society • Provided by the NASA Astrophysics Data System I showthataradiativeenvelopeinwhichtheKramersopacitylawholdscannottransportluminosity I. INTRODUCTION WHY STARSBECOMEREDGIANTS Received 1986August25;accepted1987December8 Department ofAstronomy,ColumbiaUniversity 1 James H.Applegate ABSTRACT 803 fully radiativeenvelope.Astheluminosityincreasesstar luminosity toapproachandthenexceedthemaximumfora expands andmovestotheredinH-Rdiagram.Thestar ultimately intersectstheHayashitrack,developsadeepcon- vection zonewithinitsenvelope,andascendstheredgiant clearly incompleteorfalse.Amongthesearetheideasthat into theconvectivezone. branch. Luminositiesinexcessoftheradiativemaximumare molecular weightdifferencebetweentheenvelopeandcore, accommodated byforcingmostofthemassenvelope each playaroleinstellarevolution,butnonearetheprimary stars becomeredgiantsbecauseofthevirialtheorem, cause ofredgiantformation. and becauseofthegravothermalcatastrophe.Thesefactors of theSchönberg-Chandrasekhartheorem,because throughout thestar.Thevirialtheoremgivesnoinformation gravity andpressuregradientsbalanceoneanotheronaverage hydrostatic equilibrium.Assuch,itcantellyouwhetherornot mass ofanisothermalcoreto10%thestar. In about thedetailsofdensityorpressuredistributionswithin giant, orwhitedwarf,satisfiesthevirialtheorem. stars. Anystarinhydrostaticequilibrium:main-sequence,red changes oftheradiusstar,andtheydidnotpredictwhat the growthofanisothermalcorewasaccompaniedbymodest their computationsSchönbergandChandrasekharfound that exceeded. Ifthelimitisexceeded,hydrostaticequilibrium the radiusofstarwoulddoiflimitingcoremass was requires atemperaturegradientinthecore,andcorecon- (Iben 1967),andinthissensetheSchönberg-Chandrasekhar luminosity ofthestarbecausetemperatureinhydrogen resulting energyloss.Corecontractiontendstoincrease the tracts onitsKelvin-Helmholtztimescalebecauseof the limit isintimatelyrelatedto giantformation.However,the associated withredgiantformationinthecomputedtracks burning shellisincreased.Thisphaseofcoreevolution is increase. Theresponseofthe envelopetotheluminosity core contractionreallyonly requiresthattheluminosity cause theenvelopeofstar toexpand(CoxandGiuli1968), increase dependsonthephysics oftheenvelope. Some ofthesimplerexplanationswhyredgiantsformare The virialtheoremisanintegratedformoftheequation The Schönberg-Chandrasekhartheorem(1942)limitsthe Increasing themolecularweight inthecoreofastartendsto 1988ApJ. . .329. .803A pressure arejustified. ever theuseofKramersopacity andtheneglectofradiation total massandpresentdetailed modelsforthattotalmassonly. My discussionofthephysics of giantformationisvalidwhen- type starsand,accordingly,I choose 1Masarepresentative opacity law.Theseapproximations limitthediscussiontosolar that theenvelopeopacityis wellrepresentedbyKramers polytropic indextoapproachn=5. tropic index,andthenarguethattheburningshellforces the to describethewholeinhomogeneousstarwithasinglepoly- approach thanthatofEggletonandFaulkner(1981),who try ates. Ifindthisdivisionofthestartobeamuchmorenatural does becauseitmustcarrytheluminositythatcoregener- what isgoingoninthecore,andenvelopedoes it generated.) Iassumethatthecoredoeswhatitbecause of refers totheoverlyingregionsinwhichnonuclearenergy is 0 produces) areinsensitivetotheconditionsinenvelope.(For the restofthispaper“core”referstohydrogen-exhausted the surroundingburningshell(particularlyluminosityit region plusanyshellsourcethatmaybepresent;“envelope” once aheliumcorehasformedthepropertiesofand the responseofenvelopetochangesincoreparameters R, andenvelopemassM=—.Mypurposeistostudy total massM=1luminosityL,core,radius drop areleftunexplained. with consideringhydrostaticequilibriumonlyisthatthevalue of thepolytropicindexandoriginabruptdensity the modelswhichIcomputeinnextsection.Thedrawback solution tocrossthecut.Bothofthesefeaturesarepresentin cut, andthesteepdensitydropisneededtoforceenvelope index mustexceed3inorderthattheU—Vplanecontaina density discontinuityattheedgeofcore.Thepolytropic n >3,andthattheremustbeasteepdensitygradientor find thattheradiativeenvelopemusthaveapolytropicindex ce ditions requiredforgiantformationinpolytropicstars.They 0c den Horn(1985)havepresentedadetailedanalysisofthecon- equation ofhydrostaticequilibrium.RecentlyYahilandVan giants canbegainedbystudyingpolytropicsolutionstothe bilities indeterminingthecourseofstellarevolution. port ismoreimportantthanglobalthermodynamicinsta- they descendtheHayashitrack.Thephysicsofenergytrans- tract homologouslythroughaseriesofn=3/2polytropesas tostars, forexample,derivetheirenergyfromgravitybutcon- which derivetheirenergyfromgravitationalcontraction.Pro- toward acore-halostructureisnotgeneralfeatureofsystems on theirwaytocorecollapse(Cohn1980).However,evolution evolve towardevermorecentrallycondensedconfigurations This instabilityhelpstoexplainwhyglobularstarclusters self-gravitating systemshaveanegativeheatcapacityinbulk. molecular weightdifferencedidnotcausetheexpansion. Lynden-Bell andWood1968)isaconsequenceofthefactthat exhaustion, andtheexpansionoccursconsiderablylater, as largeitwillevergetatthemomentofcorehydrogen ence inmolecularweightbetweenthecoreandenvelopeis the expansiontoredgiantdimensionsoccur.Sincediffer- has contractedandahydrogenburningshellignited,does while onthemainsequence.Onlylater,afterheliumcore tracks (Iben1967)showthatstarsexhausttheircorehydrogen but thisisnotwhyredgiantsform.Computedevolutionary 804 I neglectradiationpressureinthesemodels,andassume In thenextsectionIcomputemodelenvelopesforastarof Insight intotheconditionsneededforformationof The gravothermalcatastrophe(Antonov1962,1985; and L.Thisapproachisbasedontheassumptionthat © American Astronomical Society • Provided by the NASA Astrophysics Data System APPLEGATE thermal conductivityandopacity arerelatedby ty insteadoftheopacityforreasons thatwillbecomeclear.The sion equation,equation(2),in termsofthethermalconductivi- have theirusualmeanings.Ichoose towritethephotondiffu- where Kisthethermalconductivityandrestofsymbols envelope intotheconvectionzone. can beaccommodatedbyforcingmostofthemass excess ofthemaximumthataradiativeenvelopecantransport outer convectionzonemaynotbeneglected.Luminositiesin constant effectivetemperatureforthereasonsgivenby Hayashi (1966).Oncethestarintersectstrack section becomeinappropriate,andthestarexpandsatnearly sects theHayashitrackapproximationsmadeinthis outer convectiveregioncanbeneglected.Oncethestarinter- expansion isdrivenbytheradiativepartofenvelope, move totheredinoncetheyleavemainsequence,and Since mygoalinthispaperistoexplainwhystarsexpandand ative zerosolutiontotheexact(Schwarzschild1958). ary conditionsisjustifiedbytherapidconvergenceofradi- P =0,TOatthesurfaceofstar.Theusethesebound- mass, asgivenbythevolumeintegralofdensity,isrequired to havethepropervalue.Theouterboundaryconditionsare in hydrostaticandradiativeequilibrium,theenvelope models ofradiativeenvelopes.Theenvelopesarerequiredtobe diverges asthemaximumluminosityisapproached,whichcor- responds toW(M)=0,whereMisthemassofstar. change inlogrcanbeproducedbyasmalllumin- osity. ThisisexactlywhatIfindinmycalculations.Theradius is satisfiedintheenvelope.Thisconditionmeansthatalarge when thecondition tight. Renzinifindsthatthethermalinstabilityistriggered equilibrium, expand,andcrosstheHertzsprunggaponits Renzini findsinhismodels.Thisargumentcanbemadevery Kelvin-Helmholtz timescale.Thisisexactlythebehavior quickly, theenvelopewilldepartsignificantlyfromthermal maximum, theenvelopemustundergodramaticchangesinits structure. Aslongastheluminosityisincreasingrelatively envelope. Astheluminosityapproachesandthenexceeds Renzini’s thermalinstability.Considerastarwithradiative the maximumluminositythatIfindisintimatelyrelatedto finds responsibleforgiantformation.However,Ibelievethat instability thatRenzini(1984;seealsoIbenand1984) prung gapiscrossed,andIcannotreproducethethermal this reasonIcannotreproducethedetailsofhowHertzs- The structureequationsfortheenvelopeare In thissectionIcomputesimpleanalyticandnumerical My modelsareinhydrostaticandradiativeequilibrium.For II. MODELSANDINTERPRETATION dM W(M) =t^±0 -^ =4nr, r P : A2 : dT dP dr dr K = 3 4acT 2 3k P o logr 4nrK ’ GMp Vol. 329 (4) (3) (2) (1) 1988ApJ. . .329. .803A 9 3 corresponds toathermalconductivityoftheform mass arelistedinTable1.Themass-radiusrelationR(M) A Kramersopacity, was takenfroma1MmodelofStrömgren(1965)atanage luminosities. Thecoremassesandtheradiusassumedforeach a staroftotalmass1Musingvarietycoremassesand I assumethatthepressureisgivenbyperfectgaslaw give thecorrectsolarradius;Iobtainedlogk=23.436with 4.6 x10yr.Ideterminedtheopacitycoefficientkinequa- and Iusep=0.618,appropriateforthesolarenvelope. tion (5)bydemandingthattheM=0.3,L1model same qualitativebehavior.Astheluminosityisincreasedfrom the redinH-Rdiagramasmaximumluminosityis No. 2,1988 luminosity LandtheratioJL^,where approached. NotethatthissamebehaviorwasfoundbyHoyle T inKelvinandpgcm“. and Schwarzschild(1955;seetheirFig.2).Themaximum 1 Lthestarexpands,firstmovingupandthenabruptlyto C are givenforeachcoremassinTable1.TheM=M©,L1 0 0 maximum luminositysolutionhasapolytropicindexn=3.20 L© modelisgiveninTable2.Themaximumluminositysolu- 10 0 deep insidethestarandapolytropicindexn=3.25in LJL* is2370aslongthecoredeepwithinstar.The tion forM=0.3M©isgiveninTable3. c0 factor of1.75andthedensityfallsbya19.1as luminosity isincreasedfrom1L©to7.3L©. than inthemain-sequencemodel.Thetemperaturefallsbya the corearebothlowerinmaximumluminositymodel outermost layer.Thedensityandtemperatureattheedgeof envelope, theexistenceofamaximumluminosityand its expansion canallbeunderstoodfromsimplemodels,many of numerical value,andthephysicalmechanismbehind the maxma 0 c m¡i c r I havecomputednumericalsolutionstoequations(IH?)f° Envelope solutionsforthevariouscoremassesallhave The tableshaveseveralinterestingfeatures.ratio The polytropicindicesintheinnerandouterpartsof 0.4 .. 0.3 .. 0.2 .. 0.7 .. 0.6 ., 0.5 .. 0.99 0.9 . 0.8 . 50 5 Note.—= 1(M/M)-(R/R L. CQ G M/ © American Astronomical Society • Provided by the NASA Astrophysics Data System cg L* =1 Parameters ofEnvelopeModels 13/2 K =Tp-. 0 10 112 RJ10 cmL/L K =pT~, maxG 0 P = 4.8 2.0 2.6 2.3 3.2 1.5 1.3 1.0 1.7 TABLE 1 pkT pm H 6 1.8 x101.58 16500 3130 940 328 114 0.9 7.3 33 -o » WHY STARSBECOMEREDGIANTS 20000 2370 2370 2550 2370 2920 6530 3850 (6) (5) (7) (8) constant. Equation(9)canbeintegratedintheoutermost where nistheeffectivepolytropicindex.Ingeneral,nota tures Ifirstcombineequations(1),(2),and(7)toobtain layers ofthestar(Schwarzschild1958).Iset M(r) =Mconstantanduseequation(6)toobtain which arewellknownintheliterature.Toexplainthesefea- envelope solution(HernquistandApplegate1984).Thevalue in equation(6)isrelatedtotheopacitycoefficientkbyequa- tion (4).Thethermalconductivityisconstantalongtheouter the outerenvelopeisann=3.25polytrope.ThecoefficientK of Kisdeterminedbythestar’smassandluminosity: energy. increase toallowtheenvelopetransportadditional If theluminosityincreasesthermalconductivitymust looks likeapointsourceofenergy.Thissuggestslookingfor infinite radius;fromthepointofviewenvelopecore power-law solutiontoequations(l)-(7).IputL=constant solution. Fromthepointofviewcoreenvelopehas 0 0 Next Iconsidertheinteriorofmaximumluminosity 23.306. 24.000. 20.181. 21.917. 16.016. 18.089. 14.627. 11.503. 12.544. 13.586. 10.114. 10.461. 10.442. 10.551. 10.660. 10.842. 10.114. 10.223. 10.332. 10.733. Note.—All quantitiesincgsexceptM inM. G log R Details oftheM=( Note.—All quantitiesincgsexceptM. r G log R Details oftheM=0.3Main-SequenceModel cG din T\kJ d lnP/pm\ H =[ 0.995 0.999 0.938 0.982 0.705 0.827 0.493 0.598 0.323 0.400 0.300 1.000 0.988 0.998 0.455 0.641 0.816 0.935 0.300 M 1.000 M 167T GMK(pm 0H 17 15.981 log KT 15.983 15.982 16.008 15.989 16.200 16.130 16.061 16.283 16.467 16.373 16.499 15.121 15.118 15.118 15.268 15.186 15.140 log KT 15.523 15.384 TABLE 2 TABLE 3 7~\~k~ Ion \pmJGM H -6.634 -4.553 -5.940 -0.754 -2.822 6.685 6.499 6.241 6.927 6.828 5.982 3.748 7.020 4.556 2.614 3.592 6.459 6.775 5.510 1.284 1/2 GMK 'Y 13/4. -0.756 -1.594 -8.856 -43.030 — 36.266 -40.774 -17.334 -23.933 -30.642 -13.048 log p -3.757 -6.813 -9.904 -0.718 logp 0.657 0.074 1.576 1.374 1.078 0.293 2.244 2.566 2.847 3.180 3.236 3.248 3.250 3.054 3.250 3.249 3.250 3.242 3.247 3.218 3.230 3.204 3.210 3.200 3.200 3.201 (10) (11) 805 (9) 1988ApJ. . .329. .803A opacity tobeinappropriateand forfractionalhydrogenioniza- Hayashi tracktheenvelope has cooledenoughforKramers real star,oncetheexpansion has broughtthestarcloseto envelope solutionswiththesame MandRfromabove.Ina power-law solutionbounds the luminosityofallfinitemass the massofstarand Kramers opacitylawholds,the Thus, aslongtheenvelopecontainsasignificantfraction of L, buttheenvelopemassdivergesasisapproached. equal. Envelopesolutionsexistforallluminositieslessthan two polytropicindices,n=3.20and3.25,arealmost smoothly inthecaseofKramersopacitylawbecause the finite envelopemass.Thetwosolutionsjoinparticularly 1 maximum luminositysolutionfollowsthepower-lawsolution, but deviatesfromitintheouterenvelopeordertohave a difference comesfromthefactthatdeepinsidestar the than theempiricalLforcoresdeepwithinstar. The c pl or, numerically, in equation(9)toobtain I substitutetheseintoequations(6)and(9)usen=32/10 max and (14) and (13)toobtain predicted fromMandR.Icombineequations(1),(3),(12), tant, theluminosityofpower-lawsolution,L,canbe density dependenceoftheopacity.Second,andmoreimpor- for radiation.First,thefullsetofstructureequationspredicts the polytropicindex,n=3.20,fromtemperatureand law solutioncannotberealizedwithoutthediffusionequation den Horn(1985).However,thefullsignificanceofpower- topology oftheU—VplanewasemphasizedbyYahilandVan the singularpowerlawwhoseimportanceindetermining order tohavefiniteenvelopemass.Thispower-lawsolutionis been traversed.Itthendeviatesfromthepower-lawsolutionin power-law solutionuntilabouthalftheenvelopemasshas The interiorofthemaximumluminositysolutionfollows power-law solution,butthedivergenceisveryslow;Mocr^. c pl the numericalmodels.Theenvelopemassdivergesfor b =32/103.20,whichexplainsthepolytropicindexfoundin The polytropicindexofthepower-lawsolutionisn=c/ I findthesolution and write 806 The luminosityofthepower-lawsolution,L,is5%larger pl © American Astronomical Society • Provided by the NASA Astrophysics Data System a =fi>^=11,c’d-fa■(13) ^PL L =9.443KI pl0 GM c T --Í^iEHä\ 42\R)\kJ- c Pc ~44nR? 15/2 {Gnm\ Ml' H 2 1 M c k )Rl' (15) APPLEGATE 1,2 changes aswellKinresponse toLincreasing,butthe a significantfractionofthe star’s masstomoveoutward(n mass ofthestarthanin case whentheenvelopemassis negligible. Inthiscaseadensity decreaseintheenvelopeforces luminosity iftheenvelopecontains asignificantfractionofthe or, numerically, The maximumluminosityforanegligiblemassenvelopeis the envelopesolutionmatchingontoann=3.25polytrope.) the reasonthatenormousradiiareneededinTable3toshow equation (18).(Notethattheconvergenceisveryslow;this is envelope radiusgoestoinfinityinthedensityprofilegiven in The integralinequation(19)convergestoM=16nRfifthe L~. Theradiusoftheenvelopeisdeterminedby As longasR>thedensityatedgeofcorevaries tion (10)and(16)toobtain expand. Thisargumentmaybequantifiedbycombiningequa- density decreasesatconstantenvelopemass,themust must bebroughtaboutthroughadensitydecrease.Ifthe dependence throughR),sothethermalconductivityincrease envelope T(r)isindependentofL(exceptforaweakimplicit ec must increaseinproportiontoL.Foranegligiblemass increases, equation(11)showsthatthethermalconductivity mined bythegravitationalpotential(r).Ifluminosity which holdsforpolytropicstars.Thetemperatureisdeter- ec the relation where Ristheradiusofenvelope.Thisaspecialcase models. Equations(2)and(11)give driving theexpansionofmorerealisticmassiveenvelope physics drivingtheexpansionisexactlysameas e a negligiblemassenvelope.Thiscaseisusefulbecausethe expansion oftheenvelope.Considerfirstsimplifiedcase of thecorewithincreasingluminosityisdirectlyrelatedto L isreached. zone. ArealstarmustintersecttheHayashitrackbefore e by forcingmostofthemassenvelopeintoconvection real starsluminositiesgreaterthanLcanbeaccommodated solution wouldboundallenvelopesolutionsfromabove.In mean molecularweightwasconstantthemaximumluminosity Kramers lawheldforalltemperatureanddensitythe grow toasignificantfractionofthemassenvelope.If tion tobeimportant.Thesecausetheouterconvectionzone max max The densityintheenvelope is muchmoresensitivetothe The decreaseofthetemperatureanddensityatedge T(r) = r _(16) 2 17k\r~RJ M =4n^p(r)rdr.(19) e 4 GMfimfl1\ (n +l)k H Vol. 329 (17) 1988ApJ. . .329. .803A No. 2,1988 envelope lowersthemagnitudeofbindingenergy change ofnisaminorcorrection).Theexpansionthe locally fromequation(17)iftheenvelopeisapproximately perature decrease.Thetemperaturedecreasecanbeseen polytropic sincetheexpansionlowersmagnitudeof(r). star, andthevirialtheoremdemandsthatmeantem- envelope. luminosity goesupthanwouldbeneededinanegligiblemass larger densitydecreasesareneededtosatisfyequation(9)ifthe decreases ifTdoes.Thisnegativefeedbackmeansthatmuch forced acrosstheHertzsprunggaptoredgiantbranch. maximum thataradiativeenvelopecancarry,andthestaris leaves themainsequenceitsluminosityeventuallyexceeds Lowering thetemperatureactsasnegativefeedbacksinceK luminosity ofthestarmustultimatelyexceedmaximuma There arethreecrucialingredientsinmyexplanation.First,the envelope thatitiscapableofwithstandingafactor35drop density dependenceoftheenvelopeopacitymustbesuchthat radiative envelopecantransport.Second,thetemperatureand for L=,andthattheexistenceofdivergencedepends And third,thecoremustbesufficientlydecoupledfrom the polytropicindexofpower-lawsolutionexceedsn=3. derivative oftheradiuswithresponsetoluminositydiverges luminosity increase.Inthelanguageofpolytropicsolutions, density dropattheedgeofcore.However,Itraceorigin physical), andthatgiantformationistriggeredbyanabrupt I findthattheenvelopemusthaveaneffectivepolytropicindex by Renzini(1984).InagreementwithYahilandVandenHorn in thepressureatitsedgewithoutsignificantlychanging n >3(neededforthepower-lawsolution,eq.[12],tobe Horn (1985)andwiththeratherdifferentargumentpresented structure. the Eddingtonlimit,orwithgeneralizationsofittoaccountfor the envelope’smaximumradiativeluminosity. addition, Ibelievethattheonsetofthermalinstabilityin on thetemperatureanddensitydependenceofopacity.In Horn 1985).IncommonwithRenzini(1984)Ifindthatthe uity ratherthanamolecularweightdiscontinuity(Vanden this correspondstoadensitydropdueanentropydiscontin- of thedensitydroptoresponseenvelope James H.Applegate:Department ofAstronomy,ColumbiaUniversity,538West120th Street, NewYork,NY10027 Renzini’s modelsistriggeredbythecoreluminosityexceeding .1985,inDynamicsofStarClusters,ed.J.GoodmanandP. Hut Cohn, H.1980,Ap.J.,242,765. Antonov, V.A.1962,Vestn.Leningr.Gros.Univ.,7,135. Cox, J.P.,andGiuli,R.T.\96&,PrinciplesofStellarStructure(NewYork: Iben, I.,Jr.1967,Ap.J.,147,624. Hoyle, F.,andSchwarzschild,M.1955,Ap.J.Suppl.,2,1. Hernquist, L.,andApplegate,J.H.1984,Ap.J.,287,244. Hayashi, C.1966,.4nn.Rev.Astr.Ap.,4,171. Eggleton, P.,andFaulkner,J.1981,inPhysicalProcessesRedGiants, ed. max (Dordrecht: Reidel),p.525. Gordon andBreach). I. Iben,Jr.,andA.Renzini(Dordrecht:Reidel),p.179. My explanationofgiantformationissimple:afterastar My resultssharefeatureswiththoseofYahilandVanden My maximumluminositysolutionshavenothingtodowith © American Astronomical Society • Provided by the NASA Astrophysics Data System WHY STARSBECOMEREDGIANTS REFERENCES 2 feels onlyitsowngravitationalpotential.Thisdecouplingis ity ofthesestars.Thecentralcondensationbroughtaboutby coupling ofcoreandenvelopethatisresponsibleforthestabil- clearly wrongformain-sequencestars;indeed,itistheclose sure throughoutmycalculation. the differentopacity,becauseIhaveneglectedradiationpres- whose opacityiswellapproximatedbytheKramerslaw.As formation ofredgiants. ples thecorefromenvelopebecauseincreasingly the slowaccumulationofheliumincoregraduallydecou- unaffected bychangesintheenvelope.Thisapproximationis down andthestarexpands.IfKramerslawheldforall hydrogen exhaustionhasapredominantlyradiativeenvelope tion ofgiantformationhastwoimportantparts:theresponse nontrivial, probleminstellarenvelopestructure.Theexplana- the realcontributionofmolecularweightgradientto zone grows.Bythetimeouterconvectioncontainsa expansion forcestheenvelopetocool;eventuallyKramers temperatures anddensitiestherewouldbeamaximumlumin- the star’sluminosityincreasesenvelopedensityisforced of theHayashitrack.Asolartypestaratpointcore of theenvelopetoanincreaseinluminosity,andexistence exceeds theradiativemaximum,starisforcedto luminosity solution,althoughanidealization,isusefulin in thestar’smovingupHayashitrack.Themaximum begins toaffecttheequationofstateandouterconvection law becomesinappropriate,partialionizationofhydrogen for Physicstheirkindhospitality.ThisiscontributionNo. luminosity. giants isexplainedbyL=4tigTh,where7^theeffective understanding thebehaviorofenvelope:ifluminosity the Hayashitrack,andanyfurtherluminosityincreaseresults significant fractionofthemassenvelopestarhashit osity whichtheenvelopecouldtransport.Inrealstars useful discussion,andIwouldliketothanktheAspenCenter temperature oftheHayashitrackappropriateforstar’s Hayashi track.Onceonthetrack,largesizeofred Iben, I.,Jr.,andRenzini,A.1984,Phys.Repts.,105,332. 321 oftheColumbiaAstrophysicsLaboratory. Schönberg, M.,andChandrasekhar,S.1942,Ap.J.,96,161. Renzini, A.1984,inObservationalTestsofStellarEvolutionTheory, ed. Lynden-Bell, D.,andWood,R.1968,M.N.R.A.S.,138,495. Van denHorn,L.1985,privatecommunication. Strömgren, B.1965,inStarsandStellarSystems,Vol.5,Structure, ed. Schwarzschild, M.1958,StructureandEvolutionoftheStars(NewYork: Yahil, A.,andVandenHorn,L.1985,A.J.,296,554. A. MaederandRenzini(Dordrecht:Reidel),p.21. p. 262. Dover). L. H.AllerandD.B.McLaughlin(Chicago:UniversityofChicagoPress), The centralapproximationinthispaperisthatthecore The formationofredgiantsisastraightforward,although I wouldliketothankLeoVandenHornforaparticularly in. CONCLUSIONS 807