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198 4ApJ. . .211 . .7 91N © 1984.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. The AstrophysicalJournal,277:791-805,1984February15 there existtwomajortypesofsupernovaexplosion(see Fowler (1960)andColgateWhite(1966),hasshownthat of singlestars,followingpioneeringpapersbyHoyleand more massivethan10Mevolvetodevelopironcoreswhich Rees andStoneham1982;Nomoto1982dforreviews): Sugimoto andNomoto1980;Wheeler1981;Trimble1982; collapse toleaveneutronstarsorblackholesbehind,and stars of6(+2)-8(+l)Mformelectron-degeneratecarbon- degenerate coreduetoelectroncaptures(Rakavy,Shaviv, leavingnoneutronstarremnant. (C+O)coresandexplodeasacarbondeflagration for supernovaehasbeensuggested,i.e.,collapseof a however, onlyalittleattentionhasbeenpaidtothistype of and Zinamon1967;FinziWolf1967).Untilrecently, nonexplosivelycouldevolveuptotheformationof the supernova; manypeoplehaveimaginedthatstarswhichburn core.Thismightbeduetothelackofdetailedcalculations of thepresupernovaevolution8(+1)-10(±1)M©beyond carbon burning.Onlypreliminaryresultswerereported by 0 hydrodynamical calculation evolvingadegenerate Barkat, Reiss,andRakavy(1974)whoshowedthattheir8 M© burning. developsadegeneratecoreafternonexplosivecarbon 0 1 The studyofpresupernovaevolutionandsupernovamodels Besides thesetwomajortypes,anothertriggeringmechanism Following theirsuggestion, Miyaji etal.(1980)dida OnleavefromDepartmentofPhysics, IbarakiUniversity,Japan. {0) 2420(0) edge ofthecoreisfittedtoboundaryconditionsatbottomhydrogen-richenvelope.Two cases withinitialheliumcoremassofM=2.6M(case2.6)and2.42.4)arestudied.Both cores spendthecarbonburningphaseundernondegenerateconditionandleaveO+NeMgcores.Further evolution dependsonthemassofO+NeMgcore,M. flash isignited.Theneonandoxygenflashinglayermovesinward. zonestarts.Furtherevolutionupthroughthecorecollapsewhichistriggeredbyelectron O +NeMgcorebecomesstronglydegenerateandadredge-upofheliumlayerbythepenetratingsurface captures onMgandNemustbecommontocaseswithMh=2.0-2.5(i.e.,stellarmassof 8-10 M). to thehelium-burningshellisclose1.4M.Possibleformationofhydrogen-deficientcarbonstarsand HG0 Subject headings:stars:evolution—interiorssupernovaewhitedwarfs O +NeMgwhitedwarfsbothinsinglestarsandclosebinarysystemsarediscussed. However, thesestarswouldnotcontributemuchtothenucleosynthesisinGalaxysincemassinterior c 0 0 0 coresinstarswithmassesnear10Mareevolvedfromtheheliumburningphase;outer For case2.6,Mexceedsacriticalmassforneonignition(1.37)sothatstrongoff-center For case2.4,ontheotherhand,neonisnotignitedbecauseMsmallerthan1.37.Then Therefore 8-10MstarswouldgiverisetoTypeIIsupernovaexplosionswhichleaveneutronbehind. © American Astronomical Society • Provided by the NASA Data System 0 c0 c0 0 EVOLUTION OF8-10MSTARSTOWARDELECTRONCAPTURESUPERNOVAE. 0 I. INTRODUCTION I. FORMATIONOFELECTRON-DEGENERATEO+NeMgCORES Laboratory forAstronomyandSolarPhysics,NASAGoddardSpaceFlightCenter Received 1983March4;acceptedJuly12 1 Ken’ichi Nomoto ABSTRACT 791 24 20 flash isignitedduringthecollapse,electroncapturesarerapid O +NeMgcoreandfoundthatelectroncapturesonMg initiation ofexplosiveoxygenburning.Althoughthe and Netriggerthecollapseofcorepriorto be observedasaTypeIIsupernova. enough toovercometheoxygendeflagrationsothatcore continues tocollapseupneutronstardensity.Thistypeof evolution ofstarswhichformdegenerateO+NeMgcores supernova iscalledan“electroncapturesupernova”andwould Nomoto etal.1982;Woosley,Weaver,andTaam1980; of thesestars. after carbonburningandtodeterminetheexactmassrange Weaver, Axelrod,andWoosley1980).Preliminaryresults of masses around~10M©havebegun(Nomoto1980a,1981; these calculationshaverevealedthattheevolutionof8-12 M© in viewofthefollowingpoints: Therefore, varioustypesofsupernovaexplosionareexpected stellar massbecausethisrangeisatransitionregion (in particular8-10M©)starsisworthinvestigatingindetail to occurinthesestars. from -degeneratetonondegeneratestellarcores. the lowestmassofstarswhich giverisetoTypeIIsupernova explosion andproduceneutron stars.Thisisimportantfor comparison withstatisticsof supernovaeandpulsars(e.g., Tammann 1982;Lyne1982). This resultsuggestedtheimportanceofexploring Recently, detailedevolutionarycalculationsforstars of 2. Throughthestudyofthis mass range,wecandetermine 1. Theevolutionof8-12M©starsisquitesensitivetothe 198 4ApJ. . .211 . .7 91N (0) (0) 24 20 M starsmaybesignificantlydifferentfromthatformore bounce hasbeenshowntopropagatemoreenergeticallyfor massive starsbecauseashockwhichformsatthecore the smallermasscore(Hillebrandt1982a,b;Arnett1982). 792 progenitor inviewofelementalabundancestheejecta Woosley, Weaver,andTaam1980;Hillebrandt1982h). (Davidson etal1982;Nomoto1982c;al. enough massisaccreted,arelikelytocollapseformneutron et al.1919;Nomoto1980a,1981).Thesewhitedwarfs,if 0 +NeMgwhitedwarfsinclosebinarysystems(Nomoto q stars. M =2.6,2.4,and2.2,respectively.Thesecoresare the calculationsofevolution8-10Mstarsasmentioned and 2.2)whichcorrespondtotheinitialheliumcoremassof starting fromcentralheliumburningforthreecases(2.6,2.4, above (Nomoto1981,1983).Ihaveevolvedheliumcores 1 reporttheevolutionofcases2.6and2.4through Mq withanapproximaterelationofM^4Malthough supposed tobeformedinstarsoftotalmassesMä8.5-11 the exactcoremass-totalmassrelationdependson composition (BeckerandIben1979).Inthispaper(PaperI), evolution fromheliumburningthroughthephaseof physics aregiven.In§III,thegravitationalcontractionof evolved evenfurther,fromthedredge-upofaheliumlayer ignition ofanoff-centerneonflash(case2.6)andtheformation through thecollapsetriggeredbyelectroncapturesonMg of adegenerateO+Ne-hMgcore(case2.4).Case2.2hasbeen hypothetical pureneonstarsisdiscussedinordertoclarify evolution, whichiscrucialfor8-10Mstars.In§IV,the the effectofelectrondegeneracyonneonignitionandstellar concluded thatthemassrangeforelectroncapturesupernovae and Ne,whichwillbereportedinNomoto(1984,hereafter is 8(±l)-10(±l)M. Paper II)(seeNomotoetal.1982;1983).Itis developing O+NeMgcoreissummarized.Neonignition for case2.6isdescribedin§V,andtheformationofa h0 0 In §VII,thecompositionstructureatendofcalculation strong degeneratecoreforcase2.4isdiscussedin§VI. H does notdependonthetotal massofthestar(Paczynski example, thecoremass- relationfor3-8Mstars envelope (e.g.,Hayashi,Hoshi, andSugimoto1962).For Accordingly, thecoreevolves almostindependentlyofthe -burning shellissosteepthattheouteredge of and thehydrogen-richenvelope.Thedensitygradientat the separated intotwoparts,i.e.,thehydrogen-exhausted core is shown.Furtherevolutiontowardsupernovastages,possible the corecanberegardedas aquasisurfaceofthecore. § IX,asummaryisgiven. formation ofO+NeMgwhitedwarfs,andtheorigin of hydrogen-deficient carbonstarsarediscussedin§VIII. In 0 o 0 3. Hydrodynamicalbehaviorofthecorecollapsein8-10 4. The8-10MstarsaresuggestedtobetheCrabnebula’s 5. The8-10Mstarsarepossibleprogenitorsof In thepresentseriesofpapers,Ishallgivedetails 0 In thenextsection,methodofcomputationandinput 0 In theadvancedphaseofstellarevolution,star is II. METHODOFCALCULATIONANDINPUTPHYSICS © American Astronomical Society • Provided by the NASA Astrophysics Data System a) Models NOMOTO (0) (0) envelope atthehydrogen-burningshell,wherehydrogen the evolutioniscalculatedinfollowingway. concentration isassumedtochangediscontinuously.Theouter by subscripts1/andle,respectively.Whenaquantityis edge ofthecoreandbottomenvelopearedenoted continuous acrossthecore-envelopeinterface,iorewillbe burning stagewithacompositionofT=0.98andZ0.02, omitted. 1970). Inviewofthesecharacteristicstheevolvedstars, of thecore.Thestartingmodelisincorehelium where YandZdenotetheconcentrationofhelium heavier elements,respectively.Twocasesfortheinitialcore mass werechosen,i.e.,M=2.6(case2.6),and2.4 where X =0.602,andZ0.02wereintegrated(Nomoto hydrogen-rich envelopeswithhydrogenconcentrationof assumed tobeM=4Mandthecorrespondingstatic with theboundaryconditionsatouteredgeofcore. reach thecore-envelopeinterface,envelopeisreplaced These conditionsareobtainedfromthecentrallycondensed (case 2.4).Totalmassesofstarsforthesecoreswere expressed as: Sugimoto 1972)tobefittedtheouteredgeofcore. Nomoto 1980,Appendix).Theluminosityofthehydrogen type ofenvelopesolution(Chandrasekhar1939)andare shell burning,L,iscalculatedbyusingathin-shell processes theenvelopematerialintohelium.Theseboundary is allowedtoincreasewhenthehydrogenshellburning L isthefluxcomingfromcore.Thuscoremass approximation (Hayashi,Hoshi,andSugimoto1962) Here, thesymbolshavetheirusualmeanings(Sugimotoand been foundtobeaccurateenoughbyintegratingstatic conditions, whichdonotdependonthestellarmass,have by applyingtheseconditions(SugimotoandNomoto1975). Nomoto 1974)andtheirvaliditywasdiscussedindetail by hydrogen-rich envelope(NomotoandSugimoto1972; mass-luminosity relationfor3-8Mstarisreproduced well H0 envelope fittingisdoneandthecoremassallowed to decrease duringthedredge-up oftheheliumlayer. the surfaceconvectionzoneintocore,anactualcore- Sugimoto andNomoto(1980,p.165).Infactthe core H thermal andhydrodynamical equations(Sugimoto,Nomoto, H r U 0 The starisdividedintothecoreandhydrogen-rich A Henyey-typecalculationiscarriedoutfortheevolution When thesurfaceconvectionzoneisnotdeepenoughto When theheliumlayerexpandstoinducepenetrationby The Henyey-typecalculational methodincludingboth 4 ld\n P\_16nacGTM^ \d\n T)3P/cL r ler V _d\nPGMp r = (2U +V—4)=0, le L,le + r li b) InputPhysics 3 dlnM_47irp r d\nr M r din rrP Vol. 277 (3) (2) (1) (4) (5) 198 4ApJ. . .211 . .7 91N 23 2 4-3 No. 2,1984 burning, thetreatmentbyEndal(1975h)ismodifiedto same asusedbyNomoto(1982a,b)exceptforthefollowing. and Eriguchi1981)theinputphysicsaremostly treated asdescribedbyArnett(1972a,1974a,b).Forcarbon include Nawithsteadystateabundance(A.S.Endal masses around10M.Thisdependsonthemassof or notiscrucialfortheevolutionaryfateofstarswith The Weinberg-SalamtheorywithaWeinbergangleof are photo,pair,andplasmonneutrinoemission(Beaudet, thermodynamics ofthestellarcore. the centralregion,sinceelectrondegeneracyaffectsglobal electron, muon,andtauneutrinosareincluded.Forconvec- sin 0=0.23isappliedusingDicus’s(1972)formulaeand Petrosian, andSalpeter1967)neutrinobremsstrahlung pure neonstaranddemonstratetherelationbetween O +NeMgcoreandthedegreeofelectrondegeneracyin 0.08 M,0.31and1.06forhydrogen,helium, ignition andthemassofneonstars.Suchasimplemodel by Unno(1967)isappliedtothecore. while atime-dependentmixing-lengththeoryasformulated pressure isemployedforthestatichydrogen-richenvelope, with themixinglengthputequaltoaunitscaleheightof tion, theusualmixing-lengththeory(cf.CoxandGiuli1968) (Dicus etal1976;Cazzola,DeZotti,andSaggion1971). et al.1968).Ithasbeenfoundthatthereexistsacritical calculation hasbeenmadeforstarsofhydrogen(Kumar mass belowwhichnuclearfuelisnotignited;thevaluesare 1979, privatecommunication).Theneutrinoprocessesincluded carbon burning,respectively. The behaviorof(p,T)duringthecontractioncanbe 4 x10gcm.Evolutionarytracksinthe(p,T)plane masses of1.30M,1.351.365>1-37and understood asfollows: degenerate andthusthegravothermalspecificheatofa star are showninFigure1for1.30M,1.365and1.37. is negative(e.g.,Kippenhahn1970;Sugimoto,Eriguchi, and 1963), helium(CoxandSalpeter1964),carbon(Murai 0 dominates overheatloss. as heatisremovedfromthestarbyradiativetransport W and neutrinoemission;i.e.,thegravitationalenergyrelease Hachisu 1981).Therefore,thecentraltemperatureincreases 1.39 M.Theinitialcentraldensityisassumedtobepæ 0 temperature dependenceofthepressurebecomesweak. This negative topositive(Kippenhahn1970;Sugimoto,Eriguchi, makes thegravitationalenergyreleaseinferiortoheat loss and thesignofgravothermalspecificheatchangesfrom and Hachisu1981);thecentral temperatureofthecontracting c c 0G plotted asafunctionofthe starmass,M,inFigure2; star beginstodecrease. it isshownthatthepeakvalue ishigherforlargerM. 0 qc Nucleosynthesis duringhelium,oxygen,andneonburningis As summarizedin§I,whetherneonburningisignited Therefore, itisusefultoemployaverysimplemodelof I computedgravitationalcontractionofpureneonstarswith During earlyphaseofcontraction,electronsarenot During laterphases,electronsbecomedegenerateandthe As seeninFigure1,Tattains apeakvaluewhichis c III. GRAVITATIONALCONTRACTIONOFNEONSTARSAND © American Astronomical Society • Provided by the NASA Astrophysics Data System THE CRITICALMASSFORNEONIGNITION 8-10 MoSTARS become significant,atemperatureinversionappearsand curves showthechangeinmaximumtemperature,T,throughout the centraltemperature,T,asafunctionofdensity;dashed M =1.30,1.365Mq,and1.37.Solidcurvesshowthechangein dependence oftheplasmon-neutrinoemissionrateand star asafunctionofthedensity,p,atshellT.Stagenumbers pressure. weak temperaturedependenceoftheelectron-degenerategas outward duringthecontraction.Thisisduetodensity shell ofmaximumtemperature,T,withinthestarshifts neon ignitionoccursonlyforstarsofM>1.37. correspond tothepeakvalueofTforM=1.30and1.365, at stages1and4forM=1.301.37,respectively;26 respectively. Thedottedlineistheignitionforneonburning.Off-center (1-6) attachedtothesecurvesindicatethesamestages.Tattainsitspeakvalue max c 0 max max 0 max0 0 c neon burning.Thedottedlineindicates theneonignitionline. function ofneonstarmass,M.Dashed curvesareobtainedbysuppressing When botheffectsofelectrondegeneracyandneutrinoloss Fig. 1.—Gravitationalcontractionofpureneonstarswithmasses Fig. 2.—ThepeakvaluesofTand TgiveninFig.1areshownasa c max 793 198 4ApJ. . .211 . .7 91N 7-3 794 critical massdoesnotdependonthedetailsof core Curled regionsareconvectiveowingtonuclearburning.Theupperpartofthehydrogen-richenvelopeisomittedfromfigure. structure. M <1.365,7^risesduringtheearlyphaseofcontraction changes in7^againstthedensityofshellatlocation dependence ofComptonscatteringopacityhasbeenfound the luminositynearsurfaceapproacheslocalEddington and probablycausesmasslossfromthestar.Thisisbecause these hypotheticalneonstars,thesurfacelayerexpandsgreatly with differentstagesalongthecurvesinFig.1indicate line forTextendingthroughM=1.39isobtainedby value of7^isplottedagainstMinFigure2.Thedashed of TforM=1.30M,1.365and1.37.For starsassociatedwith X-raybursters(Ebisuzaki, by D.Sugimoto(1970,private communication)inhisneon outward astemperaturedecreasesand,thus,L limit, L,sincetheComptonscatteringopacityincreases defined bye=6,whereanddenotethenuclearenergy p =3x10gcm.InFigure2,theneonignitionlineis same stagesfor7^andT). artificially suppressingneonburning.(Thenumbersassociated and thenstartstodeclinebeforeneonignition.Thepeak star calculationandapplied to amodelformasslossfrom outward. Thismechanismofmasslossduetothetemperature because ofelectrondegeneracyandneutrinocooling.This neon ignitionis1.37Mbelowwhichnotignited flash andsothetemperaturerisesrapidly. will bediscussedin§V,neonshellburningisunstabletoa generation rateandneutrinoemissionrate,respectively.As of neonburningattheshellM=0.84with Hanawa, andSugimoto1983).] result isconsistentwiththeconclusionbyBoozer,Joss,and ignition tobe1.37-1.39M.Thisagreementimpliesthat the and estimatedthecriticalmassofC+Ostarsforneon Salpeter (1973)whocalculatedtheevolutionofC+Ostars 0 ax max0 ax max0 Ed Ed nv max 0 r0 0 Fig. 3.—Chemicalevolutionforcase2.6fromheliumburningthroughneon-oxygenshellflashes.Thetime,i,ismeasuredtheendof calculation. [It isnoteworthythat,inthelatestagesofevolution of In Figure1,thedashedcurvesshowevolutionary For M=1.37,Treachestheignitiontemperature From Figure2,wecanconcludethatthecriticalmassfor 0max © American Astronomical Society • Provided by the NASA Astrophysics Data System NOMOTO general featuresoftheevolutionpresentmodels.In is shownforcases2.6and2.4,respectively.Thetime,i, change inthechemicalcompositionduetonuclearburning, Figures 3and4,thechemicalevolutionofcore,i.e., used hereafter.Thecurledregionsareconvectiveowingto measured fromtheendofcalculationandi*=—iis phases ofheliumburning,carbonandthegrowth nuclear burning. Table 1,wherei*andthemassescontainedinteriorto The characteristicevolutionarystagesaresummarizedin the O+Ne+Mgcorewithconvectivecarbonshellburning. shell withthemaximumenergygeneration,£„,forhydrogen the onsetofdredge-uphelium layer. Before discussingthedetailsofresults,Ishalldescribe The coresofthetwocasesevolvesimilarlythrough IV. EVOLUTIONTHROUGHO+NeMgCOREFORMATION Fig. 4.—Sameas3butforcase 2.4fromheliumburningupto a) GeneralPictureofEvolution Vol. 277 198 4ApJ. . .211 . .7 91N No. 2,1984 M asseenfromTable1ati*=0yr(case2.6)and59.6 ignition. Suchadifferenceisduetothesmallin the O+NeMgcoreofcase2.4coolsdownwithoutneon cases; i.e.,neonisignitedintheoutershellforcase2.6,while later phases,however,theevolutionisdififerentforthesetwo mass (1.37M)forneonburning,whileissmallerthan (M), heliume),andcarbonburningaregiven.At Time, t,isthesameasinFig.1. discussed phasebyphase.Timescalesforthesephasesare obtained fromTable1.Evolutionarychangesinthephoton (case 2.4):Forcase2.6,Mislargerthanthecriticalcore the radiusofcoreedge,rareshowninFigures5 luminosity attheouteredgeofcore,L=,and generation rate(withoutincludingL),andtheneutrino (case 2.6)and62.4).Changesinthenuclearenergy 1.37 Mforcase2.4. c 0c hHc c u phr le H e Fig. 5.—Evolutionarychangesinthe photonluminosity,L,andtheradialdistancefromcenter,r at theouteredgeofheliumlayerforcase2.6. In thefollowingsections,evolutionofbothcasesis ph1 © American Astronomical Society • Provided by the NASA Astrophysics Data System Carbon ignition Beginning ofheliumburning... Carbon shellburning: Carbon exhaustion Helium exhaustion Neon ignition respectively. Onset ofdredge-upHelayer. 3 b 4th peak 2nd peak 3rd peak 5th peak 1st peak Time,t=—t*,ismeasuredfromtheendofcalculation. SubscriptsH,He,andCdenotetheshellofmaximumenergygenerationhydrogen,helium,carbonburning, Stage a Evolutionary StagesandCoreMasses 2.52 E3 9.73 E3 3.45 E4 8.48 E3 0.0 1.69 E5 1.68 E6 1.02 E3 1.96 E2 1.19 El Time (yr) t* 8-10 MSTARS 0 2.79 2.60 2.791 M h b Case 2.6 TABLE 1 Core Mass(M) 0 1.49 1.48 1.44 1.44 1.10 1.45 1.49 1.37 1.449 M h luminosity, L,areshowninFigures7(case2.6)and8 potential ofanelectroninunitskt,\\t(electrondegeneracy density, p,centraltemperature,Tandchemical changes inthephysicalquantitiescore,i.e.,central 2.4). Figures9(case2.6)and102.4)showevolutionary parameter). Whenatemperatureinversionappearsinthe plotted inFigures9and10.Finallytheevolutionarypath core, themaximumtemperaturewithin7^,isalso of (p,7^)isshowninFigures11(case2.6)and122.4). processed intoheliumbyhydrogenshellburning.(The 2.40 Mto2.59forcase2.4astheenvelopematerialis increases from2.60Mto2.79forcase2.6and luminosity, L,ofhydrogenshellburningduringtheearly v c c ax c 0 0 H During thecoreheliumburningphase,mass,M, H 0.67 0.85 0.67 0.97 1.21 1.38 1.403 3 2.49 E5 4.38 E3 6.68 E4 8.51 E2 3.47 E2 1.89 E6 5.96 El 1.52 E4 1.41 E4 1.91 E3 Time (yr) r* b) HeliumBurning 2.58 2.40 M 2.581 h Case 2.4 b Core Mass(M) 0 M h 1.25 1.02 1.35 1.36 1.37 1.38 1.38 1.35 1.343 0.61 0.84 0.62 0.96 1.13 1.26 1.339 M r 795 198 4ApJ. . .211 . .7 91N 46 -3 796 shell appearsattheouteredgeofformerconvective gravitational contractionoftheC+Ocore.Ahelium-burning extinguish hydrogenshellburningandtheincreaseinMis with previousresults(see,e.g.,Arnett1972a). stages isobtainedfromFigs.5-8asL=—).Inthe core ofheliumburning(Table1).Theshelladvances stopped. later stagesofthisphase,theoutercoreedgeexpandsto core evolveswithincreasingTandp(from~10to10 value giveninTable1(seeFigs.3and4).Thecontracting in masstoincreasetheC+Ocoremass,M,up g cm)asseeninFigures9-12. H included inL„. and theneutrinoluminosity,L,for case2.6.Hydrogenshellburningisnot Hphn c Ue v After exhaustionofhelium,thestarentersphase The quantitativefeaturesofthisphaseagreeapproximately Fig. 7.—Evolutionarychangesinthe nuclearenergygenerationrate,L„, © American Astronomical Society • Provided by the NASA Astrophysics Data System c) ContractionofC+OCore NOMOTO 6-3 6-3 è-3 5-3 8 for case2.4(Fig.6).Thisisessentiallythesamephenomenon r increasesbyafactorof6forcase2.6(Fig.5)and10 L, increasesbyafactorof2(Figs.5and6). the expansion,entropyinheliumlayerincreasesby to aredgiant(SugimotoandNomoto1980,p.166).During as occursinastarwhichevolvesfromthemainsequence becomedegeneratenearthecenterasseenfrom neutrino emission,althoughtheintegrated absorbing heatfromtheinteriorandphotonluminosity, luminosity, L,isstillsmallerthan(Figs.7and8).Then increase ini¡/(Figs.9and10). neutrino emissivityislargerforhigherdensityandbecause p iscloseto10gcm(Fig.13).Thisbecausethe slight temperatureinversionnearthecenterforcase2.4when case 2.6,suchatemperatureinversiondoesnotappearduring neutrino bremsstrahlungprocessaroundp~10gcmis this phasebecauseoftheweakerelectrondegeneracy.The temperature inversiontoexistasmentionedin§III.For the weaktemperaturedependenceofpressureallows not negligibleindeterminingsuchatemperatureprofile. core (M)exceedsthecriticalmassfornondegeneratecarbon results intheignitionofcarbontoyieldaspikeL in ignition (1.06M)forbothcases.Eventuallythecontraction x ph Figures 7and8. e —atp=1.0x10gcminFigure11. evolutionary pathofip,T)reachestheignitionline of vph c c the coreisalmoststationary atp~4x10gcmand to reducepbyafactorof2 byabsorbingheat.Afterwards, concentration ofcarboninthe convectivecoretoX=0.001. c n 0 T~l x10Kuntilthe carbon burningreducesthe nvc c c c c c 5 At thesametime(i*~10yr),heliumlayerexpands; The entropyinthecentralregiondecreasesmainlyby Such neutrinocoolinginasemidegeneratestateyields During thegravitationalcontraction,massofC+ O For case2.6,carbonisignitedatthecenterwhen A convectivecoredevelopsand thecentralregionexpands Fig. 8.—Sameas7butforcase2.4 d) CarbonBurning i) Case2.6 -t(yr) Vol. 277 198 4ApJ. . .211 . .7 91N 86 -3 (0) M =0.045withT5.7x10Kandp1.5g profile atthisstageisshownagainstMinFigure13 cm becauseofthetemperatureinversion.The of anelectronatthecenter,\Jj,inunitskTforcase2.6. No. 2,1984 Vilhu 1978;BeckerandIben1980). previously (Muraietal.1968;KutterandSavedoff1969; core ofMh=2.2Mat0.21(PaperII;Nomoto ignition lineinFigure12. because ofelectrondegeneracy(^~3),andLreaches et al1982).Theseresultsareconsistentwiththosefound (dashed line)andagainstpbythedashedlinenearcarbon Sugimoto 1971;Boozer,Joss,andSalpeter1973;Ergma rq r c 0r cn Fig. 9.—Evolutionarychangesinthecentraldensity,p,temperature,Tmaximumthroughoutcore,andchemical potential For case2.4,carbonignitionoccursattheoutershellof The carbonshellburningisweaklyunstabletoaflash Such anoff-centercarbonignitionoccursalsoforthe cmax © American Astronomical Society • Provided by the NASA Astrophysics Data System ii) Case2.4 8-10 MSTARS 0 6 6-3 8 5-3 4 4 0.69 M.Thetemperatureoftheflashingshellrisesupto by afactorof4.Thiscausesanexpansionthecentral 2 x10L(Fig.8).TheconvectiveshellextendstoM= Then Tincreasesandmakesalooparoundp~10gcm Afterwards thecarbon-burninglayershiftsinwardas region andthedecreaseinpTFigures1012. center andproceedsundernondegenerateconditionatp~ temperature oftheinnerlayerisraisedbyheattransport. 7.2 x10Kandthedensityofoverlyinglayerdecreases 4 x10gcm.Furtherevolutionisquitesimilartocase2.6. in Figure12.Eventuallycarbonburningisignitedatthe emission andtheelectrondegeneracyincreases.Suchcooling enters aphaseofgravitationalcontraction. in theelectrondegeneratestatecausesasubstantialdecrease After exhaustionofcarbon,acoreO+NeMgisleftwhich from theincreaseinparoundi*^10yr(Figs.9and10). reduced toA=0.001,thecorestartscontractasseen in 7^ati*~10yr(Figs.9and10). shell appearsatM=asgiveninTable1.The shell the centralcore;inotherwords,alargetemperatureinversion the centertocarbon-burningshellbecauseofcooling of of maximumtemperature,T,withinthecoremovesfrom appears asseenfromTandinFigures910. q dependence ofethanthat£,however,exceedsat the bottom ofthecarbon-burning shell.Accordingly,aconvective 0r c c disappears. Aradiativephase thenfollowsandtheshellof shell developstocarrytheexcess energyoutward. c (Figs. 7and8).Becauseofthestrongertemperature c c rc max maxc nv When thecarbonconcentrationincentralregionis The entropyinthecentralregionisreducedbyneutrino At theformationofO+NeMgcore,acarbon-burning As carbonisdepleted,the convective shellshrinksand During thecarbonshellburningphase,L^holds nv e) Growthof0+Nq+MgCore ii) ConvectiveCarbonShellBurning i) TemperatureInversion 797 198 4ApJ. . .211 . .7 91N 2 798 against logpisshownbythedashedcurve,whereopenandfilledcirclesindicatecenterburningshellsofneon,carbon, andhelium, ignition linesofcarbon,neon,andoxygen. respectively. Evolutionarypathsof(p,T)attheneonflashingshellandoxygenareshownbysolidcurve.Dotted lines indicatethe maximum eadvancesinLagrangiancoordinate,M,the burning shellreachesawithrelativelylargeX\then region ofsmallXwhere)ati*æ300yras 0 cmax nv The convectiveshellnevertouchesthetailofheliumlayer. For case2.6,somecarbon,whichremainsunburnedinthe The energygenerationrateofcarbonshellburningdoesnot Vol. 277 198 4ApJ. . .211 . .7 91N 8 4 5- 3 No. 2,1984 increase. Ontheotherhand,maximumtemperature,T, carbon-burning shell.AlsoTstaysaround4x10Kwhich since TisdeterminedbythebalanceofLæat in theoutershellincreasesonlyslightly(Figs.9and10), O +NeMgcoregrowsgradually(Table1),andp carbon intheO+NeMgcoreburnsrelativelypeacefully central regionandtheneutrinoenergyloss. is determinedbythebalancebetweencompressionof without developingaconvectiveshell,andsmallamountof As aresult,Tdecreasesbecauseoftheneutrinoemission. burning: theexpansionofcarbonburningshellmakes the peakvalue(solidcurve;t*=140yr).Filledandopencirclesindicate core contractionslowerandcompressionalheatingsmaller. and 10areduetotheonsetofconvectivecarbonshell carbon remainsunburned. burning shellsofheliumandcarbon,respectively. ignition (dashedcurve;f*=6.68x10yr)andatthestagewhenTattains Me =1-45Mq(i*^50-10yr).Thisimpliescanexceed the massofO+NeMgcore,M,growstoapproach the criticalmassof1.37Mforneonignition.Thecore is from thecarbonburningshell (M=M)to1.0. as Mincreases. relativistically degeneratesothatpincreasesratherrapidly max c maxnv c density dependenceoftheneutrino lossrate(mainlypairand photo neutrinoprocesses)which islargerneartheouteredge c m&x Such achangeinthetemperature profileiscausedbythe of theO+NeMgcore(p~ 10gcm)thaninthelayerof Hc c 0 rc0 c c Through thephaseofcarbonshellburning,mass Fig. 13.—TemperatureprofilesagainstMforcase2.4atoff-centercarbon After thefourthconvectivecarbonshellburningforcase2.6, Small irregularitiesinthesequantitiesasseenFigures9 When Mbecomes1.30,theshellof7^shiftsinward r c0ax © American Astronomical Society • Provided by the NASA Astrophysics Data System V. OFF-CENTERNEONFLASHFORCASE2.6 iii) TemperatureChangeduringtheCoreGrowth a) TemperatureProfile 8-10 MSTARS 0 7-3 9 6-3 613 7 3 8 -4 M ~1.0.Instilldeeperlayersofthecore, compressional heatingduetocontractionislesseffective. shown againstpinFigure11(dashedline)andM i.e., i//~5,neonshellburningisunstabletoaflash.The at theshellwithM,.=0.88M(p1.8x10gcm). Figure 14. temperature risesupto2x10Kandthendeclinesbecause burning (L„).Thentheshellstartstoloseentropyandcontract. Since electronsattheignitedshellaresomewhatdegenerate, Accordingly, theevolutionarypathofdensityandtempera- loss rate(L)exceedstheenergygenerationofneon shell expandstop=1.4x10gcm,theneutrinoenergy of theexpansionflashingshell.Whenburning ture oftheneonburningshellmakesaloopin(p,7)planeas r0 L increasesbyafactorof10andreaches3.5x seen inFigure11. much higherthanforheliumshellflashes(e.g.,L~10 burning rate.ThepeakvalueofLforthisneonflashis because ofthestrongtemperaturedependenceneon for a1.39MC+Ocore;SugimotoandNomoto1975) r flashing shellis—dt/dInp—400swhich10timeslonger dynamical effects.Infact,thetimescaleofexpansionat and carbonshellflashes(e.g.,L~6x10for1.1M than thedynamicaltimescale. C +Ocore;Sugimoto1971),butitistooweaktoinduce 0 extent whenneonhasbeenalmostdepleted(X=0.03), from M=0.88to1.34.Atthestageofitsmaximum carbon layerwithX<5x10.Flowever,theamountof the outeredgeofconvectiveshelltouchestail neon ignition(i*=11.9yr).Filledcircles indicatethenuclearburningshells. to appreciableeffect.Inorderfortheconvectiveshellreach carbon mixedintotheconvectivelayeristoosmalltogiverise v n0 HeG n 0 c+c0 Ne r0 c The temperatureprofileatneonignition(see§YVb)is When Mreaches1.38(i*=12yr),neonisignited Although theincreaseintemperatureisonlyafactorof1.5, c0 A convectiveshellofneonburningdevelopsextending Fig. 14.—Densityandtemperature profilesagainstMforcase2.6at r b) NeonShellFlash 799 198 4ApJ. . .211 . .7 91N 17_ 9 to ignitethenextneonflashatshellwithM=0.87. and anoxygenflashfollows.Thisdevelopsaconvective Because ofitssmallconcentration,neonisquicklyexhausted the oxygen-flashingshellisshowninFigure11.Thetempera- layer extendingtoM=1.28.Thechangein(p,T)at flash andthusLissmallerthan(Fig.11).Therefore ture peakisonlyslightlyhigherthanforthepreviousneon Thus itisunlikelythatthecarbonandneonlayersare more neonwithX~0.2intheshellshouldhaveburned. release ofmorethan10ergsgwouldberequired,i.e., the carbonlayerwithX>0.1,anadditionalnuclearenergy mixed enoughtoaffectfurtherevolution. attains apeakvalueof1.11 x10Katf*=160yr.The be needed,thecalculationwasstoppedwhenneon-flashing These neon-oxygenshellflashesoccursuccessivelyinthe after thepeaktemperatureisattainedandflashmakesa L exceeds(i.e.,theentropystartstodecrease)shortly both pandTasshowninFigures914.Whenthe pressure theredecreasesbyafactorof17duringtheneon phase withL>),theshell-burninglayerexpandsand inner shellasseenfromFigures3and7. smaller loopinthe(p,T)planethandoesneonflash. flash. Suchapressuredecreaseintheshellcausesanalmost shell reachedthewithM=0.7. sensitively onzoning.Sinceamorecarefultreatmentseemsto burning shellcontracts(i.e.,L<),thecentralcorealso adiabatic expansionofthecentralcoreanddecreasein contracts. 800 gravothermal specificheat,c,whichisacombinationofthe convective shell,aradiative carbon-burningshelladvances repeatedly fivetimes(Fig. 4 andTable1).Afterthefifth to thosethatoccurina15Mstarwith=1.7 r0 from M=1.26to1.339 M(Table1). described aboveisnowwellunderstoodinageneralized temperature attheflashingshellaredescribedintermsof fashion (SugimotoandMiyaji1981):therisedeclineof r0 usual thermodynamicspecificheatandatermduetohydro- (Sparks andEndal1980)ina10Mstarwith= 0+0Ne Ne c negative becauseelectrondegeneracybecomesasweak ture peakofthepresentflashes,cchangesfrompositiveto static readjustmentuponachangeofentropy.Atthetempera- v0+0 c nv the nuclearfuel(neonoroxygen)butdependsonlyon^ and that thepeakvalueoftemperaturedoesnotdepend on spherical ;thesetwoeffectsmakeexpansioneasier.Thisimplies \¡/ =1.1andthegeometryatburningshellgetscloser to 1.5 Mq(Woosley,Weaver,andTaam1980). r0 the geometryatburningshell.(SeeSugimotoandMiyaji nv g 0 r0 0 0He 1981 formoredetail.) g VI. FORMATIONOFASTRONGLYDEGENERATEO+NeMg After thefirstneonflashceases,heatistransportedinward The shellatTliesthecarbon burningshelluntilT During thedevelopingphaseoftheseshellflashes(i.e., The abovefeaturesofneon-oxygenshellflashesaresimilar However, thebehavioroftheseflashesmaydepend For case2.4,aconvectiveshellofcarbonburningappears It isnoteworthythatthebehaviorofshellflashes max max © American Astronomical Society • Provided by the NASA Astrophysics Data System a) CoolingoftheO+NeMgCore c) OxygenShellFlash CORE FORCASE2.4 NOMOTO 9 Afterward, bothTanddecreaseaspincreases(Fig.10); electron degeneracygetsstrongerandasthecore location ofTthenshiftstotheO+NeMglayer. between cases2.6and2.4clearintermsofneonstarmass. core isformed,i.e.,logp=8.82,T8.59,i//92,and contracts further.Ati*=0,astronglydegenerateO+NeMg Subsequent contractionofthecoreraisesTupto Thus thecoremassof1.34Mcanberegardedasaneon by afactorof1.2.Thusneonignitionneverdoesoccurfor the ignitiontemperatureofneonburning(definedbye=) The peakvalueofTattainedatthisstageissmallerthan the temperaturedistributionsagainstpandMati*=140yr. l°g fnax=8.95atM1.29(logp6.76). separates thestarintoacoreandanextendedheliumenvelope. at theheliumburningshellwithM=1.34clearly is shownbythesolidline.Averysteepdensitygradient In Figure15,thedensitydistributionofcase2.4ati*=0yr of 1.37Mqforneonignition.Wecanmakethedifference because thecarbon-burningshelladvancesratherrapidlyin during thephaseofincreasingM(i.e.,L);thisis equivalent toaneonstarmass. case 2.6inFigure14indicatesthatM=1.45is to aredgiantsizeisseen. curve) tof*=0yr(solidfor case 2.4.Expansionoftheheliumlayer mass ataratedeterminedbybalanceofLäwhile star massforcase2.4.Similarlythedensitydistribution case 2.4,whichisdistinctfrom2.6. L ismuchsmallerinthehelium-burningshellbecauseof 1.04 x10KattheshellwithM=1.18(i*140yr). cmax max c max 0 nv max r r0 He0 cHcc+c Hq0 c+cv v r0 As discussedin§III,thereexistsacriticalneonstarmass In Figure12(dashedcurve)and13(solidareshown Fig. 15.—Changesinthedensitydistribution fromt*=256yr(dashed It shouldbenotedthattheincreaseinMisnegligible He b) CriticalMassofHeliumCoreforNeonIgnition Vol. 277 198 4ApJ. . .211 . .7 91N (0) (0) (0) -3 -6 which correspondsapproximatelytothestellarmassof10M, but notforcase2.4.ThecoremassofM=2.5, neon ignition,andthisiswhyignitedforcase2.6 constant. No. 2,1984 is aboutthecriticalmassforneonignition. lower temperature.ThereforeMremainspractically evolution fromamain-sequencestartored-giant(Sugimoto with thecorecontractionisessentiallysameasof zone intotheheliumlayer,expansionwascalculatedup hydrogen-rich envelopehasadeepsurfaceconvectionzone, 4.4 Rq(i*=256yr)to38R(t*59.6yr).Sincethe starts toexpandgreatly.AsseenfromFigure6,theradius such anexpansionoftheheliumlayerleadstopenetration at theouteredgeofheliumlayer,rincreasesfrom double shellburning.Theearlyphaseofdredge-upduring vestigated byJossetal(1973). expansion dependsonthemassofheliumlayerasin- and Nomoto1980,p.166).Theextentoftheheliumenvelope to i*=0yrwhenreaches190R,almostred-giantsize. of theconvectionzoneintoheliumlayerati*=59.6yr. these casesissimilartothatwhichoccursforC+Ocores sistent withthecaseofM=2.2.Thedredge-upin the growthofO+NeMgcorewascalculatedfora9M afterward, thestarentersphaseofhydrogen-helium is dredgedupandthehydrogenshellburningbecomesactive; convection zonecontinuesuntilalmostalltheheliumlayer of Mh=2.2M(Nomotoetal1982;Nomoto1983). calculated yetforcase2.4butshouldbesimilartothe The changeinthedensitydistributionfromf*=256yr clearly showsatransitionoftheheliumlayertored-giant- in 6-8Mstars(e.g.,Paczynski1970;Sugimoto1971; star byWeaver,Axelrod,andWoosley(1980),whichiscon- It hasbeenfoundthatthepenetrationofsurface the expansion. L decreasesbyafactorof3(Fig.6)sinceenergyisusedfor the increaseinpbyafactorof5.5.Duringexpansion, cm. Thisexpansionisdrivenbythecorecontractionwith by afactorof10for1.40M