SantiCassisi
INAF-OsservatorioAstronomicodiCollurania,viaM.Maggini,64100Teramo,Italy;
cassisi@te.astro.itMax-Planck-Institutf¨urAstrophysik,Karl-Schwarzschild-Strasse1,85741Garching,
Germany
MaurizioSalaris
AstrophysicsResearchInstitute,LiverpoolJohnMooresUniversity,TwelveQuaysHouse,
EgertonWharf,BirkenheadCH411LD,UnitedKingdom;ms@astro.livjm.ac.ukMax-Planck-Institutf¨urAstrophysik,Karl-Schwarzschild-Strasse1,85741Garching,
Germany
AlanW.Irwin
DepartmentofPhysicsandAstronomy,UniversityofVictoria,P.O.Box3055,Victoria,
BritishColumbia,Canada,V8W3P6;irwin@uvastro.phys.uvic.ca
ABSTRACT
RecentprecisedeterminationsoftheprimordialHe-abundance(Yp)fromcos-micmicrowavebackground(CMB)analysesandcosmologicalnucleosynthesiscomputations,provideYp=0.248±0.001.Ontheotherhand,recentworksontheinitialHe-abundanceofGalacticglobularcluster(GGC)stars,makinguseoftheRparameterasHe-indicator,haveconsistentlyobtainedYGGC∼0.20.
InlightofthisseriousdiscrepancythatcastsdoubtontheadequacyoflowmassHe-burningstellarmodels,wehaverederivedtheinitialHe-abundanceforstarsintwolargesamplesofGGCs,byemployingtheoreticalmodelscomputedusingnewandmoreaccuratedeterminationsoftheEquationofStateforthestellarmatter,andoftheuncertain12C(α,γ)16Oreactionrate.OurmodelsincludesemiconvectionduringthecentralconvectiveHe-burningphase,whilethebreathingpulsesaresuppressed,inagreementwiththeobservationalconstraintscomingfromthemeasurementsoftheR2parameterinasampleofclusters.
arXiv:astro-ph/0301378v1 18 Jan 2003–2–
BytakingintoaccounttheobservationalerrorsontheindividualR-parametervalues,aswellasuncertaintiesintheGGC[Fe/H]scale,treatmentofconvec-tionand12C(α,γ)16Oreactionrate,wehaveobtained,respectively,ameanYGGC=0.243±0.006andYGGC=0.244±0.006forthetwostudiedGGCsamples.TheseestimatesarenowfullyconsistentwithYpobtainedfromCMBstudies.Moreover,thetrendoftheindividualHe-abundanceswithrespectto[Fe/H]isconsistentwithnoappreciableHe-enrichmentalongtheGGCmetallicityrange.Subjectheadings:cosmicmicrowavebackground–globularclusters:general–stars:abundances–stars:evolution–stars:horizontalbranch
1.Introduction
GalacticGlobularCluster(GGC)starsaretheoldestobjectsintheGalaxy,andtheirinitialHeabundance(YGGC,whereYdenotesthemassfractionofHe)issupposedtobeapproximatelyequaltotheprimordialHeabundance(Yp)producedduringtheBigBangNucleosynthesis(BBN).
ThevalueofYpderivedfromspectroscopyoflow-metallicity,extragalacticHIIregions,appearstobestillsubjecttosystematicuncertainties(see,e.g.,thediscussioninBonoeta.2002andreferencestherein);asanexample,Olive,Steigman&Skillman(1997)havedeterminedYp=0.234±0.002,whileIzotov&Thuan(1998)obtainedYp=0.244±0.002,con-sistentwithearlierfindingsbyKunth&Sargent(1983).
Ontheotherhand,recentdeterminationsofthecosmologicalbaryonicmatterden-sity(Ωb)fromthecosmicmicrowavebackground(CMB)powerspectrumobtainedbytheBOOMERANG,DASIandMAXIMAexperimentsprovideconsistently(e.g.Prykeetal.2002,¨Odmanetal.2002;Sieversetal.2002)avalueΩbh2=0.022±0.003(histheHubbleconstantinunitsof100KmMpc−1s−1).ThisbaryondensitycoupledwithstandardBBNcalculations(Burles,Nollett&Turner2001)providesYp=0.248±0.001;Suchanindependentdetermina-tionofYpisclosetothespectroscopicdeterminationbyIzotov&Thuan(1998).
AsforthevalueofYGGC,empiricalestimatesarenecessarilyindirect,sinceHe-linesarenotdetectableinGGCstarspectra,apartfromthecaseofhotHorizontalBranch(HB)objects,whoseatmospheresarehoweveraffectedbygravitationalsettlingandradiativelev-itation,whichstronglyaltertheinitialchemicalstratification(see,e.g.,Michaud,Vauclair&Vauclair1983;Moehleretal.1999).YGGCestimatesmakeuseofresultsfromstellarevolution,takingadvantageofthefactthattheevolutionoflowmassPopulationIIstarsisaffectedbytheirinitialHe-content.Moreindetail,theso-calledR-parameter(Iben1968),
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definedasthenumberratioofHBstarstoRedGiant(RGB)starsbrighterthantheHBlevel(R=NHB/NRGB),canbeemployedinordertodetermineYGGCandthereforeYp.ThebasicideabehindtheuseofthisparameterasHeindicatoristhat,atagivenmetallicity,ahigherinitialHe-contentimpliesabrighterHBand,inturn,alowervalueofNRGB(NHBisonlyslightlyaffected),withtheconsequentincreaseofR.Otherparametersderivedfromstellarevolutioncanalsobeemployed(see,e.g.,thediscussioninSandquist2000andZoccalietal.2000),buttheyarebettersuitedtodeterminerelativeHe-abundancesthanabsoluteones.
FollowingearlieranalysesbyBuzzonietal.(1983)andCaputo,MartinezRoger&Paez(1987),Sandquist(2000–S00)andZoccalietal.(2000-Z00)haverecentlyestimatedYGGCbymeasuringthevalueofRinlargesamplesofGGCs(43objectstakenfromvarioussources,incaseoftheS00paper;26objectswithhomogeneousHSTphotometryfortheZ00paper),andcomparedtheobservationalvalueswithresultsfromstellarevolutionmodels.InbothcasesavalueYGGC∼0.20wasfound,inseveredisagreementwiththeCMBresultandspectroscopicdeterminationsforHIIregions.ThislargediscrepancybetweenCMBandR-parameterresultscastsdoubtontheabilityofstellarmodelstoaccuratelypredicttheevolutionarytimesofthesecrucialphasesofstellarevolution.ReasonsforthislowHe-contentinferredfromtheRparameterhavebeenascribedtotheuncertaintyofthe12
C(α,γ)16Oreactionrate,whichisrelevantduringthelatestagesofcentralHe-burning,butmayinprinciplealsobedue,forexample,toanimpropertreatmentofthemixinginthecentralconvectiveregionofHBstars,whichaffecttheHBevolutionarytimescale,andhenceNHB(see,e.g.,Z00).
InthispaperwepresentanewdeterminationofYGGCfromtheR-parameter,employinginthemodelcomputationthemostrecentdeterminationofthe12C(α,γ)16Oreactionrate(Kunzetal.2002)togetherwithanimprovedequationofstate(EOS)forthestellarmatter(Irwinetal.2002).WeshowthatwiththesetwoimprovedphysicalinputsthediscrepancybetweenCMBandR-parameterresultsalmostcompletelydisappears.ThisgivesstrongsupporttotheaccuracyofpresentHBstellarmodelsandtheadequacyofourconvectivecoremixingtreatmentduringtheHBphase.In§2webrieflydiscusstheobservationaldataandthetheoreticalstellarmodels,whilein§3and§4theresultsarepresentedanddiscussed.Conclusionsfollowin§5.
2.Observationaldataandtheoreticalmodels
WehavedeterminedYGGCmakinguseofourtheoreticalmodelsandtheobservationaldatabasespresentedbyZ00andS00.Z00provideempiricalRvaluesfor26GGCsspanning
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alltherelevantmetallicityrange.ThenumberofRGBstarsiscomputedstartingfromtheleveloftheobservedVmagnitudeoftheZeroAgeHB(ZAHB–lowerenvelopeoftheobservedHBstardistribution).ThismeansthattocomparethemodelswithZ00dataonehasfirsttotransformtheoreticalbolometricluminositiestoVmagnitudes,andthendeterminethetheoreticalRvaluesfollowingthedefinitionofRusedbyZ00.
S00providesRvaluesfor43GGCs,butemploysaslightlydifferentdefinitionofR,thesameasBuzzonietal.(1983):thereferencelevelfortheRGBcountsisthebolometricluminositycorrespondingtotheaverageV-magnitudeofHBstars.InordertodeterminetheleveloftheRGBcorrespondingtothisbolometricluminosityS00haveappliedtotheobservationaldataarelationshipforthedifferenceinbolometriccorrectionsbetweenHBandRGBstars(whichisdifferentfromtheoneweappliedtoourmodelswhenusingtheZ00definitionfortheR-parameter).
WehavecomparedourtheoreticalresultswiththesetwodatabasestakingintoaccountthesetwodifferentdefinitionsofR.ThederivedHe-abundanceswillthusreflectthetheo-reticaluncertaintiesrelatedtothethedifferentbolometriccorrectionsemployedinthetwomethods,andthedifferentobservationalsamples.
InordertotakeintoaccountcurrentempiricaluncertaintiesontheGGCmetallicityscalewehaveusedfortheindividualclusters[Fe/H]valuesgivenbyRutledge,Hesser&Stetson.(1997)onboththeCarretta&Gratton(1997–CG97)andZinn&West(1984–ZW84)scales(theinternalaccuracyofthese[Fe/H]valuesisoftheorderof0.10dex).ForclustersnotlistedbyRutledgeetal.(1997)wehaveusedtheoriginalZW84valuestransformedthentotheCG97scaleusingtheconversionformulagivenbyCG97.
WehavedeterminedtheexistenceofpossibleHe-abundancevs[Fe/H]correlationsbycomputingthecorrespondingcorrelationcoefficientandevaluatingitssignificance.Incaseofnocorrelation,wehavedeterminedYGGCbymeansofaweightedmeanoftheindividualvalues,withweightsinverselyproportionaltothesquareoftheindividualerrors.WenoticethatinZ00thevalueofYGGCwasdeterminedbysimplyconsideringtheconstantvalueoftheHe-abundancebestfittingtheindividualdatapoints,withouttakingintoaccountindividualerrors.TheexistenceofasignificantspreadintheindividualclusterHe-abundancehasbeenstudiedbymeansofthestatisticalF-test.
WecomputednewtheoreticalmodelsandisochronesforY=0.23,0.245,0.26,using–likeinZ00analysis–thesamecode,inputphysicsandbolometriccorrectionsasinCassisi&Salaris(1997),withthefollowingmodifications:
•Wehaveupdatedtheenergylossratesfromplasma-neutrinoprocessesusingthemostrecentandaccurateresultsprovidedbyHaft,Raffelt&Weiss(1994).
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•WehaveupdatedthenuclearreactionratesusingtheNACREdatabase(Anguloetal.1999),withtheexceptionofthe12C(α,γ)16Oreaction.ForthisreactionweemploythemoreaccuraterecentdeterminationbyKunzetal.(2002),basedonγangulardistributionmeasurementsof12C(α,γ)16OandaconsistentR-matrixanalysisoftheprocess.Theclaimedrelativeuncertaintyofthisnewrateishalfoftheuncertaintyquotedinpreviousdeterminations.
•AnimprovedEOSwhosecodeismadepubliclyavailableatftp://astroftp.phys.uvic.ca/pub/irwin/eos/code/eosfortran.tar.gzundertheGNUGeneralPublicLicense(GPL),hasbeenused.AfulldescriptionofthisEOSisinpreparation(Irwinetal.2002)sowewillonlysummarizeitsprincipalcharacteristicshere.TheEOSiscalculatedusinganequilibrium-constantapproachtominimizetheHelmholtzfree-energy.Forrealisticabundancemixtures,thisapproachgreatlyreducesthenum-beroflinearequationsthatmustbesolvedperiterationsothatthesolutioncanberapidlyobtained.ThisspeedmakesitpracticaltocalltheEOSdirectlyfromthestellar-interiorcodewithoutintroducingtheerrorsassociatedwithinterpolatingEOStables(Cassisi&Irwin2002,seealsoDorman,Irwin,&Pedersen1991).Theequilibrium-constantapproachgivesnumericalsolutionsofhighqualitywiththermo-dynamicconsistencywhichistypicallybetterthan1partin1011.The“EOS1”modeofthefree-energymodelthatisusedforthepresentcalculationsincludesthefollowing:arbitrarilyrelativisticanddegeneratefreeelectrons(Eggleton,Faulkner,&Flannery1973);apressure-ionizationoccupationprobabilitysimilartothatofMihalas,Dap-pen,&Hummer(1988);aPlanck-Larkinoccupationprobability(Rogers1986);theexchangeeffectforarbitrarilyrelativisticanddegenerateelectrons(Kovetz,Lamb,&VanHorn1972);andtheCoulombeffect.TheCoulombeffectistreatedwiththeDebye-H¨uckelapproximationintheweakcouplinglimitandanapproximation(Polsetal.1995)ofthemulticomponentcombinationoftheone-componentplasmaresult(DeWitt,Slattery,&Chabrier1996)inthestrong-couplinglimit.Asplinefitisusedtointerpolatebetweentheweakandstrongcouplinglimits.Thesizeoftheintermediatecouplingregionandthesizeoftheinteractionradiithatcharacterizethepressure-ionizationoccupationprobabilityareadjustedtofittheOPALEOStablesdistributedatftp://www-phys.llnl.gov/pub/opal/eos/.
•Wehaveexplicitlytakenintoaccounttheα-enhancedchemicalcompositiontypicalofPopulationIIstars,usingthesameinitialmetalmixtureemployedbySalaris&Weiss(1998),andtheirsameopacitytables;theheavyelementdistributionhasanaverageα-enhancementequalto0.4dex.ThisispotentiallyimportantfortheuppermetallicityendoftheGGCs,sinceinthatregimethewellknownequivalencebetweenlow-massscaled-solarandα-enhancedmodelswiththesametotalmetallicity(Salaris,
–6–
Chieffi&Straniero1993)isnolongervalid(Salaris&Weiss1998,VandenBergetal.2000).Wejustnotice,inpassing,thatanaverageenhancementof0.4dexisinfullagreementwithabundancedatafromHalofieldstars(e.g.,thediscussioninSalaris&Weiss1998),whiletheGGCsdatacompiledbyCarney(1996)seemtopointouttoanα-enhancementslightlylower,ofabout0.3dex.Suchasmalldifference–ifreal–doesnotintroduceanyseriousbiasinourfinalYGGCestimates,becausesuchasmalldifferenceintheα-enhancementbetweenclustersandmodelscanbefullycompensated(alsoathighmetallicities)byasmallrescalingoftheZ-[Fe/H]relationshipofthetheoreticalmodels,whichintroducesasystematiceffectoflessthan0.001ontheindividualclusterHe-abundanceestimates.
•WehaveaccountedforthedifferentevolutionarytimescharacterizingtheredandbluepartsoftheHB.Ordinarily,thetheoreticalvaluesofRarecomputed–asinZ00–byconsideringtheHBevolutionarytimeofastarpopulatingthemiddleoftheRRLyraeinstabilitystrip(log(Teff)=3.85).ThisisstrictlyadequateonlyforthoseclusterswithanHBpopulatedattheRRLyraeinstabilitystripandredward(increasingtotalstellarmass),sincetheHBevolutionarytimescalesarebasicallyunchangedwhenmovingfromtheinstabilitystriptowardsthered(seethediscussioninZ00).However,starspopulatingtheHBbluewardoftheinstabilitystripdoshowdifferentevolutionarytimes,whichincreasefordecreasingtotalstellarmass.AtthebluestendofatypicalblueHBtheincreaseoftheHBevolutionarytimewithrespecttotheRRLyraestripcounterpartcanamounttoabout20%(Z00).Wewilldiscussin§3.3theimplicationsforthederivedHe-abundanceinGGCswithablueHB.
3.
ThevalueofYGGCfromtheR-parameter
InthissectionwepresentseparatelyourdeterminationofYGGCusingtheZ00andS00samples.
3.1.TheZ00sample
Fig.1displaystherunoftheempiricaldatabyZ00togetherwiththeoreticalpredictionsforagesof11and13GyrandY=0.245(solidlines),asafunctionof[Fe/H].AtfixedageandYthetheoreticalvalueofRisveryslowlydecreasingupto[Fe/H]∼−1.15.Between[Fe/H]∼−1.15and[Fe/H]∼−0.85Rincreasessteeply;thisincreaseisduetothefactthattheRGBbump,previouslylocatedatbrightnesseslargerthantheZAHB,movesbelowthe
–7–
ZAHBlevelwithincreasingmetallicity,thuscausinganabruptdecreaseofthenumberofRGBstarsbrighterthantheZAHB(see,e.g.,thediscussioninZ00).Athigher[Fe/H]valuesRisagainonlyverymildlydecreasingwithincreasing[Fe/H].ItisalsointerestingtonoticehowthedependenceofRonageisrestrictedtotheintervalrangingfrom[Fe/H]∼−1.15to[Fe/H]∼−0.85,whichisexactlythemetallicityrangewheretheRGBbumpcrossestheZAHBlevel.ThisiseasilyexplainedbythefactthattheRGBbumpluminositydoesdependonthestellarage(e.g.,Cassisi&Salaris1997)whiletheZAHBlevelisbasicallyunaffectedforagestypicalofGGCs;ingeneralhigheragesshifttheRGBbumplocationtowardslowerluminosities.
InthesameFig.1wealsodisplaythetheoreticalRvaluesforanageof13GyrandYGGC=0.23,toshowthesensitivityofRtothemodelinitialHe-content.TheaveragevalueofthederivativeδR/δYis∼10.
WehaveestimatedYGGCandtheassociated1σdispersionoftheindividualabundances(thelatterwillbethoroughlydiscussedattheendofthissection),byfirstassuminganaverageageof13Gyrfortheclusters(see,e.g.,theanalysesbyVandenBerg2000;Salaris&Weiss1998,2002);theerrorontheindividualclusterYvalueshasbeenobtainedfromthequotederrorsonthevalueofR.Whenconsideringall26clusterstogetherwiththeCG97[Fe/H]scale,weobtainedaweightedmeanYGGC=0.240±0.003(1σerror).However,wefoundaclearcorrelationbetweenYGGCand[Fe/H]inthesensethatthemeanYGGCobtainedforclusterswith[Fe/H]between−1.15and−0.85(themetallicityrangeinfluencedbytheassumedclusterage)isYGGC=0.231±0.005,whilefor[Fe/H]<−1.15and[Fe/H]>−0.85wefound,respectively,YGGC=0.247±0.005andYGGC=0.244±0.003(nocorrelationoftheindividualYestimateswith[Fe/H]hasbeenfoundintheselattertwometallicityranges).ItisevidentthatthemeanvaluesofYdeterminedfor[Fe/H]<−1.15and[Fe/H]>−0.85areingoodagreementwhileasubstantiallowervalueisobtainedfortheclusterswhoseRparameterisaffectedbytheprecisevalueoftheage.WehavethereforerederivedtheHe-contentwithadifferentassumptionabouttheclusterages.Rosenbergetal.(1999)andSalaris&Weiss(2002)haveshownhowclusterswith[Fe/H]largerthan∼−1.2(ontheCG97metallicityscale)displayalargeagespreadandareonaverageyoungerby≈2Gyrthanthemoremetalpoorclusters.WehavethereforerecomputedthevaluesofYassuminganageof13Gyrfortheclusterswith[Fe/H]<−1.2and11Gyrformoremetalrichones.Asexpected,themeanYvaluesfor[Fe/H]<−1.15and[Fe/H]>−0.85areunchanged,butthistime,inthe[Fe/H]rangebetween−1.15and−0.85,weobtainameanYGGC=0.239±0.004which,withinthe1σerrorbar,isinbetteragreementwiththeresultsathigherandlowermetallicities.ItisthereforeimportanttonoticethatthepreciseindividualclusteragesdomatterwhendetermininganaccurateYGGCvalueforclustersinthis[Fe/H]range.
–8–
WehaverepeatedthepreviousanalysisbyemployingtheZW84[Fe/H]scale.Adoptinganageof13Gyrforallclusters,wederiveameanYGGC=0.242±0.003forthewholeclustersample,andwedonotfindanycorrelationbetweenYand[Fe/H].However,Salaris&Weiss(2002)haveshownthat,whenconsideringtheZW84metallicityscale,clusterswith[Fe/H]>−1.6showalargeagespreadandareonaverageyoungerthanthemoremetalpoorones(seealsoVandenBerg2000).Wethereforerepeatedthepreviouscalculationconsideringanageof13Gyrwhen[Fe/H]<−1.6and11Gyrathigher[Fe/H];weobtainameanYGGC=0.243±0.003,consistentwiththevaluedeterminedforaconstantageof13Gyr.Thisresultcomesfromthefactthat,whenusingtheZW84metallicities,therearenoclusterspopulatingthe[Fe/H]rangewhichisstronglyaffectedbyage.
AnotherimportantquestiontobeaddressedisthesignificanceofthedispersionoftheindividualclustervaluesaroundthemeanYGGC.Inparticular,itisimportanttoknowiftheobserved1σdispersion,oftheorderof0.02,isentirelyduetotheerrorontheindividualclusterestimates.ToaddressthispointwehaveappliedthestatisticalF-test(see,e.g.,anapplicationtothecaseofGGCagesbyChaboyeretal.1996;Salaris&Weiss1997,2002)tooursampleofHedeterminations.IncaseoftheCG97[Fe/H]scalewehaverestrictedtheanalysistotheclusterswithinthemetallicityrangeunaffectedbyage,sothatanagespreadwillnotaffecttheobservedHe-abundancedispersion.ForeachindividualclusterwehavecalculatedasetofsyntheticHe-abundancesbyrandomlygenerating–usingaMonteCarloprocedure–10000abundancevalues,accordingtoaGaussiandistributionwithmeanvalueequaltotheobservedmeanYGGC,andσequaltotheindividualHe-abundanceerror.Thisisrepeatedforallclustersintheselectedsampleandthe10000valuesforeachindividualclustersarejoinedtoproducean“expected”YGGCdistributionfortheentireclustersample,ontheassumptionthatthedetectedHe-abundancespreadisnotintrinsic,butduejusttotheindividualerrorbars.TheF-testhasbeenthenappliedinordertodetermineifthis“expected”distributionhasthesamevarianceastheobservedone.WestatethataYGGCrangedoesexistiftheprobabilitythatthetwodistributionshavedifferentvarianceislargerthan95%.Incasethisconditionisverified,thesizeofthetrueYGGCrange(σY)canbe
22
estimatedaccordingtoσY=(σobs−σexp)0.5,whereσobsandσexpare,respectively,the1σdispersionoftheactualdataandofthe“expected”distribution.
TheresultofthistestappliedtotheZ00samplewithourtwochoicesofthe[Fe/H]scaleindicatesthattheobserveddispersionaroundthemeanYGGCisentirelyduetotheformalerrors(theprobabilitythattheobservedandthesyntheticdistributionshavedifferentvari-anceisbelow70%inbothcases)ontheindividualdeterminations;thereforenostatisticallysignificantspreadintheindividualHe-abundancesisfound.
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3.2.
TheS00sample
Fig.2displaystherunoftheempiricaldatabyS00togetherwiththeoreticalpredictionsforagesof11and13GyrandY=0.245(solidlines),asafunctionof[Fe/H].AtfixedageandYthetheoreticalvalueofRisveryslowlydecreasingupto[Fe/H]∼−0.85.AthighermetallicitiesthevalueofRincreases,dueagaintothefactthatthebolometricluminosityoftheRGBbumpcrossesthereferenceHBbolometricluminosity.TheshifttohighermetallicitiesofthiscrossingregionwithrespecttotheRdefinitionpreviouslyused,arisesfromthefactthatthebolometricluminosityoftheRGBreferencelevelcorrespondstoVmagnitudesfainterthantheVmagnitudeleveloftheHB.ThisimpliesthatRGBbumpstarsareincludedintothedeterminationofRuptohighermetallicitiesthanthecaseofZ00definition.Thishighmetallicityregionisalsotheonlyoneaffectedbyage(seediscussionin§3.1);thereforetheprecisechoiceoftheGGCagesdoesnotaffectatalltheresultswhenusingtheS00definitionoftheR-parameter,sinceonlyveryfewclustersshowthesehighvaluesof[Fe/H](seeFig.2),andonlyontheZW84scale.
Byassumingt=13GyrforallGGCwefindagainaYGGCdistributionuncorrelatedwith[Fe/H].AmeanvalueYGGC=0.246±0.005isobtainedwhenconsideringtheCG97[Fe/H]scale,whileYGGC=0.241±0.004isderivedwhentheZW84metallicitiesareemployed.AsforthedispersionoftheYGGCvaluesaroundthesemeans,weobtaininbothcasesσY=0.04;wehaveappliedtheF-testalsointhiscaseandderivedthatthedispersioncan’tbecompletelyexplainedbytheformalerrorsontheindividualdeterminations(theprobabilitythatthevarianceoftheHe-abundancedistributionintheobservedsampleandinthesyntheticonearedifferentislargerthan99%),andhasanintrinsiccomponentequaltoσY=0.03(analogousconclusionswerereachedbyS00).
3.3.ClusterswithablueHB
AlltheYGGCvaluesgivenbeforehavebeenobtainedbyconsideringtheevolutionarytimeofHBstarsintheinstabilitystripwhencomputingthetheoreticalvalueofR;thisisalsowhathasbeendonebyZ00andS00.
WhilethisassumptioniswellfoundedincaseofHBspopulatedatthestripandredward,itislessadequateincaseofveryblueHBs(seethediscussioninCaputoetal.1987andZ00);thisisparticularlytruewhenthelocationoftheaveragemasspopulatingtheobservedHBcorrespondstoVabout0.5-1.0magfainterthantheinstabilitystriplevel.Thisisduetothefact,asdiscussedinprevioussection,thattheHBevolutionarytimesincreasewhenonegreatlyreducesthetotalstellarmasswithrespecttothevaluesattainedattheinstability
–10–
strip.Tocorrectforthispossiblesystematicuncertaintycausedbyourassumptionwehaveappliedthefollowingprocedure.
ForboththeZ00andS00sampleswehaveidentifiedthoseclusterswhoseHBismainlypopulatedatthebluesideoftheinstabilitystrip;amongtheseclusters,throughcomparisonswithourHBmodels,wehaveidentifiedtheobjectswhoseaverageHBmassislocatedmorethan0.7magbelowtheRRLyraelevel.Fortheseclusters,wehaverecomputedthetheoreticalR-valuesbytakingasrepresentativeoftheHBevolutionarylifetimethecorrespondingvaluefortheaveragemass.Thereareonly6clustersintheZ00sample,and8clustersintheS00samplethatsatisfythiscondition.
WhenapplyingthesecorrectedevolutionarytimestotheblueHBclusterswefindthattheYGGCvaluesobtainedinthepreviousanalysisarereducedbyonly0.001-0.002.ThesizeandsignificanceoftheYGGCspread,andthebehaviourwiththerespectto[Fe/H],areunchangedwithrespecttothepreviousresults.InTable1wesummarizetheYGGCresults,withandwithoutthecorrectionfortheblueHBclusters.IncaseoftheZ00sampleandtheCG97[Fe/H]scalewedisplaytheresultsforthemetallicityrangethatisinsensitivetothechoiceoftheclusterages.Fig.3displaysthedistributionoftheindividualGGCHe-abundance,forbothsamplesandbothchoicesofthe[Fe/H]scale,takingintoaccountthecorrectionfortheblueHBGGCs.TheZ00sampleclearlyhasasignificantlynarrowerabundancerangethantheS00sample.
4.Discussion
IntheprevioussectionwefoundthattheRparameterprovidesvaluesofYGGCbetween∼0.240and∼0.245,independentof[Fe/H];theexactvaluesaresummarizedinTable1,togetherwiththesizeoftheintrinsicspreadσYoftheindividualclusterHe-abundances.ThemeanvaluesofYGGCdeducedfromtheZ00andS00sampleareinexcellentagree-mentwithintheassociated1σerror,inspiteofthe-inprinciple-differentbolometriccorrectionsappliedtothedataanalysisandthedifferentphotometricsamplesemployed.ItishoweverimportanttomentionthefactthattheZ00datadonotprovideanyindicationofastatisticallysignificantspreadofYGGC,whiletheoppositeistruefortheS00data.OnepossiblereasonforthisoccurrencemaybetheinhomogeneityoftheS00sample,whichismadeofphotometriestakenwithverydifferentinstrumentsanddetectors(photographic,photoelectricandCCDphotometries),reducedwithdifferentproceduresinthecourseofthelast25years,andwithpossiblydifferentmethodstocorrectforincompleteness,asopposedtothehomogeneouslyobserved,reducedandanalyzedHSTsamplebyZ00.
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AnotherpossibilitytoexplainthisHe-abundancespreadisrelatedtotheexistenceofpopulationgradientswithintheobservedclusters,coupledwiththefactthattheHSTdataemployedbyZ00mainlysampleregionsoftheclusters’cores,whereasthegroundbasedphotometriesadoptedbyS00samplemoreexternalregionslocatedatvariousdistancesfromthecores.
Wehavealsoperformedanothertest,bycomparingtheindividualHe-abundancefor13clustersincommonbetweentheZ00andS00samples.InFig.4wedisplaytheabun-dancesforthese13clustersderivedfromtheZ00andS00data(wehavechosentouseinthisfiguretheZW84[Fe/H]scale,butthischoicedoesnotaffecttheresultofthecompar-ison),consideringthecorrectionsfortheblueHBclusters.TheZ00dataprovideameanYGGC=0.237±0.004,inverygoodagreementwiththeresultfromthewholesampledisplayedinTable1(YGGC=0.240±0.003);thedispersionaroundthemeanisagaindue(asforthewholesample)onlytotheerrorontheindividualdeterminations.IncaseoftheS00dataforthesame13clusters,themeanYGGC=0.224±0.006issmallerthanfortheZ00data,andalsosignificantlysmallerthanthemeanvalueforthewholesample(YGGC=0.240±0.004);thedispersionaroundthemeanYGGCislargerthanintheZ00case.Therefore,whereasasomewhatrandomselectedsizableclustersubsample(the13commonclustersspanalltherelevant[Fe/H]rangeaswellasshowbothredandblueHBs)showthesamepropertiesofthewholesampleincaseoftheZ00data,theoppositeistruefortheS00data.ThismaylendsomesupporttotheideathatthesignificantdispersionofYGGCforthewholeS00sampleisduetosomeinhomogeneityintrinsictothedatausedfordeterminingtheobservedRvalues.Ontheotherhand,whenthe4mostmetalrichclusters([Fe/H]>−1)areexcludedfromthecomparisonshowninFig.4,thedispersionoftheS00databecomescomparablewiththeZ00one,whilethemeanHe-abundanceissimilartothevalueforthewholeS00sample.Thisseemstopointtosomemetallicity-relatedeffect,whichhoweverdoesnotexplainthedispersionforthewholeS00sample.Infact,ifweapplytheF-testdiscussedinSections3.1and3.2totheS00samplewithouttheclusterswith[Fe/H]>−1,westillobtainastatisticallysignificantdispersionoftheindividualHe-abundances.
InspiteofthisdifferenceregardingthespreadintheclusterHe-abundancefortheZ00andS00samples,ourresultsclearlyindicateameanvalueofYGGCwhichisnotinsignificantcontradictionwiththeCMBconstraint.ThisisverydifferentfromtheconclusionsreachedbyZ00andS00analyses,whichderivedanunrealisticallylowHe-abundance,namelyYGGC∼0.20,completelyinconsistentwiththeCMBconstraint.WhenweredetermineYGGCbyusingthesameobservationaldataandtheoreticalscenarioadoptedbyZ00,butusingthesameweightedaveragemethodemployedinouranalysisandconsideringthemetallicityrangeunaffectedbytheselectedclusterage,weobtainYGGC=0.21,stilllargelyincompatible
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withtheCMBconstraint.
Thenew12C(α,γ)16OreactionrateandthenewEOSarethetwophysicalingredientsthathavestronglymodifiedthetheoreticalRvalueswithrespecttotheresultsbyZ00,whoseemployedstellarmodelswehaveupdatedforthiswork.Inparticular,therecentestimateofthe12C(α,γ)16Oreactionrate(Kunzetal.2002)hasreducedtheHBevolutionarytimes(atafixedcoremassandenvelopecomposition)by∼7−8%,andthenewEOShasfurtherreducedtheHBevolutionarytimesby∼10%.Ontheotherhand,thenewEOSalsoslightlyreducesthevalueoftheHe-coremassattheHe-flashforagivenage,whichhastheeffectofincreasingby∼2%theHBevolutionarytime;thereisalsoafurtherincreaseby∼4%forthevalueofNRGBbecausealargerportionoftheRGBisconsideredintheevaluationoftheR-parameter.TheseeffectscauseatotalreductionofRby∼20%which,foratypicalaverageobservedvalueofR(i.e.,withtheZ00definitionofR)equalto∼1.4−1.5,correspondstoanincreaseoftheestimatedYGGCbyabout0.03.
The12C(α,γ)16OreactionratebyKunzetal.(2002)hasarelativeuncertaintyofabout±30%,whichtranslatesintoasystematicuncertaintyofabout±0.008aroundthevaluesobtainedinouranalysis.ItisalsoveryinterestingtonoticeatthispointthatMetcalfe&Handler(2002)find,fromasteroseismologydatafortwolocalwhitedwarfs,centralOxy-genabundancesconsistentwithvalueobtainedbyusingthe12C(α,γ)16OratebyKunzetal.(2002)duringtheprogenitorHe-burningphase.
WebelievetheEOScalculationsforthecurrentsetofstellarmodelsdonotcontributesignificanterrorstothefinalresults.ThenewEOShasbeenadjustedtofitthetabulatedOPALresults.Thequalityofthefitisquitegood.Forexample,theresidualsforsolarconditionsarelessthan0.06%inthepressure,andthisgoodagreementshouldalsoextendtoallbutthehighestdensityportionsofevolvedmodelswheretherearesomeEOSuncertaintiesinthetreatmentoftheCoulombandelectronexchangeeffects.However,thelargevarietyofeffectsonthemodelscausedbythesenon-idealeffectslargelycanceleachothersothecalculatedRvaluesareinsensitivetotheseuncertainties.
TheCoulombeffectarisesbecausetheattractiveCoulombforcebetweenionsandelec-tronstendtocorrelatethetwokindsofparticles.Theexchangeeffectarisesbecausethetotaleigenfunctionofelectrons,whichisantisymmetricwithrespecttoelectronexchange,anti-correlatestheelectronswitheachother.Forfixeddensityandtemperature,boththeCoulombcorrelationandtheexchangeanti-correlationreducetheamountofpressurere-quiredtoconfinethegastoitsvolumeandalsoreducetheadiabaticgradient.
TodeterminehowCoulombandexchangeeffectsalterthetheoreticalRvalues,wedidsometeststellar-evolutioncalculationswithandwithouttheCoulomborexchangeeffects
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formainsequence,RGBandHBphases.ThecalculatedRvalueisequaltotHB/tRGB,wheretHBisthedurationoftheHBevolutionaryphase,andtRGBisthedurationofthatpartoftheRGphasewhoseluminosityisgreaterthantheluminosityoftheHB.Byanalyzingthevariousnumericalexperiments,wefoundthatalthoughtheevolutionaryratealongtheRGBisslightlyaffectedbyCoulombandexchangeeffects,thequantitytRGBisnotsignificantlychangedasaconsequenceofthevariationoftheHBluminositylevelwhichcompensatesthechangeintheRGBevolutionaryrate.Onthecontrary,wefoundthefollowingresultsfortHB:
•Coulombandexchangeeffectsfortheprecursorphase(i.e.theMSandtheRGB)decreasethecoremassbyasmallamount,butthisreductioninfuelismorethancompensatedbytheaccompanyingdecreaseintheheliumburningluminositysothetotalprecursoreffectforCoulombandexchangeisa4%increaseintHB.Theexchangeeffectaccountsforaboutonesixthofthistotal.
•TheCoulombeffect(whichforweakcouplingandfullionizationisproportionaltothecubeoftheatomicnumberoftheelement)isconsiderablyenhancedforlaterstagesoftheHBphasebecauseHe-burninginthecoresubstantiallyincreasestheabundanceofCandO.
•CoulombandexchangeeffectsfortheHBphaseincreasetheconvectivecoremassbyroughly5%,butthatismorethancompensatedbyaheliumburningluminosityincreaseofroughlytwiceasmuch.Thus,thetotalHBeffectforCoulombandexchangeisa6%decreaseintHB.Theexchangeeffectaccountsforaboutonethirdofthistotal.•WhenonecombinestheoppositeprecursorandHBeffectstogether,afurthercancel-lationoccurssothetotaleffectforCoulombandexchangeisonlya2%decreaseintHB.
•AnalternativesplinefittotheCoulombeffect(seetheEOSdescriptionin§2)withasubstantiallyenlargedrangeofintermediatecoupling,changedtheCoulombresultsbyabout10percentoftheirsize.Thistranslatestoa0.2%uncertaintyintHBandcalculatedR,andanegligibleuncertaintyinthederivedYGGCvalue.
ModelsofstellarinteriorsaresensitivetoEOSinterpolationerrors(Dorman,Irwin,&Pedersen1991)sothemostreliablecalculationalprocedureistoeliminateEOSinterpolationerrorsbycallingtheEOScodedirectlyfromthestellarinteriorcode.ThepresentEOSisfastenoughsothatsuchdirectuseispracticalonworkstationtypecomputers,butofcoursestillsubstantiallyslowerthancalculationsdonewithinterpolatedEOStables.Thus,in
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theinterestsofreducingtherequiredcomputertimeforthecomputationsweinterpolatedtablesofEOSresultsthatweretabulatedwiththepresentEOSfortherequiredrangesofpressure,temperature,Y,XC,XN,andXOforafixednon-CNOmetalabundancemix.TheadoptedgridspacingsaresmallenoughinallcoordinatessothattheresultingtHBvaluesgaveexcellentagreementwithonetestcalculationusingdirectEOSresults.
AsanadditionaltestfortheadequacyofourmodelsandthereforeofourinferredYGGC,wehavealsoconsideredtheso-calledR2parameter,definedasthenumberratioofAsymptoticGiantBranch(AGB)toHBstars(Caputoetal.1989).ThisparameterisstronglysensitivetotheextensionoftheconvectivecoresduringtheHBphase,whileitisfairlyinsensitivetotheinitialmetalandHe-abundanceofthemodels,andtheprecisevalueoftheage.AtestforourtreatmentoftheconvectionintheHBstellarcoresisoffundamentalimportance,sincetheextensionoftheconvectivecorestronglyaffectstheevolutionarytimealongtheHBphase.AnunderestimateofthesizeoftheHBconvectivecoreswouldcauseanunderestimateoftheHBevolutionarytimes,withaconsequentspuriousincreaseofYGGC.TocomparetheorywithobservationswehaveusedthedatabasebyS00,whichalsoprovidesthenumberofAGBstarsforeachcluster.Theseempiricaldataconfirmthenegligibleeffectof[Fe/H]onR2;themeanvalueforthe43clustersbyS00isR2=0.14±0.05.
InourmodelswehavetreatedtheconvectivemixingduringcentralHe-burningbyincludingsemiconvection,followingtheprescriptionsbyCastellanietal.(1985).WehavesuppressedtheonsetofthebreathingpulsesduringthelatestphasesofcentralHe-burningbyimposingthattheallowedextensionoftheconvectivecoredoesnotleadtoanincreaseofthecentralHeabundancefromonemodeltothenextone(Caputoetal.1989).AllourmodelshavereachedthethermalpulsesphasealongtheAGB;fromthismomentontheevolutionissofastthatneglectingthecomputationofthethermalpulsesdoesnotaffectthetheoreticalvalueofR2.OurcomputationsprovideR2=0.12,ingoodagreementwiththeempiricalresult;thisconfirmstheadequacyofourmixingtreatmentintheHBstellarcores.Ifbreathingpulsesarenotinhibited,HBevolutionarytimesarelonger,duetotheingestionoffreshHeintothecentralconvectiveregionfollowingtheonsetofthepulses.WeobtaininthiscaseR2∼0.08,indisagreementwithobservations(asimilarconclusionwasreachedbyCaputoetal.1989bycomparingtheirmodelswithdataabouttheGGCM5).
Wehavealsoexperimentedwithanalternativeproceduretoinhibittheonsetofthebreathingpulses;followingthesuggestionsbyDorman&Rood(1993)wehavesettozerothegravitationaltermintheenergygenerationequationforthecentralstellarregionsduringthelaterstageofcoreHe-burning.Inthiswaythebreathingpulsesarealsoeffectivelyinhibited(seethedetaileddiscussionbyDorman&Rood1993),andweobtainedadecreaseofbothAGBandHBevolutionarytimebyabout2%withrespecttotheprocedurefollowedinour
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referencemodels;thisleavesthevalueofR2unchanged(R2∼0.12),andcausesasystematicincreaseofYGGCby∼0.003.
TheerrorontheindividualYGGCvaluesdisplayedinTable1comesbasicallyfromtherandomerrorontheindividualHe-abundancedeterminations(i.e.,fromtherandomerrorontheindividualR-parameterestimates).InordertogiveabestestimateforYGGCincludingalsothesourcesofsystematicerrordescribedbefore(associatedtouncertaintiesinthetheoreticalmodels)andtheeffectofthestilluncertain[Fe/H]scale,wehaveresortedtoaMonteCarlotechniquebrieflyexplainedinthefollowing.WehaveconsideredasreferencevaluesforYGGC,theonesdeterminedbyadoptingtheCG97[Fe/H]scale,takingintoaccountthecorrectionsfortheblueHBclusters(lines5and7ofTable1for,respectively,theZ00andS00sample);wenoticethatincaseoftheZ00dataweconsiderthesubsampleunaffectedbytheprecisechoiceoftheclusters’age.StartingfromeachofthesetworeferenceYGGCwehavegeneratedasetof10000syntheticHe-abundancevalues,byapplying(throughaMonteCarlosimulation)toeachgeneratedabundancevalueasetofrandomandsystematicerrors,accordingtoagivenprobabilitydistribution.Inparticular,randomerrorshavebeenmodeledaccordingtoaGaussiandistributionwithmeanvalueequaltothereferenceone,andσequaltothecorrespondingrandomerroronYGGCprovidedinTable1.Thesystematicuncertaintiesduetothechoiceofthe[Fe/H]scale(whichcausesadecreaseofYGGCby0.003withrespecttothereferencevalue),12C(α,γ)16Oreactionrate(variationby±0.008),andbreathingpulsessuppressiontechnique(increaseby0.003)havebeenmodeledusinganuniformdistributionspanningtheappropriaterange.
ThemeanvaluesforthetwofinalsyntheticdistributionsofHe-abundancesareYGGC=0.243±0.006incaseoftheZ00sample,andYGGC=0.244±0.006fortheS00sample.Thesevaluesare,asexpected,inverygoodreciprocalagreementandmoreovercomparewellwiththeprimor-dialHe-abundanceYp=0.248±0.001inferredfromtheCMBinconjunctionwithprimordialnucleosynthesiscomputations.
Anotherimportantresultofouranalysisisthefactthatthereisnostatisticallysig-nificantincreaseofYGGCwith[Fe/H],atleastwithintheanalyzedGGCsamples.ThisbearsconsiderableinterestsforstudiesaboutGalacticchemicalevolution.Asatestforthereliabilityofthisresultwehaveperformedthefollowingnumericalexperiment.WehaveconsideredtheclustersoftheZ00sampleandtheZW84metallicityscale(weobtainananalogousresultwhenusingtheCG97scale);foreachclusterwehaveconsideredareferenceRvalueobtainedfromourtheoreticalmodels,usingaprimordialHe-massfractionof0.245andassumingavalueforthechemicalenrichmentratio∆Y/∆Z.Wehavethengenerated,usingaMonteCarloprocedure,10000syntheticHe-abundancesforeachindividualclusterandagivenchoiceof∆Y/∆Z,usingaGaussiandistributionwithmeanvalueequaltothe
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referencetheoreticalRvalueandσequaltotheactualrandomerroronobservedRvalue.Foreachofthesesyntheticsampleswehavethentriedtorecovertheinput∆Y/∆Zvalue;weconcludedfromthisanalysisthatratios∆Y/∆Z>1shouldhavebeenunambigu-ouslydetectedeventakingintoaccounttheactualobservationalerrorsonthedeterminationofR.
5.Summary
FollowingrecentprecisedeterminationsoftheprimordialHe-abundancecomingfromCMBanalysesandprimordialnucleosynthesiscomputations,wehaverederivedtheinitialHe-abundanceforstarsintwosamplesofGGCs(Z00andS00),usingtheR-parameterasabundanceindicator.WehaveemployedtheoreticalmodelscomputedadoptingnewandmoreaccuratedeterminationsoftheEOSforthestellarmatterandofthecrucial12
C(α,γ)16Oreactionrate.Ourmodelsincludesemiconvection,whilethebreathingpulsesaresuppressed,inagreementwiththeobservationalconstraintscomingfromthemeasure-mentsoftheR2parameterintheS00sample.
BytakingintoaccounttheuncertaintiesintheobservedindividualRvalue,aswellastheuncertaintiesintheGGCmetallicityscale,the12C(α,γ)16Oreactionrateandthemethodforthebreathingpulsessuppression,weobtainYGGC=0.243±0.006incaseoftheZ00sample,andYGGC=0.244±0.006incaseoftheS00sample.TheseabundancesareingoodreciprocalagreementandfullyconsistentwithYp=0.248±0.001recentlydeterminedfromCMBanalysesandprimordialnucleosynthesiscomputations.WithintheS00samplewefindastatisticallysignificantspreadoftheindividualHe-abundances.ThisspreadintheHe-abundancesisnotfoundintheZ00sample,andwearguethatitisduetotheinhomogeneityoftheobservationaldatabaseusedbyS00,asopposedtothehomogeneouslyobservedandreducedphotometryemployedbyZ00.
ItisimportanttoremarkthatnoneofthetwosamplesshowanystatisticallysignificantincreaseofYGGCwiththecluster[Fe/H],afactthatisrelevantinthecontextofthechemicalevolutionoftheGalaxy.
S.C.andM.S.gratefullyacknowledgethehospitalityoftheMax-Planck-Institutf¨urAstrophysik,wherealargepartofthisworkhasbeencarriedout.A.W.I.gratefullyac-knowledgespartialfinancialsupportfromanoperatinggranttoDonA.VandenBergfromtheNaturalSciencesandEngineeringResearchCouncilofCanada.S.C.hasbeensupportedbyMURST(Cofin2002).Wewishthankthereferee,E.Sandquist,forusefulremarkswhich
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helpedtoimprovethepresentationofthepaper.
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Table1.SummaryofYGGCmeanvaluesandtheassociatedintrinsicspreadσY,obtained
bymeansoftheF-test(seetextfordetails).
Sample
[Fe/H]BlueHBcorrection
GGCsselection
YGGC
σY
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Fig.1.—R-parameterversus[Fe/H]forthetwoadoptedmetallicityscales.Empiricaldata(filledsquares)andindividualerrorsarefromZ00;errorson[Fe/H]havebeensetto0.10dex.TheoreticalpredictionsforY=0.245andGGCagesof11and13Gyrareshownassolidlines.ThedashedlinedisplaysthetheoreticalpredictionforY=0.230andaGGCageof13Gyr.
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Fig.2.—AsinFig.1butfortheempiricaldatabyS00.
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Fig.3.—HistogramsrepresentingthedistributionoftheindividualclusterHe-abundancesfortheZ00(upperpanel)andS00(lowerpanel)samples.ShadedhistogramsdisplaytheabundancedistributionwhentheCG97[Fe/H]scaleisadopted;short-dashedlinesrepresentthecorrespondinghistogramsfortheZW84scale.IncaseoftheZ00dataandtheCG97[Fe/H]scalewehaveincludedonlyclusterswith[Fe/H]<−1.15or[Fe/H]>−0.85,thatis,therangesunaffectedbytheprecisechoiceoftheGGCages.
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Fig.4.—Heliumabundanceasafunctionof[Fe/H](ontheZW84scale)for13clustersincommonbetweentheZ00(filledcircles)andS00(opencircles)samples.
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