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1992ApJ. . .387. .372G 3 nosity ofstellarmodelsstrongly dependsontheinitialhelium population )determined fromthesolarmodel.Thelumi- mixing-length parameterand theheliumabundance(fordisk modeled physicstobeincluded inmodelsoftheSun. modifications totheactualstructurepredictionsofmodel. Computers haveenabledmore physicsandmoreaccurately sophistication ofthemodels,albeit,withrelativelyminor duction ofcomputersresultedinadramaticadvance the zone byBöhm-Vitense(1958).Needlesstosay,theintro- ing fromthezero-agemainsequence.Themixing-length ing decadewashighlightedbytherealizationthatproton- theory ofconvectionwasfirstappliedtothesolar (1957) calculatedthefirstevolutionarymodelsofSun,start- chain,nottheCNcycle,providesprincipalsource of nuclearenergyintheSun.Schwarzschild,Howard,&Härm discovery ofthermonuclearreactions(Bethe1939).Thefollow- energy sourcewasnotincorporatedinsolarmodelsuntil the radiation inhisstaticsolarandstellarmodels.TheSun’s (1926) toincludeadescriptionofthetransportenergyby absorption coefficientsforvariousatomsenabledEddington Advances inatomictheoryandthesubsequentcalculationof published) theSun’ssurfacetemperatureanddensity(he gas sphereinhydrostaticequilibriumtodetermine(withoutthe obtained T=30,000Kandpf0.0004gcm"). aid oftheStefan-Boltzmannlaw,whichhadnotyetbeen quencies. low-/ p-modeoscillationfrequencieswithintheerrorsassoci- of theindividualimprovementsonstructureandfre- ated withthephysicsofmodel.Wealsodiscussimpact We demonstratethatour“best”solarmodelreproducesthe layers: namely,thefrequenciesofnonradialsolaroscillations. best availablephysicswiththemostsensitivetestofouter compare thepropertiesofsolarmodelsconstructedwith ing ofstarsandstellarevolution,continuetoprovidethe models oftheSun.Solararecrucialinourunderstand- best meansofcalibratingstellarmodels.Inthispaperwe paralleled thedevelopmentofsuccessivelymoreaccurate rich history;theintroductionofnewandimprovedphysicshas surfacesurace The AstrophysicalJournal,387:372-393,1992March1 © 1992.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. Models ofotherstarsareconstructed usingtheconvective Lane (1869)firstwrotedownandsolvedtheequationsofa The enterpriseofconstructingsolarmodelshasalongand physics (primarilyopacities). p-modes ofthemodeltotheseimprovementsarediscussed.Wefindthatcombinedsolarmodel,whichis Subject headings:equationofstate—:interioroscillations observed oscillationspectrum(forlow-/)withintheerrorsassociatedwithuncertaintiesinmodel based onthebestphysicsavailabletous(anddoesnotcontainanyadhocassumptions),reproduces cities, andthetreatmentofatmosphere.Theimpactonbothstructurefrequencieslow-/ ence solarmodel.Inaddition,amodelcombiningseveraloftheimprovementshasbeencalculatedtoprovide a “best”solarmodel.Improvementsweremadetothenuclearreactionrates,equationofstate,opa- © American Astronomical Society • Provided by the NASA Astrophysics Data System A setofsolarmodelshavebeenconstructed,eachbasedonasinglemodificationtothephysicsrefer- Center forSolarandSpaceResearch,DepartmentofAstronomy,YaleUniversity,P.O.Box6666,NewHaven,CT06511 1. INTRODUCTION D. B.Guenther,P.Demarque,Y.-C.Kim,andM.H.Pinsonneault Received 1991June3;acceptedAugust29 STANDARD SOLARMODEL ABSTRACT 372 written describingvariousmodifications tothestandardsolar quence, overthepast20years, hundredsofarticleshavebeen with theDavischlorinedetector (Davis1978),isafactorof2-3 times smallerthanthepredictions ofsolarmodels.Asaconse- ence modelbyincorporatingcutting-edge physics. late severalimprovedsolarmodelswhichupgradetherefer- model oftheSunbasedonwell-establishedphysicsandcalcu- define ourreferencestandardsolarmodeltobeaconservative each improvementmadetocurrentphysics;oritcanrepresent relies onadhocadjustmentstothephysics.Inthiswork we a solarmodelwhichbestfitstheobservablesofSun but model oftheSunbasedonwell-establishedphysics;it can represent anever-improvingmodeloftheSun,upgradedwith many thingstoauthors(see,e.g.,Sackmann,Boothroyd, & Fowler1990).Forexample,itcanrepresentaconservative in theliterature. within thiscontextthatthestandardsolarmodelfirstappeared calculations ofotherstars(Demarque&Larson1964).Itis radius andluminosity;thesevaluesarethenusedinmodel The mixing-lengthparameterandtheheliumabundanceare in thephysicsofouterlayers(Demarque&Percy1964). adjusted toproduceamodelwhichhastheSun’sobserved also provides,inthestandardsolarmodel,aconvenientmeans which aconvectiveelementtravelsbeforemergingwithits of adjustingthesolarradiustocompensateforothererrors a ’ssurface,i.e.,itsradius.Thusthemixing-lengththeory superadiabatic layers,itisresponsibleforfixingthelocationof energy fluxwithasingleparameteroforderunity(a),whichis surroundings. Becauseitsetsthetemperaturegradientsin the characteristicdistance(inpressurescaleheights)over theory characterizestheefficiencyofconvectionincarrying adjustable parameterofthesolarmodel.Themixing-length determinations difficult.Becauseheliumisnotvisibleinthe effects andgravitationalsettlingofheliummakeabundance vely affectstheluminosityofasolarmodel,itisusedasan optical spectrumoftheSunandbecauseitsabundancesensiti- enough toionizehelium,where,unfortunately,non-LTE canbeseenonlyinstarswithsurfacetemperatureshot depends ontheefficiencyofconvectiveenergytransport. abundance andtheradiusoflate-typestellarmodelsstrongly It iswellknownthatthesolar neutrinoflux,asmeasured Today, though,astandardsolarmodelhascometomean 1992ApJ. . .387. .372G models. Wejudgethe“quality ”ofourmodelsbycomparing mine thesignificanceoftheir effectonthemodel,andto monly incorporated“improvements,” oneatatime,todeter- p-mode oscillationspectrumwithintheerrorsofphysics used determine theapproximateaccuracy oftoday’sstandardsolar conservative (reference)standard solarmodelandaddcom- (commonly referredtoastheOPALopacities)andeitherCox’s cities, byusingIglesias&Rogers(1991)latestopacities in closeagreementtotheSun’s.Thetrickofenhancing enhancement inconvectionzone)produceoscillationspectra models constructedwithmodifiedLAOLopacities(30% simply usesthebestoftoday’sphysicscanreproducesolar that amodeloftheSunmakesnoadhocassumptions but dard solarmodelandtheSun’sp-modespectrum. crepancy betweenthep-modespectrumpredictedbystan- errors associatedwiththephysics,thatthereisindeedadis- manner, thatthestandardsolarmodel,andseveralnon- in thecalculationofstandard solarmodelWepresenta of thestandardsolarmodeltoassumptionsandshow Korzennik &Ulrich(1989).Weshowthatsimilarendscan be Christensen-Dalsgaard etal.(1985)andthenelaboratedon by or Kurucz’slowtemperature“molecular”opacities. obtained, withoutartificiallymodifyingthecalculatedopa- opacities tohelpfixtheproblemwasfirstproposed by al. 1990;Cox,Guzik,&Kidman1989)demonstratedthatsolar given whattheybelievedtobereasonableestimatesofthe known errorsofthephysicsatthattime,toreproduce standard solarmodels,couldnotbeeasilytweakedwithinthe Ulrich &Rhodes(1983)wereablesoshow,inadefinitive the standardsolarmodeltopredictfrequenciesofp-modes. eters todeterminingtheaccuracywithwhichonecanexpect but totryunderstandtheimportanceofindividualparam- is nottoproducetheultimatefixstandardsolarmodel Sun’s oscillationspectrum.Inotherwords,theyestablished, Dalsgaard 1988;Ulrich&Rhodes1983).Theimmediategoal (Guenther 1989,1991b);composition,opacities,andequation Guenther &Sarajedini1988;Kim,Demarque, Cox, Guzik,&Raby1990;Guenther,Jaífe,Demarque1989; bles tovariationsoradjustmentsinthemodelparameters:age that ofestablishingthesensitivitymodelanditsobserva- goaded solarmodelerstoadoptamoreconservativeapproach, Woodard, &Kaufman1990)andthepersistent1%discrep- model andneutrinofluxobservationsmayultimatelyestablish is wrong(Bahcall&Pinsonneault1991).Infact,thesolar of state(Christensen-Dalsgaard,Däppen,&Lebreton1988; ancy betweentheseobservationsandmodelpredictionshas quencies oftheSun’sp-modes(forexample,Libbrecht, of nonstandardtheorieselectroweakinteractions(Bahcall the existenceofresonantneutrinooscillationsandvalidity dard solarmodeliscorrectandittheneutrinophysicswhich trino flux.Thishasledmanytoconcludethatindeedthestan- be adjustedwithintheerrorstoaccountforobservedneu- dard physicalassumptionsofthestandardsolarmodelcannot model whichattempttoreconcilethesediscrepancies.In1988, trino fluxpredictionsandconcludedthat,onceagain,thestan- two independentstudiesofthesolarmodel(Bahcall&Ulrich 1977); andsurvey(Bahcall&Ulrich1988;Christensen- 1991); envelopestructure(Guenther1991a;Ulrich&Rhodes 1989; Mikheyev&Smirnov1986;Wolfenstein1978). 1988; Turck-Chièze,Cahen,&Cassé1988)focusedonneu- In thispaperwewillcontinueouranalysisofthesensitivity In akeypieceofresearch,Coxandhiscollaborators(Coxet In recentyearstheveryaccuratedeterminationoffre- © American Astronomical Society • Provided by the NASA Astrophysics Data System STANDARD SOLARMODEL whose p-modefrequenciescan becalculatedtoanumerical accuracy of±0.5//Hzwith our pulsationcode(Guenther& the envelope(outer5%by radius),andatmosphere.This distributed equallyamongthe interior(inner95%byradius), et al.(1989). spacetime meshresolution is sufficienttoproducemodels steps. Eachmodelhasapproximately 1800shellswhichare sequence modeltothecurrentageofSunin50equaltime ete Cox&Stewartmixtureofheavyelements.Amoreexten- Note thattheCox&Stewartopacitiesaregivenforobsol- sive discussionoftheopacityformulationisgiveninGuenther perform four-pointLagrangianinterpolationofthetables. heavy elementmassfraction)andintegratetheminto the tables totheappropriateX,Y,andZ(,helium, and we constructRosselandmeanopacitiesinthree-dimensional tion andthenaveraged.Radiationpressureelectron lation andexteriorformulationareweightedwitharampfunc- elements areassumedtobefullyionized.Asmalltransition for thesingleionizationstateofHandmetals, LAOL opacityarrays.Theroutines,withinYREC, specific mixtureofheavyelements.Fortemperaturesbelow arrays oftemperature,,andhydrogenabundancefora With amodifiedversionofLAOLtapecombinationsoftware a functionofdensityandtemperature),photonfrequency. the Sunasafunctionoftemperature,electrondegeneracy(itself region isdefined(5.54.0,weusetablespreparedusingthe corrections) areignored. tines determineparticledensitiesbysolvingtheSahaequation solved separatelyforeachshellinthemodel. servation andtransportequationsofstellarstructureusingthe models andcomparethelow-/p-modeoscillationspectraof LAOL tapes(Huebneretal.1977).Thecontainabsorp- single sourceofopacitiescoverstheentireastrophysical Coulomb interactionsintheplasma(i.e.,Debye-Hückel degeneracy correctionsareincludedbutperturbationsdueto structed usingthedefaultphysicsandparametersofYale comment onfuturedirections. (MHD andDebye-Hückel),opacities(Cox,Kurucz, Rotating StellarEvolutionCode(Prather1976;Pinsonneault the modelstoSun’s.In§4wesummarizeourresultsand p-mode spectrum. OPAL). In§3wedescribethestructuraldifferencesamong sphere model,thenuclearreactionrates,equationofstate The augmentedmodelsincludeimprovementstotheatmo- reference standardsolarmodelandtheaugmentedmodels. 10,000 K,weinterpolatetheCox&Stewart(1970)opacity the frequenciesoftheirp-modespectra(oflow-/)toSun’s 1988), initsnonrotatingconfiguration,i.e.,physicsandparam- To produceoursolarmodelsweevolvedazero-agemain- In theouterregions(logT<5.5),equation-of-staterou- The opacityformulationiscomplicatedbythefactthatno Our referencestandardsolarmodel(STDmodel)iscon- In §2wedescribethephysicsandmodelingprocedureofour 2.1. TheReferenceStandardSolarModel 2. THESOLARMODELS 373 1992ApJ. . .387. .372G 1 3_1 3-1_ 10 -2 2 where wehaveusedthemeteoricabundance. very small. meeting inTucson,Arizona,1991)showthattheeffects of 10 65 helium diffusiononthefrequenciesoflow-/p-modes are (presented attheGlobalOscillationNetworkGroup’sannual tant. Also,modelsbyCoxetal.(1989)andourselves vesse 1989),whichwouldbedifferentifdiffusionwereimpor- the meteoricandphotosphericabundances(Anders&Gre- nitrogen intheSun). as isverifiedobservationallybythecloseagreementbetween their initialvalues.Theamountofdiffusionisprobablysmall, cause differencesbetweenthecurrentsurfaceabundances and remain poorlydetermined(notethatneonisasabundant accounted forinanyofthemodelspresentedhere.Thiscould specific mixture.Onlytheinertgaseshelium,neon,andargon of theabundanceselementsisslowlyconvergingtoa Z =0.02.AsnotedinGuentheretal.(1989),thedetermination equation ofstatemodelusetheRoss-Allermixture(1976)with Z =0.0188.TheRoss-AllermixturemodelandtheMHD of statemodel,isthatAnders&Grevesse(1989)with except theRoss-AllermixturemodelandMHDequation models tobeL=3.8515x10ergss. average ofthetwoandadoptsolarluminosityinallour As wecannotexpertlybaseapreference,merelytakethe measurements fromspaceonboththeNimbus7andSMM 3.846 x10ergssand3.857,respectively. satellites (Hickey&Alton1983).ERB-Nimbusmeasures t =0.001(atthelimb),weadoptasolarradiusofR6.9598 eclipse andtransitmeasurementsareseeingonlytoadepthof ture calculationsisdefinedattheopticaldepthoft=fand measurements, is6.96x10cm.AsnotedbyUlrich& 1367.7 +0.802Wmwhichyieldluminositiesof 1371.6 +0.765Wm“andSMM/ACRIMmeasures mass foralloursolarmodels. 0 Rhodes (1983),becausetheradiusofSunforstellarstruc- Universal GravitationalConstantGisknown.Weadoptthis determination dependsdirectlyontheaccuracywithwhich tainty of±0.02%(Cohen&Taylor1986);theaccuracy Gyr forthesolarageinallourmodels. Tilton). Decimalprejudice,though,hasleadustoadopt4.5 determined byGuenther(1989)(thedifferenceisduetotheuse is 4.52±0.4Gyr.Thisnumberrevisesthe4.490.03Gyrage main-sequence calculations,thebestestimateofSun’sage ö of theimprovedageestimatesoldestmeteoritesby system formation,observationsofTTauristars,andpre- 4.6-4.7 Gyr,usingthelatestdeterminationsofages meteorites. Although,theageiscommonlyquotedasbeing oldest meteorites(Tilton1988),moderntheoriesofsolar fraction. 1 x 10cmatt=§foralloursolarmodels. matched toonepartin10and,respectively,by adjusting themixing-lengthparameterandheliummass 374 Sarajedini 1988).Theradiusandluminosityofthemodelswere 3 Wehaveusedsolarphotosphericabundances forallelementsexceptiron, Other physicalcharacteristics oftheSTDmodelaresum- Settling ofheliumandheavyelementsbydiffusionisnot The mixtureofheavyelementsusedinallthesolarmodels, The luminosityhasbeendeterminedfromsolarconstant The radiusoftheSun,asdeterminedbytransitandeclipse The massoftheSunis1.9891x10gwitharelativeuncer- The ageoftheSuncanbeinferredfromagesoldest © American Astronomical Society • Provided by the NASA Astrophysics Data System GUENTHER ETAL. 6 published inBahcall, &Shaviv1968isincorrect) where Tisinunitsof10 K.Thisform(note:formula compared toXandcanbeneglected,oneobtains abundance bymassofelement/.AssumingthatZissmall where ZAandXjarethecharge,atomicmass,relative by modifies the“perfect”,P,i.e.,idealgaslaw . ThespecificformimplementedherefromClayton mean molecularweight,eisthechargeofanelectron,and kis (Clayton 1968;Cox&GiuliLandauLifshitz1959) 6 attempts totakeintoaccounttheCoulombinteractionsof Boltzmann’s constant.Thevariable£isgivenby where Pisthegaspressure,NAvogadro’snumber,p the tion ofstatemodelaredescribedindetailKimetal.(1991). in themodelcalculation,includingopacitytablesbasedon Ross-Aller mixture.ThemodelandtheMHDequa- consistency weusedtheRoss-Allermixtureofheavyelements in theMHDequation-of-statetables.Notethatforself- h routines inYRECarereplacedwithtolookupvalues p Aller mixturemodelexceptthatthestandardequation-of-state to Z=0.02. mixture, consistsofC,N,OwithRoss-Allerabundance,and g0 Fe adjustedtobringthetotalmassfractionofheavyelements the heavyelementportionofmixture,aRoss-Allerbased atomic, ionicandmolecularspecies.Asprovidedforthistest, detailed internalpartitionfunctionsforthespecifiedmixtureof routines. TheMHDequationofstateformulationuses atomic physicsthanmodeledbyourexistingequationofstate come fromanewsetofcalculations,kindlyprovidedtousby Däppen (1990),whichofferamorerealisticdescriptionofthe Hummer 1988;Mihalasetal.1991;&1988) radius andluminosity. abundance wereadjustedtoproduceamodelwiththeproper models, boththemixing-lengthparameterandhelium described intheprecedingsection.Likeallothermodified lated forthismixtureusingtheLAOLtapesandsoftware, except thattheheavyelementmixtureissettopopular Ross-Aller (1976)mixturewithZ=0.02.Opacitieswerecalcu- improvement incorporatedatatime. the physicsofreferencestandardsolarmodelwithonlyone model islistedinTable3A. marized inTable1.Asummaryoftheinternalstructure The Debye-Hückelcorrectiontotheequationofstate The MHDequationofstatemodelissimilartotheRoss- The MHDequation-of-statetables(Mihalas,Däppen,& The Ross-AllermixturemodelissimilartotheSTDmodel, All ofthemodifiedmodels,exceptcombinedmodel,use 2.3. MHDEquationofStateModel = P, 2.4. Debye-HückelCorrection 2.2. Ross-AllerMixtureModel PJ 1 C =z[(z?z,)|], + - 0.044 32 31/2 y k'r/ e (nN)p 0 1/23 /2 p(3 +X)1 T 6(5X+3)J Vol. 387 1992ApJ. . .387. .372G 5 2 Gabriel 1991,privatecommunication). (1991) betweenourstandardsolarmodel(usingthenewrates) atives, instellarmodelsdestinedto beusedinpulsationcalculations(M. mic consistencybemaintained,inthe fundamentalvariablesandtheirderiv- fundamental variables(density,, andpressure),thepulsationfre- sure, temperature,density,,andcomposition, adiabatic exponentTherefore,it is especiallyimportantthatthermodyna- quencies do.Forexample,thepulsation frequenciesdependsensitivelyonthe not dependsignificantlyonquantities derivedfromthederivativesof and Bahcall’ssolarmodelagreeto0.1%,withrespectpres- the combinedmodelandBahcallnuclearratesmodel, use with aswitchtoturnthemon,oroff,whereinthelatter case p-mode frequencies,weimplementedthenewratesinYREC the oldnuclearreactionrates. the oldratesareused.Allofmodelspresentedhere,except proton ratesofClayton(1968)andtheCNOFowler, rates, collectedandpublishedbyBahcall(1989),affect the Caughlan, &Zimmerman(1975).Totesthowmuchthenew opacities. providing aself-consistenttestoflow-temperaturemolecular are calculatedfortheAnders&Grevessemixture,thereby temperature opacitytablesareincludedinourtests,andthey cal resultstothosepresentedhere,and(2)theKuruczlow- inconsistency inthemixturesbecause(1)Kimetal.(1991)uses mixture areused.Wehavenotconcernedourselveswiththe the Ross-Allermixturethroughoutandobtainsnearlyidenti- (log T<5),theyareusedinstead.Tosummarize:attem- peratures, theCoxlow-TopacitytablesforRoss-Aller1 Anders &Grevessemixtureareused,andatlowertem- peratures above10K,thestandardLAOLopacitiesfor fall withintherangeofCoxlow-Topacitytables STD modelexceptthatfortemperaturesanddensitieswhich (1976). TheseopacitiesaresignificantlylargerthantheCox& made availabletous(Cox1990)fortheRoss-Allermixture a profoundeffectonthep-modeoscillationspectrum. molecules, Ross-Aller1(Coxetal.1989),haverecentlybeen Stewart (standard)tables,and,asisdescribedin§3,theyhave Cox &Stewart(1970)opacitieswhichdonotincludetheeffects of molecules.Newtableswhichincludethecontribution the processisiterateduntilacoveragedmodelobtained. No. 1,1992 thermodynamic consistency.Themodelisthenrerelaxed,and pressure, theseexpressionswerealsomodifiedtomaintain density, forexample,tocalculatethespecificheatatconstant depend onthederivativesofpressure,temperature,and thermodynamic expressionstocalculatequantitieswhich of stateroutinestodeterminep,P,andT.BecauseYRECuses partial . Hiickel correctionroutinesaftercallingthestandardequation general Debye-Hiickelcorrectionalsotakesintoaccount detailed equationofstatecalculationstheMHDgroup.The the generalformofDebye-Hiickelcorrectionusedin and assumesthatthegasisfullyionized.This,therefore,not applicable intheregimeofhightemperatureandlowdensities 2 6 Note thatcomparisonsmadebyBahcall&Pinsonneault EventhoughthestructureandevolutionofSun(andotherstars)does YREC’s nuclearreactionratesarebasedontheproton- The Coxlow-temperatureopacitiesmodelissimilartothe At lowtemperatures,ourstandardopacitytablesusethe For temperaturesabove10K,YRECcallstheDebye- © American Astronomical Society • Provided by the NASA Astrophysics Data System 2.5. CoxLow-TemperatureOpacities 2.6. BahcallNuclearReactionRates STANDARD SOLARMODEL 6 64 4 3 4 were alsonoted),and(3)asreported in thispaper,modelsbasedontheOPAL opacities betterlocatethebaseof Sun’s convectionzone. in theLAOLopacitycalculation(other errorsassociatedwiththeironlines is believedthatonesetoflinesassociated withheliumwereincorrectlylocated tion calculationsandmassesinferred fromstellarevolutioncalculations,(2)it because (1)theyremovethedisagreement betweenmassesinferredfrompulsa- indicate thattheOPALopacitiesare tobepreferredovertheLAOLopacities Assembly oftheIAUinBuenosAires1991aspecialsessiononopacities 4 details ofthismodel’sinternalstructure. cussed in§3,wedonotincludetheDebye-Hiickelcorrection. following modifications:(1)OPALopacities,(2)Kuruczlow- it isbasedontheRoss-AllermixtureandnotAnders & model whichincludessomeoftheabovemodifications.Wedid Bahcall nuclearrates.Table3Bcontainsasummaryof the temperature opacities,(3)KrishnaSwamyatmosphere,and (4) Grevesse mixtureusedintheopacities.Andforreasons dis- not includetheMHDequation-of-statemodificationbecause cations onthestructureandp-modefrequenciesto present, insomesense,a“best”model,wehavecalculated Cox-Stewart opacitiesareused. except thatattemperaturesabove10K,theOPALopacities and 10K,theLAOLopacitiesareused,below are usedinsteadoftheLAOLopacities.Notethatbetween10 mixture only. mixture, theOPALopacitiesareprovidedforacomplete ments sothattheycanbecombinedbytheuserinanarbitrary mixture (hereafterreferredtoastheOPALopacities).Unlike the LAOLopacities,whichareprovidedforindividualele- their latestopacitiescalculatedfortheAnders&Grevesse <10K. effects ofmolecules.Becausetheyhavebeencomputedinde- throughout mostoftheinterior.Thepredictedneutrinofluxes on theAnders&Grevesse(1989)mixture,andareapplicableat Kurucz opacities,asprovidedtousintabularform,arebased an independentcheckontheaccuracyofopacities.The pendently fromtheCoxmolecularopacities,theyalsoprovide improved opacityatlowertemperaturesbyincludingthe of thesemodelsagreeto1%. perature riseabove.Itisgivenby perature minimumatthebaseofcoronanortem- t =0.312156330,to10“.Itdoesnotmodelthetem- surface ofthephotosphere,where^ffisdefined,at lower atmosphere.TherelationfollowsthedropinTfrom based empiricalfittotheSun’sobservedT-tdependencein 3 RecentreportsmadebyC.A.IglesiasandCoxattheGeneral The combinedmodelisbasedontheSTDwith To demonstratetheadditiveeffectsofindividualmodifi- The OPALopacitymodelissimilartotheSTD Recently, Iglesias&Rogers(1991)havemadeavailabletous The Kurucz(1991)lowtemperatureopacitiesprovidean The KrishnaSwamy(1966)T-trelationisatheoretically 4-25r3Ot T =0.75T(r+1.39-0.815e’0.025^"). ff 2.8. KuruczLow-TemperatureOpacitiesModel 2.7. KrishnaSwamyAtmosphereModel 2.9. OPALOpacitiesModel 2.10. CombinedModel 375 1992ApJ. . .387. .372G list anabbreviatedsummaryofthestructureSTDmodel and thecombinedmodel. the STDmodelandcombinedmodel.AndinTable3we In Table2,discussedin§3.2,welistthep-modefrequenciesof 376 Combined model OPAL opacities Kurucz low-temperatureopacities. Cox low-temperatureopacities.... Debye-Hiickel correction Krishna Swamyatmosphere Bahcall nuclearreactionrates MHD equationofstate Ross-Aller mixture In Table1welistsomeofthecharacteristicsmodels. 41. 40. 38. 34. 32. 29. 28. 27. 26. 25. 24. 23. 21. 20. 36. 35. 33. 31. 30. 22. 39. 37. 17. 19. 18. 16. 15. 14. 13. 12. 11. 10. © American Astronomical Society • Provided by the NASA Astrophysics Data System 9. 4. 7. 2. 8. 6. 5. 3. 1. Model 3.1. StructureComparisons 4966.763 4828.180 4689.359 4550.492 4411.488 4272.616 4133.761 5758.970 5639.719 5511.801 5378.427 5104.926 2757.871 2487.124 2217.943 5242.402 3995.122 3856.661 3304.871 3030.984 2894.414 2622.163 2352.759 2081.790 3718.288 3580.213 3442.266 3167.785 1945.463 1809.197 1673.810 1536.990 1396.872 1110.182 1254.726 402.423 966.338 676.282 820.660 533.610 0.000 3. RESULTS 127.918 119.251 133.374 136.025 137.476 138.163 138.582 138.822 138.867 139.004 138.872 136.570 136.543 135.708 135.039 134.364 134.816 136.154 136.327 136.266 140.118 142.147 143.843 138.856 138.639 138.461 138.373 138.075 137.948 137.395 137.086 136.801 135.387 136.820 144.544 144.377 142.672 131.187 1 =0 0.000 0.000 0.000 Av 0.2918 0.2875 0.2876 0.2736 0.2875 0.2892 0.2875 0.2843 0.2771 0.2843 Standard Model:p-ModeFrequenciesandSpacings 10.177 10.419 10.818 11.199 11.666 12.022 12.216 12.602 12.851 13.285 13.323 13.165 13.161 4.678 9.574 0.000 0.000 6.122 6.287 6.435 6.767 5.119 5.501 5.762 5.958 6.603 6.944 7.133 7.317 7.534 7.733 7.966 9.277 9.888 0.000 0.000 0.000 8.196 8.433 8.701 8.949 ôv 2.0661 1.8942 1.5588 1.2225 1.2420 1.2477 1.4229 1.1942 1.2559 1.1949 4895.333 4756.629 4617.695 4339.736 4478.723 4200.767 4062.018 2550.774 2281.295 2009.068 5814.400 5698.778 5443.631 5171.716 2959.427 2822.747 2686.398 2416.187 2145.745 5574.780 5308.639 5033.833 3233.278 3096.369 3923.302 3784.876 3646.522 3508.423 3370.704 1872.334 1600.005 1461.714 1176.500 1736.605 1319.908 1032.608 448.229 887.493 743.612 595.069 GUENTHER ETAL. 0.000 Model Characteristics log (L/Lq)\og(R/R)log(T(p -1.4E-6 -2.7E-6 Qc — LIE—5 -2.2E-6 -7.0E-6 TABLE 2A 3.0E-6 3.9E-7 1.4E-5 1.2E-5 1.6E-7 TABLE 1 123.997 115.622 131.149 134.992 136.923 137.883 138.500 138.703 138.935 138.972 138.987 138.969 138.749 138.715 138.427 138.099 137.427 136.943 136.679 136.350 134.587 134.893 135.550 136.734 141.806 1=1 138.353 137.719 136.909 135.623 136.677 135.729 136.600 138.291 143.408 143.892 145.115 143.881 148.543 0.000 0.000 0.000 Av marily onthedetailsofsurfacelayers.TheKrishnaSwamy model, theCoxopacityandKuruczmodel quoting theSun’sinteriorheliumabundancetobe helium massfractionofthesolarmodels,varyingfrom Y =0.28±0.01. Y =0.2736to70.2918.Wearecomfortable,therefore,in The mixing-lengthparameter,asexpected,dependspri- We seefromTable1thatthereisverylittlevariationinthe -1.2E-7 -3.5E-8 -5.2E-8 -3.3E-7 -8.1E-7 20.211 20.656 20.895 22.556 24.301 26.630 21.455 23.161 24.853 28.972 10.613 31.842 10.242 10.940 11.599 12.279 11.265 11.919 12.612 12.983 13.343 13.718 17.254 14.142 14.536 14.992 15.384 15.827 16.290 16.728 17.619 18.057 18.485 19.027 19.789 6.5E-8 2.4E-7 6.0E-7 8.2E-7 1.3E-7 9.210 9.817 7.828 8.392 0.000 0.000 ¿V 4960.476 4821.745 4682.756 4543.724 4404.544 4126.444 4265.483 5868.768 5754.292 5634.600 5506.300 5236.443 5372.665 5098.803 2884.840 2611.986 2341.941 2070.123 3987.588 3848.929 3710.322 3572.017 3433.833 3296.170 3158.836 2747.983 2476.705 2206.744 3021.707 1933.441 1796.982 1661.207 1524.139 1241.403 1383.587 1097.017 953.177 660.830 807.630 513.470 379.326 7.197 7.193 7.186 7.186 7.189 7.186 7.185 7.188 7.190 7.186 1 =2 119.692 133.635 114.477 128.300 136.221 137.640 138.328 138.731 138.989 139.031 139.180 139.061 139.039 138.855 138.660 137.663 138.607 138.305 138.184 137.335 137.128 136.867 136.857 135.997 135.281 134.764 135.197 136.621 136.683 136.459 135.774 140.553 144.386 145.547 137.068 142.184 143.840 146.800 147.360 134.144 2.178 2.166 2.166 2.168 2.166 2.181 2.166 2.160 2.159 2.166 0.000 Av 0.724 0.724 0.745 0.745 0.744 0.744 0.740 0.734 0.738 0.745 4884.067 4745.030 4466.444 5806.572 4605.775 4327.124 4187.784 4048.674 5922.680 5690.386 5565.570 5298.397 5433.815 5161.103 5022.893 2942.699 2805.493 2668.779 2397.703 2125.957 3909.584 3770.734 3631.987 3493.431 3355.320 3217.451 2532.718 2262.268 3080.079 1851.678 1578.550 1296.747 1007.755 1988.858 1715.711 1439.159 1152.198 414.395 714.639 860.862 563.226 1 =3 MJM coe 0.0211 0.0155 0.0153 0.0190 0.0177 0.0210 0.0152 0.0151 0.0167 0.0153 116.108 116.186 124.816 131.755 135.418 137.294 138.210 138.825 139.037 139.255 139.331 139.320 139.340 139.109 139.090 138.851 138.747 138.556 138.111 137.869 137.372 137.381 137.205 136.715 136.061 135.435 137.099 135.015 136.311 137.180 135.967 137.161 139.391 142.411 144.549 144.443 146.893 151.413 146.223 148.832 0.000 Av logiî’conv) Vol. 387 6.330 6.331 6.283 6.284 6.284 6.285 6.294 6.308 6.299 6.284 1992ApJ. . .387. .372G inverted theSun’sp-modeoscillation frequenciestodetermine Christensen-Dalsgaard, Gough, &Thompson(1991)have the physicallocationofbase ofthesurfaceconvectionzone. to theMHDmodel. perature, anddensity,theDebye-Hiickel modelisverysimilar central densitywereeffectedbychangingtheequationofstate. With respecttothefundamentalvariables,pressure,tem- (compare thestandardmodeltoRoss-Allermodel)and by dard modeltotheOPALmodel).Thelargestchanges the changing thehightemperatureopacities(comparestan- central temperaturewereeffectedbychangingthecomposition models areremarkablysimilar.Thelargeschangesto the it dependsonthespecificmixinglengthformalismusedin the stellar evolutioncode. length parametershouldnotbegivenanysignificancebecause relative changesareinteresting,theactualvalueofmixing- have littleeffectonthemixing-lengthparameter.Although models whosemodificationsdonotaffectthesurfacelayers all requirelargermixing-lengthparameters.Equivalently, No. 1,1992 One truesuccessofhelioseismology isthedeterminationof The interiortemperatureanddensitiesofthedifferent 41.. 40.. 39.. 38.. 37.. 36.. 35.. 34.. 33.. 32.. 31.. 30.. 29.. 28.. 27.. 26.. 25.. 24.. 23.. 22.. 21.. 20.. 19. 18. 17. 16. © American Astronomical Society • Provided by the NASA Astrophysics Data System 15. 14. 13. 12. 11. 10. 9. 8. 7. 6. 4. 5. 3. 2. 1. 5773.657 5642.566 4959.645 4821.427 4683.028 5508.421 5372.413 5235.400 4544.677 5097.725 4406.326 4267.985 4129.837 3991.760 3853.996 3716.421 3578.996 3305.149 3441.961 2897.246 3168.923 3033.064 2761.731 2626.406 2491.975 2358.172 2223.620 2088.119 1951.444 1814.920 1679.126 1401.579 1541.528 1258.525 1113.756 969.300 679.252 404.202 823.045 535.340 0.000 131.092 134.145 136.007 137.013 137.675 138.079 138.218 138.399 138.351 138.351 138.341 138.148 138.077 137.764 137.576 137.425 137.035 136.812 136.225 135.859 135.818 135.515 135.325 134.431 133.803 134.552 135.501 136.675 136.524 135.794 137.598 139.950 143.054 144.770 144.455 143.794 143.912 131.138 / =0 0.000 0.000 0.000 Av Combined Model:p-ModeFrequenciesandSpacings 10.175 10.501 10.792 11.207 11.879 11.665 12.114 12.230 12.444 12.813 12.652 12.708 12.557 5.270 5.515 5.700 6.004 5.855 6.136 6.281 6.424 6.571 6.740 6.900 7.089 7.278 7.472 7.698 7.911 8.168 8.421 8.670 9.240 9.580 8.968 9.912 0.000 0.000 0.000 0.000 0.000 ôv STANDARD SOLARMODEL 5835.627 5706.689 4888.421 5573.788 5438.555 5301.978 4750.069 5164.520 4611.695 4473.239 4334.908 5026.639 4196.566 4058.379 3920.446 3782.620 3645.123 3507.743 3370.678 2961.988 3234.127 3097.844 2826.098 2690.362 2555.447 2421.023 2286.860 2151.640 2014.928 1878.289 1741.794 1605.214 1466.631 1324.359 1181.046 1036.027 448.609 891.016 745.705 596.748 0.000 TABLE 2B 128.938 132.900 135.234 136.577 137.458 137.882 138.218 138.352 138.374 138.456 138.331 138.342 138.187 137.932 137.826 136.551 137.497 137.380 137.065 136.283 135.856 135.891 135.736 134.915 134.424 134.163 135.220 136.712 136.638 136.496 136.580 138.582 142.272 143.313 145.020 145.011 145.311 148.957 1=1 0.000 0.000 0.000 Av and thenatureofmixingat belowthebaseofsurface the depthofsurfaceconvection zoneasafunctionoftime problem.” Becauselithiumis arelativelyfragileelementin stellar interiors,itsdepeletion isasensitiveindicatorofboth simply deepentheconvection zonetoresolvethe“ raising thequestion,“what aboutlithium?”Onecannot greater forthemodelswithdeeperconvectionzones,thus the nearagreement. equation ofstateandtheOPALopacitieswillfurtherimprove Our resultssuggestthatamodelincorporatingboththeMHD Aller model)andtheDebye-Hiickelcorrection(compare to 0.724 +0.005.TheMHDequationofstate(comparetoRoss- standard model)bothdeepenthebaseofconvectionzone. The baseoftheconvectionzoneinOPALmodelisx = OPAL opacitiesdo,significantlyalleviatesthisdiscrepancy. the convectionzonesignificantlyhigherupatx=0.745 its radiusfractionlocationtobex=0.713±0.003.Unfor- Enhancing theopacitiesnearbaseofconvection,as the tunately, ourstandardsolarmodelpositionsthebaseof conv ± 0.005(errorbasedonshellresolutionatthisdepth). conv conv The temperatureatthebaseofconvectionzoneisalso 10.055 10.378 10.696 11.017 11.319 11.655 11.975 12.316 12.681 13.024 13.417 13.783 14.178 14.607 14.986 15.426 15.807 16.213 16.703 20.269 20.422 21.215 21.971 17.096 17.548 17.882 18.195 22.853 23.934 24.444 18.798 19.309 19.866 26.481 27.991 31.858 9.255 8.725 9.681 0.000 0.000 <5v 5894.991 5768.388 5637.050 4953.364 4815.003 5502.720 5366.559 4676.456 5229.396 5091.588 4537.937 4399.426 4260.896 4122.560 3984.288 3846.298 3708.509 3433.540 3570.828 3296.479 3159.956 3023.824 2887.666 2751.818 2616.231 2481.473 2347.380 2212.413 2076.454 1939.565 1802.807 1666.896 1529.084 1388.766 1245.873 1101.048 956.743 810.076 663.761 514.656 382.616 1 =2 126.604 131.337 134.330 136.161 137.163 137.808 138.225 138.361 138.546 138.511 138.519 138.530 138.336 138.271 137.990 137.789 137.061 137.681 137.288 136.523 136.132 136.158 135.847 135.587 134.758 134.093 134.967 135.959 136.889 136.759 135.911 137.812 140.319 142.892 144.826 144.304 146.668 146.315 149.105 132.039 0.000 Av 5951.168 5826.901 5697.433 4877.102 5564.108 5428.500 4738.414 4599.720 5291.600 5153.824 5015.622 4460.923 4322.227 4183.541 4044.962 3906.663 3768.442 3492.758 3630.517 3355.252 3218.320 3081.632 2945.286 2809.001 2672.814 2537.565 2402.828 2268.062 2132.330 1995.061 1858.021 1721.371 1583.999 1444.660 1301.506 1157.112 1011.582 717.714 416.105 864.535 564.890 1 =3 124.266 129.468 133.326 135.608 136.900 137.776 138.202 138.521 138.688 138.694 138.797 138.696 138.686 138.579 138.299 138.221 137.505 137.925 137.759 136.932 136.688 136.346 136.284 136.187 135.250 134.736 134.767 135.731 137.269 137.041 136.649 137.372 139.339 143.154 144.394 145.530 147.048 146.821 152.824 148.785 0.000 Av 377 1 9 92Ap J . . .„aSV. .372G Shell RadiusFracMass 381 6.34062E-019.61115E-01 2.77054E+063.44585E-011.27434E+142.48093E+07 2.13191E-06 361 5.96846E-019.48688E-01 3.05216E+064.96007E-012.02098E+142.60417E+07 2.52975E-06 321 5.24153E-019.13633E-013.66983E+061.03714E+00 5.08273E+142.85619E+073.44934E-06 341 5.60156E-019.33081E-013.35039E+067.16477E-01 3.20501E+142.72872E+072.96007E-06 301 4.88938E-018.89572E-014.00870E+061.50529E+00 8.06058E+142.98569E+074.06728E-06 281 4.54574E-018.59993E-014.37275E+062.18766E+00 1.27833E+153.11901E+074.66685E-06 261 4.21011E-018.23851E-014.77405E+063.17620E+002.02736E+153.25991E+075.22855E-06 241 3.88111E-017.79975E-015.22214E+064.60212E+003.21538E+153.41064E+075.79443E-06 221 201 181 161 141 121 101 81 21 41 61 6.24068E-01 3.85150E+33 0.00000E+00 1.25323E+012.15295E-013.98465E-01 1.66433E+00 6.24068E-01 3.85152E+33 0.00000E+00 1.09607E+012.05402E-013.98549E-01 1.66442E+00 6.24069E-01 3.85154E+336.70702E-059.58331E+00 1.99434E-013.98641E-011.66453E+00 6.24069E-01 3.85158E+331.80130E-048.33605E+00 1.95192E-013.98731E-011.66463E+00 6.2 4074E-013.85161E+334.67161E-047.05749E+00 1.89052E-013.98821E-011.66473E+00 6.24081E-01 3.85162E+331.18089E-036.08121E+00 1.88749E-013.98905E-011.66483E+00 6.24100E-01 3.85155E+332.94674E-035.31783E+001.92321E-013.98973E-011.66490E+00 3.55688E-01 6.24145E-01 3.85121E+337.29397E-034.65452E+001.96947E-013.99023E-011.66493E+00 3.234 93E-01 6.24253E-01 2.91183E-01 6.24508E-01 2.58253E-01 2.24 081E-01 6.25124E-01 6.26616E-01 1.87861E-01 6.30218E-01 6.38813E-01 1.48492E-01 6.58875E-01 1.16439E-01 7.234 63E-01 6.88401E-01 2.2 6567E-02 7.70284E-01 8.32887E-01 9.03622E-02 8.44467E-01 6.25625E-02 6.5927 9E-03 Mean MolWtLuminosity © American Astronomical Society • Provided by the NASA Astrophysics Data System 3.85005E+33 3.84 648E+33 5.8 9337E-01 7.27101E-01 3.83591E+33 5.02593E-01 3.80528E+33 6.63940E-01 3.71898E+33 2.96050E-01 4.04062E-01 3.487 64E+33 2.922 72E+33 2.15162E+33 1.84555E-01 1.069é6E-01 5.7 6113E-02 1.38084E+33 2.18814E-02 3.7 6902E+31 2.97851E-05 6.107 90E+32 1.1857 6E-03 9.60367E+29 5.72 663E+06 1.794 60E-02 4.39944E-02 1.0834 9E-01 2.68752E-01 6.30103E+06 7.77 668E+06 6.97318E+06 8.73438E+06 6.64190E-01 1.614 49E+00 9.87881E+06 3.734 45E+00 1.12474E+07 1.24198E+07 1.33688E+07 6.51958E+00 9.37555E+00 1.25188E+01 1.42894E+07 1.5722 6E+01 1.51942E+07 1.6204 9E+01 1.53362E+07 Nuc RateOpacityTempGrad Temperature DensityPressure Summary ofReferenceStandardModel TABLE 3A 4.08363E+00 3.59010E+00 3.26144E+00 2.92226E+00 6.65147E+00 1.372 95E+01 2.55884E+00 2.77295E+01 1.954 77E+01 3.93712E+01 9.58143E+00 2.18116E+00 5.644 66E+01 1.80 989E+00 1.57833E+00 7.4 6674E+01 1.42807E+00 1.2 9775E+00 9.27050E+01 1.20031E+00 1.14376E+02 1.41528E+02 1.18874E+00 1.4 6442E+02 378 2.03233E-01 5.09978E+15 2.11548E-01 2.28339E-01 2.44073E-01 2.03557E+16 8.08899E+15 1.2 8312E+16 2.59052E-01 3.22979E+16 2.74113E-01 5.12591E+16 2.88539E-01 8.12809E+16 3.01008E-01 1.13816E+17 3.07819E-01 3.0 9587E-01 1.44 932E+17 3.10890E-01 2.19119E+17 3.10057E-01 2.25772E+17 1.7 9770E+17 3.99056E-01 3.57287E+07 3.99072E-01 3.99068E-01 3.74 911E+07 3.94 453E+07 3.99038E-01 4.16361E+07 3.98981E-01 3.98896E-01 4.40323E+07 4.65505E+07 3.98783E-01 4.89519E+07 5.03615E+07 5.09994E+07 3.98673E-01 3.98574E-01 3.98471E-01 5.11322E +07 3.98361E-01 5.0744 6E+07 3.98342E-01 5.06369E+07 Ad TempGradGamma1 Sound SpeedBruntVSq 1.664 94E+00 1.664 92E+00 6.2 9191E-06 1.66486E+00 6.68388E-06 1.664 75E+00 6.64578E-06 1.664 60E+00 6.5354 7E-06 1.66439E+00 6.4 7782E-06 6.674 98E-06 1.66413E+00 7.43215E-06 8.02882E-06 1.66389E+00 1.66367E+00 7.924 49E-06 1.66343E+00 1.66319E+00 6.30116E-06 1.38195E-06 1.66315E+00 1.2 9244E-07 TABLE 3A—Continued Radius Frac Mass Frac Temperature Density Pressure Sound Speed Brunt V Sq Mean Mol Wt Luminosity Nuc Rate Opacity Temp Grad Ad Temp Grad Gamma 1 401 6.71531E-01 9.70925E-01 2.4 9952E+06 2.40853E-01 8.03535E+13 2.35634E+07 1.73071E-06 6.24068E-01 3.85148E+33 0.00000E+00 1.43997E+01 2.33075E-01 3.98409E-01 1.66427E+00

421 7.08787E-01 9.78588E-01 2.2277 9E+06 1.70405E-01 5.06653E+13 2.22449E+07 1.21843E-06 6.24068E-01 3.85147E+33 0.00000E+00 1.69323E+01 2.71692E-01 3.98422E-01 1.66431E+00

441 7.44 901E-01 9.844 97E-01 1.92278E+06 1.24514E-01 3.194 61E+13 2.06659E+07 5.84363E-08 6.24068E-01 3.8514 6E+33 0.00000E+00 2.182 62E+01 3.95556E-01 3.98612E-01 1.664 60E+00

461 7.77455E-01 9.88880E-01 1.60659E+06 9.4 9891E-02 2.03581E+13 1.88906E+07 -6.07092E-12 6.24068E-01 3.85145E+33 0.00000E+00 2.95143E+01 3.98930E-01 3.9892 9E-01 1.66506E+00

481 8.07434E-01 9.92209E-01 1.33662E+06 7.20190E-02 1.28389E+13 1.72307E+07 -8.7 6175E-12 6.24068E-01 3.8514 4E+33 0.00000E+00 3.7 9198E+01 3.99180E-01 3.9917 9E-01 1.66543E+00

501 8.34105E-01 9.94 620E-01 1.Ill95E+06 5.4 6092E-02 8.097 64E+12 1.57161E+07 -1.28669E-11 6.24 068E-01 3.85144E+33 0.00000E+00 4.94 738E+01 3.99371E-01 3.99370E-01 1.66571E+00

521 8.57 616E-01 9.96331E-01 9.24 996E+05 4.14128E-02 5.107 60E+12 1.43340E+07 -5.32387E-09 6.23808E-01 3.85144E+33 0.00000E+00 6.59040E+01 3.99516E-01 3.99514E-01 1.66592E+00

541 8.78165E-01 9.97527E-01 7.69453E+05 3.14349E-02 3.22181E+12 1.30674E+07 -4.6134 9E-08 6.24382E-01 3.85143E+33 0.00000E+00 8.962 68E+01 3.99566E-01 3.99564E-01 1.66608E+00

561 8.95972E-01 9.9834 9E-01 6.4 0093E+05 2.39014E-02 2.03237E+12 1.1902 6E+07 -9.65950E-08 6.2 6023E-01 3.85143E+33 0.00000E+00 1.25181E+02 3.99388E-01 3.99384E-01 1.66611E+00

581 9.11281E-01 9.98907E-01 5.32574E+05 1.81923E-02 1.28210E+12 1.08350E+07 -1.23613E-07 6.2842 IE-01 3.85143E+33 0.00000E+00 1.81058E+02 3.98847E-01 3.98842E-01 1.66582E+00

601 9.24365E-01 9.99282E-01 4.43277E+05 1.38420E-02 8.08827E+11 9.86350E+06 -1.03858E-07 6.3082 6E-01 3.85143E+33 0.00000E+00 2.82153E+02 3.97851E-01 3.97 84 4E-01 1.664 97E+00

621 9.35506E-01 9.99531E-01 3.69168E+05 1.05145E-02 5.102 69E+11 8.98455E+06 -4.03250E-08 6.3254 4E-01 3.85143E+33 0.00000E+00 5.04 975E+02 3.96355E-01 3.96345E-01 1.66334E+00

641 9.44 973E-01 9.99695E-01 3.07706E+05 7.97181E-03 3.21923E+11 8.18884E+06 -3.01789E-10 6.33589E-01 3.85143E+33 0.00000E+00 9.17142E+02 3.94172E-01 3.94158E-01 1.66054E+00

661 9.53106E-01 9.99804E-01 2.56160E+05 6.01517E-03 2.017 95E+11 7.45127E+06 -5.31966E-10 6.34 906E-01 3.85143E+33 0.00000E+00 1.63396E+03 3.90314E-01 3.90295E-01 1.65500E+00

681 9.59883E-01 9.99874E-01 2.14320E+05 4.55051E-03 1.27324E+11 6.78316E+06 -8.7 0517E-10 6.36895E-01 3.85143E+33 0.00000E+00 2.7 6241E + 03 3.83549E-01 3.83522E-01 1.644 42E+00

701 9.65615E-01 9.99919E-01 1.8004 6E+05 3.43406E-03 8.03361E+10 6.16725E+06 -1.43950E-09 6.39930E-01 3.85143E+33 0.00000E+00 4.43861E+03 3.72463E-01 3.72426E-01 1.62585E+00

721 9.70469E-01 9.99948E-01 1.5217 9E+05 2.58121E-03 5.06886E+10 5.60385E+06 -2.4122ÔE-09 6.44342E-01 3.85143E+33 0.00000E+00 6.73862E+03 3.57370E-01 3.57319E-01 1.59914E+00

741 9.74591E-01 9.99966E-01 1.2 9545E + 05 1.93037E-03 3.19823E+10 5.10251E+06 -4.034 69E-09 6.50130E-01 3.85143E+33 0.00000E+00 1.05191E+04 3.42563E-01 3.42493E-01 1.5714 4E+00

761 9.78104E-01 9.99978E-01 1.1087 6E+05 1.43751E-03 2.01795E+10 4.67282E+06 -6.67950E-09 6.56733E-01 3.85143E+33 0.OOOOOE+OO 1.674 95E + 04 3.35003E-01 3.34 906E-01 1.55547E+00

781 9.81103E-01 9.99986E-01 9.4 9835E+04 1.06935E-03 1.27324E+10 4.31055E+06 -1.10195E-08 6.63287E-01 3.85143E+33 0.00000E+00 2.63584E+04 3.38713E-01 3.38572E-01 1.56054E+00

379

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1992ApJ. . .387. .372G MeanMolWtLuminosityNucRateOpacityTempGradAdGamma1 Shell RadiusFracMassTemperatureDensityPressureSoundSpeedBruntVSq 1181 1161 9.98982E-019.99999E-01 1.41987E+041.83695E-062.01795E+061.15042E+06 -3.99531E-05 1141 9.98703E-019.99999E-011.51595E+042.62831E-06 3.19823E+061.21455E+06-2.61008E-05 1121 1101 9.98056E-019.99999E-011.72344E+045.42790E-06 8.03361E+061.35035E+06-1.14807E-05 1081 9.97682E-019.99999E-011.84010E+047.80424E-06 1.27324E+071.42466E+06-7.67077E-06 1061 1041 9.96815E-019.99999E-012.11312E+041.60461E-053.19823E+071.59287E+06-3.43856E-06 1021 9.96312E-019.99999E-012.27637E+042.29033E-055.06886E+071.68966E+06-2.30008E-06 1001 9.95752E-019.99999E-012.46323E+043.25604E-058.03361E+071.79683E+06-1.53435E-06 981 9.95127E-019.99999E-012.67904E+044.60835E-051.27324E+081.91561E+06-1.01872E-06 961 9.94427E-019.99999E-012.92993E+046.49198E-052.01795E+082.04705E+06-6.71923E-07 941 9.93638E-019.99999E-013.22298E+049.10248E-053.19823E+082.19287E+06-4.40212E-07 921 9.92746E-019.99999E-013.56762E+041.26985E-045.06886E+082.35721E+06-2.87331E-07 881 9.90570E-019.99998E-014.48136E+042.42230E-041.27324E+092.77320E+06-1.22461E-07 901 861 9.89229E-019.99997E-015.11054E+043.30045E-042.01795E+093.04068E+06-7.92287E-08 841 9.87665E-019.99996E-015.90808E+044.45295E-043.19823E+093.34606E+06-5.00334E-08 821 9.85827E-019.99994E-016.90634E+045.96497E-045.06886E+093.66990E+06-3.05247E-08 801 9.83658E-019.99991E-018.10910E+047.97557E-048.03361E+093.98896E+06-1.82600E-08 1 1.11839E+00 1.07471E+00 3.85143E+33 0.00000E+00 2.00974E+031.45253E-011.18248E-01 1.20476E+00 1.03587E+00 3.85143E+330.00000E+002.98850E+03 1.39966E-011.21148E-011.21228E+00 1.00067E+00 9.99233E-01 9.98395E-01 9.68208E-01 3.85143E+330.00000E+006.07578E+03 1.40478E-011.30652E-011.23201E+00 9.377 98E-013.85143E+330.00000E+008.70829E+03 1.44285E-011.36998E-011.24406E+00 8.81509E-01 3.85143E+330.00000E+001.84638E+041.57254E-011.53077E-011.27298E+00 9.08992E-01 9.97270E-01 8.55215E-01 3.85143E+330.00000E+002.63576E+041.66201E~011.62983E-011.29000E+00 8.30094E-01 3.85143E+330.00000E+003.65808E+041.76640E-011.74137E-011.30857E+00 8.06224E-01 3.85143E+330.00000E+004.99525E+041.88262E-011.86306E-011.32816E+00 7.8372 6E-013.85143E+330.00000E+006.64388E+042.00603E-011.99074E-011.34810E+00 7.62 688E-013.85143E+330.00000E+008.31830E+042.13537E-012.12345E-011.36859E+00 7.43128E-01 3.85143E+330.00000E+008.96821E+042.28118E-012.27187E-011.39200E+00 7.25101E-01 7.08871E-01 3.85143E+330.00000E+007.49210E+042.70891E-012.70300E-011.46312E+00 9.91732E-01 6.94 976E-013.85143E+330.00000E+006.38410E+043.00199E-012.99720E-011.51219E+00 6.83951E-01 3.85143E+330.00000E+005.46285E+043.28710E-013.28329E-011.55885E+00 6.75752E-01 3.85143E+330.00000E+004.66387E+043.46744E-013.46454E-011.58491E+00 6.69371E-01 3.85143E+330.00000E+003.70233E+043.47727E-013.47520E-011.57968E+00 © American Astronomical Society • Provided by the NASA Astrophysics Data System 3.85143E+33 3.85143E+33 9.99999E-01 9.99999E-01 3.85143E+33 9.99999E-01 3.85143E+33 9.99999E-01 0.00000E+00 0.00000E+00 1.32472E+04 1.61625E+04 0.OOOOOE+OO 1.96894E+04 3.97891E+04 0.00000E+00 TABLE 3A—Continued 3.77437E-06 1.29275E-06 4.27092E+03 1.22862E+03 1.26542E+04 1.12044E-05 8.4 6051E+04 1.7 6077E-04 380 5.06886E+06 1.27324E+06 1.57541E-01 1.38819E-01 2.017 95E+07 1.4 9924E-01 2.46590E-01 8.03361E+08 1.08689E+06 1.17260E-01 1.25355E-01 1.28075E+06 1.50503E+06 1.44442E-01 2.54 755E+06 2.45854E-01 -6.238 92E-05 -5.134 92E-06 -1.72509E-05 -1.87590E-07 1.19944E+00 1.22142E+00 1.257 68E+00 1.42246E+00 ^ ShellRadiusFracMassTemperatureDensityPressureSoundSpeedBruntVSq

19 92ApJ ; MeanMolWtLuminosityNucRateOpacityTempGradAdGamma1 1581 1.00064E+001.00000E+00 4.86107E+031.02519E-083.15565E+037.14881E+05 9.68521E-04 1561 1.00060E+001.00000E+00 4.86166E+031.36460E-084.20042E+037.15108E+05 9.68862E-04 1541 1.00055E+001.00000E+004.86261E+031.80872E-08 5.56809E+037.15313E+059.68631E-04 1521 1.00051E+001.00000E+004.86410E+032.38867E-08 7.35522E+037.15520E+059.67638E-04 1501 1.00046E+001.00000E+004.86647E+033.14434E-08 9.68632E+037.15762E+059.65581E-04 1481 1.00042E+001.00000E+004.87021E+034.12655E-08 1.27213E+047.16082E+059.61952E-04 1461 1.00037E+001.00000E+004.87613E+035.39924E-081.66645E+047.16546E+059.55934E-04 1441 1.00033E+001.00000E+004.88546E+037.04133E-082.17738E+047.17249E+059.46233E-04 1421 1.00029E+001.00000E+004.90014E+039.14680E-082.83691E+047.18337E+059.30851E-04 1401 1.00025E+001.00000E+004.92314E+031.18206E-073.68338E+047.20026E+059.06742E-04 1381 1.00020E+001.00000E+004.95894E+031.51665E-074.76031E+047.22639E+058.69512E-04 1361 1.00016E+001.00000E+005.01415E+031.92615E-076.11284E+047.26640E+058.12902E-04 1341 1.00012E+001.00000E+005.09808E+032.41105E-077.77978E+047.32662E+057.29226E-04 1321 1.00008E+001.00000E+005.22317E+032.95953E-079.78395E+047.41494E+056.09300E-04 1301 1.00004E+001.00000E+005.40473E+033.53976E-071.21090E+057.53952E+054.42588E-04 1281 1.00001E+001.00000E+005.65969E+034.09416E-071.46666E+057.70572E+052.13061E-04 1261 9.99948E-011.00000E+006.66630E+034.78023E-072.01795E+058.23619E+05-1.10080E-03 1241 9.99820E-011.00000E+009.59173E+035.14107E-073.19823E+058.86933E+05-3.90276E-04 1221 9.99653E-011.00000E+001.11261E+046.70827E-075.06886E+059.55180E+05-1.77152E~04 1201 9.99457E-019.99999E-011.22553E+049.20911E-078.03361E+051.02235E+06-1.01068E-04 1.31364E+00 3.85143E+33 0.00000E+00 7.76067E-033.34815E-043.97261E-01 1.66029E+00 1.31365E+00 3.85143E+33 0.00000E+00 9.38651E-035.38765E-043.97724E-01 1.66133E+00 1.31365E+00 3.85143E+330.00000E+001.13742E-02 8.64760E-043.98067E-011.66210E+00 1.31366E+00 3.85143E+330.00000E+001.38075E-02 1.38498E-033.98316E-011.66266E+00 1.31366E+00 3.85143E+330.00000E+001.67913E-02 2.21376E-033.98491E-011.66306E+00 1.31367E+00 3.85143E+330.00000E+002.04593E-02 3.53164E-033.98613E-011.66333E+00 1.31367E+00 3.85143E+330.000Q0E+002.49867E-025.62273E-033.98696E-011.66351E+00 1.31368E+00 3.85143E+330.00000E+003.06113E-028.93183E-033.98751E-011.66363E+00 1.31368E+00 3.85143E+330.00000E+003.76722E-021.41506E-023.98788E-011.66371E+00 1.31368E+00 3.85143E+330.00000E+004.66911E-022.23489E-023.98810E-011.66376E+00 1.31368E+00 3.85143E+330.00000E+007.43968E-025.49227E-023.98807E-011.66374E+00 1.31368E+00 3.85143E+330.00000E+005.84743E-023.51388E-023.98819E-011.66377E+00 1.31368E+00 3.85143E+330.00000E+009.66965E-028.50149E-023.98751E-011.66359E+00 1.31367E+00 3.85143E+330.00000E+001.29405E-011.29859E-013.98572E-011.66312E+00 1.31366E+00 3.85143E+330.00000E+001.80790E-011.95852E-013.98018E-011.66170E+00 1.31363E+00 3.85143E+330.00000E+002.69684E-012.94277E-013.96370E-011.65753E+00 1.31301E+00 3.85143E+330.00000E+001.34541E+009.92572E-013.75309E-011.60690E+00 1.28205E+00 3.85143E+330.00000E+006.11592E+014.43093E-011.80196E-011.26451E+00 1.22437E+00 3.85143E+330.00000E+002.80056E+022.45575E-011.31617E-011.20745E+00 1.16815E+00 3.85143E+330.00000E+006.64830E+021.83818E-011.19857E-011.19815E+00 © American Astronomical Society • Provided by the NASA Astrophysics Data System TABLE 3A—Continued 381 1992ApJ. . .387. .372G high inyoungopencluster Gstars(post-zero-agemain- increased depletionduringthepre-main-sequencephases. zone thanispresentinoldermodels,whichmightimply this impliesthat,inadditionto pre-main-sequencedepletion,a sequence stars)andprogressively decreasewithincreasedage; lithium depletioninsolar-type stars(seePinsonneaultetal. However, lithiumburninginthepre-main-sequencecannot be in ourimprovedsolarmodelcauseadeepersurfaceconvection stellar observationsplacestrong constraintsonthetimingof the solesourceforobservedsolarlithiumdepletionbecause potential lithiumdepletionmechanisms.Thehigheropacities problem,” whichsparkedanumberofinvestigations into depletion ofLiinsolarmodelsisinsufficienttoexplain the heimer 1965)indicatedthattheamountofpre-main-sequence observed degreeofdepletion,thuscreatingasolar“lithium than itistoday.Priorevolutionarycalculations(e.g.,Boden- (e.g., seethelistofabundancesinAnders&Grevesse1989). surface convectionzoneismuchdeeperandhotteratthebase During thepre-main-sequencephaseofevolution,solar depleted byafactorof200withrespecttothemeteoricvalue .Thesolarsurfacelithiumabundanceis 1989, 1990).Theobservedlithium abundancesarerelatively 382 Shell RadiusFracMassTemperatureDensityPressureSoundSpeedBruntVSq 1821 1801 1.00121E+00 1781 1.00115E+00 1761 1741 1721 1701 1681 1661 1641 1621 1601 © American Astronomical Society • Provided by the NASA Astrophysics Data System 1.31337E+00 1.00127E+00 1.31343E+00 1.31348E+00 1.00109E+00 1.31351E+00 1.31354E+00 1.00104E+00 1.31356E+00 1.00099E+00 1.31358E+00 1.00093E+00 1.00088E+00 1.31361E+00 1.00083E+00 1.31362E+00 1.31360E+00 1.0007 9E+00 1.31363E+00 1.00074E+00 1.31364E+00 1.00069E+00 Mean MolWtLuminosityNucRateOpacityTempGradAdGamma1 3.85143E+33 3.85143E+33 1.00000E+00 3.85143E+33 3.85143E+33 1.00000E+00 3.85143E+33 1.00000E+00 1.00000E+00 1.00000E+00 3.85143E+33 1.00000E+00 3.85143E+33 3.85143E+33 3.85143E+33 3.85143E+33 3.85143E+33 3.85143E+33 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 0.00000E+00 4.86005E+03 0.00000E+00 4.86006E+03 0.00000E+00 4.86006E+03 0.00000E+00 4.86007E+03 0.00000E+00 4.86007E+03 0.00000E+00 4.86009E+03 0.00000E+00 4.86011E+03 0.00000E+00 4.86015E+03 0.00000E+00 4.86021E+03 0.00000E+00 4.86031E+03 0.00000E+00 4.8604 6E+03 0.00000E+00 4.86069E+03 GUENTHER ETAL. TABLE 3A—Continued 2.092 67E-10 3.11347E-10 1.04891E-03 1.22118E-03 2.31917E-03 1.42824E-03 4.4 9969E-10 2.72599E-03 1.67 668E-03 3.21665E-03 2.30775E-09 5.40091E-03 6.37 442E-10 1.97459E-03 8.90109E-10 1.23030E-09 3.80979E-03 3.13629E-09 5.71564E-09 1.68965E-09 4.52840E-03 4.24287E-09 7.66917E-09 6.4 6258E-03 ionization, whichaffectF!in exactlytheoppositemannerto interactions oftheionsandelectrons,butapparentlythere are model areshowninFigure2. TheOPALopacitiesare30% that ofthefullyionizedDebye-Hiickel correctionwehave full andpartiallyionizedions)totakeintoaccountCoulomb equation ofstatedoesincludeaDebye-Hiickelcorrection (for interior. Atlowertemperaturesanddensities,as one used. rection wehaveapplied(whichassumesfullionizationof all approaches thesurfaceconectionzone,Debye-Hiickel cor- et al.(1988),thedominantcorrectiontoidealgasequation we canseethatindeed,asconcludedbyChristensen-Dalsgaard main-sequence depletionmechanism(notpresentinstandard other effectsorcorrections,equally important,possiblypartial species) doesnotfittheMHDequationofstate.The of stateistheDebye-Hiickelcorrection,atleastindeep state modelcanbedirectlycomparedtotheSTDmodel.Here Aller modelareidentical;henceFfortheMHDequation-of- shown inFigure1.^forboththeSTDmodelandRoss- solar models)isneeded. x The relativeopacitydifferences withrespecttotheSTD The effectoftheequation-of-statemodificationson^are 2.73280E-06 9.44271E-07 6.57 928E+01 1.62408E-06 9.71959E+01 1.39839E+02 4.53122E-06 3.79886E+02 5.21190E+02 3.13054E-05 5.03613E-05 2.08651E-04 2.36CS3E+03 1.97510E+02 1.20546E-05 7.11331E+02 1.94392E-05 8.09451E-05 1.30003E-04 1.30664E+03 1.75975E+03 9.66214E+02 . 43622E-06 , 75235E+02 3.60204E-01 3.69015E-01 7.07156E+05 7.08004E+05 3.75601E-01 7.0897 9E+05 3.80625E-01 7.0 9934E+05 3.84521E-01 3.87590E-01 7.10815E+05 7.11607E+05 3.90042E-01 3.92009E-01 3.93589E-01 3.94853E-01 3.95859E-01 3.9664 9E-01 7.12311E+05 7.12 927E+05 7.13460E+05 7.13914E+05 7.14296E+05 7.14 615E+05 1.59057E+00 9.3207 6E-04 1.60571E+00 9.37237E-04 1.61741E+00 1.62662E+00 1.63400E+00 1.63999E+00 9.424Ü9E-04 9.4 7169E-04 9.55432E-04 1.644 90E+00 1.64895E+00 9.51409E-04 1.65227E+00 1.654 98E+00 1.65718E+00 1.65892E+00 9.58525E-04 9.61204E-04 9.634 85E-04 9.66896E-04 9.6802 6E-04 9.65380E-04 TABLE 3B Summary of Combined Model Radius Frac Mass Frac Temperature Density Pressure Sound Speed Brunt V Sq Mean Mol Wt Luminosity Nuc Rate Opacity Temp Grad Ad Temp Grad Gamma 1 1 6.53350E-03 2.97851E-05 1.57209E+07 1.504 66E+02 2.334 81E+17 5.07 986E+07 1.2 6968E-07 8.59825E-01 1.01853E+30 1.77485E+01 1.28991E+00 3.34189E-01 3.98264E-01 1.66300E+00

21 2.24504E-02 1.1857 6E-03 1.55637E+07 1.45511E+02 2.2 6582E+17 5.08882E+07 1.38316E-06 8.4804IE-01 3.97 618E+31 1.70313E+01 1.30539E+00 3.35046E-01 3.98287E-01 1.66304E+00

41 6.16068E-02 2.15380E-02 1.45774E+07 1.18048E+02 1.86181E+17 5.12186E+07 6.25333E-06 7.83050E-01 6.204 62E+32 1.30174E+01 1.40810E+00 3.314 95E-01 3.98418E-01 1.66333E+00

61 8.89534E-02 5.67163E-02 1.35837E+07 9.57266E+01 1.50244E+17 5.10985E+07 8.03229E-06 7.32255E-01 1.39050E+33 9.61160E+00 1.52349E+00 3.26695E-01 3.98541E-01 1.66360E+00

81 1.14 635E-01 1.05361E-01 1.25670E+07 7.71027E+01 1.18090E+17 5.04813E+07 8.85445E-06 6.93365E-01 2.16121E+33 6.65358E+00 1.65584E+00 3.18738E-01 3.98657E-01 1.66385E+00

101 1.4 6205E-01 1.8194 6E-01 1.13214E+07 5.844 69E+01 8.44187E+16 4.90267E+07 7.37328E-06 6.61390E-01 2.93230E+33 3.772 66E + 00 1.842 68E+00 3.02463E-01 3.98785E-01 1.66413E+00

121 1.84 971E-01 2.92302E-01 9.90618E+06 4.08477E+01 5.32282E+16 4.65712E+07 6.88369E-06 6.40260E-01 3.4 9394E+33 1.61770E+00 2.11106E+00 2.7 6454E-01 3.98909E-01 1.66441E+00

141 2.20574E-01 3.99003E-01 8.77021E+06 2.87196E+01 3.35364E+16 4.4 0885E+07 7.05329E-06 6.3137IE-01 3.72355E+33 6.74338E-01 2.39728E+00 2.51360E-01 3.98990E-01 1.664 61E+00

161 2.54275E-01 4.96311E-01 7.84 642E+06 2.01457E+01 2.11348E+16 4.17909E+07 7.25614E-06 6.27644E-01 3.81177E+33 2.892 67E-01 2.72891E+00 2.31630E-01 3.99035E-01 1.66474E+00

181 2.86966E-01 5.82162E-01 7.07 660E + 0 6 1.4 0667E+01 1.33214E+16 3.97066E+07 7.25118E-06 6.2 6071E-01 3.8424 9E+33 8.2 6555E-02 3.11643E+00 2.16570E-01 3.99050E-01 1.66483E+00

201 3.19270E-01 6.56311E-01 6.42033E+06 9.77 621E + 00 8.39743E+15 3.78162E+07 6.97305E-06 6.25402E-01 3.84 911E+33 2.44382E-02 3.58345E+00 2.05868E-01 3.99043E-01 1.66487E+00

221 3.51632E-01 7.1942 6E-01 5.84 930E + 06 6.7 6948E+00 5.2 9389E+15 3.6082 9E+07 6.48362E-06 6.25116E-01 3.85100E+33 9.88234E-03 4.13363E+00 1.98335E-01 3.99018E-01 1.66488E+00

241 3.84379E-01 7.72573E-01 5.34513E+06 4.67355E+00 3.33755E+15 3.44809E+07 5.90005E-06 6.2 4 993E-01 3.85159E+33 4.06990E-03 4.77145E+00 1.92781E-01 3.9897 6E-01 1.66486E+00

261 4.17759E-01 8.16952E-01 4.89471E+06 3.21983E+00 2.10425E+15 3.2 984 9E+07 5.257 99E-06 6.24941E-01 3.85173E+33 1.66593E-03 5.52386E+00 1.89240E-01 3.98916E-01 1.66481E+00

281 4.51951E-01 8.537 4 9E-01 4.48744E+06 2.21544E+00 1.32 673E + 15 3.15742E+07 4.61557E-06 6.24 919E-01 3.85173E+33 6.74315E-04 6.41722E+00 1.87751E-01 3.98840E-01 1.66473E+00

301 4.87079E-01 8.84067E-01 4.11533E+06 1.52375E+00 8.36532E+14 3.02302E+07 4.00116E-06 6.24 910E-01 3.85169E+33 2.68740E-04 7.45926E+00 1.87865E-01 3.9874 9E-01 1.664 62E+00

321 5.23208E-01 9.08899E-01 3.77243E+06 1.04840E+00 5.274 62E+14 2.89384E+07 3.43638E-06 6.24 904E-01 3.85165E+33 1.04 992E-04 8.68140E+00 1.89910E-01 3.98645E-01 1.66450E+00

341 5.60338E-01 9.29114E-01 3.45233E+06 7.224 85E-01 3.325 91E+14 2.7 6800E+07 2.90728E-06 6.24 904E-01 3.85161E+33 3.98931E-05 1.01514E+01 1.952 91E-01 3.98534E-01 1.66437E+00

361 5.98377E-01 9.454 63E-01 3.14825E+06 4.99638E-01 2.09719E+14 2.64302E+07 2.41202E-06 6.24 903E-01 3.85158E+33 0.00000E+00 1.19245E+01 2.0554 9E-01 3.98427E-01 1.66425E+00

381 6.37092E-01 9.58587E-01 2.85286E+06 3.47708E-01 1.32243E+14 2.51580E+07 1.93230E-06 6.24 903E-01 3.85155E+33 0.00000E+00 1.40562E+01 2.23484E-01 3.9834 6E-01 1.66416E+00

© American Astronomical Society • Provided by the NASA Astrophysics Data System TABLE 3B—Continued Radius Frac Mass Frac Temperature Density Pressure Sound Speed Brunt V Sq Mean Mol Wt Luminosity Nuc Rate Opacity Temp Grad Ad Temp Grad Gamma 1 401 6.76044E-01 9.69025E-01 2.55583E+06 2.44761E-01 8.33901E+13 2.38113E+07 1.40566E-06 6.24 903E-01 3.85153E+33 0.00000E+00 1.66892E+01 2.56943E-01 3.9832 9E-01 1.66415E+00

421 7.14407E-01 9.77228E-01 2.23441E+06 1.7 6578E-01 5.25855E+13 2.22 632E+07 5.09837E-07 6.24 903E-01 3.85152E+33 0.00000E+00 2.08314E+01 3.43312E-01 3.98458E-01 1.66436E+00

441 7.50661E-01 9.83591E-01 1.86448E+06 1.33163E-01 3.30816E+13 2.03371E+07 -9.75953E-12 6.24 903E-01 3.85150E+33 0.00000E+00 2.98941E+01 3.98803E-01 3.98803E-01 1.66486E+00

461 7.83360E-01 9.88359E-01 1.55130E+06 1.00963E-01 2.08645E+13 1.85509E+07 -9.95317E-12 6.24 903E-01 3.8514 9E+33 0.00000E+00 4.06848E+01 3.99081E-01 3.99081E-01 1.66527E+00

481 8.12 67 9E-01 9.91869E-01 1.29062E+06 7.65575E-02 1.31601E+13 1.69208E+07 -1.13530E-11 6.24903E-01 3.8514 9E+33 0.00000E+00 5.34144E+01 3.992 96E-01 3.99295E-01 1.66559E+00

501 8.38708E-01 9.94400E-01 1.07368E+06 5.80561E-02 8.30122E+12 1.54334E+07 -1.39314E-11 6.24903E-01 3.8514 9E+33 0.OOOOOE+OO 6.92420E+01 3.99459E-01 3.99458E-01 1.66583E+00

521 8.61609E-01 9.96191E-01 8.93164E+05 4.40330E-02 5.23651E+12 1.4 0757E + 07 -1.134 60E-08 6.24 650E-01 3.8514 8E+33 0.00000E+00 7.50643E+01 3.99579E-01 3.99578E-01 1.66602E+00

541 8.81589E-01 9.97 437E-01 7.42 984E+05 3.34386E-02 3.30339E+12 1.28295E+07 -5.98949E-08 6.254 68E-01 3.85148E+33 0.00000E+00 1.022 92E+02 3.99574E-01 3.99573E-01 1.66614E+00

561 8.98870E-01 9.982 92E-01 6.18101E+05 2.54378E-02 2.08397E+12 1.16830E+07 -1.117 65E-07 6.27425E-01 3.85148E+33 0.00000E + 00 1.43309E+02 3.992 90E-01 3.9928 9E-01 1.66610E+00

581 9.13703E-01 9.98872E-01 5.1432 9E+05 1.93 681E-02 1.314 72E+12 1.0 6332E + 07 -1.32855E-07 6.30071E-01 3.85148E+33 0.00000E+00 2.08086E+02 3.98583E-01 3.98581E-01 1.66566E+00

601 9.26366E-01 9.992 60E-01 4.28173E+05 1.47366E-02 8.2944 6E+11 9.67 931E+06 -1.01382E-07 6.32572E-01 3.85148E+33 0.00000E+00 3.32 963E+02 3.97363E-01 3.97360E-01 1.66455E+00

621 9.37140E-01 9.99517E-01 3.56696E+05 1.11905E-02 5.232 98E+11 8.81731E+06 -2.93135E-08 6.342 67E-01 3.85148E+33 0.00000E+00 5.97177E+02 3.95592E-01 3.95587E-01 1.66255E+00

641 9.4 62 91E-01 9.99687E-01 2.97445E+05 8.48187E-03 3.30154E+11 8.03616E+06 -1.40331E-10 6.35397E-01 3.85148E+33 0.00000E+00 1.08507E+03 3.92 972E-01 3.92 964E-01 1.65909E+00

661 9.54078E-01 9.997 98E-01 2.4 8258E+05 6.41194E-03 2.07802E+11 7.31798E + 0 6 -2.79704E-10 6.36940E-01 3.8514 8E+33 0.00000E+00 1.90831E+03 3.88401E-01 3.88390E-01 1.65242E+00

681 9.60638E-01 9.99870E-01 2.07941E+05 4.84780E-03 1.31114E+11 6.66002E+06 -4.57088E-10 6.39273E-01 3.8514 8E+33 0.00000E+00 3.19603E+03 3.80569E-01 3.80555E-01 1.64000E+00

701 9.66189E-01 9.99916E-01 1.74971E+05 3.654 95E-03 8.27277E+10 6.05440E+06 -7.71869E-10 6.42757E-01 3.8514 8E+33 0.00000E+00 5.08558E+03 3.684 45E-01 3.68425E-01 1.61947E+00

721 9.70893E-01 9.9994 6E-01 1.48174E+05 2.74390E-03 5.21977E+10 5.50386E+06 -1.33862E-09 6.47644E-01 3.8514 8E+33 0.00000E+00 7.55044E+03 3.53313E-01 3.53286E-01 1.59239E+00

741 9.74891E-01 9.99965E-01 1.2 6330E+05 2.04 989E-03 3.2 9345E+10 5.01842E+06 -2.28912E-09 6.53781E-01 3.85148E+33 0.00000E+00 1.16738E+04 3.402 60E-01 3.40224E-01 1.56752E+00

761 9.78300E-01 9.99978E-01 1.08167E+05 1.52 610E-03 2.07802E+10 4.60431E+06 -3.7 9288E-09 6.60502E-01 3.8514 8E+33 0.00000E+00 1.84 834E+04 3.35643E-01 3.35591E-01 1.55691E+00

781 9.81209E-01 9.99986E-01 9.25856E+04 1.13607E-03 1.31114E+10 4.25184E+06 -6.Í2356E-09 6.67023E-01 3.8514 8E+33 0.00000E+00 2.90005E+04 3.41222E-01 3.41147E-01 1.56642E+00

384

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1 992ApJ.. . .387. .372G Shell RadiusFracMass 1181 9.98922E-019.99999E-01 1.35008E+041.28523E-061.31114E+061.10599E+06 -3.14057E-05 1161 1141 9.98382E-019.99999E-011.52551E+042.68108E-06 3.29345E+061.22065E+06-1.37143E-05 1121 9.98072E-019.99999E-011.62104E+043.87950E-06 5.21977E+061.28224E+06-9.16044E-06 1101 9.97733E-019.99999E-011.72449E+045.61010E-06 8.27277E+061.34804E+06-6.13881E-06 1081 9.97362E-019.99999E-011.83799E+048.09947E-06 1.31114E+071.41911E+06-4.12037E-06 1061 9.96954E-019.99999E-011.96395E+041.16647E-052.07802E+071.49659E+06-2.76647E-06 1041 1021 9.96009E-019.99999E-012.26534E+042.39501E-055.21977E+071.67586E+06-1.24305E-06 1001 9.95459E-019.99999E-012.44854E+043.41060E-058.27277E+071.78033E+06-8.29924E-07 981 9.94845E-019.99999E-012.65993E+044.83421E-051.31114E+081.89634E+06-5.51456E-07 961 9.94159E-019.99999E-012.90542E+046.81907E-052.07802E+082.02494E+06-3.64065E-07 941 9.93387E-019.99999E-013.19186E+049.57245È-053.29345E+082.16762E+06-2.38742E-07 921 9.92514E-019.99999E-013.52804E+041.33702E-045.21977E+082.32783E+06-1.55898E-07 881 9.90390E-019.99998E-014.41216E+042.55900E-041.31114E+092.72929E+06-6.63757E-08 861 9.89084E-019.99997E-015.01474E+043.49507E-042.07802E+092.98674E+06-4.29377E-08 841 9.87566E-019.99996E-015.77591E+044.72705E-043.29345E+093.28395E+06-2.71266E-08 821 9.85785E-019.99994E-016.73234E+046.34256E-045.21977E+093.60614E+06-1.65477E-08 801 9.83685E-019.99991E-017.89767E+048.48154E-048.27277E+093.92983E+06-9.92831E-09 901 9.91523E-019.99999E-013.92749E+041.85658E-048.27277E+082.51213E+06-1.01745E-07 1.06526E+00 1.03259E+00 3.85148E+330.00000E+002.41483E+03 1.31296E-011.21372E-011.21295E+00 1.00177E+00 3.85148E+330.00000E+003.52011E+03 1.32821E-011.25662E-011.22198E+00 1.10042E+00 3.85148E+33 0.00000E+00 1.05696E+031.36227E-011.15998E-01 1.19905E+00 9.98664E-01 9.72365E-01 3.85148E+33Q.00000E+005.08130E+03 1.36121E-011.30880E-011.23232E+00 9.44051E-01 3.85148E+330.00000E+007.28221E+03 1.40936E-011.37049E-011.24405E+00 8.90036E-01 8.64238E-01 3.85148E+330.00000E+002.08171E+041.63791E-Ô11.62090E-011.28865E+00 8.39322E-01 3.85148E+330.00000E+002.91427E+041.74150E-011.72833E-011.30672E+00 9.16638E-01 3.85148E+330.00000E+001.03648E+041.47160E-011.44242E-011.25727E+00 8.15428E-01 3.85148E+330.00000E+005.10734E+041.85635E-011.84609E-011.32590E+00 7.92728E-01 3.85148E+330.00000E+006.81365E+041.97851E-011.97052E-011.34555E+00 7.71358E-01 3.85148E+330.00000E+008.55861E+042.10603E-012.09980E-011.36565E+00 9.96505E-01 7.51382E-01 3.85148E+330.00000E+009.01018E+042.24641E-012.24157E-011.38800E+00 7.32856E-01 3.85148E+330.00000E+008.69601E+042.41907E-012.41526E-011.41627E+00 7.15996E-01 3.85148E+330.00000E+007.79717E+042.64428E-012.64124E-011.45385E+00 7.01283E-01 3.85148E+330.00000E+006.73448E+042.92222E-012.91976E-011.50038E+00 6.89287E-01 3.85148E+330.00000E+005.85515E+043.21049E-013.20852E-011.54786E+00 6.80174E-01 3.85148E+330.00000E+005.08225E+043.42337E-013.42185E-011.58015E+00 6.73231E-01 3.85148E+330.00000E+004.10897E+043.48151E-013.48041E-011.58333E+00 Mean MolWtLuminosity © American Astronomical Society • Provided by the NASA Astrophysics Data System 3.8514 8E+33 3.8514 8E+33 9.99999E-01 9.99999E-01 0.00000E+00 1.43589E+04 2.10523E+04 0.00000E+00 Nuc RateOpacityTempGrad Temperature DensityPressure TABLE 3B—Continued 1.62059E+03 1.85408E-06 1.47 953E+04 1.674 61E-05 385 2.07802E+06 1.32081E-01 3.2 9345E+07 1.54776E-01 1.1807 6E-01 1.16224E+06 1.52560E-01 1.58173E+06 Ad TempGradGamma1 Sound SpeedBruntVSq -2.06519E-05 -1.85608E-06 1.20524E+00 1.27212E+00 1992ApJ. . .387. .372G Shell RadiusFracMass 1581 1.00053E-I-001.00000E+00 4.63703E+038.46963E-092.48270E+036.98193E+05 1.01617E-03 1561 1.00049E+001.00000E+00 4.64003E+031.08431E-083.18017E+036.98442E+05 1.01275E-03 1541 1.00045E+001.00000E+004.64475E+031.38756E-08 4.07338E+036.98809E+051.00714E-03 1521 1.00041E+001.00000E+004.65217E+031.77470E-08 5.21789E+036.99371E+059.98346E-04 1501 1.0Ö037E+001.00000E+004.66376E+032.26827E-08 6.68533E+037.00242E+059.84812E-04 1481 1.00034E+001.00000E+004.68172E+032.89623E-08 8.56864E+037.01586E+059.64394E-04 1461 1.00030E+001.00000E+004.70922E+033.69194E-081.09866E+047.03641E+059.34385E-04 1441 1.00026E+001.00000E+004.75064E+034.69280E-081.40875E+047.06723E+058.91851E-04 1421 1.00022E+001.00000E+004.81160E+035.93823E-081.80547E+047.11232E+058.34514E-04 1401 1.00018E+001.00000E+004.89888E+037.46823E-082.31182E+047.17621E+057.61561E-04 1381 1.00014E+00l.OOOOOp+OO5.01998E+039.32159E-082.95685E+047.26345E+056.74599E-04 1361 1.00010E+001.00000E+005.18272E+031.15348E-073.77755E+047.37755E+055.76318E-04 1341 1.00005E+001.00000E+005.39372E+031.40977E-074.80489E+047.51900E+054.64080E-04 1321 1.00001E+001.00000E+005.65458E+031.68426E-076.01828E+047.68260E+053.30726E-04 1301 9.99955E-011.00000E+006.01409E+032.17658E-078.27277E+047.88937E+053.33897E-04 1281 9.99855E-011.00000E+007.51707E+032.75230E-071.31114E+058.38158E+05-8.47450E-04 1261 9.99720E-011.00000E+009.60036E+033.32288E-072.07802E+058.81207E+05-2.61656E-04 1241 9.99556E-011.00000E+001.08215E+044.49449E-073.29345E+059.38751E+05-1.31286E-04 1221 9.99367E-019.99999E-011.17847E+046.29213E-075.21977E+059.95444E+05-7.74141E-05 1201 9.99156E-019.99999E-011.26552E+048.95291E-078.27277E+051.05079E+06-4.85665E-05 1.31589E+00 3.85148E+33 0.00000E +001.32218E-025.42010E-Q43.98294E-01 1.66299E+00 1.31589E+00 3.85148E+33 0.00000E+00 1.63714E-028.57440E-043.98457E-01 1.66327E+00 1.31589E+00 3.85148E+330.00000E+002.02540E-02 1.35321E-033.98572E-011.66347E+00 1.31590E+00 3.85148E+330.00000E+002.50464E-02 2.12994E-033.98652E-011.66359E+00 1.31590E+00 3.85148E+330.00000E+003.09461E-02 3.33836E-033.98709E-011.66368E+00 1.31590E+00 3.85148E+330.00000E+003.82018E-02 5.20143E-033.98748E-011.66373E+00 1.31590E+00 3.85148E+330.00000E+004.71687E-028.04398E-033.98775E-011.66376E+00 1.31590E+00 3.85148E+330.00000E+005.83580E-021.23218E-023.98791E-011.66378E+00 1.31590E+00 3.85148E+330.00000E+007.23767E-021.86114E-023.98790E-011.66375E+00 1.31590E+00 3.85148E+330.00000E+008.99715E-022.75690E-023.98747E-011.66362E+00 1.31589E+00 3.85148E+330.00000E+001.11974E-013.98007E-023.98591E-011.66320E+00 1.31588E+00 3.85148E+330.00000E+001.39889E-015.59129E-023.98119E-011.66198E+00 1.31586E+00 3.85148E+330.00000E+001.80657E-017.82954E-023.96858E-011.65877E+00 1.31581E+00 3.85148E+330.00000E+002.52854E-011.13629E-013.94078E-011.65179E+00 1.31567E+00 3.85148E+330.00000E+004.55066E-012.19682E-013.88309E-011.63760E+00 1.31207E+00 3.85148E+330.00000E+004.28404E+008.34263E-013.12555E-011.47467E+00 1.27653E+00 3.85148E+330.00000E+005.44627E+013.33022E-011.63451E-011.24171E+00 1.22800E+00 3.85148E+330.00000E+001.74731E+022.10517E-011.28836E-011.20262E+00 1.18125E+00 3.85148E+330.00000E+003.70209E+021.66056E-011.18577E-011.19448E+00 1.13881E+00 3.85148E+330.00000E+006.54960E+021.45896E-011.15694E-011.19495E+00 Mean MolWtLuminosity © American Astronomical Society • Provided by the NASA Astrophysics Data System Nuc RateOpacityTempGrad Temperature DensityPressure TABLE 3B—Continued Ad TempGradGamma1 Sound SpeedBruntVSq 1992ApJ. . .387. .372G 6 4 5 6 model’s opacity.Intheoutermost layers,bothCoxandKurucz (below 10K).Notethatinordertocalculatetherelative due totheequallyabruptchangefromOPALopacitiesabove greater thantheLAOLopacities,usedinSTDmodel,near data foreachmodelonidenticalgridsinx.Thiswasdone differences inopacityforthisplot,wehadtoputthe low-temperature opacitiesshow asignificantbumpatx=0.99 tables beginattemperaturesbelow 10K(x=0.997).TheCox temperatures below10K(x =0.98),andtheKuruczopacity opacities departfromtheSTD model’sopacities(Cox& shooting atthejumpdiscontinuityinOPALopacity using cubicsplineinterpolationandisthecauseofover- the baseofconvectionzone.Theabruptdropatx=0.85 is Stewart). TheCoxlow-temperature opacitytablesbeginat 10 K,wherethetablesaredefined,toLAOLopacities Shell RadiusFracMassTemperatureDensityPressureSoundSpeedBruntVSq .1661 1861 1.00115E+00 1841 1821 1.00103E+00 1801 1781 1761 1.00089E+00 1741 1721 1701 1681 1641 1621 1601 © American Astronomical Society • Provided by the NASA Astrophysics Data System 1.31575E+00 1.31578E+00 1.00109E+00 1.31581E+00 1.31582E+00 1.00098E+00 1.31583E+00 1.00093E+00 1.31584E+00 1.31585E+00 1.00084E+00 1.31586E+00 1.00080E+00 1.31586E+00 1.0007 6E+00 1.31587E+00 1.00072E+00 1.31587E+00 1.00068E+00 1.31588E+00 1.00064E+00 1.31588E+00 1.00060E+00 1.31588E+00 1.00056E+00 Mean MolWtLuminosityNucRateOpacityTempGradAdGamma1 3.85148E+33 1.00000E+00 3.85148E+33 3.8514 8E+33 1.00000E+00 3.8514 8E+33 3.85148E+33 1.00000E+00 3.85148E+33 1.00000E+00 1.00000E+00 3.8514 8E+33 3.85148E+33 1.00000E+00 1.00000E+00 3.85148E+33 1.00000E+00 3.85148E+33 3.85148E+33 1.00000E+00 3.85148E+33 3.8514 8E+33 3.85148E+33 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 0.00000E+00 4.63187E+03 0.OOOOOE+OO 4.63188E+03 0.00000E+00 4.63189E+03 0.00000E+00 4.63190E+03 0.00000E+00 4.63192E+03 0.00000E+00 4.63195E+03 0.00000E+0Q 4.63200E+03 0.00000E+00 4.63207E+03 0.00000E+00 4.63219E+03 0.00000E+00 4.63238E+03 0.00000E+00 4.632 69E+03 0.00000E+00 4.63317E+03 0.00000E+00 0.00000E+00 4.63393E+03 4.63513E+03 STANDARD SOLARMODEL TABLE 3B—Continued 2.19589E-10 6.69186E-04 1.44393E-10 8.172 98E-04 3.16969E-10 1.00190E-03 1.23155E-03 4.42 684E-10 2.32072E-03 2.88229E-03 1.51735E-03 8.12581E-10 3.58426E-03 6.0457 6E-10 1.87383E-03 2.40730E-09 5.54312E-03 1.07 913E-09 1.41971E-0 9 1.85394E-09 3.11246E-09 5.15424E-09 4.45703E-03 8.58063E-03 4.01064E-09 1.06607E-02 6.61117E-09 6.89729E-03 2 the Brunt-Väisäläfrequency,N. units. Thecolumnidentifiedas“BruntVSq”isthesquare of STD modelandthecombinedmodel.Allentriesarein cgs low-temperature opacitiesaresimilartotheCoxopacities, surface. above x=0.997,showingthesamesharprise the and thenanothersharprisejustabovethesurface.TheKurucz them moresensitivetothe interior structurethanhigher/ because theirlowerturning points aredeeperwhichmakes pulsation program.Wehave focusedonthelow-/modes / =0,1,2,and3forallofthe solar modelsusingDBG’sstellar Tables 3AandBlistthebasicstructuralpropertiesofour We havecalculatedtheadiabatic p-modefrequenciesfor 4.81951E-07 4.34245E+01 8.86940E-07 2.67012E-06 6.54325E+01 1.56085E-06 2.38988E+02 1.30727E+02 4.48207E-06 9.39333E+01 1.7 8109E+02 7.42681E-06 3.17005E+02 3.23175E-05 5.437 96E+02 5.21496E-05 1.22001E-05 1.99158E-05 4.16690E+02 2.14555E-04 3.41593E-04 7.05788E+02 8.38077E-05 1.34294E-04 1.17528E+03 1.51031E+03 1.93740E+03 9.12247E+02 3.2. p-ModeFrequencyComparisons 3.63298E-01 3.72883E-01 3.7 9463E-01 6.94519E+05 3.84189E-01 6.94 601E+05 3.87689E-01 3.90335E-01 3.92362E-01 3.93925E-01 6.94 993E+05 6.95451E+05 3.95135E-01 3.96070E-01 3.96791E-01 6.95893E+05 6.96293E+05 3.97342E-01 3.97757E-01 3.98067E-01 6.96641E+05 6.96938E+05 6.97186E+05 6.97394E+05 6.97568E+05 6.97720E+05 6.97861E+05 6.98011E+05 1.60391E+00 1.61915E+00 1.00061E-03 9.96961E-04 1.6298 9E+00 1.00436E-03 1.63780E+00 1.00772E-03 1.64380E+00 1.01060E-03 1.64844E+00 1.012 99E-03 1.65207E+00 1.01653E-03 1.014 93E-03 1.654 91E+00 1.65713E+00 1.01783E-03 1.65887E+00 1.01879E-03 1.66022E+00 1.01935E-03 1.66124E+00 1.0194 7E-03 1.66201E+00 1.66258E+00 1.01914E-03 1.01814E-03 387 1992ApJ. . .387. .372G model, theKuruczlow-temperature“ molecular”opacitymodel,andtheIglesias&Rogers(OPAL)model. is particularlysensitivetothelocationofouierturning the veryoutermostlayers(x>0.99),evenforlow-/modes,and The first-orderspacingisprimarilysensitivetothestructureof extensive discussionofthisbehaviorinGuenther(1991b). and thesecond-orderfrequencyspacingisdefinedby versus Suncomparisonsmadehere,wereferthereadertoan the interior.Becausethisbehaviorisnotrelevanttomodel the developmentofasteepmeanmolecularweightgradientin model. Allothermodelslieontopofthestandard (STD), theMHDequationofstatemodel,andDebye-Hückelcorrection artifacts oftheradialnodenumbernclassificationschemeand model arelistedinTables2Aand2B.The0.000entrythe / =0columnatn5andinthe12are second-order' spacingsoftheSTDmodelandcombined layers only.Thefrequencies,thefirst-orderspacings,and modes, whoselowerturningpointsareconfinedtotheupper 388 Fig. 2.Therelativedifferencesinthe runofopacityareplotted,withrespecttothestandardmodel(STD), fortheCoxlow-temperature“molecular”opacity The first-orderfrequencyspacingisdefinedby Fig. 1.—F!isplottedoppositeradiusfractionxforthestandardmodel © American Astronomical Society • Provided by the NASA Astrophysics Data System ôv(n, l)=v(n,/)—v(n1,/+2). Av(n, /)=v(n,l)—v(n1, GUENTHER ETAL. effect onthep-modefrequenciesisadditive.Theof p-mode frequencies. the combinedmodel,forexample,couldhavebeencalculated in theouterregions,itdoesnothaveaslargeaneffecton effect, butbecausetheDebye-Hückelcorrectionisnotapplied frequencies isinthesamedirectionasMHDequation’s first-order frequencyspacing. high-n modesmorethanthelow-nmodes,hence,affect temperature opacities,tendtoperturbthefrequenciesof the sameamount,whereasmodificationswhichdoaffect surface layers,i.e.,KrishnaSwamyatmosphereandlow- to increaseordecreaseallofthefrequenciesbyapproximately we drawattentiontothefollowingresults. observations, isthecombinedmodel. plotted inbothsetsoffiguresforreference.BasedonFigure3, set ofmodels.InFigure3thedataforSTDmodelare bined modelincludesalloftheeffectsmodeledinsecond model, theKuruczlow-temperatureopacities model; andthesecondset,displayedonright,contains OPAL opacitiesmodel,andthecombinedmodel.Thecom- correction model,andtheCoxlow-temperatureopacities Bahcall nuclearreactionratesmodel,theKrishnaSwamy model, theMHDequationofstateDebye-Hückel on theleft,containsSTDmodel,Ross-Allermixture have dividedthemodelsintotwosets:firstset,displayed of Jiménezetal.(1987).Forplottingandlogisticpurposeswe tables ofLibbrechtetal.(1990)whichcontainthelow-/results ±10 /¿Hz,isproducedbythedifferentopacities. the frequenciesofsolarmodelandthoseSunfor / =0,1,2,and3.Theobservedfrequenciesweretakenfromthe stellar models,seeGuenther(1991b). detailed discussionofthesesensitivities,inthecontext1M larly sensitivetothelocationofinnerturningpoint.Fora structure oftheveryinnermostlayers(x<0.05)andisparticu- point. Thesecondorderspacingisprimarilysensitivetothe 0 5. Whentheindividualmodificationsarecombined,their 4. TheeffectoftheDebye-Hückelcorrectiononp-mode 2. Arguablythebestmodel,withrespecttomatching 3. Modificationswhichdonotaffectthesurfacelayerstend 1. Thelargesteffectonthefrequencies,oforder+5to In Figures3a-3h,weshowtherelativedifferencesbetween CMo oor" r" oo ft CN

15 20 25 30 15 20 25 30 Radial order n Radial order n

Radial order n Radial order n Fig. 3.—The frequencies of the low-/ p-modes of the models minus the Sun’s low-/ p-mode frequencies are plotted opposite the radial order n of the modes. The Ross-Aller mixture model, the MHD equation-of-state model, the Debye-Hiickel correction model, and the Cox low-temperature “ molecular ” opacity model are plotted in the series on the left. The Bahcall nuclear reaction rates model, the Krishna Swamy atmosphere model, the Kurucz low-temperature “ molecular ” opacity model, the OPAL opacity model, and the combined model are plotted in the series on the right. The combined model includes all the modifications plotted in the right-hand series. The frequencies of STD model, represented by a thick line, are plotted in both series.

© American Astronomical Society • Provided by the NASA Astrophysics Data System CMo oor" r" oo ft CN

Frequency [jiHz] Frequency [jliHz]

Fig. 4.—The first-order frequency spacings are plotted opposite frequency for the models and the Sun. The plots are divided into two series as described in Fig. 3.

© American Astronomical Society • Provided by the NASA Astrophysics Data System CMo oor" STANDARD SOLAR MODEL 391 oor" °? simply by adding the perturbations to the frequencies due to according to Cox et al. (1989, 1990) the nonadiabatic effects the Krishna Swamy model, the Kurucz model, the OPAL perturb all of the frequencies equally and, thus, should have ¿ia model,’ and the Bahcall nuclear rates model. little effect on the first-order spacings. CM Finally, we present the second-order spacings of the models 2 We conclude from Figure 3 that the errors associated with and the Sun in Figures 5a and 5b. The models are similar the physics modeled here can introduce up to ± 5 to ±10 //Hz enough that they cannot be distinguished in this plot. Only in perturbations to the calculated p-mode frequencies of low-/ the frequency range, 2300-4500 //Hz for / = 0 and 2300-3600 and that the sources of the error, from the set of effects sampled //Hz for the / = 1 do the second-order spacings match the here, in descending order of importance are the low- observations. The large scatter in the observational data at temperature opacities, the atmosphere model, the equation of higher frequencies suggests that their error may have been state and intermediate temperature opacities, the chemical underestimated and that systematic errors exist. If the observa- composition, and the nuclear reaction rates. The combined tional data are correctly represented then, as discussed in model falls within the ± 5 to ±10 //Hz errors of the frequencies Guenther (1991b), the discrepancies imply that the very core of of the solar model and is, therefore, representative of the best the Sun is less concentrated, and the immediately surrounding solar model that can be constructed without making use of ad regions more concentrated than specified by the standard solar hoc assumptions. Indeed, it is very satisfying that the combined model. We note that the observations of Pallé et al. (1989), are model, using the latest opacities, a reasonable model atmo- in better agreement with the models for / = 1 (see Fig. 6 in their sphere, and a well-determined mixture of elements does fit the paper) at higher frequencies. observations within understood errors associated with the We are reassured that there is now an acceptable and rea- physics of the model. sonable agreement between the Sun’s p-mode spectrum and Figures 4a-4h show the first order frequency spacings of the the spectra of theoretical solar models which include the latest models and the Sun. Here, as noted above, the greatest physics, at least with respect to the low-/ p-modes; and we note improvement is obtained from the low-temperature opacities that the agreement is not yet perfect and that there is still work and the Krishna Swamy atmosphere. Once again, the com- to be done, particularly in the area of opacities and equation of bined model falls satisfyingly close to the observations. A state. Because the opacities and equation-of-state calculations further modification, which we have not done, can be made by are being improved at this time, our combined model, or any including nonadiabatic effects in the oscillation calculations. It other recipe for a “ best ” model, is certainly going to improve is not clear what effect the inclusion of nonadiabatic physics over the next few years. will have on the p-mode frequencies of the models. According to the calculations of Ulrich & Rhodes (1983), the first-order 4. SUMMARY AND CONCLUSIONS frequency spacings increase by approximately 0.5 //Hz (for Recent advances in the basic physics of the solar interior I = 0) when nonadiabatic effects are taken into account. But now enable us to construct solar models whose p-mode oscil-

Fig. 5.—The second-order frequency spacings are plotted opposite frequency for the models and the Sun

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1992ApJ. . .387. .372G 7 5 Anders, E.,&Grevesse,N.1989,Geochim. Cosmochim.Acta,53,197 Bahcall, J.N.,&Pinsonneault,M.H. 1991,ApJ,submitted Bahcall, J.N.1989,NeutrinoAstrophysics (Cambridge:CambridgeUniv. Bahcall, J.N.,&Ulrich,R.K.1988,Rev. Mod.Phys.,60,297 Bahcall, J.N.,N.A.,&Shaviv, G.1968,Phys.Rev.Lett.,20,1209 perform themuchmoredifficulttaskofestimatingYin the derived withprecision.Notethatifiteverbecomespossible to bulk ofthemassSun,initialYSuncan be nosity, andageoftheSun,allwell-determinedparameters. points ofviewcosmologyandGalacticastronomyisa reli- equation ofstateincloseragreementwiththeMHD most stellarevolutioncalculationsandthatthefullyionized partial ionizationform)isthedominantcorrectionin outer layeroftheSunbyseismicmeanswithsufficientaccu- the Sunisveryrobust,asitreliesprimarilyonmass,lumi- We findY=0.28+0.01.Thismethodofdetermining for able determinationoftheinitialheliumabundanceSun. the deepinterior,whereitdoesbringstandardidealgas Debye-Hückel correction(usedhere)shouldonlybeappliedin that ourexistingequation-of-stateformulationisadequatefor regions inwhichtheymadetheircomparisons.Weconclude provided bytheOPALgroup.EventhoughMHDand Since Locju,where¡iisthemeanmolecularweightof the of state. authors suggestthattheDebye-Hückelcorrection(fulland tions theywerefoundtobeinverycloseagreement.The OPAL equationsofstatearederivedfromdifferentformula- Dalsgaard etal.(1988).Wenotethattheyalsocomparedthe results areinagreementwiththeconclusionsofChristensen- by A.N.Cox(1990)andR.L.Kurucz(1991),whichyield (which includemoleculareffects)kindlycommunicatedtous independently byinversion;andtheatmosphericopacities previous discrepancywiththedepthofconvectionzone, MHD equationofstatetoanotherdetailed the morecorrectfirst-orderspacings. which atx=0.724+0.005isnowclosetothatderived markedly thetheoreticalfrequencies,andremovemostof calculated byIglesias&Rogers(1991)atLivermoreforthe Anders &Grevesse(1989)elementmixture,whichimprove when takenindividually,necessarilyimprovesthedegreeof the increasedradiativeopacitiesbelowconvectionzone, responsible forresolvingthediscrepancybetweentheoryand agreement. Inotherwords,nosinglemodificationis observation. However,twofactorsdominateinimportance: the Sun’sinternalstructure. between theobservedandcalculatedp-modespectrumfor with whichweconcurreduntilrecently(see,e.g.,Guenther 392 sion thatwasfirstreachedbyUlrich&Rhodes(1983),and Sun, suggestingapossiblyseriousflawinourunderstandingof that thisstatementisanimportantdeparturefromtheconclu- standard oradhocassumptionsinthemodels.Weemphasize known uncertaintiesintheinputphysicsandsolarparameters. fidence thattheremainingdiscrepanciesbetweenourmodels within +0.2%.Moreimportantly,wecannowstatewithcon- lation spectrumreproducesthesolarlow-/p-modefrequencies In otherwords,itisnolongernecessarytointroducenon- and observationareatthelevelwhichisexpectedfrom 1991a; Kimetal.1991),thatthereisasignificantdiscrepancy conv Press) An importantby-productofthissolarmodelingfromthe With regardtoimprovementsintheequationofstate,our It isworthnotingthatnosingleimprovementinthephysics, © American Astronomical Society • Provided by the NASA Astrophysics Data System GUENTHER ETAL. 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