3D Numerical Experimentation on the Core Helium Flash of low-mass Red Giants

arXiv:astro-ph/0512049v1 2 Dec 20053DNumericalExperimentationontheCoreHeliumFlash

oflow-massRedGiants

DavidS.P.Dearborn1,JohnC.Lattanzio2andPeterP.Eggleton1

1LawrenceLivermoreNationalLaboratory,7000EastAve,Livermore,CA94551,USA

2MonashUniversity,MathematicsDepartment,Clayton,Victoria,3168,Australia

dearborn2@llnl.gov,john.lattanzio@sci.monash.edu.au,ppe@igpp.ucllnl.org

ABSTRACT

Wemodelthecoreheliumflashinalow-massredgiantusingDjehuty,afully

three-dimensional(3D)code.The3Dstructuresweregeneratedfromconverged

modelsobtainedduringthe1Devolutionarycalculationofa1M⊙star.Indepen-

dentlyofwhichstartingpointweadopted,wefoundthataftersometransient

relaxationthe3DmodelsettleddownwithabrisklyconvectingHe-burningshell

thatwasnotverydifferentfromwhatthe1Dmodelpredicted.

Subjectheadings:stars:evolution

1.Introduction

Thecoreheliumflashisanimportanteventinthelifeofmoststarswithazero-age

massbetweenabout1and2M⊙;theminimummassesarealittlelowerformetal-poor

stars.SincetheworkofMestel(1952)andSchwarschild&H¨arm(1962)ithasbeenclear

thatsuchstarsigniteheliuminathermonuclearrunawaysituation,theheliumflash,because

theheliumcoreiselectron-degenerateatthetimeofignition.Empirically,itisclearthat

thisrunawayis(usually)notacatastrophicaffair,likeasupernovaexplosion,becausea

wholeclassofstars,thehorizontal-branchstarsofglobularclusters,iswellexplainedbythe

survivalofhelium-flashstarsinalong-livedstateofcoreheliumburning(Faulkner1966).

Nevertheless,attemptstocomputetheevolutionduringtheflashhaveaconfusinghistory:

somecalculations(both1Dand2D)havepredictedarathersevereexplosion,andothers

(both1Dand2D)arelativelybenignthoughrapidignition.

Mostcalculationsuntilfairlyrecentlyhavebeen1-dimensional(1D)simulations,in

whichturbulentconvectionhasbeentreatedbyasphericalaveragingprocessbasedlargely–2–

onthemixing-lengthconcept.Deupree(1996),whogivesanicesummaryofearlierwork,

performedsome2D(axiallysymmetric)simulations.Hefoundthatthe2Destimateswere

criticallydependentonapproximationsmaderegardingeddyviscosity.Weexpectthatby

workingin3Dwewillnotneedtomakesuchapproximations,andwesuggestthatourresults

bearoutthisexpectation.

TheDjehutyprojectoftheLawrenceLivermoreNationalLaboratoryisaneffortto

modelstarsin3D.Ourultimateaimistobeabletomodelanentirestar,uptoand

includingthephotosphere;andindeedtogeneralizethistobinarystars,includinggasflows

betweenthemin,forexample,aRoche-lobe-overflowsituation.Weareapproachingthis

goal,butitisfairlyeasytoseethatastarliketheSun,forinstance,wouldrequireatleast

1012nodesinsideitifthereistobeadequateresolutionnearthesurface.Ascomputer

powercontinuestoincreasethiswillnodoubtbecomepossible,butforthepresentwelimit

ourselvestoabout108nodes.WethereforecontentourselveswithasimulationoftheHe

flashthatincludesonlytheHecoreandtheradiativeportionoftheenvelope;weignorethe

deepsurfaceconvectionzone.

ApartfromtheintrinsicinterestoftheHeflash,ourotherreasonforpursuingthis

particularproblemisthatitpotentiallyisaverygoodtestforthestabilityandaccuracy

ofahydrodynamiccode.Thisisbecausewehave,asmentionedabove,arathergood

reasontoanticipatewhattheoutcomeshouldbe.Wedonotexpectittobecomeaviolent

supernova-likeevent.Infollowinganexplosion,itisnoteasytolookattheoutcomeand

say‘thatisclearlywhatshouldhavebeenexpected’,evenifitiswhatweexpected.Butin

anon-explosivesituationitisnotdifficulttocompare,forinstance,theheatfluxactually

carriedbyturbulentconvectionacrossasphericalshellwiththeexpectationfromasimple

mixing-lengthmodel.

Inthispaperweconsideronlynon-rotatingandnon-magneticcores,butwebelieve

thatboththeseprocessescouldbeimportantandwehopetoaddresstheminalaterpaper.

InSection2webrieflyoutlinethecode;inSection3wedescribe(a)our1Dinputmodels

and(b)our3Doutputmodels.InSection4wedescribesomeissuesandnumericaltests

regardingthestabilityofthecalculation.

2.CodeDescription

Djehutyisacodedesignedtomodelentirestarsin3D.DevelopedattheLawrence

LivermoreNationalLaboratory(LLNL),itoperatesinamassivelyparallelenvironment,

andincludesthebasicphysicsnecessaryformodelingwholestars.Earlierdescriptionsof–3–

theDjehutycodecanbefoundinBaz´anetal.(2003)andDearbornetal.(2005),buta

briefdescriptionwillbeprovidedhere.

Forastar,theDjehutymeshisformedfrom7logicallyconnectedblocksofhexahedral

cellsofvariableshape.ThereisacentralcubeofN×N×Ncells,surroundedbysixlogical

cuboids(N×N×L).OneofthetwoN×Nfacesofeachcuboidisattachedpointby

pointtothefaceofthecentralcube.TheotherN×Nfaceofeachcuboidismappedto

lieonasphericalsurfaceformingtheouterboundary.Thecuboidsarethen‘morphed’into

wedgeshapeswithsurfacesonthelong(L)radialaxistransitioningfromplanartospherical.

TheN×Lfacesaresimilarlyattachedtoadjacentcuboidsinapoint-by-pointfashion.An

explodedversion,withthelogicalstructureofthecentralcubeandsurroundingcuboids,is

showninFig.1withN=50andL=100;thelowerpartofFig.1showstwooftheblocks

connectedtogetherandinphysicalspace,withN=35andL=70.

Thismeshstructureallowsreasonableazimuthalresolutionwithoutthecoreconver-

genceproblem(tinyzonesandtinytimestepsatthecenter)endemicinsphericalcoordinate

systems.Inthecoreitself,thecellsarenearlyrectangular,andastheradiusgrowssuccessive

surfacesbecomemorespherical,matchingthepotentialsurfacesaswellaspropertieslike

temperatureandpressure.Themeshcanencompassanentirestarwithfreeouterbound-

aries,aswasdonebyDearbornetal.(2005),oraportionofthestarwithvariousfixed

boundaryconditions,aswillbedonehere.Forreasonsdiscussedlater,inthesecalcula-

tionswewilllocatetheouterboundaryattheinneredgeoftheredgiant’sdeepconvective

envelope.

Theinitial3Dstructuresaregeneratedfrommodelsproducedbya1Dhydrostatic

stellarevolutioncode.This1Dcodewasusedasaplatformtotestthephysics(equation

ofstate,nucleosynthesis,etc.)incorporatedintoDjehuty,aswellastoprovidestructure

informationforconstructing3Dmodels.Djehutyreadsthe1Dstellarmodels,mappingthe

definingphysicalparametersontoa3Dsphericalgridatanygivenstageofevolution.The

radialstructureofthe1Dmeshisusedasaguideinscalingthe3Dmesh,sothatregions

withsteepgradients,likethinburningshells,areresolved.Scalarquantitieslikepressure,

temperatureandcompositionareassignedtothecellcenters,andvectorquantities(position,

andvelocity)areassignedtonodes.Thepositionofthecellcenteristhedirectaverageof

the8radiusvaluesofthenodessurroundingit.

The1Dcodecarries6elements(1H,3He,4He,12C,14N,and16O)directly,withthe

remainderassumedtobe24Mg.Thissetofelementswasselectedtoallowaccuratetracking

oftheprincipalenergy-generationreactionsforhydrogenandheliumburningoverabroad

rangeofmassesandevolutionarystages.The7-elementsetincludesthetriple-alphareaction,

aswellasthealphacapturesoncarbonandnitrogen,toprovidetheenergyproduction,but