Pulsating sdB Stars A New Approach to Probing their Interiors

Horizontal branch stars should show significant differential rotation with depth. Models that assume systematic angular momentum exchange in the convective envelope and local conservation of angular momentum in the core produce HB models that preserve a ra

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aPulsatingsdBStars:ANewApproachtoProbingtheirInteriorsStevenD.KawalerandShelbiR.HostlerDept.ofPhysicsandAstronomy,IowaStateUniversity,Ames,IA50011USAAbstract.Horizontalbranchstarsshouldshowsigni cantdi erentialrotationwithdepth.Modelsthatassumesystematicangularmomentumexchangeintheconvectiveen-velopeandlocalconservationofangularmomentuminthecoreproduceHBmodelsthatpreservearapidlyrotatingcore.Adirectprobeofcorerotationisavailable.ThenonradialpulsationsoftheEC14026starsfrequentlyshowrichpulsationspectra.Thustheirpulsationsprobetheinternalrotationofthesestars,andshouldshowthee ingmodelsofsdBstarsthatincludeangularmomentumevolution,weexplorethispossibilityandshowthatsomeofthesdBpulsatorsmayindeedhaverapidlyrotatingcores.1.IntroductionFirststudiedindetailbyR.Petersoninthemid1980s(Peterson1983,1985a,b),therapidrotationofsomehorizontalbranchstarsremainsanenigmaticfeatureofthisphaseofevolution.ObservationsbyBehretal.(2000a,b)showagapinthehorizontalbranchofM15andM13,withtwodistinctrotationratesoneitherside.Coolerthanabout11000K,HBstarsfrequentlyshowrapidrotation(vsini≈40km/s)whilenorotationratesabove10km/sappearinthehotterstars.Oneclassof eldstarsisclearlyrelatedtohorizontalbranchstars.ThesubdwarfB(sdB)starsareveryhot(Te 25,000K),andshowhighgravity(logg≈5.3 6.1).Modelsproposedtoexplaintheiroriginincludecommonenvelopebinaryevolution(eg.Sandquistetal.,2000,Mengeletal.,1976),variationsofmasslosse ciencyontheRGB(D’Cruzetal.,1996),andevenmassstrippingbyplanetarycompanions

(Soker&Harpaz,2000).Someproposedoriginscenariosmakeclearpredictionsabouttheirrotationalproperties;forexample,thosethatinvolvebinarymergersorcommonenvelopeevolutionshouldspinupthesdBproduct.ExtrapolationoftheresultsforPop.IIstarssuggeststhatsdBstarsshouldbeslowrotators,withvsini 10km/s.TheworkbyHeberetal.(1999)hasshownthatatleastonestar,PG1605+072,isamorerapidrotator–asexpectedbasedonanasteroseismicanalysisofthepulsationsbyKawaler(1998).

Sills&Pinsonneault(2000)(hereafterSP2K)exploredHBrotationandsingle-starevolutionwithdi erentassumptionsaboutinternalan- c2008KluwerAcademicPublishers.PrintedintheNetherlands.

Horizontal branch stars should show significant differential rotation with depth. Models that assume systematic angular momentum exchange in the convective envelope and local conservation of angular momentum in the core produce HB models that preserve a ra

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gularmomentumtransport.Startingwithmodelsthatarerotatingassolidbodiesatthedeparturefromthemainsequence,SP2Kfollowedevolutionofthemodelsthroughthehelium ashandontotheHB.Theystudiedseverallimitingcasesofangularmomentumtransport,twoofwhichwefollowuponinourstudyofPop.IhotHBstars.Bothconservespeci cangularmomentum(angularmomentumperunitmass,j)inconvectivelystableregions.Inconvectiveregions,convectivemixingisassumedtoproduceeitherconstantangularvelocityintheconvectionzone(CaseB)orconstantj(CaseD);inbothcases,theangularmomentumcontainedwithintheconvectionzoneispreserved.

SP2Kconcludedthatthebest ttoobservationsofPop.IIHBrotationcamefromprecursorswithrapidlyrotatingcoresandconstantspeci cangularmomentuminconvectionzones(i.e.CaseD).ThesemodelscouldrotateattheratesseeninthecoolHBstars.TheslowrotationinthehotterHBstarsrequireschokingofangularmomentumtransportbychemicalcompositiongradientsproducedbydi usion-suche ectsareseeninHBstars(andmodels)attemperaturesabove11,000K(SP2K).AllrotatingmodelsofHBstarsshowrapidlyrotatingcores.Similarly,sdBstarsthatoriginatethroughbinarymergersorcommonenvelopeevolutionshouldshowpathologicalrotation.

AsubsetofsdBstarsaremultiperiodicpulsatingstars.Thepulsat-ingsdBVstars,iftheyrepresent“typical”sdBstarsand,byextension,typicalHBstars,shouldallowasteroseismologytoaddressthestateofinternalrotationofHBstars.Infact,onesdBVstar,PG1605+072,wasfoundtohaverapidrotationbasedontheasteroseismicanalysisbyKawaler(1998)–thiswaslaterpartiallyveri edbyHeberetal.(1999)throughmeasurementofrotationalbroadeningofspectrallines.Basedonthisinitialsuccess,itislikelythatasteroseismologycanbeausefultooltoexploreimportantquestionsofthesecondparameterprobleminglobularclustersandtheoriginsof eldsdBstars.

ThesdBstarsprobablyoriginatefromhighermassprogenitorsthanPopIIHBstars.StarsthatarenowsdBstarscouldhavehadmuchhigherinitialangularmomenta(Kawaler,1987).Therefore,wehavecomputedmodelsofrotatingRGBandsdBstarsusingPopulationIprogenitorswhichsamplethevariousstructuresthatmightberelevantforthepulsators.Welookattheasteroseismicconsequencesoftheserotationpro lesontheobservedsdBstarspulsations.

2.EvolutionaryModelCalculations

TheevolutionofourstellarmodelswascomputedusingISUEVO,astandardstellarevolutioncodethatisoptimizedforproducingmodels

Horizontal branch stars should show significant differential rotation with depth. Models that assume systematic angular momentum exchange in the convective envelope and local conservation of angular momentum in the core produce HB models that preserve a ra

InteriorsofsdBstars3

foruseinasteroseismologystudies.ISUEVOcomputesmodelsrangingfromtheZAMSthroughtheRGBandAGBandbeyondthathavebeenusedforstudiesofHBstars(Stobieetal.,1997;vanHoolstetal.,1999;Kawaler,1998;Kilkennyetal.,1999;Reedetal.,2003).

AswearemodelingPopulationIstars,weassumeametallicityZ=0.02andaninitialmixturethatissolar.Modelsequenceswerecom-putedwithinitialmassesof1.0,1.5,and1.8M⊙,andevolvedfromtheZAMSuntiltheheliumcore ash.Thecoremassesatthehelium ashforthesemodelswas0.4832M⊙,0.4798M⊙,and0.4652M⊙respectively(measuredtothemidpointofthehydrogen–burningshell).WefollowtheconventionaltechniqueofcomputingaZAHBmodel(Sweigert&Gross,1976).Forenvelopesthinnerthan0.0003M⊙,wehadtoreducethethicknessofthehydrogentoheliumtransitionzonetopreservethesurfaceabundanceofhydrogenasthepre- ashvalue.

Thetransportofangularmomentumwithinstarsthatareevolvingupthegiantbranchisdominatedbythedeepconvectiveenvelope.Thedistributionofspeci cangularmomentumjisa ectedonlybyslowdi usiveprocessesintheabsenceofconvection(Zahn,1992).Withinconvectionzones,wefollowtheprocedureofSP2Kandexaminethetwolimitingcasesofjtransportdescribedabove.Theangularvelocitypro lesinourmodelsrepresenttheextremeofnodi usionofj;inrealstars,theangularvelocitypro leswillbesimilartoorshallowerthanwhatwereport.Thetotalangularmomentumwithinradiativezonesisalsoanupperlimit,asdi usivetransportofjusuallydrainsangularmomentumfromthecoreintothemoreslowlyspinningenvelope.Wenotethatwecalculatetheangularmomentumpro leasaseparatesidecalculation,usingthelocalmomentofinertiaofeachmassshell.Sinceallangularvelocitiesaremuchsmallerthanbreakupvelocities,thestructurale ectsofrotationaresecond–orderperturbationsandcansafelybeignoredinthesepilotcalculations.

Weassumethattheinitialrotationonthemainsequencewasasasolidbody.TheinitialrotationratesforourmodelsaretakenfromKawaler(1987)–forexample,forthe1.8M⊙model,Kawaler(1987)impliesvrot=185km/s,Jinit=3.2×1050,and =2.4×10 5s 1.Giventhatourplanistousethesemodelsforasteroseismology,thestructureoftheinternalrotationpro leisthefocusofourstudyratherthanthepredictedsurfacerotationvelocities.

DuringearlyRGBevolution,theconvectiveenvelopedeepens.AsteepangularvelocitygradientdevelopsquicklyasthestarreachestheRGB,steepeningsigni cantlyinafewtimes107years.EvolutionuptheRGBmeansanincreaseinradius(andmomentofinertia)meaningthattheouterlayersslowdown.Atthesametime,thecoreiscontract-ingandthereforespinningup.ThusevolutionuptheRGBsteepens

Horizontal branch stars should show significant differential rotation with depth. Models that assume systematic angular momentum exchange in the convective envelope and local conservation of angular momentum in the core produce HB models that preserve a ra

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Figure1.Rotationpro lesforEHBmodels.Left:CaseBevolution.Right:CaseDevolution.Modelsatthreestagesofevolutionareshown;verticallinesatthebottomcorrespondingtothetransitionfrompureheliumtothehelium–depletedcore.Thepro lesshowverysubtlechangesintheouterlayerswithevolution.

theangularvelocitypro le.OntheRGBthedominantfeatureoftheinternalrotationpro leisthechangeinangularvelocitythatoccursnear0.27M⊙fromthecenterofthemodel.Therotationpro ledropsprecipitouslybeyondastheresultofdrainingofangularmomentumfromthecorebytheconvectiveenvelope.

HorizontalbranchstarsevolvedfromCaseBRGBmodelsshowmuchsmallersurfacerotationvelocitiesthanthosewithCaseDprecur-sors.Allmodelspreservearapidlyrotatingcorewithroughlyconstantangularvelocityouttoabout0.3M⊙.Therotationpro lesofmodelsdescendedfromCaseDRGBevolutioncontainrelicsofthehighspeci cangularmomentumofmaterialintheouterlayersoftheRGBprogen-itor.Theangularvelocitypro lechangeslittleduringEHBevolution-thecoredoesspindownabitastheangularmomentumismixed,buttheinnercorecontinuestospinaboutafactorof5slowerthanthefastestmaterial.Theangularvelocitydropsfromthemaximumbyafactorofabout10orsotothesurface.

Theinternalrotationpro leremainnearly xedthroughthestageofcoreheliumburning.AsshowninFigure1,therotationpro le(withradius)showsthesamegeneralfeaturesatthestartandendofHBevolution.Furtherevolutionresultsinlittleadditionalangularvelocitychangewithradius.Thisrelativelysmallchangeinangularvelocitypro leontheEHBcouldallowe cientparameterizationforstudiesoftheirseismicin uence.

Horizontal branch stars should show significant differential rotation with depth. Models that assume systematic angular momentum exchange in the convective envelope and local conservation of angular momentum in the core produce HB models that preserve a ra

InteriorsofsdBstars

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Figure2.EvolutionarytracksofEHBmodels.Modelsderivedfromthepost-helium ashcoreofa1.5M⊙.Alternatingtrackscanbedistinguishedbythedi erentlinetypes.Envelopemassesare0.00001,0.00004,0.00032,0.00122,0.00232,0.00322,0.00422M⊙,runningfromlowerlefttoupperright.DatafromKilkenny(2001).

3.AsteroseismologyProbes

NonradialpulsationsinsdBstarsprovideawindowintotheirinteriors,andmaytestthehypothesisthathorizontalbranchstarshaverapidlyrotatinginteriors.HereweillustratetheprocessbywhichwecandetectrapidrotationofthestellarinteriorusingcurrentlyobservedsdBVstars.Afuturepaperwilldescribeourasteroseismicresultsinmoredetailandapplytheanalysistoindividualpulsators.

Figure2showsanH–RdiagramthatincludestheknownsdBstarsandsampleevolutionarytracksfromoursetsofmodels.Giventheconvergenceofthetracksinthisplane,itisclearthatH–Rdiagrampositionaloneisinsu cienttojudgetheirevolutionarystate,withstarsthatareclosetotheZAEHBminglingwithstarsthathavedepletedheliumintheircores.ThelowestgravityEC14026starsfallsu cientlyfarfromtheZAEHBthattheycannotbecorehelium–burnersforcoremassesthatareconsistentwithsingle–starevolution.

Manyofthehigh–gravitypulsatingsdBstarsshowcomplexpulsa-tionsthatrequirealargenumberofperiodicities.ExamplesincludePG1047+003Kilkennyetal.(2001)andPG0014+067Brassardetal.(2001).Low nonradialmodeshavealimitedfrequencydistribution,meaningthatsomeofthepulsatorscannotbeunderstoodwithoutresortingtoeitherhighvaluesof ortorapidrotation.

Nonradialoscillationsarecharacterisedbytheeigenfrequencyσn mcorrespondingtoaspheroidalmodewithquantumnumbersn, ,andm.Thevalues andmrefertothesphericalharmonicY m.Thesign

Horizontal branch stars should show significant differential rotation with depth. Models that assume systematic angular momentum exchange in the convective envelope and local conservation of angular momentum in the core produce HB models that preserve a ra

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n=1 P=172S.Kawaler&S.Hostlern=2 P=148n=3 P=122n=4 P=108n=5 P=93.2

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Figure3.Kernelsforasampleoflow nonradialmodesinaCaseBsdBmodel.Thesolidlinesrepresentthekernels,andthedashedlineshowsthe(scaled)valueoftheangularrotationvelocityasafunctionoffractionalradius.Therotationalsplittingforeachmodeisgivenaboveeachpanel

ofmgivesthedirectionofpropagationofthesurfacerunningwave,andthereforethereare2 +1possiblevaluesofmforagiven .Forperfectsphericalsymmetry,theoscillationfrequenciesareindependentofmanddependonlyonnand .Butifrotationispresent,thevalueoftheoscillationfrequencywilldependonm.Forsolidbodyrotationtheresultofrotationistosplitmodesofdi erentvaluesofmbyaconstantfrequencythatisproportionaltotherotationrate.

Ifastarundergoesdi erentialrotation(i.e. = (r))thenthesplittingisgivenby

σn m=σn 0 m R

0 (r)Kn (r)dr(1)

whereKn (r)istherotationkernelcorrespondingtoamodewithnand .Theobservedsplittingforamodewithagivennand representsanaverageof (r)weightedbytherotationkernelKn (r)(Kawaleretal.,1999).Ifthekernelfunctionofamodeisnonzerowherethestarhasveryrapidrotation,thentheresultingsplittingcanrevealtheinnerrapidrotationdespiteslowrotationatthesurfacelayers.

Figure3showssamplerotationkernelsforansdBmodelrepresen-tativeofthepulsatingEC14026stars.ThemodesillustratedspantheperiodrangeseeninthepulsatingsdBstars.Notethathighernmodeshavemorepeaks,butthatinallmodesthekernelhasnonzerovalueclosetothestellarcore.AlsoindicatedinFigure3arelinesrepresentinglog (r)forCaseBevolutionofrotationinconvectionzones.NotethateventhoughtheamplitudeofKn issmallintheinnerregions,thelargevalueoflog (r)inthesameregionsuggeststhatthevalueoftherotationalsplittingwillbedominatedbythecorerotation.

Forseveral =2modesinthismodel,therotationkernelissig-ni cantinthecore.ThisresultsfromresonantmodetrappingbytheHe/C+Ocompositiontransitionzone.Atthisevolutionarystage,the

Horizontal branch stars should show significant differential rotation with depth. Models that assume systematic angular momentum exchange in the convective envelope and local conservation of angular momentum in the core produce HB models that preserve a ra

InteriorsofsdBstars7

modelshowssomemixed–charactermodesfor =2whichcanhavesigni cantamplitudeinthecore(Charpinetetal.,2002).

Figure3liststhecomputedsplittingsaCaseBsdBmodel.Thesurfacerotationvelocity,ifrepresentativeoftheentirestar,wouldproduceasplittingoflessthan0.01µHzforallmodes(surfacerota-tionvelocityofonly6m/s),yettherapidinternalrotationresultsinsplittingsrangingfrom1µHz( =1,n=1)to80µHz( =2,n=0).

Inallcases,therotationalsplittingismuchlargerthanexpectedfromthesurfacerotationrate.Also,thesplittingchanges(sometimesdramatically)fromonemodetothenextinasequenceofthesame butdi erentn.Thekernels(seeFigure3)sweepingacrossthesteepdropin producethese uctuations.Modetrappingbythecompositiontransitionzonesproducessomestrongcorelocalizationofthekernelsforcertainmodes,andthesealsoshowlargerrotationalsplitting.

4.Conclusions

ThesdBstarsmayretainrapidlyrotatingcoresasarelicoftheirevolutionontheRGB.Suchrapidrotationcouldserveasareservoirofangularmomentumwhich,whentapped,canproduceanomalouslyfastrotationatthesurfaceontheHB.

Nonradialpulsationscanrevealtherapidlyrotatingcoresviaro-tationalsplittingofnonradialmodes.Therotationalsplittingwillbemuchlargerthanexpectedfromthesurfacerotationvelocity.ThismaybehappeninginPG1605+072;therotatonalsplittingidenti edinthatstarbyKawaler(1998)isabout3timeslargerthanthevsinimeasuredviaspectroscopy(Heberetal.,1999).Also,thesplittingsseeninFeige48(Reedetal.,2003)aresigni cantlylargerthanexpectedfromtheupperlimitstovsinibyHeberetal.(1999).

Thefactthatthesplittingscanbequitelarge,coupledwiththelargedi erencesinexpectedsplittingsfrommodetomode,canmakeobservationalidenti cationofrotationalsplittingquitedi cult.Gen-erally,onelooksforasequenceofequally–spacedtripletsinaFouriertransform,andidenti esthemas =1forexample.Evenifallmembersofsuchtripletsarenotseen, ndingseveralpairsofmodessplitbythesameamountwouldsuggestrotationalsplitting.ForsdBVstars,withsigni cantmode–to–modedi erencesinsplittingsexpected,aseriesofrotationallysplitfrequencieswouldbeindistinguishablefrommodesofdi erent andn.Withoutconsideringdi erentialrotation,sucharichmodespectrumwouldrequireappealingtoothermechanisms.

Finally,wenotethattheperiodsseeninthePG1716+426(“Betsy”)starsareaboutthesameperiodsthatweexpectfortherotationrates

Horizontal branch stars should show significant differential rotation with depth. Models that assume systematic angular momentum exchange in the convective envelope and local conservation of angular momentum in the core produce HB models that preserve a ra

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ofthecoresofsdBstarsthatevolveassinglestarsfromtheRGB.Couldthismeanthatthemechanismresponsibleforthepulsationofthesestarsmightbeconnectedwithrotation(r-modesoroverstableconvectivemodes)?

Acknowledgements

WeappreciatesupportfromtheUSNationalScienceFoundationthroughGrantAST-0205983,andtoNASAforsupportthroughtheAstro-physicsTheoryProgramviagrantNAG5-8352toIowaStateUniversity.

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