Pulsating sdB Stars A New Approach to Probing their Interior(3)

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