Pulsating sdB Stars A New Approach to Probing their Interior(6)

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

00.5100.5

100.5100.5100.51

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