Rate of Period Change as a Diagnostic of Cepheid Properties

DavidG.Turner

DepartmentofAstronomyandPhysics,SaintMary’sUniversity,Halifax,NovaScotiaB3H3C3,Canada

turner@crux.smu.ca

arXiv:astro-ph/0601687v2 20 Sep 2007MohamedAbdel-SabourAbdel-Latif

DepartmentofAstronomyandAstrophysics,NationalResearchInstituteOfAstronomyandGeophysics

(NRIAG),Box11242,Helwan,Cairo,Egypt

sabour2000@hotmail.com

and

LeonidN.Berdnikov

SternbergAstronomicalInstituteandIsaacNewtonInstituteofChile,MoscowBranch,13Universitetskij

prosp.,Moscow1199,Russia

berdnik@sai.msu.ru

ABSTRACT

˙foraCepheidisshowntobeaparameterthatiscapableofindicatingRateofperiodchangeP

theinstabilitystripcrossingmodeforindividualobjects,and,inconjunctionwithlightamplitude,likelylocationwithintheinstabilitystrip.Observedratesofperiodchangeinover200MilkyWayCepheidsaredemonstratedtobeingeneralagreementwithpredictionsfromstellarevolutionarymodels,althoughthesamplealsodisplaysfeaturesthatareinconsistentwithsomepublishedmodelsandindicativeoftheimportanceofadditionalfactorsnotfullyincorporatedinmodelstodate.

Subjectheadings:stars:Cepheids—stars:evolution

1.Introduction

Cepheidsrepresentabriefphaseinthepost-main-sequenceevolutionofstarsthatoriginallyhadmassesinexcessof∼31

inCepheidO–Cdiagrams(plotsofthedifferencesbetweenObservedtimesoflightmaximumandthoseComputedfromalinearephemeris)havebeenrecognizedforthepasthalfcenturyasev-idencefortheevolutionofsuchstarsthroughtheinstabilitystrip(Paranego1958;Struve1959;Erleksova&Irkaev1982).AsnotedbyStruve(1959),“Itappearsthatstudiesofperiodchangearebyfarthemostsensitivetestavailabletotheastronomerfordetectingminutealterationsinthephysicalcharacteristicsofastar.”

ObservationsofperiodchangesinCepheidshavebeenmatchedwithsomeconfidencetoevo-lutionarymodelsofmassivestarsinvariouscross-ingsoftheinstabilitystrip(e.g.,Turner1998;Turner&Berdnikov2001,2004)inordertoiden-tifythedirectionofstripcrossingforindividualvariables.Whenusedforsuchpurposes,thestudyofCepheidperiodchangesbecomesanimportanttoolforthecharacterizationofindividualmembersoftheclass.

InprincipleitshouldalsobepossibletouserateofperiodchangeforindividualCepheidstoes-tablishlikelylocationwithintheinstabilitystrip.BecausestripcrossingsforindividualCepheidsoc-curatdifferentratesandatdifferentluminosi-tiesforspecificstellarmasses,theobservedratesofperiodchangemustbecloselyrelatedtostripcrossingmodeandlocationwithintheinstabil-itystrip.Potentialconstraintsareimposedbyvariationsinchemicalcompositionandpulsationmode,e.g.,fundamentalmode,firstovertone,etc.(Berdnikovetal.1997;Turneretal.1999),aswellasbyourlimitedabilitytoestablishsmallratesofperiodchangeforO–Cdatacontainingsizeableobservationaluncertainties(Szabados1983).Inthispaperwedemonstratethelinkinmoredetail.2.

BasisoftheRelationship

ThelinkbetweenrateofperiodchangeinCepheidsandlocationwithintheinstabilitystripisillustratedwiththeaidofFig.1.Thedia-gramisatheoreticalHRdiagramthatdepictsthelocationoftheCepheidinstabilitystripac-cordingtotheparametersderivedforMilkyWayCepheids(Turner2001),alongwithGenevaevo-lutionarytracksforstarsof4,5,7,and10M⊙atZ=0.008fromLejeune&Schaerer(2001).Linesofconstantstellarradiusareshowncrossing

variousportionsoftheinstabilitystrip.Accord-ingtothewellestablishedCepheidperiod-radiusrelation,theyshouldrepresentlinesofconstantpulsationperiodforindividualCepheids.

¿FromanexaminationofFig.1itisclearthat,ifoneconsidersonlyCepheidsofaspecificperiodandinacommoncrossingoftheinstabilitystrip,thoseonthehotedgeofthestripmustbe∼20%moremassivethanthoseonthecooledgeofthestrip.Sincerateofevolutionincreasesinpropor-tiontothemassofastar,Cepheidslyingonthehotedgeofthestripareevolvingfaster,andhencechangingtheirpulsationperiodsatamorerapidrate,thanCepheidsofidenticalperiodlyingonthecooledgeofthestrip.Rateofperiodchangethereforerelatesdirectlytolocationwithinthein-stabilitystripforindividualCepheids.Differencesinstripcrossingmodesareonlyaminorconcern.Cepheidswithincreasingperiodsmustbeinthefirst,third,orfifthcrossingofthestrip,whereasCepheidswithdecreasingperiodsmustbeinthesecondorfourthcrossingofthestrip.

AminorcomplicationarisesfromrestrictionsonourabilitytoidentifyperiodchangesinCepheidstiedsolelytostellarevolution.SomeCepheidsexhibiterraticperiodchangesthatap-peartooriginatefromrandomfluctuationsinpul-sationperiod.SZTau(Berdnikov&Pastukhova1995),SVul(Berdnikov1994),andV1496Aql(Berdnikovetal.2004)areexcellentexamples,althoughinthefirsttwocasesitispossibletoidentifytheunderlyingevolutionarymodificationstopulsationperiod.

AstudybyBerdnikov&Ignatova(2000)maygivetheimpressionthatstellarevolutionhasonlyaminoreffectonCepheidO–Cdiagrams,sinceitnotesthatparabolictrendsweredetectedinonly67of230Cepheidssurveyed.Thatnumberismis-leading,however,giventhataprevioussurveybyTurner(1998)hadfoundparabolictrendsin137Cepheidsfromamuchsmallersample.ItwasactuallyintendedtoindicatethepoortemporalcoverageandlackofextensiveO–Cdataavailableformanywell-studiedGalacticCepheids,asitu-ationthathasbeenremediedinrecentyearsbyourongoingprogramtoobtainarchivaldataonCepheidbrightnessvariationsusingtheHarvardCollegeObservatoryPhotographicPlateCollec-tion.AtpresenttheparabolictrendsinO–Cdia-gramstypicalofstellarevolutionarefoundtobe2

extremelycommon.AsurveybyGlushkovaetal.(2005)citesatypicalfrequencyof∼80%inbothclusterandfieldCepheids,forexample,althoughtheir“anomalous”objectsincludeCepheidslikeSVVulinwhichtheevolutionarytrendisquitedistinct(Turner&Berdnikov2004).Amorereal-isticfrequencyforMilkyWayCepheidsdisplayingevolutionarytrendsisinexcessof∼90%.Formanyoftheremainingobjects,theevolutionarytrendsmaybemoreobviousinlongertimebase-linesoflightcurvecoverage.

AsalsopointedoutbyFernie(1990)andbyBerdnikov&Turner(2004),theO–Ctrendsin-dicativeofevolutioninCepheidsneednotbestrictlyparabolic.Iftherateatwhichamas-sivestarisevolvingthroughtheinstabilitystripisnotconstantwithtime,theO–CdatafortheassociatedCepheidvariablemaybebetterde-scribedbyathirdorfourthorderpolynomial.TheCepheidsYOph(Fernie1990)andWZCar(Berdnikov&Turner2004)aretwoobjects(ofseveralhundred)wherethatappearstobethecase.Suchcomplicationsmayaffectthederivedratesofperiodchange,butinmostcasesonlybysmallamounts.InthelargemajorityofstudiesofCepheidperiodchanges,thederivedrateofperiodchangereflectstheevolutionofthestarthroughtheinstabilitystrip(seeSzabados1983).3.

StellarEvolutionPredictions

Mostcomputationalevolutionarymodelsforevolvedstarsareusedforconstructingevolution-arytracksratherthantestingforpulsationinsta-bility.But,asnotedbycitetpa58,itispossibletousethebasicinformationtheyprovideongrad-ualchangesinluminosityandeffectivetempera-turetopredictexpectedratesofperiodchangeforCepheidsofdifferentperiod.Astartingpointisthewellknownperiod-densityrelation:

Pρ

1

21

=Q,

3π)

2

wherePisthepulsationperiod,ρisthedensity,Misthestellarmass,Risthestellarradius,andQ,thepulsationconstant,hasasmallperiodde-pendence(e.g.,Kraft1961;Fernie1967)thatwe

assumeherevariesasP

1

624P

=

L

−

T

.

Thedesiredquantity,therateofperiodchangeP

˙,isobtainedfromtabulateddifferencesinstellarlu-minosityandeffectivetemperatureasafunctionofageasamodelstarevolvesthroughtheinstabilitystrip.

ForofP

˙thepresentstudywecalculatedvaluesfromtheaboverelationshipusingcom-putationalstellarevolutionarymodelsfromavarietyofavailablepublishedsources,namelyMaeder&Meynet(1988),Alibertetal.(1999),Lejeune&Schaerer(2001),andClaret(2004).Thepublisheddatawereusedtocomputedif-ferentparameters,dependingupontheavailabil-ityofthenecessaryinformation.Alibertetal.(1999)citeparametersforstarsofdifferentmassreachingthehotandcooledgesoftheinstabil-itystrip,sotheirdatayieldinformationonlyaboutratesofperiodchangenearthecenterofthestrip.Inothercases,suchasClaret(2004),thereissufficienttimeresolutionintheoutputparameterstotrackchangesinpulsationpe-riodacrossindividualinstabilitystripcrossings.Fortheremainingsources(Maeder&Meynet1988;Lejeune&Schaerer2001),includingClaret(2004),wecalculatedratesofperiodchangefortheintersectionoftheevolutionarytrackswiththeobservationallydelineatedboundariesoftheinstabilitystripdefinedempiricallybyTurner(2001),whichareclosetothosepredictedbymod-elsofpulsationinstability(Alibertetal.1999),aswellasforpointslyingwithinthestripbound-aries.Pulsationperiodswereestablishedusingtheperiod-radiusrelation(Turner&Burke2002).Thepresentresultsdifferfromthoseobtainedear-lier(Turner&Berdnikov2001,2003)inbeingtiedtoalargervarietyofmodelswithagreaterrangeofmetallicity,andbytheinclusionofaweakperioddependenceforQintheperiod-densityrelation.ThecomputedresultsonratesofperiodchangeareplottedinFig.2foralloftheaccessiblemod-els.Differentsymbolsdenotethedifferentsources.ValuescalculatedfromthemodelsofAlibertetal.(1999)areplottedusingfilledcircles,whileothersareplottedusingopencircles.Plussignsindi-cateresultscalculatedforstarsevolvingthrough3

thehotandcooledgesoftheinstabilitystrip,withtherateofperiodchangeingeneralbeinglargeronthehotedgeoftheinstabilitystrip,i.e.,formoremassivestars.Largesymbolsdenotestarsofsolarmetallicity,Z=0.02,intermediate-sizedsymbolsdenotestarswithmetallicitiesofZ=0.01andZ=0.008,andsmallsymbolsdenotestarsofverylowmetallicity,Z=0.001andZ=0.004.Lineshavebeendrawntoenclosethoseregionswithinwhichtheresultsfordifferentcrossingmodesap-peartocluster.Sequencesofpointsindicatemod-elsforwhichthetimeresolutionwasfineenoughtocalculaterateofperiodchangeovertheentirecrossingoftheinstabilitystrip.

ThedistributionofdatapointsinFig.2sug-gestsavarietyofdifferentconclusionsregardingthemodels.First,thedifferentmodelsfortherapidfirstcrossingoftheinstabilitystripareinverygoodagreement,anddisplayverylittlevaria-tionwithmetallicity.Thefirstcrossingofthestripisarapidtransitionforallstars,regardlessofin-dividualdifferencesinrotationrate,etc.,andthatisevidentfromthemodels.Evidentlythecom-putationalcodesusedforcalculatingthephasesofshellhydrogenburninginstars,whileperhapsdifferingindetailfromonesourcetoanother,gen-eratenearlyidenticalresults,thesmallvariationinrateofperiodchangeatspecificpulsationpe-riodarisingfromthefinitewidthoftheinstabilitystripandthefactthatmoremassivestarscrossthestripatagreaterluminosityandatafasterratethanlessmassivestars.Forstarsinthefirstcrossingofthestrip,highrateofperiodincreaseatspecificpulsationperiodcorrespondstostarsonthehotedgeofthestrip,lowrateofperiodincreasetostarsonthecooledgeofthestrip.Negativeperiodchangesariseduringthesecondcrossingoftheinstabilitystrip,whichoccursdur-ingtheblueloopphaseofstellarevolutionfollow-ingtheonsetofcoreheliumburning.Theextentoftheblueloopcandependuponavarietyoffactors(see,forexample,Becker1985;Xu&Li2004),suchasmetallicity,thetreatmentofcoreover-shooting,andthedistributionofCNOelementsthroughoutthestar.Allfactorsaffecthowfarastarenterstheinstabilitystripduringcorehe-liumburning,andpresumablyaffectshowrapidlyitevolveswithinthestrip.Giventhepotentiallylargedifferencesininitialconditionsforsuchstarsasmain-sequenceobjects,forexample,largevari-ationsininitialrotationrate,onemightexpectrealstarstodisplaylargevariationsinhowfartheypenetratetheCepheidinstabilitystripmascoreheliumburningobjects.Somewhatunexpect-edly,therearealsoverylargevariationsamongthemodelsstarsaswell.

Evidently,metallicityplaysonlyaminorroleingoverningtherateatwhichstarstraversetheinstabilitystrip.Thereisasmuchdependenceonthespecificsofthestellarevolutionarycodeused.ThemodelsofAlibertetal.(1999),forexample,generatefasterratesofperioddecreasethandoothermodels,despitetheuseofcommonopacitytables.Modelsfromindividualsourcesareatleastinternallyconsistentintheirpredictionsforstarsofdifferentmassesandforstarsinallportionsofthesecondstripcrossing.Theratesofperioddecreaseduringindividualstripcrossingsarealsoverysimilartothevariationspredictedontheba-sisofmassdifferences,i.e,predictedvariationsinrateofperioddecreaseataspecificpulsationpe-riodaregenerallysmall,exceptforlongperiodCepheids.

Thethirdcrossingoftheinstabilitystripoc-cursduringthelatestagesofcoreheliumburn-ing,andgivesrisetoperiodincreases,forwhichthepredictedratesaredepictedinthetopportionofFig.2alongwiththoseforthefirstcrossing.Mostofthecommentsregardingthesecondcross-ingofthestripapplyequallytothethirdcrossing.Again,metallicityseemstoplayalessimportantroleinthepredictedratesofperiodincreasethandifferencesintheevolutionarycode.ThemodelsofAlibertetal.(1999)predictfasterratesofpe-riodchange(periodincreasesinthiscase)thandoothermodels,althoughwithlessconsistencyforstarsofdifferentmass.Theratesofperiodincreaseduringindividualstripcrossingsarealsosimilartothevariationspredictedonthebasisofmassdifferences,andpredictedvariationsintherateofperiodincreaseataspecificpulsationpe-riodaregenerallysmall.

Awell-knownproblemarisesforlow-massstarsinthesecondandthirdcrossingsoftheinstabilitystrip,sincetheblueloopphasesofevolutionarymodelsforstarsofsolarmetallicity,Z=0.02,donotenterthestripforM<4.75M⊙(seeAlibertetal.1999).Modelstarsoflowermetal-licitycantraversethestripatsmallermasses,butoftenonlyonthecooledge.Byinference,

4

mostclassicalCepheidsofnear-solarmetallicityshouldhavepulsationperiodsinexcessof∼31

tonepulsators.

2

days.Manymaybeover-

TheobservationalpictureisillustratedinFig.3,whichpresentsavailabledataonpe-riodchangesforover200Cepheids,asobtainedfromtheliterature(Berdnikov&Pastukhova1994a,b,1995;Berdnikovetal.1997;Turner1998;Berdnikov&Ignatova2000;Berdnikovetal.2003)andongoingresearchstudiesbytheauthors(e.g.,Berdnikov&Turner2004;Berdnikovetal.2004).TherelationshipsplottedinFig.3depictthere-gionswithinwhichthemodelcalculationsappeartocluster.

Ithasbeenpointedoutpreviously(e.g.,Szabados1983;Fernie1984;Turner1998)thattheobservedratesofperiodchangeinCepheidsaregenerallyagoodmatchtopredictionsfromstellarevolution-arymodels.ThedataofFig.3providefurtherconfirmationofthatconclusion.Moreover,threefurtherconclusionscanbereached.First,onceconsiderationistakenoftheexpectedchangesarisingfromevolutionthroughtheinstabilitystrip,theobservedperiodchangesinCepheidsareunlikelytocontainanysizablecomponentarisingfromanothersource.Thereareonlyafewexcep-tionstosuchaconclusion,andtheyareratherunusualobjectslikeV1496Aql(Berdnikovetal.2004),whichexhibitsperiodchangesdominatedbyrandomfluctuationsinpulsationperiod.

Second,theobservedperiodchangesinCepheidsdeviateinsmallbutimportantwaysfromwhatisexpectedaccordingtopredictionsbaseduponspe-cificstellarevolutionarymodels.ThemodelsofAlibertetal.(1999),forexample,predictfastersecondandthirdcrossingsofthestripthanthoseobserved,andatmuchdifferentrates.Incontrast,theobservedperiodchangesinCepheidsareverysimilarforobjectslikelytobeinthesecondandthirdcrossings.ThemodelsofClaret(2004)aremostconsistentwithobservationsinthatregard,butitisnecessarytohaveamorecompletemassgridofmodelsconstructedinthesamemannertomakeamoredetailedcomparison.

Third,therangeinobservedratesofperiod

5

changeformostCepheidsissmallerthanthatresultingfromacomparisonoftheresultsfromdifferentevolutionarymodels.Thatissomewhatsurprising,givenourpreviousdiscussionaboutpotentiallywidevariationsininitialconditionsforCepheidpredecessors.EvidentlyrealstarsaresimilarenoughintheirinternalcharacteristicsthattheyevolveatfairlysimilarratesthroughtheCepheidinstabilitystrip.

TheproportionsofCepheidsindifferentcross-ingmodesandindifferentperiodrangesinFig.3arealsoreasonablyconsistentwithevolutionaryexpectations.Forexample,starsinthefirstcross-ingoftheinstabilitystripduringshellhydrogenburningareevolvingabouttwoordersofmagni-tudefasterthanstarsinsecondandthirdcross-ings,sotheirrelativenumbersshouldbesmall.ThetwoCepheidsinFig.3undergoinglargeratesofperiodincreaseandfallinginthepredictedre-gionforfirstcrossersarePolaris(αUMi)andDXGem.Weassumethatbotharefirstcrossers,aswasalsoarguedforPolarisbyTurneretal.(2005).Moreover,theobservedrateofperiodchangeforPolarisisnowseentobeexactlywhatstellarevo-lutionarymodelspredictforastarlyingonthecooledgeoftheinstabilitystripforfirstcrossers.TheproportionofCepheidswithdetectableparabolictrendsintheirO–Cdataalsoincreasesnoticeablytowardsshortpulsationperiods,whichisagainconsistentwiththeevolutionaryexpec-tationthatthemostabundantpulsatorsmustbethoseevolvingmostslowlythroughtheinstabilitystrip.Thereisacuriousanomalyinthedistribu-tionofshortperiodCepheids,whereessentiallynovariablesarefoundtohaveratesofperiodchangeaspredictedforstarsinsecondandthirdcrossingsofthestripatP≤3.5days(logP≤0.55).Suchstarshaveprogenitormassesoflessthan∼4M⊙(Turner1996),wherestellarevolutionarymodelsforsolarmetallicitystarspredictthattheevolu-tionarytracksforcoreheliumburningstarsshouldnolongerenterthestrip.Theshortperiodcutoffintheobservationalsampleisthereforeconsistentwithexpectationsfromstellarevolutionarymod-els.ButtheexistenceofstarsofP≤3.5dayswithratesofperiodchangeroughlyanorderofmagnitudefasterthanpredictedforstarsinsec-ondandthirdcrossingsoftheinstabilitystripisnot.Theuncertaintiesintheobservedratesofpe-riodchangeintheanomalousobjectsaregenerally

muchtoosmalltoresolvetheanomalybyinvoking

systematicerrorsinthevaluesofP

˙.AnadditionalfactorthatcanbeimportantforshortperiodCepheidsisovertonepulsation.TheCepheidsintheobservationalsamplehaveallbeenassumedtobefundamentalmodepul-sators,andrequireadisplacementof+0.15inlogPtoestablishtheirproperlocationsinFig.3iftheyareovertonepulsators.Yettheapplica-tionofsuchcorrectionstoalloftheanomalousobjectsdoesnotaffecttheirdistributionsignifi-cantly;moststillfalloutsidetheregionofP

˙-spacepredictedforstarsinthesecondandthirdcross-ingoftheinstabilitystrip.Currentstellarevolu-tionarymodelsarethereforeunabletoexplaintheexistenceofsuchstars,whichsuggeststhatthemanneroftreatingthedetailsofstellarevolutionduringblueloopstagesisveryimportant(seealsoXu&Li2004).Thatisoneareawhereimprove-mentstotheobservationalsampleonCepheidpe-riodchangescanplayanimportantroleintestingtheresultsfromstellarevolutionarymodels.4.

P

˙asaFundamentalParameterInFig.3riodchangeP

˙thedispersionintheratesofpe-observedinlongperiodCepheids(P>10d)issmallerthanwhatisobservedforthecalculateddispersioninthatparameteramongdifferentstellarevolutionaryexpectP

˙models.Onemight

tocorrelatecloselywithlocationintheinstabilitystripforCepheidsinallstripcrossings,accordingtotheresultsofFig.1.Itisinformativetoexaminetheobservationaldatamorecloselytodetermineifthatisthecase.

Asafirststep,wenotethattheobservedratesofCepheidperiodchangeplottedinFig.3fallmainlywithinspecificbandsdelineatedbylinearmarginsofslope3.0separatedbyannituderangeinP

˙orderofmag-.Fig.4isaseparateplotofthedatathatdisplayssuchempiricallydefinedmar-gins.AllbuttwoofthelongperiodCepheidswithincreasingperiodsfallwithinthelowersetofmar-gins,asdothemajorityofshortperiodCepheidswithincreasingperiods.Cepheidswithdecreas-ingperiodsdisplayagreaterdispersioninP

˙thatmaybeintrinsic,ormaybecausedbylargerun-certaintiesinP

˙forthestars,particularlythosewithsmallratesofperiodchange.

TheanomalyforCepheidswithP≤3.5days

(logP≤0.55)isagainapparentinFig.4.AllCepheidsofshorterperioddisplayfasterratesofperiodchangethanistypicalofvariablespopu-latingthelowerband,andthereareanumberofstarsoflongerperiodalsofallinginthisregionofrapidperiodchange.Presumablythosestarsrep-resentCepheidsinfourthandfifthcrossingsoftheinstabilitystrip,withfasterassociatedratesofpe-riodchange.Multiplecrossingsoftheinstabilitystripappeartobepossibleforstarsinlatecoreheliumburningstages,dependingupontheCNOabundancesinthehydrogenburningshellsofsuchstars(Xu&Li2004).

Thefiniterangeinstellarsurfacetempera-tureforstarspopulatingtheinstabilitystripatconstantpulsationperiodimpliesdistinctdiffer-encesinpulsationefficiencythatshouldcoincidewithmarkeddifferencesinpulsationamplitudeforCepheidsofsimilarperiod.Onthehotedgeofthestriptheionizationzoneisjustbeginningtoreachdepthswherethepistonmechanismforpulsationbecomesefficient,solightamplitudesshouldbesmallbutincreasingwithdecreasingsurfacetem-peratures.Onthecooledgethelowersurfacetemperaturesareassociatedwithincreasedcon-vectiveenergytransportinthestar’souterlayers(Deupree1980),sopulsationamplitudesshouldalsobesmall.

ThefirststudyofCepheidamplitudesasafunctionofpositioninthestripbyKraft(1963)wasconsistentwiththatpicture,althoughsmallamplitudeCepheidswerefoundonlyonthehotedgeofthestrip.AllsubsequentamplitudemapsoftheinstabilitystripbyHofmeister(1967),Sandage&Tammann(1971),Payne-Gaposchkin(1974),Pel&Lub(1978),Turner(2001),andSandageetal.(2004)haveproducedsimilarre-sults,namelyasharprisetomaximumamplitudeonthehotedgeofthestripfollowedbyamoregradualdeclinetowardsthecooledge.

Cepheidamplitudesdisplayaperioddepen-denceaswellasadependenceuponlocationwithinthestrip,anaturalconsequenceofaneffecttiedtosurfacegravityaswellaspulsationefficiency.Inordertoeliminatethatfactorincharacteriz-ingCepheidperiodchanges,wehavenormalizedtheP

˙resultingvaluesofbluelightamplitudeandasfollows:(i)blueamplitudes∆Bwerestan-dardizedthroughtheratio∆B/∆B(max),where∆B(max)isthemaximumvalueof∆Bforthe

6

star’spulsationperiod,and(ii)P

˙wasadjustedtotheequivalentvalueforaCepheidwithapulsa-tionperiodof10dusingtheempirically-obtainedslopeplottedinFig.4.

Fig.5plotssuchdataforCepheidswith12d≤P≤40dandincreasingpulsationperiods(P≃20d).Theupperpartofthediagramplotstheindividualdata,whilethemiddlepartofthedia-gramplotsrunningmeansforthedata.Thelowerpartofthediagramisanalternateinterpretationofthesamedata,asdescribedbelow.SimilarplotsaregiveninFig.6forCepheidswith4d≤P≤8dandincreasingpulsationperiods(P≃6d),andinFig.7forCepheidswith4d≤P≤8dandde-creasingpulsationlargevaluesofP

˙periods(P≃6d).Recallthat

shouldcorrespondtothehotsideoftheinstabilitystrip,andsmallvaluestothecoolside.

Thedatafor20dCepheids(topportionofFig.5)displayatendencyforlargeamplitudeCepheidstohaveratesofperiodincreasetypicalofstarsly-ingnearthecenteroftheinstabilitystrip,withsmalleramplitudeCepheidsfallingtowardsthehotP

˙andcooledges(largerandsmallervaluesof,respectively).Thetrendismoreobviouswhenoneplotsrunningfive-pointmeansofthesamedata,asinthemiddlesectionofFig.5.TherearetwolongperiodvaluesofP

˙Cepheidswithanomalouslylarge

,SZCasandAQPup,whicharecon-ceivablyfifthcrossers.IftheyareomittedfromtherunningmeansandaveragesoverareincludedattheextremesofP

˙smallersamples

,oneobtainstheresultsinthelowerportionofFig.5,whicharetypicalofindependentcross-sectionalamplitude

mapsoftheinstabilitystrip.ThescatterinP˙val-uesevidentinthetopportionofFig.5isintrinsictothestars,andisnottheresultoflargeuncer-taintiesinthecalculatedvalues.PresumablythereareintrinsicphysicaldifferencesfromoneCepheidtoanotherthataccountforthescatter,asnotedearlier.Differencesininitialrotationvelocityfortheprogenitormain-sequencestarsmightbethesolefactor,giventhattheywouldgeneratesuffi-cientlylargevariationsintheabundancesoftheCNOelementsthroughoutthestartoaffecttheextentoftheblueloopstages(Xu&Li2004).Thedatafor6dCepheidswithperiodincreases(topportionofFig.6)aremorecomplicated.Itappearsthatthesampleconsistsoftwooverlap-pinggroupsofobjects,afeaturethatalsoappears

intherunningfive-pointmeansdisplayedinthemiddlesectionofFig.6.WeassumethateachgroupconsistsofCepheidsdisplayinganordermagnitude(factorof10)variationinP

˙of

valuesfromthehottocooledgesoftheinstabilitystrip,asdis-playedbythelongperiodCepheidsinFig.5,andusetheresultspresentedinthelowerportionofFig.5asatemplateforthelikelyvariationsativeamplitudewithP

˙inrel-forshortperiodCepheids.WhenthetwogroupsinFig.6areseparatedinsuchfashionandaveragesoverincludedatextremevaluesofP

˙smallersamplesare

foreachgroup,oneobtainstheresultsinthelowerportionofFig.6.ThesimplestexplanationfortheexistenceoftwogroupsamongtheshortperiodCepheidsistheex-istenceofhigherstripcrossingsamongthestars,namelyfifthcrossingsforCepheidsundergoingpe-riodincreases.

Similarresultsapplytothedatafor6dCepheidswithperioddecreases(Fig.7),whenanalyzedinsimilarfashion,despitethesmallersamplesize.ThetopportionofFig.7displaysexcessivescat-ter,muchlikethatinthetopportionofFig.6,withonlymarginalimprovementthroughrunningfive-pointmeans(middleportionofFig.7).Re-strictingthedatasetsasaboveproducesthere-sultsdepictedinthelowerportionofFig.7,whichsuggestsanoverlapbetweenCepheidsinsecondandfourthcrossingsoftheinstabilitystrip.Asnotedforlong-periodsicscatterinP

˙Cepheids,thereisanintrin-valuesthatisnottheresultoflargeuncertaintiesinthecalculatedvalues.SuchscattermakesitdifficulttouserateofperiodchangeforindividualCepheidstoidentifytheirexactlocationwithintheinstabilitystrip,althoughapproximateplacementsarepossibleinmostcases.

Theconclusionsreachedhere,whileadmittedlyspeculative,provideareasonableexplanationforthecharacteristicbehaviorofpulsationamplitudeandrateofperiodchangeforindividualCepheidsatspecificpulsationperiod.Asnotedearlier,theremaybefurthercomplicationsarisingfromthepresenceofovertoneCepheidsinthesample,buttheirnumbersshouldberelativelysmallinthepresentcase,exceptatshortperiods,andtheywouldnotaltertheobserveddistributionofdatapointssignificantly.AmorecomprehensivestudyincludingrecognizedovertoneCepheidsshouldbepossibleonceO–Cstudieshavebeencompletedforsuchvariables.

7

5.Discussion

OurintenthereistodemonstratethatrateofperiodchangeforaCepheidisausefulparame-terthatpermitsonetocharacterizethevariableintermsofspecificevolutionarytiononP

˙state.Informa-foraCepheid,inconjunctionwithitsknownpulsationperiodandlightamplitude,canbeusedtoidentifythestripcrossingmodefortheobjectaswellasitslikelylocationwithinthestrip,thelatterindependentofitsobservedreddening.TheparameterP

˙colorand

mayevenbeuse-fulforestablishingifaCepheidisafundamentalmodepulsatororanovertonepulsator,althoughweleavethatasafutureexercise.

IfweinterprettheresultsofFigs.5–7asagenericindicatorofhowpulsationamplitudevariesacrosstheinstabilitystrip,thenthewidththestripinP

˙of

atconstantperiod,whichamountsto∼1˙,mustencompassarangeof∼16inP

˙.2inlogP

.Ofthat,anintrinsicdispersioninlogP˙valuesamountingtoperhaps0.4–0.5,afactorof∼3,presumablyarisesfromactualinternaldif-ferencesintheCepheidsresultingfromdifferenthistoriesfortheirprogenitorstars.SpecificstellarevolutionarymodelspresentedsmallervariationinlogP

˙inFig.2predicta

thanwhatisobserved,whichmayreflectthesimplicityofthemodels.Inthatregard,observedratesofperiodchangeinCepheidscanplayanimportantroleasacheckonhowcloselystellarevolutionarymodelsmatchrealstars.UntilnowCepheidperiodchangeshavenotbeenusedforthatpurpose.

ThepresentstudywassupportedbyresearchfundingawardedthroughtheNaturalSciencesandEngineeringResearchCouncilofCanada,theRus-sianFoundationofBasicResearchthroughtheFederalProgram”Astronomy”oftheRussianFed-eration,andtheSmallResearchGrantsProgramoftheAmericanAstronomicalSociety,inpartbyfundingfromtheCeceliaPayneandSergeiGaposchkinMemorialFund.REFERENCES

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Fig.1.—ThetheoreticalHRdiagramillustrat-ingpost-main-sequenceevolutionarytracksforstarsof4,5,7,and10M⊙(Lejeune&Schaerer2001).IncludedistheobservationallocationoftheCepheidinstabilitystrip(dottedlines,fromTurner2001)andlinesofconstantstellarradius.

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Fig.2.—PredictedratesofperiodchangeforstarscrossingtheCepheidinstabilitystripastiedtopublishedstellarevolutionarymodels.Themean-ingofthedifferentsymbolsisexplainedinthetext.Linesdenoteregionswithinwhichthepre-dictionsfromdifferentstellarevolutionarymodelsappeartocluster.Thedifferentcrossingsoftheinstabilitystripareidentified.

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Fig.3.—Observedratesofperiodchange,alongwiththeircalculateduncertainties,forwell-studiedCepheidspossessingmanyyearsofO–Cdata.DottedlinesaretherelationsdepictedinFig.3,andthedifferentstripcrossingsareidenti-fied.

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Fig.4.—ThedataofFig.3plottedalongwithsuggestedempiricaldelineationsoftheregionscor-respondingtoCepheidsindifferentcrossingsoftheinstabilitystrip,asidentified.

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Fig.5.—NormalizedblueamplitudesofCepheidswith12d≤P≤40dandincreasingpulsationpe-riodsasafunctionofnormalizedrateofperiodchange(uppersection).Themiddlesectiondis-playsrunningfive-pointmeansforthedata,andthelowersectionadjustedandextendedmeansofthedata.

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Fig.6.—NormalizedblueamplitudesofCepheidswith4d≤P≤8dandincreasingpulsationperiodsasafunctionofnormalizedrateofperiodchange(uppersection).Themiddlesectiondisplaysrun-ningfive-pointmeansforthedata,andthelowersectionadjustedandextendedmeansofthedataafterjudiciousseparationoftheoverlappingsam-plesusingtheresultsofFig.5asatemplate.

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Fig.7.—NormalizedblueamplitudesofCepheidswith4d≤P≤8danddecreasingpulsationpe-riodsasafunctionofnormalizedrateofperiodchange(uppersection).Themiddlesectiondis-playsrunningfive-pointmeansforthedata,andthelowersectionadjustedandextendedmeansofthedataafterjudiciousseparationoftheoverlap-pingsamplesasinFig.6.

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