A Global Physical Model for Cepheids

4.7 MB·English

by  Ondrej Pejcha

DRAFTVERSIONJANUARY27,2012
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
AGLOBALPHYSICALMODELFORCEPHEIDS
ONDRˇEJPEJCHA1ANDCHRISTOPHERS.KOCHANEK1,2
DraftversionJanuary27,2012
ABSTRACT
Weperformaglobalfitto∼5,000radialvelocityand∼177,000magnitudemeasurementsin29photometric
bandscovering0.3µmto8.0µmdistributedamong287Galactic,LMC,andSMCCepheidswithP>10days.
2 We assume thatthe Cepheidlightcurvesandradialvelocitiesarefully characterizedby distance, reddening,
1 and time-dependentradiusand temperaturevariations. We constructphase curvesof radiusandtemperature
0 forperiodsbetween10and100days,whichyieldlightcurvetemplatesforallourphotometricbandsandcan
2 be easily generalized to any additional band. With only 4 to 6 parameters per Cepheid, depending on the
n existence of velocity data and the amountof freedom in the distance, the models have typical rms light and
a velocity curve residuals of 0.05mag and 3.5kms- 1. The model derives the mean Cepheid spectral energy
J distributionanditsderivativewithrespecttotemperature,whichdeviatefromablackbodyinagreementwith

5 metal-line and molecular opacity effects. We determine a mean reddeninglaw towards the Cepheids in our
2 sample, which is not consistent with standard assumptions in either the optical or near-IR. Based on stellar
atmosphere models we predict the biases in distance, reddening, and temperature determinationsdue to the

] metallicityandwequantifythemetallicitysignatureexpectedforourfitresiduals. Theobservedresidualsasa
R
functionofwavelengthshowcleardifferencesbetweentheindividualgalaxies,whicharecompatiblewiththese
S predictions. In particular, we find that metal-poorCepheids appear hotter. Finally, we providea framework
. foroptimallyselectingfiltersthatyieldthesmallestoverallerrorsinCepheidparameterdetermination,orfilter
h
combinationsfor suppressing or enhancingthe metallicity effects on distance determinations. We make our
p
templatespubliclyavailable.
-
o
Subjectheadings:Distancescale—Stars: variables:Cepheids—Stars: abundances
r
t
s
a
1. INTRODUCTION 1982;Gierenetal.2005;Perssonetal.2004;Freedmanetal.
[
The classical Cepheids play a key role in the extragalac- 2008; Madoreetal. 2009; Ngeowetal. 2010; Stormetal.
2 tic distance determination, both as a direct way of ob- 2011a,b). There are, however, differences in the near/mid-
v infrared extinction law for differentsightlines in the Galaxy
taining distances of nearby galaxies and as a calibrator
8 (e.g.Román-Zúñigaetal.2007;Nishiyamaetal.2009).
for other methods (e.g. the review by Freedman&Madore
3 Cepheidpropertiesmustalsochangeasafunctionofchem-
2010a, and Madore&Freedman 1991; Feast&Catchpole
0 icalcomposition, butthe associated observationalsignatures
1997;Freedmanetal.2001;Benedictetal.2007;Riessetal.
3 have been a matter of considerable debate. While the gen-
2009a,b, 2011a,b). Thisis enabledbytherelativelytightre-
. eral conclusion of observational studies is that metal-rich
2 lationbetweenthepulsationalperiodofaCepheidanditslu-
1 minosity(thePLrelation). Accuratedistanceestimatesusing Cepheidsarebrighterthantheirmetal-poorcounterparts(e.g.
1 Cepheids require increasingly better understandings of sys- Stothers1988;Kochanek1997;Sakaietal.2004;Macrietal.
1 tematic effects such as extinction, composition (metallicity) 2006; Valleetal. 2009; Shappee&Stanek 2011), results
v: andblending. from stellar pulsation models and some observations yield
the opposite dependence (e.g. Fiorentinoetal. 2002, 2007;
i For example, the light from Cepheids is typically ex-
X tinguished by dust, which can have different properties in Marconietal. 2005; Bonoetal. 2008; Romanielloetal.
2008; Freedman&Madore 2011). Estimates of the metal-
r different galaxies (e.g. Cardellietal. 1989; Laney&Stobie
a 1993;Falcoetal.1999;Misseltetal.1999;Mottaetal.2002; licity correction vary by almost an order of magnitude, as
can be seen in Figure 14 of Gerkeetal. (2011). It is
Gordonetal. 2003; Indebetouwetal. 2005; Marshalletal.
also now clear, at least in the extragalactic context, that
2006; Laney&Caldwell 2007). Cepheid observations must
uncertainties in the appropriate metallicities and metallic-
be corrected for extinction before any subsequent analysis.
ity gradients are nearly as important as the actual metallic-
ThisusuallymakesuseofanassumptionthatCepheidswith
ity dependence of Cepheid parameters (see Bresolin 2011;
a given period have a specific color and that there is a
Gerkeetal. 2011; Shappee&Stanek 2011, and references
known, universal extinction law. This holds for the Baade-
therein). Furthermore, studies find different trends in dif-
Wesselink and related methods as well as purely photomet-
ferent filters (Bonoetal. 2008, 2010; Ngeowetal. 2011;
ric analyses. To alleviate these shortcomings, recent ef-
Stormetal. 2011b). Many of these problems arise because
forts have shifted from the optical (Hubble Key Project) to
the effects of metallicity have filter-dependent degeneracies
the infrared wavelengths (H, I and Spitzer IRAC), where
with reddening and distance, making it crucial to fully re-
the effects of reddening are reduced (e.g. McGonegaletal.
cover all covariances in the final results, as emphasized by
pejcha,[email protected] Gould (1994), Kochanek (1997), Sasselovetal. (1997), and
1DepartmentofAstronomy,TheOhioStateUniversity,140West18th Riessetal. (2011a). As a result, the question of metallicity
Avenue,Columbus,OH43210,USA effectsontheCepheiddistancescaleremainsacontroversial
2CenterforCosmologyandAstroparticlePhysics,TheOhioStateUni- issue. The problemof blending then adds further complica-
versity,191WestWoodruffAvenue,Columbus,OH43210,USA
2 Pejcha&Kochanek
tionsformoredistantgalaxies(Mochejskaetal.2000). 2.1. PhysicalModel
In manycircumstances, lightcurvetemplatesfor the indi-
For each Cepheid, the radial velocity v(t) and magnitude
vidualobservationalbandsareneededtoaccuratelydetermine
m(t)infilteriatatimet aregivenas
the mean magnitudes of extragalactic Cepheids because of i
their sparse light curves. Stetson (1996) constructedV- and R(t) T(t)
m(t)=M +µ+RE(B- V)- 5log - 2.5βlog (,1)
I -bandtemplatesfortheLMC,SMC,andGalacticCepheids i i i i
C R T
withperiodsbetween7and100daysbyfittingFourierseries (cid:18) 0(cid:19) (cid:18) 0 (cid:19)
1dR(t)
tothelightcurves.Heassumedthattheamplitudesandphases v(t)=v¯- , (2)
of individual modes vary continuously with period, thus p dt
formingthewell-knownHertzsprungsequence(Hertzsprung
whereM istheabsolutemagnitudeofa“mean”Cepheidwith
1926).Hendryetal.(1999),Ngeowetal.(2003),Tanviretal. i
radiusR ≡10R andtemperatureT ≡5400K,µisthedis-
(2005) and Yoachimetal. (2009) expanded on these mod- 0 ⊙ 0
tancemodulus,R istheratiooftotaltoselectiveextinctionin
elsusingPrincipalComponentAnalysis(PCA)techniquesto i
filteri,E(B- V)isthereddening,R(t)andT(t)aretheradius
buildnewtemplatesforanalysisofHSTCepheiddata. Allof
andtemperatureoftheCepheid,v¯isthemeanradialvelocity,
these templatesare limitedin the sense thattheyare defined
and p≡1.36 is a projection factor for converting observed
onlyforasmallsetoffilters(typicallyV andI ),andthereis
C radialvelocitiesto photosphericones(e.g.Burkietal. 1982;
nowell-definedmeansofshiftingthemtoother,similarfilters,
Nardettoetal. 2004). Although the exact value of p and its
letalonetoverydifferentwavelengths. Freedman&Madore
dependence on the pulsational period is a matter of debate,
(2010b)showthatalllightcurvesarerelatedbyalineartrans-
theseuncertaintieslargelycancelbecausewehomogeneously
formationwhichprovidesapathtowardsaddressingthisprob-
analyzethethreegalaxiesinoursamplewiththesamevalue
lem.
of p(Stormetal.2011a). Thecoefficientsβ arethelogarith-
In this work, we address these issues by constructing a i
micderivativesofthespectralenergydistribution(SED)with
global model of Cepheid light curves and radial velocity
respecttotemperatureatthereferencetemperatureT ,
curves in the Galaxy, LMC, and SMC. We self-consistently 0
determine distances, reddenings, radii, and temperatures of ∂logF
individual Cepheids along with a global reddening law, the βi= i , (3)
∂logT
meanSED ofa Cepheidandits dependenceontemperature, (cid:12)T0
(cid:12)
and the phase variations of the radii and temperatures as a whereF isthefluxinfilteri. Alllo(cid:12)garithmsinthispaperare
function of period. We self-consistently determine the un- i (cid:12)
base10.
certainties in all parameters including all their covariances.
The radii R and temperatures T depend on the time t
Ourmodelenablesusto constructa lightcurvetemplatefor through the pulsational phase φ=(t- t )/P, where t is the
0 0
anarbitraryfiltergivenjustasinglecalculableparameterfor
referencetimeandPisthepulsationalperiod.Weexpressthe
therelativecontributionofradiusandtemperaturevariations
time-dependentpartsofEquations(1)–(2)as
to that filter. We evaluate metallicity effectson Cepheidob-
servationsfirstfromatheoreticalpointofview, andthenwe R(t)
log =ρ¯+A2δρ(φ), (4)
searchformetallicityeffectsintheestimatedparametersand R
residualsofthefit. (cid:18) 0 (cid:19)
T(t)
TheapproachtotheproblemiscloselyrelatedtotheBaade- log =τ¯+A2δτ(φ), (5)
Wesselink method (Baade 1926; Wesselink 1946) and as- (cid:18) T0 (cid:19)
sumes that the magnitude of a Cepheid at a given time de-
where ρ¯and τ¯ are the mean logarithmic radii and tempera-
pends only its distance, reddening, and instantaneousradius
tures of a Cepheid with respect to the “mean” Cepheid with
andtemperature. Ratherthancarryingoutastudyofindivid-
radiusR andtemperatureT ,A2 isthedimensionlessampli-
ualCepheids,wesimplyfitaglobalmodeltoalltheavailable 0 0
tude,andδρ(φ)andδτ(φ)aretheperiodandphasedependent
photometric and velocity data. A simpler variation on this
changes in the radius and temperature. In the above defini-
ideawasrecentlyproposedbyFreedman&Madore(2010b).
tions, quantities with subscript i are different for each filter,
This paperis organizedas follows. In Section 2, we outline
ourphysicalmodel,priors,datasetandthemethodoffitting. whileµ,E(B- V),v¯,ρ¯,τ¯,A2,P,andt0 aredifferentforeach
InSection3,wedescriberesultsofthefit, discussinginturn Cepheid. Onlytheradiusandtemperaturechangesδρandδτ
thefitresiduals,globalquantitiesandindividualpropertiesof arefunctionsoftimet.
theCepheids. InSection4,wediscussourresultsandsearch NotallCepheidsin oursamplehaveenoughdata toallow
forsignsofadditionalphysics,principallymetallicityeffects. for independentestimates of ρ¯and τ¯, but they can still con-
We review our findingsand outline future directionsin Sec- tributetothedeterminationofglobalvariables. Inanalogyto
tion5. AnAppendixoutlinesageneralquantitativeapproach thenormalperiod–luminosity(PL)relations,weassumethat
the mean radius and temperature of a Cepheid are functions
toselectingfiltersforCepheidstudies.
ofthepulsationalperiod,
2. MODELANDDATA hρ¯(P)i=aρ¯+bρ¯log P , (6a)
InthisSectionwedescribeourphysicalmodelofCepheid 10d
(cid:18) (cid:19)
lightcurvesandradialvelocitycurvesindetail(Section2.1) P
alongwithitsconnectiontotraditionalmethods(Section2.2). hτ¯(P)i=aτ¯+bτ¯log , (6b)
10d
Thepriorsneededtoaddressdegeneraciesinourmasterequa- (cid:18) (cid:19)
tion,suchasthezeropointofextinction,aredescribedin§2.3. where hρ¯i and hτ¯i are the average radii and temperatures at
Thedatathatareusedforthefitaredescribedin§2.4andthe periodP,andthecoefficientsaρ¯,bρ¯,aτ¯,andbτ¯areestimated
fittingmethodisoutlinedinSection2.5. duringthefit. Theserelationshavecorrespondingwidths,σρ¯
GlobalmodelforCepheids 3
andστ¯,whichinprinciplecouldbeestimatedduringthefitas curves in one band can always be constructed as weighted
well,butforsimplicityweleavethemfixedatσρ¯=στ¯=0.02 sumsofthoseintwootherbands.Thisistruebyconstruction
basedonthedifferencesbetweencanonicalandnon-canonical forourmodels,wheretoproduceband3frombands1and2
theoretical pulsational models of Bonoetal. (1998, 2005). onechoosesa scale factorx suchthatβ =xβ +(1- x)β so
3 1 2
Equations (6a)–(6b) also allow us to connect our approach thatm (φ)=xm (φ)+(1- x)m (φ). Ingeneral,thiscanbean
3 1 2
to the traditional methods based on PL relations and to de- extrapolationsince one can choose bandssuch that x>1 or
fine template light curvesat any period or wavelength (Sec- x<0.
tion2.2).
We model the time-dependent components as a truncated 2.3. Priors
FourierseriesoforderN ,
F The parameters of the model are obtained by minimizing
NF themasterconstraintHdefinedas
δρ= cρ,jcos(2πjφ)+sρ,jsin(2πjφ) , (7) H=χ2+S, (12)
j=1
X(cid:2) (cid:3) whereχ2isthesumofthesquaresofdifferencesbetweenthe
NF
δτ= c cos(2πjφ)+s sin(2πjφ) , (8) modelandtheobservedmagnitudesmobsandradialvelocities
τ,j τ,j
vobs
j=1
X(cid:2) (cid:3)
wherethecoefficientsarenormalizedinphaseandamplitude mobs- m 2 vobs- v 2
χ2= + , (13)
suchthatcρ,1≡1andsρ,1≡0. Thenormalizationofthetem- σ σ
peraturetemplatewithrespecttotheradialtemplateisapart alldata(cid:18) (cid:19) alldata(cid:18) (cid:19)
X X
ofthesolutionsincetherelativeamplitudesoftheradiusand where σ is the measurement error, and S includes contribu-
temperaturechangesarecapturedinthemagnitudeofcτ and tions from all priors, and the sums are over all stars, filters
sτ. We use a Cartesian representation of the Fourier coeffi- andmeasurements.
cients (cρ,j,sρ,j) and (cτ,j,sτ,j) in order to avoid the coordi- Although we fit Equation (13) to a huge dataset, the pa-
natesingularitieswhichoccurina“polar”representation(i.e. rameters in Equation (13) suffer from degeneracies without
δρ(φ)∝ajcos[2πjφ+bj]) when the amplitude of a Fourier the addition of priors on the wavelength dependent vectors
modevanishes(aj→0)andthephaseisdegenerate,asisseen M,R,andβ andtheabsolutedistanceandextinctionscale.
i i i
in the Stetson (1996) Cepheid templates. In order to model
For example, we can simultaneously shift the extinction by
the period dependence of the radius and temperature varia- E(B- V)→E(B- V)+∆E(B- V)andtheabsolutemagnitude
tions,wespecifycρ,j,cτ,j,sρ,j andsτ,j atNP periodschosen vector M →M - R∆E(B- V) while keeping m the same.
insucha waythatthenumberofCepheidsinagivenperiod i i i i
More generally, M is well-determinedup to addingcompo-
binisapproximatelyconstant(exceptforthelongestperiods). i
nentsproportionalto a constant(a change in the distance or
ValuesofthecoefficientsforaparticularCepheid’speriodare
radiusscale), R (achangein theextinctionzeropoint),and
obtained by linear interpolation on the grid of templates in i
β (achangeinthetemperaturezeropoint). Thisistruealso
logP. WechooseN =20andN =19. i
F P forR. Because we assumethatβ isthesame forthemean
i i
and time-variable effects of temperature, it is not subject to
2.2. ConnectiontoTraditionalMethods
thesamedegeneracies.
InthisSectionwerelateourapproachtotraditionalPLstud- Inordertoaddressthesedegeneracies,wefirstdefinepriors
ies. Withinourapproach,themeanabsolutemagnitudeMiin forthevectorsM,β,andR. WeaddpriorsonM,β andR
i i i i i i
filteriofaCepheidcanbeconstructedfromEquation(1)as
oftheform
M =M - 5ρ¯- 2.5βτ¯, (9) M - Mprior 2
i i i S = i i , (14)
whereρ¯andτ¯aredifferentforeachCepheid.Thismeansthat Mi i σMi !
the average PL relation is defined by substituting the mean X
radii and temperatures from Equations (6a)–(6b) into Equa- whereMprior isobtainedbyconvolvingtheT =5400K,R =
i 0 0
tion(9)toobtain 10R ,logg=1.5,[M/H]=0.0“mean”Cepheidbasedonthe
⊙
Castelli&Kurucz (2004) model atmospheres with a tophat
M =M - 5hρ¯(P)i- 2.5βhτ¯(P)i=
i i i functiondefinedbythecentralwavelengthandwidthofeach
=Mi- (5aρ¯+2.5βiaτ¯)- (5bρ+2.5βibτ¯)log 1P0d .(10) ffioltrera.llTfiheltesursmwishoevreerwalelpkansoswbanthdes.zWereocphooionstes.σMFio=r0t.h1emraeg-
(cid:18) (cid:19) maining filters3, we estimate the conversionfactor based on
Thus our model leads to a PL relation with a zero point of
ourdata and then set σ =0.2mag. As we will see in Sec-
Mi- 5aρ¯- 2.5βiaτ¯andaslopeof- (5bρ¯+2.5βibτ¯). Foruncor- tion 3.2, this assumptioMni is unimportant. We constrain the
related radius and temperature deviations (σρ¯ and στ¯) from vector β by evaluating Equation (3) as a derivative in tem-
the mean trends (Eq. [6]), the scatter in the PL is (52σρ2¯+ peratureiaboutthis referencemodel. The width of this prior
2(β.5i)2,βsi2loσpτ2¯e)1a/n2,dssocatthteerreabisouatstthreonPgLc(oserreeMlataiodnoroef&wFarveeelednmgathn Cisasσtβeill=i&0.K1βuiprruiocrz. (2W0e04fi)xprtihoer cvoaleuffieciinenotrdβeVr≡to5p.r1e4vetnot iatns
2011). Similarly,themeantemplatelightcurveforafilteriis overall shift of β due to a weak degeneracy with ρ¯. Simi-
i
mi(φ)=- 5δρ(φ)- 2.5βiδτ(φ), (11) larly,weaddaprioronRi,whereRiprior isobtainedfromthe
whichhasameanofzero(hmii=0)andisscaledtoanampli- 3 Wewere unable tofindzeropoints fortheCTIOJHK filters andthe
tudeofA2=1. Freedman&Madore(2010b)arguethatlight WashingtonfiltersCMT1T2.
4 Pejcha&Kochanek
Cardellietal. (1989) galactic extinction law with R ≡3.3
V
and R ≡ R +1 is held fixed so that E(B- V) has the
B V
standard interpretation. The remaining coefficients are con-
Rstrai=ne3d.3bywaasnc“huonsceenrttaoinatgyr”eeσRwii,tih=th0e.0H5Rubipbriloer.KTeyhePrvoajleucet
V
(Madore&Freedman1991;Freedmanetal.2001),butisac-
tually an additional source of uncertainty that could be ex-
plored.
Thesecondsetofpriorsisonthemeanradiiandtempera-
tures(Eqs.[6a]–[6b]).First,weaddtwopriors
ρ¯ - hρ¯(P)i 2
k
Sρ¯= , (15a)
allstarsk(cid:18) σρ¯ (cid:19)
X
τ¯ - hτ¯(P)i 2
Sτ¯= k , (15b) Figure1. Effectofthepriorsonthehigh-orderFouriercoefficientsandtem-
allstarsk(cid:18) στ¯ (cid:19) platesatadjacentperiodsontheoverallgoodnessoffit.Thecontoursindicate
X theχ2excess(denotedwithnumbersateachcontour)overafitwithoutthe
wherehρ¯(P)iandhτ¯(P)iarethemeanradiiandtemperatures priorsShoandSaponthesmoothnessofthetemplates. Theinsetplotshows
at a period P, which are given in Equations (6a)–(6b), and χ2excessalongthelineofapproximatelyequalcontributionsofShoandSap
whichareestimatedaspartofthefit.Thewidthsoftheserela- totheχ2(logσap=1.8logσho,dashedlineinthemainplot).Theblackfilled
tionsarefixedatσρ¯=στ¯=0.02basedondifferencesbetween cciormclpeacroerdretospnoondsmstoootthheinfignoaflacbhoouicte5o%f.σhoandσho,whichgiveaχ2excess
thecanonicalandnon-canonicaltheoreticalpulsationalmod-
elsofBonoetal.(1998,2005). ThesepriorsdriveaCepheid
SMCCepheidsbecauseitleadstoperiod-dependentresiduals
ontothemeanperiod–radiusandperiod–temperaturerelations
in distance moduli and reddenings which absorb any differ-
ifthereareinsufficientdatatoindependentlyconstrainitsra-
encesbetweentherequirementsofthedataandthepriorson
diusandtemperatures. Thesecondsetofpriorsisontheco-
radiiandtemperatures.Essentially,thePLrelationisdictated
efficientsofhρ¯(P)iandhτ¯(P)iinEquations.(6a)–(6b). These
bytheradiusandtemperaturepriorsratherthanthedata.This
havetheform
problemwouldbesolvedbyincludingasampleofCepheids
S = aρ¯- apρ¯rior 2, S = bρ¯- bpρ¯rior 2, (16) tthinacttitorunlysclailee,awt aeicmompomseonapdriisotarnthcaetmthoedLuMlusC. CTeopfihexidthsehaevxe-
aρ¯ 0.02 ! bρ¯ 0.01 ! a mean extinction of hE(B- V)i=0.147mag (Udalskietal.
1999)withawidthσE(B- V)=0.02mag.
along with a similar set of priors for aτ¯ and bτ¯. The prior The effects of most of these priors on the final results are
valuesof weak, because most of the values are ultimately controlled
apρ¯rior=0.843, bpρ¯rior=0.655, (17a) bthyetLheMdCa,tath.eTdhisetaknecyeetxoctehpeticoennstearreofthtehemLeManCe,xatnindctsieotntining
apτ¯rior=- 0.019, bpτ¯rior=- 0.080, (17b) RV ≡3.3andRB- RV ≡1. Itispossibletorecognizewhere
the data dominate a prior by the final values and uncertain-
were set to match the “canonical” solar-metallicity, period–
ties–ifthefinalresultsmatchthepriorandtheuncertaintyis
radius and effective temperature–luminosity relations of
comparableto the priorwidth, then the data added noinfor-
Bonoetal. (1998) and Bonoetal. (2005). We derived the
mation,whileifthevalueshaveshiftedandtheuncertainties
temperaturepriorsassumingthestandardrelationofluminos-
aremarkedlysmaller,thenthedatadominate.
ity,radiusandeffectivetemperatureL=4πR2σT4. Theprior Thelastsetofpriorsconcernthetemplatesthemselves. In
widthsof0.02and0.01werechosentoroughlycorrespondto ordertominimizeunnecessary“oscillations”inthetemplates,
thedifferencesbetweenthesolar-metallicity“canonical”and we add a prior to keep the high-order terms of the Fourier
“non-canonical”modelsofBonoetal.(1998,2005),asanes- seriessmall
timateofthesystematicuncertaintiesinthemodels.However,
the specific choices for the prior values and their widths are 1 NF
oflittleconsequencefortheactualresults. Thezeropointsaρ¯ Sho= σ2 j4 c2ρ,j+s2ρ,j+c2τ,j+s2τ,j , (18)
andaτ¯arewell-constrainedbythedata.Theslopesbρ¯andbτ¯ hoXj=3 (cid:0) (cid:1)
are less constrained because of the limited number of long-
andtominimizedifferencesbetweenadjacentperiodbinswe
period Cepheids and our restricted period range. For these
set
coefficients,thewidthofthepriorissomewhatimportant.
Thethirdsetofpriorssetthedistanceandextinctionscales. 1 NF- 1
ICnepohrdeeidrstoocficuxptyheatdhiisntadnicskewsciathlea,nwinecalisnsautmioentohfa3t0t.h7e◦LanMdCa Sap=σa2p j=1 j2 (cρ,j- cρ,j+1)2+(sρ,j- sρ,j+1)2+
mpoosdituiolunsaantgtlheeocfe1n5te1r.0o◦f(1N8i.k5o0lmaeavge.tTahle.2d0is0t4a)nwceisthtoatdhiestianndcie- +(cτ,jX- cτ,j+(cid:2)1)2+(sτ,j- sτ,j+1)2 . (19)
vidualCepheidsintheSMCarefreetovary,butweimposea Therelative strengthsof priorsSho and Sa(cid:3)p andtheir relative
priorthatµprior =18.90magwithascatterofσ =0.10mag weightswithrespecttoothercomponentsoftheHaredeter-
SMC µSMC
(e.g. Hilditchetal. 2005; Keller&Wood 2006). The LMC minedempiricallyfollowingFigure1. Wevarythevaluesof
distancescalecannotbefixedusingapriorthatassumesadis- σ andσ ,andwerecordthechangeintheχ2relativetoafit
ho ap
persionaroundsome meanvaluelike theone we use forthe withnegligiblevaluesfortheweights. Thefinalchoiceofσ
ho
GlobalmodelforCepheids 5
and σ givesan equalweightto bothpriorsand leadsto an {M,1,R,- 5,- 5δρ,- 2.5β,- 2.5βδτ}andα ={1,µ ,E(B-
ho i i i i k k
increaseintheχ2ofthefitbyabout5%comparedtousingno V) ,ρ¯,A2,τ¯,A2}. The basis vectors e are not orthogonal
k k k k k j
smoothing. Wechoosetheseparticularvaluesofσ andσ anddo nothaveunitnormsin orderto maintaintheir physi-
ho ap
toseesomesmoothingofthetemplateswhileonlyintroduc- calmeaning. The lack of orthogonalitythen leadsto degen-
ingasmallincreaseinχ2 comparedtohavingnosmoothing. eracies,whichwemustcontrolbyintroducingthepriorsdis-
Theexactchoiceissomewhatsubjective,butalsohasnoim- cussedinSection2.3. Mathematically,however,thesimilar-
portantconsequencesfortheresults. ity to PCA means that the modelis well defined and can be
solvedbystandarditerativeorminimizationmethodstoyield
2.4. Data auniquesolutiongiventhedataandpriors.
In order to obtain well-determined templates, the prop- Tothisend,wehavedevelopedaversatileprogramthatal-
erties of individual Cepheids and the global parameters of lowsindividualvariablestobeswitchedonoroff,toreinitial-
the model solution, we require a sample of Cepheids with a izethephysicalvariablestotheirpriorvalues,andtoalternate
large quantity of photometric and radial velocity measure- fitting all variablessimultaneouslywith fitting just the prop-
ments. The data include the huge database of OGLE-III erties of individual Cepheids. We assign a weight to each
V and I measurements from Soszyn´skietal. (2008; 2010), measurementcalculatedasa maximumof thereportedmea-
the major databases of Berdnikov (1999) and Gorynyaetal. surementerror(ifavailable)andarmsscatterofallmeasure-
(1992, 1996, 1998), the large sample of near-IR measure- mentsinthegivenfilterfromthatparticulardatasource. The
ments by Perssonetal. (2004), and the Spitzer IRAC mea- fitting procedure itself can proceed in two ways. First, all
surements of Freedmanetal. (2008), Marengoetal. (2010), variables are minimized independently in each iteration us-
and Ngeow&Kanbur(2010). The referencesfor the photo- ing analytic first and second derivatives through the conju-
metric and radial velocity data are given in Tables 1 and 2, gategradientmethod(Pressetal.1992). Thismethodisfast,
respectively. but does not provide any explicit error estimates. The sec-
These data include photometry in 29 filters covering the ondoptionistoconstructthefullcovariancematrix,whichis
wavelength range from 0.3µm to 8.0µm, as summarized theninvertedusingtheCholeskydecomposition.Thismethod
in Table 3. In the optical, we include the standard John- is much slower, but provideserror estimates that include all
sonUBV, Cousins (RI) and Johnson (RI) bands. We also thecorrelationsofthemodelaswellasthe measurementer-
C J
use Hipparcos and Tycho photometry (bands Hp, B , and rors. Given any reasonable starting point, the fits are stable
T
V ), photometry in the Washington system (C, M, T , and and well-behaved for all parameters except the phase refer-
T 1
T2), and measurements in the Walraven photometric bands ence time t0, where the χ2 surface is more complex and a
((WULBV) ). TheWashingtonsystemwasdesignedtopro- manualinterventionis sometimesnecessary. While we have
W ˙
videmetallicityandtemperatureestimatesforGandKgiants keptperiodsPandperiodderivativesP≡0fixed,theycanbe
(Wallerstein&Helfer 1966). The Walraven system was de- includedwithoutdifficulty, but are notas stable because the
˙
signed for studies of early-type stars (Walraven&Walraven χ2surfaceinPandPisnotsmooth.
1960). WeconverttheWalravendatafromtheirdefaultlog
10
scale to magnitudes by multiplying the data by a factor of 3. RESULTS
2.5. In the near-IRwe includethe JHK filters of the SAAO InthefollowingSectionswepresentthedetailedresultsof
(Glass1973)andCTIO(Eliasetal.1982)systemsseparately. ourmodel. InSection3.1, wediscussresidualstotheglobal
Finally, we include the Spitzer IRAC [3.6], [4.5], [5.8], and fit. InSection3.2,wediscusstheglobalparameters:thezero
[8.0]bands. point M, temperature dependence β and extinction vector
i i
Sincewearemostlyinterestedinextragalacticuses,were- R. Section 3.3 examines the individual parameters of the
i
strict our dataset to fundamental-mode Cepheids with P ≥ Cepheids.
10d. We do not include ultra-long period Cepheids with
P>100days (Birdetal. 2009). As we proceeded with the 3.1. ResidualsoftheFit
fit,wepurgedthedatasetofobviouslywrongmeasurements
Figure 2 shows the data and the best-fit models for six
anddatabaseerrors. WealsoremovedCepheidswithtoofew
Cepheids in the Galaxy, LMC and SMC. For the left col-
data,binarysystems,stronglyblendedsystems,andCepheids
umn,wehavechosenstarswithalargenumberofdatapoints
with obvious period changes. In the end, we have 177,314
in many filters and a typical fit quality, while the right col-
photometricand5,031radialvelocitymeasurementsfor287
umn shows Cepheids with fits among the worst 5% for the
Cepheidsin29photometricbands.
givengalaxybasedonthephotometricχ2 perdegreeoffree-
2.5. FittingMethod dom. We see that the modelfits the data verywell, andthat
evensparsedatacanstronglyconstrainthefitsgiventhelight
Minimizing Equation (12) based on the physical model
curve structures required by the better sampled bands (e.g.
given by Equations (1)–(8) is somewhat similar to princi-
HV 1543,inthe lowerleftpanelof Figure2). Thecommon
pal component analysis (PCA). Specifically, we are trying
problemsarenoisydata(HV6320,inthelowerrightpanel),
to decompose the measurement m in filter i for Cepheid
ik inadequacy of the template (SU Cru, upper right panel, see
k into an unknown set of vectors M, R, β, δρ, and δτ
i i i thediscussioninSection4.2)andasmallphaseshiftbetween
whilesimultaneouslyobtainingthecoefficientsoftheexpan-
datafromdifferentsources(HV879,middlerightpanel).The
sion µk, E(B- V)k, ρ¯k, τ¯k, and A2k. In a PCA, we would de- remarkablepoint,however,isthatthissimplephysicalmodel
compose a data point mi as a sum mi = jαjeij of coeffi- ofaradiusplusatemperaturetemplatecombinedwithasingle
cientsα multipliedbyorthonormalvectorse , e e =δ , amplitudeproducesgood fits for objectsthat have photome-
j Pj i ij ik jk
simultaneously determining α and e . In this paper, we tryinmanyfiltersacrossabroadwavelengthrangeaswellas
j j
fit a measurement m of a Cepheid k in filtPer i as a sum theirradialvelocitymeasurements.
ik
m = α e , wherefromEquation(1)itfollowsthate = Figure 3 shows the residuals of the fits as a function of
ik j kj ij i
P
6 Pejcha&Kochanek
Figure2. ExamplesoflightcurvesandradialvelocitycurvesforCepheidsinthethreegalaxies(Galaxyattop,LMCatcenter,SMCatbottom). Thefitsare
shownwithgreysolidlines. TheleftcolumnshowsCepheidswithdatainmanyphotometricbandsandmedian-qualityfits,whiletherightcolumnshowsthe
Cepheidswithpoorfitstotheirphotometricdata.
GlobalmodelforCepheids 7
Figure3. Residualswithrespecttothemodelfitsforradialvelocities(toprow;inkms- 1)andthemostcommonfilters(remainingpanels).Theredlinesineach
panelareaveragesinbinsof∆φ=0.05andtheerrorbarsaretheuncertaintiesintheseaverages.Eachpanelgivesthemeanvalueoftheresiduals,itsuncertainty,
andthedispersionofthedataaroundthemean.Thisdispersionisacombinationofmeasurementerrorsandsystematicresiduals.
8 Pejcha&Kochanek
Figure4. ResidualsfortheSpitzerIRACbands.TheformatisthesameasinFigure3,butwedonotshowthephase-binnedaveragesduetothesmallnumber
ofdatapoints.Forcomparisontothedispersionoftheresidualsaboutthemean,wealsoreporttheaveragephotometricuncertaintyofthedata.
phase for the radial velocities and the most common opti-
cal and near-IR bands. Figure 4 shows the residuals for the
SpitzerIRACbands. Therearetwoseparatewaystoevaluate
theseresiduals.First,forthemeanpropertiesoftheCepheids,
we wantthe mean residualsto be consistentwith zero given
theiruncertainties.Thisisgenerallytruetohighaccuracyand
itisreassuringthattheaverageresidualsareessentiallyzero
forallfilterswith few systematictrendsin the residualsasa
functionofphase. Theyarenotexactlyzero,however,which
willbeanimportantpointinSection4.2. Second,wewould
like the dispersion of the residuals to be consistent with the
measurementuncertainties. Thisismoredifficulttoevaluate,
becausenotallthedatasourcesincludeerrorestimatesandwe
usedasaweightthemaximumoftheactualmeasurementer-
rorandthermsresidualforeachbandfromeachdatasource.
However, all measurementsin the Spitzer IRAC bands have
associated errors and therefore we can compare them to the
fit residuals in Figure 4. We see that in the LMC the mean
data error bars are similar to the scatter around the fit, sug-
gestingthatourmodelfitsthedataverywell. IntheSMCand
especially in the Galaxy, there is a noticeable overall offset
betweenthedataandthemodelinsomeSpitzerbands,which
causesformallyalargerscatteraroundthemodel. Theorigin
oftheoffsetswillbediscussedfurtherinSection4.2. Further-
more,weseeinFigure3thattheresidualsarenotentirelyfree Figure5. Thetemperatureandradiustemplatesattheperiodgridpointsas
ofstructure.ThemostobviousstructureisthatseenintheI - afunctionofphaseφ=(t- t0)/P. Thetemplatesareshiftedwithrespectto
C eachotherforthesakeofclarity. Theperiodincreasesfromthebottomto
bandLMCandSMCresidualsatphasesbetweenφ=0.4and
thetop,andtheactualperiodvaluesaregiventotheright. Theverticalline
0.7. Thesephasescorrespondtothefastrisefromminimum segmentsinbothpanelshaveunitlength.
tomaximumlight. We deferthediscussionofthesepatterns
intheresidualstoSection4. able4 includingacodetocalculatetheaveragetemplatelight
curveandradialvelocitycurveforanyperiodwithinthepe-
riodrangeandanyfilteralongwiththeradiusandtemperature
3.2. GlobalQuantities changes for that band. We clearly see the Hertzsprung pro-
Figure5showstheradiusandtemperaturetemplatesforthe gressioninthetemperaturetemplatesbyfollowingtheshifts
anchorpointsofourperiodgrid.Thevaluesofthecoefficients
are givenin Table 4. We make the templatespublicly avail- 4http://www.astronomy.ohio-state.edu/~pejcha/cepheids
GlobalmodelforCepheids 9
inthepositionofmaximumandthebumpontherisingbranch we have not convolved the extinction law with the average
to earlier phases at longer periods. The changes in the ra- Cepheidspectrumandfilterbandpass. Forexample,thevery
dial templates are more subtle, and the main feature is that broad Hp filter has anomalously low R, presumably be-
i
theminimumbecomesmoresharplypeakedastheperiodin- cause it is effectively a much redder band pass in the pres-
creases. ence of significant extinction. An advantage of the model
The magnitude zero-points M in Equation (1) represent R is that they “correctly” include all these band pass av-
i i
the flux of the “mean” Cepheid with radius R0 and temper- erages. Figure 8 shows that between 0.5 and 1.5µm- 1, the
ature T0. Comparing the fitted values to a theoretical model results fall below the prior, while for λ- 1 &2µm- 1 they are
of a Cepheid atmosphere allows us to judge the results. In above. This suggests that the “mean” extinction law of our
Figure 6, we convert the values of Mi to fluxes and com- photometric dataset is different from the canonical RV ≡
pare them to a supergiant model atmosphere (Kurucz 1979; 3.3 Cardellietal. (1989) law, falling between the R ≡3.3
V
Castelli&Kurucz2004) with ourreferenceT0=5400K and Cardellietal. (1989) law and the empirical LMC and SMC
R0=10R⊙ at10pcassumingafixedLMCdistancemodulus curvesofGordonetal.(2003). TheRi valuesarecontrolled
of18.50mag. Weseethatourfitgivesfluxesconsistentwith bythedataexceptfortheSpitzerbands,wherethevaluesand
the Castelli&Kurucz (2004) model. There appears to be a uncertainties closely follow the Cardellietal. (1989) prior.
small systematic offset in the sense that our data are about ThatthebestfitextinctioncurveisnotsimplyaCardellietal.
0.08mag too bright. Such a small shift can be caused by a (1989) R ≡3.3 law should not be a surprise. There is no
V
numberofreasonssuchasslightlyincorrectfluxcalibrations, uniqueextinctionlawevenintheGalaxy(Berryetal.2011),
a small change of the LMC distance or a slightly different and the LMC and SMC extinction laws show further differ-
meanLMCextinction. TheerrorsonM presentedinTable3 ences(e.g.Misseltetal.1999;Gordonetal.2003). Herewe
i
are much smaller than the assumed width of the prior, indi- havedeterminedthebestaverageextinctionlawforCepheids
cating that M is well constrainedby our data. We note that in these three galaxies. We can attempt to parameterize our
i
thetemperaturescale ofourmodeliscontrolledbytheprior resultswithintheframeworkofthe Cardellietal. (1989)ex-
onMi sothemeanperiod–temperaturerelation(Eq.15b)ad- tinction models. Because we keep RV fixed, we perform a
juststomatchthistemperaturescale. Thiscalibrationleadsto fit of the Cardellietal. (1989) extinction law evaluated for
Cepheidtemperaturesthataresystematicallyhigherthanthe λcenter of filter i to Ri/RV with RV as a free parameter. We
period–temperaturerelationfromtheBonoetal.(1998,2005) find that the best fit value is RV =3.127±0.002, but most
of the R/R values still show significant offsets of several
pulsationalmodels. If we removedthe M prior and instead i V
i
percentfromthe best fit value. Interestingly, the largestdif-
fixedthecoefficientsinEquation(6a)–(6b),wewouldgetper-
ferencesareforthenear-IRbands,whereK andK are
fectagreementofthemeantemperatureswiththeirpriorval- SAAO CTIO
shifted by about47% and 31% from their respective best-fit
ues(Eq.[17b])atthecostofpoormatchtothespectralenergy
Cardellietal. (1989) values, as can be seen also in the inset
distributions(M) oftheCastelli&Kurucz(2004) modelfor
i of Figure 8. We also fit the Cepheids data using R ≡3.1.
that temperature. Clearly, having a theoretical stellar atmo- V
The changein χ2 is negligible and the R/R curve tilts to
sphere model that is known to be correct would allow us to i V
a slightly steeper slope to accommodate R ≡ R +1, but
useastrongerpriortoconstrainourmodelandremovemost B V
it is still discernibly different from the R/R ratios of the
oftheseambiguities. i V
R =3.1Cardellietal.(1989)reddeninglaw.
As discussed in Section 2.3 (Eq. [3]), the temperatureco- V
After establishing that β and R represent the physical
efficientsβ arethelogarithmicderivativesofthefluxinany i i
i
quantities intended in Equation (1), we can assess whether
bandwithrespecttotemperatureattemperatureT . Figure7
0
temperatureandextinctioncanbedeterminedindependently.
compares the fitted values of β to those for a 5400K black
i In Figure 9 we show the ratio of β to R as a function of
bodyandtothe priorbasedonthe Castelli&Kurucz(2004) i i
inversewavelength.Becausechangesintemperatureandred-
atmospheremodelswithlogg=1.5andatT =5400K.Recall
0
dening can have similar effects on the observed colors for a
thatβ ≡5.14isfixedtoitspriorvaluebasedonthemodelat-
V limitedwavelengthrange,achangeinτ¯canbemimickedby
mospheres.Ourresultsclearlydeviatefromtheblackbodyin
achangeinE(B- V)ifβ/R isconstantoverthatrange. We
thesensepredictedbythemodelatmospheres.Atshortwave- i i
see from Figure 9 that the ratio β/R has an approximately
lengths,the β arelargerthantheblack-bodymodelbecause i i
oftheeffectsiofmetalopacities,andthemodelmayevenre- parabolicshapeinλ- 1,andthereforeweshouldbeabletosep-
solvesomeoftheexpectedspectralfeatures. Intheinfrared, aratetheeffectsofreddeningandtemperaturequiterobustlyif
weclearlyseetheCObandheadat∼5µm. Weobtainedes- thereisenoughwavelengthrange. Determiningbothtemper-
sentiallythesameresultwithablackbodyprioronβ,which atureandextinctionrequiresaminimumofthreefilters(two
i
means that the weakly imposed prior on β is simply over- colors)andtheefficacyofanychoicecanbeassessedbycon-
i
whelmedbythestatisticalpowerofthedata–thevaluesofβ nectingthelongestandshortestwavelengthbandsbyalinein
i
arerobustlydeterminedbythecolorchangesoftheCepheids Figure9,andthenexaminingthedistanceofthemiddleband
duringthe pulsationalcycle. There are clear systematic dif- from the line. Temperature and extinction degeneracies are
ferencesfromtheCastelli&Kurucz(2004)model,especially minimized by maximizing that distance. We generalize this
in the infrared, which are likely due to real problems in the procedurein AppendixA.Inassessing Figure9, itis impor-
molecular opacities of the Castelli&Kurucz (2004) model tant to recognize the large vertical scale. The curvature be-
(Frémauxetal.2006). tweenfilter wavelengthsneed onlybe largecomparedto the
In Figure 8 we compare the final extinction curve to the photometricerrors(<0.1mag).Forthesameerrors,however,
Cardellietal. (1989) R = 3.3 extinction curve used as a alargerwavelengthbaselineisalmostalwaysbetter.
V
prior. We focusonthe shapeofthe extinctioncurveR/R We can check for possible temperature/extinctioncorrela-
i V
because we fixed R ≡3.3 in the models. In Figure 8 one tionsbylookingattheindividualtemperaturesandextinctions
V
should also focus on the narrower band pass filters because showninFigure10.IntheLMC,thereisagroupofstarswith
10 Pejcha&Kochanek
Figure6. Theluminosityofthe“mean”CepheidwithR0=10R⊙andT0=5400Klocatedat10pcconstructedfromthevaluesofMiandconvertedtoλFλusing
theLMCcalibrationdistanceof18.50mag. Thehorizontalerrorbarsshowthewidthofeachfilterandthefilternamesaregivenatthetop. Theuncertainties
onMi areshown, buttheyaregenerally smallerthanthesizeofthesymbol. Forcomparison, wealsoshowanerrorbarcorresponding tothewidthofthe
prioronMi.Thegreylineshowstheflux(λFλat10pc)ofasupergiantmodelatmospherewithT0=5400K,R0=10R⊙,andlogg=1.5interpolatedfromthe
Castelli&Kurucz(2004)gridofatmospheremodels.
onlyV andI photometry,whichdoesnotallowforindepen- sphereschangesthe results by a negligibleamount. The de-
C
dent extinction and temperature determination. These stars parturesfrom a black bodydue to metalline blanketingand
concentrate along the prior hE(B- V)i≡0.147mag. There molecularopacitiesarearobustresult,ascanbeseenbycom-
is also a tail of LMC and SMC stars with both high τ¯ and paring the fit errors on the global parameters with the prior
E(B- V). These stars lack photometry in bands bluer than widths in Table 3 (prior widths are 5% for R, 10% for β,
i i
V, sopartoftheproblemmaylie indegeneraciescreatedby and 0.1 or 0.2mag for M, while the typicalfit uncertainties
i
smallsystematiceffectsinthephotometrycalibrationandthe aretypically1%forR,0.7%forβ and0.01to0.02magfor
i i
extinction law. For the most extreme cases the model pre- M,respectively).Ourchoiceofpriorsisalsooverdetermined.
i
dictsU-bandmagnitudesatmaximumbrighterthanin theB
Forexample,theprioronM essentiallydrivesourtempera-
i
band, which cannot be true. We consider the temperatures
turescaletobethesameasinCastelli&Kurucz(2004),but
andextinctionswell separatedformost ofthe stars, because the prior on τ¯ (Eq. 16) drivesit (weakly) to the temperature
Cepheids in the Galaxy have much higher extinctions than
scaleoftheBonoetal.(2005)pulsationalmodels. Similarly,
those in the LMC and SMC but noticeably lower tempera-
radialvelocitiesandmulti-filterphotometryimplicitlydefine
tures.Metallicitydifferencesalsointroduceapparentshiftsin
thedistancetotheLMCthroughtheBaade-Wesselinkmethod
temperatureandextinction.ArrowsinFigure10,basedonthe
andsointheorynopriorontheLMCdistancewouldbenec-
resultsofSection4.1,showtheeffectofincreasingthemetal-
essary. We discuss these issues further in the Conclusions
licityby0.5dex,whichisapproximatelythespreadinLMC (Section5).
CepheidmetallicitiesandthetypicaldifferencebetweenLMC
Finally, we check our assumption that the LMC Cepheids
andSMCCepheids(Romanielloetal. 2008). While theam-
are distributed in a thin disk with the inclination and posi-
plitudeandsenseoftheshiftsareconsistentwithmetallicity
tion angle given by Nikolaevetal. (2004) at an LMC center
effects,noneoftheoutliershasameasuredmetallicitytover-
distance of µ=18.50mag. In Figure 11 shows the I -band
C
ifythisorigin.
residuals for the LMC Cepheids as a function of their posi-
Somechoicesforourpriorsareunimportantasthedataso
tionanglerelativetothecenteroftheLMC.Anincorrectin-
stronglyconstraintheproblemthatthesamefinalresultswill
clinationor positionangleof the disk wouldproduceexcess
be obtained even with very different choices for the priors.
residualsatsomepositionangles,butweseeverylittlevaria-
For example, using a black body prior for βi instead of the tioninthescatteroftheCepheidmeanresidualsasafunction
estimatebasedontheCastelli&Kurucz(2004)modelatmo-
ofthepositionangle.Wenote,however,thatmodestinconsis-