The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

ApJSupplementSeries,inpress

APreprinttypesetusingLTEXstyleemulateapjv.4/12/04

THERRLYRAEPERIOD-LUMINOSITYRELATION.

I.THEORETICALCALIBRATION

M.Catelan

Ponti ciaUniversidadCat´olicadeChile,DepartamentodeAstronom´ ayAstrof´ sica,

Av.Vicu naMackenna4860,782-0436Macul,Santiago,Chile

BartonJ.Pritzl

MacalesterCollege,1600GrandAvenue,SaintPaul,MN55105

arXiv:astro-ph/0406067v1 2 Jun 2004

and

HoraceA.Smith

Dept.ofPhysicsandAstronomy,MichiganStateUniversity,EastLansing,MI48824

ApJSupplementSeries,inpress

ABSTRACT

WepresentatheoreticalcalibrationoftheRRLyraeperiod-luminosity(PL)relationintheUBVRIJHKJohnsons-Cousins-Glasssystem.Ourtheoreticalworkisbasedoncalculationsofsyn-thetichorizontalbranches(HBs)forseveraldi erentmetallicities,fullytakingintoaccountevolu-tionarye ectsbesidesthee ectofchemicalcomposition.Extensivetabulationsofourresultsareprovided,includingconvenientanalyticalformulaeforthecalculationofthecoe cientsoftheperiod-luminosityrelationinthedi erentpassbandsasafunctionofHBtype.Wealsoprovide“average”PLrelationsinIJHK,forapplicationsincaseswheretheHBtypeisnotknownapriori;aswellasanewcalibrationoftheMV [M/H]relation.Thesecanbesummarizedasfollows:

MI=0.471 1.132logP+0.205logZ,MJ= 0.141 1.773logP+0.190logZ,MH= 0.551 2.313logP+0.178logZ,MK= 0.597 2.353logP+0.175logZ,

and

MV=2.288+0.882logZ+0.108(logZ)2.

Subjectheadings:stars:horizontal-branch–stars:variables:other

1.INTRODUCTION

RRLyrae(RRL)starsarethecornerstoneofthePop-ulationIIdistancescale.Yet,unlikeCepheids,whichhaveforalmostacenturybeenknowntopresentatightperiod-luminosity(PL)relation(Leavitt1912),RRLhavenotbeenknownforpresentingaparticularlynote-worthyPLrelation.Instead,mostresearchershaveuti-lizedanaveragerelationbetweenabsolutevisualmag-nitudeandmetallicity[Fe/H]whenderivingRRL-baseddistances.Thisrelationpossessesseveralpotentialpit-falls,includingastrongdependenceonevolutionaryef-fects(e.g.,Demarqueetal.2000),apossiblenon-linearityasafunctionof[Fe/H](e.g.,Castellani,Chie ,&Pulone1991),and“pathologicaloutliers”(e.g.,Pritzletal.2002).

Tobesure,RRLhavealsobeennotedtofollowaPLrelation,butonlyintheKband(Longmore,Fernley,&Jameson1986).ThisisinsharpcontrastwiththecaseoftheCepheids,whichfollowtightPLrelationsboth

Electronicaddress:mcatelan@astro.puc.clElectronicaddress:pritzl@macalester.eduElectronicaddress:smith@pa.msu.edu

inthevisualandinthenear-infrared(see,e.g.,Tanvir1999).ThereasonwhyCepheidspresentatightPLre-lationirrespectiveofbandpassisthatthesestarscoveralargerangeinluminositiesbutonlyamodestrangeintemperatures.Conversely,RRLstarsarerestrictedtothehorizontalbranch(HB)phaseoflow-massstars,andthusnecessarilycoveramuchmoremodestrangeinluminosities—somuchsothat,intheircase,therangeintemperatureoftheinstabilitystripisasimportantas,ifnotmoreimportantthan,therangeinluminositiesofRRLstars,indeterminingtheirrangeinperiods.There-fore,RRLstarsmayindeedpresentPLrelations,butonlyifthebolometriccorrectionsaresuchastoleadtoalargerangeinabsolutemagnitudeswhengoingfromthebluetotheredsidesoftheinstabilitystrip—asisindeedthecaseinK.

Thepurposeofthepresentpaper,then,istoperformthe rstsystematicanalysisofwhetherausefulRRLPLrelationmayalsobepresentinotherbandpassesbesidesK.Inparticular,weexpectthat,usingband-passesinwhichtheHBisnotquite“horizontal”attheRRLlevel,aPLrelationshouldindeedbepresent.Since

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

2M.Catelan,B.J.Pritzl,H.A.Smith

Fig.1.—Upperpanels:MorphologyoftheHBindi erentbandpasses(left:B;middle:V;right:I).RRLvariablesareshowningray,andnon-variablestarsinblack.Lowerpanels:CorrespondingRRLdistributionsintheabsolutemagnitude—log-periodplane.Thecorrelationcoe cientrisshowninthelowerpanels.AllplotsrefertoanHBsimulationwithZ=0.002andanintermediateHBtype,asindicatedintheupperpanels.

theHBaroundtheRRLregionbecomesdistinctlynon-horizontalbothtowardsthenear-ultraviolet(e.g.,Fig.4inFerraroetal.1998)andtowardsthenear-infrared(e.g.,Davidge&Courteau1999),wepresentafullanal-ysisoftheslopeandzeropointoftheRRLPLrelationintheJohnsons-Cousins-Glasssystem,fromUtoK,in-cludingalsoBVRIJH.

2.MODELS

TheHBsimulationsemployedinthepresentpaperaresimilartothosedescribedinCatelan(2004a),towhichthereaderisreferredforfurtherdetailsandreferencesabouttheHBsynthesismethod.TheevolutionarytracksemployedherearethosecomputedbyCatelanetal.(1998)forZ=0.001andZ=0.0005,andbySweigart&Catelan(1998)forZ=0.002andZ=0.006,andas-sumeamain-sequenceheliumabundanceof23%bymassandscaled-solarcompositions.ThemassdistributionisrepresentedbyanormaldeviatewithamassdispersionσM=0.020M⊙.Forthepurposesofthepresentpa-per,wehaveaddedtothiscodebolometriccorrectionsfromGirardietal.(2002)forURJHKovertherele-vantrangesoftemperatureandgravity.Thewidthoftheinstabilitystripistakenas logTe =0.075,whichprovidesthetemperatureoftherededgeoftheinstabil-itystripforeachstaronceitsblueedgehasbeencom-putedonthebasisofRRLpulsationtheoryresults.Morespeci cally,theinstabilitystripblueedgeadoptedinthispaperisbasedonequation(1)ofCaputoetal.(1987),whichprovidesa ttoStellingwerf’s(1984)results—exceptthatashiftby 200Ktothetemperatureval-uesthusderivedwasappliedinordertoimproveagree-

mentwithmorerecenttheoreticalprescriptions(see§6inCatelan2004aforadetaileddiscussion).Weincludebothfundamental-mode(RRab)and“fundamentalized” rst-overtone(RRc)variablesinour nalPLrelations.Thecomputedperiodsarebasedonequation(4)inCa-puto,Marconi,&Santolamazza(1998),whichrepresentsanupdatedversionofthevanAlbada&Baker(1971)period-meandensityrelation.

InordertostudythedependenceofthezeropointandslopeoftheRRLPLrelationwithbothHBtypeandmetallicity,wehavecomputed,foreachmetallicity,se-quencesofHBsimulationswhichproducefromverybluetoveryredHBtypes.Thesesimulationsarestandard,anddonotincludesuche ectsasHBbimodalityortheimpactofsecondparametersotherthanmasslossontheredgiantbranch(RGB)orage.Foreachsuchsimula-tion,linearrelationsofthetypeMX=a+blogP,inwhichXrepresentsanyoftheUBVRIJHKbandpasses,wereobtainedusingtheIsobeetal.(1990)“OLSbisec-tor”technique.Itiscrucialthat,iftheserelationsaretobecomparedagainstempiricaldatatoderivedistances,preciselythesamerecipebeemployedintheanalysisofthesedataaswell,particularlyincasesinwhichthecor-relationcoe cientisnotverycloseto1.The nalresultforeachHBmorphologyactuallyrepresentstheaveragea,bvaluesover100HBsimulationswith500starseach.

3.GENESISOFTHERRLPLRELATION

InFigure1,weshowanHBsimulationcomputedforametallicityZ=0.002andanintermediateHBmorphology,indicatedbyavalueoftheLee-ZinntypeL≡(B R)/(B+V+R)= 0.05(whereB,V,

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

TheRRLPLrelationinUBVRIJHK3

Fig.2.—PLrelationsinseveraldi erentpassbands.Upperpanels:U(left),B,V,R(right).Lowerpanels:I(left),J,H,K(right).Thecorrelationcoe cientisshowninallpanels.AllplotsrefertoanHBsimulationwithZ=0.001andanintermediateHBtype.

andRarethenumbersofblue,variable,andredHBstars,respectively).EvenusingonlythemoreusualBVIbandpassesoftheJohnson-Cousinssystem,thechangeinthedetailedmorphologyoftheHBwiththepassbandadoptedisobvious.Inthemiddleupperpanel,onecanseethetraditionaldisplayofa“horizontal”HB,asob-tainedintheMV,B Vplane.Asaconsequence,onecansee,inthemiddlelowerpanel,thatnoPLrelationre-sultsusingthisbandpass.Ontheotherhand,theupperleftpanelshowsthesamesimulationintheMB,B Vplane.OneclearlyseesnowthattheHBisnotanymore“horizontal.”ThishasaclearimpactupontheresultingPLrelation(lowerleftpanel):nowonedoesseeanindi-cationofacorrelationbetweenperiodandMB,thoughwithalargescatter.Thereasonforthisscatteristhatthee ectsofluminosityandtemperaturevariationsupontheexpectedperiodsarealmostorthogonalinthisplane.Nowonecanalsosee,intheupperrightpanel,thattheHBisalsonotquitehorizontalintheMI,B Vplane—onlythatnow,incomparisonwiththeMB,B Vplane,thestarsthatlookbrighterarealsotheonesthatarecooler.Sinceadecreaseintemperature,aswellasanincreaseinbrightness,bothleadtolongerperiods,oneexpectsthee ectsofbrightnessandtemperatureupontheperiodstobemorenearlyparallelwhenusingI.Thisisindeedwhathappens,ascanbeseeninthebottomrightpanel.Wenow ndaquitereasonablePLrelation,withmuchlessscatterthanwasthecaseinB.

ThesameconceptsexplainthebehavioroftheRRLPLrelationintheotherpassbandsoftheJohnson-Cousins-Glasssystem,whichbecomestighterbothtowardsthenear-ultravioletandtowardsthenear-infrared,ascom-paredtothevisual.InFigure2,weshowthePLrelationsinalloftheUBVR(upperpanels)andIJHK(lowerpanels)bandpasses,forasyntheticHBwithamorphol-ogysimilartothatshowninFigure1,butcomputedforametallicityZ=0.001(theresultsarequalitativelysim-

ilarforallmetallicities).Asonecansee,asonemovesredwardfromV,wheretheHBise ectivelyhorizontalattheRRLlevel,anincreasinglytighterPLrelationde-velops.Conversely,asonemovesfromVtowardstheultraviolet,theexpectationisalsoforthePLrelationtobecomeincreasinglytighter—whichiscon rmedbytheplotforB.InthecaseofbroadbandU,ascanbeseen,theexpectedtendencyisnotfullycon rmed,ane ectwhichweattributetothecomplicatingimpactoftheBalmerjumpuponthepredictedbolometriccorrectionsintheregionofinterest.1AninvestigationoftheRRLPLrelationinStr¨omgrenu(e.g.,Clemetal.2004),whichismuchlessa ectedbytheBalmerdiscontinuity(andmightaccordinglyproduceatighterPLrelationthaninbroadbandU),aswellasoftheUVdomain,shouldthusproveofinterest,buthasnotbeenattemptedinthepresentwork.

4.THERRLPLRELATIONCALIBRATED

InFigure3,weshowtheslope(leftpanels)andzeropoint(rightpanels)ofthetheoretically-calibratedRRLPLrelation,inUBVR(fromtoptobottom)andforfourdi erentmetallicities(asindicatedbydi erentsymbolsandshadesofgray;seethelowerrightpanel).Eachdat-apointcorrespondstotheaverageover100simulationswith500starsineach.The“errorbars”correspondtothestandarddeviationofthemeanoverthese100sim-ulations.Figure4isanalogoustoFigure3,butshowsinsteadourresultsfortheIJHKpassbands(fromtop

TheBalmerjumpoccursataroundλ≈3700 A,markingtheasymptoticendoftheBalmerlineseries—andthusadiscontinuityintheradiativeopacity.ThebroadbandU lterextendswellred-wardof4000 A,andisthusstronglya ectedbythedetailedphysicscontrollingthesizeoftheBalmerjump.InthecaseofStr¨omgrenu,ontheotherhand,thetransmissione ciencyispracticallyzeroalreadyatλ=3800 A,thusshowingthatitisnotseverelya ectedbythesizeoftheBalmerjump.

1

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

4M.Catelan,B.J.Pritzl,H.A.Smith

Fig.3.—TheoreticallycalibratedPLrelationsintheUBVRpassbands(fromtoptobottom),forthefourindicatedmetallicities.Thezeropoints(leftpanels)andslopes(rightpanels)aregivenasafunctionoftheLee-ZinnHBmorphologyindicator.

tobottom).Itshouldbenotedthat,forallbandpasses,thecoe cientsofthePLrelationsaremuchmoresub-jecttostatistical uctuationsattheextremesinHBtype(bothveryredandveryblue),duetothesmallernum-bersofRRLvariablesfortheseHBtypes.IntermsofFigures3and4,thisisindicatedbyanincreaseinthesizeofthe“errorbars”atboththeblueandredendsoftherelations.

TheslopesandzeropointsfortheUBVRIJHKcal-ibrationsaregiveninTables1through8,respectively.AppropriatevaluesforanygivenHBmorphologymay

beobtainedfromthesetablesbydirectinterpolation,orbyusingsuitableinterpolationformulae(Catelan2004b),whichwenowproceedtodescribeinmoredetail.

4.1.AnalyticalFits

AstheplotsinFigures3and4show,allbandsshowsomedependenceonbothmetallicityandHBtype,thoughsomeofthee ectsclearlybecomelesspro-nouncedasonegoestowardsthenear-infrared.Analysisofthedataforeachmetallicityshowsthat,exceptfortheUandBcases,thecoe cientsofallPLrelations(at

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

TheRRLPLrelationinUBVRIJHK5

Fig.4.—AsinFigure3,butforIJHK(fromtoptobottom).

a xedmetallicity)canbewelldescribedbythird-orderpolynomials,asfollows:

MX=a+blogP,

with

a=

3 i=0

5.REMARKSONTHERRLPLRELATIONS

(1)

ai(L),b=

i

3 i=0

bi(L).

i

(2)

ForalloftheVRIJHKpassbands,theai,bicoe cients

areprovidedinTable9.

Figures3and4revealacomplexpatternforthevaria-tionofthecoe cientsofthePLrelationasafunctionofHBmorphology.While,asanticipated,thedependenceonHBtype(particularlytheslope)isquitesmallfortheredderpassbands(notethemuchsmalleraxisscalerangeforthecorrespondingHandKplotsthanfortheremainingones),thesamecannotbesaidwithrespecttothebluerpassbands,particularlyUandB,forwhichonedoesseemarkedvariationsasonemovesfromveryredtoveryblueHBtypes.Thisisobviouslyduetothemuchmoreimportante ectsofevolutionawayfromthe

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

M. Catelan, B. J. Pritzl, H. A. SmithFig. 5.— Variation in the MU log P relation as a function of HB type, for a metallicity Z= 0.001. The HB morphology, indicated by the L value, becomes bluer from upper left to lower right. For each HB type, only the rst in the series of 100 simulations used to compute the average coe cients shown in Figures 3 and 4 and Table 1 was chosen to produce this gure.

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

The RRL PL relation in U BV RIJHK

Fig. 6.— As in Figure 5, but f

or the MB log P relation.

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

M. Catelan, B. J. Pritzl, H. A. SmithFig. 7.— As in Figure 5, but for the MV log P relation.

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

The RRL PL relation in U BV RIJHK

Fig. 8.— As in Figure 5, but for the MR log P relation.

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

M. Catelan, B. J. Pritzl, H. A. SmithFig. 9.— As in Figure 5, but for the MI log P relation.

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

The RRL PL relation in U BV RIJHK

Fig. 10.— As in Figure 5, but for the MJ log P relation.

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

M. Catelan, B. J. Pritzl, H. A. SmithFig. 11.— As in Figure 5, but for the MH log P relation.

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

The RRL PL relation in U BV RIJHK

Fig. 12.— As in Figure 5, but for the MK log P relation.

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

14M.Catelan,B.J.Pritzl,H.A.Smith

TABLE1

RRLPLRelationinU:CoefficientsoftheFits

Lσ(L)aσ(a)b

σ(b)

Z=0.0005

00.000.9340.00.042 0.00.8777760.0130180.0.0.6270.0220.4970.0.5195470.034 08630.129 0.8490. 0.0.414167.1020.027.0280290.5520.0311.7990.024 0.8270.1101..0450130.0552670..01401000.8600.0791920..037.8257060.9490..058048 0.3580.0310.9930.0100.6300.046 0.5900.0250.9830.0090.5950.041 0.7650.0210.9800.0100.5860.050 0.8830.0140.9770.0130.5870.068 0.950

0.010

0.979

0.038

0.605

0.200

Z=0.0010

00.000.9400..0110.53700.070 00.87374400.0180.5800. 10.1910.070 0.0270. 0.5560.282.0370.0250340.6090..0330351.1.844.1341360.083277 0.9520.2170..10901200.9230..151.8338180.2759620..376055 0.3420.0361.1170.0110.7410.047 0.6030.0251.1070.0120.6980.055 0.7890.0221.0980.0170.6590.081 0.9060.0151.0940.0170.6460.078 0.963

0.008

1.104

0.034

0.701

0.169

Z=0.0020

00.00.9650.00900 0.89000.9107940..0150.0.6757080.22401.8530.207 0.8260.69600.4050. 0.5940.307.0230.0250.029.0360341.1550.1..2552680.2860.250.1170150.0.6430.541.8478650.9208160..383066 0.3560.0351.2580.0170.8200.069 0.6300.0321.2480.0170.7710.069 0.8220.0211.2470.0210.7680.089 0.9280.0121.2420.0300.7390.127 0.974

0.008

1.236

0.077

0.709

0.326

Z=0.0060

00.010.922.8106010.0150.1.03201.0.369 00.4390 00.298.0700.0210.034.0390391.2430.0.1281.9851.445.5355740.3830.285.1860770.0.0340.648.8719590.0.792.520210 0.4200.0361.5820.0240.9580.075 0.6930.0251.5860.0320.9540.095 0.8680.0181.5710.0790.8870.229 0.951

0.012

1.564

0.136

0.866

0.398

zero-ageHBinthebluerpassbands.InordertofullyhighlightthechangesinthePLrelationsineachoftheconsideredbandpasses,weshow,inFigures5through12,thechangesintheabsolutemagnitude–log-perioddistri-butionsforeachbandpass,fromU(Fig.5)toK(Fig.12),forarepresentativemetallicity,Z=0.001.Each gureiscomprisedofamosaicof10plots,eachforadi erentHBtype,fromveryred(upperleftpanels)toveryblue(lowerrightpanels).Inthebluerpassbands,onecanseethestarsthatareevolvedawayfromapositiononthebluezero-ageHB(andthusbrighterforagivenperiod)graduallybecomingmoredominantastheHBtypegetsbluer.Asalreadydiscussed,thee ectsofluminosityandTABLE2

RRLPLRelationinB:CoefficientsoftheFits

L

σ(L)

a

σ(a)

b

σ(b)

Z=0.000500.934000.8770.01300..124110 0.7760.0180.5530 00 0..7638380.5240.0.0220.0.1.5551.7390930.25700.2630.431 0.6270.414167.1020.0270.028.0290311.1040.0580.9330.1..1071060.0.013.0120100.8990.9400..8738510.1990440..040037 0.3580.0311.1050.0100.8360.035 0.5900.0251.1050.0100.8310.036 0.7650.0211.1070.0110.8390.047 0.8830.0141.1050.0130.8380.060 0.950

0.010

1.108

0.020

0.853

0.100

Z=0.001000.0000.9400.0.5680. 00.8737440.0110.0180250.0.123196 0.0.4370.271000.665 0.5560.282.0370.0341.6510..0330351.9941951..2022060.1.362.9667680.9440.016.0130150.0010..9739550.0.048.046051 0.3420.0361.2050.0130.9350.044 0.6030.0251.2050.0130.9230.048 0.7890.0221.2030.0150.9060.067 0.9060.0151.2000.0190.8950.076 0.963

0.008

1.207

0.036

0.930

0.164

Z=0.002000.96500.0.00900 0.910.7945940.0150.741 000.1.8220.30000.6590. 00.307.0230.0251.0.1..4420.9740.029.0360341.1160.3051.302.3153200.2940680..0170171.4660220.9729391..0240110.2140..059063 0.3560.0351.3190.0180.9900.066 0.6300.0321.3200.0190.9690.066 0.8220.0211.3200.0210.9690.080 0.9280.0121.3210.0330.9550.125 0.974

0.008

1.315

0.084

0.923

0.346

Z=0.006000.922010.8100.0.0151.0720 1.2880.41300.1 0.6010.298.0700.0210340..0390391.0.3770.3702051.1.495.5635740.0.212.0790271.9070.1101..0670680.0205890..222069 0.4200.0361.5800.0281.0600.078 0.6930.0251.5880.0351.0650.099 0.8680.0181.5730.1080.9920.317 0.951

0.012

1.571

0.145

0.987

0.420

temperatureupontheperiod-absolutemagnitudedistri-butionarealmostorthogonalinthesebluerpassbands.Asaconsequence,whenthenumberofstarsevolvedawayfromthebluezero-ageHBbecomescomparabletothenumberofstarsonthemainphaseoftheHB,whichoc-cursatL～0.4 0.8,asharpbreakinsloperesults,fortheUandBpassbands,ataroundtheseHBtypes.Thee ectismorepronouncedatthelowermetallicities,wheretheevolutionarye ectisexpectedtobemoreim-portant(e.g.,Catelan1993).Fortheredderpassbands,includingthevisual,thechangesaresmootherasafunc-tionofHBmorphology.

ThedependenceoftheRRLPLrelationinIJHKon

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

TheRRLPLrelationinUBVRIJHK

15

TABLE3

RRLPLRelationinV:CoefficientsoftheFitsL

σ(L)

a

σ(a)

b

σ(b)

Z=0.0005

00.934000.8770.00. 00.0.0130.0.038 0.9730.0.776.6274140.0180220.2030.0270.2312740.029021 0.9470.017 0.8710.126 0.8150.096065 00.167.1020.0.3050.028.0290310.0.338.3603740.0.013.013014 0.7490.049 00..7237140.0.041.045053 0.3580.0310.3850.017 0.7230.069 0.5900.0250.4150.096 0.6390.409 0.7650.0210.4480.134 0.5300.600 0.8830.0140.5100.158 0.2710.745 0.950

0.010

0.520

0.137

0.249

0.689

Z=0.0010

00.94000 00.8730.0110.21500.0.065036 1.1620.0.0.0.0180.2650.028 1.0600.188103 0.7440.556282.0370.0250340.0..0330350.3120.351.3833990. 0.9670.0880.019.016016 0.879 00..8158010.0.057.044057 0.3420.0360.4280.096 0.7410.359 0.6030.0250.4680.147 0.6270.567 0.7890.0220.5960.205 0.1460.824 0.9060.0150.6250.183 0.0370.762 0.963

0.008

0.664

0.152

0.126

0.678

Z=0.0020

00.96500.0.0090000.910.7945940.0.2760. 0.4150.015.0250290.3100.142063 1.195 1.118 00.307.02300.0.3610.041 1.0070.1720..0360340.401.4294640.0.024.023097 0.9190. 00..8657840.1070.068.061314 0.3560.0350.4770.092 0.7710.312 0.6300.0320.5330.171 0.6100.590 0.8220.0210.6300.224 0.2930.796 0.9280.0120.7330.1960.0720.715 0.974

0.008

0.737

0.170

0.073

0.654

Z=0.0060

00.000.922.8106010.0.0150.00.177 00.428454 00.298.0700.0210340..0390390.4870.065 1 1..1030610.4450..5135510.0.046.033074 10. 0.0.001.9578800.1520.113.084201 0.4200.0360.5840.117 0.8150.325 0.6930.0250.6420.201 0.6770.561 0.8680.0180.7890.273 0.2800.781 0.951

0.012

0.840

0.257

0.142

0.748

theadoptedwidthofthemassdistribution,aswellasontheheliumabundance,hasbeenanalyzedbycomput-ingadditionalsetsofsyntheticHBsforσforamain-sequenceheliumM=0.030M⊙(Z=0.001)andabundanceof28%(Z=0.002).TheresultsareshowninFigure13.AscanbeseenfromtheIplots,thepreciseshapeofthemassdistributionplaysbutaminorroleinde ningthePLrelation.Ontheotherhand,thee ectsofasignif-icantlyenhancedheliumabundancecanbemuchmoreimportant,particularlyinregardtothezeropointofthePLrelationsinallfourpassbands.Therefore,cautionisrecommendedwhenemployinglocallycalibratedRRLPLrelationstoextragalacticenvironments,inviewofthe

TABLE4

RRLPLRelationinR:CoefficientsoftheFits

L

σ(L)

a

σ(a)

b

σ(b)

Z=0.000500.934000.8770.013 0. 0.0. 00.031 10 1..1950.1080.776.6274140.018 0.1320.022027 0.10600..0650280.021 10.068 00.167.1020.0280..0290310.0.017014 0.163.0810.0.013.0470700.0110..011009 00.0490.040033 0.9960.891.8017380..033031 0.3580.0310.0870.008 0.6890.029 0.5900.0250.0980.008 0.6560.032 0.7650.0210.1070.010 0.6300.039 0.8830.0140.1100.012 0.6190.055 0.950

0.010

0.110

0.020

0.634

0.102

Z=0.001000.00. 00.9408730. 00.7440.011 00.018025 0.1190.051 1.0.0.032026 10.148089 0.5560.282.0370.03400.0700..0330350..0210.023.0650980.018 1.354.2480.0780..017015 1 0.1340.017.9068150.0.054.051047 0.3420.0360.1250.014 0.7400.045 0.6030.0250.1430.014 0.6870.049 0.7890.0220.1590.016 0.6390.058 0.9060.0150.1670.019 0.6100.070 0.963

0.008

0.167

0.027

0.623

0.116

Z=0.002000..9650 00.009 0.0710 100.0.910.7945940.0.01500.0270.0.0.092 1..401 10.268 00.307.0230.0250290..0360340.0270.0590430.074.1101450.0280..029 1.290.1580.163118024 00390..9488540.0820..082070 0.3560.0350.1650.020 0.8030.061 0.6300.0320.1910.022 0.7320.067 0.8220.0210.2010.029 0.7080.095 0.9280.0120.2240.036 0.6400.116 0.974

0.008

0.233

0.049

0.619

0.183

Z=0.006000.92200..8100.01500.0210.00.103 00.252 00.601298.0700.0.0601110.070 1 1..3320.1680.034.0390390.1470..1792170.0510..041041 1 1.2170.142.0769910.1270..106103 0.4200.0360.2460.039 0.9280.101 0.6930.0250.2700.048 0.8760.125 0.8680.0180.3150.055 0.7670.146 0.951

0.012

0.320

0.080

0.763

0.221

possibilityofdi erentchemicalenrichmentlaws.Fromatheoreticalpointofview,aconclusiveassessmentofthee ectsofheliumdi usiononthemainsequence,dredge-uponthe rstascentoftheRGB,andnon-canonicalheliummixingontheupperRGB,willallberequiredbeforecalibrationssuchasthepresentonescanbecon-sidered nal.

6.“AVERAGE”RELATIONS

InapplicationsoftheRRLPLrelationspresentedinthispaperthusfartoderivedistancestoobjectswhoseHBtypesarenotknownapriori,asmayeasilyhappeninthecaseofdistantgalaxiesforinstance,some“av-

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

16M.Catelan,B.J.Pritzl,H.A.Smith

TABLE5

RRLPLRelationinI:CoefficientsoftheFits

Lσ(L)aσ(a)b

σ(b)

Z=0.0005

00.934000.0.013 0.023 0.8770.018 0.0. 1.45300.7766270. 00.022 0.343.3220.015 1.4260. 1.3640.082046 0.4140.167.1020.027028 0 0.291.2612310.012010 1.2940.0360..029031 00..2071920.0090..008007 1.2150. 11..1501090.0300270..026022 0.3580.031 0.1820.006 1.0800.020 0.5900.025 0.1760.006 1.0600.023 0.7650.021 0.1710.007 1.0440.029 0.8830.014 0.1700.008 1.0380.037 0.950

0.010

0.171

0.014

1.050

0.070

Z=0.0010

00.0 00.9408730.011 0.0.037 00.7440.0.018 00. 10.1070.025 1..5680. 0.0.556282.0370.025034 0.318.2790..033035 0 0.2390.204.1721480.019014 10.0680570..013012 1.482.385 11.292.2081400.0420..039037 0.3420.036 0.1300.011 1.0890.034 0.6030.025 0.1190.010 1.0550.033 0.7890.022 0.1100.011 1.0280.037 0.9060.015 0.1060.014 1.0120.051 0.963

0.008

0.105

0.020

1.013

0.086

Z=0.0020

00.96500..9100. 0.2650 1.59200.7945940.0090.015025 0.2300. 0.1850.074 1.5030. 1.3900.216128 00.307.0230.0290..036034 0.1480.047035 00..1200940.0230..022018 1.2950.097 11..2251540.0.067.062052 0.3560.035 0.0800.014 1.1190.044 0.6300.032 0.0620.016 1.0700.048 0.8220.021 0.0560.020 1.0540.066 0.9280.012 0.0410.025 1.0110.080 0.974

0.008

0.036

0.037

0.999

0.137

Z=0.0060

00..9220 0 0.8106010.0.015021 0.0.088 1.5230 00.298.0700.034 0 0.142.0980.057 1.4230.0.2141390..039039 00.068.0420130.0.042.034033 1.358 11..3022370.1040..086083 0.4200.0360.0100.031 1.1880.080 0.6930.0250.0300.037 1.1450.098 0.8680.0180.0620.042 1.0680.112 0.951

0.012

0.066

0.061

1.061

0.168

erage”formofthePLrelationmightbeusefulwhichdoesnotexplicitlyshowadependenceonHBmorphol-ogy.Intheredderpassbands,inparticular,ameaningfulrelationofthattypemaybeobtainedwhenoneconsid-ersthatthedependenceofthezeropointsandslopesofthecorrespondingrelations,aspresentedinthepre-vioussections,isfairlymild.Therefore,inthepresentsection,wepresent“average”relationsforI,J,H,K,obtainingbysimplygatheringtogetherthe389,484starsinallofthesimulationsforallHBtypesandmetallicities(0.0005≤Z≤0.006).Utilizingasimpleleast-squaresprocedurewiththelog-periodsandlog-metallicitiesas

TABLE6

RRLPLRelationinJ:CoefficientsoftheFits

L

σ(L)

a

σ(a)

b

σ(b)

Z=0.000500.000.9340.013 0..012 100.8777760. 0.82600.6270.018 00.022027 0..8158000.007007 1.9020.0450.006 1.8900.024 1.8610. 0.0.414167.1020.028 0.7860..029031 0.772 00..7607540.0.005.005004 1.8250.020 1.7881..7577390.0170150..015012 0.3580.031 0.7500.003 1.7270.011 0.5900.025 0.7480.004 1.7170.014 0.7650.021 0.7470.004 1.7100.018 0.8830.014 0.7460.005 1.7060.021 0.950

0.010

0.747

0.008

1.709

0.040

Z=0.001000.9400 00.0. 0.7950. 00.873.7445560.0110180.025 0.7740.020014 1.9810.0.010 1.9340.057039 00.282.0370. 0.7510.034.033035 0.733 00..7177050. 1.8790.0300.008.007007 1.830 11..7867520.0.024.021020 0.3420.036 0.6960.006 1.7260.019 0.6030.025 0.6910.005 1.7100.017 0.7890.022 0.6870.006 1.6980.020 0.9060.015 0.6850.008 1.6890.030 0.963

0.008

0.684

0.011

1.685

0.049

Z=0.002000.00.9659100.009 0.0 0.7940. 200..5943070.015 00.042 0.748.7280. 1..0059550.123 10. 0.0230.0250290..036034 00.027 0.7020.681.6676530.0200130..012010 1..8908370.072056 11..8007630.0380..034028 0.3560.035 0.6470.008 1.7450.024 0.6300.032 0.6380.009 1.7210.027 0.8220.021 0.6350.011 1.7130.034 0.9280.012 0.6280.014 1.6910.044 0.974

0.008

0.625

0.020

1.682

0.076

Z=0.006000.9220 0..8100.0.015 00 0.6490.0.052033 10 1..9899290.0.125 00.601298.0700.0210340..039039 0.623 0.6060..5915750.0.024.019019 1.891 11..8608260.0800590..049047 0.4200.036 0.5630.017 1.8000.044 0.6930.025 0.5530.021 1.7770.053 0.8680.018 0.5360.023 1.7390.062 0.951

0.012

0.534

0.033

1.733

0.092

independentvariables,weobtainthefollowing ts:2

MI=0.4711 1.1318logP+0.2053logZ,(3)

withacorrelationcoe cientr=0.967;

2

rorsFor10 5in 10theallequations 3derivedcoe cientspresentedinarethisalwayssection,verythestatisticaler-corresponding,duetotheverylargesmall,oforderthatthesewe ts.Consequently,wenumberomitthemofstarsfrominvolvedtheequationsintheabundancesrelationsprovide.areUndoubtedly,systematicratherthemainthansourcesstatistical—e.g.,oferrora ectingheliumcientoftheperiod-mean(see§5),bolometricdensityrelation,corrections,etc..

temperaturecoe -

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

TheRRLPLrelationinUBVRIJHK

17

TABLE7

RRLPLRelationinH:CoefficientsoftheFitsL

σ(L)

a

σ(a)

b

σ(b)

Z=0.0005

00.93400.013 0.018 10 2.31100.0. 1..1360.003 2.3150.0.8777760.022 10.002 1.137.1370. 2.3170.0130.627 2.3160.009008 0.0.414167.1020.0270280..029031 10.002002 1.1371.137.1381390.0020..002002 2.3160.007 22..3153160.0.006.007006 0.3580.031 1.1410.002 2.3200.006 0.5900.025 1.1420.002 2.3200.008 0.7650.021 1.1430.003 2.3230.011 0.8830.014 1.1440.003 2.3220.013 0.950

0.010

1.146

0.005

2.327

0.027

Z=0.0010

00.940000.8730.011 0.006 00.0. 10.018 1..1010. 2.0.0190.005 20.014 0.7440.556282.0370.025034 10..033035 1.096.090 11.086.0830810.003003 2.358.3460.0..002002 2 2.3322.322.3103040.0090.009.007008 0.3420.036 1.0790.003 2.2970.008 0.6030.025 1.0790.002 2.2950.008 0.7890.022 1.0790.003 2.2950.011 0.9060.015 1.0800.004 2.2920.016 0.963

0.008

1.080

0.006

2.292

0.025

Z=0.0020

00.0 0.9659100.009 10 200.0.7940.0150. 1.0580. 1.0510.015009 2.380 2.3610.045025 0.0.594307.0230.0250.029.036034 1.0420.007 1.0351..0310270..005005 2.3390.0210.004 2.3212..3082970.0.014.013012 0.3560.035 1.0250.003 2.2910.010 0.6300.032 1.0230.004 2.2840.012 0.8220.021 1.0230.004 2.2820.014 0.9280.012 1.0210.006 2.2750.019 0.974

0.008

1.020

0.009

2.271

0.034

Z=0.0060

00.92200.0. 0000.0150. 2.0.053 0.8100.601298.0700.021034 0.9750.0220140.0..039039 0.964 0.9570..9509440.010 20..008008 2.381.355 22.338.3253120.0350.026.021020 0.4200.036 0.9390.007 2.3010.019 0.6930.025 0.9350.009 2.2920.023 0.8680.018 0.9290.010 2.2770.027 0.951

0.012

0.928

0.014

2.274

0.040

MJ= 0.1409 1.7734logP+0.1899logZ,(4)

withacorrelationcoe cientr=0.9936;

MH= 0.5508 2.3134logP+0.1780logZ,(5)

withacorrelationcoe cientr=0.9991;

MK= 0.5968 2.3529logP+0.1746logZ,

(6)

withacorrelationcoe cientr=0.9992.Notethatthelatterrelationisofthesameformastheonepresentedby

TABLE8

RRLPLRelationinK:CoefficientsoftheFits

L

σ(L)

a

σ(a)

b

σ(b)

Z=0.0005

00.000.9340.013 0. 00.8777760. 1.1680.002002 2 2..3430.0120.6270.018 1.1690.022027 1.1700.0020. 20.009 2.348.3520. 0.0.414167.1020.028 1.1710..029031 1.172 11..1731750.0020.002.002002 20.007 2.3522.355.3553580.0070060..007006 0.3580.031 1.1770.002 2.3620.006 0.5900.025 1.1780.002 2.3640.008 0.7650.021 1.1800.003 2.3680.011 0.8830.014 1.1810.003 2.3670.013 0.950

0.010

1.183

0.005

2.3730.026

Z=0.001000.9400 00.0. 1.1330. 00.873.7445560.0110180.025 1.1280.005004 2.3880.0.003 2.379 2.3670.0170130.009 00.282.0370. 1.1240.034.033035 1.121 11.118.11700..003002.002 2.359 22..3503450.0.009.007008 0.3420.036 1.1150.002 2.3390.008 0.6030.025 1.1160.002 2.3380.008 0.7890.022 1.1170.003 2.3390.011 0.9060.015 1.1170.004 2.3370.016 0.963

0.008

1.11

70.006

2.338

0.024

Z=0.002000.00.9659100.009 1.0 0.7940. 200..5943070.015 10.014 1.091.0840. 2..4103930.041 20. 0.0230.0250290..036034 10.008 1.0771.071.0670640.0070040..004004 2..3743580.022019 22..3473370.0130..012011 0.3560.035 1.0630.003 2.3330.010 0.6300.032 1.0610.004 2.3270.012 0.8220.021 1.0610.004 2.3250.013 0.9280.012 1.0590.006 2.3200.018 0.974

0.008

1.059

0.009

2.317

0.032

Z=0.006000.9220 0..8100.0.015 10 1.0110.0.021013 20 2..4133890.0.049 00.601298.0700.0210340..039039 0.001 0.9940..9889830.0.010.008008 2.374 22..3623490.0320240..019019 0.4200.036 0.9780.007 2.3400.018 0.6930.025 0.9740.008 2.3310.021 0.8680.018 0.9680.009 2.3170.025 0.951

0.012

0.968

0.013

2.315

0.037

Bonoetal.(2001).ThemetallicitydependencewederiveisbasicallyidenticaltothatinBonoetal.,whereasthelogPslopeisslightlysteeper(by0.28),inabsolutevalue,inourcase.Intermsofzeropoints,thetworelations,atrepresentativevaluesP=0.50dandZ=0.001,provideK-bandmagnitudeswhichdi erbyonly0.05mag,oursbeingslightlybrighter.

Inaddition,thesametypeofexercisecanprovideuswithanaveragerelationbetweenHBmagnitudeinthevisualandmetallicity.Performinganordinaryleast-squares toftheformMV=f(logZ)(i.e.,withlogZastheindependentvariable),weobtain:

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

18M.Catelan,B.J.Pritzl,H.A.Smith

TABLE9

CoefficientsoftheRRLPLRelationinBVRIJHK:AnalyticalFits

Metallicitya0a1a2a3V

b0b1b2b3

ZZZZ====0.00600.00200.00100.00050.52760.44700.39760.3668 0.0420 0.0232 0.0416 0.03140.10560.06660.0482 0.0053 0.2011 0.2309 0.2117 0.1550R

0.9440 0.8512 0.8243 0.7455 0.04360.0122 0.02310.04070.32070.31070.32120.1342 0.5303 0.7248 0.7169 0.4966

ZZZZ====0.00600.00200.00100.00050.21070.14690.10200.0659 0.0701 0.0566 0.0808 0.0811 0.0204 0.0633 0.0784 0.0832 0.0764 0.0995 0.0744 0.0537I

1.0050 0.8523 0.8086 0.7553 0.1507 0.1500 0.2397 0.2482 0.0390 0.1519 0.1809 0.1769 0.1723 0.2521 0.1608 0.0677

ZZZZ====0.00600.00200.00100.0005 0.0162 0.0913 0.1445 0.1939 0.0540 0.0393 0.0545 0.0515 0.0216 0.0568 0.0680 0.0679 0.0640 0.0788 0.0614 0.0450J

1.2459 1.1499 1.1340 1.1192 0.1155 0.1058 0.1647 0.1640 0.0442 0.1397 0.1614 0.1474 0.1466 0.2002 0.1378 0.0655

ZZZZ====0.00600.00200.00100.0005 0.5766 0.6517 0.7023 0.7546 0.0278 0.0181 0.0250 0.0209 0.0150 0.0335 0.0383 0.0343 0.0378 0.0452 0.0355 0.0238H

1.8291 1.7599 1.7478 1.7438 0.0581 0.0498 0.0786 0.0728 0.0318 0.0809 0.0891 0.0706 0.0874 0.1167 0.0824 0.0379

ZZZZ====0.00600.00200.00100.0005 0.9447 1.0262 1.0798 1.1385 0.0110 0.0041 0.00370.0037 0.0070 0.0124 0.0108 0.0027 0.0160 0.0152 0.00790.0012K

2.3128 2.2953 2.3019 2.3169 0.0231 0.0138 0.01700.0016 0.0149 0.0292 0.0242 0.0029 0.0375 0.0414 0.01860.0056

ZZZZ====0.00600.00200.00100.0005 0.9829 1.0630 1.1159 1.1742 0.0101 0.0032 0.00240.0051 0.0066 0.0113 0.0093 0.0012 0.0148 0.0132 0.00610.0028 2.3499 2.3357 2.3427 2.3580 0.0210 0.0115 0.01310.0061 0.0140 0.0268 0.0212 0.0004 0.0346 0.0361 0.01420.0090

MV=1.455+0.277logZ,(7)

logZ=[M/H] 1.765.(9)

withacorrelationcoe cientr=0.83.

Theaboveequationhasastrongerslopethanisoftenadoptedintheliterature(e.g.,Chaboyer1999).ThisislikelyduetothefactthattheMV [M/H]relationisac-tuallynon-linear,withtheslopeincreasingforZ 0.001(Castellanietal.1991),wheremostofoursimulationswillbefound.Aquadraticversionofthesameequationreadsasfollows:

MV=2.288+0.8824logZ+0.1079(logZ)2.

(8)

Thee ectsofanenhancementinα-captureelementswithrespecttoasolar-scaledmixture,asindeedobservedamongmostmetal-poorstarsintheGalactichalo,canbetakenintoaccountbythefollowingscalingrelation(Salaris,Chie ,&Straniero1993):

[M/H]=[Fe/H]+log(0.638f+0.362),

(10)

Asonecansee,theslopeprovidedbythisrelation,atametallicityZ=0.001,is0.235,thusfullycompatiblewiththerangediscussedbyChaboyer(1999).

Thelasttwoequationscanalsobeplacedintheirmoreusualform,with[M/H](or[Fe/H])astheindependentvariable,ifwerecall,fromSweigart&Catelan(1998),thatthesolarmetallicitycorrespondingtothe(scaled-solar)evolutionarymodelsutilizedinthepresentpaperisZ=0.01716.Therefore,theconversionbetweenZand[M/H]thatisappropriateforourmodelsisasfollows:

wheref=10[α/Fe].NotethatsucharelationshouldbeusedwithduecareformetallicitiesZ>0.003(Vanden-Bergetal.2000).

Withtheseequationsinmind,thelinearversionoftheMV [M/H]relationbecomes

MV=0.967+0.277[M/H],

whereasthequadraticonereadsinstead

MV=1.067+0.502[M/H]+0.108[M/H]2.

(12)(11)

ThelatterequationprovidesMV=0.60magat[Fe/H]= 1.5(assuming[α/Fe] 0.3;e.g.,Carney1996),in

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

TheRRLPLrelationinUBVRIJHK19

Fig.13.—E ectsofσMandoftheheliumabundanceupontheRRLPLrelationintheIJHKbands.Thelinesindicatetheanalytical tsobtained,fromequations(1)and(2),forourassumedcaseofYMS=0.23andσM=0.02M⊙forametallicityZ=0.001(thickgraylines)orZ=0.002(dashedlines).Inthetwoupperpanels,theresultsofanadditionalsetofmodels,computedbyincreasingthemassdispersionfrom0.02M⊙to0.03M⊙,areshown(graycircles).Asonecansee,σMdoesnota ecttherelationinasigni cantway,sothatsimilarresultsforanincreasedσMareomittedinthelowerpanels.Theheliumabundance,inturn,isseentoplayamuchmoreimportantrole,particularlyinde ningthezeropointofthePLrelation.

verygoodagreementwiththefavoredvaluesinChaboyer(1999)andCacciari(2003)—thussupportingadistancemodulusfortheLMCof(m M)0=18.47mag.

Toclose,wenotethatthepresentmodels,whichcoveronlyamodestrangeinmetallicities,donotprovideusefulinputregardingthequestionofwhethertheMV [M/H]relationisbetterdescribedbyaparabolaorbytwostraightlines(Bonoetal.2003).Toillustratethis,weshow,inFigure14,theaverageRRLmagnitudesforeachmetallicityvalueconsidered,alongwithequations(11)and(12).Notethatthe“errorbars”actuallyrepresent

thestandarddeviationofthemeanoverthefullsetofRRLstarsinthesimulationsforeach[M/H]value.

7.CONCLUSIONS

WehavepresentedRRLPLrelationsintheband-passesoftheJohnsons-Cousins-GlassUBVRIJHKsys-tem.WhileinthecaseoftheCepheidstheexistenceofaPLrelationisanecessaryconsequenceofthelargerangeinluminositiesencompassedbythesevariables,inthecaseofRRLstarsusefulPLrelationsareinsteadpri-marilyduetotheoccasionalpresenceofalargerangeinbolometriccorrectionswhengoingfromtheblueto

The RR Lyrae Period-Luminosity Relation. I. Theoretical Calibration

20M.Catelan,B.J.Pritzl,H.A.Smith

Fig.14.—PredictedcorrelationbetweenaverageRRLV-bandabsolutemagnitudeandmetallicity.Eachofthefourdatapointsrepresentstheaveragemagnitudeoverthefullsampleofsimulationsforthatmetallicity.The“errorbars”actuallyindicatethestandarddeviationofthemean.Thefullanddashedlinesshowequations(11)and(12),respectively.

therededgeoftheRRLinstabilitystrip.ThisleadstoparticularlyusefulPLrelationsinIJHK,wheretheef-fectsofluminosityandtemperatureconspiretoproducetightrelations.Inbluerpassbands,ontheotherhand,thee ectsofluminositydonotreinforcethoseoftem-perature,leadingtothepresenceoflargescatterintherelationsandtoastrongerdependenceonevolutionarye ects.Weprovideadetailedtabulationofourderivedslopesandzeropointsforfourdi erentmetallicitiesandcoveringvirtuallythewholerangeinHBmorphology,fromveryredtoveryblue,fullytakingintoaccount,forthe rsttime,thedetailede ectsofevolutionawayfromthezero-ageHBuponthederivedPLrelationsinalloftheUBVRIJHKpassbands.Wealsoprovide“av-erage”PLrelationsinIJHK,forapplicationsincases

Bono,G.,Caputo,F.,Castellani,V.,Marconi,M.,&Storm,J.,2001,MNRAS,326,1183

Bono,G.,Caputo,F.,Castellani,V.,Marconi,M.,Storm,J.,&Degl’Innocenti,S.2003,MNRAS,344,1097

Cacciari,C.2003,inNewHorizonsinGlobularClusterAstronomy,ASPConf.Ser.,Vol.296,ed.G.Piotto,G.Meylan,S.G.Djorgovski,&M.Riello(SanFrancisco:ASP),329

Caputo,F.,DeStefanis,P.,Paez,E.,&Quarta,M.L.1987,A&AS,68,119

Caputo,F.,Marconi,M.,&Santolamazza,P.1998,MNRAS,293,364

Carney,B.W.1996,PASP,108,900

Castellani,V.,Chie ,A.,&Pulone,L.1991,ApJS,76,911Catelan,M.1993,A&AS,98,547Catelan,M.2004a,ApJ,600,409

Catelan,M.2004b,inVariableStarsintheLocalGroup,ASPConf.Ser.,Vol.310,ed.D.W.Kurtz&K.Pollard(SanFrancisco:ASP),inpress(astro-ph/0310159)

Catelan,M.,Borissova,J.,Sweigart,A.V.,&Spassova,N.1998,ApJ,494,265

Chaboyer,B.1999,inPost-HipparcosCosmicCandles,ed.A.Heck&F.Caputo(Dordrecht:Kluwer),111

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wheretheHBtypeisnotknownapriori;aswellasanewcalibrationoftheMV [Fe/H]relation.InPaperII,wewillprovidecomparisonsbetweentheseresultsandtheobservations,particularlyinI,whereweexpectournewcalibrationtobeespeciallyusefulduetothewideavail-abilityandeaseofobservationsinthis lter,incompar-isonwithJHK.

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