The peculiar motions of early-type galaxies in two distant regions - V. The Mg-sigma relati

A(MNLTEXstylefilev1.4)

Thepeculiarmotionsofearly-typegalaxiesintwodistant

regions–V.TheMg–σrelation,ageandmetallicity

MatthewColless1,DavidBurstein2,RogerL.Davies3,RobertK.McMahanJr4,5,R.P.Saglia6andGaryWegner7

arXiv:astro-ph/98110v1 5 Nov 1998StromloandSidingSpringObservatories,TheAustralianNationalUniversity,WestonCreek,ACT2611,Australia

ofPhysicsandAstronomy,Box871054,ArizonaStateUniversity,Tempe,AZ85287-1504,U.S.A.

3DepartmentofPhysics,SouthRoad,DurhamDH13LE,UnitedKingdom

4DeptofPhysicsandAstronomy,UniversityofNorthCarolina,CB3255PhillipsHall,ChapelHill,NC27599-3255,U.S.A.5P.O.Box14026,McMahanResearchLaboratories,79AlexanderDrive,ResearchTriangle,NC27709,U.S.A.6Universit¨ats-SternwarteM¨unchen,Scheinerstraße1,D-81679M¨unchen,Germany

7DepartmentofPhysicsandAstronomy,6127WilderLaboratory,DartmouthCollege,Hanover,NH03755-3528,U.S.A.

2Department

1Mount

Accepted—.Received—;inoriginalform—.

ABSTRACT

WehaveexaminedtheMg–σrelationforearly-typegalaxiesintheEFARsampleanditsdependenceonclusterproperties.AcomprehensivemaximumlikelihoodtreatmentofthesampleselectionandmeasurementerrorsgivesfitstotheglobalMg–σrelationofMgb′=0.131logσ−0.131andMg2=0.257logσ−0.305.Theslopeoftheserelationsis25%steeperthanthatobtainedbymostotherauthorsduetothereducedbiasofourfittingmethod.TheintrinsicscatterintheglobalMg–σrelationisestimatedtobe0.016maginMgb′and0.023maginMg2.TheMg–σrelationforcDgalaxieshasahigherzeropointthanforEandS0galaxies,implyingthatcDsareolderand/ormoremetal-richthanotherearly-typegalaxieswiththesamevelocitydispersion.

WeinvestigatethevariationinthezeropointoftheMg–σrelationbetweenclusters.WefinditisconsistentwiththenumberofgalaxiesobservedperclusterandtheintrinsicscatterbetweengalaxiesintheglobalMg–σrelation.WefindnosignificantcorrelationbetweentheMg–σzeropointandtheclustervelocitydispersion,X-rayluminosityorX-raytemperatureoverawiderangeinclustermass.Theseresultsprovideconstraintsformodelsoftheformationofellipticalgalaxies.HowevertheMg–σrelationonitsowndoesnotplacestronglimitsonsystematicerrorsinFundamentalPlanedistanceestimatesduetostellarpopulationdifferencesbetweenclusters.

WecomparetheintrinsicscatterintheMg–σandFundamentalPlane(FP)re-lationswithstellarpopulationmodelsinordertoconstrainthedispersioninages,metallicitiesandM/Lratiosforearly-typegalaxiesatfixedvelocitydispersion.WefindthatvariationsinagealoneormetallicityalonecannotexplainthemeasuredintrinsicscatterinbothMg–σandtheFP.WederivethejointconstraintsonthedispersioninageandmetallicityimpliedbythescatterintheMg–σandFPrelationsforasimpleGaussianmodel.Wefindupperlimitsonthedispersionsinageandmetal-licityatfixedvelocitydispersionof32%inδt/tand38%inδZ/Zifthevariationsinageandmetallicityareuncorrelated;onlystronglyanti-correlatedvariationsleadtosignificantlyhigherupperlimits.ThejointdistributionofresidualsfromtheMg–σandFPrelationsisonlymarginallyconsistentwithamodelhavingnocorrelationbetweenageandmetallicity,andisbetter-matchedbyamodelinwhichageandmetallicityvariationsaremoderatelyanti-correlated(δt/t≈40%,δZ/Z≈50%andρ≈−0.5),withyoungergalaxiesbeingmoremetal-rich.

Keywords:galaxies:distancesandredshifts—galaxies:ellipticalandlenticular,cD—galaxies:stellarcontent—galaxies:formation—galaxies:evolution

c0000RAS󰀁

2

Collessetal.

1

INTRODUCTION

TheprimaryaimoftheEFARproject(Wegneretal.1996;

Paper1)istousethetightcorrelationsbetweentheglobalpropertiesofearly-typegalaxiesembodiedintheFunda-mentalPlane(FP:Djorgovski&Davis1987,Dressleretal.1987)tomeasurerelativedistancestoclustersofgalaxiesinordertoinvestigatepeculiarmotionsandthemassdistri-butiononlargescales.Howevertheseglobalrelationsalsoconstrainthedynamicalpropertiesandevolutionaryhisto-riesofearly-typegalaxies.Forexample,Renzini&Ciotti(1993)showthatthetiltoftheFPimpliesarangeinmass-to-lightratioM/Lamongellipticalsoflessthanafactorofthree,whilethelowscatterabouttheFPimpliesascatterinM/Latanylocationintheplaneoflessthan12%.Simi-larreasoninghasbeenusedtoconstrainthestarformationhistoryofclusterellipticalsusingthecolour–magnituderela-tion(Boweretal.1992,Kodama&Arimoto1997).RecentlytheFP,Mg–σandcolour–magnituderelationshavebeenfollowedouttohigherredshiftsandusedtoshowthattheearly-typegalaxiesseenatz∼1differfrompresent-dayearly-typegalaxiesinamannerconsistentwithpassiveevolution-aryeffects(vanDokkum&Franx1996,Ziegler&Bender1997,Kelsonetal.1997,Ellisetal.1997,Kodama&Ari-moto1997,Stanfordetal.1998,Kodamaetal.1998,Benderetal.1998,vanDokkumetal.1998).

Inthispaperweconsidertherelationbetweenthecen-tralvelocitydispersionσandthestrengthofthemagne-siumlinesatarestwavelengthof5174˚Afortheearly-typegalaxiesintheEFARsample.ThisMg–σrelationconnectsthedynamicalpropertiesofgalaxycoreswiththeirstel-larpopulations.Theremarkablysmallscatteraboutthisrelation(Bursteinetal.1988,Guzm´anetal.1992,Benderetal.1993,Jørgensenetal.1996,Benderetal.1998),anditsdistance-independentnature,makeitapotentiallyusefulconstraintonmodelsofthestarformationhistoryofearly-typegalaxiesandatestforenvironmentalvariationsintheFP(Bursteinetal.1988,Benderetal.1996).

Thereare,however,someproblemswithusingtheMg–σrelationforprobinggalaxyformation.Twooftheseprob-lemsareapparentfromthestellarpopulationmodels(e.g.Worthey1994,Vazdekisetal.1996):(i)bothageandmetal-licitycontributetotheMglinestrengthsincomparablede-gree,sothataspreadinlinestrengthscouldbeduetoeitherarangeofagesorarangeofmetallicitiesorsomecombina-tion;(ii)theMglinestrengthsarenotparticularlysensitiveindicators—atfixedmetallicityadifferenceinageofafac-toroftenonlyresultsinachangeof0.05-0.1mag,whileatfixedageadifferenceof1dexinmetallicitygivesachangeof0.1–0.2mag.ThusMglinestrengthmeasurementsmustbeaccurateinordertoyieldusefulconstraintsontheagesandmetallicitiesofstellarpopulations,andtheMg–σrelationonitsowncanonlysupplyconstraintsoncombinationsofageandmetallicityandnotoneortheotherseparately.

RecentlyTrager(1997)hassuggestedthatthetight-nessoftheMg–σrelationmaybetheresultofa‘conspir-acy’,inthatthereappearstobeananti-correlationbetweentheagesandmetallicitiesofthestellarpopulationsinearly-typegalaxiesatfixedmasswhichactstoreducethescatterintheMglinestrengths.TragertakestheaccurateHβ,MgandFelinestrengthsfromGonz´alez(1993)andappliesthestellarpopulationmodelsofWorthey(1994)toderiveages

andabundancesfromlineindiceswithdifferentdependencesonageandmetallicity.Hefindsthatatfixedvelocitydis-persiontheagesandabundanceslieinaplaneofalmostconstantMglinestrength,leadinghimtopredictlittlescat-terintheMg–σrelationevenforlargedifferencesinageormetallicity—afactorofteninage(from1.5Gyrto15Gyr)givesaspreadinMg2ofonly0.01–0.02mag.Thisconclusiondependsontheappropriatenessofthesinglestellarpopu-lationmodelsandrequiresconfirmationfromfurtherhigh-precisionlinestrengthmeasurements.Itcanalsobetestedusingthehigh-redshiftsamplesnowbecomingavailable.

Inasimilarvein,anumberofauthors(Ferrerasetal.1998,Shioya&Bekki1998,Boweretal.1998)haverecentlyre-examinedwhethertheapparentpassiveevolutionofthecolour–magnituderelationouttoz∼1reallyimpliesahighredshiftforthebulkofthestar-formationinellipticalgalax-ies.Theyconcludethatinfactsuchevolutioncanbecon-sistentwitharatherbroadrangeofagesandmetallicitiesifthegalaxiesassemblingmorerecentlyareonaveragemoremetal-richthanoldergalaxiesofsimilarluminosity.

Aswellasstudiesfocussingontheevolutionofthegalaxypopulation,therehavealsobeeninvestigationsofpossiblevariationswithlocalenvironment.Guzm´anetal.(1992)havesuggestedthattherearesystematicvariationsintheMg–σrelationwhichaffectestimatesofrelativedis-tancesbasedontheFP.TheyreportasignificantoffsetinthezeropointoftheMg–σrelationbetweengalaxiesinthecoreoftheComaclusterandgalaxiesintheclus-terhalo.Jørgensenandco-workers(1996,1997)examineasampleof11clustersandfindaweakcorrelationbe-tweenMglinestrengthandlocaldensitywithintheclus-terwhichisconsistentwiththisresult.SimilaroffsetsareclaimedbetweenfieldandclusterellipticalsbydeCar-valho&Djorgovski(1992)andJørgensen(1997),althoughBursteinetal.(1990)findnoevidenceofenvironmentalef-fects.Suchsystematicdifferencescouldresultfromdifferentstar-formationhistoriesindifferentdensityenvironments,producingvariationsinthemass-to-lightratioofthestellarpopulation.FPdistancemeasurementswouldthenbesub-jecttoenvironment-dependentsystematicerrorsleadingtospuriouspeculiarmotions.Wheredataforfieldandclusterellipticalscomefromdifferentsources,however,thepossi-bilityalsoexiststhatanyzeropointdifferencesareduetouncertaintiesintherelativecalibrationsratherthanintrin-sicenvironmentaldifferences.

TheMg–σrelationhasthusbecomeanimportantdi-agnosticfordeterminationsofboththestarformationhis-toryandthepeculiarmotionsofellipticalgalaxies.HereweexaminetheMg–σrelationintheEFARsample,whichincludesmorethan500early-typegalaxiesdrawnfrom84clustersspanningawiderangeofenvironments.In§2wesummarisetherelevantpropertiesofthesampleandthetechniquesusedtodeterminetheMgbandMg2linestrengthindices,thecentralvelocitydispersionsσ,andtheerrorsinthesequantities.WepresenttheMg–σrelationin§3andinvestigatehowitvariesfromclustertoclusterwithinoursample,andwithclustervelocitydispersion,X-rayluminos-ityandX-raytemperature.In§4wecompareourresultswiththepredictionsofstellarpopulationmodelsinordertoderiveconstraintsontheages,metallicitiesandmass-to-lightratiosofearly-typegalaxiesinclusters.Inparticular,weconsidertheconstraintsonthedispersionintheagesand

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c0000RAS,MNRAS000,000–000metallicitiesfromtheintrinsicscatterintheMg–σrelationonitsown,andincombinationwiththeintrinsicscatterintheFP.Ourconclusionsaregivenin§5.

2THEDATA

Herewegiveashortdescriptionofoursampleanddataset,withemphasisonthevelocitydispersionsandlineindicesusedinthispaper.Theinterestedreadercanfindmorede-tailonthesampleselectioninPaper1(Wegneretal.1996);onthemeasurement,calibrationanderrorestimationproce-duresforthespectroscopicparametersinPaper2(Wegneretal.1998);andonthestructuralandmorphologicalprop-ertiesofthegalaxiesinPaper3(Sagliaetal.1997).2.1

Thesample

TheEFARsampleofgalaxiescomprises736mostlyearly-typegalaxiesin84clusters.TheseclustersspanarangeofrichnessesandlieintworegionstowardHercules–CoronaBorealisandPerseus–Pisces–Cetusatdistancesofbetween6000kms−1and15000kms−1.Inadditiontothisprogramsamplewehavealsoobserved52well-knowngalaxiesinComa,Virgoandthefieldinordertoprovideacalibrat-inglinktopreviousstudies.

TheEFARgalaxiesarelistedinTable2ofPaper1,andcompriseanapproximatelydiameter-limitedsampleofgalaxieslargerthanabout20arcsecwiththevisualappearanceofellipticals.Photometricimaging(Paper3)showsthat8%arecDs,12%arepureEsand49%arebulge-dominatedE/S0s;thus69%ofthesampleareearly-typegalaxies,withtheremaining31%beingspiralsorbarredgalaxies.Wehaveobtainedspectroscopyfor666programgalaxies,measuringredshifts,velocitydispersionsandlinestrengthindices(Paper2).Wehaveusedthered-shiftsweobtainedtogetherwithliteratureredshiftsforothergalaxiesintheclustersinordertoassignprogramgalaxiestophysicalclusters.Wehaveusedthecombinedredshiftdataforthesephysicalclusterstoestimateclustermeanredshiftsandvelocitydispersions.Theearly-typegalaxiesinoursamplespanawiderangeinluminosity,sizeandmass:theyhaveabsolutemagnitudesfromMR=−24to−18(󰀌MR󰀋=−21.6;H0=50kms−1Mpc−1),effectiveradii

from1h−1−1(󰀌Re󰀋=9.1h−1

50kpcto70h50kpc50kpc)andcen-tralvelocitydispersionsfromlessthan100kms−1toover400kms−1(󰀌σ󰀋=220kms−1).Thesampleisthusdominatedbyearly-typegalaxieswithluminosities,sizesandmassestypicalofgiantellipticals.2.2

Themeasurements

Wesummariseheretheproceduresusedinmeasuringtheredshifts,velocitydispersionsandMglinestrengths;fullde-tailsaregiveninPaper2.

RedshiftsandvelocitydispersionsweremeasuredfromeachobservedgalaxyspectrumusingtheIRAFtaskfxcor.LinestrengthindicesontheLicksystemweredeterminedusingtheprescriptiongivenbyGonz´alez(1993).TheMgbandMg2indiceswerebothmeasured:Mg2becauseitistheindexmostcommonlymeasuredinpreviouswork,andMgbbecauseitcouldbemeasuredformoreobjects(asitrequires

Peculiarmotionsofearly-typegalaxies–V

3

Figure1.Asummaryoftheerrorsinlogσ,Mgb′andMg2.Theupperpanelshowsthecumulativedistributionsoftheestimatederrors,withthemedianand90th-percentileerrorsindicated.Thelowerpanelshowsthecalibrationagainsttherepeatobservations:thedistributionoftheratioofrmserrortoestimatederrorforobjectswithrepeatmeasurementsiscomparedtothepredicteddistributionassumingtheestimatederrorsarethetrueerrors.

anarrowerspectralrange)andisbetter-determined(beinglesssusceptibletovariationsinthenon-linearcontinuumshape).Wefinditmoreconvenienttoexpressthe‘atomic’Mgbindexinmagnitudeslikethe‘molecular’Mg˚2indexratherthanasanequivalentwidthinAngstroms,sincethisputsthesetwoindicesonsimilarfootings.TheconversionisMgb′

=−2.5log10󰀁

1−

Mgb

4Collessetal.

Figure2.TheMg–σrelationsforallearly-typegalaxieswithlinestrengthmeasurementsintheEFARsample:(a)TheMgb′–σrelation;(b)theMg2–σrelation.InbothpanelsEgalaxiesaremarkedascircles,E/S0galaxiesassquaresandcDgalaxiesasasterisks.Errorbarsrepresentingthemedianerrorsineachquantityareshownintheupperrightofeachpanel.ThesolidlineistheMLfitandthedashedlineistheunweightedregressionoflinestrengthondispersion.ThedistributionsofresidualsinMgabouttheMLfitforallobjectsareshownintheinsetsasopenhistograms.TheGaussiansdescribedbythemedianresidualandtherobustly-estimatedscatteraresuperimposed.ThesolidhistogramsarethedistributionsofresidualsforthecDgalaxies,showingtheiroffsetfromtheoverallrelation.

K-Stest)tothermserrorsfromtherepeatmeasurements.Are-scalingbyfactorsof0.85and1.15respectivelygivesgoodagreementfortheerrorsinσandMgb;adding0.005maglikewisegivesgoodagreementfortheerrorsinMg2.

Acomparisonwiththeliterature(Paper2,Figure13)showsthatourdispersionsareconsistentwithpreviousmeasurementsbyDaviesetal.(1987),Guzm´an(1993),Jørgensen(1997),Luceyetal.(1997)andWhitmoreetal.(1985).Forthesubsetofgalaxiesincommon,wecomparedourlinestrengthswiththedefinitiveLicksystemmeasure-mentsofTrageretal.(1998)inordertoderivethesmallze-ropointcorrectionsrequiredtocalibrateourmeasurementstotheLicksystem(Paper2,Figures14&15);theoverlapofourMg2measurementswiththoseofLuceyetal.(1997)alsoshowsconsistency(Figure16,Paper2).

100kms−1orselectionprobabilitieslessthan10%,andalsooutlierswithlowlikelihoods,theMLfitstotheMgb′–σre-lation(490galaxies)andtheMg2–σrelation(423galaxies)are:Mgb′Mg2

==

(0.131±0.017)logσ−(0.131±0.041),(0.257±0.027)logσ−(0.305±0.0).

(2)(3)

33.1

THEMg–σRELATIONTheglobalrelation

ThesefitsareshowninFigure2assolidlines.TheratiooftheslopesoftheserelationsisconsistentwiththeMg2-Mgb′relationweobtainedinPaper2:Mg2≈1.94Mgb′−0.05.MonteCarlosimulationsofthedatasetandfittingprocess,theresultsofwhicharedisplayedinFigure3,showthatthereisnobiasintheMLestimatesoftheslopesandzero-points,andprovidereliableestimatesoftheuncertaintiesinthefit.

TheMLfitscanbecomparedtosimpleregressionsof′

MgbandMg2onlogσ.Theseregressionsareshowninthefigureasdashedlines,andare:Mgb′Mg2

==

(0.104±0.011)logσ−(0.067±0.026),(0.199±0.016)logσ−(0.168±0.038).

(4)(5)

InthissectionweinvestigatetheglobalMg–σrelationfoundamongsttheentiresampleofEFARgalaxieswithearly-typemorphologicalclassifications(cD,EorE/S0;seedefinitionsinPaper3)forwhichweobtainedlinestrengthmeasure-ments.TheMgb′–σrelationisshowninFigure2aandtheMg2–σrelationinFigure2b.

Inordertofitlinearrelationswithintrinsicscatterinthepresenceofsignificantmeasurementerrorsinbothvari-ables,arbitrarycensoringofthedatasetandabroadsam-pleselectionfunction,wehavedevelopedacomprehensivemaximumlikelihood(ML)fittingprocedure(Sagliaetal.,inpreparation).Excludinggalaxieswithdispersionslessthan

Asexpected,thesimpleregressionsyieldslopeswhicharebiasedlowduetothepresenceinthedataoferrorsintheabscissaaswellastheordinate,andalsotheintrinsicscatterintherelation.Slightlyless-biasedresultsareobtainedbyleastsquaresregressionminimisingtheorthogonalresiduals(cf.Jørgensenetal.1996):Mgb′Mg2

==

(0.109±0.012)logσ−(0.078±0.027),(0.215±0.017)logσ−(0.205±0.041).

(6)(7)

Theseleastsquaresfitsandtheiruncertaintiesareob-tainedusingtheslopesregressionprogramwrittenby

c0000RAS,MNRAS000,000–000󰀁

Figure3.MonteCarlosimulationsoftheMLfitstotheMg–σrelations.ThetoppanelisforMgb′–σandthebottompanelforMg2–σ.IneachpanelweshowthedistributionofMLfitstotheslopeandzeropointforasetof99simulationsoftheEFARdataset.Theinputparameters(fromthefitstotheEFARdata)andthemeanandstandarddeviationofthefittedparameters(fromthesimulations)aregivenineachcase.

E.D.FeigelsonanddescribedinIsobeetal.(1990)andFeigelson&Babu(1992).Theuncertaintiesareunder-estimatedbecausetheseregressionsdonotproperlyaccountforthemeasurementerrorsortheselectionfunctions.WeconcludethatpreviousdeterminationsoftheslopeoftheMg–σrelationarelikelytobebiasedlowwheneverthedatasetbeingfittedhadsignificanterrorsinthevelocitydispersions(asisgenerallythecase).HereafterweadopttheMLfitstotheMg–σrelation.

ThedistributionsoftheresidualsinMgb′andMg2abouttheMLfitsareshownintheinsetstoFigures2aand2b.Inordertominimisetheeffectsofoutliers,wero-bustlyestimatethescatterabouttheMg–σrelationsashalftherangespannedbythecentral68%ofthedatapoints.Wefindanobservedscatterof0.022±0.002magabouttheMgb′–σrelationand0.031±0.003magabouttheMg2–σre-lation.Excludingoutliers,thedistributionsofresidualsareverywellfittedbyGaussiansparametrisedbythemedianresidualandtherobustlyestimatedscatter.ThereisnoevidenceforatailofnegativeresidualssuchasnotedbyBursteinetal.(1988)andJørgensenetal.(1996).Asthelat-terauthorspointout,thepresenceofsuchatailissensitivetotheadoptedslopeoftheMg–σrelation.Somegiantellip-ticalsdo,however,haveintrinsicallyweakMglinestrengthsfortheirvelocitydispersions(Schweizeretal.1990).

Theestimatesoftheintrinsicscatterabouttherela-tionsthatareprovidedbytheMLfitmaybeexaggeratedbyoutliersorbydeviationsoftheunderlyingdistributionofgalaxiesintheMg–σplanefromabivariateGaussian.WethereforedroptheassumptionofanintrinsicbivariateGaus-siandistributionintheMg–σplaneanduseMonteCarlo

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c0000RAS,MNRAS000,000–000Peculiarmotionsofearly-typegalaxies–V5

Figure4.Thenormalisedlikelihooddistributionsfortheintrin-sicscatterinMgb′andMg2atfixedvelocitydispersiongiventheobservedscatterabouttheMgb′–σandMg2–σrelations.Thesearederivedfromsimulationsbasedontheobserveddistributionofdispersionsandlinestrengths(andtheirerrors),assumingonlythatthereisagloballinearMg–σrelationaboutwhichthereisGaussianintrinsicscatter.

simulationsbasedontheobserveddistributionofdispersions

andlinestrengthsandtheirestimatederrors(accountingforbothmeasurementerrorsandruncorrectionerrors).ThesesimulationsassumeonlythatthereisagloballinearMg–σrelationaboutwhichthereisGaussianintrinsicscatter.Wevarythisintrinsicscatterandcomputetherobustestimateoftheobservedscatteraboutthefit(thehalf-widthofthecentral68%oftheresiduals)forthesimulateddistributions.TheresultsofthesesimulationsarepresentedinFigure4,whichshowsthenormalisedlikelihooddistributionsfortheintrinsicscatterinMgb′andMg2giventheobservedscat-ter.Wefindthattoaccountfortheobservedscatterintherelationswerequireanintrinsicscatterof0.016±0.001magforMgb′–σand0.023±0.002magforMg2–σ.TheratiooftheintrinsicscatterinMg2totheintrinsicscatterinMgb′isslightlylowerthanexpectedfromtheobservedMg2–Mgb′relation,Mg2≈1.94Mgb′−0.05(seePaper2).

Table1comparesourfitstotheMg–σrelationsob-tainedbyotherauthors,andgivestheobservedscatterδMgobsandtheintrinsicscatterδMgintintherelationsob-tainedineachcase.ForbothMgb′–σandMg2–σtheslopesweobtainareabout25%steeperthanthoseobtainedbymostpreviousauthors.Thisisnotduetoadifferenceinourdata,butstemsfromouruseoftheMLmethodratherthanregressions.Inthissituationregressionsarebiasedtowardsflatterslopesthanthetruerelationbecausetheyignoretheintrinsicscatter,thepresenceoferrorsinbothvariablesandtheselectionfunctionofthedataset.Thestandardoror-thogonalregressionfitstoourdata,whichoursimulationsshowunder-estimatetheslopeoftherelations,giveresultsverysimilartothoseobtainedbyotherauthors.

Ifwedividethesamplebymorphologicaltype,wefind

6Collessetal.

Figure5.Cluster-to-clusteroffsetsintheMg–σrelation.TheleftpanelsareforMgb′–σandtherightpanelsforMg2–σ.ThetoppanelsshowthemedianoffsetsfromtheglobalMg–σrelationforclusterswiththreeormorelinestrengthmeasurements.ThemiddlepanelsshowthedistributionofoffsetscomparedtoaGaussianwiththesamemeananddispersion;thescatteris0.012maginMgb′and0.019maginMg2.Thebottompanelsshowtheoffsetsasafunctionofredshift.

Table1.ComparisonofMg–σrelationfitsMgb′–σ

Slope

relation...

0.1060.131±0.0200.1750.260±0.0270.2000.196±0.0160.257±0.028

−0.079−0.131±0.048−0.110−0.316±0.003−0.166−0.155−0.305±0.067

—0.022±0.0020.0160.0160.0250.0250.031±0.003

—0.016±0.0010.0130.0130.0180.0200.023±0.002

Intercept

δMgobs

δMgint

Ziegler&Bender(1997)EFAR(thiswork)Mg2–σrelation...Bursteinetal.(1988)Guzm´anetal.(1992)Benderetal.(1993)Jørgensenetal.(1996)EFAR(thiswork)

fromwhichoursampleisdrawn.ThetworegionshaveMg–σrelationswithslopesandzeropointswhichareconsistentbothwitheachotherandwiththeoverallMg–σrelations,providingacheckthattherearenogrosssystematicenvi-ronmentaldifferencesbetweenthesetworegions.3.2

Cluster-to-clustervariations

thatthecDshaveazeropointwhichis0.009maghigherthanthatoftheotherearly-typegalaxiesinMgb′,and0.014maghigherinMg2.ThesedifferencesinthezeropointsarereadilyapparentinthedistributionsofresidualsabouttheglobalMg–σrelations(seetheinsetstoFigures2a&b),andaresignificantatthe3σ-level.Despitethesezeropointoffsets,includingorexcludingthecDschangesthescatterabouttheMLfitbylessthanitsuncertainty,astheymakeuponly10%ofthewholesample.

Wefindnosignificantdifferences,however,ifwecom-paretheMg–σrelationsforthetwovolumesofspace,theHercules-Corona-BorealisandPerseus-Pisces-Cetusregions,

WedonothaveenoughgalaxiesperclustertofitboththeslopeandthezeropointoftheMg–σrelationsonacluster-by-clusterbasis,eveninourbest-sampledclusters.WethereforelimitourselvestoinvestigatingthevariationintheMg–σzeropoint.TothisendwemeasuredthemedianoffsetinMgb′andMg2fromtheglobalfitsgivenabovefortheclusterswiththreeormorelinestrengthmeasurements(75clustersforMgband72forMg2).Notethatweonlyusedgalaxiesthatareclustermembersbasedontheirredshifts(seePaper2).Theresultsarenotchangedsignificantlyifweuseallclusters,oronlyclusterswithfiveormoremeasure-ments.

ThetoppanelsofFigure5showthesezeropointoff-setsasafunctionofclusterIDnumber(CID),whilethemiddlepanelsshowthedistributionsoftheoffsetvalues.Therobustly-estimatedscatterinthezeropointoffsetsis0.012±0.002maginMgb′–σand0.019±0.004maginMg2–σ,showingthattherelationsareremarkablyuniformamongtheaggregatesofgalaxiesintheEFARsample.Thebottompanelsinthefigureplotthesameoffsetsasafunctionofred-c0000RAS,MNRAS000,000–000󰀁

Peculiarmotionsofearly-typegalaxies–V7

Figure6.VariationsintheMg–σrelationwithvariousindicatorsofclustermass.TheleftpanelsareforMgb′–σandtherightpanels

2;themiddlepanelsshowforMg2–σ.ThetoppanelsshowthemedianoffsetsfromtheglobalMg–σrelationasafunctionoflogσcluster

theoffsetsasafunctionoflogLX;thebottompanelsshowtheoffsetsasafunctionofX-raytemperature,kT.

shift,showingthatthereisnodependenceoftherelationsonrelativedistancewithinthesample.

Thisscatterinthezeropointoffsetscouldpurelybeaconsequenceofagalaxy-to-galaxyscatterinaglobalMg–σrelation,oritcouldalsorequireavariationinthezeropointoftherelationfromclustertocluster.Thesepossibilitieswereexaminedbyextendingthesimulationsdescribedintheprevioussection,addingafurthersourceofscattertotheMg–σrelationintheformofanintrinsicvariationbetweenclustersinthezeropointoftherelation.ForsimplicityweassumethatthisvariationalsohasaGaussiandistribution.

Wefindthatifwemaketheextremeassumptionthatthereiscluster-to-clusterscatterbutnointrinsicscatterbetweengalaxieswithinacluster,thenzeropointvaria-tionsbetweenclusterswithanrmsof0.009maginMgb′and0.015maginMg2arerequiredtorecovertheob-servedcluster-to-clusterscatter.Howeverthismodelunder-predictstheobservedscatterabouttheglobalrelation,giv-ing0.017±0.001magforMgb′and0.025±0.002magforMg2comparedtotheactualvaluesof0.022±0.002magand0.031±0.003mag.Ontheotherhand,ifweassumethatthereisnozeropointvariationbetweenclusters,thentheintrinsicscatterbetweengalaxiesrequiredtorecovertheobservedscatterintheglobalrelation(0.016maginMgb′and0.023maginMg2;seeprevioussection)predictsascat-terintheclusterzeropointsof0.012±0.001maginMgb′and0.016±0.002maginMg2,whichisconsistentwiththeobservedvaluesof0.012±0.002magand0.019±0.004magwithinthejointerrors.

c0000RAS,MNRAS000,000–000󰀁

Weconcludethatthereisnoevidenceforsignificantin-trinsiczeropointvariationsbetweenclusters,sincesampling

agalaxypopulationdrawnfromasingleglobalrelationwithintrinsicscatterconsistentwiththeobservationscanaccountforthezeropointdifferencesbetweenourclusters.3.3

Variationwithclusterproperties

AsthereisverylittlechangeinthezeropointoftheMg–σrelationfromclustertocluster,itfollowsthattherecanbeatmostonlyaweakdependenceofthezeropointontheprop-ertiesoftheclusters.HereweinvestigatetheeffectofclusterpropertiesonthestellarpopulationsasreflectedintheMg–σzeropoints,consideringclustervelocitydispersions,X-raylu-minositiesandX-raytemperatures(allindicatorsofclustermass).TheclusterdispersionscomefromTable7ofPaper2,usingredshiftsbothfromEFARandfromtheZCATcata-logue(Huchraetal.1992;versionof1997May29).X-rayluminositiesandtemperaturesareavailablefor26ofour84clustersinthehomogeneousandflux-limitedcatalogueofX-raypropertiesofAbellclustersbyEbelingetal.(1996)basedonROSATAll-SkySurveydata.TheX-rayluminosi-tiesaredeterminedtoatypicalprecisionofabout20%.Inordertohavecomparableprecisionintheclustervelocitydispersions,weonlyuseclusterswithdispersionscomputedfromatleast20galaxyredshifts;thisalsoleaves26clusters,17ofwhichareincommonwiththeX-raysubsample.

Figure6showstheoffsetsintheMg–σrelationsasfunc-2

tionsoflogσcluster,logLXandkT.ApplyingtheSpearman

8Collessetal.

rankcorrelationstatistic,wefindthatthereisnosignificantcorrelationbetweentheMg–σoffsetsandanyofthesequan-tities,andthusnoevidenceforatrendinthezeropointoftheMg–σrelationwithclustermass.Weightedregressionsgivebest-fitrelationsandtheiruncertainties:∆Mgb′=(−0.002±0.009)logσ2

cl+(0.016±0.050),(8)∆Mg2

=

(−0.001±0.011)logσ2cl+(0.002±0.060).(9)

Ifwetakeacomplementaryapproach,splittingtheclusters

intotwosubsamplesaboutthemedianvalueofLXandfit-tingtheMg–σrelationstothegalaxiesofthehigh-LXandlow-LXclustersseparately,weagainfindnosignificantdif-ferencesintheslopesorthezeropointsofthefits,whicharecompatiblewiththeglobalfitsobtainedabove.

4DISCUSSION

Thereareatleastfourmainquestionswhichcanbead-dressedusingtheaboveresults.

(i)WhatarethetheoreticalimplicationsofthelackofcorrelationbetweenthemassofaclusterandthezeropointoftheMg–σrelationforclustergalaxies?

(ii)Whateffectdothestellarpopulationdifferencesim-pliedbytheobservedvariationsintheMg–σrelationhaveonFundamentalPlaneestimatesofdistancesandpeculiarvelocities?

(iii)WhatconstraintdoestheintrinsicscatterabouttheMg–σrelationplaceonthespreadinage,metallicityandmass-to-lightratioamongstearly-typegalaxiesinclusters?

(iv)WhatfurtherconstraintsonthesequantitiesresultfromcombiningthescatterabouttheMg–σrelationwiththeintrinsicscatterabouttheFundamentalPlane?4.1

Mg–σzeropointandclustermass

ThesmallscatterinthezeropointoftheMg–σrelationfromclustertocluster,andinparticularthelackofcorrelationbetweentheMg–σzeropointandtheclustermass,seemstoimplythatthemassover-densityonMpcscalesinwhichanearly-typegalaxyisfoundhaslittleconnectionwithitsstellarpopulationandstar-formationhistory.

ThevariationoftheMg–σrelationwithclusterprop-ertieshaspreviouslybeenstudiedinasampleof11nearbyclustersbyJørgensenetal.(1996)andJørgensen(1997).FollowingGuzm´anetal.(1992),theseauthorslookforatrendinMg2–σoffsetswiththe‘localdensity’withinclusters.Theestimatoroflocaldensityusedisρcluster=

logσ2

cluster−logR,whereRistheprojecteddistanceofthegalaxyfromtheclustercentre.SinceRisonlyalowerlimitonthegalaxy’struedistancefromtheclustercentre,thisisaratherpoorestimatorofthetruelocaldensity.Jørgensenetal.findthattheresidualsinMg2showaweaktrend⋆withlocaldensity,∆Mg2∝0.009ρcluster.Sincetheresidualsdonotcorrelatewithradiuswithinthecluster(seeFigure3ofJørgensen(1997)),butdoshowasignificantcorrelationwith

clustervelocitydispersion,∆Mg2∝0.02logσ2

cluster(least-squaresfittothedatainFigure5ofJørgensen(1997)),we

⋆Thesignofthetrendinequation6ofJørgensen(1997)isin-correct;thecoefficientshouldbe+0.009(Jørgensen,priv.comm.)

wouldarguethatamorestraightforwardinterpretationoftheirresultsisacorrelationofMg2–σzeropointwithtotalclustermassratherthanlocaldensity.

Acorrelationofthisamplitudeisformallyconsistentatthe2σlevelwiththedistributionofMg2offsetsversus

logσ2

clusterfortheEFARdata(seeequation9);transformingJørgensen’sresultviatheMg2–Mgb′relationgivesacorre-lationwhichisconsistentatthe1.4σlevelwithequation8.WeconcludethatanycorrelationbetweentheMg–σrela-tionzeropointandtheclustermassissufficientlyweak(of

order∆Mg2∝0.02logσ2

clusterorless)thatitisnotreliablyestablishedbytheexistingdata,whichareconsistentwithnocorrelationatall.

Semi-analyticmodelsfortheformationofellipticalgalaxies,whichpreviouslyneglectedmetallicityeffects(seeKauffmann1996,Baughetal.1996),areonlynowbegin-ningtoincorporatechemicalenrichmentandsuccessfullyre-producethegeneralformoftheobservedcolour–magnitudeandMg–σrelations(Kauffmann&Charlot1998).Incon-sequence,thereareasyetnoreliablepredictionsforthevariationoftheMg–σrelationzeropointwithclustermass.Thelimitsgivenabove,togetherwithlimitsonthedifferenceinMg–σzeropointsforfieldandclusterellipticals(Bursteinetal.1990,deCarvalho&Djorgovski1992,Jørgensen1997),shouldprovidevaluableadditionalconstraintsandencour-agefurtherdevelopmentofchemicalenrichmentmodelswithinahierarchicalframeworkforgalaxyandclusterfor-mation.4.2

SystematiceffectsonFPdistances

WenowconsidertheeffectsonFPdistanceestimatesof

systematicdifferencesinthestellarpopulationsofearly-typegalaxiesfromclustertocluster.In§3.2wefoundthattheobservedcluster-to-clustervariationsintheMg–σzeropointwereconsistentwithsamplingasingleglobalMg–σrelationwithintrinsicscatterbetweengalaxies,anddidnotrequireintrinsicscatterbetweenclusters.Hereweturnthequestionaroundandaskhowmuchintrinsiccluster-to-clusterscatterisallowedbytheobservations.

Fromsimulationsusingthemodeldescribedin§3.2,in-corporatingintrinsicscatterbothbetweengalaxiesandbe-tweenclusters,wefindthatthemaximumcluster-to-clusterscatterallowedwithinthe1σuncertaintiesinthescatterintheglobalMg–σrelationandtheclusterzeropointsisap-proximately0.005maginMgb′and0.010maginMg2.Forourbest-fitMLMg–σrelationsandaFPgivenbyR∝σαIβwithα≈1.27,thislevelofcluster-to-clusterscatterwouldleadtormserrorsinFPdistanceestimatesofupto10%.Thesesystematicerrors,resultingfromdifferencesinthemeanstellarpopulationsbetweenclusters,wouldapplyeventoclustersinwhichtheFPdistanceerrorsduetostellarpopulationdifferencesbetweengalaxieshadbeenmadeneg-ligiblebyobservingmanygalaxiesinthecluster.

Weemphasisethatourresultsheredonotrequireanycluster-to-clusterscatter,butareconsistentwithcluster-to-clusterscattercorrespondingtosystematicdistanceerrorsbetweenclusterswithanrmsofupto10%.WethereforecannotdeterminefromtheMg–σrelationalonewhethersys-tematicdifferencesinthemeanstellarpopulationsbetweenclusterscontributesignificantly(oratall)totheerrorsinFPestimatesofdistancesandpeculiarvelocities.Amore

󰀁

c0000RAS,MNRAS000,000–000Figure7.TherelationbetweenMgb′andlogM/LRasafunc-tionofageandmetallicityinthemodelofWorthey(1994).Thesolidlinesarecontoursofconstantage(1.5,2,3,5,8,12and17Gyr),withincreasinglinethicknessindicatingincreasingage.Thedashedlinesarecontoursofconstantmetallicity(−0.5,−0.25,0,0.25,0.5),withincreasinglinethicknessindicatingin-creasingmetallicity.Thedottedgridisthelinearfittothemodel.

effectivewayoftestingforsuchsystematicdifferencesisbydirectlycomparingeachcluster’szeropointoffsetfromtheglobalMg–σrelationtotheratioofitsFPandHubbledistanceestimates;thisapproachwillbeinvestigatedinafuturepaper.

4.3Stellarpopulationmodels

Toanswerthequestionsconcerningthetypicalage,metal-licityandmass-to-lightratioofearly-typegalaxieswhichwereraisedatthestartofthisdiscussion,weneedtoem-ploystellarpopulationmodels.WeusethepredictionsfromthesinglestellarpopulationmodelsofWorthey(1994)andVazdekisetal.(1996),notingthemanycaveatsgivenbytheseauthorsregardingtheirmodels.Tosimplifyouranal-ysis,wefitMgb′,Mg2andlogM/LRaslinearfunctionsoflogarithmicage(logt,withtinGyr)andmetallicity(logZ/Z⊙),forgalaxieswithagesgreaterthan4Gyrandmetallicitiesintherange−0.5to+0.5.ForthemodelofWorthey(1994;SalpeterIMF)weobtain

Mgb′≈0.058logt+0.086logZ/Z⊙+0.077(10)Mg2

≈0.107logt+0.182logZ/Z⊙+0.147

(11)

logM/LR

≈

0.825logt+0.184logZ/Z⊙−0.169.(12)

Figure7comparesthisfittoWorthey’smodelinthecaseofthepredicteddependenceofMgb′andlogM/LRonageandmetallicity.Thefigureshowsthatforagesof5Gyrorgreaterthefitandthemodelareinsatisfactoryagreementforallmetallicities.

ForthemodelofVazdekisetal.(1996;bimodalIMF,

󰀁

c0000RAS,MNRAS000,000–000Peculiarmotionsofearly-typegalaxies–V

9

µ=1.35)wehave

Mgb′≈0.051logt+0.083logZ/Z⊙+0.084(13)Mg2

≈0.115logt+0.187logZ/Z⊙+0.137

(14)

logM/LR

≈

0.673logt+0.251logZ/Z⊙−0.216,(15)

inagreementwiththefitobtainedbyJørgensen(1997).ThereisgoodagreementbetweenthepredictionsofthetwomodelsforthedependenceofMg2andMgb′onageandmetallicity,andmoderatelygoodagreementforthedepen-denceofM/LR.

NotethatthesamechangeintheMgindicesisproducedbychangesinage,∆logt,andmetallicity,∆logZ/Z⊙,if∆logt/∆logZ/Z⊙≈3/2.Thisisthe‘3/2rule’ofWorthey(1994),whichappliestomanyoftheLicklineindices,leav-ingthemdegeneratewithrespecttovariationsinageandmetallicity.HoweverageandmetallicityproducethesamechangeinlogM/LRonlyif∆logt/∆logZ/Z⊙≈1/3or1/4,sothatmeasurementsofmass-to-lightratioscaninprinciplebecombinedwithMglinestrengthstobreaktheage/metallicitydegeneracy.

4.4Dispersioninage,metallicityandM/L

Inthefollowinganalysisweinferthedispersionintheages

andmetallicitiesofearly-typegalaxiesbycomparingthescatterintheMg–σrelationwiththepredictionsofthesinglestellarpopulationmodelsdescribedintheprevioussection.Thisanalysisusesthestellarpopulationmodelstopredictdifferentialchangesinthequantitiesofinterest,andnotabsolutevalues.Itisalsoimportanttorememberthatbythedispersioninageormetallicitywemeanthedispersioninthesequantitiesatfixedlogσor,equivalently,thedisper-sionaftertheoveralltrendwithlogσisaccountedfor.Thusthedispersioninageormetallicityweinferisthedispersionatfixedgalaxymass,notthedistributionofagesandmetal-licitiesasafunctionofgalaxymass(whichisrelatedtotheslopeoftheMg–σrelationandthedistributionofgalaxiesalongit).

Singlestellarpopulationsmodelsspecifiedby(amongstotherparameters)auniqueageandauniquemetallicitycanonlyprovideanapproximationtorealgalaxies,whosestel-larcontentsmustnecessarilyspanarange(thoughperhapsanarrowone)ofagesandmetallicities.SincetheglobalMgindicescanbequitesensitivetothedetailedmetallicitydis-tribution(Greggio1997),someofthescatterweobservemaybeduetogalaxy-to-galaxydifferencesintheshapeofthemetallicitydistributionratherthanadispersioninthemeanmetallicityorage.

Afurthercomplicationispresentedbytheover-abundanceofMgwithrespecttoFe(comparedtothesolarratio)inthecoresofearly-typegalaxies(e.g.Peletier19,Gorgasetal.1990,Wortheyetal.1992).AsacomparisonofFigures2&7shows,themodelsdiscussedintheprevi-oussectionareunabletoaccountforthehighestobservedMglinestrengths.Tantaloetal.(1998)haveproducedsin-glestellarpopulationmodelsincludingtheeffectsof[Mg/Fe]variationsandfindthat

∆Mg2≈0.099∆[Mg/Fe]+0.0∆logt+0.166∆logZ/Z⊙(16)Comparingthisequationwiththoseabove,weseethatthedifferentialdependenceonageandmetallicityissimi-

10Collessetal.

lartothatpredictedbyWorthey(1994)andVazdekisetal.(1996).However,anyintrinsicscatterinthe[Mg/Fe]–σre-lationwillcontributeadditionallytotheintrinsicscatterintheMg2–σrelationandreducethedispersioninageandmetallicityrequiredtoaccountfortheobservations.

Forthesereasons,andalsobecauseofotherpotentialsourcesofintrinsicscattersuchasdarkmatter,rotation,anisotropy,projectioneffectsandbrokenhomology,thees-timatesofthedispersioninageandmetallicityderivedheremustbeconsideredasupperlimits.

Withthesecaveatsinmind,weproceedtousethemodelfitsgivenintheprevioussectiontoinferthedispersioninageormetallicitybasedontheobservedintrinsicscatterof0.016maginMgb′–σand0.023maginMg2–σ.Foreaseofinterpretationwequotethedispersionsinageandmetal-licityasthefractionaldispersionsδt/t≡δlogt/logeandδZ/Z≡δlogZ/loge.Inapplyingthemodelsinwhatfol-lows,weadoptthemeanofthecoefficientsforthetwomod-elsandgivethedispersionsinageandmetallicitycorre-spondingtotheintrinsicscatterabouttheMgb′–σrelation.UsingtheintrinsicscatterobtainedfromtheMg2–σrela-tionwouldgiveresultsthatare∼30%smaller,sincetheobservedratiooftheintrinsicscattersisδMg2/δMgb′≈1.4,ratherthanabout2aswouldbeexpectedeitherfromtheobservedMg2–Mgb′relationorfromthemodels.WeusethescatterinMgb′ratherthanMg2becauseourgoalistoestablishupperlimitsonthedispersionsinageandmetal-licity.Theestimatederrorsintheintrinsicscatterleadtouncertaintiesinthedispersionsof5–10%.

Ifagevariationsinsinglestellarpopulationsaretheonlysourceofscatterthenthedispersioninageisδt/t=67%,whereasifmetallicityvariationsarethesolesourcethenthedispersioninmetallicityisδZ/Z=43%.Similarly,theob-serveddifferenceintheMg–σrelationzeropointforthecDgalaxiesimpliesthattheseobjectsareeitherolderormoremetal-richthannormalEorE/S0galaxies.Ifthezeropointdifferencesareinterpretedasagedifferences,cDsareonav-erage40%olderthantypicalEorE/S0galaxies(i.e.asoldastheoldestearly-typegalaxies);ifthezeropointdifferencesareinterpretedasmetallicitydifferences,cDshavemetallic-itiesonaverage25%higherthantypicalEorE/S0galaxies(i.e.ashighasthemostmetal-richearly-typegalaxies).

Wecanalsousethemodelfitstoestimatetheapprox-imatechangeinM/LRcorrespondingtoachangeintheMglineindices.Ifthesechangesarecausedbyagevaria-tionsalone,thenwefindthat∆logM/LR≈7∆Mg2and∆logM/LR≈14∆Mgb′;if,however,theyaredueonlytovariationsinmetallicitywehave∆logM/LR≈1.2∆Mg2and∆logM/LR≈2.6∆Mgb′.ThusthechangeinlogM/LRisabout5timeslargeriftheobservedchangeintheMgin-dicesisduetoagedifferencesratherthanmetallicitydiffer-ences.TheintrinsicscatterintheMgb′–σrelationimpliesadispersioninmass-to-lightratioof50%ifduetoagevaria-tions,butonly10%ifduetometallicityvariations.

ThispredictedscatterinM/Lisinfactascatterinluminosityorsurfacebrightness(sincethatisallthemodelsdealwith).WecanthereforereadilyestablishtheeffectofthisscatterondistancesestimatedusingtheFundamentalPlane(FP)ifthescatterinM/Lisuncorrelatedwiththegalaxies’sizesanddispersions,asindeedisthecasefortheEFARsample(atleastforgalaxieswithσ>100kms−1).ForaFPgivenbyR∝σαIβ,withRtheeffectiveradiusandIthe

meansurfacebrightnesswithinthisradius,ifthescatterinM/LissimplyascatterinIwehave∆logR=β∆logM/L.MostdeterminationsoftheFP,includingourown,yieldβ≈−0.8(e.g.Dressleretal.1987,Jørgensenetal.1996,Sagliaetal.1998).

CombiningthisrelationwiththedependenceofM/LontheMglineindicesobtainedabove,wefindthatthescat-terintheMg–σrelationcorrespondstoanintrinsicscatterinrelativedistancesestimatedfromtheFPof40%ifduetoagevariations,or8%ifduetometallicityvariations.AstheintrinsicscatterintheFPisfoundtobeintherange10–20%(Djorgovski&Davis1987,Jørgensenetal.1993,Jørgensenetal.1996),onecannotexplainboththescatterintheMg–σrelationandthescatterintheFPasthere-sultofagevariationsaloneormetallicityvariationsalone(unlessthesinglestellarpopulationmodelsareincorrectortherearesignificantgalaxy-to-galaxydifferencesinthemetallicitydistributions).Suitablecombinationsofagevari-ationsandmetallicityvariationscan,however,accountforthemeasuredintrinsicscatterinboththeMg–σandFPrelations.

4.5CombinedMg–σandFPconstraints

Asasimplemodel,weassumethatthescatterintheFPandtheMg–σrelations(atfixedlogσ)isentirelyduetovariationsinageandmetallicity(atfixedgalaxymass).ThesevariationsarefurtherassumedtohaveGaussiandis-tributionsinlogtandlogZ/Z⊙withdispersionsδlogt≡δt/tlogeandδlogZ≡δZ/Zlogeandcorrelationcoeffi-cientρ(−1≤ρ≤1).WhileaGaussiandistributionofmetal-licitiesatfixedgalaxymassisareasonableinitialhypothesisfordescribingvariationsinthechemicalenrichmentprocess,thesingle-peakedshapeoftheassumedlognormaldistribu-tionforthemeanagesmaynotrealisticallyrepresentthestar-formationhistory(evenforgalaxiesofthesamemass).Thedispersioninageinferredunderthismodelshouldthere-forebeconsideredonlyasageneralindicationofthetime-spanoverwhichearly-typegalaxiesoffixedmassformedthebulkoftheirstellarpopulation.

WritingthescatterinMglinestrengthsandFPresidu-alsasδMgandδFPandthedispersioninlogtandlogZ/Z⊙asσtandσZ,thissimplemodelrelatesthescatterintheobservedquantitiestothedispersioninageandmetallicityby:

δ2Mg=a2tσ2t+2ρataZσtσZ+a2Zσ2Z(17)δ2FP

=

b2tσ2t+2ρbtbZσtσZ+b2Zσ2Z.

(18)

HereatandaZarethecoefficientsoflogtandlogZ/Z⊙for

Mg,andbtandbZthecoefficientsforlogR=−0.8logM/L,derivedfromthemeanofthelinearfitstothetwostellarpopulationmodelsgivenin§4.3.

Figure8showstheconstraintsonthevariationsinageandmetallicity(assumedfornowtobeuncorrelated)whichareimposedbythemeasuredintrinsicscatterintheMg–σrelationsandtheintrinsicdispersioninlogM/LRinferredfromtheintrinsicscatterintheFP.TheintrinsicscatterwefindabouttheMgb′–σandMg2–σrelationsisthenconsis-tentwithdispersionsinageandmetallicityonanellipticallocusdefinedbyequation17(withρ=0)intheδt/t–δZ/Zplane.ThedifferentlociforMgb′andMg2(thesolidlines

󰀁

c0000RAS,MNRAS000,000–000Figure8.Constraintsonthedispersioninage,δt/t,andmetal-licity,δZ/Z,fromtheintrinsicscatterintheMgb′–σrelation(δMgb′=0.016mag;thicksolidline)andtheMg2–σrelation(δMg2=0.023mag;thinsolidline),andfromanintrinsicscatterindistanceabouttheFPof10%,15%and20%(dashedlines).

inFigure8)resultfromthedifferencebetweentheobservedratioofthescatterinMg2tothatinMgb′andthepredictedratiofromthemodel,andgivesomeindicationofuncertain-tiesbothintheintrinsicscatterabouttheMg–σrelationsandinthemodelpredictions.Asecondconstraintissimi-larlyobtainedfromtheintrinsicscatterindistance(i.e.inlogR)abouttheFPusingequation18(againwithρ=0).ThedashedlinesinFigure8correspondtointrinsicscatterabouttheFPof10%,15%and20%.

Theimportantpointtonoteaboutthefigureisthat,asmentionedin§4.3,thedependencesoftheMglinestrengthsandmass-to-lightratioonageandmetallicityarequitedif-ferent,sothat(ifvariationsinageandmetallicityareun-correlated)thetwosetsofconstraintsarenearlyorthogonal.Thustheregionoftheδt/t–δZ/ZplanethatisconsistentwiththescatterinboththeMg–σrelationandtheFPisquitelimited.IfweusetheintrinsicscatterintheMgb′–σrelationandassumea20%intrinsicscatterinlogRabouttheFP(attheupperendofthequotedrange—see,e.g.,Djorgovski&Davis(1987)orJørgensenetal.(1996)),weobtainapproximateupperlimitsonthedispersionsinageandmetallicityofδt/t=32%andδZ/Z=38%.If,however,weusetheintrinsicscatterintheMg2–σrelationandadopt

󰀁

c0000RAS,MNRAS000,000–000Peculiarmotionsofearly-typegalaxies–V11

Figure9.Constraintsonthedispersioninage,δt/t,andmetallic-ity,δZ/Z,asafunctionofthecorrelationcoefficientρ.Forvaluesofρbetween−1and1(instepsof0.2)theconstraintsimposedbyanintrinsicscatterintheMgb′–σrelationofδMgb′=0.016magareshownbysolidlines,whiletheconstraintsfromanintrinsicscatterindistanceabouttheFPof20%areshownbydashedlines.Thejointconstraintsolutionsforeachvalueofρaremarkedbycircles.Theuncorrelatedcase(ρ=0)isindicatedbyafilledcircleandthickerlines.

anintrinsicFPscatterof10%(asobtainedforComaby

Jørgensenetal.1993),thenweobtainapproximatelowerlimitsofδt/t=15%andδZ/Z=27%.

SimilarargumentsallowustoevaluatetherelativecontributionsofthedispersionsinageandmetallicitytotheerrorsindistanceestimatesderivedfromtheFP.Forthefiducialcase(δFP=20%,δMgb′=0.016magandρ=0),whereδt/t=32%andδZ/Z=38%,themeanstellarpopula-tionmodelimpliesthatthedispersioninagegivesanintrin-sicFPscatterof19%whilethedispersioninmetallicitygives7%.InfactformostoftheplausiblerangeofdispersionsinageandmetallicityshowninFigure8,itisthedispersioninagewhichdominatestheintrinsicscatterabouttheFP.Onlyforthelowestplausibleagedispersionandthehighestplausiblemetallicitydispersion(δt/t=11%andδZ/Z=43%,correspondingtoδFP=10%andδMgb′=0.016mag)doesthecontributiontotheFPscatterfromthedispersioninmetal-licityachieveequalitywiththecontributionfromthedis-persioninage.

Theconstraintsonthedispersionschangeifthereisasignificantcorrelation(oranti-correlation)betweenthevari-ationsinageandmetallicity.Figure9showshowthecon-straintscorrespondingtotheupperlimitsδFP=20%andδMgb′=0.016mag(correspondingtothethicklinesinFig-ure8)aremodifiedasthecorrelationcoefficientρvariesoveritsfullrangefrom−1to1.Notethatforρ=±1wehaveδt/t∝∓3/2δZ/Z.Themainpointtoextractfromthisfigureisthatifthevariationsinageandmetallicityhaveacorrelationcoefficientintherange−0.5<ρ<1,thenthedispersionsinageandmetallicityvarybyonly±6%

12Collessetal.

Figure10.ThejointdistributionofresidualsfromtheMgb′–σandFPrelations:(a)theresidualsfortheEFARdataset;(b)asimulationwithδt/t=32%andδZ/Z=38%(thevaluesderivedfromtheintrinsicscatterintheMgb′–σrelationandaFPscatterof20%assumingthevariationsinageandmetallicityareuncorrelated);(c)asimulationwithδt/t=67%andδZ/Z=0(fromtheMgb′–σintrinsicscatterassumingadispersioninageonly);(d)asimulationwithδt/t=0andδZ/Z=43%(fromtheMgb′–σintrinsicscatterassumingadispersioninmetallicityonly).ThebottompanelshowssimulationsconsistentwiththeintrinsicscatterintheMgb′–σrelationandaFPscatterof20%,andwithvariousassumedcorrelationsbetweenageandmetallicity:(e)asimulationwithδt/t=26%,δZ/Z=27%andρ=1;(f)asimulationwithδt/t=28%,δZ/Z=31%andρ=0.5;(g)asimulationwithδt/t=38%,δZ/Z=50%andρ=−0.5;(h)asimulationwithδt/t=57%,δZ/Z=80%andρ=−1.Thedottedlinesaretheexpectedcorrelationsforadispersioninagealone(∆logZ=0)ormetallicityalone(∆logt=0).Thedashedlineisthecorrelationexpectedifthedistributionisdominatedbytheerrorsinlogσ.

and±12%respectivelyaboutthevaluesinferredintheun-correlatedcase.Onlyiftheageandmetallicityvariationsarestronglyanti-correlated(ρ≈−1;i.e.youngergalaxiesaremoremetal-rich)doweobtainsignificantlydifferentsolu-tions,withabroaderallowedrangeinbothageandmetal-licity(δt/taslargeas57%andδZ/Zaslargeas80%).ThisconclusioniscomplementarytothatreachedbyFerrerasetal.(1998),whofindthattheapparentlypassiveevolutionofthecolour–magnituderelationobservedinhigh-redshiftclustersdoesnotnecessarilyimplyacommonepochofma-jorstar-formationifyoungergalaxiesareonaveragemoremetal-rich.

Wecantestthedegreeofcorrelationbetweenthevari-ationsinageandmetallicitybyexaminingthejointdistri-butionofresidualsabouttheMgb′–σandFPrelations.ThisdistributionisshownfortheEFARdatasetinFigure10a.Thereisnoevidenceforacorrelationbetweentheresidualsinthisfigure;theSpearmanrankcorrelationcoefficientbe-tweentheresidualsis0.084,andisnotsignificantatthe2σlevel.

Inordertoinvestigatetheexpecteddistributionofresidualsinthepresenceoftheestimatedmeasurementer-rors,wehaveperformedMonteCarlosimulationsoftheEFARdatausingthemodelsforthedispersioninageandmetallicitydiscussedabove.Figure10bshowsasimulationwithδt/t=32%andδZ/Z=38%;thesearethevaluesde-rivedfromtheintrinsicscatterintheMgb′–σrelationandaFPscatterof20%whenthereisnocorrelationbetweenageandmetallicity.Thesimulateddistributionresemblestheobserveddistribution,althoughthereisaweakbutsig-nificantanti-correlationbetweentheresiduals(duetothedominanceoftheagevariationsintheFPresiduals)whichisnotapparentintheEFARdata.Over100suchsimula-tions,atwo-dimensionalK-Stest(Pressetal.1992)givesamedianprobabilityof0.3%thatthisdistributionandtheobserveddistributionarethesame.

Figures10c&dshowsimulateddistributionsforthecaseswheretheintrinsicscatterinMgb′–σisduetoagealoneormetallicityalone.Neithercaseisconsistentwiththeobserveddistribution,supportingtheclaimthatneither

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agenormetallicitycanbesolelyresponsibleforthescat-terinboththeMgb′–σrelationandtheFP.Figures10e–hshowsimulateddistributionsforfourcaseswherethevari-ationsinageandmetallicityarecorrelated(withρ=+1,+0.5,−0.5and−1respectively).Theperfectlycorrelatedandperfectlyanti-correlatedcasesarenotconsistentwiththeobserveddistribution.HoweverFigure10gshowsthatadistributionwithnosignificantcorrelationbetweentheMgb′–σandFPrelationresidualsisproducedwhenρ=−0.5.Atwo-dimensionalK-Stestgivesamedianprobabilityover100suchsimulationsof1.7%thatthisdistributionandtheobserveddistributionarethesame.Thisrelativelylowprob-abilitymayreflectaproblemwiththemodel,althoughitmaysimplybeduetosamplinguncertainty(theprobabili-tiesunderthistestvarybetweensimulationswithanrmsofafactorof6)ornon-GaussianoutliersintheEFARresid-uals.Thepointtobeemphasisedisthatamodelwithamoderatedegreeofanti-correlationbetweenageandmetal-licityappearstogivesignificantlybetteragreementwiththeobserveddistributionthanamodelinwhichageandmetal-licityareuncorrelated.

5CONCLUSIONS

WehaveexaminedtheMg–σrelationforearly-typegalaxiesintheEFARsample.WefitglobalMgb′–σandMg2–σrela-tions(equations2and3)thathaveslopesabout25%steeperthanthoseobtainedbymostpreviousauthors.Thisdiffer-enceresultsnotfromthedataitselfbutfromanimprovedfittingprocedure:weapplyacomprehensivemaximumlike-lihoodapproachwhichcorrectlyaccountsforthebiasesin-troducedbyboththesampleselectionfunctionandthesig-nificanterrorsinbothMgandσ.TheobservedscatterabouttheMg–σrelationsis0.022maginMgb′and0.031maginMg2;theintrinsicscatterintheglobalrelations,estimatedfromMonteCarlosimulations,is0.016maginMgb′and0.023maginMg2.

Withtoofewgalaxiesperclustertoreliablydeterminethefullrelationforeachclusterseparately,wefixtheslopesoftherelationsattheirglobalvaluesinordertoinvesti-gatethevariationinthezeropointfromclustertocluster.Wefindthatthezeropointhasanobservedscatterbetweenclustersof0.012maginMgb′and0.019maginMg2,andthatthisobservedscatterisconsistentwiththesmallnum-berofgalaxiessampledineachclusterbeingdrawnfromasingleglobalrelationwithintrinsicscatterbetweengalax-iesasgivenabove—i.e.theobservationsdonotrequireanyscatterintheMg–σzeropointbetweenclusters.Theallowedrangefortheintrinsicscatterbetweenclusterscorrespondstocluster-to-clustersystematicerrorsinFundamentalPlanedistancesandpeculiarvelocitieswithanrmsanywhereintherange0–10%.WethereforecannotdeterminefromtheMg–σrelationalonewhethersystematicdifferencesinthemeanstellarpopulationsbetweenclusterscontributesignif-icantly(oratall)totheerrorsindistancesandpeculiarvelocitiesobtainedusingtheFundamentalPlane.

WehavealsoexaminedthevariationintheMg–σrela-tionwithclusterproperties.OurclustersamplerangesfrompoorclusterstoclustersasrichasComa,havingvelocitydispersionsfrom300kms−1to1000kms−1andX-raylu-minositiesspanning0.3–8×1044ergs−1.Wedonotdetecta

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significantcorrelationofMg–σzeropointwithclusterveloc-itydispersion,X-rayluminosityorX-raytemperature,nor

isthereanysignificantdifferenceintheMg–σrelationsob-tainedbyfittingthegalaxiesinthehigh-LXclustersandlow-LXclustersseparately.ThepredominantfactorintheproductionofMgintheseearly-typegalaxies(andpresum-ablyotherα-elementsandperhapstheirmetallicityandstar-formationhistoryingeneral)isthusgalaxymassandnotclustermass.Theseobservationsplaceconstraintsonsemi-analyticmodelsfortheformationofellipticalgalaxies,whicharenowbeginningtoincorporatechemicalenrichmentandshouldsoonbeabletomakereliablepredictionsforthevariationoftheMg–σrelationwithclustermass.

WeapplythesinglestellarpopulationmodelsofWorthey(1994)andVazdekisetal.(1996)toplaceupperlimitsontheglobaldispersionintheages,metallicitiesandM/Lratiosofearly-typegalaxiesofgivenmassusingtheintrinsicscatterintheglobalMg–σrelation.WeinferanupperlimitonthedispersioninM/LRof50%ifthescatterinMg–σisduetoagedifferencesalone,or10%ifitisduetometallicitydifferencesalone.Thesecorrespondtoupperlim-itsonthedispersioninrelativegalaxydistancesestimatedfromtheFundamentalPlane(FP)of40%(agealone)or8%(metallicityalone).SincetheintrinsicscatterintheFPisfoundtobe10–20%,onecannot(withinthecontextofthesinglestellarpopulationmodels)explainboththescatterintheMg–σrelationandthescatterintheFPastheresultofagevariationsaloneormetallicityvariationsalone.

Wethereforedeterminethejointrangeofdispersionsinageandmetallicitywhichareconsistentwiththemea-suredintrinsicscatterinboththeMg–σandFPrelations.ForasimplemodelinwhichthegalaxieshaveindependentGaussiandistributionsinlogtandlogZ/Z⊙,wefindupperlimitsofδt/t=32%andδZ/Z=38%atfixedgalaxymass.Ifthevariationsinageandmetallicityarenotindependent,buthavecorrelationcoefficientρ,wefindthatsolongasρisintherange−0.5to1theselimitsonthedispersionsinageandmetallicitychangebyonly±6%and±12%respectively.Onlyiftheageandmetallicityvariationsarestronglyanti-correlated(ρ≈−1)doweobtainsignificantlyhigherupperlimits,withδt/taslargeas57%andδZ/Zaslargeas80%.ThedistributionoftheresidualsfromtheMg–σandFPrelationsisonlymarginallyconsistentwithamodelhavingnocorrelationbetweenageandmetallicity,andisbetter-matchedbyamodelinwhichageandmetallicityvariationsaremoderatelyanti-correlated(δt/t≈40%,δZ/Z≈50%andρ≈−0.5),withyoungergalaxiesbeingmoremetal-rich.

Strongerboundsonthedispersioninageandmetallicityamongstearly-typegalaxiesofgivenmasswillrequiremoreprecisemeasurementsofthedeviationsfromtheMg–σrela-tionandtheFundamentalPlaneandalsoimprovedmodelsforthedependenceofthelineindicesandmass-to-lightratioonageandmetallicity.FurtherpowerfulconstraintscanalsobeobtainedbymeasuringtheintrinsicscatterintheMg–σandFPrelationsathigherredshifts,sincethelinestrengthsandmass-to-lightratiohavedifferentdependencesonage.

ACKNOWLEDGEMENTS

MMCacknowledgesthesupportofaDISTCollabora-tiveResearchGrant.DBwaspartiallysupportedbyNSF

14Collessetal.

GrantAST90-16930.RLDthankstheLorenzCentreandProf.P.T.deZeeuw.RKMreceivedpartialsupportfromNSFGrantAST90-208.RPSacknowledgesthefinan-cialsupportbytheDeutscheForschungsgemeinschaftunderSFB375.GWisgratefultotheSERCandWadhamCollegeforayear’sstayinOxford,totheAlexandervonHumboldt-StiftungformakingpossibleavisittotheRuhr-Universit¨atinBochumandtoNSFGrantsAST90-17048andAST93-47714forpartialsupport.Theentirecollaborationbenefit-tedfromNATOCollaborativeResearchGrant900159andfromthehospitalityandmonetarysupportofDartmouthCollege,OxfordUniversity,theUniversityofDurhamandArizonaStateUniversity.SupportwasalsoreceivedfromPPARCvisitorsgrantstoOxfordandDurhamUniversi-tiesandPPARCrollinggrant‘ExtragalacticAstronomyandCosmologyinDurham1994-98’.Wethankthereferee,Prof.AlvioRenzini,foracritiquewhichresultedinsubstantialimprovementstothepaper.

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