画鸵萌宠网
CORRELATIONFUNCTIONSOFTHEABELL,APM,ANDX-RAYCLUSTERS
OFGALAXIES
SUNGHOLEEANDCHANGBOMPARK
AstronomyProgramE-mail:leesh@astro.snu.ac.krinSEES,SeoulNationalandUniversity,cbp@astro.snu.ac.kr
Seoul,151-742Korea
(Received???.??,2002;Accepted???.??,2002)
ABSTRACT
WehavemeasuredthecorrelationfunctionsoftheopticallyselectedclustersofgalaxiesintheAbellandtheAPMcatalogs,andoftheX-rayclustersintheX-ray-BrightestAbell-typeClustersofgalaxies(XBACs)catalogandtheBrightestClustersSample(BCS).Thesameanalysismethodandthesamemethodofcharacterizingtheresultingcorrelationfunctionsareappliedtoallobservationalsamples.WehavefoundthattheamplitudeofthecorrelationfunctionoftheAPMclustersismuchhigherthanwhathasbeenpreviouslyclaimed,inparticularforrichersubsamples.Thecorrelationteam)isfoundtober0=25.4+3−3..1lengthoftheAPMclusterswiththerichnessR≥70(asdefinedbytheAPM
0h
−1
Mpc.Theamplitudeofcorrelationfunctionisabout2.4timeshigherthanthatofCroftetal.(1997).ThecorrelationlengthsoftheAbellclusterswiththerichnessclassRC≥0and
1aremeasuredtober0=17.4+1−1..21and21.0+2.8−1
Mpc,respectively,whichisconsistentwithourresultsfortheAPMsampleatthesimilarlevelofrichness.−2.8h
Therichnessdependenceofclustercorrelationsisfoundtober0=0.40dc+3.2wheredcisthemeaninterclusterseparation.ThisisidenticalinslopewiththeBahcall&West(1992)’sestimate,butisinconsistentwiththeweakdependenceofCroftetal.(1997).TheX-raybrightAbellclustersintheXBACscatalogandtheX-rayselectedclustersintheBCScatalogshowcorrelationlengthoftheXBACsclusterswithLxweakerthanxwhat≥0.70is×expected1044h−2ergs−1is30≥.20.2−strong1
clustering.The
+9.65×1044h−2ergs−1is30.3+8−6.5h
Mpc,andthatoftheBCSclusterswithLfromtheoptical−8.clusters.
.89h−1
Mpc.TheclusteringstrengthoftheX-rayclustersismuchKeyWords:cosmology:large-scalestructureoftheuniverse–galaxies:clusters:general
I.INTRODUCTION
Richclustersofgalaxiesareusedtostudythelarge-scalestructureoftheuniverseonscaleslargerthanabout10h−1Mpc(h=H0/100kms−1Mpc−1).Tomeasuretheclusteringstrengthofclustersthespatialtwo-pointcorrelationfunction(CF)oftheAbellclus-ters(Abell1958;Abell,Corwin,&Olowin1989)hasbeenestimatedbymanyauthors(Bahcall&Soneira1983;Huchraetal.1990;Postman,Huchra,&Geller1992;Peacock&West1992;Milleretal.1999;seeBahcall1988forareview),andhasbeenfoundtobeconsistentwiththepowerlaw
ξcc(r)=
r
2LEEANDPARK
Fig.1.—Distributionofclustersofgalaxiesontheskywithmeasuredredshifts;351Abellclusters,374APMclusters,276XBACs
clusters,and201BCSclusters.Thedottedcurvesrepresentthegalacticlatitudesb=±40◦,±30◦,and±20◦.
ousstudiesindividualobservationalsampleshavebeenanalyzedbydifferentauthorswhouseddifferentmeth-odsofanalysis.Inthecurrentsituationwheretheclus-terclusteringresultsfromdifferentstudiesdonotagree,itisimportanttoreanalyzeandcompareallpreciouslystudiedsamplesinanimpartialway.InthispaperweapplymethodsofcalculatingandcharacterizingtheCFstoallclustersamplesinasamewaytoremoveanyrelativebiases.Itishopedthatinthisobjectiveandconsistentwayofanalysiswecouldfindthecommoncharacteristicsof,andintrinsicdifferencesinthespa-tialdistributionsofvariousclustersamples.
Ontheotherhand,richclustersofgalaxiesoftenhavediffuseintraclustergastrappedintheirpotentialwells.ThethermalX-rayfluxfromtheintraclustergas,whichisheatedtotemperatureofafew107K,isproportionaltothesquareoftheiondensity,andthusismoreconfinedtothecenteroftheclustersthantheprojectedgalaxydistribution.Therefore,theX-rayselectedclustersareexpectedtohavenegligibleprojec-tioneffects(Briel&Henry1993;Ebelingetal.1996).
Ebelingetal.(1996)havecross-correlatedtheclustersintheACOcatalog(Abelletal.1989)withtheROSATX-raysources(Tr¨umper1993)andcreatedtheX-ray-BrightestAbell-typeClustersofgalaxies(XBACs)cat-alog.Morerecently,Ebelingetal.(1998)havepub-lishedtheBrightestClustersSample(BCS)whichisatrueX-rayselectedsamplefromtheROSATAll-SkySurvey.Itshouldbeinterestingtocomparetheclus-teringstrengthsoftheseX-rayclustersampleswiththoseofopticallyselectedsamplestofurtherextendourknowledgeontheclusteringpropertiesofclustersofgalaxies.II.(a)
DATA
TheAbellClusters
Postmanetal.(1992)havepublishedacompleteredshiftcatalogof351AbellclusterswithrichnessclassRC≥0,declinationδ≥−27◦.5,andtenth-rankedgalaxymagnitudem10≤16.5.Werestrictourfull
CORRELATIONFUNCTIONSOFCLUSTERSOFGALAXIES3
Table1
RichnesssubsamplesoftheAbellandtheAPMclusters
nc3(hMpc−3)
r0
−1(hMpc)
RichnessNcγ
RC≥0RC≥1
232110
1.74×10−50.68×10−5
.217.4+1−1.1.821.0+2−2.8
.15
1.89+0−0.16.341.91+0−0.36
R≥50R≥60R≥70R≥80
2181397342
2.62×10−51.66×10−50.93×10−50.57×10−5
.8
15.4+0−0.8.622.3+1−1.6.125.4+3−3.0.026.8+7−5.7
.17
2.49+0−0.16.161.96+0−0.17.292.01+0−0.30.582.03+0−0.67
1+z)h−1Mpc.
(2)
Sinceweusethesamed(z)relationtoallclustercat-alogs,theresultingCFsarenotbiasedwithrespect
tooneanotherduetothechoiceofcosmology.Fur-thermore,mostotherauthorshavealsoadoptedthestandardEinstein-deSittermodelintheircalculation.Toestimatetheselectionfunctionofthesurveyintheradialdirection,wefirstcomputetheabsolutemag-nitudeM10correspondingtom10ofeachclusterataredshiftzfromtherelation
m−M=5log[d(z)(1+z)]+25+K(z).
(3)
sampletothedistancerangefromdin=50h−1Mpctodout=500h−1Mpc,andtheresultingsamplein-cludes232and110clustersforRC≥0andRC≥1,respectively(seeTable1).Theregioncloserthandin=50h−1Mpcisremovedtoimprovethecomplete-nessoftheAbellclustersampleandtoavoidthepossi-blelocaleffects.Largevariationofthelocationsoftheinnerandouterboundariesmakesnegligibleeffectonourresults.
WeestimatethecomovingspacedensityofclustersineachsampleusingthemethoddescribedinSection2.3ofEfstathiouetal.(1992).ThemeanspacedensityoftheAbellclustersamplewithRC≥0is1.74×10−5h3Mpc−3whilePostmanetal.(1992)’sestimateis1.2×10−5h3Mpc−3andEfstathiouetal.(1992)’sis1.9×10−5h3Mpc−3.ThedensityofourRC≥1subsample,nc=6.79×10−6h3Mpc−3,agreeswellwithBahcall&Soneira(1983)’s6×10−6h3Mpc−3andwithEfstathiouetal.(1992)’s7×10−6h3Mpc−3.
Figure1showsthedistributionsofclustersontheskywithmeasuredredshiftsineachcatalogusedinourCFanalysis.(b)
TheAPMClusters
TheK-correctionK(z)is1.122zfortheredmagnitudesofthenorthernAbellclustersand4.14z−0.44z2forthevisualmagnitudesofthesouthernACOclusters(Ebel-ingetal.1996).ThenthemaximumdistanceDmax,itowhicheachclustercouldbeobservedforthesurveymagnitudelimitm10,lim=16.5canbecalculated.Andtheselectionfunctionisgivenby
S(d)=
3
3Dmax,i
Daltonetal.(1997)havepublishedacatalogof
APMclusterswithrichnessrangeR≥40.ThisisthecatalogthatDaltonetal.(1994a)haveusedtomea-suretheAPMclusterCF,whichisconsistentwiththeresultsofDaltonetal.(1992)andCroftetal.(1997).ThesurveyregionisshownbyMaddoxetal.(1990)∗andcontains957clusterswithmx≤19.4(mxistheap-
.(4)
Usingthedistributionofclusterssmoothedinthered-shiftspaceastheradialselectionfunctiondoesnotmakemuchdifferenceinourresults.Welimitthe
4LEEANDPARK
parentmagnitudeofx-thbrightmembergalaxywherex=R/2.1;thisisanalogoustoAbellcluster’smofclustersintheAPMsampleshows10).Thenumberaratherstrongdependenceondeclination(Park&Lee1998).InadditiontotheAPMsurveylimitswehaverestrictedourAPMsamplestothedeclinationrangeof−25◦≥δ≥−65◦toreducethepotentialeffectsofthisdeclinationdependence.Butthischangeofdecli-nationboundariesturnedouttomakeaninsignificantdifferenceintheCF.
SincetheAPMgalaxycatalogiscompleteoverthemagnituderange17.0≤bandtheclusterrichnessJcounting≤20.5(Maddoxsliceiset[mal.1990)miscompletexx+1.0],theAPMclustercatalog−over0.5,magnituderange17.5≤msamplecontains927clusters.x≤But,19.in4.thisThissample,completethenumberofclusterswithmeasuredredshiftsisonly350.ForclusterswithR≥50and17.5≤mhowever,thecompletenessbecomes83%andxincreases≤19.2,forhigherrichnesslimits.Wethereforemakefourmagnitude-limited(17.5≤msampleswithR≥50,60(completenessx≤19.2)richness93%),sub-70(96%),and80(100%)(seeTable1).
TocalculatetheselectionfunctionoftheAPMclus-ters,thebrightmagnitudelimitmaccountaswellasthex,low=17.5shouldbetakenintofaintlimitmx,upp=19.2.TheabsolutemagnitudeMxcorrespondingtomxofeachclusteriscomputedbyequation(3),whereK(z)=3zisadopted(Daltonetal.1997).ThentheselectionfunctionoftheAPMsurveyisgivenby
S(d)=
3
D3−3(5)
max,i
Dmin,iwhereDmax,iisthemaximumdistancetowhichani-thclustercanbeobservedbythefaintendofthesur-veymagnitudemdistancetowhichx,upp=19.2,Dtheclustercouldmin,iistheminimumbeincludedinthesampleofmx,low=17.5,andΩ=1.02sristhesolidangleofoursurveyregionoftheAPMsample.Again,usingthedistributionofclusterssmoothedinredshiftspaceastheselectionfunctionmakeslittlechangeinourresults.Thesampleislimitedtothedistancein-tervalfromdin=50h−1Mpctodout=500h−1Mpc.ThemeanspacedensitiesofourAPMR≥70and80subsamplesagreewellwiththoseestimatedbyCroftetal.(1997).ButthedensityofourR≥50sampleissomewhatlowerthan3.4×10−5h3Mpc−3ofCroftetal.(1997)becauseoflowercompletenessofclus-terswithpublishedredshifts.WeadoptCroftetal.(1997)’svalueinthestudyoftherichnessdependenceofcorrelationamplitudesin§IV.(c)
TheXBACs
TheXBACscatalog(Ebelingetal.1996)isacom-plete,−2all-sky,X-rayflux-limited(fs−1inthe0.1∼2.4keVband)x≥5.sample0×10−12ergcmof276
Fig.2.—X-rayluminosityLxofversusredshiftzdistribution
lines276limit;representXBACsclustersand201BCSclusters.Thedashed10−125.0×10−the12Lxerglimitcm−2correspondings−1tothesurveyfluxshowergcm−2s−1fortheBCS,respectively.fortheXBACsTheandsolid4.4×ples.
theLanddistancelimitsofvolume-limitedsubsam-linesxlimitsAbellclustersofgalaxiescompiledfromtheROSATAll-SkySurvey.Weconstrainoursampleto|b|≥30◦toreducetheeffectsofthegalacticobscuration.AftertheLMCandtheSMCregionsareremoved,theremainingsurveyareacovers6.27srofthesky.Thecomplete-nessoftheXBACscatalogisabout80%duetovari-oussourcesofincompletenesswhichareverydifficulttoremove(Ebelingetal.1996).Wehavefurtherre-movedtheX-raysourcescontaminatedbypointsourcesorwithoutmeasuredredshiftsfromtheXBACscatalog.ThesixdoubleclustersintheXBACscatalogarecon-sideredassinglesystemswhichhaveundergonemerg-ing.Theresultingfinalsampleincludes186clusterswiththeflux-limitoffwith|b|≥30◦inthex≥redshift5.0×10−12ergcm−2s−1andrange0 TheBCS Recently,theROSATBCScataloghasbeenpub-lished(Ebelingetal.1998).UnliketheXBACscatalogwhichisbasedontheAbellclustercatalog,theBCS CORRELATIONFUNCTIONSOFCLUSTERSOFGALAXIES5 Table2 Volume-limitedsubsamplesoftheXBACsandtheBCSclusters LX−h2ergs−1) nc 3(hMpc−3) r0 −1(hMpc) .725.7+3−3.8.125.2+4−4.3.230.3+8−6.5+26.445.8−12.5 (1044 Ncγ .51 2.55+0−0.43.552.48+0−0.54.722.21+0−1.27.751.84+0−0.99 Lx≥0.24Lx≥0.48Lx≥0.65Lx≥1.00 49675951 3.53×10−61.84×10−61.07×10−60.53×10−6 Lx≥0.24Lx≥0.48Lx≥0.70Lx≥0.90 33494034 3.02×10−61.70×10−60.84×10−60.51×10−6 .2 33.0+6−5.9.424.9+5−5.6.830.2+9−8.9+21.836.3−13.8 .49 1.82+0−0.50.912.68+0−0.77.942.22+0−1.26.242.05+1−1.76 havecalculated.(a) DD.Table3liststheCFdatawe OpticallySelectedClusters 4NcNr DR2 Least-squarefitsofapowerlawequation(1)tothe bestestimatesofCFarepresentedinTable1fortherichnesssubsamplesoftheAbellandtheAPMclus-ters.Wedeterminethe68.3%confidencelimitsofeachparameterindividuallyassumingthatthedistributionfortheuncertaintiesofξccisGaussiananddifferentξccbinsareindependentofeachother,whichisapopular,butstrictlyspeaking,impreciseassumption. Figure3ashowstheCFsoftheAbellclusterswithRC≥0(circle)andRC≥1(square).ThedashedlineisPostmanetal.(1992)’spowerlawfitwithr0=20.6h−1Mpcandγ=1.86estimatedfortheAbellRC≥0sample.TheCFsinFigure3barethoseoftheAPMclusterswithR≥50(filledcircle),R≥60(triangle),R≥70(square),andR≥80(opencircle).Thedottedlineisapowerlawwithr0=14.3h−1Mpcandγ=2.05(R≥50)andthedashedlineiswithr0=16.6h−1Mpcandγ=2.10(R≥70)bothde-terminedbyCroftetal.(1997).ItcanbenotedthattheamplitudeofCFsteadilyincreasesastherichnessincreases. Daltonetal.(1992)havecomparedtherichnessoftheACOclusterswiththatoftheAPMclusters(hereafterAPM92)andfoundthatthemeandifferenceinrichnessisRACO−RAPM92=10.1.AndDal-tonetal.(1994a)havecomparedanewversionoftheAPMclustercatalog(whichisthecatalogweuse)with 6LEEANDPARK dashedlineisξ(r)=(r/20.6h−1Mpc)−1.86ofPostmanetal.(1992)forRC≥0Abellclusters.(b)Magnitude-limitedsamplesoftheAPMclusterswithR≥50,60,70,and80.Thedottedlineisξ(r)=(r/14.3h−1Mpc)−2.05ofR≥50sampleofCroftetal.(1997)andthedashedlineisξ(r)=(r/16.6h−1Mpc)−2.10oftheirR≥70sample.(c)Volume-limitedsamplesoftheXBACsclusterswithLx≥0.24,0.48,and1.00×1044h−2ergs−1.Thedashedlineisξ(r)=(r/24.6h−1Mpc)−1.80ofAbadietal.(1998)forclusterswithLx≥0.54×1044h−2ergs−1.(d)Volume-limitedsamplesoftheBCSclusterswithLx≥0.24,0.48,and0.90×1044h−2ergs−1. Fig.3.—Correlationfunctionsofclustersofgalaxies.(a)Magnitude-limitedsamplesoftheAbellclusterswithRC≥0and1.The theiroldversioncatalog,andfoundarelationRAPM=1.2RAPM92+24.8withascatterof16inRAPM.WethengetaroughrelationRAPM=1.2RACO+12.7.Therefore,thecountsoftheAbellclusterswithR=30(RC=0)and50(RC=1)correspondtotherich-nessesoftheAPMclusterswithroughlyR=49and73,respectively.InTable1wefindthatthecorrelationlengthsoftheAPMclustersaregenerallysimilartothoseoftheAbellclustersatsimilarlevelsofrichness.ContrarytoCroftetal.(1997),theAPMsubsampleswithhigherrichnesslimitsshowclusteringasstrongasorratherstrongerthantheAbellclustersamples.Itshouldbenotedthattherearelargeuncertaintiesbothintherichnessrelationandintherichnessitself(theuncertaintiesofRACOandRAPMisabout18and5,respectively;Daltonetal.1997),andthattheAbellRC≥0sampleisanincompletesamplewithfewerpoorerclusters. Park&Lee(1998)havecalculatedtheCFsoftheAPMclustersusingvariousCFcalculationmethodsandsamplecorrections.Theyhaveuseddifferentselec-tionfunctions,variedsampleboundariesbothinangleanddepth,andappliedgalacticlatitudeordeclinationdependencecorrections.Evenwiththesevariations theyhaveconsistentlyobtainedCFswithamplitudehigherthanthatreportedbyDaltonetal.(1994a)orbyCroftetal.(1997),andcouldnotfindthesourcesfordiscrepancy.Dalton(1998)haspointedoutthatourredshiftsamplecontainstheclusterswithredshiftsdrawnfromtheLasCampanasRedshiftsurveyorfromtheEdinburgh-Durham-MilanoclusterredshiftsurveywhichcoveronlypartsoftheAPMGalaxySurveyre-gion,andthattheseclusterscouldproduceaninho-mogeneityinthedistributionofclustersonthesky.Ourstatisticalsamples,however,containveryfewofthoseclusters,andwehavefoundnoeffectontheCFbythem.Forexample,ourmagnitude-limitedsamplewithR≥70doesnotcontainanyoftheclustersinquestion.(b) X-rayClusters Figure3cshowstheCFsoftheXBACssampleswithLx≥0.24(triangle),0.48(circle),and1.00×1044h−2ergs−1(square).ThedashedlinerepresentsAbadietal.(1998)’sfitξ(r)=(r/24.6h−1Mpc)−1.80fortheXBACsclusterswithLx≥0.54×1044h−2ergs−1.ThecorrelationlengthsreportedbyAbadietal.(1998) CORRELATIONFUNCTIONSOFCLUSTERSOFGALAXIES7 Table3 Two-pointcorrelationfunctionsofclustersofgalaxies Abell RC≥0ra ξcc(r) r RC≥1 ξcc(r) r R≥50 ξcc(r) r R≥60 ξcc(r) r APM R≥70 ξcc(r) r R≥80 ξcc(r) XBACs Lx≥0.24br ξcc(r) r Lx≥0.48 ξcc(r) r Lx≥1.00 ξcc(r) r Lx≥0.24 ξcc(r) r BCS Lx≥0.48 ξcc(r) r Lx≥0.90 ξcc(r) ab Separationinunitsofh−1Mpc. X-rayluminosityinunitsof1044h−2ergs−1. agreewellwithourresults,exceptforthehighestLxsubsamplewithLx≥1.00×1044h−2ergs−1.Itshouldbeconsidered,however,thattheuncertaintiesinthefittingparametersareverylargebothfortheirandourhighestLxsubsamples.TheCFsoftheBCSclustersareshowninFigure3d. InTable2welisttheparametersofCFsoftheXBACsandtheBCSclusters.OurX-rayclustersub-sampleshaveverylownumberdensityandconsistoftherarestclusterswhichmightbeverymassive.Asexpected,thecorrelationlengthisverylargeandtendstoincreaseastheluminositylimitincreases.Bor-gani,Plionis,&Kolokotronis(1999)calculatedtheCFsfromtheXBACsandtheirresultingr0fromthesamplewiththesameflux-limitasours,fx≥5.0×10−12ergcm−2s−1,is26.0h−1Mpc,whichagreesex-cellentlywithourresultofthelowestLxsubsample;r0=25.7h−1Mpc.IV. RICHNESSDEPENDENCE WecomparetheamplitudesofCFsoftheAbellandtheAPMclustershavingthesamelevelofrichness.Richnesslevelofaclustersampleisoftenrepresented −1/3 bythemeanseparationdc=ncratherthanrichnessitself(Szalay&Schramm1985;Bahcall1988;Bahcall&West1992;Croftetal.1997)becausethedefinitiononrichnessisdifferentfordifferentcatalogs. Ourresultsontherichnessdependenceoftheclus-terCFareshowninFigure4.FilledcirclesaretherelationsmeasuredfromtheAbellsample,andfilledsquaresarefromtheAPMsample.WefindthattheclusteringstrengthoftheAPMclustersisconsistentwiththatoftheAbellclusters,contrarytowhatCroftetal.(1997)haveclaimed.Ontheotherhand,ourre-sultsagreewellwiththepreviousstudiesontheAbellclusters(seeTable1ofNicholetal.1994andotherref-erencespresentedinFig.4),butdisagreewiththoseoftheAPMteamandNicholetal.(1992).ThesolidlineinFigure4isalinearfittother0versusdcdistributionfoundfromouropticalclustersamples.Therelationwefoundisr0=0.40dc+3.2.ThelongandshortdashedlinesaretherelationsofBahcall&West(1992)andCroftetal.(1997),respectively.OurrelationisverysimilartoBahcall&West(1992)’slawexceptforaconstantoffset.However,ourresultstronglydisagreeswiththeslopefoundbyCroftetal.(1997). Ther0versusdcrelationsoftheXBACsandtheBCSclustersareconsistentwitheachother,butaremuchlowerthantherelationforopticalclusters.Theslopeoftherelationisuncertain,buttendstoin-creaseasdc>80h−1Mpc.ThecorrelationlengthscalculatedbyBorganietal.(1999)fromtheXBACsandbyCollinsetal.(2000)fromtheROSAT-ESOFlux-LimitedX-ray(REFLEX)galaxyclustersurvey(B¨ohringeretal.2001)areconsistentwithourre- 8LEEANDPARK Fig.4.—Dependenceofthecorrelationlengthr0ontherichnessorthemeaninterclusterseparationdc.Thesolidlineisr0= 0.40dc+3.2,whichisabestfittoourresultsfortheAbellandtheAPMsamples.ThelongdashedlineistherelationreportedbyBahcall&West(1992),andtheshortdashedlineisbyCroftetal.(1997). sultsandtheirdistributionsshowratherweakdepen-dencewithLxlikeours,orlooklikeevenrandomlydistributed.Asdiscussedinthefollowingsection,wethinktheX-rayluminosityofrichclustersisnotaverygoodmeasureofrichnessormass.Itshouldbealsonotedthattheuncertaintiesinr0areratherlarge.V. DISCUSSION Wehavemeasuredthecluster-clustercorrelationfunctionbyusingfourcatalogsofopticalandX-rayclusters.WehaveappliedthesameanalysismethodandthesamemethodofcharacterizingtheresultingCFstoallobservationalsamples.OurCFfortheAbellclustersampleisconsistentwiththepreviousworksbyBatuski&Burns(1985),Ling,Frenk,&Barrow(1986),Retzlaffetal.(1998),andMilleretal.(1999).Thecor-relationlengthsreportedbyPostmanetal.(1992)andBahcall&Soneira(1983)areslightlylargerthanours.ThoseestimatedbyHuchraetal.(1990)andPeacock&West(1990)areveryclosetooursfortheAbellclus-terswithRC≥1,butaresomewhatlargerthanoursinthecaseofRC≥0.Ontheotherhand,ourCFfortheAPMclustersamplehasthecorrelationlength significantlylargerthanthosereportedbyDaltonetal.(1992,1994a)andCroftetal.(1997),inparticularforrichersubsamples.Wehavenotbeenabletofindthereasonforthisdiscrepancy(Park&Lee1998). Sutherland(1988)andEfstathiouetal.(1992)havesuggestedthatprojectioneffectsoftheAbellclustercatalogcausedanartificialradialcorrelation,whichcouldleadtoanoverestimationofCFrelativetotheAPMsample.Theyhaveattemptedtocor-recttheAbellsamplefortheprojectioneffects,andreportedthatthecorrelationlengthwasreducedtor0≃14h−1Mpc.However,severalotherstudieshavepointedoutthattheuncertaintiesinherentinthiscor-rectionareverylarge,andthatthecorrectionmayhavebeenoverdone(Nicholetal.1992).Milleretal.(1999)foundthattheexcessradialcorrelationisnotartificialbutduetotwoelongatedsuperclusters,UrsaMajorisandCoronaBorealis.Postmanetal.(1992)haveesti-matedthemagnitudeoftheprojectioncontaminationandhavefoundthatonly17clusters(5%)intheir351Abellclustersmaybecontaminated.TheyalsohavemeasuredtheeffectsofthesecontaminatedclustersontheCFandhavefoundthattheamplitudeandslopeoftheCFdonotchangesignificantly.Ourstudyindi- CORRELATIONFUNCTIONSOFCLUSTERSOFGALAXIES 9 catesthattheAPMteam’sCFsoftheAPMclustersareunderestimatedforrichersubsamples,andtheCFoftheAbellsampleisconsistentwiththatoftheAPMsampleatsimilarrichness. Wehavedetectedaratherstrongdependenceoftheclusteringamplitudeontherichnessofclusters.Thecorrelationlengthrhavefoundfrom0versusmeanseparationdtionwetheAbellandtheAPMcrela-sam-ples,r0=0.40d&Westc+3.2,hasaslopeequaltothatfoundbyBahcall(1992),buthasafinitecorrelationlengthatzeroseparation.Wethinkthisismorerea-sonablebecausethecorrelationlengthshouldbefiniteatthecontinuumlimit,namelyforthematterfield.ByusingtheX-rayselectedclustersonecangreatlyovercometheprojectioneffectswhichmighthaveaf-fectedtheopticalclustersamplesselectedfromthedis-tributionofgalaxiesprojectedonthesky.AlthoughtheXBACsclustersarenotgenuineX-rayselectedclustersandhaveincompletenessduetothemissingpoorACOclusters,theyareindeedthree-dimensionallyboundsystemsandthesamplingisnearlyfreefromfalsede-tectionoroverestimationofrichness.TheBCSclustersareselectedpurelyfromtheirX-rayproperties,andalsohavenegligibleprojectioneffects.However,sincethenumberofX-rayclustersisnotyetlargeenoughtomakeasufficientlylargevolume-limitedsample,theCFsmeasuredfromtheXBACsandtheBCSclustershavelargeuncertaintiescomparedtothoseofopticalclusters.TheclusteringstrengthoftheseX-rayclus-tersisingenerallowerthanwhatisexpectedfromther0versusdcrelationofopticalclusters.Thisisproba-blyduetothefactthattheX-rayluminosityofX-rayclustersisnotanexcellentmeasureoftheclusterrich-nessormass.TheX-rayluminosityofaclusterdependsontheenvironmentandactivityaswellasitsdynami-calmass.Infact,Figure4ofDavid,Forman&Jones(1999)showsthattheclustersverybrightinX-raybe-longtovariousrichnessclasses.Gilbank,Bower,&Ca-stander(2001)alsofoundthatX-rayselectionmissessomesignificantrichclusterswhileopticalselectioncandetectallX-rayclusters.However,thereisatrendoftheclusteringstrengthoftheX-rayclustersincreasingasthemeanseparationincreases.Despitethelargeun-certainties,thecorrelationlengthoftheX-rayclustersishigherinamplitudecomparedtoCroftetal.(1997)’sexpectation. WeconcludethattheAbellandtheAPMclustersdohaveCFsstatisticallyconsistentwitheachotherandshowasimilarrichnessdependencerelationoftheclus-teringstrength.Itshouldbeagainemphasizedthatourresultshavebeenobtainedbyapplyingthesamemethodofanalyzingtheobservationaldataandcalcu-latingtheCF. ACKNOWLEDGEMENTS WethankDr.EbelingforsupplyinguswiththeBCScatalogattheearliesttime.WealsothankDrs.MichaelS.Vogeley,NetaA.Bahcall,GavinB.Dalton, andR.A.C.Croftforvaluablecomments.ThisworkwassupportedbytheBasicScienceResearchInstituteProgram,MinistryofEducation1995(BSRI-98-5408). 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