1991MNRAS.252...19S 3 123 3 2 1 A statisticallycompletesurveyforarc-likefeaturesinimagesofdistant H. E.Jorgensen I. Small,R.S.Ellis,M.J.Fitchett,H,U.Norgaard-Nielsen,L.Hansenand rich clustersofgalaxies Mon. Not.R.asir.Soc.(1991)252,19-29 UniversityObservatory,Ostervolgáde,Copenhagen DanishSpaceResearchInstitute,Lyngby,Copenhagen DepartmentofPhysics,UniversityDurhamDH13LE Accepted 1991April24.Received23;inoriginalform1990October22 Tyson, Valdes&Wenk1990)arisefromgravitationallensing of backgroundsources,theirstatisticsmightbeusedtocon- If thearc-likefeaturesseeninlong-exposureimagesof (which isdifficulttoobtainbyothermethods)andthered- moderate redshiftclustersofgalaxies(Fortetal1988; C12244-02 (Lynds&Petrosian1989),whicharesuffi- giant arcs,e.g.thoseinAbell370(Soucailci«/.1988)and arcs asacosmologicaltool. a demonstrationofgravitationallensing,butouraimistouse menon issufficientlynewthatithasbeenregardedlargelyas unobtainable byconventionalspectroscopy.Thearcpheno- strain boththemassdistributioninlensingclusters ciently extendedthattheycannoteasilybeunderstoodas shift distributionofthepopulationfaintfieldgalaxies normal galaxyimages.Althoughthenumberofgiantarcsis small, inthosefewcaseswhere spectroscopyhasbeenpos- 1 INTRODUCTION Fort 1990). Tyson etal1990), wheretheirstrikinglyblue coloursand sible, thelensinghypothesis appears tobeconfirmed(see deep framestakenindifferent passbands(Fortei<2/.1988; Such arcletsareoftenapparent onlyafterthesubtractionof The arcsdetectedfallintotwocategories.First,thereare Secondly, therearethesmaller featurestermed‘arclets’. © Royal Astronomical Society • Provided by the NASA Astrophysics Data System Using datafromanearliersearchforsupernovaeindistantclusters,wehave SUMMARY constructed ahomogeneoussetofVimagesfor19richclustersmeanredshift the usefulnessofsuchacatalogueforderivingbackgroundsourceredshift lensing ofbackgroundsources.Alist20candidatearcsispresented.Weexamine analysed thedataforextendedarc-likefeatureswhichmightarisefromgravitational ¿ =0.32.Byconsideringtheimagesaboveafixedsurfacebrightnesslimit,wehave galaxies tofaintlimits.Weshowhow,inprinciple,alikelihoodratiotestbasedonthe distribution, N{z).Whilstthenumberandshapedistributionofarccandidatesis vides ameansofdeterminingaccurateclustervelocitydispersionsatanyredshift. radial distributionofarcsinasinglewell-studiedclustercoulddeterminewhether consistent withthelensinghypothesis,clustervelocitydispersionsofveryhighpreci- significant fractionofthefaintgalaxypopulationisathighredshift.Thistestalsopro- sion wouldbeneededtoprovideusefulconstraintsonthefractionofhigh-redshift further evidenceofalensingphenomenon.Withoneexcep- orientation withrespecttotheclustercentrearetakenbe tion (theisolatedarcA5inAbell370-Soucailetal1990), firmed. Forexample,asignificantfractionofarcletscouldbe the lensinghypothesis,whilstplausible,isnotstrictlycon- edge-on bluespiralsatanyredshift. spectroscopy isnotyetavailableforthearcletpopulation,so which hasbeensystematically searchedforarcs.Theresult- ments inthecontextofanewsampledeepclusterimages tional sampleprecludeddefinitiveconclusions. of thenaturebackgroundsourcepopulation.These (1989) demonstratedthatforeachgiantarcthereshouldbe field galaxies.Wealsocompare thisstrategyofsurveyinga ing catalogueofcandidatearcs allowsustoexaminetheir articles exploredthevariousprobabilitydistributionsin a newprobeofthemassdistributioninclusterlensesand sizes wouldtestthelensinghypothesisfurther,andmightadd many arcletsandthattheoccurrencerateforarcsofdifferent some detail,buttheabsenceofanywell-definedobserva- potential forconstrainingthe redshiftdistributionoffaint the alternativeapproachwhere thesurfacedensityofback- reasonably largenumberofclusters tomoderatedepthswith ground galaxies israisedtoastatisticallyuseful levelina Our aiminthispaperistoexaminesuchtheoreticalargu- Grossman &Narayan(1988)andNemiroffDekel 19 91MNRAS.252. . .19S -2 20 I.Smalletal. been constructedtomaintainasuniformadetectionlimitfor single clusterviamuchdeeperintegrations(c/.Tysonetal found isconsistentwiththelensinghypothesis,butourclus- strates theirapplicationtooursample.Thenumberofarcs new observationaldatabaseofdistantclusterswhichhas we considertheimplicationsofthisworkandprospective tion forthestatisticaltestswehavedevelopedanddemon- As partoftheDanishdistantsupernovasearch(Hansen, role whicharcsmayplayinfaint-objectcosmology.Wesum- more detailedanalysisofoneorclusters.InSection4 cation andbrieflydiscusshowthiscouldbeappliedtoa that ourcurrentsampleistoodiverseforaneffectiveappli- new techniquewhichmightresolvethisproblembutshow detail beforestrongconstraintscouldbeplacedonthepro- ter propertieswouldneedtobeunderstoodinmuchgreater arcs andarcletsaspossible.Section3discussesthemotiva- in asearchforfaintsupernovae(SNe).SinceTypelasuper- (Abell 1958),itssoutherncounterpart(Abell,Corwin& taken ontheDanish1.5-mtelescopeatLaSilla,Chilefor~ Norgaard-Nielsen &Jorgensen1987;Couchetal1989), marize ourmainconclusionsinSection5. portion offaintgalaxiesbeyondtheclusters.Wedevelopa discovered andtheimplicationsforcosmologySN nova couldoccuratanylunation,limitingmagnitudesofF~ photometric observationsoftheclustersatdifferentepochs logue (Couchetal1990).Theprincipalaimwastocompare Olowin 1989)andthesouthernAAOdistantclusercata- many VCCDframesofaverageexposuretime~1hrwere those whoseseeingmatchedmostcloselythecurrentcon- varied. IntheSNsearch,newframeswerecomparedwith rate. OverthemanyrunsatLaSilla,observingconditions Norgaard-Nielsen etal(1989)discussthefaintSNe 60 clusterswithredshifts0.24hr.The19clustersthat can reliablyreachalimitingsurface brightnessoï/¿y~26.0 determined thata4-hrVintegration intypicalconditions extended object.Fromphotometryinseveralclusters,we that, providedtheframeswererestrictedtothosewith detection ofthearcletA5(Soucailetal1990).Wefound cluster Abell370(Plates1and2,Fig.1),testingforthe comparison. one imageusingfiducialstarsascalibratorspriortoeventual matches werethenallowedforbyrescalingandsmoothing mag arcsec. scope wassecured,thearcletcouldbereadilydetectedasan measured seeing<1.5arcsecFWHMandthataneffective arcs, weregisteredandmedianfilteredseveralframesofthe 1987-89. ClusterswereselectedfromtheAbellcatalogue A planofthepaperfollows.InSection2wedescribea As atrialapplicationoftheDanishframesforfindingfaint Our clustersampleforthisstudy consists,therefore,of © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 2 with 1pixel=0.47arcsec. /¿j/=26.0 magarcsecandthearcletA5.Thescaleisinpixels, Figure 1.ContourmapofPlate2,showingalowestisophote o Table 1.Danishclustercatalogue. AC 122 J2183.27TL Abell 370 Abell 222 AC118 J2001.21C 0346-45 J2175.23C Cluster J1834.2TC J2090.7CL Abell 1942 J1836.14RC J1836.23T AC103 AC106 AC113 Abell 2397 1141-28 AC114 RA (1950)DecRedshiftExposure(hrs) 00 1148 03 3108 02 3720 01 3504 05 1214 04 0054 03 4649 03 4537 20 5245 20 0600 22 5600 22 5043 21 5337 10 0212 13 4111 13 4035 11 4140 10 4214 14 3555 A370(V) 5HRMEDIAN -13 1448 -30 4200 -28 1824 -07 0612 -48 2154 -27 2049 -45 2430 -34 46 -39 1649 -01 4751 -64 5122 -53 1850 -00 1546 -33 5927 -35 0444 00 1320 00 1412 03 5224 01 0832 0.38 0.38 0.42 0.21 0.33 0.28 0.43 0.37 0.21 0.31 0.42 0.54 0.24 0.22 0.28 0.24 0.31 0.31 0.22 1991MNRAS.252...19S Plate 1.Plates1and2showexamplesoftheco-additionDanish CCDframestakenatdifferentepochs.Single1-hrexposure. © Royal Astronomical Society •Provided bytheNASA Astrophysics Data System A370(V) 1hrRaw 19 91MNRAS.252. . .19S © Royal Astronomical Society •Provided bytheNASA Astrophysics Data System A570(V) 5hrMedian 1991MNRAS.252...19S arcsec. (d) Abell222and(e)2397. ThearcIDscorespondtothosegiveninTable2.Northisupand eastisleft.Thescalebarsrepresent10 Plate 3.Examplesofcandidatearcs infiveclustersprocessedaccordingtothepreceptsofSection 2:(a)AC118,(b)Abell1942,(c)114, © Royal Astronomical Society •Provided bytheNASA Astrophysics Data System 19 91MNRAS.252. . .19S © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 19 91MNRAS.252. . .19S -1 -1 2 -2 2 dispersion intherestframeisa=1483±566kms, from theiropticalcontrastagainstthebackgroundat The typicalrichnessofaclusterenteringthissubsetthe The individualVframeswerefirstregisteredbyageneral- weaker thanComa,sothemeanisreducedslightly.Overall, slightly inexcessofthatforComa.Therichnessesthe Velocity dispersionsareavailableforthreeofthesixsouth- Abell clustersarealldistance6,richnessclass1clusters, (z) =0.32±0.09. (cr ~800-1000kms)atameanredshiftof of Couch&Sharpies(1988).Forthesethemeanline-of-sight Silla during1987-89. is statisticallyequivalentto19clusterscomparableComa it seemsreasonableatthisstagetoconcludethatthesample known redshift(cf.Couchetal.1990).Severaloftheseare seven AAOclusters(Jsequencenumbers)canbeestimated ern Abelletal.clustersfromtheextensivefibrespectroscopy none hasareliablevelocitydispersionorX-rayluminosity. comparable toorexceedingComainrichness.However, mately fromthesourcematerialforclusters.Thefour supernova searchclustersamplecanbecalculatedapproxi- frames of1.5-arcsecseeingwilloccupy~10arcsec,we in thefigure,clearlyemerges. centres donotalignperfectlybutthecommonfieldisatleast deep imagefreefromcosmicraysanddefects.Theframe ized rotationusingthefiducialpositionsoftypicallysevento image andasingle1-hrintegrationarecomparedatthesame addition isillustratedinFig.1whereboththefinalAbell370 sky count,stackedandmedianfilteredtoproduceasingle choose tocutallcombinedframesatathresholdof2the might varyfromframetoframe.Sincethesmallestarcwith searched forarc-likefeatures,weneedtodefineaself-con- 2.2 HomogeneityoftheCCDsurvey useful forourpurposes.Ofcourse,byvirtueofaveragingat This cannotbeeliminatedentirelywithoutaccuratephoto- per centofthebackgroundsky. sky brightnessonthesescales:thiscorrespondstypicallyto1 an axialratioof,say,a/b>2thatwecanhopetorecognizein sistent limitingsurfacebrightnessvalueanddiscusshowthis 2cr thresholdofthebackgroundsky.ThearcletA5,labelled brightness thatcorresponds to aconstantsignal-to-noise sky brightnesswillapproachtheoverallaveragevalueatLa least fourframestakenatdifferenttimesforeachcluster,the ing surfacebrightnessissmallenoughforthecataloguetobe Nevertheless, wedemonstratethatthevariationinlimit- checked forthreeclusterscontaining goodVphotometric meridian, implyaspreadofless than0.15maginthesurface per centwhich,togetherwith airmassdifferencesonthe metric zeropointswhicharenotavailableineverycluster. the varyingnightskybrightnessanddifferenceintotalexpo- zero points(A1942,A370and AC118)andtheseconfirma ratio. Theabsolutevalueof theskybrightnesscanbe limiting surfacebrightnesscutvariesfromclustertocluster. sure timefromoneclustertoanotherwillensurethatsuch a 0 10 stars.Theresampledimageswerescaledtothesametotal mean skyvalue of=21.0±0.3magarcsec . Wetakethis 0 ~ 2x3arcmininallclusters.Thegaindepthbythisco- Before describinghowtheco-addedclusterframesare The meanexposuretimeof6.4hrhasalascatter~20 Although theclusterswerealwaysobservedindarktime, © Royal Astronomical Society • Provided by the NASA Astrophysics Data System axis iselongatedalongthenormaltolinejoining equals theundistortedradiusofgalaxy,andmajor mal cluster,anintrinsicallycircularbackgroundsource angular separationofthesourceandlens,0.Foranisother- (1989) showedindetailhowlensingelongatesanimageby Both Grossman&Narayan(1988)andNemiroffDekel lensed image,andtheangulardistance,d,ofarccandi- crucial toanystatisticalanalysis:theaxialratioA=a/bof appears elongatedsuchthattheminoraxisofimage sight velocitydispersion,a,insimplecases),andthetrue z), thelensmass(whichcanberepresentedbyline-of- an amountthatdependsonthesourceandlensdistances(z, date fromtheclustercentre(whichwillusuallybequotedin source andlenscentre.Twoobservationalparametersare Lavery &Henry1988).Secondly,byexaminingtheframes the skylevel(asdiscussedinSection2.2)andexaminedvisu- arcsec). during thesameperiod.Insummary,therefore,bycuttingat range asrepresentativeofthe15otherclustersobserved using theestimatedopticalcentreofcluster(ineverycase of defectsorweakfeatures. Finally, inconfusingsituationstheframescanbereadily visually, thecurvatureofarcscanalsobeassessed. centres wherearcshaveoftenbeenfound(e.g.Abell963, crowded fieldsandextensiveenvelopesnearthecluster more appropriatethananautomaticimageanalysisfor 2.3 Identifîcationofarccandidates was doneafterselection.SinceAbell370includedinthe the brightestmember),butitshouldbeemphasizedthatthis given inTable2.Inthislist,theradiusvectoriscalculated pushed toalowerthresholdclarifytherealityorotherwise several reasons.First,suchalgorithmsarenotsuitableforthe ally forobjectswithaxialratiosA>2.Thiswasdeemed survey shouldmaintainastrictsurfacebrightnesslimitof 2 aofthemedianskyvalueacrossco-addedimage,our S c cl cl s Each processedframewascutatathresholdof2above The completelistofcandidatearcsandtheirparametersis -2 26.0 ±0.3magarcsec. Table 2.Arccatalogue(/I>2). AC113 A1942 A1942 A1942 J2090-7CL A370 AC 114 AC114 AC113 AC113 A2397 J2090-7CL J2183-27T A370 A222 A222 AC114 AC118 AC118 AC 118 Cluster RedshiftArcIDAd\(arcsec) Arc-like featuresindistantclusters21 1 Radiusfromcluster centre. 0.22 0.38 0.38 0.22 0.22 0.24 0.22 0.22 0.28 0.37 0.37 0.21 0.21 0.31 0.31 0.31 0.22 0.31 0.31 0.31 A1942-3 A1942-2 A1942-1 J2090-2 J2090-1 J2183-1 A370-2 A370-1 A222-2 AC 114-1 AC 113-2 AC 113-1 A2387-1 A222-1 AC 118-3 AC 118-2 AC 114-3 AC 114-2 AC 113-3 AC 118-1 12.3 2.3 2.2 4.0 4.7 4.6 2.2 2.4 2.6 2.0 3.0 2.7 2.6 3.6 4.1 2.3 2.6 2.2 2.5 2.9 63.7 26.3 32.4 23.2 55.4 61.9 25.8 27.1 25.3 30.5 21.4 14.6 14.1 12.2 10.9 13.7 3.1 6.1 8.1 5.1 19 91MNRAS.252. . .19S 5 -2 22 /.Smailetal. besides A370,containthearccandidateswithlargest arc, weconsideritappropriatetoincludethisclusterinthe Danish SNsearchwithoutanyregardforitsspectaculargiant vector, itisclearthatsomearenot.Thesecouldbespurious distortions (A>3.5).Whilstthemajorityofcandidates sample. detections, orcaseswherethesymmetryofcluster are, infact,approximatelynormaltotheclusterradius few innumberandconsequentlytheirmeandepthcannot potential isfarfromcircularorelliptical. these arcsareinprogress,atopticalandinfraredwave- A ATprimefocusbjplatescannedbytheCOSMOS contamination rateinoursamplehasbeenestimatedby match thatoftheprimeF-bandmaterial.Furtherstudies images wereoccasionallytakenfortheSNsearch,theyare examine imagesinotherpassbands.AlthoughBandR density ofobjectscomplyingwithourdistortionandsize analysing theellipticitydistributionof10objectsina1 lengths. ForexamplethearcinA1942hasbeendetected we havenotemphasizedtheorientationofarccandi- i.e. 2percentofthoseinTable2.Furtherevidenceforthe criteria forthearcsissuchthatwewouldexpecttofindone machine toalimitingmagnitudeofbj~24.5.Thesurface square degreerandomlychosenfield.Thefieldusedisan dates, themajorityaretangentiallyaligned. lack ofcontaminationisgivenbythefactthat,eventhough mag arcsecwouldgiveoneinterloperinevery40clusters, number countstooursurfacebrightnesslimitofju=26 edge-on interloperinevery200clustersforthislimiting K (Aragon-Salamancaetal,inpreparation).Theexpected Their 58clusterscoveredamuchwiderrangeinredshiftand surface brightness.ExtrapolationusingTyson’s(1988) they pointedoutthatbothofthesearcsoccurinclusterswith clusters, theyfoundonlytwoexamplesofsuchgiantarcs. plete sampleofarccandidateswasthatLynds&Petro- A majorhopewithgravitational lensingisthatthearcstatis- fied above)butisprobablywithinthestatisticaluncertainties. and 02244-02giantarcsthat,duringanexaminationof58 asian (1989)whocommentintheiranalysisoftheAbell370 tions, e.g.about thelensingclustersthemselves. to addressthisquestion,using ournewsampletodemon- of backgroundsources.Inthis sectionwedeveloptechniques tics willprovidenewconstraints ontheredshiftdistribution that oneisseenin«8percentoftheAbellclusters.This with 0.2redshifts wellabovethemedianvaluefortheirsample strate thesensitivityofvarious statisticalprobestoassump- homogeneous andmoreusefulfortheanalysisbelow. passbands withamuchgreaterrangeofexposuretimes Since Lynds&Petrosian’simagesweretakeninavarietyof already dependsheavilyonthepresenceofAbell370justi- somewhat higherthanourrateforthelargestarcs(which 3.1 Motivation 3 ANALYSIS depending ontheclusterredshift,oursurveyshouldbemore v Plate 3showstheVimagesforthosefiveclusterswhich, To testthereliabilityofcandidatelist,wewouldliketo The onlysimilarattempttogenerateastatisticallycom- © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 2 2 form ofN(z)canbeestimatedforanymagnitude-limited galaxies (Efstathiouetal.1988)andcertainassumptions which isconservedinthelensingprocess.Formagnitude- number countsandspectroscopicstudies,thecandidatearcs Whereas galaxiesareselectedbyapparentmagnitudeinthe fibre opticsystemattheAATtodefineN(z)with85percent few hundredgalaxiestosuccessivelyfainterlimitsofappar- galaxies haspresentedadilemma.Multipleobjectspectros- on thebasisofpresent-dayluminosityfunction limited samples,aselectionfunction^(z)isusuallymodelled in Table2arefoundbyvirtueoftheirsurfacebrightness, Tyson 1988).Somesupportforthisevidenceistheprepon- tion functionforasamplelimitedbysurfacebrightness,this the redshiftsurveys. about theeffectsofredshiftandevolution.Inthisway 3.2 Approximatingtheselectionfunction what fainter,issignificantlymoredistant(Cowieetal.1989; this population,whichrisestodominatethecountssome- ent magnitude.Broadhurstetal.(1988)usedtheAUTOFIB copy on4-mtelescopeshasgeneratedredshiftsurveysofa would demandmoreinformationonthepresent-daypro- sample, includingoneslightlyfainterthanthecurrentlimitof faint bluepopulationhasbeenprovidedbytheirdetectionin derance atthisstageofarcswithbluecoloursandTysonet used theLDSSmultislitspectrographwithasimilarcom- completeness toalimitofb=21.5,andCollessetal.(1990) ness profiles(theircentralvaluesandforms)forwhichthere ing wouldberequiredofthedistributionsurfacebright- perties ofgalaxiesthaniscurrentlyavailable.Forexample,in However, ifthishypothesisiscorrect,theonsetofsucha the ultraviolet(Guhathajurta,Tyson&Majewski1990). deep clusterimages.Anupperredshiftlimit(z^3.0)forthis CFHT (seeKoo1989;Cowie&Lilly1990). conclusions havebeenderivedfromsurveysatKittPeakand of thefaintpopulationtobj=23liesbelowz~0.5.Similar increases withapparentmagnitudeandimpliesthatthebulk pleteness tobj=22.5.Theresultingmedianredshiftzhardly mag arcsec",wouldbebasedonhighlyuncertaininitial dicted N(z)forasample,example,limitedatp=26. Q remains considerablecontroversybecauseofobservational luminosities ofgalaxiesperunitvolume,agoodunderstand- addition toknowledgeofthedistributionintegrated distant populationmustbeverysuddenintermsofapparent if themagnificationisa/b,asample limitedat/¿=26.0mag meters areconcerned,wecan alsoestimatestatisticallythat, is notrequired.Asfarasthe limitingobservationalpara- are behindorinfrontoftheclusters; thepreciseformofA(z) are requiredtotestwhetherthe majorityoffaintblueobjects assumptions. selection effects(Phillipps,Disney&Davies1990).Anypre- simpler approach.Atthemost basiclevel,thearcstatistics rently seeninthespectroscopicsurveys(Ellis1990). magnitude, sincenohigh-redshift(z>l)objectsarecur- arcsec" in1.5-arcsec seeingapproximates anapparent- al.'s (1990)claimthatonlythebluepopulationisdistortedin trum sourcesemergingatbj=23hasledtospeculationthat 7 v K Recent workontheredshiftdistribution,N{z),offaint The discoveryofapopulationextremelyblueflatspec- Although itispossible,inprinciple,tocalculatetheselec- Rather thanintroduceuncertainmodels,weadoptamuch 19 91MNRAS.252. . .19S -2 1 example Turner,Ostriker&Gunn1984).Intrinsicallycircu- face densityof13.6objectsarcminwhich35percent made byNemiroff&Dekel(1989).Inthiscasethelensing The simplestclusterpotentialthatmightbeconsideredisthe ing thiswouldimply(z)~0.55andthebulkof hurst etal(1989)andCollessal.(1990).Roughlyspeak- of varioussubsetsthefaintgalaxypopulationwefirstpro- can placeusefulconstraintsonthelikelyredshiftdistribution defined byCowie&Lilly(1990).Toexaminewhetherarcs have coloursintheflat-spectrumcategory¿?-/<1.0as magnitude-limited samplewithV<26~2.5log(1.5a/b).For this population.Thenullhypothesis,//,impliesthatthe these redshiftlimits. explore variousdistributionsthatplacethesegalaxieswithin 4 assuggestedoriginallybyCowieetal.(1989).Wewill population havingz<1. pose thefollowingtwoillustrative(andextreme)hypotheses. verse withmatterinsidethebeamtocluster(seefor where r(z)=2c//~(l-1/y/l+z),assuminganQ=1uni- tance ajl-r(z,)/r(zj]awayfromtheclustercentre, geometry isstraightforward-alllightraysfromthesource their redshift,thesurfacedensityofthisfractionblue a/b =2,ourminimumcriterion,thisbecomesV<25. Every pointonthesourceplaneismovedanangulardis- where a potentialontheshapesofbackgroundgalaxieshavebeen singular isothermalsphere.Predictionsfortheeffectofsuch 3.3 Lensmodelsandrobustarcparameters tion. Ofcourseahostofintermediatepossibilitiesexist.Our flat-spectrum componentisacosmologicallyimportantone, fied lensinggeometrydoesnot affectthewidthofsource. lar sourceswhoseimpactparameterliesclosetothecluster are deflectedthroughaconstantangleatthecluster, the overallN(z)beingno-evolutionprediction.The nature (e.g.byvirtueofshort-termburstsstarformation); population co-existswiththeremainderandistemporaryin sources impliesthatthebulkofmetalswereformedin A(z) asthenon-evolvingdistribution,suggestedbyBroad- The singularisothermalsphere modelisusedinthispaperto centre willbedistortedconsiderably, whilesourceswith these extremehypotheses. purpose hereistoexaminethesensitivityofvarioustests alternative hypothesis,amountstothestatementthat remote past.Inthiscasethefraction(1-/),)withoutflat- perhaps associatedwithgalaxyordiscformationinthe larger impactparameterswill be distortedless.Thissimpli- spectrum coloursisassumedtobeanon-evolvingpopula- tions. 4 wewilldiscussitsvalidityand otherpossiblemassdistribu- show thetypeofanalysisthatmight becarriedout.InSection a =4jt y 0 0 F=25 asf.Cowieeta/.(1989)claimthat,regardlessof () () h At thislimit,Tyson’s(1988)bjandRcountsimplyasur- (ii) H:theflat-spectrumgalaxiesallliebetweenz=1and (i) //:theflat-spectrumgalaxiesbroadlysharesame We parametrizethefractionofflat-spectrumobjectsto { 0 © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 3 \l0kms \ ( ac) arcsec. (1) 2 J(Ñ,zJ = for comparingtheorywithobservationssincelittleisknown forward. FollowingNemiroff&Dekel(1989),forasingle their lengths(withsomeassumptionontheintrinsicsource tribution radiallyrelativetotheclustercentreorcentresand number ofarcsasafunctiontheirdistortionA,dis- distance fromtheclustercentre,d,asafunctionof unknown quantities. however, fairlyeasytomeasureandnotsensitivesuch about theintrinsicsourcesize.ThequantitiesAanddare, able data.Thearclengthdistributionisprobablynotuseful and ahypothesizedN(z),thismodelpredictstheobserved parameter, 0,andtheclustervelocitydispersion,o.Givena source redshiftz,thelensZ/,impact cluster withasingularisothermalspheremassdistribution size). Anyofthesequantitiescanbecomparedwiththeavail- set ofclusterswithknownorassumedvelocitydispersions N(A >A)=jtaUAc-1)-27iV,z,(2) where Therefore amixtureofpopulationswithfractionf brightness limitofthesampleandA(z)isnormalized thesized A(z).Noticethattheintegralislinearin^(z trivial tocalculatefromthisformulatheexpectednumberof arcs foranensembleofclusters,anddifferenthypo- redshift distributionofthesesources[i.e.JA(z)=1].Itis Here 2isthesurfacedensityofobjectsdownto c Figure 2.Thepredictednumber ofarcswith4>2asafunction c Sd s the fractionofsources,/,„placed atz,,=2,plottedforvarious cluster velocitydispersions. aKCscl h s s s S This modelgivesthefinalshapeofimage,A,andits Calculation oftheexpectednumberarcsisstraight- J Z c[ Arc-like featuresindistantclusters23 2 Ñ(z)[l -r(z/r(z]dz. sd 19 91MNRAS.252. . .19S -1 l = 24 I.Smalletal. from Ñ{z)wouldgive galaxies drawnfromA^(z)andtheremaining(1-f Figure 3.Predictionsfortheradialdistributionofarcswithy4>2aclustervelocitydispersion1000kmsandvarioushypothesized distribute theremainderaccordingtono-evolutionprediction.ThehistogramofradialdistancesforDanishdataisalsoshown. If aisknown(sayfromX-raydataorlargesampleof tribution isdescribedbythesingularisothermalsphere measured velocitiesinthecluster)andclustermassdis- N(z)\ thesolidcurveisforano-evolutionA^(z),whileothercurvesplacevariousfractions,f,ofsourcesatredshifts, z,and than somespecifiedvaluecoulddetermine/. model thenthenumberofarcsseenwithdistortionsgreater N.{A>) first steptonoteisthatimagesofcircularsourcesatzwith distortions A>Amustlieintheannulus s ®(d) =ß sh bution ofdforarcswithdistortionA>Aas cluster, onecanthencalculatetheprobabilitydistributionfor Given thatsourcesuniformlypopulatetheregionbehind source redshiftdistributiongivesthefinalprobabilitydistri- cl respectively thesolutionsof equations b h d withinthisannulus,andconvolutionofthatwiththe ft where ßisthenormalizingconstant andZi(d),z{dare àïCc and s d =[l+(A-l)]A(z,o ) c d A(zz,o'd). c c c c2 chl d ch2 2 Predicting thedistributionofdisnotquitesosimple.The = nal{A-\)^fJ{Ñ,z+,zl(3) c chcX . ^i(^c)Z,^cl) s Z2Ídc) © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 2Ñ{z)[d-A(z,Öd)] sch ¿fes (4) - 2 In thispaperwehavesolvedintegralbyMonteCarlo There aretwopossibleteststhatcanbecarriedoutonthe population. simulation oftheeffectourclustersonabackground Danish arcsample.Thesimplestistocomparethenumber under thetwohypothesesdiscussedabove.Thesecondis based onthedistributionofarcdistances,^(¿/),from 3.4 ComparisonofmodelswiththeDanishsample their clustercentres. galaxies atz=2withtheremainingdistributed ^4 >2expectedforourensembleofclusters,whereeach of arcsdetectedwithA>,N{\thatexpected according tothestandardno-evolutionN(z)predictedfor a ferred. However,theseclusters areveryunlikelytohave The observednumberofarcs,20,isconsistentwiththese cluster hasbeenassumedtohavethesamevelocitydisper- previous sectionwehavecalculatedthenumberofarcswith plot thatasmallfractionofhigh-redshiftgalaxiesispre- (Tyson 1988),theresultsforvariousa,areshowninFig.2. predictions. Indeedonemightbetemptedtoinferfromthis sion. Adoptingasourcesurfacedensityof13.6arcmin the clustervelocitydispersion. Thepredictednumberofarcs the relationbetweenN(A> A) andfdependsstronglyon equal velocitydispersions,and onecanseefromFig.2that but N2variesbyafactor ~ 2asfincreasesfrom0to1, c measurement of theclustervelocitydispersion caneasily carc F=25 maglimitedsample.Usingtheformulafrom c arc cb arc b Suppose forexamplethatoneplacesafractionfof b 1 991MNRAS .252,. . . 19S -1 2 2 2 figure theDanisharcdataprobabilitydistributionisshown. the same(unknown)velocitydispersionandliesatmean redshift population. mask theincreasednumberofarcsexpectedduetohigh- therefore, withalargesampleofarcsinonecluster,beable of thesourcesareplacedatvariousfixedredshiftsz.These tion andfourotherdistributionswhereavariablefractionf redshift ofthesample,0.32.Theprobabilitydistribution becomes. the fractionofhigh-zobjects,moreskewthisfunction to constrainN{z)usingtheshapeofthisfunction.Thelarger cluster velocitydispersionofo=1000kms.Onthesame probability distributions,<&(d),areshowninFig.3fora range ofredshiftdistributionscomprisingthenulldistribu- dispersion. Themostlikelycauseofthisdiscrepancyisthe and thedata.Indeedthereisnoagreementforanyvelocity <3>(d) isdifferentforeachofthesehypothesesandonemight give risetolargevaluesofd,andwithlowvelocitydisper- likely tocontainclusterswithhighvelocitydispersion,which range ofclustervelocitydispersions.Thesamplewillbe Clearly thereislittleagreementbetweenanyofthesemodels d forthosearcswithA>2hasthenbeencalculateda sample thedistributionofddoesnotacttoconstrain sion, whichwillnotactassignificantlensesatall.Forthis N(z) ofthefaintsources.Wenote,however,thatshape that expectedfromlensingandarethenumberofarcsin each clusterconsistentwiththeclustershavingequalvelocity sample; forexample,isthedistributionofAconsistentwith we shouldfindN(>A)<*(A-l)~,andthelackof dispersions? Independentoftheclustervelocitydispersion that theprobabilityitwasdrawnfrom dependence ona,meansthatthisshouldholdfortheentire h b cl c sample. AKStestontheobserveddistributionofshows of 20/19.Thenumberarcsintheseclustersvariesfrom0 cluster shouldhaveaPoissondistributionwithmeanvalue probability. Removal ofthelargearcinA370haslittleeffectonthis low value,itremainsconsistentwiththelensinghypothesis. c c ^íLíhmíT' the errorono(5crimustsatisfy bution isnotconsistentwiththePoissondistributionat in severaltomaximally8.Axtestshowsthatthisdistri- same velocitydispersionthenthenumberofarcsseeninany c then ôa,/a,<0.045.This accuracy wouldrequirethe c /V( >A)oc(v4-l)"distributionis~9percent.Whilsta why theabovetestswillnotworkonoursamplein For instance,ifR~2andf= 0Aandwerequire<5/),<0.2, where absence ofdetailedknowledgeabouttheclustersthemselves. measurement of~1000redshifts (Dáñese,deZotti&Tullio arc c chc c R =1)-AU/„0)}/N (f=0). < 1percentlevel.Wehavethereforeeffectivelydemonstrated arc 1980), aformidable taskeveninalocalcluster, letaloneone h arch <7ci 4L\R0fI h For thesecondtestweagainsupposethateachclusterhas There areothersimplequestionsonecanaskaboutthis On thesecondquestion,ifallofclustersdidhave One canshowthat,inordertoobtainfaccuracyôf, h © Royal Astronomical Society • Provided by the NASA Astrophysics Data System (5) _ 1 -1 -1 -1 bution <&(d\Ho)underthehypothesisofano-evolution cluster centre.Foranassumeda,thesingularisothermal chosen criticalvalueAatdistances{d,d-.)fromthe Suppose aclusterhasarcswithdistortionslargerthansome sphere modelcanbeusedtopredicttheexpectedddistri- 3.5 Alikelihoodratiotestbasedon(d) different shapesofthepredicted0>{d)fortwohypo- for 1075kms),themostsensiblemethodcompar- theses neverthelesssuggestthat,forasingleclusterwith at z~0.4.X-raytemperaturemeasurementstohighpreci- estimate theclustervelocitydispersion.Theawhichyields (d, £/.d)p(d\Ha=/?,forshort.Usingthe (//J. This®{d)yieldstheprobabilityofobservingdata is themostlikely.Thisapproachwillbediscussedirinext sion mightallowadeterminationofftothisaccuracy.The ing thehypothesesistoconstructratioA=p/p, distinction isunlikely(forA370Hyields1200kmsand the otherhypothesiswouldbefavoured.Sincesuchaclear the inferredvelocitydispersionsisclearlyunreasonablethen hood methodtoeachhypothesiswillyieldtwoestimatesof the sourcesatz=2withremainderdistributedaccording the greatestlikelihoodofobservingsetarcsis might enableustotestwhichofthesehypothesesaboutN{z) to theno-evolutionN(z).Applicationofmaximumlikeli- A> 1.5.Forano-evolutionN(z)theradialpositionsofthese persion. ThisestimatorwilldependontheassumedN{z). standard maximumlikelihoodmethod(Silvey1975),onecan N(z)(H) oronewithafraction,/),,ofgalaxiesathighredshift section. against another.Ifthisratiois large,onecansafelyrejectthe standard statisticallikelihoodratiotestofonehypothesis are distributedaccordingtotheno-evolutionmodel there issomeevidencethatthesearesubclusteredinvelocity A370 [thebluegalaxiesyieldalargervelocitydispersionbut alternate hypothesis(alargefraction ofsourcesathighred- ter velocitydispersions,althoughthecalculationsrelyon tion isthereforeanindependentmethodforestimatingclus- determined valueof1367!1Í4kmsfortheredgalaxiesin arcs yieldd=1200kms,gratifyingclosetotheoptically many arcs,ananalysisbasedontheshapeofthisdistribution theses. Inordertoanswerthis question,severalsimulations needed todistinguishconfidently betweenthesetwohypo- shift) infavourofthenull(noevolution) hypothesis. space, seeHenry&Lavery(1987)].Theradialarcdistribu- have beenmade. Thesearedescribedstepwise below. the estimateofvelocitydispersionbythismethodcanbe of A370since,forasmallnumberarcs(~6),theerrorson have beenfortunateinourestimateofthevelocitydispersion assumed clusterpotentialandN(z).Additionallywemight maximum likelihoodestimatorôoftheclustervelocitydis- a, denotedâandcq,twoprobabilitiesp,p^Ifoneof be veryaccurate,andcouldprovideausefulmeansofdeter- we willshow,withalargersampleofarcsthisestimatorcan mining velocitydispersionsofclustersathighredshift. as largetheerrorsindynamicalestimate.However, chd c cx2n c c c {2nflhc c h {)l {) V= 25,andthealternatehypothesiswillplaceafraction//,of 0 0{) In thissectionthenullhypothesiswillbethatsources The importantquestionnow is howmanyarcswouldbe As anexampleconsiderthefivearcsinA370with Arc-like featuresindistantclusters25 19 91MNRAS.252. . .19S 1 likelihood ratiosandestimatesof the clustervelocitydispersionforN=40.Noticethereisamuch clearerseparationofthecomponents. arcs generatedfromano-evolution N(z)whilethestarscorrespondtoarcsgeneratedwithf= 0Aandz=2.0.(b)Thedistributionof Figure 4.(a)Thedistributionof likelihoodratiosandestimatesoftheclustervelocitydispersion for N=6.Squaresymbolscorrespondto 26 I.Smalletal. arc b b arc © Royal Astronomical Society • Provided by the NASA Astrophysics Data System < en o (a) (b) < cr> o 0.8 0.9 0.85 0.95 0.9 0.95 D D * °.9*.„wF, *.3*^* m * N6 A15 Narc =40A>1.25 arc =>- LJu * □ a ** * ? * i1 . **° *rP □D D * Í1***^ □ °o * # S D * * g t cpn n 1.05 ^ □ D □B ° □ c % □□□ B 1.05 □ NoEvolutionModel □ NoEvolutionModel * f.=0.4 1.15 2.0 1.2 19 91MNRAS.252. . .19S 1 _2 [ og(Po/Pi)and5 =<7/f. generate thesetofAarcs(stage2)usingAf(z) a setof(S,logA)pairs. this velocitydispersionandthenullN(z)distribution(no critical A)usingasingularisothermalspheremodelwith and placetheclusteratz=0.4. S =ÔJCTandlogA\og(p/pi)asbefore. evolution). the simulationswithredshiftdistributionofandthus, tions shouldbegreaterthanthoseinthesecondset,and alternate hypothesis.InthiscasetheratioSisdefinedas hood estimatesofo(âandô)CalculatelogA= hypothesis N(z)andforarangeofclustervelocitydisper- arcs forarangeofclustervelocitydispersionswiththenull values ofSineithercaseshouldbeclosetounity.Fig.4(a) sions withthealternateredshiftdistributiondescribedabove. data setconsistingofsixarcswouldnotyieldatestH each hypothesis.ThereforeameasurementoflogAfrom terms ofdefiningprecisesignificancelevelsforthistesta ter velocitydispersion(typicallytobetterthan5percent).In this approachalsoprovidesaverygoodestimateoftheclus- clearly differentiatebetweenthetwohypotheses.Moreover, against Fig.4(b)showstheresultsforiV=40,which simulation withN=6.ThevaluesoflogAaresimilarfor shows aplotofthe100(S,logA)pairsforeachtype larger simulationwouldbeneeded,butbyinspectiononecan their effectonouranalysiswill dependontheirextent.Sup- only atmosta1percentchancethatwewouldbeinerror if thedatahaveameasuredvalueoflogA>1.0,thereis see thatavalueoflogA>1.0arisesforonly1percent tion p°cr.Weneedtocompare theangularscaleof cluster approximatestheisothermal spheredensitydistribu- criminate withsomeconfidencebetweenano-evolutionN{z) choosing thenulloveralternatehypothesis.Thus,witha lncl cluster core0 =rjr(z)withthelensing scale 0.The pose thecorehasasizer,and thatexteriortothisregionthe which fallasr~.Thisfindingisconsistentwithasingular distribution. Ontheotherhand,Beers&Tonry(1986)have difficult (seeFitchett1990forareview).Eventhebest- first isouruseofthesingularisothermalspheremodelas There areseveralassumptionswhichmustbediscussed.The set of40arcsinasinglecluster,itshouldbepossibletodis- distribution inclustersawayfreefromassumptionsisvery larger, thenthistestbecomesevenmorepowerful. and anN(z)with40percentofobjectsatz=2.lfforare centres ofclusters. isothermal spheremassdistributioniftraceslightinthe shown thatclustershavegalaxysurface-densitydistributions studied richcluster,Coma,hasapoorlyconstrainedmass lens model.Itiswellknownthatdeterminationofthemass arc i0 c d 4 DISCUSSION C|10w c[0{:1 0 10 arc arc10 10 10 10 corc d lens c hb (5) Repeattheaboveprocedureseveraltimes,resultingin (1) Choosetheclustervelocitydispersiona,atrandom, (6) Nowrepeattheabovesimulationsbutinthiscase (2) Generate7Varcs(withdistortionsaboveaspecified (4) Foreachhypothesisdeterminethemaximumlikeli- (3) Nowcalculatethelikelihoodofobservingthisset Ideally thenthevaluesoflogAinfirstsetsimula- If clustersreallydohavecores ofconstantdensitythen c arc 10 © Royal Astronomical Society • Provided by the NASA Astrophysics Data System 2 {2-1 <12 1 A =1.25,gives0,~108(a/1200kms~*)arcsec.Now,for This isthescalethatmustbecomparedto0.Foracluster A. Foranintrinsicallycircularsourcetobedistorted>A,its of theimagestohavebeenlensedbycore.Thisfraction, (r/250 kpc)arcsec.Thisisapproximatelyhalfthelensing gives 0~27(a/1200kms”){A-1)arcsec,which,for impact parametermustlieat lens scaledependsontheclustervelocitydispersion,itsred- cation ofourmethod.Thisisbecausethecoreregionwill while non-negligible,wouldnotsignificantlyaltertheappli- scale, andthusonewouldexpectapproximately25percent a clusteratz=0.4,theangularscaleofcoreis0~55 shift, theredshiftofsourcepopulationandonsize distort backgroundobjectslessthaninthesingularcase,and at zi=0.4andatypicalsourceredshiftof,say,2.0,this be preserved.Weintendtocheckthisanalysisusingnon- which theddistributionisbased.Sinceskewnessof so somelensedimagesclosetotheclustercentrewillnot cluster, thelikelihoodratiotestcouldbestraightforwardly course, iftheactualmassdistributionwereknowninone singular isothermalspheremodelsinournextpaper.Of satisfy thedistortioncriteriaandsonotbeinsampleon test. Inprinciplethetestcanbeappliedwithsimilareaseto perature anddensityprofileofthehotX-rayemittinggas will mostlikelybedeterminedbymeasurementsofthetem- ^(d) distributionistowardslarged,however,thisshould sitting inequilibriumtheclusterpotential. applied withconfidence.Themassdistributioninclusters ö0ien =A(A-l)-«42(a/12OOkms’)A-1)" ens core c lens Figure 5.Therangeintheexpected numberofarcswithA>1.25 core c from 0to0.4atz=2.0withthe remainingfractionofgalaxies, c s~ andtherangeinnumberof arcscorrespondsto/^increasing seen inaclusteratvariousredshifts. Theclusterhas<7,=1200km (1-/,), distributedaccordingto theusualno-evolutionN(z)to c sS h C J/-25. We nowdiscusstheoptimalstrategyforcarryingoutour Arc-like featuresindistantclusters27 x [1-r(z/)/r(z)]arcsec. s Z CI 19 91MNRAS.252. . .19S -2 -1 = f andzastheyareforhigherredshiftclusters.Thissuggests 28 I.Smailetal. clusters atarangeofredshiftssincearcdetectionrequires which suggestthatthetestisclosetooptimalforaclusterat population. Nevertheless,thereareseveralcompetingeffects securing acertainsignal-to-noiseratioforthebackground to alargefractionofarcsatdistancesfromthecluster high-redshift componentofthefaintgalaxypopulationleads tion abouttheclustercentreofalargersamplearcsas gests that,unlessaveryrichclusterisavailable(recall using thehighestredshiftclustersknownforsuchatest.At of the0>(d)curvesarenotasdifferentforvalues the shapeofradialdistributionarcsaroundcluster centre, andaconsequentskewnessinthedistributionofarc possible probeofthefaintgalaxyredshiftdistribution.A found. Wehavemodelledthegravitationallensingeffectof We havesearchedforarc-likefeaturesinahomogeneousset number ofarcsproducedinclustershigherredshiftsug- the sourcesurfacedensityistakentobe27.6arcmin.This high purportedvelocitydispersions provideastrongargu- claimed thattheabundance of high-redshiftclusterswith theories ofgalaxyformation, sinceithasrecentlybeen velocity dispersion,whichcanbefoundforaclusteratany be foundinonecluster. distributions forthebackgroundobjectsif>40arcletscan centre. Thistestcandiscriminatebetweendifferentredshift distances. Wehavedevisedalikelihoodratiotestbasedon candidate arcswithA>2andsix3.5havebeen of deepVimages19richclustersmoderateredshift.20 limit isreadilyachievableona4-mtelescope.Thedeclining model clusterhasavelocitydispersionof1200kmsand with A>1.25asthefraction,f,ofsourcesatz=2varies in Fig.5wheretherangenumberofarcsproduced duced decreaseswhateverthepostulatedN(z).Thisisshown higher redshift,ontheotherhand,numberofarcspro- z =0.4.Foralow-redshiftcluster(say,0.1),theshapes persions, aresubject totheusualprojection-effect problems. (Evrard 1989;Peebles,Daly &Juszkiewicz1990).These ment againstthecolddarkmatter theoryofgalaxyformation prove extremelyinterestingasameansofconstraining approach todeterminingclustervelocitydispersionscould redshift (providedonlythatthereareenougharclets).This measure (typicallytobetterthan5percent)ofthecluster background sources.Wehaveshownthatthenumberofarcs our ensembleofclustersonabackgroundpopulation analyses, basedonspectroscopically determinedvelocitydis- discovered abovesomecriticaldistortion,ortheirradial used tosetconstraintsontheredshiftdistributionoffaint type (i.e.afewarcsineachofmanyrichclusters)couldbe from 0to0.4isplottedagainsttheclusterredshift.Here h city dispersionswouldberequiredtocarryoutthesetests. were wellknown,butverypreciseestimatesofclustervelo- redshift distributionofthesourcesifclusterproperties distribution abouttheclustercentres,couldconstrain and orientationsareconsistentwiththelensinghypothesis. sources andfindthatthenumberofarcstheirshapes 5 CONCLUSIONS N. °ca^i),theoptimalclusterforthistesthasz~0.4. c h clc ux Additionally thismethodshouldyieldaveryaccurate We havethereforeexploredtheuseofradialdistribu- One aimofthispaperwastoexplorewhetherdata © Royal Astronomical Society • Provided by the NASA Astrophysics Data System ACKNOWLEDGMENTS We acknowledgeusefuldiscussionswithG.Efstathiou,B. 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