1989ApJ. . .339. .904C operated byAURA,Inc.,undercontract withtheNSF. now clearthatcorecollapsedoes notrepresentacatastrophic authors. Hénon(1961)predictedthatthecorecollapseshould arrest thecollapse.Whilesignificant uncertaintiesremain,itis subsequent evolutionhasbeenaddressedbyanumber of even suggestedthattheformation ofcentralbinariesmay King 1986;Djorgovskietal.1986). clusters differsignificantlyfromKingmodels(Djorgovski and ultimately occurinallsufficientlyconcentratedclusters, and carried outrecentlywhichshowedthatapproximately20% of plete anduniformstudiesoftheGalacticclusterhavebeen case ofM15(King1975;NewellandO’Neil1978),nearlycom- observations andKingmodelfitshadbeennotedbeforein the stantial power-lawregions.Althoughdiscrepanciesbetween well fittedbyKing(1966)models,andinsteadmayhave sub- lished thatclusterprofilesinandnearthecoresarenotalways observational side,increasinglyaccuratestudieshaveestab- Inagaki 1987andSpitzerreferencestherein).Onthe Grindlay andPhilip1987,thereviewsbyElson,Hut, years (cf.thevolumeseditedbyGoodmanandHut1985, understanding ofglobularclusterevolutioninthepastfew 2 1 VisitingAstronomer,CerroTololo InteramericanObservatory,NOAO, The AstrophysicalJournal,339:904^918,1989April15 PresidentialYoungInvestigator. © 1989.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. On thetheoreticalside,problemofcorecollapseand There havebeenanumberofimportantadvancesinour CeStherhandthecc clusters efind thatthes Subject headings:clusters:globular—galaxies:TheGalaxystructurestars:stellardynamics cluster density,setbythemeantidalfieldofGalaxy,and,perhaps,strengthshocks. tric radius,suggeststhatsomeexternaleffectsareimportantinclusterevolution.Theseincludetheaverage but thereisnosignificanttrendwiththeconcentrationwithinKMfamily. average metalhcitiesofthetwofamilies,insensethatPCCclustershaveslightlylowermetalabundances point towardtheGalacticcenter;KMclustersshownosucheffect.Thereisaveryslightdifferencein KM clusters.ThePCCclustersshowsignsoftidaldistortionsmtheirenvelopes:majoraxestendto K MHnTr^Thpiï^°,’.Parelessluminousthanthehighestconcentration are marginallymoreconcentrated.ThedataindicatethattheconcentratedKMclusterstendtohave hxed distancefromthecenter,clustersatsmallerheightsaboveplane(andthuslessinclinedorbits) systems; theyareeachconsistentwithasymmetricaldistributionoftheclustersaboutGalacticcenterAt oped andpublishedbyFrenkWhitein1982.WeanalyzedtheshapesofKMPCCcluster deal withtheproblemofGalacticobscuration,weusedadistance-independentanalysis,similartoonedevel- similar trendexists:centrallycondensedKMclustersarefound,onaverage,atsmallergalactocentricradiiTo about theGalacticcenterthandistributionofKingmodel(KM)clusters.WithinKMfamilya IhZTth' ?|ditnbutionofthepost^ore-collapsed(PCC)clustersismuchmoreconcentrated from thecompletesurveypublishedbyDjorgovski,King,andcollaboratorsin1984,1986,1987 t © American Astronomical Society • Provided by the NASA Astrophysics Data System The factthatsomeinternalpropertiesofclusterscorrelatewellwithglobalvariables,suchasthegalactocen- We presentanewcompilationofstructuralparametersforGalacticglobularclusters,basedonthedata AN ANALYSISOFTHEDISTRIBUTIONGLOBULARCLUSTERSWITHPOSTCOLLAPSE I. INTRODUCTION Center forRadiophysicsandSpaceResearch,CornellUniversity Received 1988May6¡acceptedSeptember20 CORES INTHEGALAXY California InstituteofTechnology 1 David F.Chernoff 2 S. Djorgovski ABSTRACT 904 AND 2,22 treatments whichusetheFokker-Planck equationwitheffec- which thecorecollapsesignals. Itisuncertainwhethercurrent bly tiedtothebreakdown of theFokker-Planckequation proper, detailedtreatmentof the clusterreexpansionisinevita- complications associatedwith thereexpansionphase.The number ofuncertainties.First, thereareahostofpossible behavior inthesurfacebrightness. the presenceofsubstantialinnerradialrangespower-law fore, itmaybesufficienttoidentifycore-collapsedclusters by fine distinctionsirrelevantforthepresentdiscussion.There- racy andespeciallythetheoreticalsimplificationsmakethese region. However,boththelimitationsinobservationalaccu- varying betweenp~randintheinneroscillating The detailedbehaviorofthedensityprofileiscomplicated, core oscillationsinvolveonlytheinner1%or2%ofmass. with equalmassstarsandagenericheatingsourceshows that is theoccurrenceofgravothermaloscillations(Bettwieser and Sugimoto 1984).Goodman’s(1987)analysisofsimplemodels question. Aninterestingcomplicationinthepostcollapsephase clusters retainduringtheperiodofcoreexpansionisakey during thecollapsephase.Howmuchofthispower-lawprofile power-law densityprofiledevelopsoveralargeradialrange lated clusters,composedofsingle-massstars,showthata have notdonesoalready.Specificmodelsofcollapseiso- many clusterswillreachandpassthroughinthefuture,ifthey event inthelifeofacluster,butrathernormalpointwhich Although thisisasimpleand attractivepicture,therearea 1989ApJ. . .339. .904C issue ofwhethersuchaspectrumalters,insignificantfashion, modifies therateofbinaryformation.Likewise,thereis tive heatingtermsareadequate.Itisalsotriviallytruethata whether acontinuumdistributionofmassesundergoesgravo- spectrum ofstellarmasstypesispresent,whichdoubtless is dominatedbytheKgiants,whichdonotnecessarilytrace thermal oscillationsasasingle-componentsystemdoes. the system’sresponsetoheating;inparticular,itisunclear in §II.Weassumethattheclassificationofclustertype(PCC the clustermass.Realisticstudyofinternaldistribution cluster isfundamentallydifferentfromaKingmodel;Djor- vs. KM)isnearlyindependentofselectioneffects.APCC the observedpropertiesofKingmodel(KM)andPCCclusters stellar typeshasonlyjustbegun. Finally, thereisthepracticalcomplicationthatclusterlight complete catalogofclusterswithclassificationshasbeen data aresufficientlyaccuratetodifferentiatethealternatives govski andKing(1986)havearguedthattheobservational from thetwo-dimensionaldistributiononsky,without unambiguously forpracticallyallknownclusters.Anearly measurements. Weshowthattheprojectedclustersample,asa introducing uncertaintiesassociatedwiththeclusterdistance properties oftheclusterdistributionwhichcouldbeinferred use ofthetechniqueFrenkandWhite(1982),whostudied of PCCclusterstonormalKingmodelclusters.Wehavemade assembled. In§IIIwecomparetheGalaxy-widedistribution whole, isdistributedapproximatelysymmetricallyaboutthe be membersofanonsphericaldistributionortheymay Galactic centeronthesky,i.e.,withoutagrossflattening.The near theGalacticcenter.Finally,althoughobserverselec- there isaverystrongtendencyforPCCclusterstobelocated tion ofclustersabouttheGalacticcenteronskyshowsthat subject tosignificantobservationalbiases.Theradialdistribu- situation forthePCCclustersislesscertain:thesemay dimensional distribution,weshowthatthereisanincreasein tion effectsarelesswellunderstoodinthecaseofthree- also includedananalysisofthetrendsinmass,concentration, the fractionofPCCclustersnearGalacticcenter.Wehave centrally increasingfractionofPCCclusters.In§IVwemake and metalicitywithgalactocentricposition.Thesetrendsare suggestive, butnonearenearlyasstrongthetrendtowardsa cores collapsedandexploreevolutionaryscenarioswhich the assumptionthatPCCclustersareclusterwhichhavetheir cluster evolution.Thisapproximatetreatment,which only could explainthetrendusingasimplifiedtreatmentofglobular mass lossacrossatidalboundary,stellarbyevolving most significantmechanismsofprecollapseclusterevolution : applies toclustersoncircularorbits,includesanumberof the (Djorgovski andKing1984, 1986;DjorgovskiandPenner extensive power-lawdistribution oflightintheircentralparts. model (KM)statesasdiscussedbyDjorgovskiandKing(1986). used toclassifyclustersaspost-core-collapsed(PCC)orKing emphasized. passages. Anumberofshortcomingstheapproach are stars, relaxationviaevaporation,andtidalshockingby disk The datacomefromtheBerkeley globularclusterssurvey by theprofilesfromKron,Hewitt, andWasserman(1984),the PCC clustersaredistinguished fromKMclustersbyhavingan 1985; Djorgovskietal1986; Kingetal1989),supplemented The planofthepaperisasfollows.Webeginbydiscussing Surface brightnessprofilesofglobularclustershavebeen © American Astronomical Society • Provided by the NASA Astrophysics Data System II. KINGMODELSANDPCCCLUSTERS GLOBULAR CLUSTERSWITHPOSTCOLLAPSECORES 1/2 1/2 2 estimates ofqualityandweightingdifferentprofilesthe cedure, sinceitwasoftennecessarytomakecomplexdecisions done deliberately,ratherthanthroughanautomaticpro- set ofKing(1966)modelswereplottedonthesamelog-log and King(1986).Allavailableprofilesforeachcluster,a several othersources,summarizedinthereportbyDjorgovski star countsbyKingetal(1968)andPeterson(1976), cluster profile.Theinnerlimitissetbythequalityofseeing delimit theregionofreasonablemeasurementaccuracy cluster observationshaveinnerandouterangularradiiwhich collapsed coresmainlyfollowtheclassificationbyDjorgovski cluster), andcomparisonswiththenumberspublishedbyother internal comparisons(independentestimatesforthesame determined; therelevantresultsarelistedinTable1.From here. Theangularcoreradiusr,thetidaland the centralpartsofclusters,whichareprimaryinterest highest weight,sincetheycontainthemostinformationabout same cluster,etc.Generally,thenewCCDdataweregiven profile, aboutrecognizingandignoringbaddatapoints,the and judgmentsabouttheradialrangetobetrustedforeach scale, andtheKingmodelcurveswerefittedbyeye.Thiswas cluster members.Itistypically0~150".Overtherangeof contamination andconfusionofbackgroundforeground sample supersedeothersimilarcompilationscurrentlyavail- and King(1986).Webelievethatourmeasurementsforthe about 0.1.Theclustersclassifiedashavingorprobably surement isabout0.1-0.2,andthemeasurementsoflogr authors, weestimatethattheaverageaccuracyofourcmea- the Kingmodelconcentrationparameterc=log(r/r)were files maybemeasuredtoanaccuracythatislimitedprimarily and, typically,6~1";theouterlimitissetbylevelofsky PCC andKMclusters(seealsoDjorgovskiKing1986).All logical parameterscominginthenearfuture(Kingetal1989). produce mostofthelight. by Nfluctuationsinthenumberofredgiants,which stars complementedbythedecreasingnumberdensityof able intheliterature,butweanticipateabettersetofmorpho- (assumed tobeNinorigin)arederivedbydividingeach least-squares technique,weightingthepointsbyobserva- eters, acentralsurfacedensity,cr(0),anangularcoresize, r, formal erroraboutthewholeannulusresult(seeDjorgovski tionally determineduncertainties.Theprofileuncertainties scheme istofittheprofileoverrange[0,0]usinga annulus intoseveralangularregionsandthencalculating the [0n, 0x]>underoptimumobservingconditions,clusterpro- and thenadjusting<7(0)tominimizexwhere angular tidalradius.Thefitiscarriedoutbychoosingrand c, and aconcentrationparameterc=log(r/r),whereris the ct max 19870). ThefitfortheprojectedKingmodelhasthreeparam- c tc min c minmax mima King c tc where <7(0^isthemeasuredvalue,0o’(0)uncer- contours oftheformalerrorin thefitsofKingmodelsforNGC tainty and<7istheKingmodel. In thesum,irunsoverdatapointswith0<0, , and Trz2,classifiedasaPCC cluster.Figure1illustrates the aboveprocedureonNGC 6388,classifiedasaKMcluster, 6388, with0=1"5and =122",asafunctionofcandr. I King min max minmax c We beginbydiscussingindetailthedistinctionbetween It istraditionaltofitclusterprofileswithKingmodels.One To illustratethecharacteristic results,wehavecarriedout x yLrjl. The that r>0andofallfitsr,.Veryfewclusters formal qualityofthemodelfitisreasonablewith%=24, for 350". Itistypicalofthewell-determinedfitstoKingmodels, 22 datapoints.TheprobabilityQthatthevalueofyis that poor isQ«0.17,sothatonecannotrejectitoutofhand.For a profile rangewhichisverysmall, itisusuallypossibletofinda formally goodfit,however,the fitmaynotbewell-determined. (See dashedlinesinFigs1and 2,whichillustratetheformal c error andbestfitwhen0= 8"5.) t cminmax has beenclassifiedasPCC.Figure 3illutratesthexsurfacefor King modelfitsovertherange 0"25fromZinnandWest(1984), tic center.WelistthesePAdifferencesinTable1.Finally, we cluster majoraxes,andthevectorspointingtowardGalac- from whichwederivetheabsolutePAdifferencesbetween the are notdirectlyrelevantforourpurposes. clusters abouttheGalacticcenter. FollowingFrenkandWhite major axispositionangles(PA)byShawlandWhite(1987), axes andvectorspointingtowardsthe NorthernGalacticpole.Theseangles Table 1forconvenience.Weusedthemeasurementsofcluster the GalacticplanelistedbyWebbink(1985),repeatedherein Galaxy. WeusethegalactocentricradiiandZ-distancesfrom correlated withtheclusterconcentrationoritspositionin in detailwhetherthedifferentobserverselectioneffectsare the worstknownnumberforeachcluster.Wehavenotstudied derived massismerelyaroughapproximationandprobably probably swampthephotometricerrors.Inanycase, constant M/L,theuncertaintiesindistanceandextinction foreground stars.Theerrorsintroducedbytheassumptonof most availableaperturemagnitude.Anerrorintheopposite sense ismadebytheextralightcontributedunremoved known; oftenthetotalluminosityissimplybasedonouter- the structureofouterregionsmostclustersispoorly underestimate theluminosities(andthusmasses),because stant M/L=3forallclusters.Theremaybeatendencyto them totheclustermassesbyassumingauniversalandcon- data wederivedtheabsolutevisualluminositiesandconverted or Webbink(1985)forafewremainingclusters.Fromthese on thedatacompiledbyPeterson(1986),whereveravailable, visual extinction,andheliocentricdistancesofclusters,based morphological classificationswiththetotalvisualmagnitudes, surements oflograbout0.1.Wesupplementour racy ofourcmeasurementsisabout0.1to0.2,andthemea- is oneestimateoftheerror.Wethataverageaccu- extreme Kingmodelsandtherangeofderivedconcentrations rigorize. Instead,forthemajorityofclusterswesimplyfit combining thedatafromavarietyofsourcesarenoteasyto cluster, becausethecomplexdecisionsthatmustbemadein v more concretefashion. project istodealwiththevariousobservationalbiasesina effect aregivenbyLuggereta/.(1987).Animportantfuture c density profiles.Additionalexamplesandargumentstothis and PCCclusterscanbedistinguishedonthebasisofsurface models, discussedabove.) conclusion regardingfitstoclusterswhichareaprioriKing either rorc.(Notealsothatthisisconsistentwiththegeneral and tidaldata,itisimpossibletoconstrainmorestrongly value, thatofaisothermalsphere,whichis-1.0.Withoutcore the slopeofinnerhalotendstowardasinglecharacteristic King modelincreasesandthesizeofcoredecreases,then very flatforr<0"25andc>3.6.Astheconcentrationofa 3 c 910 c We beginbyanalyzingtheazimuthaldistributionof ShawlandWhite(1987)computed the PAdifferencesbetweencluster In practice,onedoesnotperformtheaboveanalysisforeach The pointofthisanalysisistomakeitplausiblethatKM © American Astronomical Society • Provided by the NASA Astrophysics Data System III. DISTRIBUTIONOFPCCCLUSTERS a) Distance-independentAnalysis a) ObservedClusterProperties CHERNOFF ANDDJORGOVSKI are duetoselectioneffects.We willarguethatPCCclustersare tuations intheregionenclosed bythe50%levelatsmallb approximately sphericallysymmetrically, thenthelargefluc- changes inb(seeFig.6).If thePCCclustersaredistributed assumed cutoffwith0. data arefoldedintoasinglequadrant(0b,where=5?7and10° lim which reemphasizestheneedtodealcarefullywithGalactic obscuration. FrenkandWhiteassumedthattheclustercata- “hidden” veryneartheGalacticplane(Djorgovski1987a), selection effects.Recentlythreenewclusterswerelocated, tributed uniformly,or,inotherwords,thattheclusterswere Table 1forbothKMandPCCclusters. Frenk andWhite(1982)testedthehypothesisthatv'wasdis- the intrinsicasymmetryofthree-dimensionaldistribution. distributed inv'foranarbitrarye.The“stretching”removes (v', co')generatedby(b\/')=[arctan(tanb/e),/]areuniformly We haveperformedasimilaranalysisfortheclusterslistedin randomly drawnfromanintrinsicallyflatteneddistribution. Um Um generated by(h,l)arenotuniformlydistributedinv,thepoints limUm and aareprincipia!axesinFig.5).Althoughthepoints(v,co) be thecharacteristicflattening,oroblateness(e=b/a,whereb included toprovideorientation. axis alignedwiththeperpendiculartoGalacticdisk.Lete distribution ofclustersisellipsoidorprolate,withasymmetry constant galacticlatitude(b)anddeclinationare angles; PCC/PCC?arefilledtriangles).Severallinesof trates thedistributionofclusters(KM/KM?areopentri- angle vwithauniformrandomdistribution.Figure5illus- statistical testforsymmetry:comparethedistributionof uniformly distributedinthevariablev.Thisleadstoasimple trary sphericallysymmetricwayabouttheGalacticcenter,is three-dimensional distributionofclusters,arrangedinanarbi- the latitudewhereGalacticplane(b=0)definesv0.A is theangulardisplacementfromGalacticcenter,andv The newcoordinates(v,co)havethefollowinginterpretation:co and galactic coordinates(b,l)to(v,co),definedas The dashedlineinFigure6isthe50%confidencelevelfor A keypointoftheanalysisisdealingwithobserver Another possibilitythatcanbetestedistheintrinsic tan v=bescl. cos co=bl 198 9ApJ. . .339. .904C clusters abouttheGalacticcenterhashorizontalsemiaxisaandverticalb,withellipticity,e=b/a. are \b\=27?5,45?0,and72?5.TheGalacticplaneisthehorizontalline.skewedcircles<5—75.0,0.0,75.0.assumedellipsoidal distributionof clusters (level0.5). Galactic center,withanassumedobservational cutoffofb.ThesolidlinesrefertoKingmodelclusters(levels 0.1,0.3,0.5,0.7,and0.9)thedashedlinestoPCC lim F. 6.—KSresultstorejectthehypothesis thatthetwo-dimensionaldistributionofclustershavebeenchosen fromasamplewithintrinsicellipticityeaboutthe Fig. 5.—Thedistributionofclusters(KMandKM?:opentriangles;PCCPCC?;solidtriangles)abouttheGalacticcenter.Thetwosetsconcentric circles IG © American Astronomical Society • Provided by the NASA Astrophysics Data System 911 198 9ApJ. . .339. .904C P UmtlVe dStlbUtn0 fKm0bTthat0.21331 0.064 1.291.404.934.84 2. 313kpc.Althoughcisknownonly±0.1 the concentrationdecreasesmonotonicallyfrom1.59atR<3 much smallerthanthesizeoftrendactuallyseen.Thesame mean forasampleofNclustersis0.1/iV«0.02,which from apurelyempiricalpointofview,PCCclusterscanoften concentration andinPCCfractionmayberelated.Inaddition, or soforanindividualcluster,theexpecteddeviationin behavior withradialdisplacementfromthecenter.Themass is trend occursinthemedianvalueofc,whichisamorerobust clusions canbedrawnatthistime. not welldeterminedforamajorityofclusters,sonofirmcon- statistic. Ontheotherhand,massesdonothaveasimple As wewilldiscussinthesectiononevolution,trends c (KM)logM/Mq 913 198 9ApJ. . .339. .904C esting tolookatthebehaviorofZ/Rindifferentconcentration may simplyreflectthetrendsinR.However,itisalsointer- of thesestatistics. plane (Z)arenecessarilycorrelated,andthusthetrendsin Z simple extensiontohigherconcentrationKingmodelsinterms height abovetheplaneexist.ThePCCsappeartofit as a line). Obvioustrendsinbothgalactocentricdistanceand (the meanisgivenbythedottedline,mediansolid interest ontheconcentrationclassisillustratedinFigure11 for thePCCclusters.Dependenceofsomequantities concentration intothreeclassesfortheKMclusters,plusone concentration objects.Table3dividesclustersonthebasisof so itmaybeappropriatetoconsiderthemassimplyveryhigh be successfully“fitted”byhighlyconcentratedKingmodels, The PCCclustersaremoreconcentratedtowardtheGalacticcenterandplane. (solid lines),andtheclusterswithKingmodel(KM)morphology(dottedlines). Galactic plane,Z,fortheclusterswithpost-core-collapse(PCC)morphology 914 Galactocentric radius(R)andthedistancefromGalactic Fig. 10—Distributionsofgalactocentricradii,R,anddistancesfromthe o O c D © American Astronomical Society • Provided by the NASA Astrophysics Data System All Kingmodels94 15.7 8.08.72.6523 4. PCCandPCC?33 4.6 3.01.90.7512 ^c>l.7 28 8.5 6.93.31.95.657 2. 1.2index,formostoftheknownGalactic concentration classes.Wefindnotrendwith globulars. Wetestedthebehavioroffordifferent within theKMfamily,butthereisamarginaldifference between thePCCandKMclusters,inthatones persists whenwecomparethe[Fe/H]values,whichareavail- are ontheaveragemetalpoorer(Table4).Thissmalldifference able forasmallersetofclusters:themeanvalues[Fe/H] the PCCandKMclustersare—1.68+0.13 between theclusterconcentrationandgalactocentric sense fromwhatmightbeexpectedthecorrelation radius (reportedabove),andtheknownmetallicitygradient among theglobulars:moremetal-richclustersareon for ourinnermostradialbin).Inordertoavoidthe index changesfrom0.127inouroutermostradialbin,to0.210 average closertotheGalacticcenter(e.g.,mean metallicity gradientamongtheclusters,wecomputedsta- (R< 3kpc);thetrendwithmetallicitypersists.Theseresults tistics ofthecoinafiniteamountof by thetidalfield,whichoccursveryquickly(thatis,withina times tocomplete,soclustersmayeitherbeincollapseorhave collapse. Corecollapsetakesabout330centralrelaxation sequence, someclustersbecomedenseenoughtobegincore dissolved models(D).AstheclustersevolvealongKing (2) collapsingmodels(C),(3)collapsed(CC),and(4) four generalareas.Clustersareclassifiedbytheirfateafterthe Hubble timeofevolutionas(1)Kingmodels(denotedbyK), tion (orW)forthecluster.Notethatpanelisdividedinto represent thegalactocentricradiusandinitialconcentra- 14.0 M,respectively.Theabscissaandordinateofthepanel 0 g Vol. 339 198 9ApJ. . .339. .904C 46 No. 2,1989 dynamical time)whentheKingconcentrationparameter reaches W=0.Someclustersdonotreacheitherextreme after thegivenlengthoftimeandarestillmembersKing sequence afteraHubbletime.Thetheoreticalquestionofwhat process haltsthecollapseisquitecomplicated;binaryforma- tion isthoughttoplayacrucialrole.Inanycase,wedonot instead weassumeimplicitlythataclusterhascompleted attempt todescribetheevolutionbeyondcorecollapse,and core collapsecanbedistinguishedbyitspower-lawinner from otherclusters.Otherpanelscoverdifferentinitialmasses orbits ofagivenradiusabouttheGalacticcenter.Itisquite (10 and10M)differentIMFs(a=2.354.5),as but howmanyareonellipticorbitsisunknownandmaybea 0 crucial factorintheevolutionofglobularsystem.The indicated inthefigurecaption. lation isunknown,andweareunawareofanyreasonthatthey actual distributionofinitialconditionstheclusterpopu- likely thatsomeclustershavecircularornearlyorbits, in clusterevolutionthefigurearehighlysuggestive.Mass should startoutasKingmodels.Nonetheless,thebasictrends loss fromclustersbystellarevolutionisimportantforallclus- by relaxationquicklybecauseofthehighaveragedensity;clus- ters withsmalla.ClustersclosetotheGalacticcenterevolve intermediate radii(5kpc)tidalshockcanunbindthelessdense ters farfromthecenterevolvebyrelaxationverylittle.At 0 clusters. Detailedanalysisshowsthattidalshockscanalso increase therateofcorecollapseforclustersatintermediate radii; however,theprimarydeterminantofrateevolu- tion formostclustersisgalactocentricradius. function ofclustersininitialmass,concentration,IMF,and galactocentric radiusforwardintimeandcomparewith today’s distribution.However,wefeelthatatthemoment would appearpremature.(Onealternativeapproachhasbeen are soignorantoftheclusterinitialconditions,suchaproject ed foraself-consistentdescriptionofthepresent-daycluster taken byAguilar,Ostriker,andHut[1988],whohavesearch- distribution functionusingournewdata[Table1]andknown evolution fornoncircularorbitshasonlyjustbeguntobe destructive mechanisms.)Inaddition,thephysicsofcluster found intheclusterdata. explored. Instead,letusexaminesomeofthetrendswehave cluster mass,concentration,andIMFpowerlaw,that core Galactic center.Thediagramsillustratethatforfixedinitial collapse proceedsmostquicklyatthecenterofGalaxy. In addition, forclusterswhichdonotcollapse,tidaldissolution occurs morequicklynearthecenterofGalaxy.Sincethis is true foreverysetofinitialconditions,itmustalsobe for any distributionofinitialconditions,whichareindependent of conditions, PCCclusterswillbefoundpreferentiallyat the the galactocentricradius,i.e.,foraverybroadrangeofinitial Galactic center.Aglanceatthediagramshowsthat very concentration, and/orIMFarenecessarytochangethistrend. strong variationswithgalactocentricradiusofinitialmass, An immediatecorollaryisthat Kingmodelclustersatsmall galactocentric radius,withsmall mass,areunexpected.Those that existmaybeinterlopers,i.e., clustersnotoncircularorbits, examination oftheircoresand theirthree-dimensionalveloc- ities areimportantobservational projectsforthefuture. or perhapstheymayhavecollapsed andreexpanded.Abetter All resultsinFigure13areforKingmodelsoncircular In principle,itispossibletoevolveanarbitrarydistribution 1. ThedistributionofPCCclustersispeakedtowardthe 2. Kingmodels aremoreconcentratedatsmall galactocen- © American Astronomical Society • Provided by the NASA Astrophysics Data System GLOBULAR CLUSTERSWITHPOSTCOLLAPSECORES 7 9 model todayistiedtotheinitialconcentrationandvariationin tric radiusthanatlargeradius.Thereasonsarethesameas above, butthetrendislessstrong.Theconcentrationofa day values.(Incontrast,allclustersatsmallgalactocentric the initialconcentrationsisreflectedinvariationpresent- radius mayhavecollapsedforasimilarrangeofinitialcondi- effect concentrationevolution.Thus,evenifallclustersstarted tions.) Inaddition,variationsininitialmassandIMFstrongly with exactlythesamecbutdifferentmasses,thenthere would bevariationtodayinc. core collapsemorequicklythanclusterswithshallowIMFs mass, concentration,andgalactocentricradius.(Thelikelihood because ofthedegreemassloss.Thisistrueforanyinitial IMF isshallow.)PCCclustersaremetal-poor,and,ifthe of survivalduringtheperiodsmasslossisenhancedif steep IMF.Thiswouldbeconsistentwiththeirsurvivaland their earlycollapse. McClure etal.(1986)trendisgeneral,thenPCCsmayhavea morphology, buthaveshortenoughdynamicaltimesto that therearemanyclusterswhichshownotracesofthePCC collapsed inasmallfractionoftheHubbletime(Djorgovski models haveshortenoughrelaxationtimes(t<3x10yr, and King1986;Djorgovski1987h).Inoursample,between and/or i<3x10yr;thesenumbersmaybeuncertainby about afactorof2or3),tohavecollapsedbynow.Some 6341,6440,6528, or6864.Themeanandmedianconcentration outstanding examplesinclude,e.g.,NGC1851,5824,6093, of theseclustersisaboutc«1.8,andtheydonotdifferintheir evolution ofsinglespeciescoressuggestthattheclustersessen- in thesameconcentrationrange.Thetheoriesofpostcollapse galactocentric radialdistributionfromtheremainingclusters 10% and20%ofclustersunambiguouslyclassifiedasKing tially stayinthePCCstate“forever,”thatis,theydonot possibility isthatthereexpansiontheorywrong,and reassume aKingmodelshapeinHubbletime(cf.Goodman verge ofcollapsehasneverbeenadequatelyanswered.One rc clusters reexpandtoaKingmodelstatemorequicklythanthe rh clusters collapseandreexpandinastochasticmanner,dueto ejection andformationofcentralbinaries,whichstabilize the single-species modelspredict.Anotherpossibilityisthatthe clusters andprovidetheenergyforreexpansion(McMillan and 1984 orCohn1985).Whytherearesomanyclustersonthe findings: iftheclustersrecoverfromcorecollapsedue to are notrecognizableatmosttimes(BettwieserandSugimoto Lightman 1984).Possiblythepostcollapsecoresoscillate and some internalmechanism,theobserveddependencesof the It ispossiblethatthesepuzzlingclustersonthevergeof col- composed ofdark,heavystellarremnants,whosedistribution lapse dohavepower-lawdensitycuspsintheircenters,but are is notfollowedbythelow-massredgiantswhichprovideprac- PCC clustersonthepositioninGalaxyarehardtoexplain. tically allthelight(Larson 1984). Theexplanationofthese 1984). Alltheseexplanationsappearunlikelyinthelightof our “ uncollapsed”clustersremains achallengetothetheory. al propertiesofglobularclusters arestronglydependenton (having asmallcore)follows theoreticallyfromtheshorter their positionwithintheGalaxy. Thisrelationbetweena global variable(galactocentric radius)andaninternalproperty 3. ClusterswithsteepIMFs(fewmassivestars)undergo One outstandingproblemimpliedbytheobservationsis In anycase,wehavedemonstrated thatsomeofthestructur- V. CONCLUSIONS 917 198 9ApJ. . .339. .904C 987in IAU x asadena,CA91125 S. Djorgovski:AstronomyDepartment,PalomarObservatory,105-24RobinsonLaboratory,CaliforniaInstituteof Technology, Goodman, J,andHut,P.,eds.1985,IAUSymposium113,DynamicsofStar .1987,Ap.J.,313,576. David F.Chernoff:602SpaceSciencesBuilding,CornellUniversity,Ithaca,NY14853 Goodman, J.1984,Ap.J.,280,298. Frenk, C.S.,andWhite,S.D.M.1982,M.N.R.A.S.,198,173. Elson, R.,Hut,P.,andInagaki,S.1987,Ann.Rev.Astr.Ap.,25565 Djorgovski, S.,andPenner,H.1985,inIAUSymposium113,DynamicsofStar — .1986,Ap.J.(Liters),305,L61. Djorgovski, S.,King,I.R.,Vuosalo,C,Oren,A.,andPenner,H.1986,inIAU Djorgovski, S.,andKing,I.R.1984,Ap.J.(Letters),277L49 Djorgovski, S.1987a,Ap.J.(Letters),317,L13. 77-i.?,’Symposium126,GlobularClusterSystemsinGalaxies, Cohn, H1985,inIAUSymposium113,DynamicsofStarClusters,ed. Chernoff, D.F.,andWeinberg,M.1989,Ap.J.,submitted. Chernoff, D.,andShapiro,S.1987a,Ap.J.,322,113. Chernoff, D.,Kochanek,C,andShapiro,S.1986,Ap.J.309183 Aguilar, L.,Ostriker,J.,andHut,P.1988Ap.335,720 Bettwieser, E.,andSugimoto,D.1984,M.N.R.A.S.,208,493. may indeedbesmallerintheClouds. parts. PreliminaryresultsbyMateo(1987)andMeylan Djorgovski (1987)indicatethatthefrequencyofPCCclusters ages anddynamicaltimescalesthandotheirGalacticcounter- In addition,theMagellanicclustershavealargerrangein tidal shocksareweakerthaninourownGalaxy,willbeuseful. action ofaclusterwiththedisk.Thestudyclustersin galactocentric radius,andtoalesserextent,fromtheinter- Magellanic Clouds,wheretheaveragetidalfieldstrengthand relaxation timesthatatidallylimitedclusterhasatsmall 918 Clusters (Dordrecht:Reidel)(GH85). scopes, ed.J.HearnshawandP.Cottrell(Dordrecht:Reidel)p281 Clusters, ed.J.GoodmanandP.Hut(Dordrecht:Reidel),p.73. Symposium 118,InstrumentationandResearchProgrammesforSmallTele- ed. J.GrmdlayandA.G.D.Philip(Dordrecht:Reidel),p.333 J. GoodmanandP.Hut(Dordrecht:Reidel),p.161 ed. 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Krön, G.,Hewitt,A.,andWasserman,L.1984,Pub.A.S.P96198 King, I.R.,etal.1989,inpreparation. —75, inIAUSymposium65,DynamicsofStellarSystems,ed.A.Hayli King, I.R.,Hedemann,E.,Hodge,S.,andWhite,R.1968,A.J.,73,456. King, I.R.1966,A.J,71,64. Hénon, M.1961,Annd’Ap.,24,369. Grmdlay, J.,andPhilip,A.G.D.,eds.1987,IAUSymposium126,Globular California InstituteofTechnology. acknowledge partialsupportfromHarvardUniversityand Hut, GerryQuinlan,IraWasserman,andMartinWeinberg. reductions. WeacknowledgehelpfulconversationswithPiet 15162 andAST-86-57467atCornellUniversity.S.D.wishesto Oren, CarlVuosalo,andHowardPennerfortheirhelpindata advice andmanystimulatingdiscussions.WealsothankAbe use thedataforthisworkandprovideduswithmuchgood collaboration withIvanR.King,whogenerouslyallowedusto here. TheBerkeleyglobularclusterssurveywasconductedin american Observatoryfortheirhelpinobtainingthedataused J. GoodmanandP.Hut(Dordrecht:Reidel),p.541. (Dordrecht: Reidel),p.99. Cluster SystemsinGalaxies(Dordrecht:Reidel),press. D. C.acknowledgessupportbyNSFgrantno.AST-84-