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1977Apj. . .213. .327A the Astrophysical Journal, 213:327-344

1977ApJ. . .213. .327A 1 2m The AstrophysicalJournal,213:327-344,1977April15 © 1977.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. luminosity functionbyaGaussiancurvewithAf= investigated theabsolutemagnitudesof134late luminosity functionwasbyHubble(1926),who that theluminosityfunctionrisesmonotonicallywith not inconspicuousclusters).Herepresented_their spirals and11irregulargalaxies(mostofwhichare the numbersofvisiblegalaxiesinclustersvarious numbers ofgalaxieslowluminosity,andsuggested Later, Zwicky(1942)summarizedevidenceforlarge if =50kms“Mpc",Hubble’sMbecomes—19.4). increasing magnitude.Stilllater,fromananalysisof angular sizes,Zwicky(1957)foundthenumber luminosity functionpermitsthedeterminationof particular tofindwhetheranyfeatureinthecluster luminosity functioniscorrectandgeneral,in N(

U> to H-* to u> to 1977ApJ. . .213. .327A © American Astronomical Society •Provided bythe NASAAstrophysics Data System in \ ro V \. to N \ \ CD CVJ 1 \

Fig. 1.—Identification chart of the photoelectric standard stars described in Table 1 1977ApJ. . .213. .327A © American Astronomical Society • Provided by theNASA Astrophysics Data System 1977ApJ. . .213. .327A be determinedbytheextrafocaltechniquewithcon- correction tothebrightermagnitudemeasured siderable confidence,providedthatmagnitudesare fore, errorscanarisefrom(1)comparisonofthe measures areavailableforonlyasmallrangeofimage available forextrafocalimagescoveringalargerange be rathersensitivetoerrorsofmeasurement.There- obtain atotalmagnitudecanbequitelarge,andthus size—say, thetwosmallestextrafocalsteps—the of size—say,afactor5to10.Ontheotherhand,if magnitudes arisingfromthedensitycomparisons. images, and(2)inappropriatenessofthesurface- densities oftheextrafocalgalaxyandstandardstar Images ofselectedgalaxies,andnearlyallthe brightness modelforsomegalaxies. with thoseofthestandardstars,arecording brightest onesinthecluster,wereactuallytraced,along extrafocal photographs.Thus,inpractice,most many hundredsofgalaxiesoneach10different microdensitometer. Thisprocedureisfartootedious, extrafocal images,underamagnifyinglens,to density comparisonsaremadebymatchingthe however, tobepracticableforthemeasurementof density, themicrophotometermeasuresandstep- density stepscale.Exceptfortheimagesofverylow tenth magnitude. the writer’sluminosity-functionprogram,andsetsof scale estimatesareusuallyinagreementtowithina Palomar Schmidtplatesandphotoelectricstandards cluster (A2065).ClusterA2065isanotherin used tocompareextrafocalimagesof548starswith A2065, however,therearesevenextrafocalsteps:in standard starsinthefieldofCoronaBorealis are availableforit,astheComacluster.For of thedensitycomparisons,step-scaleestimateswere magnitude derivedforanextrafocalstarimage plates 0.5and1.0mmoutoffocus.Becausethe addition tothoseusedfortheComaclusterthereare magnitudes inthesequenceforanyparticularstar should notdependontheextrafocaldistance, are allofthoseinasmallregiontheclusterfield, should allbethesame.Theforegroundstarsselected tion witha=0.21mag. First letusconsidertheerrorsinextrafocal Nevertheless, todeterminetheoverallconsistency Fig. 2.—Residualsaboutthemeans ofindividualstarmagnitudesmeasuredonextrafocalplates. Solidline,thenormaldistribu- © American Astronomical Society • Provided by theNASA Astrophysics Data System LUMINOSITY FUNCTIONOFCOMACLUSTER -2 and theyrangeinderivedmagnitudefrom11.1to pared toaGaussianerrorcurvewith

338 ABELL Vol. 213 (Mihalas and Routly 1968). The space density of r"r- galaxies, p(r), at a distance r from the cluster center, is given by 1 dF(r) p(r) = 2nr dr ’ (7) where F(r) is the count of galaxies in a strip with a projected distance r from the cluster center. The derived densities are shown in Figure 9, both for all galaxies to the limit of the survey (mv — 18.3), and for those with mv < 16.5. If the cluster has undergone even a moderate amount of dynamical evolution, its inner portion should have a structure resembling that of an iso- thermal gas sphere. The radial density distribution of the isothermal polytrope is tabulated, for example, by Chandrasekhar (1957). Following Emden (1907), the ratio of the density to the central density is given as e~*, where 0 is a dimensionless variable that is a function of the dimensionless radial variable f, £ being related to the actual radial distance, r, by r = Fig. 8.—Normalized counts of galaxies in 1 cm strips, in several different magnitude ranges, corrected for the field, and «£. The solid line in Figure 9 is the density distribution averaged in the north, south, east, and west directions about given by Chandrasekhar, shifted to match the points the cluster center. representing the distribution of galaxies brighter than mv = 16.5. The match of the isothermal density curve to the points determines the parameter a, relating f to and 15 cm from the cluster center; the excess is prob- r. The value we obtain for a (called the “structural ably due to the galaxies in the northwest part of the index” by Zwicky (1957) is 7Í05. The fit of the distri- 6.4 deg field shown in Figure 6. Aside from this feature bution of the brighter galaxies to the isothermal model there is no convincing case for segregation between is quite good over the range 1-10 cm from the cluster the distributions of bright and faint galaxies in the center on the plate scale (about 10' to 100'), but the fit cluster, nor, presumably, for marked mass segregation. is less good for the fainter galaxies. To obtain a quantitative measure of the radial distri- Bahcall (1973) also compares the galaxy distribution bution of galaxies of different magnitudes, we calculate to the isothermal polytrope. She quotes a core radius the dispersion of galaxies in the different magnitude for the cluster (the radial distance at which the pro- intervals about the orthogonal north-south and east- jected density of galaxies drops to one-half the central west lines; to eliminate the possibly large contribution value) of 5!2 to 6', depending on the magnitude limit of noncluster galaxies not removed in the field correc- of the counts. These core radii correspond to structural tion, only galaxies with projected (plate scale) co- indices of about 2' and 2!3, similar to the value a = 2' ordinates within 10 cm of that of the cluster center are found by Zwicky (1957). Bahcall’s and Zwicky’s small included in the calculations. The standard deviation of the mean distribution of galaxies north, south, east, or west of the cluster center in the different magnitude intervals is as follows:

mv < 15.6 , a = 3.81 cm ;

15.6 < mv < 16.5 , a = 3.98 cm ;

16.5 < mv < 18.3 , or = 4.12 cm . (6) The magnitude (and presumably mass) segregation is possibly significant, but is far too small to be com- patible with statistical equilibrium. At best a relatively small amount of dynamical evolution has taken place in the Coma cluster. Bahcall (1973) obtained similar results. Fig. 9.—Derived volume density distribution of galaxies in c) The Volume-Density Distribution the Coma cluster. The open circles are for all galaxies to the The volume or space density of galaxies can be limit mv = 18.3, and the filled circles are for all galaxies to the limit mv = 16.5. The smooth curve is the isothermal polytrope derived from the strip counts described in the last (right hand and lower scale), shifted to match the points for subsection very conveniently with Plummer’s method the brighter galaxies.

© American Astronomical Society • Provided by the NASA Astrophysics Data System 1977ApJ. . .213. .327A 2 1 25 No. 2,1977 a’s resultfromtheirinclusionofcountsverynearthe cluster center,wheresmalldifferencesinthenumbers trope doesnotimplytruestatisticalequilibrium,or the clustergalaxydistributiontoisothermalpoly- those foundhere.Inanycase,theroughsimilarityof there wouldbeaverypronouncedsegregationofbright densities. Thustheirresultsarenotincompatiblewith of galaxyimagesresultinlargedifferencesderived evolution ofclustersbyPeebles(1970)predicta fact, morerealistictheoreticalmodelsofthedynamical and faintgalaxies,contrarytoobservation(Fig.8).In potential energyas distribution ofgalaxiessimilartothatfoundhere. J q(r)dq(r)/rwiththedensitydistributionofgalaxies cluster hassphericalsymmetry,wecanwritethe where M(r)isthetotalmasscontainedwithinaradial where Jtisthetotalclustermass,expressionfor found here.Thevolumedensitydistributionof which isingoodagreementwiththevalue0.986 m =18.3,andnumericallyintegratethequantity We nextassumethatthemassdistributionissame the potentialenergybecomes the maximumclusterradiusestimatedin§III6from from theclustercenter.Thisvalueis7%higherthan numerical factorintheaboveequationtobe0.92, as describedinthelastsubsection.Thereresults as thenumericaldistributionofgalaxiestolimit galaxies appearstoapproachzeroatadistanceof2?9 Rood (1965),fromasimilaranalysis,foundthe probably irregulargalaxies). Inall,the40.16deg inspected allofthegalaxyimagesonin-focus tude ofaboutm=15.Thewriterhascarefully we regarditasanupperlimitthatisprobablytoo estimated maximumandminimumclusterradii.Thus distance rofthecenter.IfwenowwriteM(r)=q(r)Jt, a spiral(someofthegalaxies calledspiralsare which onescouldbedefinitelyclassifiedasspirals.If, plate ofthe6?4fieldComaclustertodetermine large. IftheHubbleconstantis=50kms"Mpc, Figure 4,andnearly20%greaterthanthemeanof region contains137galaxies thatwereclassedas be distinguishedfromthoseoftypeEorSOtoamagni- then theouterclusterradiusRis2.144x10cm. spiral structureorabsorption features,itwascalled on theSchmidtplate,agalaxy showedrecognizable v v If wemaketheapproximationthatComa On thePalomarSchmidtplates,spiralgalaxiescan © American Astronomical Society • Provided by theNASA Astrophysics Data System e) SpiralGalaxiesintheComaClusterField d) PotentialEnergyoftheCluster _ =a^j2(!Ä).(9) Q 2 -0. =0M6GJt!R.(10) LUMINOSITY FUNCTIONOFCOMACLUSTER ) (8 certain orprobablespirals.Theirsurfacedistribution region oftheComacluster.Somehaveradialvelocities is shownonFigure10,whichshouldbecomparedwith on theComacluster.ComparewithFigs.4and6. on theSchmidtplatecovering6?34squareregioncentered the coreofComacluster. these spiralsareinthegeneralneighborhoodof would seemtobegeometricallycloseit.Although concentration ofthecluster.Somearedoubtlessfore- their distributionbearsnorelationtothatofthemain gators wouldthereforecallthemclustermembers.Yet, Figures 4and6. inhomogeneity orregionofspacecontaininggalaxies, gravitationally bounddynamicalunitthatcomprises cluster, thereisnoevidencethattheybelongtothe least thosewithvelocitiesnearthatofthecluster, ground orbackgroundfieldgalaxies;butmost,at similar tothatoftheclusteritself,andsomeinvesti- politan area.StatisticalstudiesbyPeeblesandhis tion, ratherlikeanurbancenterinalargemetro- called theComaclusterissimplyadenseconcentra- groups, andclusters,inwhichwhatiscommonly positions ofgalaxiesandclustersinspacearecorre- that arefellowmembersofamuchlargersuper- Abell (1961).ThedensecoreoftheComacluster lated witheachotheroverdistancesofatleast associates (HauserandPeebles1973;1973, another suchsystem,andthere isstrongevidencethat Local Superclustermaywell beatypicalexampleof ( >3?5)foundforthecluster byRoodetal.(1972).The cluster thatmaynotbebound. Itissuggestedthat must begravitationallyboundoritcouldneverhave such asituationaccounts for theverylargeradius and clusters(includingmostofthespiralsinFig.10) space therearequitelikelyothergalaxiesandgroups survived overitspresumedage.Butsurroundingitin one ofthespecificsuperclustersidentifiedandlistedby 100 Mpc.TheComacluster,infact,isamemberof 1974; PeeblesandHauser1974)haveshownthat There areunquestionablysomespiralgalaxiesinthe Fig. 10.—Thedistributionofrecognizablespiralgalaxies The picturethatsuggestsitselfisofalarge i o \thetotalluminosityis822.Ofthis,497 it is1094.Alloftheseluminosities,course,arein v units ofthelightanindividualgalaxymagnitude model (2)thetotalluminosityis636,andfor(3) v the Comacluster.WithH=SOkms"Mpc, m*. nosity, correspondingtoeachofourthreemodels, et al.(1972),ameanradialvelocity6888kmsfor ofM=4.8.Further,weadopt,fromRood is then distance modulusoftheclusteristhen35.7,and becomes L*/L=2.512x10.Theclusterlumi- v 0 We shalladopt,fortheclusterluminosity, greatest slopeatthefaintend, andthereforepredicts v 0 There are21galaxiesbrighterthanm=13.3. We cancalculatetheclusterluminosityfrom To calculatetheluminositycontributedbyfainter v We assumeanabsolutevisualmagnitudeforthe The luminosityfunctionused inmodel(1)hasthe © American Astronomical Society • Provided by theNASA Astrophysics Data System = —21.2.Ourunitofvisualluminositythus (1) m*inequation(13)isthemeanvelocity dispersion inradialvelocities,.Thus, of galaxiesthroughtheclustercorearemostlyradial, =3isdeterminedfromgalaxiesnearthe the uncertaintiesinvolvedininequalityofequation We maynowsolvefortheclustermasswithequations values (10), (13),(14),andeither(12)or(15).Weincorporate (14), andwhetherequation(12)or(15)ismorenearly We adoptthevaluex861kms",givenby appropriate, intotheconstant£,where£canhave mass, Rood etal.(1972),andobtain,forthetotalcluster With theluminositygiven by equation(9),wethen r r r obtain forthemass-to-light ratio(insolarunits), r If theclusterisincompletestatisticalequilibrium, If theclusterisnotinstatisticalequilibrium,but b) TheMass-to-LightRatio 2 15 <3,(13) 2T +£1=0,(12) *#/L =61£. (18) T +Q<0.(15) Í <£3.(16) Vol. 213 1977ApJ. . .213. .327A 2 -2 No. 2,1977 radius adoptedin§IVrfistoolarge. motions ofgalaxiesprobablydodominateinthe be lessthantheabovevaluesif,asislikely,cluster mass-to-light ratio,weobtainJt\L=122.Thisvalue parameters thathavegoneintothecalculationof cluster, sothatthetruevelocitydispersionisprobably we haveseenintheprevioussection,andisprobably visible galaxies.Hereistheoriginofso-called ratio forlargeandgiantellipticals,hencethanthe is largerthanmostvaluesfoundforthemass-to-light between equations(12)and(15).Moreover,radial somewhat lessthan3.Ifweadopt£=2,which ratio onewouldestimatefortheclusterfrom described byasituationsomewhatintermediate is likelytobeanestimateasaccuratetheother missing massmystery.Somuchhasbeenwritten the 5mtelescopeatPalomar.BecauseSandagehadobservedmostofgalaxiesthroughseveralapertures of electric observationstototalmagnitudes,Fp,theAbellmagnitudesbasedonextrafocalphotographic photometry. Sandagethusverykindlyprovidedtherawdatapriortotheirpublication. Mihalas (1966)techniquetoprovideacheckonthetotalmagnitudesobtainedbyextrafocalphotographic photometry, Vandthedifferencesbetweenthem.Incolumnheaded“Galaxy”RBreferstoidentification magnitude containedwithintheisophotalcontourof26magarcsec,V6\Abellreductionphoto- different sizes,itoccurredtothewriterthatobservationscouldbereducedtotalmagnitudesbyAbell- number givenbyRoodandBaum(1967). not berepeatedhere.Thereduceddatafor20galaxiesaregiveninTable6.Given,eachgalaxy,Sandage’s NGC 4889. NGC 4921. NGC 4874. IC 4042... NGC 4923. IC 4012... NGC 4886. NGC 4881. IC 4021... IC4026... IC 3998... NGC 4873. NGC 4883. IC 4011... NGC 4906. RB 45 e RB 46 eU 2 RB 37 RB 74 RB 85 Note thattheclustermass,andJt\Lratio,may In fact,theclusterisnotinstatisticalequilibrium,as In 1961SandageobtainedphotoelectricobservationsofanumberEandSOgalaxiesintheComaclusterwith The magnitudesobtainedthroughvariousapertureshavesubsequentlybeenpublished(Sandage1972),sowill Standard deviation. Mean © American Astronomical Society • Provided by theNASA Astrophysics Data System Galaxy TOTAL MAGNITUDESOFGALAXIESMEASUREDPHOTOELECTRICALLY 11.8 12.4 14.1 12.8 13.8 14.5 14.4 14.4 14.9 15.0 Ï5.0 15.4 14.5 15.1 17.0 16.0 15.8 15.3 17.4 LUMINOSITY FUNCTIONOFCOMACLUSTER Galaxies ObservedbySandageandReducedAbell 11.55 11.84 13.60 14.15 13.68 12.80 14.55 14.92 13.96 14.18 13.53 15.16 14.16 14.94 14.36 16.64 15.48 15.36 14.97 16.65 APPENDIX A TABLE 6 11.55 12.75 12.3 14.25 13.5 14.55 14.55 14.5 14.45 14.3 14.0 13.75 16.85 15.3 14.6 14.6 15.95 15.7 14.8 17.15 kef certainly notbetween10and100,asoftensuggested. discrepancy maybeonlyafactorof2or3,and be discussedfurtherhere,excepttonotethatthe about it(forexample,RoodetaL1972)thatwillnot estimates inthelargeclusterregionweremadeby graduate assistantatUCLA.Thecountsandmagnitude Dimitri Mihalas,whenhewasemployedasanunder- were obtained.Theearlyworkontheprojectwas Hale Observatoriesforpermissiontousethetelescopes acknowledged. ThewriterthankstheDirectorof student. Thecarefulworkofeachisgratefully at PalomarandMountWilson,wheretheobservations Michael Stone,whenhealsowasanundergraduate supported inpartbyagrantfromtheOfficeofNaval Research. Much oftheextrafocalphotometrywasdoneby V -K 2QP( + 0.37 0.34 0.25 0.42 0.00 0.56 0.80 0.27 0.24 0.16 0.35 Ó.08 0.32 0.25 0.36 0.52 0.44 0.33 0.64 0.75 0.21 v 2i -0.1 + 0.24 0.25 0.05 0.1 0.45 Ó.5 0.2 0.15 0.4 0.6 0.1 0.5 0.35 0.05 0.15 0.05 0.1 0.5 0.25 0.2 v ei Fpe -Vei -0.46 -0.65 + 0.05 -0.34 -0.14 + 0.17 -0.44 -0.19 + 0.42 -0.49 -0.12 + 0.15 -0.22 +0.18 -0.15 -0.21 -0.47 + 0.34 -0.50 0.00 0.00 0.30 341 1977ApJ. . .213. .327A 2 2 342 used byAbell.BecausemanyofthegalaxiesarefainterthanfainteststarsobservedphotoelectricallyAbell, calibrated theastrophotometerreadingswithmeasuresonstandardstars(Table1)andsecondarystandards He measuredeachimageonthein-focusvisualplateofclusterwithaniris-diaphragmastrophotometer,and Dobias added,asstandards,thestarsobservedbyStebbins,Whitford,andJohnson(1950)Baum(private 7.05 degfieldofextrafocalphotometry,Dobiasidentifiedallfaintgalaxiesthatcouldbedistinguishedfromstars. magnitude range,Dobias,whoidentifiedthegalaxiesindependentlyofAbell,anddidnothaveaccessto In all,hemeasured1478galaxiestom=19.9;regardshisidentificationofbecomplete19.4. communication) inSelectedArea57,whichappearsonthesameSchmidtplate,althoughoutsidefieldof In the2.76degfield,Abellhasextrafocalmagnitudesfor877galaxiesinrangem=17.0to18.3.same Dobias’s galaxyphotometry. Abell’s identificationcharts,observed878galaxies.Inthisrange,thetwoindependentmagnitudecalibrationsappear provide agoodestimateofthefaintextensiontoluminosityfunction. to besurprisinglysimilar.Forgalaxiessuccessivelyfainterthanm=18.3,theastrophotometermagnitudesshould be increasinglyfreerofapertureeffects.Consequently,weregardDobias’sfaintsamplegalaxyphotometry to 12.3.. 12.8.. 12.6.. 11.6.. v 13.0. . 13.3.. v 13.2.. 17.0, thedataareAbellextrafocalmagnitudesofgalaxiesincentralfield;form> 13.5.. 13.4.. 13.1.. v 13.8. , 13.7.. 15.0. . 14.9.. 14.6. 14.5.. 14.2.. 14.1.. 14.0. . 13.9.. 13.6.. 15.5.. 15.4.. 15.3.. 15.2.. 15.1.. 14.8.. 14.7.. 14.4.. 14.3.. 15.7.. 15.6.. 16.0. 15.9. 15.8.. v 2 2 Dobias recordedallgalaxiesforwhichastrophotometerreadingsindicatedvisualmagnitudesof17.0andfainter. Table 7givestheluminosityfunctionforgalaxiesininner2.76degfield.Atmagnitudeslessthanm— In anapproximatelyrectangularregionof2.76deg,centeredontheclustercenterandcompletelyinside v © American Astronomical Society • Provided by theNASA Astrophysics Data System m v Cluster + Photometric DataforGalaxiesintheCentral2.76-SquareDegreesofComaCluster Field 24 15 11 14 11 10 12 19 13 2 2 2 4 0 0 0 3 1 6 3 1 9 9 5 2 0 7 7 6 8 8 8 1 5 1 EXTENSION OFTHELUMINOSITYFUNCTIONTOM=19.4 v Field Department ofAstronomy,UniversityCalifornia,LosAngeles Cluster 24 15 11 13 10 12 19 11 2 2 2 4 0 0 0 9 6 3 3 5 1 1 0 7 7 2 9 8 8 1 6 8 5 3 1 Jan DobiasandG.O.Abell Integral Count APPENDIX B 204 218 207 168 104 144 153 146 136 136 124 123 195 189 176 158 20 48 42 23 24 96 72 63 31 31 13 13 16 4 6 3 8 8 1 TABLE 7 ABELL 16.2. 16.4. 16.1. 16.3. 16.8. 16.7. 16.6. 16.5. 18.1. 18.0. 17.9. 17.7. 17.5. 17.4. 17.3. 17.1. 17.0. 16.9. 18.5. 18.3. 17.8. 17.6. 17.2. 18.9. 18.8. 18.6. 18.4. 18.2. 19.4. 19.2. 19.1. 19.0. 18.7. 19.3. m v Cluster+ Field 139 125 102 198 189 174 153 27 27 28 25 21 28 79 77 34 10 90 91 75 71 57 55 18 17 84 19 19 2 9 7 8 3 8 Field 28 26 21 48 41 24 20 44 37 34 10 31 57 12 11 52 18 16 14 13 2 2 4 4 4 2 3 3 9 6 7 6 5 8 Cluster 141 126 109 137 23 21 41 22 63 56 13 98 58 51 51 39 63 16 12 88 68 59 18 13 12 17 0 4 7 6 5 3 1 6 Vol. 213 Integral Count 1054 1665 1524 1387 1261 1152 225 230 225 219 276 253 250 237 280 458 411 301 417 354 325 313 966 776 718 667 372 338 611 560 521 389 898 839 1977ApJ. . .213. .327A No. 2,1977 from Dobias’sastrophotometerreadings.Successivecolumnsgivethemagnitude,totalnumberofgalaxies, the (2) Forgivenrangesofredshiftz,therelativenumbersgalaxiesarebyanassumedcosmologicalmodel. logarithmic integralluminosityfunctionisplottedinFigure11. number ofexpectedfieldgalaxies(fromAppendixC),theclusterobjects,andintegralcount.The theory and/orobservations.(5)Thusforagivenz,therelativedistributionofgalaxiesbyapparentmagnitude is assumptions: (1)Theuniverseexpandsuniformly,andishomogeneousisotropic(thecosmologicalprinciple). determined. Integrationoverallredshiftsthenleadstotheobservedmagnitudedistributionoffieldgalaxies. (3) Atanygivenrangeinz,therelativenumbersofgalaxiesvariousabsolutemagnitudesaredeterminedby an models andluminosityfunctions.HefindsthatfortheusualFriedmannwithzerocosmologicalconstant, assumed luminosityfunction.(4)Givenz,thedimmingduetoredshiftand^-correctionaredeterminedfrom porated. Integrationoverredshiftgives,then,thedistribution ofapparentmagnitudes.Thisdistributionhasbeen field, andtheonlyimportantcontributiontofieldcomes fromgalaxieswithM

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