1957AJ 62 . .2483 248 spectral typeAocomparedtotheirappearance Abt, HelmutA.Thevariabilityofsupergiants. Walker, M.F.1956,Ap.J.Suppl.2,365. in NGC2264notearlierthanGo. 2. TheTTauri-typeobjectsarepresentat TITLES OFADDITIONALPAPERSPRESENTED Bidelman, WilliamP.Spectralclassificationofsome Chamberlain, JosephW.Evidenceforalargevelocity Blanco, V.andGrant,G.Thecolorsspectraltypes due totheunavoidableselectionofavailableclustersforobservation.Thepresentstudyislimitedeffectsonly Hemenway, CurtisandFullam,Ernest.Asearchfor the interpretationofmagnitude-redshiftrelation.Itisfoundthatnearthresholdinstrumentsused, of theeffectsconditions{a)and{b)isdevelopedappliedtoproblemestimatingdistancesclusters two conditions,whicharecertainlyinvolvedintheselectionofclusters:(a)Theclustermusthaveatleastacertain Estimates oftheabsolutemagnitudeadistantgalaxywilltendtobetoofaintanditsdistanceunderestimated Duncombe, R.L.andClemence,G.M.Theaccuracyof effects ofselectionbiasarestrong. number ngalaxiesbrighterthanalimitmi,(b)therthbrightestgalaxymustbemÚmi.Anexacttheory brightest (orofthefifthbrightest,etc.)galaxy brightest galaxyisapproximatelythesamein in aclusterasdistanceindicatorrestsonthe how thebiasarises.Wethendeveloptheoretical The samebiasaffectsthestudyofrelation clusters leadstoestimatesoftheabsolutemag- every cluster.Aspointedout,heuristically,by assumption thattheabsolutemagnitudeof distance toagalaxy. such asredshift.Similarremarksapplytothe any quantityassumedtodependondistance, nitude ofthebrightestgalaxythataretoofaint, bias anddistance.Thelast partsofthepaper sampling experiment,inordertoseeintuitively use ofthebrighteststarasanindicator between apparentmagnitudeanddistance,or and underestimatesthedistancetocluster. illustrate thebiasinquestionbymeansofa Behr (1951)thisprocedureappliedtodistant formulae establishingtherelationbetween present thenumericalconsequences ofthebias 2 visual binarieswithprimariesabovethemainsequence. dispersion inauroralprotons. of theveryredstarsinyoungclusterNGC6530. micrometeorites. the solarephemeris. Abstract. Thecustomaryuseofthebrightest(orfifthbrightest)galaxyinaclusterasdistanceindicatorisstudied. The purposeofthepresentpaperis,first,to I. Introduction.Thecustomaryuseofthe AT THEMEETINGINURBANA,ILLINOIS © American Astronomical Society • Provided by theNASA Astrophysics Data System THE BRIGHTESTGALAXYINACLUSTERASDISTANCEINDICATOR THE ASTRONOMICALJOURNAL Northwestern University, Dearborn Observatory, By ELIZABETHL.SCOTT Evanston, III. Hoag, ArthurA.Acard-punchingphotoelectricobserving Herget, Paul.Generalperturbationcalculations. Hiltner, W.A.Three-colorphotometryofM13. Searle, Leonard.Thestatisticalequilibriumofahydrogen O’Keefe, A.AnapplicationofJacobi’sintegraltothe Kraft, RobertP.ThebinarysystemNovaTCoronae Meinel, A.B.ReportontheKittPeaksiteforNa- McNamara, D.H.TheHalineinthespectrumofRZ Van denBergh,Sidney.Interstellargasandstarcreation. Spitz, Armand.Increasingtheobservabilityofsatel- Segre, Sergio.AstellarinteriormodelforAlgolA. Minnaert, M.TherevisionoftheRowlandtableand Wilson, RaymondH.,Jr.Theoryofapolyhedralheli- Wyatt, S.P.,Jr.ThedoublecloudofgalaxiesinHercules. ent paperisconcernedsolelywiththebiasin- difficulties. Forexample,thereisthequestionof tion betweenapparentmagnitudeandredshift. est, etc.)galaxyasadistanceindicator,andall correcting theapparentmagnitudeforeffects the distancetoaclusterandinstudyingrela- in thetwoproblemsmentioned,estimating duced byusingthebrightest(orfifthbright- of redshift.Wewanttoemphasizethatthepres- particular, itisassumedthaterrorlesscorrections redshift areavailable. of theapparentmagnitudeforeffects other difficultiesareleftoutofconsideration.In clusters areavailabletotheobserverbutonly selection ofobjectstobeobserved,sincenotall depend ontheobserver.Inpresentpaperwe these conditionsareisnotclear;probablythey consider thecasewhereobservationsofmagni- those satisfyingcertainconditions.Exactlywhat tude, andofredshiftifnecessary, areaccumu- lated subjecttothefollowing twoconditions: system. tional AstronomicalObservatory. Scuti. background ofthesolarspectrum. Borealis. motion ofanearthsatellite. otrope onanartificialsatellite. lite bystructuralchangesinthespecularsurface. nebula. These twoproblemsaresurroundedbymany 2. Natureofthebias.Thebiasarisesin 62, No.1252 1957AJ 62 . .2483 as acluster.Thissomewhatvaguerequirement are brightenoughtobevisibleonthephoto- may bestatedas:atleastnclustermembers servational setup. graphic plateonwhichtheobserverissearching 1957 October m, tobespecifiedlater,withámi. say thattheapparentmagnitudeofrth cluster (orthefifthbrightest,etc.)mustbe magnitude oftheplate.Weassignvarious available totheobserver.Theformulaedeveloped numerical valuestomi,dependingontheob- members mustbebrighterthanmi,thelimiting for clusters.Weshallsaythatatleastncluster conditions {a)and(6),withmoreorlessstringent brightest galaxymustbebrighterthanalimit ess ofselection.Obviously,thisislikelytobean magnitude andredshift,asnecessary.Weshall bright enoughformeasurementsofapparent since variousobserversprobablyimposeaddi- oversimplification oftheprocessselection, values ofmiandm,completelydescribetheproc- cluster containingfewergalaxies.Thus,aswe in latersectionsarebasedontheassumptionthat cluster mustbeunusual.Inparticular,thisfar- cluster isavailabletotheobserver,thenthis shall showquantitatively,ifaverydistant distance, aclusterwithmanymembersismore both conditions(a)and(è).Also,atanygiven tional restrictions. (two dozenorperhapstwicethismanygalaxies) away clustermustcontainatleastnextra- likely tobeavailabletheobserverthana the smallerisprobabilitythatitwillsatisfy very distantclustersmusthaveatendencyto least r)isextremelyluminous.Thismeansthat the brightestgalaxiesactuallyobservedin luminous galaxiesofwhichatleastone(or average brightestgalaxiesinthenearerclusters. 2 possess brighterabsolutemagnitudesthanthe (6), itwillbeconvenienttosaythattheclusteris brightest galaxyinaclusteratdistance£will where x(£)istheadditionaldimmingduetored- apparent magnitudembytheequation have itsabsolutemagnitudeMrelatedto this biasinabsolutemagnitude.Atypical dening orothercauses.However, when£islarge 2 a typicalbrightestgalaxywill notbeobservable; t t Condition (a).Theclustermustbeidentifiable If aclustersatisfiesbothofconditions(a)and Condition (b).Thebrightestgalaxyinthe It isobviousthat,themoredistantacluster, Let usexaminetheconsequencesofignoring © American Astronomical Society • Provided by theNASA Astrophysics Data System m =M-$+5log£x(£),(i) t THE ASTRONOMICALJOURNAL we have be observable.Forthisunusualbrightestgalaxy only theunusualbrightestgalaxywithextra- have anapparentmagnitudembrightenoughto luminous absolutemagnitudeM
increases, only those brightest galaxies for which r- this curve would be invisible on a photographic LO plate that has limiting magnitude 19 mag. We the luminosity is more and more unusual will be notice that, for all the nearby clusters, the bright- observable. Eventually, the brightest galaxy in a est and usually even the fifth brightest galaxies cluster of 250 members will be invisible unless it lie above the limiting line. However, for the should happen to be extremely luminous, much clusters at about 300 million parsecs we note brighter than would be expected for such a that, occasionally, even the brightest galaxy is cluster. too faint to be seen. As the distance to the cluster Figure 3 is similar to Figure 2 except that it
Figure 2. Synthetic observations of absolute magnitude of brightest (dot) and of fifth brightest galaxy (circle) in each of 100 clusters of 250 members each, as a function of distance. Arrows indicate expected absolute magnitude. Curve marks limiting apparent magnitude of 19.
pertains to clusters of 1000 members each. Com- galaxies each. Thus, at any given distance, it is parison of the two figures illustrates the number- more likely that the brightest galaxy among of-cluster-members selection effect. The expected 1000 members will be brighter than 19 mag. than luminosity of the brightest and of the fifth was the case for clusters of 250 members only. brightest from a cluster of 1000 galaxies is Therefore, a larger proportion of the symbols in — 20.24 mag. and —19.61 mag., respectively, Figure 3 lie above the limiting curve than in noticeably brighter than from clusters of 250 Figure 2. Even so, only the unusually luminous
© American Astronomical Society • Provided by the NASA Astrophysics Data System CO 00 CM CDCM 252 THE ASTRONOMICAL JOURNAL 62, No. 1252
brightest galaxy in a distant cluster will be ob- Now that we have illustrated heuristically the C"< LO servable according to condition (b). source of the bias in the absolute magnitude of Similar figures could be drawn for the twenty- the brightest galaxy in a faraway cluster, we fifth brightest galaxy in a cluster and similar proceed to the theoretical evaluation of its im- conclusions would apply. Only in the ususual portance. faraway cluster will at least 25 galaxies be visi- 4. Theoretical formulae for the apparent magni- ble, and thus only the unusual distant cluster will tude of the rth brightest galaxy in an available satisfy condition (a). cluster at distance ¿.If the stated conditions (a)
Figure 3. Synthetic observations of absolute magnitude of brightest (dot) and of fifth-brightest galaxy (circle) in each 2 c us ers °/limiting >5- l apparentt of 1000 magnitude members 19. each, as a function of distance. Arrows indicate expected absolute magnitude. Curve marks
and {b), required for a cluster to be available to rth brightest galaxy in a cluster. The apparent the observer, correspond to reality, then the magnitude is in the particular system of magni- formulae derived below can be used to obtain tude applicable to the actual observational corrections for the bias. First, we introduce some program, with no corrections made for the effects notation and basic assumptions. of redshift or other causes of dimming. Let v Let br denote the apparent magnitude of the equal the number of galaxies in the cluster and
© American Astronomical Society • Provided by the NASA Astrophysics Data System 1957AJ 62 . .2483 v 6 We maythinkofv*asthenumbergalaxiesin magnitude brighterthanaspecifiedlimitnti. let v*bethenumberofgalaxieswithapparent earlier (NeymanandScott1952)ofthedistribu- the clusterthatarevisibleonphotographic plate. Inlinewiththegeneraltheorydeveloped cluster. Itwillbeconvenienttointroducethe galaxies inaclusterisrandomvariableinde- tion ofgalaxies,weassumethatthenumberv defined for|¿^1astheexpectedvalueoft, pendent ofthelocationcluster,andinde- 1957 October distance tothebrightestgalaxyincluster. pendent ofthenumbergalaxiesinanyother secs, thepositionofgalaxieswithinacluster probability generatingfunctionofv,sayG{t), effect oftheactualvariation(Neyman,Scott However, fordistancesexceeding5millionpar- change theapparentmagitudebylessthan0.05 parsecs wemayassumethatallthegalaxiesin be lessthananynumbermdependsonlyon£ and Shane1953)ofdistancewithinaclusterwill the clusterhavesamedistance£because may beignored.Inotherwords,for£>5X10 ject totheconditions{a)and(b)setforthin distance £toacluster,theconditionaldistribu- factors comingunderstudy.Wedenotethisprob- other galaxiesinthesameclusterandofall and m,thusisindependentofthenumber parent magnitudeofagalaxyatdistance£will ability ingeneralby mag. Section 2.Weabbreviatetheseconditionsas: brightest galaxyinaclusteratdistance£,sub- tion oftheapparentmagnitudebfth mi. Herenandmiarenumberstobespecified at leastw.Inotherwords,ngalaxiesin the numberv*ofvisiblegalaxiesinclusteris v the clusterhaveapparentmagnitudelessthan than thelimitm.Asmentioned, wetakem the rthbrightestgalaxyinclusterisbrighter later, withr^n. r 2 2 ^ mi. 0(£,m) =Pfapparentmagnitude S when the function Q(£,m), prescribing the prob- of the rth brightest galaxy in the apparent mag- S ability that a galaxy at distance £ will have ap- nitude and to shift the mean as indicated by the ^ parent magnitude brighter than m, is assumed arrows. to be the particular function believed to corre- The middle panel still corresponds to the hy- spond most closely to present observations (de- pothetical situation of no selection but now the scribed in detail in the next section as Case i). number v of galaxies in the cluster is no longer Thus, the first curve is a plot of the derivative of assumed to be a fixed known number but is al- 0(3.5 x 108, m) as a function of m. The three lowed to vary from one cluster to another in remaining parts of the top panel of Figure 5 accordance with formula (17), illustrated in show the probability density of the apparent Figure 4. The three parts of the panel corre- magnitude of the brightest and of the fifth spond to three assumed values for the param- brightest galaxy in a cluster of exactly 25, 250 eter y, namely, 50, 100 and 200, and are ob- and exactly 1000 galaxies, respectively. The tained by differentiating expression (26) with effect of increasing v is to decrease the dispersion 0(3.5 X 108, m) as specified for Case 1 in the next APPARENT MAGNITUDE Figure 5. Implications of the theoretical formulae. Upper row: Distribution of apparent magnitude of brightest and of fifth brightest galaxies if there were no selection and a known number of galaxies per cluster. Middle row: No selection and random number of galaxies. Bottom row: Selection subject to conditions (a) and (¿>). © American Astronomical Society • Provided by the NASA Astrophysics Data System 1957AJ 62 . .2483 8 cluster. galaxy, r=1), section andwithr=1(solid)5 a completelydefinedfunctiontorepresentthe Also, wemustsubstituteforthesymbol0(£,m) The differentpartsshowtheeffectofchanging entering thetheoreticalformulaeofSection4: numerical valuesaregiveninthenextsection. differentiating thedistributions(25)and(20), somewhat morerealisticsituationofselection as totheappropriatevalues ofn,miandm2. observer andupontheobservationalsetupem- r, dependupontheconventionsadoptedby apparent magnitudebrighterthanm.Thisfunc- probability thatagalaxyatdistancefwillhave to measureredshift,magnitude,etc.,ofagalaxy, plate whereclustersareidentified, parent magnitudeinducedbyselection.Some and ofmused. obtained. Theeffectofselectionisquitepro- toward brightermas7increases. magnitude, Siand<£5fromformula(32),shift Humason andtoDr.N.U.Mayallfordiscussion tion dependsinanobviouswayontheluminos- the middleandbottompanelsisbiasinap- nounced, andisverysensitivetothevalueofmi quired torecognizeacluster,andofmakingm n, thenumberofgalaxiesbrighterthanmire- and forthesame0(3.5X10,m)usedabove. available clusters.Allcurvesarefor7=100 ployed. TheauthorisindebtedtoDr.M.L. in excessoftheinversesquarelaw. ity functionofgalaxiesandonx(0>thedimming recognize acluster, particular numericalvaluestotheparameters to obtainnumericalresultsweneedassign from (33)and(34)usingthedensitiesalready tude, JErandFrespectively,werecomputed rows indicatingtheexpectedapparentmagni- respectively, asdescribedearlier,whilethear- more stringent.Thedensitiesareobtainedby 1957 October In ordertocovertherange of possiblevalueswith (dashed). Noticethatthemeansofapparent 2 2 r m2, thelimitingapparentmagnituderequired 7 +1,theaveragenumberofgalaxiesper r, therankofgalaxyused(forbrightest mi, thelimitingapparentmagnitudeof The differenceinthepositionofarrows Finally, thebottompanelcorrespondsto Some oftheparameters,namelyn,mi,mand n, thenumberof“visible”galaxiesrequiredto 5. Numericalvalueofthebias,Case1.Inorder £, thedistancetocluster,and 2 © American Astronomical Society • Provided by theNASA Astrophysics Data System THE ASTRONOMICALJOURNAL want offurtherinformation,followNeyman, different observersandtelescopes,we we shallconsider of : shall considersetsofpossiblevalueschosenout Scott andShane(1953)inestimating7.Thus, difficult toestimate.Asmentionedearlier,we where, forbrevity, parsecs shall evaluatethebiasasafunctionof£and,for to theirluminosityfunction,andaremuchmore refer tothespatialdistributionofgalaxiesand cation andalsotoDr.J.Neymanforaprelimi- We aregratefultoDr.A.E.Whitfordforprovid- 0(£,m) = distribution. Accordingly,weadopt such aboundikf*exists,thedataindicatethat nary determinationoftheluminosityfunction normal distribution.Wenotethatbyputting selection. Neyman’sresultsconfirmtheoriginal and Sandage(1956)correctedfortheeffectsof field galaxiesfromthedataofHumason,Mayall ing informationabout%(£)inadvanceofpubli- terially. Weconsider a cumulativenormaldistribution,withcrrep- it mustbequitedistantfromthemodeMoof not beexcludedbythisanalysis.However,if possible absolutemagnitudeofanygalaxy,can- point Af*,whichwouldrepresentthebrightest bility ofatruncationnormaldensityatsome luminosity functionoffieldgalaxies.Thepossi- density providesareasonablygoodfittothe assumption ofHubblethatanormalprobability because theywouldnotchange theresultsma- resenting thedispersionofpossiblytruncated truncated normaldistributionarenotconsidered tions oftheintrinsicallyfaintgalaxiesfroma truncated normaldistribution.Possibledevia- M* =—coinformula(35)weobtainanon- m The otherparametersandthefunction0(^,m) 7 =50,100and200. £ =anyfixedvaluebetweenoand700million G(a,b) m2 =mi,mi—Iand2 mi =anyfixedvalue r =iand5. G[M*, m+5—logf-x(£)] = 5,25and50 G[M*,oo] 2 q)/2(Tm ¿M, (36) (35) 257 00C/} CM UOCM 258 THE ASTRONOMICAL JOURNAL 62, No. 1252 3 ikfo = any fixed value in the neighborhood of x(£) = any fixed function of £ (including zero). £ —17 mag. We first give in some detail the numerical ^ &M — i and 2 mag. values of the theoretical formulae in Section 4 If* = ikfo — 3, ilfo — 4 and — co and the consequent biases in the estimated dis- 0 100 200 300 400 1000 1500 2000 3000 Basic Distance £ (million parsecs) Figure 6. Expected apparent magnitude 81, Ei and Fi and bias in magnitude Èi — Si and Fi — Si for Case I. Figure 7* Estimated distance to a galaxy as a function of true distance under Case I when selection bias is ignored. tance and in the estimated magnitude-redshift spond most closely to the observational data relation for a particular set of the parameters now available (Humason, Mayall and Sandage, enumerated above, the set believed to corre- 1956). This choice of parameters is described as © American Astronomical Society • Provided by the NASA Astrophysics Data System 00w CM CM 1957 October THE ASTRONOMICAL JOURNAL 259 than mi = 21.5 while Fi is bounded by m — r-< Case I. In Section 6 are several figures summariz- 2 O'!to ing the results for other choices of the parameters 20.5 whereas the hypothetical mean Si can in- compared with the standard Case i. In the nu- crease without limit. merical illustrations we have employed photo- The lower part of Figure 6 shows the differ- graphic magnitudes, unless otherwise specified. ences Ei — Si and Fi — Si which are the bias in The first set of parameters to be considered is apparent magnitude induced by selection ac- cording to condition (a) and according to condi- Case i: n = 25, mi = 21.5, ra = 20.5, r = 1, 2 tion {a) plus {b). The bias is always negative, 7 = 100, Mq =- 17, (tm = i, df* = - co, x(£) -9 since Ei and Fi are always brighter than Si, and = 5.15 X io £. The bias will be shown as a the bias increases as £ increases. For any particu- function of the distance £. lar distance £, the bias Ei — Si or Fi — Si is The upper part of Figure 6 gives the expected just the expected difference in absolute magni- apparent magnitude corresponding to Case I tude Mu — Mt between the unusual brightest for three conditions of observation: (i) Si, no galaxy that is observable and the typical bright- selection of clusters to be observed, (ii) Ei, selec- est galaxy, as described in Sections. tion of clusters subject to the single condition (a) One consequence of the bias is shown in Figure which requires in Case 1 that at least 25 galaxies 7 which portrays the estimated distance when in the cluster be brighter than 21.5, and (iii) Fi, the effects of selection are ignored. The ordinate selection of available clusters, that is, according is computed from the fact that the difference to conditions (a) plus (b), which requires in addi- between the logarithms of the estimated dis- tion that the brightest galaxy be brighter than tance and the true distance is 0.2 times the bias 20.5. We notice that as the distance £ increases in magnitude. For large £ the underestimate in the difference between the three expected values the distance is severe. becomes appreciable since Ei cannot be fainter Another consequence of the selection of avail- Figure 8. Estimated relations under Case 1 between expected photographic apparent magnitude and redshift, and between expected bolometric apparent magnitude and redshift, for no selection and for selection according to conditions {a) and (b). © American Astronomical Society • Provided by the NASA Astrophysics Data System 1957AJ 62 . .2483 6 The newordinateislabeledexpectedbolometric slope 5.Wealsoassumethattheredshiftisa we haveassumedthatthetruerelationbetween able clustersisgiveninFigure8whichshowson respectively. selection ofavailableclusters.Fordefiniteness, graphic apparentmagnitudeandredshiftunder the leftrelationbetweenexpectedphoto- 260 The curvesresultingunderselectionarenot excess dimmingx(£)assumed;thiscurveis H —180km/secper10parsecs.Therelation the expectedbolometricmagnitudeand tion accordingtocondition(a)only,and(iii) the threeconditionsof(i)noselection,(ii)selec- the excessdimming%(£)fromeveryordinate. structed fromtheleft-handgraphbysubtracting straight lineseitherbuttendtocurvedown between theexpectedphotographicmagnitude known linearfunctionofthetruedistancewith logarithm oftheredshiftisastraightlinewith not exceedthelimitsmi=21.5andm220.5, from thelineofslope5sinceordinatescan- labeled “noselection”intheleftofFigure8. rithm oftheredshiftwillcurveupdueto £1, astransferredfromFigure6,andtheloga- The right-handgraphinFigure8wascon- © American Astronomical Society • Provided by theNASA Astrophysics Data System Figure 9.ObservationsofHumason, MayallandSandageplottedalongwiththeoretical THE ASTRONOMICALJOURNAL curves correspondingtoCaseI. Alsom—19.5. 2 Thus, underthehypotheticalconditionofno dots) includeanaperturecorrectioninthemag- z =0.2.Furthermore,thedeviationsfrom correction forexcessdimmingisincreasingwith before. Wecometothebizarreconclusionthat selection willnowturndownmoresharplythan halves ofFigure8,thecurvescorrespondingto slope 5.Sincethedifferencesbetweencurves, redshift isassumedtobeastraightlinewith bolometric magnitudeandthelogarithmof selection, thetheoreticalrelationbetween called i£-correction,isknownandequalsx(£)- served deviationsfromastraightline(Humason, for anygivenredshift,arethesameintwo bolometric magnitudeduetoredshift,theso- Figure 9.Inthisplottheobservations(shownby in thesamedirectionandverysimilartoob- theoretical straightlineinducedbyselectionare nounced forCase1whentheredshiftisabout redshift. Theeffectsofselectionbecomepro- photographic magnitudeisboundedwhilethe ally decreasewithincreasingredshiftbecausethe the expectedbolometricmagnitudewilleventu- increase inphotographicmagnitudeoverthe magnitude whichamountstoassumingthatthe Mayall andSandage1956),asisillustratedin 62, No.1252 00 CM UOCM 1957 October THE ASTRONOMICAL JOURNAL 261 nitude of all but the last six clusters. A double dip in the expected bolometric magnitude for r- LO symbol, consisting of a dot and an open circle larger values of the redshift z. 0.20 mag. brighter, is plotted for the six fainter We conclude that at least in Case I, which is objects for which the aperture correction is not characterized by the particular set of parameters known. Following Sandage, the empirical ob- believed to correspond to the observational data servation is presumed to be near or above the now available, the numerical value of the bias open circle. induced by the selection of available clusters is If the assumed theoretical curve is not a too large to be ignored. The effect of the bias straight line, or if the assumed corrections for must be taken into account before the observa- dimming due to redshift are in error, the resulting tions can be interpreted. plot would be somewhat different from either All the above refers to the conventional use of part of Figure 8. However, the differences be- the brightest galaxy in a cluster as a distance tween the curves would be maintained so that indicator. Figure io, analogous to Figure 6, was the conclusions concerning the effect of selection constructed using the same set of parameters would be unchanged: selection causes a severe as in Case I excepting r = 5 in order to illustrate Basic Distance ( (million parsecs) Figure 10. Expected apparent magnitudes 85, £5 and F5 and bias in magnitude £5 — 85 and F5 — 85 as in Case 1. the selection bias which would result from the to the distribution of galaxies in space, is rather similar conventional use of the fifth-brightest vague. The question arises: How strongly does galaxy. It will be seen that the bias persists and the bias depend upon the true values of these that it is occasionally smaller and occasionally parameters? This question is studied in the next larger than that connected with the use of the section. fifth-brightest galaxy. Similar figures can be 6. Numerical values of the bias, other cases. made for any value of r. We investigate the effects on the bias of changes If the value of the parameters characterizing in the assumed values of the parameters and of Case I were established with reasonable pre- the luminosity function. The effects of the changes cision, then the curves in Figures 6 to 8 could be are studied with reference to Case 1 so as to show used with confidence to correct the observations the role of each parameter taken separately. In of distant galaxies for bias due to selection. How- each of the panels in Figure 11 the curves drawn ever, the present determination of the parameters are the bias in apparent magnitude, £1 — Si and involved, particularly of the parameters referring F\ — S\, based on parameters having the same © American Astronomical Society • Provided by the NASA Astrophysics Data System CO 00 CM CDCM 262 THE ASTRONOMICAL JOURNAL 62, No. 1252 values as in Case I with the exception of one selection under both of conditions (a) and (b), C"< LO single parameter, the one whose effect on the bias we see that as the distance increases, the depend- is under study. ence on n decreases so that the three curves giv- The first panel in Figure n shows the effect ing F\ — &i combine into one curve. This is due of changes in n, the number of galaxies required to the fact that for. large £ condition (6) tends to to be brighter than mi in order that the cluster of be more stringent than condition (a) ; if the galaxies be recognizable as a cluster. In the brightest galaxy in the cluster is brighter than standard Case i we assume n = 25. Here, the m2 = 20.5 then almost surely there will be n effects are contrasted with those of w = 5 and galaxies brighter than mi = 21.5 whether n is n = 50. Clearly a smaller value of n will induce 5, 25 or 50. less bias due to selection. As might be antici- The second panel of Figure 11 illustrates the pated, the numerical value of the bias induced dependence of the selection bias on the assumed by condition (a) is quite sensitive to the value value of m2 ^ mi. Naturally, the closer the value of n. However, when we consider the effect of of m2 to the adopted mi = 21.5 mag., that is, the i——i 1—j—i—.1 i 1 1 i i 1 i i i i i i 0 100 200 300 500 800 1000 1500 2000 3000 0 100 200 300 500 800 1000 1500 2000 3000 BASIC DISTANCE £ (MILLION PARSECS) Figure II. Effect on selection bias of changes in the parameters entering the theory, that is, on n, m2, mi, 7, © American Astronomical Society • Provided by the NASA Astrophysics Data System 1957AJ 62 . .2483 = ==r effect ofcondition(b).Also,forafixeddistance, ever, andthisisimportant,forlargedistancesthe more powerfulthetelescopeused,milder effect ofachangeinthevaluem2isverystrong. 1957 OctoberTHEASTRONOMICALJOURNAL mi. Inadditiontothevaluemi=21.5corre- sponding toCaseI,thecurvesindicatebias the biasdecreaseswithanincreaseinw.How- corresponding tomi=20.5and22.5.Asmight distance £issmallerwithanincreaseinmi. changes intheassumedvaluesofmi,Moand be expected,thebiascorrespondingtoafixed m anddistance£isassumedtobeknown,a Since therelationbetweenapparentmagnitude through thefunction^(£,mi),whichrepresents have apparentmagnitudebrighterthanmi. the probabilitythatagalaxyatdistance£will fact thatthethreequantitiesenterformulae x(£). Thepossibilityofdoingsodependsonthe due toselectionasthesamenumericalchangein change inmiwillhavethesameeffectonbias nation ofthebiascorrespondingtoanydesired 2 different valueofmi,butmerelyentertheavail- of £.Asimilarremarkappliestochangesinthe able graphswiththeappropriatelyalteredvalue recompute thebiasE—SandF&for tain theeffectofachangeinmi,weneednot 10) areconstructedsoastoalloweasydetermi- attained iftheoriginalscaleof£isconstructed ming x(0- distance £”correspondingtotheassumptions mean absolutemagnitudeMointheexcessdim- 5 log£+x(£)-Inotherwords,inordertoascer- responding tox(£)o,allotherparameters for x(£)F°thisreason,wehaveinserted being asinCase1.Toobtainthebiascorrespond- made inCase1,andthe“basicdistance£”cor- two distancescalesonthegraphs:‘‘standard scale labeledbasicdistanceshouldbeenteredat ing toanygivenvaluesoftheparametersmi, soon as£becomeslarge,thenumericalvalueof rather thanaty.Aswouldbeanticipated,as Mo andx(£)foraparticulardistancey,say,the r the biasisverysensitivetovalueof—-mi be rememberedthat7+1 represents theaverage of theparametersy,gmandM*,allwhichare the dependenceofselectionbiasonvalues entirely outoftheastronomer’s control.Itwill + Mox(?)- The thirdpanelshowstheeffectofchangesin The panelsinFigure11(alsoFigures6and The appropriatealterationof£ismoreeasily The remainingpanelsofFigure11illustrate © American Astronomical Society • Provided by theNASA Astrophysics Data System 238,5— j: =yXlO-[(37) 00 value of7maybe.Therefore,thedependence number ofgalaxiespercluster.Atthepresent moment thereareonlyguessesastowhatthe est. Thecurvesdrawncorrespondto7=50, the biasuponvalueof7isparticularinter- that thebiasdecreaseswithincreasing7.The 7 =100(Case1)and200.Itwillbeseen intuitive explanationofthisphenomenonisthat with large7thansmallsothat,atafixed clusters withmanymembersaremorefrequent distance, clustersaremorelikelytobeavailable effect. Thus,withlarge7theselectioneffect must bemilderthanwithsmall7. to theastronomerdueso-callednumber value of(7m. density functionusedtoapproximatethelu- that theselectionbiasisrathersensitiveto minosity functionofthegalaxies.Itwillbeseen in deviation from linearity is an effect of selection r" that even when ikf* is assumed to be finite, the LO effect of bias due to the unavoidable selection of or is indicative of temporal changes in the veloci- available clusters is substantial. ties of galaxies may perhaps be solved by the 7. Concluding remarks. The results presented accumulation of further data using instruments in this paper indicate that in studies of clusters corresponding to larger values of both m\ and’ of galaxies involving their distance, the selection m%. In fact, as illustrated by the curves in Figure effect may be very considerable and cannot be 9, due to the logarithmic scale employed, the neglected. In particular, an interpretation of the selection effect on the magnitude-redshift rela- deviations from linearity in the magnitude- tion begins to be noticeable only relatively near redshift relation that occur near the threshold of the threshold imposed by the values of m\ and available instruments cannot be made with con- m2. fidence without appropriate allowance for selec- Other examples of the effect of selection bias tion bias. Unfortunately, the evaluation of the can be given. We indicate one briefly. If we as- bias and the development of unbiased estimates sume that the distribution of energy from a gal- of distances require the now unavailable knowl- axy is related to its absolute magnitude in such edge of certain details of the spatial distribution a way that intrinsically bright galaxies tend to be and luminosity function of galaxies, since the bluer, then the unavoidable selection of available amount of the bias is rather sensitive to the value objects to be observed induces a bias in favor of of each of the parameters entering. selecting bluer objects. If the joint distribution The question as to whether or not the apparent of color and luminosity of galaxies (and the other Figure 12. Effect of changes in on the bias in the magnitude-redshift relation. © American Astronomical Society • Provided by the NASA Astrophysics Data System