1988ApJ. . .331. .5833 The AstrophysicalJournal,331:583-604,1988August15 © 1988.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. the Gauss-Riemannspace-time scalarcurvature(kc/R)isnot mental geometrybegunbyGauss andbySchwarzschild.He directly fromgalaxycounts, following themethodofexperi- yet accuratelyknown.Hubble (1936c)attemptedtomeasureit relate apparentmagnitudes and redshiftstodistancesina used asemi-heuristicmethod (HubbleandTolman1935)to After ~50yearsofeffort—somesporadic,someintense— 1 1 luminosity function.Theself-consistencycanbebrokenbyaddingdatafromafainterflux-limitedsample, the formofvelocityfield(i.e.,whetherredshift-distancerelationislinear),orondispersion yet oneortheothergiveswrongmeanHubbleconstantunlessexternalinformationisknown,eitheron method ofassigningafixedforeachvalueoftheparameter,againzero-pointedlaterfromCepheidcalibratinggal- distances tolocalcalibratinggalaxies.Inthesecond,oneusessomeparametersuchas21cmlinewidth,or from thevelocities.Asystemofabsolutemagnitudesisobtainedthereby,laterzero-pointedusingCepheid might beadoptedasadistanceindicator)producesanartificiallycompressedscale,imitating the internalvelocitydispersion,ordeVaucouleursA-index,etc.,towhichisassignedafixedabsolute galaxies. Inthefirst,oneassumesalinearvelocity-distancerelation,fromwhichrelativedistancesarefound list thatapproachesavolume-limitedcatalogforredshiftssmallerthan~4000kms,fromwhichitis varying Hubbleratiothatappearstoincreaseoutward.However,addingthebrightandfaintsamplesgivesa axies. demonstrated that(1)thelocalvelocity-distancerelationislinearoverthisredshiftrange(2),Scllumin- errors of(1)thecalibrationrelevantSclzeropointfromonlythreelocalgalaxies,and(2)uncer- where theerrorisestimatedbyassigninganabsolutemagnitudeuncertaintyof0.6magforcombined brators, gives osity functionisbroadwitha(My=0.7mag,and(3)thevalueofHubbleconstantlow. from therecentsubdwarfdata.Fromglobularcluster age,plusthegestationperiodofgalaxies,age 47 Tue(Hesseretal)usingVandenBergisochronesthatpermit[O/Fe]tovarywith[Fe/H],nowrequired limited) surveyforsuchgalaxieswillbeneededtoimprovethevalueofHviathismethod. the M,logvdiagram.CepheiddistancestomanymoreSclcalibratinggalaxies,andacomplete(volume- tainty oftheapexmagnitude(0limit,usingM31,M81,M101ascali- The ageoftheglobularclustersisadoptedtobe13.5+1Gyrfromprecisionmeasurement 0 A CASEFORH=42ANDQ1USINGLUMINOUSSPIRALGALAXIESTHE 0 Center forAstrophysicalSciencesintheDepartmentofPhysicsandAstronomy,TheJohnsHopkinsUniversity; I. INTRODUCTION COSMOLOGICAL TIMESCALETEST Received 1987May29;acceptedSeptember9 1- H =42±11kms"Mpc, 0 Space TelescopeScienceInstitute Allan Sandage Oo =L2ÍS.9- ABSTRACT 583 from thatexpectedinEuclideanspace. program aimedatfindingadeviationofthemeasuredvolume Johnson 1950),aninadequate definitionoftotalgalaxymagni- inconclusive foravarietyofreasons. Someweretechnicalcon- cerning magnitudescaleerrors (Stebbins,Whitford,and appendix A),impreciseknowledge oftheK-effectredshifts tudes (Humason,Mayall,and Sandage1956,hereafterHMS, -1 Hubble’s 1936conclusionconcerning thevalueofRwas 1988ApJ. . .331. .5833 1 1 1 2 -1 2 1,2 1/2 with theageofuniverse,T^,determinedfromsomeinde- be determinedbythecomparisonofTandT^. strains theacceptablerangeofQandA(Robertson1955, Fig. pendent agedatingmethod.Thenumericalvalueof/con- and Hq.ThetestismadebycomparingT=/(Q,A)Ho are completeovertherelevantparameterrangeofT,Q , A, we onlycomparetwotimescales,eachofwhichisindependent common tothesethreeclassicaltestsisthevariationwithtime stant, HoforthoseFriedmanmodelswithA=0(Sandage only onthedensityparameterQandHubbletimecon- redshift, 0(z),test(cf.Miley1971;Kapahi1975,1987;Swarup diagram. (3)Evolutionoflinearsizeswithtimeorthe 8). If/=fandA0,thenQ1exactly.Evenifnot, can 3; SandageandTammann1984,Figs.69;1986 and Refsdal, Stabell,anddeLange(1967),followingSolheim(1966), of secularvariationsthemeasuredparameters. On theotherhand,time-scaletestisevolution-freebecause of somemeasuredpropertythegalaxiesthatmarkspace. absolute radiopoweralsocomplicatestheangulardiameter- bling blockforthebestknownoftestsviam(z)Hubble impossible toobtain,giventheavailablesamplingprocedures counts inanyredshiftbinzAz.Thisknowledgeisalmost (1986) requirespreciseknowledgeoftheincompleteness ent magnitude,Hubbletype, 21 cmlinewidth,etc.)tofindH. absolute luminosities.(2)Luminosityevolutionisthestum- using galaxieswiththeirlargerangeofsurfacebrightnessand classical cosmologyandparticlephysics,broughtaboutbythe mally closeto1exceptinhighlycontrivedcircumstances.The primary aimofobservationalcosmologyisthentomeasurean inflation intheearlyuniversesoastoachievebothpresent- constant, A.Fromtherequirementofgrandunificationfor some distance-indicatingparameter (e.g.,angularsize,appar- trasting thetwomethodsof usingredshiftscombinedwith tory criticalanalysishasyet appearedintheliteraturecon- or 100kmsMpcisintolerable forthetest.Nosatisfac- Tinsley 1974).TheN(z)testrecentlyappliedbyLohandSpillar to firstorderinz(Sandage1961a;RobertsonandNoonan grand unificationhypothesis. dynamics totestthesepredictionsoftheconnectionbetween Gamow-Alpher-Herman 3Kradiation,Qmustbeinfinitesi- day near-flatnessandtheobservedhomogeneityof 2q +2/3Ac/Hlifwewishtoretainanonzerocosmological closure density3Hl/SnG.TherelationbetweenthemisQ= eration parameterq[relatedtoRqbykcRQ=H(2q closed form. The twogeneral methodsofusingthedata differonlyin tive toluminosityevolutioninthelook-backtime(Brownand scale testisrobust.(1)TheN(m)countdegenerateinq accurate valueofeitherqfromgeometry,orQ different geometries(HubbleandTolman1935comparedwith theory wasunknownuntilMattigderivedtheequationsin tions oftheproperdistance-magnitude-redshiftrelationsfor 1961a, §V).FormodelswhereA#0,thecalculations by 1975; Hickson1977;BruzualandSpinrad1978).Theproblem Oke andSandage1968;Lasker1970;Whitford1971),etc. on magnitudes(Greenstein1938;HMS1956,AppendixB; Mattig 1958,1959;Sandage1961a,1962)becausetheprecise Some wereconceptualsuchastheuseofinappropriatedefini- 1968; Misner,Thorne,andWheeler1973),ishighlysensi- E 0 E0 E0 584 0 0 0 0 0 0 0 0 0 — 1)]orbytheratio,Q,ofpresent-daydensityto 0 The timesincethebeginningofexpansion,T,depends There arefourknowndirectwaystoq,butonlythetime- Clearly, thepresentdichotomous valueofHneareither50 The geometryofspace-timeismeasuredeitherbythedecel- E 0 0 © American Astronomical Society • Provided by the NASA Astrophysics Data System SANDAGE -1- -1 absolute magnitude A/canbecalculatedfor anygalaxyifrom adopt aworkingvalueofH (latertobedetermined),the redshifts wecanthenobtain relative distances.Further,ifwe distance relationexists,justified bythedataonnearbyclusters, current literaturetoobtaindistancesfarbeyondtheLocal groups, andlocalgalaxieswith Cepheiddistances(Sandage The methodsareasfollows. with increasingdistance,atleastoneofthemmustbewrong. Group. Becausethesetwodistancescalesprogressivelydiverge cient range,onlymethodsusingsecondarybrighterindicators for galaxiesinacatalogsuchastheRSA(Sandageand Sandage, Tammann,andHardy 1972).Fromtheratiosof are available.Twodifferentmethodsbeingusedin the bright resolvedstarshavebeenexhaustedbecauseofinsuffi- TYS). known (cf.Tammann,Yahil,andSandage1979,hereafter to agivenfluxlimit,orwhoseincompletenessfunction is Tammann 1981)thatiscompleteinredshiftand also ent magnitude(oranyothersupposeddistanceindicatorsuch as angulardiameter,orHubbletype,luminosityclass)exist is anextensionofapreviousargument(Sandage,Tammann, low. giving toolargeanHvalueifuncorrectedforbias. local HubblevelocityfieldislinearandthatthevalueofH The conclusionsofthethreesetsstudiesagreethatvery korpi (1975a,b;1984)andappliedbyBottinellietal(1986). formal proofviatheMalmquistbiasequationsgivenbyTeeri- here thattheapparentincreaseofHwithdistance(Hawkins 1972a, b,c,1975,1986;Sandage andTammann1975a,1985; brighter thanapparentmagnitude~13,andisparalleltothe and Yahil1979,hereafterSTY)thatusedonlyfieldgalaxies to anygivendistanceindicatorissubjectsystematicerror, values ofHderivedbythemethodassigninganvalue not thepurposeofthispaper.Rather,itistoshowthatall low (~55kmsMpc^isgiveninthefollowingpaperusing Peters 1986;Giraud1985,1986a,b)isaresultofselectionbias Segal 1982inanswertoSoneira1979;deVaucouleursand 21 cmradiolinewidthdata.Theexactvaluethatwederiveis s Mpc.AsimilaranalysisandaresultthatHis Bergh luminosityclass.CalibrationusingM31,M81,and optical dataonfieldspiralgalaxiesofthebrightestvanden are showndirectly.Inthefollowingsectionsweanalyze different apparentfluxlevels.Inthisway,theselectioneffects The proofismadebyanalyzingtwosetsofcatalogsthatreach flawed byselectioneffectswhenusingflux-limitedcatalogs. 0 M101 whichhaveCepheiddistancesgivesH=42±11km treating thedataandshowthereinthatoneroutetoHis 1962; deVaucouleurs1972;Segal1975,1981,1982;Nicolland the biasisgivenbyTammann(1987). distance scaletothecalibratinggalaxiesandproblemof brators. Areviewofthegeneralagreementforverylocal rather thananydifferenceintheinputdataforlocalcali- trivial choiceisthecauseofpresentdisagreementoverH this andthefollowingpaperistoshowthatseemingly line width,etc.)isusedasindependentvariable.Thepurposeof whether theredshiftorotherparameter(magnitude,21cm 0 0 0 0 0 II. TWOSECONDARYMETHODSTODETERMINEDISTANCERATIOS 0 0 0 After theprimarydistanceindicatorssuchasCepheidsor We supposeinwhatfollowsthatdataonredshiftandappar- As totheformofexpansion,empiricalproofgiven 1. Inthefirstitisassumedthatanideallinearredshift- In thisfirstpaperwecomparethetwoprincipalmethodsof Vol. 331 1988ApJ. . .331. .5833 an 2 No. 2,1988 calibrated usingCepheiddistancestosuitablelocalgalaxies. its apparentmagnitudeandfromthedistancegivenbyr= adopt alinearvelocity-distancerelationandfindpropertiesof corrected forthebiascausedbyflux-limitednatureof (2) weusetheindividual/z=i;*/^values,whichthenmustbe luminosity classIsystems.Tofindthepropervalueof{M,), have CepheiddistancessuchasM31,M81,andM101for compared with(M,)fromthelocalcalibratinggalaxiesthat for thesampleasfollows.Inmethod(1)meanMvaluesare distance moduli—toeachgalaxyi,fromwhichindividual type, oragiven21cmlinewidth,hasstablemeanvalue. such asthevandenBerghluminosityindexforagivengalaxy Vi/H. Thezeropointoftheabsolutemagnitudescaleisthen number tobeadjustedlater(§VI)viathelocalcalibrators. priori waytodecidebetweenthemethodsexceptbycomparing either (a)usinganincorrectlynarrowluminosityfunction0>(M) with nii)istakentobetheindependentvariable. galaxy inthesubsetandfromresultingparameterdistance data beforeacorrectvalueisfound. Vi-+0 whichdefinesthevolumeelementoflocalcali- (or mi—M)forthetotalsample. Thisdiagramisusefultotest distance fromtf/Hodthen calculateM,-fromM=+5 made eitherfrom(1)anindependentknowledgeofthe true external dataonO(M),adecisionbetweenthemcanonlybe in method(2),or(b)byignoringthebias,whichisequivalentto distance-limited sample.However,themethodsdonotgive later (Fig.9)thattheresultsofeachmethodconvergeto between themethodsisinwhichparameter(Viorcoupled r oneattemptstofindpropertiesoftheHubbleflowitselfasif v ratherthanrissuspect.Noteagainthatthedifference local calibrators(§VI)intheredshiftlimitofv-►0.Inmethod where theincreaseofradiopower aszfor3CRradiosources if biasispresent.Examples are Figure7ofSandage(1972c) limited sampleforatleastpartofthedataset—ourpresent nitudes inageneralfieldcatalogtoapproachdistance- great clusterfunction)or(2)bysamplingfainterapparentmag- luminosity function(themethodofTYS1979,byadopting the a faintandbrightcatalog. the luminosityfunctionisnotknownapriori,thereno assuming aninfinitelynarrowdeltafunctionforQ>(M).Because same resultwhenagivenflux-limitedcatalogisanalyzedby sively decreasedtoformadatasubsetthatapproaches same valueofH,asthefluxlimitsampleisprogres- route. thereby.Inthesecondweadoptafixedforevery t f f 0 t 0 0 t f h t 0 a) ExpectedDistributionofMinaFluxLimitedSampleUsing - 5logr. f t The stepstobefollowedinmethod(1)arethefollowing. The twomethodsaddressdifferentquestions.Inthefirstwe 2. Inthesecondmethoditisassumedthatanyindicator a. Assume=vJHqwhereHqisanarbitrarilyadopted If thelocalvelocityfieldisindeedlinear,wedemonstrate For eachmethodoneproceedstothemeanHubbleconstant b. Foreverygalaxyinthesample,calculatekinematic c. Constructa“biasdiagram ”byplottingversuslogy* Because eachmethodisself-consistentintheabsenceof © American Astronomical Society • Provided by the NASA Astrophysics Data System a LinearVelocity-DistanceRelation CASE FORH=42 0 1 centroid andforVirgocentricinfallifdesired)toobtain the metric distancerfrommi—+5=log; either seemingtobecomeevenbrighterwithincreasingdis- fainter galaxieswillcontinuetheapparenttrendof(M,), with distancewhichistheMalmquistbias. ent fractionsof0(M)aresampledatdifferentdistances(Fig. is appropriateforthecompletedistributiontoeverygalaxyin corresponding changeinthescaleof theordinate.Figure11laterhereshows course, dependsontheo(M)value.The pointisthatdiagramsisomorphicto tion withdistancethatisshown inFigure1ispresenteverysuch principle ofthebiasprobleminflux-limited samples.Thesametypeofvaria- ate forSclgalaxies.This,however,isofnoconsequenceinillustrating the the localvelocityfield. Hubble ratio=i^/r,foreachgalaxy; assumption ontheformofvelocity-distancerelation,are : trend inFiguresIhorAb. tance orwilltackoninacontinuousmannertotheendof velocity-distance relationisnotlinear,thentheapparently m =13andafainterlineat15.5,say.However,ifthe relation ifitisreallynonlinear,thenthefaintergalaxies i.e. isnotcausedbyincorrectlyusingalinearvelocity-distance ent increaseinwithdistanceFigureTbisduetobias, current literatureallweneedtodoischangethelimitlinein the fluxlimitofcatalogs,withconsequencethatdiffer- such asetofisomorphicdiagramsfor differentvaluesofcr(M). sample whenever=f(D)variation, analyzed usingmethod(1)willbegintofilltheregionbetween this istheoriginofdivergencedistancescalesin regard forthebias,thenprogressivelyincorrectvaluesof However, influx-limiteddatasetsthesampleistruncatedby the completesamplewouldnotbeafunctionofdistance. toward thebrightendasfaint,andmeanisappliedtoallgalaxieswithout toeachgalaxyi; The stepstobefollowedinmethod(2),makingnoapriori b) form—that Theavalueof2maginFigure1islargerthanthe=0.7appropri- A demonstrationoftheusefulnessM,logvdiagramis t Method (2)withoutBiasCorrection 585 1988ApJ. . .331. .5833 absolute magnitude.Bottom:Thedistributionthatwouldbeobserved inaflux-limitedsample,cutatm=13mag.Theobservedbecomesbrighterwith increased distanceduetotheimposedflux-limitoncompletesample. Iftheabscissahadbeengivenaslogdistance,lowerfluxlimit-linewouldhave is available.ThiswhyaddingafainterdatasettotheRSA straight. Top: DistributionofMforacompletesamplegalaxiesingivenvolume. Theupperandlowerenvelopelinesopensymmetricallyabouttheassumedmean results ofmethod(1)and(2)iftheluminosityfunction is unknown andifonlyasingledatasampletogivenfluxlimit ity fieldinwhichHincreasesoutward,orwhether{M)distributionexistsforgalaxies intrinsic luminosities.Hubble(1936a,b)wasthefirsttosuggest (M).Itwasthendemonstratedfrom defended byKennicutt(1982),andhisresultwasdiscussed (especially forluminosityclasses>II)wasthenquestionedby D the distance-limitedsampleinVirgoCluster(Bingelli, showed thatthelocalvelocityperturbationhadnegligible Kraan-Korteweg, Sandage,andTammann(1984,KKST)who TYS (1979)intheiranalysisoftheRSAdatatoobtain continued tohavenearlyaGaussiandistributionwith for theRSAsample,combinedwithfainter fainter samplethantheRSAcontinuedtodefinearestricted adopted velocity,reducedtothecentroidofLocalGroup the RSA(SandageandTammann1987).Column(3)shows made hereinwhichweaddafaintersample. we requirenosuchassumptiononO(M)inthedemonstration explained bytheMalmquistbiasiffield0>(M)isclosely (M)for intheflux-limitedsampledefinedbyHMScatalog 1975h, Fig.1)ratherthantomerelyincreasetheapparentmag- 1956, Figs.3-12;Hawkins1962).Inthissectionweshowthat 3 2 ThedeVaucouleursAindexisageneralizationofthevariation ThediscussionbyKKSTcenteredonwhethertheinclusionofaVirgo However, theideaofamoderatelynarrow0>(M)forspirals Data forSb,Sbc,andScgalaxiesofluminosityclassesI-I.3 The nextstepwasthedemonstrationthatSclgalaxiesina It wasknownfromHMS(1956)thatapparentmagnitudes © American Astronomical Society • Provided by the NASA Astrophysics Data System a) DatafromtheRSA CASE FORH=42 0 -1 -1 made byRubinetal.(1976)in theirsearchforlarge-scaledevi- made bySandageandTammann (1975b),hereaftercalledthe galaxies betweenm=13and ~15.7,chosenbyinspectingthe S-T sample.Anindependent catalog of202suchgalaxieswas axies. Twosamplesofsuchgalaxies exist.Acatalogof69Scl bias ornonlinearvelocityfield)wenowaddfainterScl gal- POSS originalplates,withredshifts measuredatPalomar,was This isanimportantpointinthefinaldeterminationof H using thelocalcalibrators(§VI). due toanassumedsymmetricalO(M),meetintheu-►0limit. apex wherethetwoenvelopelines,assumedtobesymmetrical is theapproximatecataloglimitcorrectedforaninternal absorption of~0.4magforScgalaxies. apparent magnitudelimitoftheRSA,putatm=12.5which in Figure4bthatenclosethedistribution.Alsoshownis the from theassumedlinearity. (Fig. lb),orisduetoarealdepartureofthelocalvelocityfield either whatisexpectedduetobiasinaflux-limitedsample progressively withlogvforgalaxiesinthesample,shown Figure 4a.Thevariationisclearand,aspreviouslydiscussed, and 3isthattheMwhichcalculatedusingr=varies line throughthedatainbothFigures(2)and(3)later for Virgoinfall.Thepurposeistoillustratethatthesecorrec- internal absorption(CIA)andusingvvelocitiesuncorrected (STY 1979). 0 Figure (6)hasasmallerslopethan5. tions areminoranddonotalterthesituation.Aleast-squares eye, haveaslopeof5,requiredifthelocalvelocityfieldislinear HMS (Figs.3-10)andintheESOgalaxiesofRSA velocities relativetotheridgeline.Thisissamesenseasin deviate fromthisslope,showingbrightermagnitudesathigher and iftherearenoselectioneffectsinthedata.Thedataclearly distribution ofpointsrelativetotheScldatasimplybysliding were determinedbyplottingthev,Bdiagramsforeachtype one diagramoveranyother. separately fromthedatainTable1,andfindingoffset galaxies toreducethemthesamemeanluminosityasfor magnitude correctionshavebeenappliedtotheSbcandSb (9) orincolumn(5)forthelisting(10). luminosity classesI-I.3andI-IIisshowninFigure2.Small (9), and(10)withwithoutinternalabsorptioncorrections, 0 i Scl-I.3. Thesecorrections,showninthecodetodiagram, and usingthevelocitiesincolumn(3)forlistings(8) culated asifH=50kmsMpc,arelistedincolumns(8), column (7)iscorrectedforgalacticandinternalabsorption taken fromcolumn(15)oftheRSA.Absolutemagnitudes,cal- column (12)oftheRSAisin(6).Themagnitude columns (3)plus(4).The“total”bluemagnitudeBfrom infall inthedensitymodelsheadopts.Column(5)islogof correction forVirgocentricinfallcalculatedbyKraan- 0 Korteweg (1986)usingv=220kmsfortheLocalGroup as listedincolumn(20)oftheRSA.Column(4)isvelocity Btt 0 T 0 T 20 To decidebetweenthetwoexplanationsofFigure4a(i.e., The arrowinFigure4bputat(H=50)—21.0isthe For laterreferenceweshowupperandlowerenvelopelines The consequenceofthisslopedifferencefrom5inFigures2 In Figure3weshowdatawithandwithoutcorrectionfor The envelopeandridgelinesinFigure2,drawntoguidethe 0 The Vq,BYHubblediagramforSb,Sbc,andScgalaxiesof c) DatafromtheFainterSclSamples b) TheM,logvBiasDiagram Bt0 587 1988ApJ. . .331. .5833 NGC 1241. NGC 309 IC 764 4939. 4030. 6699 . 2713 . 2369 . 2336. 2223 . 2120-46 . 4653 4535 4321 4303 4254 5905 . 3963 . 3344. 3259 . 3145 . 3124. 3054. 6878 6118 5427 . 5426 . 5351 . 5324. 3720. 3687 . 3486 . 3433 . 5660 5457 5364 5230 5161 3938 3893 2998 2997 2989 2955 2942 2835 2280.... 958 3735 .... 3614 3478 3464 3294 2776 2207 628 521 1365 . 1566 . 1376 1232 .... Galaxy (1) Redshift andOpticalLuminosityDataforIntrinsicallyBrightSpiralsofHubbleTypeSc,Sbc,Sb © American Astronomical Society • Provided by the NASA Astrophysics Data System Sc(s)1.2 Sc(s)I Sc(r)1.3 Sbc(s)1.2 SBbc(rs)I Sbc(s)I Sbc(rs)1.2 Sbc(r)1.3 Sbc(rs)I Sbc(r)I Sbc(r)1.2 Sbc(s)I SBbc(r)I.2 Sbc(r)1.2 Sbc(r)1.3 Sbc(rs)1.2 Sbc(r)I SBbc(rs)I SBbc(r)I SBbc(s)I Sbc(s)I Sbc(s)Ipec SBbc(r)I SBbc(r)I.3 Sbc(s)1.2 SBbc(s)I SBbc(rs)I.2 Sc(s)1.3 Sc(s)1.2 Sc(s)I Sc(r)I Sc(s)I Sc(s)I Sc(rs)1.3 SBc(s)I.3 Scl Sc(s)1.2 Sc(s)1.3 Sc(s)1.2 Sc(s)I Sc(s)1.2 Sc(s)1.3 Sbc(rs)1.2 Sc(sXI) Sc(r)I Sc(s)I Sc(rs)I Sc(rs)I Sc(s)1.3 Sc(s)I Sc(s)I Sc(s)I Sc(s)1.3 SBc(rs)I.2 Sc(rs)I Sc(s)1.2 Sc(s)I Sc(rs)I Sc(s)1.2 SBc(rs)I Sc(r)I Type (2) x 4072 2565 2455 2903 2456 2566 2005 2424 2529 2600 2433 6755 4813 3357 3663 2853 3295 3416 3307 3690 3016 2113 2433 2301 2836 2362 4399 4198 3544 5831 5791 MIOlGrp 6730 7051 2673 3571 3916 2590 5837 861 5223 5786 1923 1303 1486 1535 1322 1140 1818 1464 1404 1851 1026 1566 1709 1775 (km s) 636 627 844 799 624 (3) {Sbc, SBbc} {Sbc, SBbc} 20 Virgo Virgo Virgo Virgo A»? -115 -137 -161 -180 n {1-1.3} {1-1.3} -32 -48 -57 -96 -63 -42 (4) TABLE 1 474 265 254 268 226 250 425 254 264 256 296 256 290 214 220 225 291 332 362 353 384 217 263 221 222 201 236 335 343 330 320 305 321 356 247 165 196 151 103 64 72 22 26 63 19 18 10 588 20 log vl 2.811 2.811 3.463 3.503 3.510 3.255 3.550 3.785 3.450 3.465 3.355 3.566 3.552 3.341 3.593 3.488 3.413 3.415 3.121 3.163 3.597 3.407 2.460 2.998 3.529 3.576 3.447 3.597 3.758 3.252 3.436 3.195 3.846 3.381 3.450 3.074 3.074 3.074 3.025 3.125 3.486 3.429 3.842 3.584 3.284 3.702 2.889 3.258 3.074 3.337 3.617 3.860 3.666 2.913 3.466 3.424 3.613 3.233 3.756 3.704 3.748 (5) 11.91 12.78 10.48 12.91 12.13 12.60 12.68 10.21 12.66 12.73 12.33 12.07 13.00 12.43 11.56 11.07 12.38 13.70 12.85 10.85 12.28 12.35 12.35 11.12 12.15 10.21 12.56 14.07 12.3 11.05 12.75 11.98 12.82 10.51 10.11 10.43 10.91 11.1 12.50 12.21 12.95 12.82 12.2 12.65 10.32 12.42 10.17 12.35 13.45 12.79 10.95 12.20 11.96 11.35 12.79 10.50 12.95 12.5 12.40 8.18 9.77 Bj’ (6) 11.76 12.28 12.25 12.14 13.39 10.47 11.97 10.14 12.12 11.73 11.87 11.47 11.87 11.77 10.48 11.48 12.21 11.78 13.58 11.24 11.99 10.64 10.65 12.22 11.77 11.05 10.69 12.07 12.57 12.43 11.24 12.51 10.12 10.11 11.24 10.60 11.62 11.75 12.38 12.31 11.69 12.08 12.87 12.90 12.39 10.25 10.60 11.84 11.04 12.43 10.18 12.32 12.19 12.07 Ij y 9.79 9.45 9.79 7.89 9.86 9.62 9.43 ÏÏOyi (7) (RSA) -M° 21.62 21.92 21.48 20.68 21.32 21.35 22.26 21.71 21.63 20.59 21.27 20.01 20.17 21.92 21.89 21.02 21.88 21.46 22.45 21.72 21.91 22.19 21.89 21.07 21.36 20.65 21.59 20.23 21.16 21.04 21.14 21.22 20.74 22.90 21.32 20.62 21.19 21.53 21.27 20.61 20.46 21.32 22.70 21.55 20.28 22.30 21.03 21.22 22.34 21.97 20.23 21.50 22.54 21.14 21.89 22.25 22.39 21.44 22.57 22.92 19.67 Bt (8) (RSA) 21.92 22.48 21.79 21.18 22.07 21.64 22.77 21.42 22.02 21.94 20.89 20.05 21.58 20.35 20.91 22.44 22.23 21.46 22.47 22.13 22.94 22.04 22.27 22.95 22.34 21.80 21.74 21.20 21.45 21.31 21.15 23.22 21.92 20.93 21.58 21.84 21.60 20.54 20.91 22.15 21.62 23.27 21.96 20.79 22.84 21.40 21.91 21.59 21.60 22.85 22.33 21.80 21.63 22.97 22.19 22.57 23.03 21.75 22.88 23.25 19.81 (9) -ms; 20.42 22.60 21.93 22.61 22.05 21.46 22.24 21.88 23.00 22.09 22.18 22.03 21.31 20.09 21.86 21.16 22.39 21.97 22.60 22.17 23.09 22.10 22.32 22.87 22.51 21.76 21.71 21.52 21.69 20.91 21.84 23.30 22.17 21.24 21.03 21.48 22.31 21.90 22.11 22.15 22.08 22.01 21.76 21.95 23.33 21.23 22.93 21.87 21.72 22.90 22.44 20.70 21.99 21.75 23.02 22.49 22.14 22.97 21.64 22.83 23.17 (220) (10) 1988ApJ. . .331. .5833 NGC 210.. IC 1788. 7606. 6951 . 6753 . 6384. 4999 . 5985 . 4814. 5878 . 5792 . 5533 . 5406. 5371 . 5172 . 5033 . 3992 . 3642 . 3347 . 2935 . 2633 . 2523 . 3200. 772 .. 7755 . 7678 . 7531 . 7479 . 7392 . 7171 . 7124. 6984. 1832 . 1512 . 1417 . 1300. 7038 . 6925 . 6814. 6780. 4947 . 4891 . 4603 . 4412 . 4123 . 5921 . 5592 . 5430. 5350. 5248 . 4045 . 5194. 3981 . 3953 . 3506. 3430. 3338 . 2545 . 2347 . 976 .. 3162 . 3001 . 289 .. 214.. 1640. 1625 . 1097 . Galaxy (1) © American Astronomical Society • Provided by the NASA Astrophysics Data System Sb(r)I Sb/SBb(rs)I.3 Sb(r)I Sb(r)1.2 Sb(s)I SBb(r)I Sb(s)1.2 SBb(s)I.3 Sb(s)I SBb(r)I Sb(rs)I/SBb(rs)I Sbl SBb(rs)I Sb(s)I SBb(rs)I Sb(r)I SBb(r)I Sb(r)I SBb(s)I.2 SBb(s)I.3 SBb(r)I SBb(r)I SBb(rs)Ipec Sb(s)1.3 SB(s)I.2 Sb(rs)I Sb(rs)I SBbc(r)/Sbc(r)I-II SBbc(s)I-II Sbc(r)I-II SBbc(s)I-II Sbc(s)I-II Sbc(r)I-II Sbc(rs)I-II Sbc(s)1.8 Sbc(r)1.8 Sbc(r)I-II Sbc(rs)I-II Sbc(rs)I-II SBbc(s)I-II Sbc(s)I-II SBbc(s)I.8 SBbc(rs)I-II Sbc(s)I-II Sbc(s)I-II Sbc(s)I-IIpec SBbc(r)I-II Sbc(s)I-II SBbc(s)I-IIpec SBbc(rs)I.8 Sbc(s)I-II Sbc(s)I-II SBbc(r)I-II Sbc(s)I-II Sbc(rs)I-II Sbc(s)I-II Sbc(s)1.8 SBbc(s)I-II SBbc(r)I-II Sbc(r)I-II SBbc(r)I-II Sbc(s)I-II RSBbc(rs)I-II Sbc(r)I-II Sbc(s)I-II SBbc(rs)I-II Sbc(r)I-II Type (2) x 2323 2694 3001 2616 3903 5241 2954 2650 4139 3960 2626 2003 3313 2416 3638 2645 1735 1974 2969 4957 1889 2630 2758 4785 4435 1134 1733 1855 3756 3035 2780 4190 1526 1875 3381 2305 1607 3016 2222 2418 2073 6348 4692 1643 2171 4550 1428 4757 1049 3312 3032 1577 1159 1765 1554 3366 1036 1555 1171 1226 1600 1284 1834 710 897 (km s) 760 220 541 v o (3) TABLE 1—Continued (Sbc, SBbc} 20 {Sb, SBb} -120 -115 Virgo -119 -109 -124 At;? -116 -100 -121 {1-1.3} -93 -53 -93 -62 -98 -95 -87 -80 -60 -35 -23 -52 -99 {1-11} 225 291 (4) 277 252 332 249 276 320 309 350 297 243 260 264 310 113 182 157 189 225 251 468 323 281 442 337 267 351 295 257 517 368 241 383 398 356 196 126 589 23 24 43 12 52 8 6 3.343 3.261 3.481 3.283 3.465 3.355 3.347 3.621 3.740 3.468 3.630 3.059 3.519 3.466 3.159 3.308 3.458 3.553 3.355 3.416 3.274 2.888 3.579 3.607 3.168 3.407 3.246 3.455 3.562 3.188 3.403 3.464 3.425 3.687 3.672 3.641 3.438 2.774 3.288 3.227 3.530 3.244 3.645 2.514 3.182 3.422 3.399 3.442 3.369 3.074 3.224 3.344 3.284 3.124 3.820 3.196 3.199 3.382 3.545 3.683 3.205 3.478 3.090 3.648 3.512 3.239 3.666 (5) 11.55 12.2 11.93 11.29 11.80 12.3 12.96 11.72 12.65 11.40 12.60 10.63 12.64 12.8 10.64 11.53 12.27 12.29 12.00 12.85 12.65 12.1 11.38 12.75 11.10 11.10 11.65 12.10 12.8 12.14 11.7 12.65 13.00 13.10 12.36 13.33 12.10 12.02 13.15 11.53 13.27 10.80 12.78 12.2 12.74 12.61 12.09 13.07 11.84 12.65 12.44 10.79 13.44 12.15 11.32 12.15 12.72 13.20 13.30 12.45 13.2 10.16 13.21 13.10 11.81 12.95 8.98 (6) 10.65 11.31 11.25 10.42 11.01 11.35 10.54 11.98 12.58 10.81 11.85 10.11 12.12 11.01 12.21 11.29 11.16 11.25 12.09 11.86 11.32 10.77 12.02 10.43 10.33 11.01 11.54 11.73 12.33 11.27 12.19 12.25 12.48 11.84 12.84 11.36 11.37 12.64 11.14 12.74 12.10 11.84 10.42 12.11 12.25 11.43 12.76 11.47 12.26 11.39 10.28 12.97 11.68 10.89 11.82 11.99 12.57 12.54 11.96 12.32 12.79 12.53 11.41 12.48 9.95 Jj rp 8.57 9.75 DO, i (7) 21.79 20.85 22.14 21.65 21.90 20.77 21.21 21.81 22.14 22.19 21.89 20.64 21.22 20.82 21.14 21.17 21.59 21.94 21.22 20.67 21.78 20.89 21.90 21.32 22.59 21.22 21.77 21.67 (RSA) 20.40 21.97 21.27 20.76 21.92 22.60 21.48 21.73 20.91 -m° 19.58 21.16 20.80 21.48 21.12 21.12 20.81 21.24 20.65 20.85 21.22 20.09 20.09 20.79 22.08 20.31 20.53 20.78 21.07 21.72 20.17 20.83 21.89 21.68 21.04 21.01 19.42 22.03 19.99 19.80 t (8) 22.69 21.36 22.64 22.28 22.64 21.63 22.35 22.48 22.52 22.78 22.64 21.16 21.74 21.41 21.83 21.69 22.31 22.95 21.76 21.33 22.45 21.53 20.14 22.57 21.99 23.29 21.86 (RSA) 22.14 22.05 21.00 22.34 21.73 21.46 22.50 23.06 21.90 22.37 21.21 21.51 21.14 21.88 21.80 21.48 21.19 21.65 21.13 21.17 21.66 20.36 20.48 21.07 21.30 22.55 20.78 20.96 20.13 21.20 21.54 22.32 20.57 21.59 22.30 22.01 21.61 21.41 22.41 19.73 (9) 22.57 21.50 22.65 22.50 22.82 21.93 22.70 22.63 22.62 23.03 21.49 22.81 21.69 21.98 21.62 21.72 22.35 22.03 22.50 23.11 22.03 22.54 21.55 20.17 22.52 21.91 23.20 22.05 21.98 21.23 22.25 21.63 21.38 22.46 23.02 21.87 22.33 21.27 21.51 21.58 21.99 21.97 21.80 21.77 21.99 21.39 21.65 21.92 21.15 20.96 21.53 21.84 22.63 21.26 21.59 20.68 21.42 21.66 22.38 20.57 21.57 22.20 21.85 21.53 21.51 19.11 22.35 (220) (10) 1988ApJ. . .331. .5833 1 4 with boththeS-Tand Rubin faintsamplesaddedare added inFigure5a.Thesubset oftheRubinsamplethatis shown inFigure5.Onlythe RSA spiralsoftypeSb,Sbc,and Sc ofluminosityclassI-I.3are plotted.ThetotalS-Tsampleis galaxy. from theRSAtobeA=0.28+0.88loga/bforthistype of magnitudes (basedontheZwickyetal1961/68catalogs),axial rected forgalacticandinternalabsorption,thelatteradopted The magnitudesincolumn(4)arethe(2)values cor- ratios a/b,andredshiftsrelativetotheLocalGroupcentroid. m ~15intheiroriginalpaper. magnitudes arelistedforthesegalaxiesbetweenm~14 and original catalog(ST1975h).Columns(2),(3),and(5)list m name isincolumn(1);identificationdatacanbefound the ations fromtheHubbleflow.Redshifts,21cmlinewidths, and 590 pg 4 Seepage591. The B,logvHubblediagramsforthebrightRSAspirals The datafortheS-TsamplearelistedinTable2.galaxy t0 NGC 23... © American Astronomical Society • Provided by the NASA Astrophysics Data System IC 4351. 4902 . 7782 . 7723 . 7552 . 7331 . 7329 . 7083 . 6887 . 4679 . 4593 . 4548 . 4394 . 4050. 5850. 5740 . 5347 . 5156 . 5150. 5054. 3705 . 3681 . 3673 . 3504. 3169 . 2815 . 2712 . 2642 . 2551 . 986 .. 3223 . 3147 . 3031 . 615 .. 224.. 782 . 779 .. 670.. 1964. 1433 . 1228 . Galaxy (1) Sb(s)I-II SBb(rs)I-II SBb(s)I-II Sb(rs)I-II SBb(r)I-II Sb(s)I-II Sb(s)I-II SBb(sr)I-II Sb(s)I-II SBb(s)I-II Sb(s)I-II SBb(rs)I-II Sb(r)I-II Sb(s)I-II SBb(s)I-II Sb(s)I-II SBb(rs)I-II SBb(rs)I-II SBb(sr)I-II Sb(r)I-II Sb(r)I-II SBb(r)I-II Sb(s)I-II Sb(s)/SBb(s)I-II Sb(s)I-II Sb(r)I-II Sb(s)1.8 Sb(r)I-II Sb(s)I-II SBb(s)I-II SBb(rs)I-II Sb(r)I-II Sb(s)I-II SBb(s)I-II Sb(r)I-II SBb(rs)I-II SBb(r)I-II Sb(rs)I-II Sb(s)I-II Sb(r)I-II Sbl-II Sbl-II Type (2) 4127 4509 5584 3043 2951 2938 2430 2394 2367 2670 2426 2505 4262 4461 2619 2899 2333 4023 4836 2484 2006 1976 1565 1114 5881 1490 1524 1661 1135 1662 1480 1067 1892 1579 1492 1971 366 853 923 870 (km s (3) TABLE 1—Continued SANDAGE {Sb, SBb} Virgo Virgo -158 -111 -104 -106 -119 Ae““ -67 -39 -30 -17 -46 -73 -97 -83 -94 {1-11} 477 284 212 410 304 362 348 241 211 238 380 347 368 315 231 263 383 327 197 185 182 (4) 49 4 1 fainter samplealone(opencirclesfortheS-Tand tri- would notbepresentifthe velocityfieldwerenonlinear, diagram ofFigure7,similarto Figureslaand4abutnowwith the faintS-Tsampleadded in Figurelb.Thisdiscontinuity the twosamplesinFigure6 can beunderstoodfromthebias residuals asmagnitudedifferences. Thediscontinuitybetween from Figure6whereleast-squaresregressionsareputsepar- course, istheexpectedbehaviorifmethod(1)correctas seen ately throughtheRSAand fainterS-Tsample,takingthe compared withthefaintsamples.Thisisbecauseeachshows samples), butthebiasstartsatdifferentmagnitudes.This, of the Malmquistbiasseparately(eachbeingflux-limited distribution ofpointsbetweentheenvelopelinesforbright angles fortheRubin)isalsolessthan5,i.e.,thereadifferent most interestingdifference.AsinFigures2and3,theslope of envelope andridgelinesaredrawnwithaslopeof5. the RSAsamplealoneisagainlessthan5,butslopeof the restricted totheirtypeI1.3isplottedinFigure5b. The 3.734 3.271 3.176 3.047 3.478 2.460 3.466 3.465 3.437 3.268 3.438 3.423 2.640 2.916 3.280 3.443 3.674 3.458 3.074 3.207 3.298 3.276 3.456 3.144 3.491 2.967 3.074 3.310 3.409 3.333 3.648 3.212 3.426 3.639 3.286 3.762 3.149 3.594 3.271 3.674 The biaseffectisagainseenineachplot,butnowwith a (5) 13.1 11.85 11.40 10.39 12.32 11.80 12.46 11.71 12.62 13.40 12.30 12.87 13.26 11.51 11.90 12.95 11.72 11.76 12.40 11.8 10.98 12.25 11.77 12.41 11.88 11.28 11.45 12.66 12.70 12.54 13.05 11.60 10.68 12.80 11.8 13.17 12.83 11.86 12.3 12.80 7.86 4.38 b (6) t 12.36 11.23 10.99 11.84 11.04 11.43 11.20 10.75 11.86 12.86 10.86 12.15 12.72 11.39 12.23 11.15 11.94 11.65 11.25 10.43 11.28 11.58 10.88 10.91 10.56 10.88 11.34 11.89 11.88 12.31 10.49 10.18 12.33 11.21 12.36 10.80 12.19 11.39 12.07 9.14 0l 7.01 2.71 Bj (7) 22.14 21.13 21.08 21.59 21.65 22.12 21.49 21.76 20.00 21.23 21.20 21.43 20.98 21.57 22.03 21.78 20.72 20.94 20.41 20.32 20.56 21.98 20.42 22.44 21.01 20.99 20.26 21.33 20.54 21.06 20.65 21.95 21.22 22.52 20.51 21.47 20.68 20.50 22.22 19.75 19.38 19.43 (RSA) -m° Bt (8) 22.88 21.75 21.49 22.60 22.08 22.81 22.42 22.23 20.51 20.54 22.55 21.55 21.86 21.69 22.04 22.55 22.35 21.27 21.42 20.32 20.96 21.11 22.69 21.09 22.94 21.79 21.03 22.00 21.00 21.77 21.17 22.01 21.15 22.42 21.81 22.99 21.57 22.34 21.59 21.53 22.86 19.84 (RSA) (9) 22.81 21.62 21.39 22.60 22.28 22.78 22.40 22.49 20.98 20.83 22.76 21.65 21.98 22.15 22.33 22.64 22.64 20.60 21.34 21.64 21.44 20.59 21.48 20.20 22.87 21.66 23.08 21.79 22.21 21.28 22.86 21.32 22.07 21.16 22.37 21.72 22.95 21.45 22.28 21.46 21.53 22.80 (220) (10) Vol. 331 1988ApJ. . .331. .5833 found fromthedatainTable1.Theridgeandenvelopelinesaredrawnwithaslopeof5. types andvandenBerghluminosityclassesthatdifferfromthoseofScl-I.3arereducedtoabsolutemagnitudesystembythecorrectionsshown inthecode, (Figures 8and9here)filltheM,logvdiagram,asexpectedif= f{D) differs by0.12magfromBforHolmbergcolorsofC=0.5inthesense dent ofthediameter.ThiscorrectionwasdeterminedbyS-Tfromtheir rected forameanzeropointdifferenceofm—=—0.1,appliedindepen- column (3).TheyarebasedontheZwickyetal(1961/68)magnitudes,cor- distinction inthefollowingdiagrams betweenBandthemvaluesthatare proof wouldrequireameanmagnitudecorrectionfarinexcessofany the correlation ofFigure4aisduetotheflux-biasedRSAsample.Todestroy the proof giveninFigure7thatthefainterS-TsampleandRubinetal. photoelectric aperturephotometryofBothunetal.(1984,Table2)asreduced conversion ofmtoBuntenableatpresent. ing ontheestimateddiameters.Thesemagnitudecorrectionsdependcritically from Am=B—m—0.9to+0.9magfortheTable2galaxies,depend- the mvaluesincolumn(2)ofTable2wouldbe0.12magbrighterthan being fainterthanm(Table11oftheRC2). m system,whichisthesametowithin0.04±0.2magaszeropointof aperture photometryof11galaxiesfainterthanm=13.8(ST1975a,Table No. 2,1988 tions toBare,themselves,small. listed eitherbyS-TorRubinet al. (1976),becausethemagnitudecorrec- only forqualitativetestsbiasin the remainderofpaper,wemakeno used here(seefootnote5).Becausethe magnitudesofthefaintgalaxiesareused suggested smallcorrectionsdiscussed aboveand/oragrossmisidentificationof scatter inFigures5a,6,7,10,and12.Moreimportantly,itisnegligiblefor the to theRC2systembyreferee,givesacorrectionof(B—m(Table2)) = on thediametervalues,whicharequiteuncertain,makingthisrouteto the stronger apertureeffect,consistentinsignwithwhatwasknownbefore (ST, Holmberg (1958)system(HMS,FigureA5).Inturn,thezeropoint Scl typegalaxiestowardlaterluminosity classesforthefaintgalaxysamples RC2 BvaluesinTable1here.However,Graham’s(1976)photometryshows a 3). TheS-TphotoelectricvalueswerereducedtotheHMS(1956,AppendixA) 1975h; §Illh)butverymuchlargerinamplitude,givingcorrectionsthatrange —0.3 +0.1.Buteventhiscorrectionissmallcomparedwiththeintrinsic T pgzw Tpg zwT Tzw pg Ho pg zw t Tpg t 4 Fig. 2.—Themagnitude-redshiftdiagramforSbandScgalaxiesofthebrightestluminosityclassesthatarecontainedinRSAcatalog.Galaxies withHubble Themmagnitudesincolumn(2)arethoselistedbyS-TtheirTable4, Indeed, comparisonoftheTable2mvalueswithBdeterminedfrom If thiswerethecorrectreductionofmmagnitudestoBsystem,then pg pgT zwT © American Astronomical Society • Provided by the NASA Astrophysics Data System o 4.2 3.4 3.8 2.6 3.0 CASE FORH=42 0 B^iScl-LS) -1 1 fill thespacebetweentwolimitlinesbutwouldtackon (Figs. 11and12)becausethetrianglesinFigurelbwouldnot calculating hiforgalaxiesinTable1ofthetypesindicated. We previous sections.Fordefiniteness(M)distributionwithwhatisbecominganapprox- Figs. 4etc.),andexplained as biasbyTeerikorpi(1975a,b, For theotherwenowfollowmethod(2)toreachacontradic- 7000 kms“.Ifthisvariation weretobereal,thefainterScl 1984), STY(1979),TYS andBottinellietal(1986). 0 — 21.2,tobeadjustedlater(§VI)usingthelocalcalibrations of — +5]mustvarywithi;,-.Figure8ashowstheresult of -1 The stepsformethod(2)havebeengivenin§lib.We It isclearfromFigureslb,4a,andlathatifwerequire Instead, whatweseeinFigurelbisthefillingoutof The mean{/if)valuesvaryfrom ~35to90kmsMpc A FIXEDTOEACHDATASAMPLESEPARATELY 591 1988ApJ. . .331. .5833 -1 5 l 592 samples mustgivethesamevaluesof/iatanygivencommon effective Hubbleconstant6found byusingmethod(2)ismulti- sample ofTable2.Themeanregressionthroughthepoints of redshift. Figure86showsthecalculationusingS-Tfaint but nowdisplacedtowardlargervelocities.Saiddifferently, the show avariationof<6^overthesamerangefrom~30to 100 Figure 8aisshownasasolidline. valued atagivenredshift—clearly acontradiction.Thesameis volume limitedsetforv<3000 kms.Thisissimilartothe shown inFigure8cfromtheRubin etalsample. of the6„Viplaneusinga total samplethatapproachesa f magnitudes Themeanmagnitudedifferencebetweenthedataherefrom thoseinFig.2isAB°j=0.15;thedataherebeingfainter.Thebottompanelshows f correlation withoutthemagnitudecorrectionforinternalabsorption. The datapointsforthefaintS-Tsample,inFigure86,again What, infact,weareseeing Figure8isaproperfillingout Fig. 3.—Themagnitude-redshiftdiagramforSbandScgalaxiesoflower luminosityclassthaninFig.2.Toppanelshowsthecorrelationusingfullycorrected © American Astronomical Society • Provided by the NASA Astrophysics Data System SANDAGE J IL -1 (S-T 1975h,§II)plates.Theexperiment consistedofinspectingselectednearby more, theexperimenttheyperformedoneffectofreducedspatialresolution (1975h) intheir§V,listingsixpointssupportofthissupposition.Further- galaxies wehaveaddedareScltypes,similarintheirrelevantproperties with gives adirectdemonstrationofthebias,andwhymethod(2)multi- conclusion thatthegalaxiesinfaint samplearepredominantlySblandScl resolution testthatwasmadelater byRubinetal.(1976),withthesame galaxies fromTable1ontheLickObservatory SkyAtlasprintswhosescalefor types withverylittleconfusionastothe morphologicalclassification. these brightSclgalaxiesmatchedthe muchlargerfocallengthscaleofthe remote Sclgalaxiesatt;~10,000kmscouldberecognizedonthePOSS of faintgalaxiesrelativetotheRSAsamplewasconclusiveinshowing that the brightRSAgalaxieslistedinTable1.Thequestionwasdiscussedby S-T valued HubbleconstantinFigure8dependsonthesuppositionthat faint POSS fromwhichtheremoteScl samplewaschosen.Thisisthesame 5 ThevalidityoftheargumentwhyaddingfaintsampleinFigure lb Vol. 331 C/} 00 00LO ï—I No. 2, 1988 CASE FOR H0 = 42 593 00 'o a 00 CO00
220 LOG v0 Fig. 4.—Top: The apparent variation of the absolute magnitude with redshift for the RSA sample of Sb to Sc galaxies of bright luminosity class, reduced to the magnitude system of the Scl-L3 sample by the corrections shown in the code. Bottom: Same as the top panel but with envelope lines and the m = 12.5 flux limit line superposed. The apparent increase in mean luminosity with distance is an artifact of the flux limitation of the sample.
1 1 effect seen in Figures 4b and lb in the different representation = -21.2 for the of the same result. This is shown more clearly in Figure 9 set. The correct value of to use is the subject of § VI where the separate samples of Figure 8 have been combined. where data for M31, M81, and M101 are applied to obtain a The top panel is the same as Figure Sa but with the m = 13 direct calibration of and therefrom a proper value for H0. limit line of the RSA shown, and with an upper envelope line drawn by eye to accommodate most of the data. The beginning 6 1 The distribution of points in Figures lb, 9, and later in Figures 10 and 12 of a lower envelope is shown, truncated at i; ~ 2000 km s~ by is not uniform, showing gaps between the bright and faint samples. This is the m = 13 limit line. Figure 9b shows the S-T sample added most likely due to the incompleteness of the RSA near its flux limit of m ~ 13 from Figure Sb, now with a m = 15.5 limit line drawn that (STY 1979, Fig. 6), and even greater incompleteness of the two faint surveys, truncates the lower envelope at p ~ 5000 km s" ^ The same for especially near their bright boundary between mpg ~ 13 to 14. These discontin- uities are especially visible in Figures 10 and 12. It is for this reason that we the Rubin et al sample is shown in Figure 9c. prefer the upper and lower boundary argument of the next section to estimate a The sum of the top three panels is in Figure 9d which con- MBr( pex) rather than an analysis of the distribution of points within the tains the principal conclusion. The hh vt plane is filled in a more boundary lines. proper way as the sample approaches a volume-limited set, Figure 9 with its upper and lower envelope lines and the flux cutoff lines is similar to Figure 9 of de Vaucouleurs and Bollinger (1979), but with a different showing that the apparent variation of with vt using RSA conclusion drawn here as to the meaning of the apparent increase of H0 with sample alone is not real. To the extent that the statistics of the redshift for the incomplete sample that is flux-limited. The similarity of the two present Scl total sample approximates a proper volume- diagrams, but their different interpretation, was recalled by de Vaucouleurs as limited set,6 the mean Hubble constant from these data is a helpful comment on an early draft of this paper.
© American Astronomical Society • Provided by the NASA Astrophysics Data System 1988ApJ. . .331. .583S 0950 +43.. NGC 1094 0229-01 .. 0228 +01.. NGC 926.. 0158 +08.. 0152 +06.. IC 1516 NGC 7782 NGC 7780 NGC 7756 NGC 7750 NGC 7428 NGC 3470 NGC 3408 NGC 3202 NGC 3191 NGC 1085 NGC 1019 IC 211 NGC 840.. IC 198 IC 173 NGC 706.. 0145 +12.. NGC 673.. NGC 664.. 0135 +07.. 0116 +01.. 0112-00 .. NGC 99... NGC 36... 0008 +02.. IC 1515.... 2348 +00.. 2346 +05.. 2342 +06.. NGC 7460 2255 +02.. IC 1743.... NGC 658.. IC 1706.... NGC 521.. NGC 497.. 0117 +07.. 0115 +11.. 0054-01 .. NGC 257.. 0037 +02.. NGC 180.. 0033 +12.. 0028 +13.. NGC 7816 NGC 182.. NGC 173.. NGC 105.. 1111 +57.. 1049 +59.. 1013 +05.. 1002 +51.. 1001 +14.. 1111 +56.. 1103 +57.. 1051 +56.. 1022 +55.. 1014 +53.. 1012 +55.. 1001 +13.. Name (1) Magnitude andRedshiftDatafortheFaintS-TScIGalaxySample © American Astronomical Society • Provided by the NASA Astrophysics Data System (13.7) 13.50 14.20 13.80 14.70 14.10 15.67 14.40 13.40 14.50 14.50 14.40 14.60 14.30 14.80 14.40 13.90 13.10 13.90 13.20 13.80 13.50 14.70 14.10 14.20 15.30 13.80 14.70 14.80 13.70 15.54 15.20 14.20 15.67 14.80 14.00 15.00 13.80 15.20 14.50 14.50 13.50 15.10 12.80 14.00 14.80 15.67 14.20 14.40 15.20 14.20 14.40 15.20 13.90 13.90 14.70 13.10 14.70 14.30 13.70 14.10 14.90 13.60 13.70 15.20 14.00 14.40 15.54 m pg (2) 2.34 2.10 2.00 2.57 2.19 2.08 2.34 2.18 2.93 2.11 1.26 1.25 1.23 1.07 1.29 1.25 1.06 1.36 1.43 1.00 1.13 1.50 1.23 1.85 1.24 1.43 1.07 1.06 1.50 1.29 1.59 1.82 1.52 1.42 1.29 1.50 1.00 1.19 1.50 1.75 1.50 1.14 1.64 1.03 1.39 1.85 1.16 1.25 1.36 1.30 1.42 1.33 1.00 1.05 1.14 1.60 1.18 1.35 1.35 1.04 1.23 1.00 1.56 1.59 1.38 1.04 1.33 1.56 a/b (3) 13.90 15.27 14.38 13.72 14.67 13.73 13.36 14.84 13.98 14.14 15.25 13.13 14.79 14.10 12.96 13.14 14.12 13.99 13.16 13.96 14.00 14.14 13.76 14.39 14.02 12.66 13.92 14.35 12.66 14.21 13.61 14.86 13.47 13.23 14.41 13.69 14.49 13.18 15.16 14.83 14.56 13.74 13.32 13.54 12.80 13.42 12.94 14.29 13.71 12.52 13.40 15.34 14.86 13.12 14.50 13.74 14.12 13.15 13.24 13.80 14.00 14.90 14.80 13.61 13.60 13.83 15.09 13.54 (4) TABLE 2 594 (km s 14575 13658 13586 14092 11268 10432 10079 13947 13412 10209 15124 10205 10081 12874 3270 4798 4955 9670 9146 9476 4819 4646 4974 4307 9886 7669 6729 2577 8828 6986 7507 6856 6737 8489 7301 6284 7258 7407 6510 3355 7196 5383 5473 6461 9564 4332 5307 8371 3996 5507 3482 3195 5251 5326 8176 5519 3053 5100 5170 5420 5244 5363 5343 6241 3078 5377 5439 5285 (5) log v 4.133 4.149 4.052 4.164 4.135 4.018 4.003 4.144 4.127 4.180 3.929 3.985 3.885 3.828 3.961 3.863 3.411 3.681 3.798 3.844 3.870 3.526 3.977 3.720 3.697 3.731 3.738 4.009 4.009 4.004 3.602 3.515 3.741 3.695 3.504 3.995 3.828 3.946 3.861 3.814 3.857 3.683 3.667 3.726 3.488 3.634 3.810 3.637 4.110 3.875 3.836 3.742 3.725 3.923 3.485 3.542 3.708 3.913 3.981 3.713 3.734 3.720 3.729 0 3.731 3.736 3.728 3.795 3.723 (6) (H =50) 0 -21.32 -21.45 -21.55 -21.14 -21.26 -21.68 -20.43 -21.82 -19.22 -20.28 -20.23 -20.51 -20.12 -20.18 -20.33 -21.06 -21.58 -21.66 -21.35 -22.43 -20.93 -22.51 -21.98 -22.67 -21.32 -21.58 -20.06 -20.51 -22.10 -22.23 -21.31 -22.57 -21.36 -19.74 -21.19 -21.69 -20.62 -20.71 -21.89 -21.94 -21.40 -20.45 -19.98 -21.46 -22.07 -21.47 -20.99 -22.12 -21.78 -22.43 -20.94 -22.25 -21.61 -20.87 -22.15 -22.21 -21.58 -21.41 -21.45 -20.96 -20.29 -21.33 -21.19 -21.25 -21.35 -21.08 -21.52 -21.22 (7) (H =50) 0 -21.96 -19.66 -20.80 -20.49 -21.69 -21.75 -22.06 -22.71 -21.92 -23.19 -22.00 -21.45 -22.59 -23.03 -21.87 -20.18 -21.79 -22.25 0,i -21.62 -22.34 -21.78 -21.01 -20.39 -21.85 -22.67 -21.80 -21.34 -22.56 -20.92 -22.51 -20.61 -20.70 -20.53 -20.85 -21.44 -21.92 -21.77 -21.26 -22.95 -22.34 -21.68 -20.43 -20.81 -22.54 -21.69 -22.42 -21.16 -22.84 -21.09 -21.52 -22.33 -22.53 -21.91 -21.33 -22.43 -22.55 -21.99 -21.91 -21.36 -21.55 -22.03 -20.69 -21.65 -21.73 -21.58 -21.65 -21.97 -21.58 x M (8) pg 1988ApJ. . .331. .5833 l 0 0 0l give astandarddeviationof B°f =5logv—5.30.Themagnitude residualsfromthisline flux levels(seefootnote6).Imposing aslopeof5onthedatain each ofwhichareonlysemicompletethemselvestoparticular Figure 6givestheridgeline that isdrawnwhoseequation surprising becausethedataconsistoftwomergedcatalogs, that thesampleisnotentirelyvolume-limited,butthis not sample is=0.70magaboutthelineB/4.445log v mag fortheS-TfaintsamplealoneaboutequationB°/ = mean relationB°/=3.186logv0.657,anda(B/) 0.56 2.338 logv=4.995.Thestandarddeviationforthecombined G(Bf) =0.53magforthebrightRSAsamplealoneabout the v inFigure6.Theleast-squaresregressionlinesthere give osity functioncouldbefoundfromthescatterinB°/atagiven volume-limited sample,thestandarddeviationoflumin- 0 with aslopeof5. galaxies. ThefaintS-Tsample(opencircles)isinthetoppanel.Rubinetal.(triangles)bottomridgeandenvelope linesaredrawn 0 0 0 0 — 3.38.Theslopeofthislastregressionisnotyet5.0,showing I V. THEABSOLUTEMAGNITUDEDISPERSIONFORSCIGALAXIES ^?‘ ^1,diagramforSbandScgalaxiesofthebrightestluminosityclassinRSA(closedcircles)combinedwithtwofainter catalogs ofScl If thedatainTables1and2were,indeed,toformacomplete 1 © American Astronomical Society • Provided by the NASA Astrophysics Data System FROM ANEARVOLUME-LIMITEDSAMPLE = 0.72mag. CASE FORH=42 0 2 between valuesMof-20.8 and-20.4,asusedin§VI. shown, givingtheapexof upper andlowerenvelopestobe galaxies intoaccount,would be drawn~0.7magbelowthat priate lowerenvelopelinein Figure10,takingthefainter nal absorptionratherthanB°/ asinFigure4.Amoreappro- this neglectandalsoinusingBjvaluesuncorrectedforinter- galaxies nearlogv=2.8.Figure10differsfrom 4 in to encompassmostofthepoints,neglectingfournearby similar toFigures4band7bbutwiththeenvelopelinesdrawn metrical about=constantmeanline. factor, makingtheupperandlowerenvelopeabsolutemagni- tudes becomebrighterandfainter.IftheformofO(M)issym- Bt0 of (M)=1conditionineachredshiftinterval is aGaussianwiththeo-(M)dispersionvaluesshownalong 596 data byeye. is reasonable.AfitofFigure 11 curvestotheFigurelbdatais scale values,calculatedfromm—M=5logt;+16.50 ordinate orabscissadoesnotapplytothemagnitudelimit and abscissasoastoenvelopeanyparticulardatasetinthe M, they arezero-pointfree.Theycanbeshiftedbothinordinate second setofenvelopecurvesthatareidenticalwiththosein It canbeshownthatforanyothernormalizationwegeneratea the redshiftintervalofAlogv=0.2centeredat3.9. requires anabsolutevolumenormalizationfactorforO(M). upper curves.TocalculatetheandlowerMvaluesthat of Figure11bothinabscissa and ordinateforabestfittothe shown inFigure12wherewe haveshiftedtheenvelopefamily that theassumptionofasymmetrical GaussianshapetoO(M) Figure 11butwithaconstantdifferenceinthelogvabscissa. For Figure11wehaveassumedthatthereare1000galaxiesin 11 agreeswellwiththelinesdrawn byeyeinFigure10showing 0 0 0 0 0 0 0 0 Fig. 6.—TheHubblediagramusingfullycorrectedBymagntitudesandvelocities,uncorrectedforVirgoinfall.Separateleast-squaresregressions foreach Although thedata pointsdonotfilltheenclosed areapartic- The generalshapeofthecalculatedenvelopelinesinFigure © American Astronomical Society • Provided by the NASA Astrophysics Data System SANDAGE 1 -1 (i.e., inthev-+0limit).Thisvalueofoagreeswith 3.5 (orv=3200kms~).AstudyofthedistributionMin footnote 6again).Clearlyitwillbeofthegreatestinterestto in Tables1and2areexpectedtobemostincomplete(cf. increase ofwithredshiftinthetoppanelsFigures8and 9, large MalmquistbiasimpliedinFigure4a,theapparent directly, whichissurelyanextstepinthisproblem. sample thatisflux-limitedatm=15outtoredshiftsoflogv~ is adirectconsequenceofthe large o(M)ofO(M). bottom panelsofFigure9intheapproximatevolume-limited broadness ofthe^distributionatagivenredshiftin the bottom twopanelsofFigure8usingmethod(2),and the the apparentmultivalued^valuesatagivenredshiftin the residuals inFigure6.Itisthislargeovaluewhichleadsto the shows thatwithincreasingdistancehasnowdisappearedforthetotalsample. Fig. 7.—TheSpaenhauerbiasdiagramofM,logv,similartolb,forthesampleslistedinTables1and2.Top:TheRSAsampleSb Scgalaxiesof © American Astronomical Society • Provided by the NASA Astrophysics Data System M31 .. M81 .. M101. Galaxy (1) Sbl-II 7.86 Sbl-II 4.38 Scl 8.18 Type (2) (3) 0 0.64 0.00 0.07 A (4) Data fortheThreeCalibratingGalaxies 7.79 3.74 8.18 B° (5) t 1 0.29 7.89 0.78 7.01 1.03 2.71 (6) A TABLE 3 597 By (m—M)° ab (7) (8) 29.2 28.7 24.24 Means -20.91 -20.50 -20.81 -21.02 M° Bt (9) -21.51 -21.31 -21.69 -21.53 My (10) -20.85 -21.02 -21.26 -21.04 (11) s -21.88 -21.31 -22.04 M¿, -21.74 (12) C/} 00 00LO : \—I 598 SANDAGE Vol. 331 00
o 4000 8000 12000 16000
V0 Fig. 8.—The apparent Hubble ratio vi/ri calculated by method (2) of the text in which a fixed value is assigned to each galaxy in the sample. Top: The data for Sb, Sbc, and Sc galaxies from the bright flux-limited catalog of the RSA. Middle: Same as the top but for the faint Scl sample from S-T. The mean relation from the top panel is shown as the solid line. Bottom: Same as the middle panel but using the faint Scl sample from Rubin et al (1976). The multivalued Hubble constant at a given distance for the combined sample is an artifact of the analysis that assigns to each galaxy.
the RSA. The original source for these values is either Cepheid variables as taken from a review by Tammann (1987). Holmberg (1958) or de Vaucouleurs (1958), based on Combining columns (5) and (8) gives column (9) for M2t. photographic integrations over the area covered by these large Applying a correction of 0.35 mag to convert the Sbl-II galaxies. Column (4) is the adopted correction for galactic absolute magnitude system to that of Scl galaxies (discussed in absorption7 from column (13) of the RSA. Column (5) is § II and shown in the code to Figs. 2 and 7, based on data in c column (3) corrected by column (4). The adopted mean Table 1) gives the M% *0 values listed in column (11). correction to B°T for internal absorption is in column (6), taken The galaxian absolute magnitudes corrected for the internal from column (14) of the RSA using the precept given on page 8 absorption A1 are listed in column (10), found by subtracting of that catalog. column (6) from column (9), a procedure requiring a comment. The adopted apparent blue distance moduli, corrected for The A1 values are those which are expected to apply to an galactic absorption alone, are given in column (8), based on average taken over the entire galaxy face so as to reduce BT to l what would be observed, J3 r, in the absence of all internal 1 7 The listed value of A0 = 0.64 mag for M31 in the RSA requires special absorption. The A value must not be applied to (m — M)°B to comment. This is obtained from the observed color excess of E(B—V) = 0.16 obtain the true modulus because Cepheids or brightest stars do 1 1 in the Baade-Swope (1963) field IV of M31, giving A°B = 4E(B — V) = 0.64. The not suffer the total A values. For example, A » 0 in the value differs from the cosec equation on p. 8 of the RSA that has been adopted Baade-Swope field IV of M31, hence their (m — M) value for all other galaxies in the catalog. This equation would have given A0 = 0.21 mag for M31. However, the reddening in M31 field is well (corrected for galactic absorption alone) is the true modulus for determined from the Swope photometry that was calibrated photoelectrically. Andromeda. The Cepheids and the brightest stars that have We adopt E(B—V) = 0.16 to be the total reddening due to galactic absorption been observed in M81 and M101 are the brightest found and, alone. This is because field IV is far removed into the outskirts of M31, in the absence of additional information we make the same resembling fields in IC 1613, SMC, LMC, Sextans A, etc., where the internal assumption as for M31 that the (m — M)%B value in column (8) reddening is known to be negligible over the face of most of these galaxies. The 1 statement that the listed value of A0 for M31 in the RSA “appears to be a must not be corrected by the A value (col. [6]) that applies in misprint” (de Vaucouleurs and Corwin, 19866, footnote 2) is, then, not the correcting the BT value to Bj. Justification to take the (m — M) explanation of this entry for M31. value in column (8) as the presently best value for the true
© American Astronomical Society • Provided by the NASA Astrophysics Data System C/} 00 00LO ï—I No. 2, 1988 CASE FOR i/o = 42 599 00 'o a 00 CO00
Fig. 9.—Top: Same as Fig. 8 for the RSA sample alone but now with an upper envelope line and the line for a flux limit of m = 13.0 drawn, (b) Same as top panel but with the S-T sample of Table 2 added, calculated by method (2) which assigns a fixed value to every galaxy. The m = 15.5 flux limit line is shown, (c) Same as panel (b) but using the Rubin et al. Sel sample from data listed in their catalog. Bottom: The RSA sample of the top panel to which the S-T- and the Rubin et al. Sel samples have been added. The m =1 15.5 flux limit line is drawn. Note that the data approach a distance-limited sample for v0 < 5000 km s \ from which a bias-free value of H0 near 50 km s" ^'Mpc" can be deduced if = — 21.2.
modulus that can be defended on principle is not only based on b) Table 3 Values Applied to the Table 1 and 2 data the M31 case (data from field IV), but also from NGC 2403. The Cepheid apparent modulus of (m — M)AB = 27.6 for The most transparent way to use the calibrations of Table 3 NGC 2403 (Tammann and Sandage 1968) has been found to to find H0 would be to apply the column (9) or (11) values to be close to the true modulus (McAlary and Madore 1984) the ridge lines of Figures 5a and 5b (no CIA), and/or columns (actually is somewhat smaller) rather than to be only the (10) or (12) to the ridgeline of Figure 6 (has CIA) as if there was apparent blue modulus as initially claimed by Madore (1976). no bias effect (cf. Sandage and Tammann 1984, 1985, 1986). The magnitudes in column (12) of Table 3 for M31 and M81 The equation of the ridge line in Figure 5a is = 5 log v0 are those listed in column (10) but again corrected by 0.35 mag - 4.85 which, with = -20.81 from Table 3 (col. [9]) -1 -1 brighter to put them on the Scl system. gives H0 = 64 km s Mpc . Using the reduced value in
© American Astronomical Society • Provided by the NASA Astrophysics Data System C/} 00 00LO
00 'o a 00 CO00
LOG v0
Fig. 10.—The M, log v0 bias diagram for the RSA sample (closed circles) and the S-T fainter Scl sample (Triangles). Envelope lines are shown, put by eye. The flux limit line is put at mB = 15.5. A more realistic lower envelope line that encompasses the faintest seven galaxies with log v0 < 3.1 gives MBt (apex) between —20.8 and — 20.4, used in Table 4.
Fig. 11.—Theoretical envelope lines in the M, log v0 bias diagram calculated using symmetrical Gaussian 0)(M) luminosity functions with marked © American Astronomical Society • Provided by the NASA Astrophysics Data System C/} 00 00LO CASE FOR H0 = 42 601 00 'o a 00 CO00
Fig. 12.—Superposition of Fig. 11 on the data of Fig. 4b, showing an apex absolute magnitude of -1 -1 (220) column (11) of —21.04 gives H0 = 58 km s Mpc , again considered, the apex point is at M^ = —21.0 (if H0 = 50). neglecting the bias. Using the data corrected for internal In Figure 12 where the three faintest galaxies are ignored, the absorption with the ridge line of Figure 6 whose equation is apex point is drawn at = —21.4 (again if H0 = 50). 0 1 B ¿ = 5 log v0 — 5.35, and the calibrations of = Because both these values are fainter than the mean values in — 21.51 and = —21.74 from columns (10) and (12), columns (10) and (12) of Table 3, the true value of H0 must be give H0 = 59 and ff0 = 53, respectively. smaller than our arbitrarily assumed value of 50. The range of However, the Malmquist bias is present even in the ridge H0 from these apex values and from the calibration of —21.51 lines of Figures 5 and 6b (see Sandage and Tammann [1974] and —21.74 for from Table 3 is listed in Table 4. The for a similar discussion with an independent calibration of mean value from this section of Table 4 is i/0 = 41.3 as if all and a different correction procedure for the bias, that gave entries using are equally probable. H0 = 57). Merely applying the mean magnitudes from Table 3 Results of the same procedure applied to the data in Figure to these ridge lines gives only an upper limit to H0. 10, uncorrected for internal absorption (Figs. 3 and 5), using With only three calibrators we have no way of knowing the different lower envelopes mentioned in the last section, are where within the scatter of the luminosity function the mean listed in the second part of Table 4 using the two calibrations values in Table 3 lie. Because all three galaxies are nearby, they of Table 3, columns (9) and (11). define an average for a sample as the volume approaches zero. There is no formal way via statistics to put a rigorous error -1 -1 Hence, the Table 3 values must be applied to the data in budget on the final mean value of H0 = 42 km s Mpc . Figures (4), (10), and (12) at the apex of the envelope lines, i.e., in There are, however, reasonable limits on the uncertainties in (1) the Vq —► 0 limit. This apex point is not well defined by our the Table 3 calibration of , and (2) the apex value of M as present data set because the envelope lines can be drawn in if H0 = 50 in Figures 4,10, and 12. several ways that give a range for the apex position. Figures 4, The outside limits of these errors are taken to be 10, and 12 show different ways to encompass the points.8 ~ ± 0.4 mag for each, suggesting a total error of ~ ± 0.6 mag.9 In Figure 4 where the five faint galaxies near log v = 2.8 are 9 To determine the range of error of the envelope fits to Figures 10 and 12, the referee ran Monte Carlo simulations of the M, log v diagram using a 8 Several commentators (Koo, Faber, Green) on an early draft of this paper Gaussian (¡)(M) with © American Astronomical Society • Provided by the NASA Astrophysics Data System 1988ApJ. . .331. .5833 1 4, thenthe0.6maguncertaintycorrespondstoafactorof±1.3 If frombothcalibratorsand 0.81 and0.47usingthe29%errorgivenbyequation(3). where wehavetakentheupperandlowerlimitonQtobe equation (3)gives polation inthistableforthe“dustuniverse”withHTfrom the valuesarelistedelsewhere(Sandage1961a,Table8).Inter- where thelistederroriscombinationofquoted26% uncertainty inHandthequoted13%Tu ian gestationperiod. together withequation(2)givesthen for 47Tueandhavedoubledittheuncertaintyofgalax- where wehaveadoptedtheHesseretal.uncertaintyof1Gyr No. 2,1988 0 age oftheuniversetobe (15) ~1.4Gyr.This,then,takenatitsupperlimitandaddedto 0v 0 the ageofglobularclustersystemGalaxy,gives galaxy gestationtimethatisshorterthan~(0.09) v0 0 0 0v TJ; æ15Gyrfortheageofuniverse,requiringtherebya 0 10 (Dordrecht: Reidel),p.281. Catalog ofBrightGalaxies(Austin:UniversityTexasPress)(RC2). 1 Objects, ed.D.S.Evans(Dordrecht:Reidel),p.353. Ap. J.,278,475. 156,157. Danziger, F.Mateucci,andK.Kjar(Garching:ESO),p.49. AdoptingalesspessimisticerrorbudgettoHbyassigning0.3mag The timeratioHTisafunctiononlyofQifA=0,where Equation (1)fortheHubbletimeofHo=(23.2±6)Gyr Later inthissectionweobtainQ«1byaniterationand 0 0v 0 © American Astronomical Society • Provided by the NASA Astrophysics Data System H 7^=0.64±0.19(3) 0 Tjj =14.9±2Gyr,(2) Qo =1.2í¿;°(4) CASE FORH=42 0 REFERENCES 1 1 Oke, B.J.,andSandage,A.1968,Ap. J., 154,21. Nicoll, J.F.,andSegal,I.E.1982,Ap. J.,258,457. Refsdal, S.,Stabell,R.,anddeLang,F.-G. 1967,Mem.R.A.S.,71,143. .1959,Astr.Nach.,285,1. Misner, C.W.,Thorne,K.S.,andWheeler, J.A.1973,Gravitation(SanFran- Miley, G.K.1971,M.N.R.A.S.,152,477. Melnick, J.,Terlevich,R.,andMoles, M.1987,preprint. McAlary, C.W.,andMadore,B.F.1984, Ap.J.,282,101. Mattig, W.1958,Astr.Nach.,284,109. Matteucci, F.1986,Pub.A.S.P.,98,973. Madore, B.F.1976,M.N.R.A.S.,111,157. Loh, E.D,andSpillar,J.1986,Ap.(Letters),307,Ll. .1987,inIAUSymposium124,ObservationalCosmologyed.A.Hewitt, Leep, E.M,andWallerstein,G.1981,M.N.R.A.S.,196,543. Lasker, B.M.1970,A.J.,75,21. Kraft, R.P.1985,inProductionandDistributionofC,N,OElementsed. I.J. Lambert, D.L.,Sneden,C,andRies,L.M.1974,Ap.J.,188,97. Kraan-Korteweg, R.C,Sandage,A.,andTammann,G.A.1984,Ap.J.,283,24 Kraan-Korteweg, R.C.1986,Astr.Ap.Suppl,66,255. .1936c,Ap.J.,84,517. Kennicutt, R.C.1982,Ap.J.,259,530. .19366,Ap.J.,84,270. Kapahi, V.K.1975,M.N.R.A.S.,111,513. Iben, I.,andRood,R.T.1970,Ap.J.,159,605. Humason, M.L.,Mayall,N.U.,andSandage,A.1956,A.J.,61,97(HMS). Hubble, E.,andTolman,R.C.1935,Ap.J.,82,302. Hubble, E.1936a,Ap.J.,84,158. Holmberg, E.1958,MeddLundObs.,Ser.II,No.136. Hickson, P.1977,Ap.J.,217,964. Hesser, J.E.,Harris,W.E.VandenBerg,D.A.,Allright,B.,Shott,P.,and and commentingonearlydraftsofthemanuscript. leurs, SandraFaber,RichardGreen,DavidKoo,VeraRubin, script forpress.Finally,IamgratefultoGerarddeVaucou- Telescope ScienceInstituteinthefinalizationofmanu- important incoordinatingthepublicationactivitiesofSpace Sidney vandenBergh,andtheunknownrefereeforreading press, forwhichIamgrateful.BarbaraEllerwasparticularly Carl SchuetzandStuartSimpsonoftheSpaceTelescope diagram inFig.1.ItisaparticularpleasuretothankJanet corrections forVirgocentricinfallandtoA.Spaenhauerhis Science Institutegraphicsofficepreparedthediagramsfor Krupsaw forherpreparationofsomanydraftsthispaper. to R.C.Kraan-Kortewegforuseofhercatalogvelocity lems ofbiasasitaffectsthedeterminationH.1amgrateful aration ofthispaper.G.A.Tammannhasdiscussedtheprob- closed withoutaviolationofthetimescaletest,boughtnow, universe innonbaryonichotdarkmatter. however, atthepriceofaccepting99%mass equations (1)and(2)thereisnownoneedtoresurrectA#0 is stilltoolarge.Butclearly,ifoneacceptstheagesfromboth obtain Q=1,suggestingthattheuniversecan,infact,be scale testnowrequiresQ=1;theerrorinequation(1)onH However, toemphasize,itisnotbeclaimedthatthetime Gyr, thenfHo«15givingthepossibilitythatQ=1. with A=0.If,now,H42(Ho23Gyr)andT15 time at19.5Gyr(H=50)therewasnopossibilityforQ1 0 of theglobularclustershadbeenputat17GyrandHubble now atallpossiblefromtheastronomicaldata.Whenage quite soifweareonlyaskingÍ2=1requiredbyGUTis usefully constrainQinequation(4).This,however,isnot assigned errorof26%tofffromequation(1),seemnot 0 0 0 0v 0 0 0 0 cisco :Freeman)Exercise29.6. Danziger, F.Matteucci,andK.Kjar(Garching:ESO),p.21. (KKST). C. Burbidge,andL.Z.Fang(Dordrecht:Reidel),p.251. Stetson, P.1987,Pub.A.S.P.,99,739. It isapleasuretothankmanypeopleforhelpintheprep- These formallimitsonQ,dictatedmostlybythelarge 0 603 1988ApJ. . .331. .5833 — .19856,A.J.,90,1759(STB). .1975a,Ap.J.,196,313. .19616,Ap.J.,134,916. .1986,inInnerSpace-OuterSpace,Proc.FermilabConference,May .1985,inSupernovaeed.N.Bartel,(LectureNotesPhysicsNo.224), .1984,inLarge-ScaleStructureoftheUniverse,Cosmology,andFunda- .1981,ARevisedShapley-AmesCatalogofBrightGalaxies(Carnegie .19756,Ap.J.,197,265. .1972c,Ap.J.,178,25. .19726,Ap.J.,178,1. .1972a,Ap.J.,173,485. .1962,Ap.J.,136,319. 604 .1987,ARevisedShapley-AmesCatalog(2ded.;No.635). Sandage, A.,Bingelli,B.,andTammann,G.A.1985a,J.,90,395(STB). .1986,Ap.J.,307,1. .1982,Ap.J.,252,553. .1975,Ap.J.,202,563. Sandage, A.,andTammann,G.A.1974,Ap.J.,194,559. Sandage, A.1961a,Ap.J.,133,313. Rubin, V.C.,Ford,W.K.,Thonnard,N.,Roberts,M.S.,andGraham,J.A. Rood, R.T.,andCrocker,D.A.1985,inProductionDistributionofC,N,0 Rood, R.T.1978,paperpresentedattheNATOAdvancedStudyInstituteon Robertson, H.P.,andNoonan,T.W.1968,inRelativityCosmology Robertson, H.P.1955,Pub.A.S.P.,67,83. Sandage, A.,Tammann,G.andHardy,E.1972,Ap.J.,Ill,253. Allan Sandage:MountWilsonObservatories,CarnegieInstitutionofWashington,813SantaBarbaraStreet, Pasadena, CA 91101-1292 (Philadelphia: Saunders),§15.7. p. 61. Elements ed.I.J.Danziger,F.Matteucci,andK.Kjar(Garching:ESO), Globular Clusters,Cambridge,England. p. 1. mental Physics,Proc.CERNSymposium,(Geneva:CERN),p.127. Inst. WashingtonPub.635). 7954, (Chicago:UniversityofChicagoPress),p.41. 1976, AJ.,81,687. © American Astronomical Society SANDAGE Provided bythe NASA Astrophysics Data System .19756,Observatory,95,105. .1986,inIAUSymposium136,GlobularClustersSystemsAroundGal- — .1985,inProductionandDistributionofC,N,OElements,ed.I.J. .19606,Ap.J.,131,558. Twarog, B.,andWheeler,J.C.1982,Ap.J.,261,636. Zwicky, F.,andcollaborators.1961-1968,CatalogofGalaxiesClusters Zwicky, F.1957,MorphologicalAstronomy(Berlin:Springer),pp.220-228. Whitford, A.E.1971,Ap.J.,169,215. van denBergh,S.1960a,Ap.J.,131,215. VandenBerg, D.A.,andBell,R.A.1985,Ap.J.Suppl,58,561. VandenBerg, D.A.1983,Ap.J.Suppl,51,29. .1982,Ap.J.,252,37. .1981,Proc.Nat.Acad.Sei.,77,10. .1984,Astr.Ap.,141,407. .1970,Ap.J.Suppl,22,81. Teerikorpi, P.1975a,Asir.Ap.,45,117. Tammann, G.A.,Yahil,andSandage,A.1979,Ap.J.,234,775(TYS). Tammann, G.A.,andSandage,A.1968,Ap.J.,151,825. Tammann, G.A.1987,in1AUSymposium124,ObservationalCosmologyed. Swarup, G.1975,M.N.R.A.S.,111,501. Stebbins, J.,Whitford,A.E.,andJohnson,H.L.1950,Ap.112,469. Spaenhauer, A.M.1978,Astr.Ap.,65,313. Soneira, R.M.1979,Ap.J.{Letters),230,L63. Solheim, J.-E.1966,M.N.R.A.S.,133,321. Sneden, C.,Lambert,D.L.,andWhitaker,R.W.1979,Ap.J.,234,964. Sneden, C.1985,inProductionandDistributionofC,N,OElementsed.I.J. Simoda, M.,andIben,1.1968,Ap.J.,152,509. Segal, 1.1975,Proc.Nat.Acad.Sei.,72,2473. Sandage, A,Tammann,G.A.,andYahil,A.1979,Ap.J.,232,352(STY). axies,^. J.GrindlayandA.G.D.Philip,(Dordrecht:Reidel). Danzinger, F.Matteucci,andK,Kjar(Garching:ESO),p.73. Galaxies, Vols.I-VII(Pasadena:CaliforniaInstituteofTechnology). Danziger, F.Matteucci,andK.Kjar,(Garching:ESO),p.1. Hewitt, C.Burbidge,andL.Z.Fang(Dordrecht:Reidel),p.151.
