1990ApJ. ..349 .... IF -1 hI1 -1 -1 -1 -1 -1 (classical correction300sin/cosb),—117(Yahil,Tammann, correction AV=—100kms.Butotherformulaegive —78 formula (Richter,Tammann,andHuchtmeier1987)gives a center ofthecluster(1225!0+13°00'),bestdetermined mean heliocentricvelocityoftheclusterisnotsameas corrections leadtoa“Hubble flow”velocityoftheVirgo will adoptaprovisionalvalue of300kms.Theseadopted velocity oftheLocalGrouptoward theVirgoCluster.Thereis We adoptAF=-100±25kms. and Sandage1977),—169(deVaucouleursPeters1981). centroid oftheLocalGroupaddsanewuncertainty.At the velocity ofthecentralgalaxyM87,whichagreeswithmean (50 +7X21.6/D)kmsMpc,whereDisthedistanceto still alargeuncertaintyabout theactualvalueofthisinfall.We latter. Correctingthemeanheliocentricvelocityto Virgo Clusterisnotwelldeterminedfortworeasons.First,the Virgo ClusterinMpc.Infact,therecessionvelocityof instance, TammannandSandage(1985)havewrittenH= been atraditionalsteptowardtheHubbleconstant.For Cluster ofF=1300±100km s. values fromBinggeli,Tammann,andSandage(1987) are velocity ofthedwarfellipticalpopulation.Thebestdetermined 1094 +42kmsfortheformerand125810 the The AstrophysicalJournal,349:1-21,1990January20 © 1990.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. 0 The secondreasonisthatwe havetocorrectfortheinfall The determinationofthedistancetoVirgoClusterhas -1 able calibrators,evenwithdistancesnotsowelldetermined. by PierceandTully,jl=30.96.Suchaccidentalerrorsarereducedtheuseoflargestnumberavail- but correctedparametersstillsubjecttosmallvariations,isresponsibleofthelowerdistancemodulusfound Tammann, =31.60,comesonlyfromdifferentdistancesadoptedforthecalibrators,andnot Subject headings:cosmology—galaxies:clusteringdistancesphotometry systems ofcorrectionstotheobservables.Theuseonlythreecalibrators,withgooddistancedeterminations uncertainties isstudiedfromanalmostcompletesampleofspiralgalaxiesbelongingtotheVirgoCluster.The and S'cloudsareshowntolieatsignificantlydifferentdistances. includes allknownsourcesoferrors(accidentalandsystematic).ThecorrespondingHubbleconstantisH= on themagnitudes,whichpreventsapplyingsameuniversalrelationtodifferentdatasamples. sample islimitedinlinewidth;moreover,theslopeofthisinverseregressionstronglydependsonerrors strong influenceofthelimitingapparentmagnitudewhenusingdirectTully-Fisherrelation(allerrors on magnitude)isconfirmed.Theinverseregression(alltheerrorslinewidth)leadstoasimilarbiasif 68 +8kmsMpc,assumingacosmologicalvelocityoftheclusterV—1300100.TheVirgoS 0 THE EXTRAGALACTICDISTANCESCALE.II.UNBIASEDTOVIRGO The behavioroftheB-bandTully-Fisherrelationwithrespecttoobservationalbiasesandparameters The differencewiththehigherdistancemodulusrecentlyobtainedbyKraan-Korteweg,Cameron,and Both regressionsappliedtothecompleteVirgosamplegiveadistancemodulusof31.4+0.2.Theerror © American Astronomical Society • Provided by the NASA Astrophysics Data System I. INTRODUCTION CLUSTER FROMTHEB-BANDTULLY-FISHERRELATION Observatoire deParis,sectionMeudon;andUniversitéParis-Sud Received 1989January3;acceptedJuly17 L. BottinelliandGouguenheim Observatoire deParis,sectionMeudon Observatoire deLyon ABSTRACT G. PATUREE P. Fouqué AND Virgo Cluster.Itistheaimofthispapertoestimatedis- cities. SectionIIIpresentsanewcalibrationoftheblueTully- the completesampledefinedin§IItoderivedistance the to thenearestspiralgalaxies.SectionIVappliesthisrelation to homogenized Hilinewidths,andmaximumrotationvelo- complete sampleofspiralgalaxiesbelongingtothecluster. properties ofgalaxies,theblueTully-Fisherrelation,usinga Fisher relation,basedonrecentdeterminationsofthedistances types, inclinations,correctedasymptoticB-bandmagnitudes, derivation oftheusefulparameters,namelymorphological Hubble constanthasremainedtillnowthedistanceto inverse Tully-Fisherrelations.Anappendixevaluatestheeffect Virgo Cluster,and§Vconsidersbiasesinthedirect and tance fromoneofthebestdistanceindicatorsbasedonglobal nation. the observablestoestimateexternalerrorinourdetermi- on theadopteddistanceofadifferentsystemcorrections to mean by“spiralgalaxies”all thegalaxieswithmorphological type fromSatoSm(towhich wehaveaddedforcompleteness belonging totheVirgoCluster. Hereandinthefollowing,we Nevertheless, themainuncertaintyinthisroutetoward Section IIdescribestheconstitutionofthissampleand We wishtobuildasamplecontaining allthespiralgalaxies II. DEFINITIONOFTHESAMPLE a) PreliminarySample 1990ApJ. ..349 .... IF C hm hm -1 classified Imgalaxies. not massiveenoughtogenerateaspiralpatternandwillbe are mixedtogether.Itmeansthatnospiralgalaxyfainterthan total widthofabout7mag,whenallthemorphologicaltypes peaked at12.8mag(inourBsystem,seebelow)andwitha Virgo ClustercanberepresentedbyaGaussianfunction Vaucouleurs 1959a,Fig.2;Sandage,Binggeli,andTammann Tammann (1985,hereafterBST),whichisclaimedtobecom- about 16thmagexistsintheVirgoCluster.Fainterobjectsare galaxies classifiedamorphousorpeculiar).Thisgoalisachiev- plete downtothe18thmag.Weextractedfromitallspiral galaxies intheVirgoClusterarea,byBinggeli,Sandage,and able becausetheredonotexistdwarfspirals(Shapley1943;de galaxies classifiedastruemembersoftheVirgoCluster:there are 142suchobjects.Weaddedtothissamplesomegalaxies the VirgoCluster.Thechoiceofthe-value7°,instead outside theareasurveyedbyBST,butwithin7°ofcenter classical value6°,isjustifiedbytwofacts:M87,oftentakenas leurs 1961),northeNgroup(Ftaclas,Fanelli,andStruble Vaucouleurs 1961),orwithaheliocentricvelocitygreaterthan two reasons:someofthegalaxiesareinfactbackground members mayinfactbetruemembers.Torecoverthese center areclassifiedbythemastruemembers.Dwarfsofthe the centerof6°,liesinfact0?87fromadopted 1985). Theluminosityfunctionofthespiralgalaxiesin have beengivenintheliterature(deVaucouleurs1961;Paturel objects, andsomegalaxiesclassifiedSabyBSTareprobably morphological type(1Galaxy Reason c) MorphologicalTypes Excluded Galaxies TABLE 1 Vol. 349 1990ApJ. ..349 .... IF No. 1,1990 (hatched parts).InFig.la,thetype0isusedtocodeamorphousandpeculiargalaxies.Figs,lb,1c,Id,symbol“<”inabscissameans thattheabscissa maximum rotationvelocities(Fig.Id)forthe178galaxiesformingcompletesample(totalcolumns)andsubsampleof139retained inourstudy pova (1964),andthenexcluded.Noneofthesesevenobjectsis phological descriptionfromVorontsov-VelyaminovandArhi- fied SabyBST,buthaveanearliertypefromtheothersources. smaller than26°.Theabscissavaluesoficorrespondtoconstantintervalscosi. values refertotheupperlimitofcolumn.Forinstance,firstcolumnFig.lbshowsthateightgalaxiesout139,and14178,have aninclination A fewotherearly-typegalaxies,thatwouldbeexcluded revised type.ThesegalaxiesenterTable1asexcludedobjects. detected intheHiline,whichgivessomesupporttoour earlier) byBSTandImothers.Inthesamespiritas for reported inthenotesofTable2. according toRC2classification,arekeptBSTand One more,notinBST,isclassifiedSO,interpretingthemor- Table 2.Ahistogramofthemorphologicaltypesfor this Virgo Clusterspiralgalaxies,containsthe178objectslisted in in thenotesofTable2. plete sample,deferringourdecision.Theseobjectsarereported background objects,wechosetokeeptheseobjectsinthecom- sample isgiveninFigurelabythetotalcolumns.Compared to similarhistogramsforfieldgalaxies,itappearsthat Sm galaxies areundersampledinthefield. is neededtoevaluatefromits axisratiotheinclinationto line ofsightthepolaraxis thegalaxy.Forthispurpose,we Fig. 1.—Histogramsofthemorphologicaltypes(Fig.la),inclinationslb),absorption-correctedapparentbluemagnitudes1c),andlogarithm ofthe The problemisharderforgalaxiesclassifiedSm(sometimes The completesample,whichweregardasthelistof Aside fromitsusetodefineour sample,morphologicaltype © American Astronomical Society • Provided by the NASA Astrophysics Data System 0 12345678926374654606773798590 d) Inclinations T EXTRAGALACTIC DISTANCESCALE.II. -2 use theclassicalformula(see,e.g.,Hubble1926) intrinsic axisratio(thicknessovermajoraxis).Theparameter morphological typehasledustoareexaminationof this q isobtainedfrompost-RC2logR5(FouquéandPaturel where q=5istheobservedaxisratio(minorovermajor problem, inthespiritofSandage,Freeman,andStokes(1970), axis) atthe25magarcsecisophotallevel,andqis morphological type.Thepreliminaryformulaobtained,which preliminary resultshavebeendrawnfromthisstudy(Mazan using allthegalaxies,andnotonlymostinclined.Two we willadoptinthisstudy,is type isfarfromnegligible,andthemeanvalueofqvaries with 1983, 1985).Thelongdebateaboutthevariationofqwith 1988): theintrinsicdispersionofqwithinamorphological 25 2 most inclinedgalaxiesintheir study,BGPVsampledthetailof 0 BGPV), thevariationwithmorphological typeissimilar,but Compared toanolderformula (Bottinellietal.1983,hereafter 0 our valuesofqareshiftedtoward highervalues.Usingthefive 0 0 0 For peculiarandamorphous galaxies,weadoptq=0.3. 0 -log 4=0.43+0.053Tfor-58.(2) 0 i < 3 1990ApJ. ..349 .... IF © American Astronomical Society NAME VCC566 N4321 N4316 N4313 N4312 N4309 N4307 N4305 VCC479 N4330 N4302 VCC453 U7424 U7421 N4299 N4298 N4294 N4254 U7423 N4293 N4237 N4222 N4216 N4212 N4207 VCC132 VCC124 13229 N4206 VCC87 13239 N4197 1776 13115 13105 Z98-132 N4192 N4180 13099 13094 U7249 U7239 N4178 13074 U7223 N4152 N4082 N4067 N4037 13066 13063 U7200 U7186 U7016 13061 13059 13044 13033 12990 Z69-10 1755 12 2010. 12 2005. 12 2005. 12 1959. 12 1938. 12 1932. 12 1931. 12 2044. 12 2037. 12 2023. 12 2019. 12 1925. 12 1923. 12 1922. 12 1910. 12 1908. 12 1900. 12 18 12 18 12 18 12 18 12 16 12 16 12 1526 12 1501 12 1454 12 1438 12 14 12 14 12 1349 12 1320 12 1311 12 1306 12 57 12 43 12 43 12 33 12 32 12 31 12 22 12 12 12 12 1136 12 1115 12 1107 12 1053 12 1028 12 1015 12 1013 12 0942 12 0848 12 0837 12 0804 12 0242 12 0206 12 0137 11 5948 11 5849 11 5836 11 5700 a (1950)ÔVT 56, 44, 41, 33 22 30 37 04 04 14 0 113843 8 12 4 8 8 11 9 13 9 13 4 14 8 13 8 13 0 8 4 15 8 15 Complete Sampleof178VirgoClusterSpiralGalaxies 36 33 04 51 25 20 25 42 00 18 06 00 34 24 48 58 01 03 57 24 19 17 46 53 53 03 47 18 39 36 52 24 08 06 57 06 14 36 43 42 52 43 41 46 55 53 25 38 40 00 55 00 36 08 43 54 54 13 35 11 58 36 10 45 21 54 51 46 18 10 45 06 17 42 18 42 18 30 44 12 05 01 05 24 03 12 10 23 43 54 03 00 23 05 19 01 15 16 08 30 45 54 18 00 51 54 18 42 57 00 19 00 08 00 07 00 40 48 10 00 1254 1443 2536 1407 1092 1898 2467 2410 1570 1575 1540 1258 1150 5 1136 2132 -162 -158 1979 8 2378 2085 2317 2084 2065 1225 2075 2082 2165 -141 -134 2329 2425 -184 1450 1507 232 8 153 869 851 360 746 157 948 910 732 230 860 138 3 -83 597 263 699 375 623 264 378 -53 931 FOUQUE ETAL. TABLE 2 Provided bythe NASA Astrophysics Data System 84 13. 20 14. 58 13. 52 13. 79 13. 30 10. 90 12. 64 15. 90 12. 90 12. 78 15. 85 12. 72 15. 18 12. 56 12. 59 16. 90 15. 60 14. 69 12. 55 14. 32 10. 32 13. 70 14. 65 11. 90 16. 41 14. 51 12. 83 14. 33 14. 84 13. 81 11. 49 11. 59 13. 84 12. 76 15. 90 14. 34 16. 86 14. 46 14. 24 13 62 14. 84 13 82 15 23 15 78 10 60 15 72 13 74 11 36 14 50 14 40 12 54 14 46 13 83 14 30 12 59 14 66 13 90 13 90 15 64 14 0 16. .47 .41 .68 .79 .21 .60 .35 .65 .75 .98 .61 .43 .98 .80 .18 .79 .70 .69 .63 .32 70 24 07 73 80 04 08 50 59 79 70 20 a 10 30 50 98 88 10 a 00 j 01 c 43 a 80 a 60 j 53 a 35 21 60 44 60 90 91 41 76 48 86 46 81 72 40 30 00 S WQlogVBc 205Mt 266 219 266 320 219 266 278 226 263 234 110 142 117 152 265 187 127 375 145 106 381 101 115 242 135 229 282 206 291 540 163 308 108 280 135 131 475 277 231 269 241a2.177 214 113 352 144 125 95 46 38 58 77 93 95 261 206 244 308 223 a 249 230 d 161 a 134 135 371 106 a 133 a 194 a 358 b 117 a 247 208 220 106 208 259 518 275 265 187 140 122 116 209 454 243 320 114 131 195 a 76 89 37 c 75 c 37 a 97 a 88 a 28 a 20 97 79 a 88 a 95 b 58 a d c a a a b a d a 2.140 2.047 2.337 2.234 2.266 2.062 1.567 1.943 1.946 2.296 1.591 1.753 1.540 1.932 1.973 1.943 2.145 1.632 1.967 1.940 2.396 2.191 2.090 1.990 1.957 1.973 2.113 2.069 2.077 1.795 1.741 2.340 1.704 2.195 2.068 2.152 1.674 1.925 1.770 1.785 1.843 1.924 1.467 ,647 ,815 216 199 107 484 541 12.53 14.29 11.34 10.45 11.56 14.22 14.24 12.40 14.84 13.36 11.96 13.88 14.19 15.06 11.34 12.91 11.62 12.61 11.45 15.92 11.70 13.27 13.76 13.81 12.00 13.73 13.93 12.75 13.25 11.18 12.80 10.05 11.70 14.46 14.10 15.89 13.13 14.15 15.54 12.30 14.25 13.19 14.54 15.11 12.43 13.66 14.64 14.06 12.16 13.96 13.60 12.47 13.21 10.85 14.00 12.73 12.45 13.76 9.56 9.91 9.72 Vol. 349 1990ApJ. ..349 .... IF No. 1,1990 © American Astronomical Society C FAME a(1950)8VTiBSWoQlogV T205M VCC740 *12 N4343 12 N4336 12 N4470 12 N4451 12 N4445 12 VCC989 12 N4390 12 N4388 12 N4383 12 N4380 12 N4356 12 N4344 *12 N4351 *12 N4353 12 N4492 *12 N4491 12 N4466 *12 N4450 12 U7579 12 N4440 12 N4438 *12 N4430 12 VCC952 12 N4411A *12 N4405 *12 N4402 *12 N4396 12 N4394 *12 N4376 *12 U7590 12 N4424 *12 N4423 12 N4419 12 N4411B *12 N4416 12 N4413 12 U7567 12 U7557 12 N4498 12 13322 12 13311 12 13298 12 13268 *12 13267 12 13259 12 N4502 12 N4501 12 13418 *12 1796 *12 13392 12 13391 12 13371 12 13355 *12 I3322A 12 13258 12 13446 12 13414 12 13365 *12 13356 *12 1797 12 25 46.2900 25 22.8559 20 58.11942 28 24.41145 27 11.41140 27 05.3805 26 12.01516 26 08.1932 25 58.01721 25 55.61841 25 43.0942 25 21.21234 21 42.1849 21 27.4803 21 16.3727 21 06.11749 21 05.0713 28 52.21146 28 27.4821 26 58.0758 26 56.2702 26 54.81640 25 13.51317 24 56.4755 24 53.6632 24 49.21108 23 17.61044 23 09.9729 23 00.61232 22 53.91644 22 49.61017 22 32.417 22 07.2846 21 34.7653 21 32.8719 21 29.51229 21 11.91245 24 45.0756 24 40.21610 24 39.0941 24 38.4732 24 36.2609 24 25.11519 24 21.61150 23 35.81627 23 27.51556 23 24.71829 23 21.3749 23 14.81256 22 45.1601 29 08.81707 24 22.81009 24 18.01327 24 14.7909 24 14.0811 23 59.71253 23 56.1909 23 35.813 29 22.91524 29 32.41657 29 28.11441 EXTRAGALACTIC DISTANCESCALE.II. 05 8645 48 3633 54 22526 23 14435 23 -4319 11 861 24 9339 13 11035 49 17101 06 11375 01 23065 44 11258 58 14209 05 11419 58 10133 16 11341 40 16873 40 19563 32 17035 18 11199 10 7231 18 8749 36 9285 23 11067 28 -2741 26 17471 22 2353 30 9163 55 11956 18 25133 36 9976 12 -1425 06 11389 30 24545 42 8759 00 7279 10 12319 42 9 22 7592 01 12847 55 -1278 33 9712 00 12519 13 17721 35 4971 56 23405 50 5348 50 1556A 54 18469 30 23519 51 4561 18 9855 18 11009 12 -109 11 1022 38 12696 44 13937 00 20978 46 15047 50 22783 48 16269 TABLE 2—Continued Provided bythe NASA Astrophysics Data System 79 15.30b 27 12.78a 28 12.48b 20 11.76a 78 14.14b 41 13.27b 80 14.55b 84 14.04b 78 13.30a 52 13.32a 48 10.86a 40 14.04b 82 11.81a 59 12.92a 57 13.69b 74 14.91h 44 12.90a 54 14.00a 65 13.61b 47 13.04b 90 14.74c 90 13.60a 83 10.82a 75 14.17b 16 13.58e 90 13.76b 62 12.70a 69 15.70j 33 14.22b 38 13.31a 71 14.75h 18 13.20a 50 14.00j 67 13.30c 69 14.85d 90 15.32c 65 12.36a 81 14.28b 23 12.89b 52 12.99b 80 12.56a 67 14.30a 75 14.62b 58 13.70b 66 14.23b 77 12.05a 52 16.50j 58 14.49e 30 13.50a 90 13.09a 61 13.50a 73 15.46a 48 12.97b 58 12.62b 14 12.93a 50 14.01c 61 10.13a 64 14.57b 0 15.80j 0 14.20a 0 13.21b 205 a 143 129a 296 280a 221 206a 293 b 230 224c 239 207a 286 250a 170 161a 381 376c 292 278b 118 104b 184 169c 178 165a 351 326a 228 208a 160 a 251 237c 190 182c 171 d 126 115b 105 80a 215 b 202 187b 180 162a 144 124a 155 150a 147 85c 156 139b 196 172a 126 89a 150 133a 306 280a 138 109a 151 118a 153 124c 183 146a 159 136a 171 145b 167 154a 127 103c 131 104a 105 96b 148 129a 532 502a 197 183a 181 171a 72 59a 75 59a 70 47a 80 67a 36 28d 86 65c 62 38b 86 80b 345 c 2.229 10.63 2.091 12.87 2.188 12.21 2.096 12.59 2.282 11.28 1.907 12.72 2.157 12.23 1.700 14.86 1.791 12.33 1.941 13.37 2.069 11.97 1.913 13.05 1.821 13.41 1.996 11.97 1.889 13.95 1.893 13.70 1.871 13.51 1.982 14.17 2.205 9.66 2.074 11.40 1.839 13.02 1.768 12.78 2.061 12.70 2.235 10.27 1.973 12.44 1.760 14.02 1.780 14.16 2.325 13.10 1.874 11.94 2.165 12.70 1.877 12.45 1.835 13.58 2.025 11.01 2.109 12.45 2.176 12.39 1.762 12.99 1.778 12.37 1.943 13.64 1.625 13.46 1.884 13.50 1.736 13.21 1.931 12.39 1.826 13.87 1.737 13.03 1.932 12.50 1.829 13.11 1.427 15.89 1.510 13.82 2.436 9.42 1.852 13.41 1.957 11.96 1.893 13.84 12.75 12.87 13.73 12.28 14.55 15.33 11.61 12.79 13.41 5 1990ApJ. ..349 .... IF 6 © American Astronomical Society • Provided by the NASA Astrophysics Data System C NAME a(1950)8VTiBSWQlogV N4746 12 N4758 12 N4698 12 N4694 *12 U8032 12 N4689 *12 N4651 12 VCC1933 12 N4654 12 N4647 12 N4641 12 N4639 *12 N4634 *12 N4633 *12 N4607 12 N4606 *12 VCCI725 12 N4595 12 N4584 *12 N4580 12 N4579 12 N4571 12 VCC1675 12 13742 12 13718 12 U7802 12 U7795 *12 N4569 *12 N4568 *12 VCCI644 *12 VCC1605 12 N4561 12 N4567 *12 N4548 12 N4540 *12 N4539 12 N4523 *12 U7739 *12 N4535 12 N4532 *12 N4531 *12 N4522 *12 N4519 12 13617 *12 13611 12 T205eM 13589 *12 N4506 12 13583 *12 13576 *12 13521 12 13520 12 13517 12 1800 12 13483 12 13476 12 13466 12 52 12.01330 50 14.81607 49 25.21221 45 51.88 45 44.01115 45 15.31402 43 01.21336 42 15.01237 41 25.71323 41 12.51640 41 01.11151 40 36.01219 40 21.71331 40 12.6736 40 09.71434 40 06.61437 38 40.91209 37 20.91534 36 52.8814 36 32.41338 35 48.6809 38 26.41211 35 46.41323 35 15.6538 35 12.61205 35 12.0722 35 09.0850 34 29.9711 34 25.51429 34 18.71326 34 12.51332 34 04.8653 34 02.4819 34 01.11132 34 03.01130 33 38.41935 33 21.01408 32 55.11446 32 07.0726 32 42.61042 32 19.91549 32 12.1634 32 04.41828 32 00.01346 31 47.9828 31 46.7644 31 25.81537 31 17.21526 29 39.31341 29 33.01205 31 58.8925 31 44.61321 31 07.8927 30 58.1855 30 38.61137 30 10.81419 28 1176A 00 5 10 12433 18 17813 37 10072 24 9658 21 14165 24 20129 13 16185 56 9804 36 850A 58 10366 05 8045 42 24092 13 1576 23 6367 47 2898 36 22573 08 16641 00 10689 12 20839 18 2738A 54 17885 06 17371 38 10341 40 15213 42 619 59 16359 33 3427 18 -2282 01 11249 48 10769 45 22334 20 4883 24 10775 25 19595 48 1795P 01 21864 00 7569 41 20539 13 5969 40 12871 56 14038 41 12866 54 7996 54 4349 06 81 38 2608 02 23256 48 12197 43 20079 51 23354 31 1293 36 -1719 51 6801 36 902P FOUQUE ETAL. TABLE 2—Continued 77 13.87d 83 13.67e 77 13.34d 51 11.55a 77 13.68c 28 15.80j 37 11.60a 56 11.12a 53 11.31a 38 12.16a 32 14.12c 52 12.21a 64 12.05a 66 13.86c 76 13.20a 73 13.80a 48 12.92b 59 13.85c 77 14.99d 49 13.72c 42 12.61d 39 10.40a 41 14.54e 58 14.51g 23 14.02a 27 14.62a 90 13.70a 63 12.70a 63 14.30a 56 14.11i 30 11.70a 26 14.42a 66 10.12a 48 14.47i 44 12.10a 37 10.85a 37 12.49b 48 13.98b 73 12.80a 42 14.87i 63 13.91c 66 11.70a 32 12.95a 90 17.00j 53 14.51e 72 12.30a 41 14.05b 78 12.73b 37 12.30a 37 15.70a 45 14.78i 90 17.50j 44 10.39a 49 12.58b 37 13.20a 46 13.64c 204 181a 417 412a 280 253d 207 185a 235 215b 351 331a 117 86b 198 179b 176 165a 304 284a 306 276a 385 358a 192 181c 177 157a 196 178c 363 359a 194 146c 153 150b 130 95a 100 70c 106 69a 112 90a 370 340a 118 104a 265 225a 173 154a 294 267a 227 156b 233 201a 209 165a 148 121b 131 111b 123 74a 127 84b 107 c 109 c 135 121a 112 95b 140 99c 141 118a 75 57a 55 38a 87 63c 68 56a 59 45a 86 78a 56 17c 94 82a 67 41a 12.84 2.190 12.29 2.380 10.94 2.103 11.07 2.189 10.47 2.321 10.69 2.085 11.62 2.208 11.60 1.899 12.51 1.650 11.30 1.397 12.63 2.074 12.18 2.054 12.04 2.404 9.85 2.116 11.20 2.243 9.35 1.870 13.08 1.374 13.61 1.670 15.30 1.924 12.87 1.971 12.52 2.244 10.32 1.893 11.96 1.898 12.34 1.700 13.57 1.593 13.18 1.839 13.96 1.644 13.99 1.613 13.84 1.679 13.46 1.686 13.19 1.615 13.53 1.491 15.84 1.896 11.96 1.937 14.12 2.243 9.82 2.058 13.93 2.110 11.77 1.997 12.44 1.529 16.34 1.758 13.40 1.765 14.32 1.703 13.89 1.929 11.40 1.782 13.49 1.966 11.64 1.901 15.17 1.929 12.67 1.417 14.21 13.11 10.91 13.89 11.53 11.87 11.99 13.06 Vol. 349 1990ApJ. ..349 .... IF No. 1,1990 These inclinationsarelabelediinTable3. comparing totheformulausingq=0.2independentoftype. than thepreviousone,andsameresultisobtainedwhen q. Therevisedformulaleadsgenerallytohigherinclinations the distributionofq,andunderestimatedmeanvalues Notes.—Key togalaxiesdenotedbyasteriskfollows: (1988, meanaxialratioofCCDisophotesoverthedisk:iin intensity profilesalongellipses:iinTable3),PierceandTully mic spirals:iinTable3).Thereaderisreferredtothesepub- retained fivesourcesforourVirgosample:Grosbol(1985, mean oftheHicatalog(Bottinellietal.1989). lications fordetailsaboutthemethods.InTable3,welist velocity fields:iinTable3),andDanver(1942,fitoflogarith- Table 3),Warmels(1988)andGuhathakurtaetal(Hi results fromthesefourmethodsandoursfor56spiralsin erbork (Warmels1988a)andattheVLA(Guhathakurtaetal. Virgo. ThemeasuresfromHivelocityfieldsmadeatWest- R al. 1981andrejectstheHiprofileofHuchtmeier,Tammann,Wendker 1976. 0 0 0 by CorwinfromPalomarSkySurveyglasscopiesandSmIV:BST. rejected. 4548), wehaveretainedthemeaninclination. (19866) rejected. son; forthetwogalaxiesincommon(NGC4501andNGC Arecibo (V=2332;Lewis,Helou,andSalpeter1985),isnotunderstood. Therefore, NGC4424isconsideredhere undetected. Huchtmeier andRichter(1986n)possibly referstoadwarfcompanionlocated0:6south.Thiswouldexplainwhy thesignalisstrongeratEffelsbergthanArecibo. T G 1988) havebeenputtogetherforthepurposesofcompari- D H 19866) andrecentlyatArecibo{V=733;Hoffmanetal.1988).Thehighopticalvelocity(V2261;Huchra1983),confirmedbyaprevious Hidetectionat + W Other methodsexisttomeasureinclinations,andwehave NGC 4152:possiblememberoftheMgroup(Ftaclas,Fanelli,andStruble1984;Huchra1985),butoutsideourlimits. NGC 4178:HimeasurementsofDaviesandLewis1973rejected. UGC 7186:DDO112;classifiedImbyFisherandTullyfromPalomarSkySurveyprints(RC2,T=10±1.2);reclassifiedSminRC3. NGC 4180:HilinewidthsmeasuredbyHuchtmeierandRichter19866Hoffmanetal.1988disagree,butnoneisconvincing,sowehavekept theweighted UGC 7223:positionandvelocitywouldputitintheMgroup,butBSTgiveasamemberofVirgoCluster. NGC 4192:weadopti=78°fromDanver1942andvisualinspection. VCC 87:classifiedImIVbyCorwinfromPalomarSkySurveyglasscopies(deVaucouleursand19866)SmIIIBST. NGC 4197:positionandvelocitywouldputitintheWcloud,butBSTgiveasamemberofVirgoCluster.HimeasurementsDavis Seaquist1983 NGC 4376:classifiedImIVbyCorwinfromPalomarSkySurveyglasscopies (deVaucouleursandCorwin19866)ScdIIIbyBST. VCC 740:classifiedImIVbyCorwinfromPalomarSkySurveyglasscopies (deVaucouleursandCorwin19866)SmIIIbyBST. IC 3268:classifiedImIV?byCorwinfromPalomarSkySurveyglasscopies (deVaucouleursandCorwin19866)ScIII-IVorSmIIIbyBST. NGC 4351:anewobservationofthisgalaxywiththeNançayradiotelescope confirmstheHiparametersofHelou,Salpeter,andKrumm1979,Helouet NGC 4344:classifiedSObyCorwinfromPalomarSkySurveyprints(RC2, T=—2±1.0)andSpec/BCDbyBST. NGC 4309:classifiedS0bydeVaucouleursfromMountWilson2.5mplate (RC2,T=—1±0.8)andSapecbyBST. UGC 7421:classifiedImVbyCorwinfromPalomarSkySurveyglasscopies (deVaucouleursandCorwin19866)SmIVbyBST. IC 3059:DDO115;classifiedImbyFisherandTullyfromPalomarSkySurveyprints(RC2,T=10±1.1)SdBST. UGC 7249:DDO114;classifiedImbyNilsonfromPalomarSkySurveyprints(RC2,T=10±1.2)andScBST. NGC 4405:classifiedSO/abyCorwin fromPalomarSkySurveyprints(RC2,T=0±1.1);classifiedSc/SOby BST.HimeasurementsofHuchtmeierandRichter NGC 4402:theBvalueisfromdeVaucouleursandPence19796. NGC 4394:HimeasurementsofDavisandSeaquist1983atAreciborejected. NGC 4298/4302:thispairofgalaxiescanbeseparatedonlyatArecibo.So weonlyrelyonArecibomeasurements. NGC 4294/4299:thispairofgalaxiescanbeseparatedonlyatArecibo.So weonlyrelyonArecibomeasurements. NGC 4212:Hiparametersuncertainduetogalacticconfusion. IC 3063:positionandvelocitywouldputitintheMgroup,butBSTgiveasamemberofVirgoCluster. VCC 132:positionandvelocitywouldputitintheMgroup,butBSTgiveasamemberofVirgoCluster. VCC 124:positionandvelocitywouldputitintheMgroup,butdeVaucouleursCorwin19866giveasamemberofVirgoCluster.Itis classifiedImV NGC 4411A/B:thispairofgalaxiescan beseparatedonlyatArecibo.SowerelyonArecibomeasurements. NGC 4237:HimeasurementsofChamaraux,Balkowski,andGérard1980rejected. IC 3099:positionandvelocitywouldputitintheMgroup,butBSTgiveasamemberofVirgoCluster. NGC 4216:weadopti=81°fromDanver1942andvisualinspection. NGC 4424:detectionattributedto NGC 4424byHuchtmeier(1982),Giovanardi,Krumm,andSalpeter (1983),Helou,Hoffman,andSalpeter(1984), IC 3355:DDO124;classifiedImIV-V byCorwinfromPalomarSkySurveyglasscopies(deVaucouleursand Corwin19866)andSmIIIbyBST. NGC 4254:possiblememberoftheMgroup(Ftaclas,Fanelli,andStruble 1984),butoutsideofourlimits. IC 3115:anewobservationofthisgalaxywiththeNançayradiotelescopeconfirmslowvelocitymeasuredatEffelsberg(K=731;Huchtmeier andRichter IC 3356:classifiedImVbyCorwinfrom PalomarSkySurveyglasscopies(deVaucouleursandCorwin19866) andSmIVbyBST. NGC 4293:classifiedSO/abydeVaucouleursfromMountWilson2.5mplate (RC2,T=0±0.7)andSapecbyBST. IC 776:possiblememberoftheWcloud(deVaucouleursandCorwin19866; PierceandTully1988),butoutsideofourlimits. IC 3365:classified Im IV-VbyCorwinfromPalomarSky Surveyglasscopies(deVaucouleurs andCorwin19866)ScdIIIby BST. T IC 3074:backgroundobjectaccordingtoPierceandTully1988,butBSTgiveitasamemberoftheVirgoCluster. © American Astronomical Society • Provided by the NASA Astrophysics Data System EXTRAGALACTIC DISTANCESCALE.II. Notes forTable2 method, whichleadstoinclinationstoosmallinthemeanby inclinations, andweadoptthemeaninclinationsfromfour § IVc.Therefore,weprefertoexcludetheHivelocityfields formula (BGPV)ortheconstantvalueq=0.2onother shifts fromonetoanother,exceptfortheHivelocityfields (through theirformulawhichusesaconstantq=0.2)under- other methodsforthe56galaxiesinTable3(labeled<¿».The other methods.Apossibleexplanationofthiseffectisgivenin 6°, whencomparedtothemeansystemdefinedbyfour estimate trueinclinationsby3°.Thesameeffectisobtained Huchra (1980)thatinclinationsderivedfromaxisratios hand, isrespectively,3°and2°.Thisgivesapossibleexplana- derived throughformula(1)adoptingqfrom(2)on about 5°. accuracy ofthefivemethodsiscomparable,withanrms tion fortheeffectfirstobservedbyAaronson,Mould,and one hand,andinclinationsobtainedadoptingtheolder 0 0 0 A comparisonamongthesefivemethodsgivesnegligible Over thissample,themeandifferencebetweeninclinations 7 1990ApJ. ..349 .... IF _1 // NBG), whicharecorrectedforthe3°effect. new formulafortheothers.Ahistogramoftheseinclinationsis Table 3,andinclinationsderivedfromaxisratiothroughour tions fromthefourmethodsretainedfor56galaxiesin nations intheNearbyGalaxiesCatalog(Tally1988,hereafter couleurs andPence(1979a).Asthecorrectionsappliedtothese accurate Bmagnitudesderivedfromaperturephotometryby these inclinationsarerandomlydistributed. given bythetotalcolumnsinFigurelb.Abscissavaluescorre- have kepttheweightedmeanofHicatalog(Bottinellietal1989). estimate (m=14.9)bydeVaucouleurs,andButa1981, which givesB=15.16. value, wehavepreferredthereductionofZwickyestimatebydeVaucouleurs andPence19796(B=14.54)thanthereductionofFisherTully1975 Ames, Reiz,andZwicky,reducedtotheBsystembydeVau- For theremaining,wehavetousevisualestimatesmadeby R. ButaforRC3.Themeanerroronthesemagnitudesis0.10. size ofthesampleineachsubclass,itcanbeconcludedthat spond toconstantintervalsofcosi.Takingintoaccountthe Pierce andTally(1988),whoseinclinationsagreewiththeincli- mean ofZwickyandReizestimates,reduced accordingtodeVaucouleursandPence1979a;a=0.14mag.(e) Zwicky estimate,reducedaccordingtodeVaucouleurs former classification. 8 Vaucouleurs andPence1979a;a= 0.11 mag.(c)meanofZwickyandAmesestimates,reducedaccordingto deVaucouleursandPence1979a;o=0.12mag.(d) from anestimatedm=14.9byFisherandTully1975. confusion. WeonlyrelyonthemeasurementsofHelouetal1981. according todeVaucouleursandPence 1979awithoutdiameterinformation;o=0.37mag.(j)BSTestimate,not reduced;a=0.50mag. of ZwickyandAmesestimates,reduced accordingtodeVaucouleursandPence1979awithoutdiameterinformation; a=0.30mag.(i)Zwickyestimate,reduced a =0.21mag.(g)meanofZwickyand Reizestimates,reducedaccordingtodeVaucouleursandPence1979a without diameterinformation;o=0.27mag.(h)mean and Pence1979a;o=0.20mag.(f)mean ofZwicky,Ames,andReizestimates,reducedaccordingtodeVaucouleurs andPence1979awithoutdiameterinformation; (de VaucouleursandCorwin19866)SmIIIbyBST. inspection ofthephotographpublishedbyMargonetal1972. be separated.SowehaverejectedallHimeasurementsforbothobjects.Recently,NGC4568hasbeenstudiedattheVLA(Guhathakurtaetal 1988),butwitha Vaucouleurs andCorwin19866)SmIVbyBST. Huchra etal.1983isF=8kms,sotheHiprofileprobablyconfusedbyourGalaxy. 1981 rejected. and Corwin19866)SmIIIbyBST.HimeasurementsofFisherTully(1981)rejected. spatial resolutionof45,sobothgalaxiesarestillnotseparated. of a15.2magstarsuperposed. and Corwin19866,thegalaxyisinfactIC35913589anearby14thmagstar.TheZwickyestimateformagnitudegivenheremayinclude acontribution T pg T T T pg _1 -1 The adoptedinclinationsinTable2arethemeaninclina- NGC 4639:HimeasurementsofHuchtmeier,Tammann,andWendker1976 rejected. NGC 4633/4634:thispairofgalaxiescanbeseparatedonlyatArecibo.So weonlyrelyonArecibomeasurements. NGC 4606:HilinewidthsmeasuredbyGiovanardi,Krumm,andSalpeter 1983andHelou,Hoffman,Salpeter1984disagree,butnoneisconvincing,sowe IC 3617:DDO140;classifiedImbyNilsonfromPalomarSkySurveyprints (RC2,T=10±1.2)andSmIII/BCDbyBST.HimeasurementsofFisherTully NGC 4584:detectionofHaynesandGiovanelli1986ismarginal;therefore, theydonotgiveHilinewidth. UGC 7795:DDO139;classifiedImVbyCorwinfromPalomarSkySurvey glasscopies(deVaucouleursandCorwin19866)SdmIVbyBST.FortheB Almost halfofthegalaxiesinourcompletesamplehave NGC 4694:classifiedSObydeVaucouleursfromMountWilson2.5m plate (RC2,T=—2±0.8)andAmorphousbyBST.TheHiprofiledisagreeswiththe NGC 4689:HimeasurementsofChamaraux,Balkowski,andGérard1980 rejected. NGC 4466:HimeasurementsofHuchtmeierandRichter19866rejected. IC 796:HimeasurementsofGiovanardi,Krumm,andSalpeter1983rejected. NGC 4438:classifiedSO/abydeVaucouleursfromMountWilson2.5mplate(RC2,T=0±0.7)andSb(tides)BST.Hiparametersuncertain duetogalactic NGC 4522:HimeasurementsofFisherandTully1981rejected. NGC 4492:HimeasurementsofHaynesandGiovanelli1986rejected. IC 3418:DDO130;theB=14.0estimateofBSTisuncertainanddisagreeswithvalue15.13deVaucouleurs,Buta 1981reduced Sources ofB—.(a)fromaperturephotometry(R.Buta,RC3);<7 = 0.10mag.(b)meanofZwicky,Ames,andReizestimates,reducedaccordingtode NGC 4532:classifiedImIII:bydeVaucouleursfromMountWilson2.5mplate(RC2,T=10±0.8),SmIIICorwinPalomarSkySurvey glasscopies NGC 4531:Helou,Hoffman,andSalpeter1984seemtohavedetecteditatK=195kms,buttheydonotgiveHilinewidth;moreover,theoptical velocityof NGC 4523:DDO135. IC 3576:DDO138;fortheinclination,wehaveadoptedvaluederivedfromaxisratio(i=23°)andrejectedofGrosbol1985, i=58°,after NGC 4567/4568:backgroundobjectsaccordingtoHuchra1984,butBSTgivethemasmembersoftheVirgoCluster.EvenatArecibo,thispair galaxiescannot VCC 1644:classifiedImVbyCorwinfromPalomarSkySurveyglasscopies(deVaucouleursand19866)SmIVBST. NGC 4540:HimeasurementsofHuchtmeier,Tammann,andWendker1976rejected. UGC 7739:DDO137;classifiedImVbydeVaucouleursfromMountWilson2.5mplate,CorwinPalomarSkySurveyglass copies(de NGC 4569:meanHivelocitygivenheredifferssignificantlyfromtheopticalV=—313kms. IC 3583:classifiedImbydeVaucouleursfromPalomar5mplate(RC2,T=10+0.8),SmIVCorwinSkySurveyglasscopies(de Vaucouleurs IC 3589:classifiedImV:byCorwinfromPalomarSkySurveyglasscopies(deVaucouleursand19866)SmIIIBST.Accordingto de Vaucouleurs T T tT © American Astronomical Society • Provided by the NASA Astrophysics Data System e) B-BandTotalMagnitudes Notes toTable2—Continued FOUQUÉ ETAL. can thusbeusedasqualitycodes. racy ofthecorrespondingsource(abest,jworst),and 2. Thesecodes(fromatoj)areorderedaccordingtheaccu- mag. Detailsaboutthesourceofmagnitudeforeachgalaxy are giveninTable2,withcodesexplainedthenotesof Zwicky andHerzog1963),forthesewehaveretainedthe meter andaxisratio.Thisisgenerallythecase,mean catalog (Nilson1973),whichgivesagoodestimateofthedia- uncorrected estimatesofBST,withameanrmsabout0.5 to theCGCGcatalog(Zwicky,Herzog,andWild1960; the meanrmsisabout0.33.Finally,14galaxiesdonotbelong rms isthen0.13.For13galaxieswithoutdiameterinformation, reduced magnitudeisbetterifthegalaxybelongstoUGC estimates dependonsurfacebrightness,themeanerror the RC2valueA=0.19magforVirgoCluster.Different The debateaboutthevalueofGalacticabsorptionforhigh nal absorption.TheK-correctionforredshiftisnegligiblehere. Galactic latitudeobjectsisnotclosed,andwewilladopthere B The BmagnitudesmustbecorrectedforGalacticandinter- t Vol. 349 1990ApJ. ..349 .... IF No. 1,1990 for inclinationslessthan80°,andA=0.96higherinclina- tions. Thisrelationgivesthetotalcorrection,asinSandage rection ofTullyandFouqué(1985,hereafterPaperI),givenby the Appendix.Forinternalabsorption,weadoptcor- assumptions aboutthisandothercorrectionsarepresentedin t tion isthereforeonly tentative. nations inbrackets haveonlytwodiscrepantmeasurements; adoptedinclina- N4037 0.87 N4192 0.26 N4178 0.37 N4152 0.79 N4212 0.62 N4206 0.24 U7249 0.42 N4237 0.66 N4222 0.20 N4216 0.22 13074 0.21 N4305 0.57 N4298 0.58 1776 0.62 N4254 0.88 N4380 0.58 N4321 0.80 I3322A 0.16 N4351 0.72 N4450 0.79 N4411B 1.00 N4413 0.66 N4411A 0.93 N4402 0.34 N4394 0.96 N4388 0.29 N4540 0.81 N4535 0.76 N4531 0.66 N4523 0.97 N4522 0.30 N4501 0.55 N4498 0.59 N4492 0.99 N4451 0.64 N4440 0.90 N4430 0.88 13356 0.60 N4571 0.91 N4568 0.48 N4567 0.76 N4548 0.84 N4532 0.43 N4519 0.86 N4606 0.51 N4580 0.76 N4579 0.84 N4569 0.47 13576 0.93 N4654 0.59 N4651 0.66 N4647 0.84 N4639 0.66 13617 0.57 N4689 0.82 N4698 0.73 Note.—Explanations ofcolumnsgiven in§lid.Galaxieswithmeanincli- Name qTi 250HDGTR © American Astronomical Society • Provided by the NASA Astrophysics Data System (1 ^-0.55seci\ 05seci Inclinations ofVirgoSpirals 0.25(1+e--) +———,(3) 4 0.23 9 0.42 4 0.2341... 4 0.23...78 2 0.297078 7 0.1662... 3 0.26 5 0.20 4 0.233031 7 0.16...86 8 0.42 2 0.29 5 0.20 6 0.18 5 0.20...29 3 0.266381 9 0.42 6 0.18 2 0.29 6 0.18 9 0.42 3 0.263753 7 0.16 3 0.26...82 3 0.26...21 3 0.26...81 5 0.20 4 0.234367 4 0.23...45 6 0.18 6 0.18 7 0.16...26 3 0.265764 7 0.16 5 0.20 5 0.20 4 0.23 9 0.42 5 0.204042 8 0.42 1 0.33 9 0.42 7 0.16 2 0.295664 3 0.2638... 6 0.1849... 3 0.2636... 2 0.29 5 0.2042... 5 0.2036... 1 0.33 1 0.33 1 0.33 5 0.2027... 1 0.33 1 0.33 TABLE 3 EXTRAGALACTIC DISTANCESCALE.II. 49 4454 41 ...3940 ... 869078 30 ... 44 ...60[52] 40 2932 ... 90 ... 908584 ... 749082 ... 787074 ... 9081 48 454144 43 ...4644 25 3830 55 ...56 ... 575355 8384 42 353437 45 ...5349 45 443237 28 ...0[14] 43 ...5248 51 ...... 6184[72] 28 ...9[18] ... 5152 27 ... 27 ...2928 ... 536258 7780 22 ...1820 ... 8382 ... 90 58 6159 41 ...4442 46 ...4244 ... 797678 58 6160 57 635558 50 4840 39 ...21[30] 40 3438 6466 37 ... 36 ...16[26] 46 6251 60 ...6663 ... 616563 39 4235 6766 58 ...23[23] 54 555153 52 5550 31 352530 36 393537 56 5855 1.1 seciJ c -1 _1 with spiralgalaxies),asinRC2(theapparentdependenceon rection forsuchagalaxyamountsto0.27mag.Ourformula for theVirgospiralgalaxieshasaGaussianshape,withmean in Figure1cbythetotalcolumns.Itisclearthatdistribu- in theAppendix. and lessforSmgalaxies.Amoredetailedcomparisonisgiven sic axisratioq).TheRSAformulacorrectsmoreforSa-Sb type inRC2onlyreflectsthedependenceonofintrin- does notdependonmorphologicaltype(aslongaswedeal seen face-ondependsonthedustdistribution.Ourmeancor- and Tammann(1981,hereafterRSA).TheRC2formulagives morphological typesareputtogether. histogram ofthesemagnitudesforthecompletesampleisgiven which istheabsorption-freemagnitudegiveninTable2.A only aface-oncorrection,becausetheabsorptioningalaxy value B=12.8andadispersion1.4mag,whenallthe tion ofapparentmagnitudes(andthusabsolutemagnitudes) catalog ofBottinellietal.(1989),whichpresentsmeanHiline widths atdifferentlevelshavebeenhomogenizedintheHi in theline.Wehavetakenthesevalues,exceptafewcases widths attwolevels:20%and50%ofthemaximumintensity several timesbyvariousauthors.ThemeasurementsofHiline to noncircularmotions.Itdependsonthelevel/according to produces asmoothtransitionbetweenlinearandquadratic This formulaisvalidforthewholerangeofspiralgalaxiesand mation fordwarfs.WefollowhereTullyandFouqué(1985)in where discrepantlowsignal-to-noiseratiomeasurementshave This hasbeenconfirmedbyLewis(1987).Fromhisstudy, it the formula observed linewidthatlevel/(20%or50%).Wisthe due summation foracharacteristiclinewidthW.is the defining arotationalprofilewidthparameterWby linear summationforluminousspiralsandaquadraticsum- circular motions.Theconvolutionofbothmotionsresultsina Table 2. code totheHilinewidths,fromad(goodbad),givenin We haveHilinewidthinformationfor90%ofoursample. been rejected.TheserejectionsaregiveninthenotesofTable2. circular motions{g=aginplaceof(jl.5a ). The maindifferencerelativetoBGPVisthatTully and between thevarioussources,wehaveattributedaquality According tothequalityofprofilesandagreement place of(t=10kms(BGPV). Notethatthisisotropycon- Fouqué (1985)admitanisotropicdistributionofthenon- we willadopt the values1.96and1.13(see §II#fordetails). from isotropy.Thevaluesofk(t) are1.80atlevel20%and1.18 warps), notonlytheturbulent motions,whichprobablydepart cerns thenoncircularmotions asawhole(includingmoderate galaxies (excludingextreme warps). Weadoptthisvaluein appears thatthemeanvalueof includesthezero-pointerrorfromcalibrators,but and takethemeanovercluster.Theresultis corresponding distancemodulusoftheVirgoCluster,noting cussion asinSandageandTammann(1988),letusevaluatethe Virgo ClusterandM31.Withoutenteringintoadetaileddis- absorption canbeneglected;thatis ness ofthesample,aswillbeshownin§Va.Wethusadopta agreement between<^>and)isdueonlytothecomplete- Here thesubscript/standsforinverseregression.Thegood that thenovaeareinbulgesofgalaxies,whereinternal distance modulusoftheVirgoCluster31.41,ora (1988). Wewillonlycompareourvaluetotherecentdetermi- modern determinations,andmorerecentlybyTammann because theseareprimaryindicators.Theseauthorsfinda nation byPritchetandvandenBergh(1988),basedonnovae, the scopeofpresentdiscussion.Areviewhasbeengivenby tance modulusissolargethatadetailedcomparisonoutof 0.17 (NBG)to0.64(RSA)(seedeVaucouleursandCorwin where 24.2isouradoptedtruemodulusforM31(seeTable 5) differential apparentdistancemodulusof6.8+0.4betweenthe Richter (1985),whogivesvaluesbetween30.5and31.9for the Appendix. and Virgo.PublishedvaluesofAAspanalargerange,from and AAisthedifferenceinGalacticabsorptionbetweenM31 calibrators wouldreducethislargeuncertainty. have adopted0.22,whichgives¡i=31.2±0.5.Clearlymore cloud. Definingthelimitbetween bothcloudsat<5=10?5(de sample arethedifferenceindistancebetweenVirgoS and — 3.16+0.18.Thenwecanevaluateindividualinversemoduli I) moduli : 19 Mpc.Thesystematicerrorsonthisvaluewillbediscussedin Vaucouleurs andCorwin1986h), wegetthefollowingmean Sandage 1987),andthestructure invelocityoftheVirgoS S' clouds(calledVirgoAandBinBinggeli,Tammann, and D7 1986a, forareview,andSandage1988a,note7,p.598). We B b To derivethedistancetoVirgoCluster,wehaveuse If wenowwanttoderiveadistancemodulusfromthe The numberofpublishedvaluesfortheVirgoClusterdis- Other problemsthatwecaninvestigatewiththepresent © American Astronomical Society • Provided by the NASA Astrophysics Data System b) StructureoftheVirgoCluster <^> =31.63± 0.15(N=54). =31.28±0.14{N85) s fi =24.2+6.8AA, b {fiy =31.40±0.13. ifij} =31.45±0.20. D EXTRAGALACTIC DISTANCESCALE.II. -1 -1 _1- o inverse moduli(thedirectassumeaslopeof—5.5and These valuesaretheweightedmeansbetweendirectand difference of0.35±0.13mag,confirmingthe0.4mag insignificant difference—98+116kms(inthesamesense compare Figs.4aand4b).Includingellipticallenticular (his Table5).Concerningthemeanrecessionvelocitiesof distance indicator,andthe0.31magdifferencereportedby reported bydeVaucouleursandCorwin,basedontheA the inversemoduliaslopeof—8).Theindividualvaluesare definition oftheS'cloudisneeded. galaxies, Binggeli,Tammann,andSandage(1987)reportan two clouds,wefind1281+77kmsforS'and99690 tances arenotdifferentfromfivedistinctdistanceindicators KKCT. Nevertheless,Tammann(1988)concludesthatthedis- 31.64 and(gj}=31.59fortheS'cloud.Wefindasignificant tion oftherelativemotiontwoclouds.Moreover,aclear s forS,i.e.,asignificantdifferenceof285+118km(also S' —S).Theseuncertaintiesprecludeanyphysicalinterpreta- rounded byanescapinghalo.”Figure4ashowsheliocentric the Scloud,interpretedas“acollapsingcore(orshell)sur- cloud. ThetrendfoundbydeVaucouleursisnotconfirmed regression) forthe85spiralgalaxiesassignedtoVirgoS velocity versusTully-Fisherindividualmodulus(direct with ourcompletesample.Asimilardiagramisdisplayedin is surroundingthemorecondensedVirgoEcloud(e.g.,de c (rms 0.76vs.0.46).Threecomplementaryexplanationswill be more dispersedintheVirgoClusterthanforcalibrators distribution ofvelocitiessimilartotheonefoundinHydra central galaxies(withinIofthecenter).Theseseemtobe cluster. Nospecialfeatureappears,exceptperhapsforthevery trend isvisiblehereeither,but,forbothclouds,wenotealack Figure 4bforthe54galaxiesassignedtoVirgoS'cloud.No explanation ofthelargedispersion.Alinearregressionmade distances fortheVirgoSandS'clouds,couldbeanatural differences, andthequalityofdata. I cluster(Fouqué1987),ifanythingcanbeconcludedfrom closer tousthanthemeanandappearhaveabimodal Fisher distancemoduliversusthetocenterof respectively, thedistributionofvelocitiesandindividualTully- Vaucouleurs andCorwin1986h).Figures4c4dshow, same spirit,wecaninvestigatetheideathatVirgoScloud of galaxieswithlowvelocityandlargedistancemodulus.Inthe whole cluster.Thisapparentcontradictionisduetoaslightly tested: thestructureofVirgoCluster,morphological type only sevenpoints. smaller dispersionfortheS'cloud. of 0.76mag,whichdoesnotdifferfromthedispersionfor the on the85galaxiesassignedtoScloudleadsadispersion angular distancefromthecenter givesthecontributionto spiral galaxiesarewellmixedwiththeellipticals,mean 0.12 mag.Forthewholecluster (completesample),thecorre- total dispersion.FortheScloud, themeanangulardistanceis mixing iscorrect. IftheScloudisconcentric withtheEcluster, 0.15 mag.Thisisasmallcontribution ifthehypothesisofgood sponding figuresare4?0,which gives1.3Mpcat19Mpc,or 3?2, whichatadistanceof18 Mpccorrespondsto1Mpc,or De Vaucouleurs(1982)hasreportedavelocitystructurein We willnowinvestigatewhytheTully-Fisherrelationis The depthoftheclusteralsoentersdispersion.If The doublestructureoftheVirgoCluster,withdifferent = 31.25and{¡llj}31.36fortheScloud, D c) DispersionoftheRelation 13 1990ApJ. ..349 .... IF luminous inthemeanthanearliertypegalaxies,andthusmore clearly demonstratedthatitwasanartifactduetotheMalm- has oftenbeenclaimed.Inthecaseoffieldgalaxies,it contribution of0.23mag. we mustusethepeak-to-peakangulardistance,whichisabout the explanation,becausewedealwithavolume-limited quist bias(Bottinellietal.1986h):latertypegalaxiesareless 6°, or1.9MpcatthedistanceofScloud,leadingtoa affected bythebias.Forclustergalaxies,suchabiascannotbe corrections (Fig.5b).See§IVcforinterpretation. from thecenter,forVirgoScloudgalaxies. galaxies assignedtotheVirgoS'cloud,(c)Velocityvs.radialdistancefromcenter,forScloudgalaxies,(d)Individualmoduli vs.radialdistance 14 Fig. 5—Morphologicaltypedependence oftheTully-Fisherrelationwithourinternalabsorptioncorrections tothemagnitudes(Fig.5a),andwithRSA Morphological typesegregationintheTully-Fisherrelation Fig. 4.—StructureoftheVirgoCluster(see§IVh).(a)Velocity-distancerelationfor85galaxiesassignedtoScloud,(b)Samediagram forthe54 © American Astronomical Society • Provided by the NASA Astrophysics Data System FOUQUÉ ETAL, Virgo Clusterdata,asshowninFigure5a,wherethemean Cluster mustbefortuitous.Infact,thereisnotypeeffectinthe follow theTully-Fisherrelation(whenNGC4424isexcluded), Appendix andFig.5b).WealsoemphasizethatSagalaxiesdo internal absorptioncorrectiontothemagnitudes(see each oftheninemorphologicaltypes.Theeffectfoundby field resultfromGiraud(1986)andtheirfortheVirgo KKCT probablycomesfromthetypedependenceoftheir distance modulusfromthedirectregressioniscomparedfor sample. Thus,theagreementreportedbyKKCTbetween Vol. 349 1990ApJ. ..349 .... IF -2 No. 1,1990 C no longertoformdifferentHubblesequencestages(G.de pensated bythehighvalueforT=8,asthesetwotypesseem the faintestnot.TheabnormallylowvalueforT=7iscom- Vaucouleurs 1988,privatecommunication). type isprobablyduetothelargeincompletenessofourdatafor and thatthelowdistancemodulusshowninFigure5forthis Sa galaxies(50%:seeFig.la),thebrightestbeingdetectedand explain theincreaseddispersionbetweencalibratorsand persion, iftherangeofmorphologicaltypesandinclinationsis B andlogVvaluesforbothsamples.Theabsorptioncorrec- the Virgogalaxies,wewillcomparemeandispersionof because ofthenecessaryusereducedZwickymagnitudes. 0.08 magforacalibratorand0.16Virgospiral,mainly the sameforbothsamples.ThemeanerroronaBvalueis tions onlygivesystematicerrorsthatcannotincreasethedis- for awell-observedgalaxy.Wetakethisfigureasrepresenta- The correspondingincreaseindispersionamountsto0.14mag. we willderivethecorrespondingmeanerrorbycomparingour log VvaluestothosederivedfromHivelocityfieldsatthe tive ofthemeanerrorforacalibrator.ForVirgogalaxies, VLA (Guhathakurtaetal.1988;Skillmanal1987)and in Table6.Itisevidentthatourvaluesaresystematically Westerbork (Warmeis1988a).Thepublishedvaluesarecor- rected toouradoptedinclinations.Therelevantdataaregiven values derivedattheVLAandWesterborkareinagreement. smaller thanthosefromtheVLAorWesterbork,while to thestatementin§lid,thatinclinationsderivedfromHi Before concludingthatonesystemisinerror,letuscomeback tM T estimated. FortheWesterborksample,Warmelsexplainsthat, M from thefitsforonlyfivegalaxies.IfwecompareourlogV due tobeamsmearingeffects,theinclinationscouldbederived velocity fieldsfortheVirgospiralgalaxiesareclearlyunder- values totheWesterborkforfivegalaxieswithincli- nation measuredsimultaneously(flaggedasfootnoteeinTable 6), wefindagoodagreementafteradjustmentoftheinclinations. Therefore, itisourfeelingthatthedisagreementbetween Hi than ours.SuchaneffectisnotseenintheVLAdata,probably agreement withours,thelogVvaluesareconsequentlyhigher adopted inclinationistheopticallydeterminedone,inbetter For theothergalaxiesmeasuredatWesterbork,where velocity fieldsandsingle-dishmeasurementsisduetoinclina- appear tobeinthesamesystemastheirHiinclinations. H ivelocityfields,arestillsmallerthanours,sothatthey because theiradoptedinclinations,whennotderivedfromthe mag arcsecisophote,wheretheaxisratioismeasured, due where theHiinclinationisderivedmorecentralthan 25 tion. Apossibleexplanationofthiseffectisthattheregion contradictory. Theeffectofbeamsmearingcannotberuled galaxies. Thisisnotthecaseforgalaxiesusedin§III, so to thedepletionofHiinexternalregionsVirgo the agreementbetweenbothsystemsforthesegalaxiesis not beams alongtheminoraxisissmall.Comingbackto the out, especiallyforhighlyinclinedgalaxies,wherethenumber of Virgo Cluster.Thiscorresponds toanincreaseofonly0.014 problem ofdispersion,andassumingthat,apartfromthe sys- M mag, relativetothecalibrators. Takingaccountoftheslope tematic shift,bothsystemsshare equallytheerrors,weareled to anrmsdispersionofabout 0.028forourlogVvaluesinthe from thecalibrators issmallerfortheHi datathanforthe log V,whileitwas0.14due toB.Thus,thelossinquality the relation,weseethatincrease indispersionis0.08dueto M M M T In ordertotesttheideathatqualityofdatacould In §III,wehaveseenthatthemeanerroronlogVis0.024 M © American Astronomical Society • Provided by the NASA Astrophysics Data System EXTRAGALACTIC DISTANCESCALE.II. magnitudes. Thisexplainswhythedirectslopesareclosefor quality ofthedata,isthen0.16mag. are moredifferent.Thetotalincreaseindispersion,duetothe the calibratorsandVirgogalaxies,whileinverseslopes depth oftheclusterandqualitydata.Wearefarfrom be reducedby0.22magto0.73mag,takingintoaccount the eters. Itmeansthatwewillextractfromourfinalsamplea set these estimatesofthereductiondispersionarerather the 0.46magobservedforcalibrators.Infact,itseems that (codes aorb,bothfVandWmeasured),excluding Sa from aperturephotometry,Hilinewidthsofgoodquality uncertain, andwepreferamoredirectapproach,which con- 0.55 maginthe B-band. persion thatwecanreachfor the VirgoClusterspiralsisabout of galaxiesbelongingtotheScloud,withmagnitudescoming sists inselectingonlytheobjectswithbestqualityparam- N4178 . N4152 . persion anyfurther.Inthat case, theonlyfurtherreduction mag. Anadditionalcutininclination doesnotlowerthedis- inclinations smallerthan30°.This“bestquality”sample con- galaxies thatarenotcompletelysampledand with N4206 . N4192 . that wecaninvokeisthedepth oftheScloud,corresponding tains 28galaxies.Thermsdispersion isnowreducedto0.56 N4222 . N4216 . to 0.12mag.Therefore,ourconclusion isthatthelowestdis- N4321 . N4299 . N4294 . N4254 . N4237 . N4388 . N4351 . 13258 .. N4501 . N4413 . N4402 . N4394 . N4519 . N4498 . N4450 . 13365 .. N4548 . N4535 . N4569 . N4579 . N4571 . N4561 . N4639 . 2050 N4654 . N4651 . N4647 . N4689 . N4698 . a e d c b Inclinationsusedtohomogenizevalues. Discussedin§IVc. VLAprofiles. Westerborkprofiles. Single-dishprofiles. We concludethattheobserveddispersion0.76magcanonly Name Comparison ofVfromSingleDishesandInterferometers m 40 74 48 84 87 84 85 48 44 69 32 51 44 75 80 20 30 61 82 38 58 37 37 18 30 66 32 52 39 38 37 56 53 51 hcd log V(LW)F(WES)K(VLA) Mm 2.068 2.152 2.145 2.391 2.090 2.332 2.296 2.074 2.337 2.199 2.235 2.282 2.229 2.110 2.436 2.244 2.243 1.990 1.973 2.404 2.116 2.243 1.791 1.768 2.208 1.957 1.736 1.931 2.085 2.380 2.103 2.189 2.321 1.997 TABLE 6 e e e e e 2.243 2.189 2.024 2.432 2.195 2.311 2.435 2.081 2.242 2.303 2.184 2.082 2.307 2.431 2.015 2.244 2.320 2.181 1.985 2.190 2.334 2.150 1.828 1.869 2.263 2.450 2.172 2.187 2.275 2.321 2.446 2.077 2.413 2.049 2.375 2.331 2.131 2.060 2.158 2.389 2.330 2.306 2.454 2.289 2.447 2.356 2.347 2.059 1.766 2.133 2.260 15 1990ApJ. ..349 .... IF persion ofthe5-bandTully-Fisherrelationisabout0.5mag. for 27degreesoffreedomand0.4617is cor’s testshowsthatthedifferencebetweenadispersionof0.55 the correspondingoneforcalibratorsisartificial.ASnede- (direct andinverse)onthederiveddistances. for clustergalaxiesofabiassimilartotheMalmquistbias, far fromsignificant.Thisresultsuggeststhattheintrinsicdis- called theclusterpopulationincompletenessbias,was latter applyingonlytofieldgalaxies.Theneglectofthisbias, magnitude ofthesample(Schechter1980),ifcoveragein must beemployedtoderivedistances(see§III)andtheinverse responsible forthehighvalueofHubbleconstantfoundby that thedispersioninTully-Fisherrelationwasresponsible widths. AllthiscanbesimplyunderstoodbylookingatFigure log Viscomplete(Teerikorpi1984).Letusrecallthatthe of amagnitude-limitedsampleontheslopesrelation sample fortheVirgoCluster,wecanstudydirectlyeffects Aaronson etal(1986).Nowthatwehaveanalmostcomplete and theinverserelationisonewitherrorsassignedtoline direct relationistheonewitherrorsassignedtomagnitudes, regression leadstomeandistancesindependentofthelimiting 16 log V.(c)Individualinversedistance modulivs.bluemagnitudes,(d)Individualinversedistancelog V. M M M Our feelingisthatthediscrepancybetweenthisfigureand Teerikorpi (1987)andBottinellietaldemonstrated We studybothregressionsbecausethedirectregression Fig. 6.—Theclusterpopulationincompleteness bias(see§Va),(a)Individualdirectdistancemodulivs.bluemagnitudes, (b)Individualdirectdistancemodulivs. a) VerificationoftheBiasTheoryonVirgoCluster V. THEINCOMPLETENESSBIASFORCLUSTERGALAXIES © American Astronomical Society • Provided by the NASA Astrophysics Data System FOUQUÉ ETAL. C correct valuecanbedeterminedfromamagnitude-limited from Figurelbthattheinverseslopeisconstant,so derived theinverseslopefromresultingsample.Itisclear from thedataforcluster,notcalibrators,ascanbe diagram, wehaveadoptedthevalue—8in§IVa.Asimilar sively brighterlimitingmagnitudes,andateachstep,wehave slope of—7.8±0.4.Thenwehavecutthissampleatprogres- diagram, wehavetakentheVirgofinalsample,whichisrepre- complete sample,canbederivedfromamagnitude-limited correct inverseslope,theonethatwouldbederivedfroma is magnitude-limited.Theproblemthentoknowifthe diagram similartoFigure6a,andabiasappearsifthesample the calibrators,whichisclosetodirectslope,wegeta sample, ifthenumberofpointsislargeenough.Fromthis sentative ofthecompletesample,andwhichgivesaninverse sample. seen fromFigure6c.Ifweusetheinverseslopederived the cutinmagnitude,wemustuseinverseslopederived leads tosystematicallyunderestimatedmeandistancesifone inverse relation(Fig.6c).Symmetrically,ifthecoverageisnot tion, versusBorlogV.Itisclearthatacutinmagnitude Cluster spirals,derivedthroughthedirectorinverserela- 6, whichshowstheindividualdistancemoduliofVirgo complete inlogV,Figure6dshowstheeffectonmean uses thedirectrelation(Fig.6a),whichisnotcasefor inverse distancemodulus. TM M The answerisyes,ascanbeseeninFigurelb.Tocreatethis To obtainameaninversedistancemodulusindependentof Vol. 349 1990ApJ. ..349 .... IF if theseshallowslopesareforcedtothecalibrators,corre- cut atprogressivelybrighterlimitingmagnitudes.Nevertheless, the directslopegetsslighterandslighter,whensampleis study forthedirectregression,showninFigurela,showsthat No. 1,1990 galaxies inthedeVaucouleurs sample(smallsquaresinFigs. sarily correct,ifthecoverage in logVisnotcomplete.Figure tance moduliderivedfroman incompletesamplearenotneces- lated withtheinverseslopederivedfromdata. value, allthemoresosincecutismadeatbrightermagni- 8 isthesameasFigure6,but with specialsymbolsshowingthe Nevertheless, theseindividualdistancemoduliarebiased, tance modulus,whentheindividualdistancemoduliarecalcu- sponding meandistancemodulistillunderestimatethetrue Figs. 8candSd). Straightlinesrepresent the meandistance 8a andSb) inthePierce-Tullysample (largesquaresin 4567, andNGC4568havebeenrejectedbyus)givesadirect account istheapplicationofourresulttofieldgalaxies which mustbeusedtoevaluatedistances.Therefore,onno because theinverseslopeisnotsameasdirectslope, modulus. errors (see§III),andthisslopeleadstoameandirectdistance tudes. Whenthesampleiscomplete,directslope distance modulusof31.01+0.11,inagreementwith his moduli thanoursarewellsuitedforthisapplication.Thede plete samples.Twodeterminationsleadingtolowerdistance straightforward. modulus inagreementwiththemeaninversedistance same asforthecalibrators,apartfromdifferencesdueto if werestricttowhattheycallVirgocore).Buthavedemon- adopted value30.88+0.09,takingintoaccountthatwe use Vaucouleurs (1982)sampleof34galaxies(NGC4189,NGC nations ofthedistancetoVirgoClusterbasedonincom- both) leadstoadirectdistancemodulusof31.01+0.11(30.86 rejected byus,IC776and3074them,769 by sample of29galaxies(NGC4189andIC3356have been our dataandcalibration.ThePierceTully(1988) confirmed ifwenowlookattheinversemoduli:same for theVirgocore). strated thatthesedeterminationsarelowerlimits,andthis is The inverseslopedoesnotchangewithcutsinmagnitudes. samples, respectively,give31.60+0.14and31.56(31.48 M In summary,theinverserelationgivesacorrectmeandis- It isinterestingtoseehowtheseresultsapplydetermi- Nevertheless, letusemphasize thatthemeaninversedis- Fig. 7.—VariationoftheslopeTully-Fisherrelationforsampleslimitedinapparentmagnitudes,(a)Thedirectbecomesslighterbrightercuts,(b) © American Astronomical Society • Provided by the NASA Astrophysics Data System EXTRAGALACTIC DISTANCESCALE.II. intrinsic scatterandobservationalerrors,onthelimitingmag- larger thanagivenvaluep.Thisdependson the developed inTeerikorpi(1987),predictsthatanunbiased dis- errors inthederiveddistancemoduli. in ourzero-pointisonly0.07mag.Thisresultconfirms communication) and—1.63+0.22withourdataforthesame mag withourB-bandinversedistancemoduli(31.48and31.56, B-band distancemodulusof31.02(Virgocore)or31.11(Virgo: relation deservesadditionalcomments,becausetheyfinda moduli (directandinverse)derivedfromthecompletesample. nitude ofthesamplemi(weassumeasharpcut-ofi),andon the tance moduli,onlyforthegalaxieswhichhavelogVvalues sample, ifwecalculatethemeanofindividualdirect dis- tance modulusoftheclustercanbeobtainedfromalimited largest numberofcalibratorsavailablereducestheaccidental eters arenecessaryforthecalibrators,andthatusing 0.45 magisnotsignificantbecauseofthelargedispersionfrom data fortheirthreecalibrators(R.B.Tully1988,private point oftheB-bandinverserelationis—1.18+0.19withtheir see theirpaperfordetails).Theapparentdiscrepancyof0.45 Tully-Fisher relationestablished onthecalibratorsm,isgiven between theobservedmeanmagnitude matagivenp=log distance moduluspofthecluster. Indeed,thedifferenceAm^ observed dispersionoftherelationa,whichincludesboth statement thatnotonlygooddistances,butalsoparam- three calibratorsinourcalibration,thecorrespondingchange three calibrators(itisworthnotingthatthelargedifferenceof respectively) isentirelyduetodifferentcalibrations.Thezero- the incompletesamplesareclosertoactualdistancemoduli It isclearthatthemeaninversedistancemoduliderivedfrom by one calibratortotheother).Ifwenowputtheirdataforthese than thedirectones. V, andthecorrespondingvalue calculatedfromtheadopted 0 M c 0 M b) TheCorrectWaytoDeriveanUnbiasedDistanceModulus The theoryoftheclusterpopulationincompletenessbias, The determinationofPierceandTullybasedontheinverse Am„ =(m-m ) = 0 cp from aLimitedSample ■y 7t^1+erf(A) ’ 2 Í2 exp(—A) (9) 17 1990ApJ. ..349 .... IF where curved relation,whichislinearonlyifAm=0.This increases whenpdecreases,thisbiasleadstoapredicted versus A'fordifferentobserveddispersionsa.SinceAm 18 which givesAm=—0.11,consideringthatabiassmallerthan 0.7 mag,thebiasvanishesforvaluesofA'largerthanA' = 1, Teerikorpi (1987),weseethatforobserveddispersionsabout relation appearstotheobserverlikealinearwith a superposed (largesquares).Thestraightlinecorrespondstogj=31.45. the PierceandTully(1988)samplesuperposed(largesquares).Thestraightlinecorrespondstog=31.40.(d)Fig.6dwith (1988)sample corresponds toii=31.40(b)Fig.6dwiththedeVaucouleurs(1982)samplesuperposed(smallsquares).ThestraightlineHj31.45. (c)Fig.6awith value oflogV,pisthengivenby slighter directslopethanforthecalibrators.UsingTable 1 in Table 1inTeerikorpi(1987)givesthevaluesofbiasAm where aandbaretheslopeinterceptofdirectTully- Fisher relationadopted,thevaluesofwhichare All thisleadsto 10%-20% of<7isdifficulttodetect.Thecorrespondinglimiting p p p 0 D D M0 p Fig. 8.—Sameas6,butwithincompletesamplessuperposed,(a)6athedeVaucouleurs(1982)samplesuperposed(smallsquares).The straightline © American Astronomical Society • Provided by the NASA Astrophysics Data System A =—-j=iandA'm—(m).(10) lcp a =—5.5andb8.0+ . 2