1977ApJ. . .217 . .903Y The AstrophysicalJournal,217:903-915,1977November1 © 1977.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. for verynearbygalaxiesitisnecessarytocorrectthe measured radialvelocitiesforthesolarmotionwith respect toacoordinateframethathasrelevancethe Universe itself.Forexample,the“localstandardof the peculiarmotionofGalacticcenter,andisnot rest” (LSR)thatisdefinedbythemotionofnearby stars inourGalaxyisrotatingandtranslatingwith as fundamentaltheframedefinedbycentroidof imaginary Robertson-Walkermanifoldofanideal the LocalGroup(LG)ofgalaxies.Thiscentroid isotropic andhomogeneousUniverse.Ithasbeen LG may,infact,approachfirstapproximationthe expanding Friedmannframe(i.e.,theHubbleflow does nothavealargemotionrelativetoanideally argued onobservationalgroundsthat(1)theLGitself galaxies outsidetheLG;e.g.,SandageandTammann appears tobeisotropicwhendeterminedfromnearby troid, issmall(deVaucouleurs1958;Sandageand motions ofnearbyfieldgalaxies,relativetothecen- properties aretrue,thenthemotionofLGrelative Tammann 1975;FisherandTully1975).Ifthesetwo to theunderlyinginertialmanifold wouldbesmall,and fundamental. the solarmotionrelativeto thecentroidwouldbe 1975; Sandage1975),and(2)themeanrandom To understandthevelocityfieldofexpansion * OnleavefromTelAvivUniversity, TelAviv,Israel. © American Astronomical Society • Provided by theNASA Astrophysics Data System -1o _1 -1 -1 precepts formembershipwithintheLG.ThismotionofLSRcorrespondstoabest-fit solar motionrelativetotheLGcentroidof^()=308kmstoward/105°,b—I. Local Group(LG)ofgalaxies,basedon21cmredshiftsforanumbercandidates,gives i;(LSR) =300kmstoward/107°,b—8°.Threeothersolutionsaregivenusingdifferent are certainlynotmembers.Likelymembers,onthebasisofkinematicsalone,IC10, Pegasus dwarf,WLM,DDO210,LeoA,andIC5152.Possible,butunlikely,members,again shows thatthesometimes-mentionedgalaxiesIC342,NGC6946,404,andMaffei12 based onkinematicsalone,areDDO187,GR8,SextansAandB,NGC3109.Allfiveof these lattergalaxieshavepositiveresidualsofabout125kmsrelativetothesolution,andmay be thenearestgalaxiesthatshowcosmologicalexpansion. 0 be ofthesamemagnitudeasvelocitydispersionothermembersLG,orrequiringv of theLSRrelativetocentroidLGintosumarotationvelocityvandmotion to becollinearwithM31permitlimitsputonv.Inpracticethesearesowideatthe of thecenterGalaxyv.Inprinciplefurtherrestrictionsonsuchasrequiring|^|t° 90% confidencelevelthatfromtheseconsiderationsalonevisonlyrestrictedtothealready known rangeof200^\v\300kms. Subject headings:galaxies:clustersof—radiosources21cmradiationstarsstellardynamics G c c G c c A newsolutionforthemotionoflocalstandardrest(LSR)relativetocentroid Consideration ofthevelocityresidualsfromridge-linesolutioneachcandidategalaxy A discussionoftheerrormatrixisgiven,withspecialemphasisonbreakingupvelocityv 0 THE LOCALGROUP:SOLARMOTIONRELATIVETOITSCENTROID Hale Observatories,CarnegieInstitutionofWashington,CaliforniaInstituteTechnology I. INTRODUCTION A. Yahil,*G.Tammann,andAllanSandage Received 1976October29;accepted1977April29 ABSTRACT 903 The mostseriousareduetodeVaucouleurs(1958, claimed. Iftrue,thenknowledgeofthesolarmotion relative totheframeofLGisevenmoreimportant, (1976), wherelocalanisotropiesintheexpansionare because itisonlybysubtractingsuchmotionfromthe and thentestingforanisotropiesthatsuchdeviations observed velocitiesofnearbyfieldandgroupgalaxies can befound. 1964, 1966,1976),Peebles(1976),andRubinetal radial velocitiesofgalaxieswereavailablefromSlipher, field hasalonghistory.Soonafterthefirstseveral introduction ofaKrterm,precededHubble’s(1929) independent ofdistance,andbyWirtz(1924),Land- from alargermaterialbyWirtz(1921)withtheintro- solutions byTruman(1916),Wirtz(1918),andSlipher mark (1920,1925),andStromberg(1925)withthe duction ofaATtermtoaccountforanexpansion relative togalaxiesthanstars.Furthersolutions (1917) showedasolarmotionthatwasmuchlarger clear demonstrationofthelinearexpansion.Much partake oftheexpansion,andthattheirapparent later, Hubble(1939)showedthatLGmembersdonot motions aredeterminedlargely bytherotationof to meanthatitisbound,and hencethatitformsa Galaxy. dynamical unitwithnegative totalenergy.Withthis There are,ofcourse,challengestothispointview. The problemoftherestframelocalvelocity The nonexpansionoftheGroup isgenerallytaken 1977ApJ. . .217 . .903Y o hm 904 point ofview,anewdefinitionmembershipcanbe if theypartakeoftheexpansionwillshowalarger if theGrouphasreacheddynamicalequilibrium,then nearby, low-luminosityresolveddwarfs(WLM,Sex- immediate applicationbecausethereexistmanyvery rather thanonthesomewhatarbitrarycriterionof made onkinematicgrounds(includingonlythose they shouldbewithintherandomscatterofknown bers isindoubt.IftheyareboundtotheGroup,and tans A,SextansB,LeoPegasusdwarf,NGC3109, galaxies thatdonotpartakeofthegeneralexpansion), justify anewdiscussionofanoldproblem. and positivedeviation. members oncethesolarmotionisremoved,whereas spatial proximitytotheGalaxy.Thequestionhasan GR 8,andIC10)whosestatusasLocalGroupmem- (cf. GunnandGott1972),hencebemembers.But but neverthelesswithnegativetotalenergyfortheLG since someoutlyinggalaxiescouldstillbeexpanding, the GalaxyandtoM31couldbelargeenoughslow will, asaworkinghypothesis,treatoutliersunbound. because ofthedifficultydeterminingenergy,we the expansioninimmediateneighborhoodof the LG. dwarfs couldyieldinformationonthemasswithin marily fromnew21cmobservations,andtheprevi- LG andastudyofthedecelerationoutlying solution forthemotionofSun.Thelargeavailable certain, probable,orpossiblemembersoftheLocal that have,atonetimeoranother,beenconsideredas ously unavailabledataforsomeofthefaintergalaxies sample ofaccurateradialvelocities,determinedpri- There areothermemberssuchasthedEgalaxiesof These arenotlistedbecausetheirvelocitiesun- (cf. Hodge1971)thataresatellitesoftheGalaxy,and Group, mainlyonthebasisofestimateddistance. the twodEcompanionsofM31(vandenBergh1974). Tammann 1974],which,intheabsenceofkinematic members. (ThemostdistantoftheseisM33with known. AdditionalpossibledEmembershavebeen Sculptor, LeoIandII,UrsaMajorDracosystems data forothercandidates,hadledinthepastto mentioned bydeVaucouleurs(1975). unwarranted conclusionthatallmembergalaxieshad [m —M]^24.56,orr=820kpc[Sandageand coincides withthecentroid ofextremePopulationI tory. TheGalacticlongitudeandlatitudearelistedin to bewithin~1Mpc.) differs fromthecenterof barby~I,butnearly (1950) fromKerrandde Vaucouleurs (1955).This the radiocenterofrotation at a=520,6—68?8 columns (3)and(4).FortheLMCwehaveadopted objects (cf.Westerlund1974), andwiththesymmetry 0 In fact,thiskinematiccriterionmaybeconservative But evenso,thegravitationalaccelerationdueto With theseproblemsinmindwehavemadeanother Table 1lists26galaxies(inorderofrightascension) Only 10galaxiesinTable1(col.[12])arecertain The columnsofTable1aregenerallyself-explana- © American Astronomical Society • Provided by theNASA Astrophysics Data System II. THEDATA YAHIL, TAMMANN,ANDSANDAGE 1- -1 _1 1 -1 -1 1 1 point oftheopticalrotationcurve(Feast,Thackeray, is sensitivetotheadoptedpositionofcenterbecause centric radialvelocitiesandtheirsources.(Thesyste- the projectedrotationcomponentofLMCitself matic velocityoftheLargeMagellanicCloud[LMC] and Wesselink1961;Cheriguène1975). in itsinnerpartis~24kms-deg[Cheriguène positional velocitydependenceexistsforSMC,but refer tothecenterofbar.)Inprinciple,same heliocentric velocityanditsmeanerrorfromalllisted 2; andtheSCBG,150+5. much smaller.TheRCBGgives,forSMC,v=163± here therotationalvelocitycomponentiscertainly SCBG [deVaucouleurs,andCorwin velocities agreesatisfactorilywithinthequotederrors, giving confidencethattheadoptedmeanerrorsare optical andradioobservations.The values toaccountforsomeofthedifficultiesdescribed realistic. ForLMCandSMCwehaveadoptedsome- 1964] givesv=265±1.6kmsforLMC,whilethe 1975] .TheRCBG[deVaucouleursandde members oftheLocalGroup. what largererrors(±5kms)thantheformal circular rotationvaboutthegalacticcenter,and(2) peculiar motionwithrespecttotheLSRof16.5kms“ above. 1976] givesv=260±5kms;butthesevelocities the velocityv=—ofgalacticcenterrelative toward /=53°,b+25°,correspondingto£/ field oftheLocalGroup.Satellitegalaxiescannotbe velocities (listedincol.[11]ofTable1)forthose apparent anisotropyoverthecelestialsphereof s (Delhaye1965).Column(12)identifiesthecertain s “(towardthedirectionofrotation),andZ=+7km standard ofrest(LSR)incolumn(11)byusingasolar used becausetheyarelikelytobeboundthe galaxies thataremovingrandomlyinthegravitational of theLGcanbedecomposedintotwovectors:(1) rather thanpeculiarvelocitiesrelativetothecentroid. primaries, andwillreflectorbitalmotionsaboutthem to thecentroid.Wecansolveforvbyfinding because itsorbitalstatusisnotsoclear). NGC 205fromthesolutionfori;(butkeepFornax problems, twointhissection, andthethirdin§VI. this lattermotionwiththevelocitydispersionofother used todeterminet>,and toestimatethevelocity (a) Galaxiesthatarecertain membersoftheLGare and onthemotionofcenter(v),tocompare useful toconsiderlimitsontherotationvelocityv, Hence, weexcludeLMC,SMC,M32,NGC185,and dispersion awithintheLG itself,{b)Thepossibility LG members. c G0c + 9kms"(towardthegalacticcenter),F=+12 Q 0 0 G c Columns (5)-(8)listtheopticaland21cmhelio- Columns (9)and(10)givetheweightedmean The velocityincolumn(9)isreducedtothelocal The motionvoftheLSRrelativetocentroid The datainTable1are used todiscussthree Although weareprincipallyinterestedinr,itis 0 0 III. MOTIONOFTHELOCALSTANDARDREST 1977ApJ. . .217 . .903Y DDO 187 Sextans A Sextans B LMC Leo A IC 342 Maffei 2 SMC DDO 210 GR8 NGC 3109 Maffei 1 IC 1613 IC 10 IC 5152 NGC 6946 NGC 6822 NGC 598 NGC 404 NGC 224 NGC 221 NGC 205 NGC 185 WLM Pegasus Fornax 20. 22. 21. 11. 10. 16. 14. 12. 19. 18. 15. 13. 4. 3. 9. 8. 6. 2. 1. Solution Galaxy © American Astronomical Society • Provided by theNASA Astrophysics Data System One sigmaerrorsofindividualparameters. Forthecompletevariancematrixsee§VI. (1) Volders, L.,andHögbom,J.A.1961,B.A.N.15,307. Mayall, N.U.,andVaucouleurs,A.de1962,A.J.67.360. Huchtmeier, W.K.1973,Astron.Astrophys.22,91. Burley, J.1963,A^J.68,274. Humason, M.L.,Mayall,N.U.,andSandage,A.R.1956,J.61,97(Mt.Wilson) Warner, P.J.,Wright,M.CH.,andBaldwin,J.E.1973.M.N.163,163. Rubin, V.C.,Ford,W.K.,andKumar,C.K.1973,Ap.J.181,61. Rubin, V.C.andd'Odorico,S.1969,Astron.Astrophys.¡2,484. ibidem (Lick) Bergh, S.vanden1969,Ap.J.Suppl.19,145. Shostak, G.S.1974,Astron.Astrophys.31,97. Dieter, N.II.1902,A.J.67;,317. Gouguenheim, L.1969,Astron.Astrophys.3,281. Bottinelli, L.,Gouguenheim,andHeidmann,J.1972,Astron.Astrophys. Rogstad, D.IL,Shostak,G.S.,andRots,A.II.1973,Astron.Astrophvs. Rogstad, D.IL,andShostak,E.S.1972,Ap.J.170,315. Bottinelli, L.,Gouguenheim,L.,Heidmann,J.andN.196s. Rogstad, D.IL,Rougoor,G.W.,andWhiteoak,J.B.1967,Ap.J.150.9. Roberts, M.S.1962,A.J.67,431. Burbidge, E.M.,andG.R.1965,Ap.J.142,634. Epstein, E.E.1964,A.J.69,490. Evans, D.S.1963.M.N.A.S.S.A.22,140. Ford, W.K.,Rubin,V.C.,andRobeils,M.S.1971,A.J.70,22. _22, 111. Ann. d'Astrophys.31,205. LS,121. Solutions forMotionoftheLocalStandardRestusing FourDifferentDataSamples DDO 75,A1009 DDO 236 DDO 70,A0957 A 0524 DDO 8 A 0051 DDO 155 A0956 Leo III,DDO69 Other Name DDO 221,A2359 M33 M32 DDO 216,A2326 M31 2 +SMC,LMC, 2 +GR8,SexA, 3 +DDO187 1 +WLM 1 +110,Peg,WLM, M31, M33,11613, (2) N185, N205,N221 15152 Sex B,N3109 DDO 210,LeoA, N6822, Fornax Sample 310. 70 246.23 262.10 233.19 279.33 237.31 302.80 343.91 138.16 136.87 133.63 127.02 129.87 121.17 121.16 120.71 120.77 196.91 136.09 118.96 25.60 95. 71 25.38 34.10 94. 79 75. 71 (3) + 70.50 + 77.00 +39.92 +23.07 +43.76 + 52.41 + 10.58 +11.66 LOCAL-GROUP MEMBERSANDSOMETIMECANDIDATES -18.38 -33.45 - 0.06 -65.72 - 0.17 -31.33 -27.01 -44.30 -21.57 -21.97 -21.13 -14.48 - 3.32 -31.30 -60.55 -73.60 -43.54 -50.22 (4) WITH KNOWNRADIALVELOCITIES SOURCES FORREDSHIFTSINTABLE1 -1 + 270±2 + 10±20 + 1±20 + 49±10 + 166±3 - 10±50 - 178±4 - 23±15 - 298±3 - 193±4 - 240±12 - 252±34 - 238±10 - 343±12 ^ 257+30 t- 370±30 ^ —50±45kmsMpc km s* v (opt) 287 ±15 264 ±301137 334 ±30 343 ±30 300 ±22 287 ±33* 32 ±20 16 ±35 78 ±30 (km s“^ 78 ±20 Vo (LSR) (5) TABLE 1 TABLE 2 33 +216±5 25,26,27,28 +270±2 40 -18±5 35 6 +323±5 6,7,21 +31±3 6, 34 see 4-180±1 25 +161±2 6,7,8 6,8,13 6 -235±3 see 1,2,3,45-302±2 6,7,8,41,45 23, 24 6,7 7 +403+1 6,7,41 6,7,8 6 -343±2 6 (6) 45. Sai'gent.W.L.W.,Scheehter,P.L.,Boksenbci’g,A.,andShoi-ti’idge,K. 44. Dean,J.F.,andDavies,R.D.1975,M.N.170,503. 43. Huchtmeier,W.K.1973.Astxxm.Astx’ophys.22,27. 42. Seielstad,G.A.,andWhiteoak.J.B.1965,Ap.J.142,616. 41. Vaueouleux’s,A.de,Shobbi'ock,R.R.,andStl'obel,1976,J.81,219. 40. Spini'ad.IL,Bahcall,J.,Becklin,E.E.,Gunn,J.E.,Kxdstian, 39. Love,R.1972,Natui’e235,53. 37. Shostak.G.S..andWeliachew.L.1973,Ap.J.Lettex’s169,L71. 36. Bottinelli,L..Chamaraux.P.,Gei'axxl,E.,Gouguenheim,Heidmann,J,, 33. Hodge.P.W.1974,P.Â.S.P.86,645. 31. Hindman,J.V.1967,Austral.J.Phys.20,147. 24. Evans,D.S.,andWayman,P.A.1958,M.N.A.S.S.A.17_,137. 38. Lewis.B.M.,andDavies,R.D.1973,M.N.165,213. 35. Spinrad.IL,Sargent.W.L.W.,Oke.J.B..Neugebauer,G.,Landau,R., 34. Bei'gh.S.vanden1909.Ap.J.Suppl.19.145. 32. Fisher,J.R..andTully,R.B.1975.Astron.Astrophvs.44,151. 30. McGee,R.X..andMilton,J.A.1966,Austral.J.Phys.19,343. 29. Kerr,F.J.,andVaucouleurs,G.de1955,Austral.J.Phys.8j508. 28. Prdvot,L.1973,Astron.Astrophys.28,165. 27. Smith,M.G.,andWeedman,D.W.1972,Ap.J.177,595. 26. Feast,M.W.1964,M.N.127,195. 25. Feast,M.W.,Thackeray,A.D.,andWesselink,A.J.1961,M.N.122,433. 1 + 295±10 + 26±10 + 153±5 + 42±3 - 59±3 - 181±10 - 131+10 - 124±9 km s v (21cm) 105 ±5 105 ±6 101 ±6 107 ±5 106° ±8° Kazès, I..andLauque,R.1971,Astron.Astx’ophys.12,264. Ap. J.Letters163,L25. (7) Neugebauer, G.,Sargent,W.L.W.,andSmith,H.1973,Ap.J.180,351. 1976. pi'eprint. King. I.R.,Gunn.J.E.,Garmire,G.,andDieter,N.H.1971, / 43,44 (36),37,38, (11), 12,(14), 32 32 39 32 32 32 11.12.32 29,30 29,31 (15), 16,17 9,10,11,12 11,32,42, 11,16,17,22 9,11 see 1 14.32 18,19,20 11,12,14,15, 12.32 (8) v (adopt) -16 ±5 -11 ±5 -13° ±8° + 78 + 153 + 217 + 324 + 403 + 295 + 26 + 270 + 31 + 49 + 42 - 58 - 17 - 10 - 180 - 235 + 163 - 181 - 131 - 23 - 301 - 216 - 240 - 252 - 116 - 343 -4 ±8 -7 ±4 — 8±4 (9) km s 30 10 50 15 34 10 12 10 10 10 (10) 3 5 3 2 5 5 1 km s* v (LSR) + 394 + 257 + 54 + 164 + 223 + 317 + 289 + 24 + 34 + 36 + 154 + 76 - 15 - 8 - 182 - 22 - 239 - 298 - 178 - 123 - 48 - 119 48 (+19,-12) 43 (+9,7) 72 (+15,-11) 45 (+12,-9) 68 (+15,-11) (11) 66 (+28,-16) -1 (km s) certain certain certain certain certain certain certain (12) 1977ApJ. . .217 . .903Y x 906 least squaresfit,minimizingthestatistic centroid oftheLocalGroup,visdeterminedbya hypotheses. trial memberships,(c)Theerrorsinourestimateofv that othergalaxiesinTable1maybemembersis upper indices,e.g.,v,thex-axisistoward radial velocityofgalaxy/relativetotheLSR(col.[11] breakup intovandunderdifferentdynamical are discussedindetail,withspecialemphasisonits performing additionalsolutionsforvwithvarious analyzed onthebasisofkinematicargumentsby galaxy /.(Componentsofvectorsarecharacterizedby as crosses;allothergalaxies, open circles.Theabscissaisthecosineofanglefromgalaxy inquestiontotheapex(Table2). lines arethe±1ordeviationsfrom theridge-linesolution.Closedcirclesarefivecertainmembers. LMCandSMCareshown with thevelocitydispersionor,andareignored.A in thefit,buttheyturnouttobenegligiblecompared vational errorsofvshouldalsobetakenintoaccount number ofdegreesfreedom.Inprincipletheobser- of Table1)andw*istheunitvectorindirection detailed discussionoftheconfidencelimitsisgivenin where Nisthenumberofdatapoints,andiV—3 and thez-axistowardnorthgalacticpole. Galactic center,they-axisindirectionofrotation, residuals and thevelocitydispersionaisestimatedfromrms 0 0 0 cG 0 t a) MotionoftheLocalStandardRestDetermined The velocityoftheLSRrelativeto The notationinwhatfollowsisthat^theobserved Fig. 1.—Solution1(Table2)based onthefivecertainmembersofLocalGroup(LMCand SMCexcluded).Thetwosolid © American Astronomical Society • Provided by theNASA Astrophysics Data System from KnownMembersoftheLocalGroup vn2 2 A =2(i+o'i)>(1) * =Amin/(A-3),(2) YAHIL, TAMMANN,ANDSANDAGE -1 _1 1 -1 is the1adispersion.Thedetailsofdeviations(D) the centralsolutionaredrawnat±66kmswhich from thequotedapex.Thetwoparallellinesbordering the residualsofindividualgalaxiesfromridge-line § YI.Inthissectionweareprimarilyconcernedwith from theridge-linesolutionforthisandother are plottedagainsttheangleofgivengalaxy where theVi(LSR)velocitiesfromTable1(col.[11]) next section.ThesolutionisillustratedinFigure1 tions, aresummarizedinTable2tobediscussedthe ions totheGalaxy[LMCandSMC])areVq=287km within reasonableconfidencelimits,thebest-fitvalues residuals arenotsignificantlyalteredwhenvisvaried The fivegalaxiesnearcosA=+0.85(IC342,Maffei1, ized residuals(D/S),whicharethenumberofstandard to M31[NGC185,NGC205,M32]andthecompan- NGC 6822,andFornax,butexcludingthecompanions certain membersoftheLG(M31,M33,IC1613, the confidencelimitsismade. may beusedforthispurpose,beforethediscussionof candidates formembershiponkinematicgrounds.(2) to thefivecertainmembers{filledcircles)usedfor ridge-line. o- =66kms.Thesevalues,alongwithlatersolu- solution asindicatorsofmembership.Becausethese deviations (S)thatanygivengalaxyfallsfromthe solutions arelistedinTable3termsofthenormal- s" towardb=—13°,/106°,withadispersionof scatter band(Ai;=±66kms),andhenceare DDO 210,IC5152,andLeoA)fallwithinthe1a solution, sixadditionalgalaxies(IC10,Pegasus,WLM, 0 The parametersvandor,determinedonlyfromthe Noteworthy featuresofFigure1are:(1)Inaddition 0 Vol. 217 1977ApJ. . .217 . .903Y -1 -1 IC 5152.... IC 10 DDO 210.. Leo A Pegasus IC 1613.... WLM NGC 6822. Fornax IC 342 NGC 404.. Sextans B.. Sextans A.. GR 8 NGC 6946. M31 No. 3,1977 DDO 187.. NGC 3109. M33 within ~2aofthemeanrelation,butallhavepositive relative totheLGcentroid.(3)Fivegalaxies(DDO They evidentlyhaveHubblevelocitiesof~250kms and 4afromthesolution,areclearlynotmembers. Maffei 2 Maffei 1 by morethan1a(66kms),itseemslikelythatmost toward negativevelocitiesrelativetothemeanline Maffei 2,NGC6946,and404)standbetween3a residuals. Becausenogalaxyinoursampledeviates 2 ofTable2. 187, GR8,SextansAandB,NGC3109)fall Fig. 2.Sameas1butwith sixadditionalgalaxiesaspostulatedmembersoftheLocalGroup. Theridge-lineissolution © American Astronomical Society • Provided by theNASA Astrophysics Data System Name Angle fromtheApexandNormalizedResidualsforFourSolutions -0.053 + 0.857 + 0.639 + 0.959 + 0.222 + 0.377 -0.192 + 0.851 -0.154 + 0.960 + 0.459 -0.584 -0.424 + 0.915 + 0.843 + 0.768 -0.721 -0.908 + 0.836 + 0.890 -0.164 cos A Solution 1 + 0.99 + 0.32 -0.85 -0.35 + 0.25 -0.21 -0.47 + 1.01 -0.94 + 0.20 + 0.44 + 3.88 + 3.58 + 1.83 + 1.54 + 2.03 + 3.42 + 2.84 + 1.68 + 4.73 + 1.78 DIS LOCAL GROUP + 0.832 -0.131 + 0.946 + 0.178 -0.111 + 0.800 + 0.574 + 0.976 + 0.376 + 0.320 + 0.808 + 0.895 -0.519 -0.344 -0.237 + 0.860 + 0.921 -0.665 -0.881 + 0.867 -0.088 cos A TABLE 3 Solution 2 1 galaxies (fivecertainmembersusedinFig.1,andthe for membership.Figure2showsasolutionfrom11 different galaxysubsamples,mightaddtotheevidence an expansioncomponentoftheorder~100kms^. of thefivearenotboundtoLG,butrathershow + 1.50 -0.32 -0.06 -1.47 + 0.13 -0.98 -0.13 -0.57 -0.19 + 1.34 + 0.09 + 6.07 + 5.15 + 2.87 + 2.62 + 5.32 + 7.27 + 2.58 + 2.85 + 3.74 + 3.02 DIS b) OtherSolutionsforvfromanIncreasedSample 0 These pointssuggestthatothersolutions,using + 0.810 -0.110 -0.138 -0.214 + 0.842 + 0.595 + 0.930 + 0.288 -0.384 + 0.944 + 0.438 + 0.427 + 0.798 + 0.916 + 0.877 -0.609 -0.742 + 0.806 + 0.735 -0.105 -0.929 cos A Solution 3 + 1.42 -0.52 + 0.30 + 0.75 -0.02 + 0.39 -0.20 + 0.46 + 0.35 -0.72 + 1.62 + 1.35 + 1.18 + 0.93 + 3.19 + 4.22 + 5.43 + 4.01 + 1.11 + 3.80 + 1.89 DIS -0.015 + 0.863 -0.104 + 0.414 -0.239 + 0.876 + 0.669 + 0.959 + 0.253 -0.464 + 0.947 + 0.502 -0.621 -0.753 + 0.816 + 0.823 + 0.739 + 0.873 + 0.919 -0.201 -0.923 cos A Solution 4 + 0.43 + 1.12 + 1.48 -0.77 -0.23 + 0.30 + 0.51 -0.30 + 0.67 +0.21 + 1.58 + 0.95 + 0.53 + 4.80 + 3.86 + 0.92 + 1.19 + 3.56 + 3.04 + 3.89 + 1.34 D/S 907 1977ApJ. . .217 . .903Y -1 _1 1 _1 1 expected ratioforaGaussiandistribution.Thesmaller the 11residualsarein1-2arange,whichis 0.015 forGaussianresiduals.Ontheotherhand,in in agreementwiththepreviousvaluewithinerror, the latitudeofapex,andareducedradialvelocity The parameters,listedinTable2,solutionshowan six additionalcandidatesdiscussedinpoint1above). ridge line,andreinforcesthesecondthirdcon- gives a=66kms,all11residualswouldbewithin defined. Thelowervalueofaforthissolution,while increase in|*;|o(LSR)to300kms,a5°change nonmembership ofDDO187,GR8,SextansAandB, a increasesthesignificanceofdeviationsfrom seems morereasonable.Inthefirstsolutionwhich dispersion ofcr=45kms". 908 clusions (above)concerningtheIC342groupand second solutionwherea—45kms,threeoutof and NGC3109,allofwhichhavepositiveresiduals including fourofthefivelastmentionedgalaxies 2 alone,andwehavemadetwoothersolutions:first above 2.5o-. 1.0 or,andtheprobabilityofsuchaneventualityisonly third solution,showanincreaseof¿?(LSR)to listed inTable2andillustratedFigure3forthe (solution 3),andthenallfive4).Theresults, their radialvelocitiesreflectessentiallyorbital were excludedinsolutions1-4onthegroundsthat motion abouttheGalaxyandM31solar LMC, NGC185,205,and221),which as expected,anincreaseinor. solutions: (1)Thefivecompaniongalaxies(SMC, 0 ~ 340kms“,onlyamoderatechangeintheapex,and, Figure 2andTable3showthatthesolutioniswell However, thislastpointisstillnotsettledbyFigure For completenesswehaveconsideredtwoadditional © American Astronomical Society • Provided by theNASA Astrophysics Data System Fig. 3.—SameasFigs.1and2 but forsolution3ofTable2 YAHIL, TAMMANN,ANDSANDAGE -1 -1 1 motion, weretreatedinsolution5asindependent members oftheLG.Theresult,setoutinTable2,is is restrictedtothegalaxiesforwhichreliabledistances close agreementwithsolution2.Thisshowsthatvis with Hubble’sresult,wefindnoevidencefora members (M31,M33,IC1613,NGC6822,and does notapplywithintheLG,wehavemadeasixth about thestatusofcompaniongalaxies.(2)Totest quite independentofwhateverassumptionsaremade are known(SandageandTammann1971,1974;Hodge solution includingaKrtermusingthefivecertain Hubble’s suppositionthatthevelocity-distancerelation 45 kmsMpc“,whichclearlyisanullresult. significant Krterm.OurformalvalueofKis—50± Fornax) andthepossiblememberWLM.Thissample to amotionoftheSunrelativeLGcentroid. toward /=107°,ô—8°(solution2),canbechanged The resultisasolarmotionof308kmstoward 1971; SandageandKatern1976,1977).Inagreement adopted presentsolutioncorrespondstoacorrection precepts (e.g.,inclusionorexclusionofLMCand earlier determinationslistedinTable4.Thevaluesare solar motionofA*?=300sin/cosèkms“.Our Sun relativetotheLSRadoptedin§II. / =105°,b—T,foundbyusingthemotionof 0 Group galaxieshavenormallybeencorrectedforthe all surprisinglyconcordantinviewofdiscordant SMC). _1 Our adoptedmotionoftheLSR300kms The observedheliocentricredshiftsofnon-Local This solution,whichweadopt,canbecomparedwith IV. THESOLARMOTIONRELATIVETO CENTROID OFTHELOCALGROUP Vol. 217 1977ApJ. . .217 . .903Y 1 _1 -1 _1 -1 by significantamountsincertaindirections.The which, althoughclosetotheconventionalvalue,differs This paper308*±231055-74 Byrnes 1966280±231075-74 Humason and maximum deviationfromtheoldsolutionis±87km de Yaucouleursand b =—24°.Thisdifferencemaybeimportantfornear- by galaxieswithsmallredshifts.Thechangeofthe s“ inthedirectionsl=3°,b24°,and/183°, of Ap=—79cos/b+296sin36 May all1946300±2593°6°-14°4° No. 3,1977 with thenewmotionofSunwithinLG,leaves toward /=135°,—8°(Rubinetal1976),which, effect requiresasolarmotionof600±125kms local expansionfield.Forinstance,theRubin-Ford also someinfluenceonthepossibleanisotropyof a componentofanisotropyonly370±125kms. solar motionrelativetotheLGcentroidhas,ofcourse, in someofthenearergalaxiestheywouldexceed Tammann 1975,andtoFisherTully1975).The components (<50kmsaccordingtoSandageand galaxies outsidetheLGcanhaveonlysmallrandom line ofthesolution.Furthermore,independentevidence expansion component,andthenetvelocity(relativeto point isthatifthepeculiarmotionswerelarge,then far beyondtheLocalGroup.Theangularsizeof from thestellarcontentinIC342andNGC6946,both very muchmorechaotic. not happeninFigures1-3,orthelistofdwarf the LGcentroid)couldbenegative.Becausethisdoes current beliefthatobservedradialvelocitiesofnearby uals inanyofthesolutions(Figs.1-3)strengthensa are clearlyfieldgalaxies.Theystandfarfromtheridge <7 £lOOkms;otherwiseFigures1-3wouldbe clear thatthelocalvelocityfieldismoreregularthan galaxies byFisherandTully(1975),alimitcanbe While itisdifficulttocalculateexactlythislimit, placed onthecomponentofrandomvelocitya. Scl galaxies,showthatatleastthesetwosystemsare where (SandageandTammann 1974),showbeyond doubt thatthesetwogalaxies aremoredistantthan H iiregions,togetherwith the calibrationgivenelse- Although thecasesofMaffei 1and2aremoredifficult any thatliewithinthe1ascatter linesinFigures1-3. 1-3 isagainthatthefivepointsclusteredaboutIC342 r Y Peters 1968315±15956-83 Wahlquist 1955..292±32106±6-74 The factthattherearenosignificantnegativeresid- * Forthecompletevariancematrixsee§VI. The moststraightforwardconclusionfromFigures © American Astronomical Society • Provided by theNASA Astrophysics Data System Our AdoptedSolutionComparedwithPrevious 1 V. DISCUSSIONONTHEDIFFERENTSOLUTIONS Source (kms~)/b Solutions fortheSolarMotion TABLE 4 LOCAL GROUP -1 1- because ofthehighextinctioninthislow-latitudefield, in WLM(SandageandKatern1977)gives(m—M)= is unresolved,andtherenoreasontodoubtthatits them tobemoredistantalso.TheS0galaxyNGC404 inspection oftheavailablephotographsclearlyshows the availableplatematerial,resolutionintobrightest 210). Furthermore,photometryofthebrighteststars within thescatter(IC10,Pegasus,WLM,DDO210, IC 10,WLM,and5152,somewhatfainterfor this galaxywithinthespatialconfinesofLG.From 26.0 ±0.2,orr=1.60.2Mpc,whichclearlyputs (we havenolongfocal-lengthreflectorplatesofDDO large-scale platematerialexists,arehighlyresolved distance islargerthanr^3Mpc. resolution beginsatbrightermagnitudesthanfor Pegasus andLeoA.Butevenforthesegalaxies,the Leo A,andIC5152).Allofthese,forwhichadequate members oftheNGC2403group(HoI,HoII,IC stars occursatnearlyequallybrightmagnitudesfor 2574, NGC2366,etc.)whichisatadistanceofr= five membersarenotoftheLG.Thenthey velocities themselves.First,letusassumethatthese the fivegalaxiesarecloserthanmembersofM81- 3.25 Mpc,or(m—M)=27.56(Tammannand their Hubbledistancesof(r+50)1Harestillvery random velocitiesoffieldgalaxiesare<7=±50kms must showsomeHubblemotion.Hence,ifindeedthe this identification. NGC 2403group,andhencecouldbemembersofthe scale Halereflector5mplatesisthatatleastfourof Sandage 1968).Hence,thedirectevidencefromlarge- all approachinguswiththisrandomvelocity,then and evenundertheextremeassumptionthattheyare kinematic evidence,summarizedinTable3,supports LG, basedonproximityalone;and,clearly,the most ofthesegalaxiesmustbemembers,becausethey (The calculationsaresetoutinTable5,withH= small—in fiveoutofthesixcaseslessthan1.6Mpc. ever, thisargumentisweakenediftheLGconsiderably 0 3 various parametersforthe five galaxiesinquestion, are soclose,andtheinitialhypothesisisinvalid.How- based onthesecondsolution (Fig.2),whichisthe flow, althoughperhapsdecelerated. members oftheLGandareexpanding withtheHubble that atleastsomeofthesecandidatesindeedarenot five candidateslieonthepositivesideofridgeline, for thethirdsolution),madeonbasisthatmostor the intervalbetween1.5and2forfirstsolution, well abovethe1ascatterline.TheD/Svaluesarein difficult decisiononthestatusofDDO187,GR8, decelerates thelocalexpansionflow,orifoyislarger. 50 kms“Mpc.)Therefore,ifcr