aa r X i v : . [ a s t r o - ph . GA ] N ov Galaxy Groups
R. Brent Tully,
Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
ABSTRACT
Galaxy groups can be characterized by the radius of decoupling from cosmic expansion, theradius of the caustic of second turnaround, and the velocity dispersion of galaxies within thislatter radius. These parameters can be a challenge to measure, especially for small groups withfew members. In this study, results are gathered pertaining to particularly well studied groupsover four decades in group mass. Scaling relations anticipated from theory are demonstrated andcoefficients of the relationships are specified. There is an update of the relationship between lightand mass for groups, confirming that groups with mass of a few times 10 M ⊙ are the most lit upwhile groups with more and less mass are darker. It is demonstrated that there is an interestingone-to-one correlation between the number of dwarf satellites in a group and the group mass.There is the suggestion that small variations in the slope of the luminosity function in groupsare caused by the degree of depletion of intermediate luminosity systems rather than variationsin the number per unit mass of dwarfs. Finally, returning to the characteristic radii of groups,the ratio of first to second turnaround depends on the dark matter and dark energy content ofthe universe and a crude estimate can be made from the current observations of Ω matter ∼ . Keywords:
Galaxies: groups; mass and luminosity functions; dark matter
1. Introduction
In the imaginary world of simulations, re-searchers have a well developed picture of thecollapse of matter into halos. Over time, smallhalos are absorbed into larger units. Collapsedregions filled with substructure can be defined byhundreds or thousands of particles. Halos can beidentified with precision within those simulations(Boylan-Kolchin et al. 2009).In the real world, most galaxies are observedto lie in groups or clusters. However, member-ship may be so limited that the structure is illdefined. There is no concensus among observersabout what is meant by the terms ‘group’ and‘cluster’. For example, it is commonly acceptedthat we live in something called the Local Group.A quantitative boundary that would include thetraditional members is the zero–velocity surface(Karachentsev et al. 2009). Galaxies inside thissurface are infalling or on more complex boundorbits. Galaxies outside this surface are partic- ipating in the Hubble expansion. But now con-sider the Virgo Cluster. This entity would tradi-tionally be taken to include the region within 2Mpc where several thousand galaxies are follow-ing randomized orbits (Binggeli et al. 1987). Thezero–velocity surface around the Virgo Cluster liesat a radius of 7.2 Mpc (Karachentsev et al. 2014),almost half of our distance of 16.4 Mpc from thecluster. The implicit traditional definitions of theLocal Group and the Virgo Cluster are inconsis-tent.It does not seem useful to draw a distinctionbetween the terms ‘cluster’ and ‘group’. Clusterscontain a lot of galaxies and groups contain onlya few, but there is no clear demarcation betweenthe two. In this article, the two terms will be usedinterchangeably.So how should an observer define a group?Structure forms as sufficient matter accumulatesthrough gravitational attraction to decouple fromthe expansion of the universe. The matter fallstogether on nearly radial orbits and, given enough1ime, evolves toward dynamic equilibrium. Twonatural dimensions to describe a collapsed struc-ture are the gravitational radius, r g and the radiusat 200 times the ‘critical’ density for a matter-dominated closed universe, r . Here, lower case r implies a 3-dimensional radius and an upper case R will be used to denote a projected radius. Ac-cordingly, if all galaxies in a group sample aregiven equal weight (ie, galaxies of whatever lumi-nosity are considered to be test particles within anenvironment dominated by distributed dark mat-ter) then the gravitational radius is defined as R g = N P i
2. From Big to Small
The questions to be addressed are whetherthere are observational features that distinguishthe infall and quasi-virialized regions of halos and,if so, how these features scale with halo mass.Most of the discussion in this paper will focus onmodest to very small groups of galaxies, the mark-ers for the overwhelming majority of identifiablehalos. The sites of intermediate mass halos, thosein the range 5 · − M ⊙ , are investigatedwith a wide field imaging survey at the Canada-France-Hawaii Telescope (CFHT), initially usingthe 0.3 sq. deg. 12K CCD camera and then the1 sq. deg. Megacam detector. Follow up spec-troscopy was undertaken with Subaru and Kecktelescopes. The study is extended to very smallhalos in the range 10 − · M ⊙ by giving con-sideration to the region within 4 Mpc where, out-side the zone of obscuration, almost every galaxybrighter than M B = −
11 has probably been iden-tified (Karachentsev et al. 2004). Almost all ofthese nearby galaxies have now been sufficientlyobserved with Hubble Space Telescope (HST) thatan accurate distance is available from a measure-ment of the luminosity of the tip of the red giantbranch, the TRGB method (Jacobs et al. 2009).There is now detailed information on the group-ing properties of galaxies down to the scales of as-sociations of dwarfs (Tully et al. 2006). However,before giving attention to those intermediate andlow mass structures the discussion will begin witha few words about a couple of large and familiarsystems.
The Coma Cluster , at a distance of 100 Mpc,is the nearest high latitude example of a richand evolved cluster of galaxies. A binary oftwo dominant ellipticals lies at the center andmost galaxies in the core are early morphologi-3al types. X-ray emission arising from hot in-tracluster gas is relatively uniformly distributed,indicative of a relaxed system, although there isthe clear marker of the infall of a sub-unit withhot gas around NGC 4839, 45 ′ = 1 . M b = − σ giant = 979 ±
30 km s − and σ dwarf = 1096 ±
45 km s − . Extending to a moreextreme range of low surface brightness dwarfswith − . < M R < − .
5, Chiboucas et al.(2010) find a dispersion σ LSB = 1269 ±
126 km s − (but confined to the center of the cluster!). Eachof these differences is at the marginal level of 2 . σ .Our interest is in exploring the boundaries ofthe cluster so we give attention to a velocity sur-vey that is complete over a wide angle. We sum-marize with Figure 1. The top left panel showsthe distribution of all galaxies in the 2MASS Ex-tended Source Catalog (Jarrett et al. 2000) withina large region around the cluster. Here, and withmost other groups/clusters that will be discussed,the plots are in supergalactic coordinates becausethe entities tend to lie near the supergalactic equa-tor. Colors are coded by velocity with galaxieswithin 1000 km s − of the cluster mean in greenand yellow. The panels to the right and immedi-ately below the first panel illustrate the velocitiesof galaxies within 3 ◦ strips through the center ofthe cluster. It can be seen that the cluster is be-ing fed by filaments that extend in the SGB di-rection. The cluster is abruptly bounded in SGL.This characteristic is demonstrated most clearly inthe lower right panel. There is an edge to the clus-ter at 1 . ◦ = 3 Mpc which is tentatively associatedwith the projected second turnaround radius R t .The mass of the system can be estimated fromapplication of the virial theorem with a sample of367 galaxies in the 2MASS Extended Source Cat-alog within 1 . ◦ of the cluster center. The dimen-sion and velocity parameters are calculated usingthe bi-weight estimators discussed by Beers et al.(1990). Here, and with all the groups, velocitiesare deprojected assuming α = 2 .
5. The projectedgravitational radius is R g = 2 .
15 Mpc, the veloc-ity dispersion is σ p = 954 ±
50 km s − , the virial mass is M v = 1 . × M ⊙ , and the mass to Kband light ratio is 127 M ⊙ /L ⊙ . Here, the K bandluminosity of 1 . × L ⊙ includes a correctionof 60% for lost light, a detail that will be discussedin Paper 2 of this series. The Virgo Cluster , at a distance of 16.4 Mpc, isthe nearest entity to attain Abell richness classzero (though not in the Abell catalog). Because ofits proximity, the Virgo Cluster has received par-ticularly detailed attention (Ferrarese et al. 2012).The interior of the cluster is composed of multiplecomponents as manifested by the X-ray distribu-tion (B¨ohringer et al. 1994) and is now somewhatresolved in three dimensions with accurate surfacebrightness fluctuation distances (Mei et al. 2007;Blakeslee et al. 2009). It has long been knownthat the velocity structure of the cluster is com-plex. Huchra (1985) reports that the early typegalaxies lie in close proximity to either M86-M87or M49 and have a roughly gaussian distributionof velocities with dispersion 581 ±
33 km s − whilethe late type galaxies somewhat avoid the core andhave a roughly boxcar distribution of velocitieswith dispersion 817 ±
46 km s − . The most ex-treme blueshifted galaxies, surely legitimate mem-bers of the cluster, are distributed in an east-westband, evidence of a retained memory of a recent in-fall event. Tully & Shaya (1984) discussed the in-fall of galaxies into the Virgo Cluster and pointedout that there will be an influx of spirals into thecluster over the next Hubble time that is compa-rable with the population that is already there. Amajority of these will arrive in the next 3 Gyr.There is an attempt with Figure 2 to illustratethe structure in the vicinity of the Virgo Clus-ter. The sample is an extract from the LyonExtragalactic Database (LEDA) of all galaxiesin the plotted region with velocities less than5000 km s − . The sample was further restrictedto B <
16 to avoid the very large contaminationfrom spurious objects, particularly galactic stars,in a magnitude unlimited sample. The few spuri-ous objects remaining with the
B <
16 restrictionwere removed by hand. Velocities, V LS , are givenin the rest frame of the Local Sheet (Tully et al.2008). The Local Sheet reference frame is similar to the severalalternative variations of the Local Group reference frame.
Top left:
Distribution of galaxies in the vicinity of the Coma Cluster in supergalactic coordinates.Colors are coded by increasing redshift in 1000 km s − intervals from blue through red to black. Top right:
Velocities of galaxies within the central 3 ◦ band in SGL of the top left figure as a function of SGB. Bottomleft:
Velocities of galaxies within the central 3 ◦ band in SGB as a function of SGL. The dotted box in thisand the top right panel enclose galaxies within the collapsed Coma Cluster. Bottom right:
Radial numberdensity gradient in SGL from the Coma Cluster core in the central 3 ◦ band in SGB.5 Fig. 2.— Galaxies in the region of the Virgo Cluster with
B <
Top left: V LS < − in blackand 0 < V LS <
500 km s − in blue. Top right:
Add 500 < V LS < − in green. Mid left:
Add1100 < V LS < − in orange. Mid right:
Only 1100 < V LS < − with those above1500 km s − in red. Bottom left:
Immediate background 3000 < V LS < − . Bottom right:
Entiresample with V LS < − . The circle in each panel is centered on M87, with 6 . ◦ radius.6he plot in the top left panel shows the dis-tribution of galaxies with velocities less than500 km s − . It is seen that most galaxies in thispanel are strongly clustered. The circle is centeredon M87 and has a radius of 6 . ◦ = 2 . R t .The black symbols identify galaxies with negativevelocities. These galaxies cluster tightly over aregion within the circle from the core to the lowerright. The most negative high confidence velocityin the B <
16 sample is V LS = −
592 km s − forIC 3492. The most negative velocity in the unre-stricted sample is V LS = −
820 km s − for VCC846 (Binggeli et al. 1993).The same galaxies are plotted on the top rightpanel and we add those with 500 − − in green. Together, these are the galaxies in thesample that have velocities less than roughly themean for the cluster. Within the 6 . ◦ circle a denseclump of objects is seen at SGL=107, SGB= − − − window picks up a large numberof objects near the supergalactic equator to theleft of the 6 . ◦ circle, part of what is called theVirgo Southern Extension (de Vaucouleurs 1961).The other features that make an appearance inthis panel are parts of what are called the ‘UrsaMajor Cloud’ (center right), the ‘Leo Spur’ (lowerright), and the ‘Virgo-Libra Cloud’ (upper left) inthe Nearby Galaxies (NBG) Atlas (Tully & Fisher1987).The same objects are shown in the mid-leftpanel, now with the addition of galaxies in therange 1100 − − . The purpose for draw-ing attention to this small incremental step in ve-locities is to illustrate that galaxies in the South-ern Extension have modest velocity dispersionswith respect to their immediate neighbors. Thegreen and orange points separate in this region.The structure that has now appeared to the lowerleft is the ‘Crater Cloud’ in the Nearby GalaxiesAtlas.The mid-right panel shows the distribution ofgalaxies with velocities in the range from roughlythe mean to the high velocity extreme of the clus-ter. The main new feature to note is the ap-pearance of de Vaucouleurs’ Virgo W Cluster atSGL=108, SGB= −
7. This significant entity isknown to be in the background at twice the dis- tance of the Virgo Cluster. Its mean velocityis V LS = 2261 km s − and distance is 34 Mpc.Virgo W lies at radii 6 ◦ − ◦ from M87 so itsurely contaminates the quasi-virialized region ofthe Virgo Cluster. Other background contami-nants are known. Both Cepheid and SNIa dis-tances for NGC 4639 and 5 surface brightness fluc-tuation distances places the Virgo W ′ group at23.7 Mpc, 7.3 Mpc directly behind the Virgo Clus-ter. Then the M Cloud (Ftaclas et al. 1984) is de-termined to lie at 35 Mpc, similar to the distancefor Virgo W. Details on current distance measure-ments for these background entities are given byTully et al. (2013) and Karachentsev et al. (2014).In Table 2 of Tully et al., W is group icnt = 56(NBG catalog 11-24). W ′ is group icnt = 5 (NBG11-5), and M is group icnt = 177 (NBG 13+12).In the mid-right panel of Fig. 2 there is a hori-zontal dashed line with a dog-leg. Almost all theknown background contaminants within the 6 . ◦ circle lie below this line. The Virgo Cluster suffersfrom contamination that cannot be distinguishedwith velocity information but known cases lie overa restricted part of the cluster.The lower left panel shows the distribution ofbackground galaxies in the range 3000–5000 km s − .It is seen that the cluster is relatively clear of con-taminants from the near background and whatexists is confined to a band at SGB ∼ −
6, belowthe dogleg dashed line introduced in the previouspanel. These objects can be distinguished withredshifts. The void behind Virgo extends to theGreat Wall at 6000 km s − (and probably accountsfor an apparent peculiar velocity of the cluster to-ward us). Finally, in the lower right panel all theinformation from the previous panels is combined.Conveniently, as with Coma, the current infallinto the Virgo Cluster is along a cardinal axis insupergalactic coordinates. The constraints on thedimension of the cluster is investigated by consid-ering the distribution of galaxies along a strip 12 ◦ wide in SGL centered on M87 and extended inSGB (with V LS < − to minimize con-tamination). The results are shown in Figure 3. Itis seen that there is an abrupt edge to the clusterat 6 . ◦ ± . R t .At larger radii, the Virgo Southern Extensionat more positive values of SGL is well popu-lated and the velocity dispersion is large if one7ig. 3.— Surface density of galaxies ( B <
16 and V LS < ◦ wide centered on M87 in SGLand running from −
29 to +25 in SGB. A discon-tinuity is seen at 6 . ◦ ± . × M ⊙ (Mohayaee & Tully2005). A consistent value is found from a spher-ical infall model (Karachentsev et al. 2014). Nu-merical Action models suggest that infalling galax-ies have orbital angular momentum acquired fromlarge scale tides, arriving in the cluster with radialvelocities of 1500 km s − and tangential compo-nents of ∼
100 km s − (Shaya et al. 1995).The essential point for the current discussion is that the infall and quasi-virialized domains ofthe Virgo Cluster can be distinguished, charac-terized by a projected second turnaround radius R t = 6 . ◦ ± .
4. If, now, a virial analysis iscarried out, with elimination of known or sus-pected contaminants, one finds the parameters < V LS > = 1083 km s − , σ p = 732 ±
45 km s − ,and M v = 7 × M ⊙ . The virial and infallmodel mass estimates are in good agreement. The NGC 5846 Group , at an assumed distance of27.5 Mpc, is dominated by two giant ellipticals.Most known members are early types. This groupwas selected for intensive study as part of theCFHT wide field imaging program (Mahdavi et al.2005) because it is relatively rich (250 galaxiesbrighter than M R = −
11) and lies within a fil-ament that projects near the plane of the skyin front of a substantial void. The following in-formation is drawn from Mahdavi et al. withadjustments due to a revised distance. In thethree panels of Figure 4 one sees the distributionof known galaxies at roughly the distance of thegroup, the density distribution as a function ofradius from the group center, and the velocity dis-persion as a function of this radius. There are cleardensity and velocity dispersion discontinuities at R t = 890 kpc. The group velocity dispersion of σ p = 322 ±
35 km s − is well established by 83measurements. The inferred virial mass for thegroup is 8 × M ⊙ . The NGC 1407 Group , at 26.3 Mpc, almost qual-ifies as a ‘fossil group’ (Ponman et al. 1994) witha difference in magnitude between the dominantelliptical and the second brightest member of 1.4mag (a difference of 2 mag is required to meetthe qualification of a fossil group). ExtendedX-ray emission is seen from a hot intra-groupplasma but it is faint. The group was selected forstudy during the CFHT wide field imaging surveyas a dense accumulation of overwhelmingly earlytype galaxies nicely isolated from foreground andimmediate background confusion (Trentham et al.2006). Some 240 galaxies are associated with thegroup brighter than M R = −
11. Velocities arenow available for 69 of these, including materialfrom Romanowsky et al. (2009) and two veloci-ties given below. Figure 5 illustrates the run ofsurface density and, particularly interesting, of8
Fig. 4.— it Top: Distribution of galaxies inthe vicinity of the NGC 5846 Group. Red:E/S0 galaxies; open blue: spiral/irregular galax-ies. Stars: unknown velocities.
Middle:
Run ofsurface density with radius.
Bottom:
Run of veloc-ity dispersion with radius. Each data point repre-sents the dispersion in the local proximity to eachgalaxy in the sample as described by Mahdavi etal. (2005). Beyond 2 ◦ , the local velocity disper-sion is low. A discontinuity at 1 . ◦ = 890 kpc isseen in each of the two bottom panels. velocity with radius from the dominate centralgalaxy. The group has received considerable at-tention because of the large relative blueshift ofthe second brightest galaxy, NGC 1400 (Gould1993; Quintana et al. 1994). With our recent spec-troscopy with Subaru Telescope we have foundtwo dwarfs with relative blueshifts that are al-most as extreme: Trentham et al. (2006) ID 79with V helio = 656 km s − and ID 96 with V helio =603 km s − . The likely explanation is the re-cent infall of NGC 1400 and a small entourage.NGC 1400 must be near the potential minimumwith a velocity vector fortuitously pointing al-most at us. Inferred properties for the group are R t ∼
900 kpc (uncertain because at the limit ofthe Trentham et al. survey), σ p = 365 ±
44 km s − , M v = 6 × M ⊙ , and M v /L R = 240 M ⊙ /L ⊙ .The NGC 1407 Group is part of the EridanusCloud, itself in close proximity to the Fornax Clus-ter. The dynamical status of the Eridanus regionhas been studied by Brough et al. (2006) who rea-sonably conclude that the structure is bound. Infact, the entire region from Fornax Cluster to theNGC 1407 Group is probably destined to collapsewithin another Hubble time. Figure 6 is a plot ofthe projected distribution of galaxies in the region.Figure 7 compares velocities and distances. Thepoint to the lower left in Fig. 7 stands for the NGC1097 Group which is probably to the foreground.Otherwise, there is little hint of a correlation of ve-locity with distance. Unfortunately, only the dis-tance to the Fornax Cluster (Blakeslee et al. 2009)is known with the sort of precision required for aproper dynamic analysis of the region.In the region, in addition to NGC 1407 andFornax, the Eridanus (NGC 1395) and NGC 1332groups are significant. The Fornax Cluster, at18.8 Mpc, with σ p = 311 ±
44 km s − , and M v = 9 × M ⊙ , has indications of an infallzone at radii less than 5 Mpc (Fig. 6). The Eri-danus and NGC 1332 groups have velocity dis-persions 228 and 205 km s − , respectively, andmasses in the range 1 − × M ⊙ . Globallythe environment includes 3 − × M ⊙ . Fig. 6includes estimates of the radii of first turnaround, R t , around the various groups (justified in a latersection). These circles correspond to spheres thatare strongly overlapping in three-dimensions, evi-dence that the entire region is bound and collaps-ing. The NGC 1407 Group may be a (near) fossil9 Fig. 5.— it Top: Velocities as a function of dis-tance from NGC 1407. The extreme blueshiftedvelocity of the second brightest galaxy in thegroup, NGC 1400, is shared by two dwarfs.
Bot-tom:
Run of the surface density of galaxies withdistance from NGC 1407. Ratings 0 − h m < RA J < h m , − < DE J < − V h < − .Black: 500-800 km s − ; blue: 800-1300 km s − ;green: 1300-1900 km s − ; red: 1900-2400 km s − .Red/blue rings: second and first turnaround radiiaround principal groups. The NGC 5353/4 Group was studied during theCFHT campaign and discussed by Tully & Trentham(2008). Again, the group was selected in part be-cause of its apparent isolation. The panels ofFigure 8 illustrate the situation. The group con-tains 126 galaxies brighter than M R = −
11, 53with known velocities. A discontinuity identifiesthe second turnaround at R t = 690 kpc if thedistance is 34.7 Mpc. The velocity dispersion is209 ±
29 km s − and the virial mass is 3 × M ⊙ .A particular interest with this group is themixed morphology. A core of early type galax-ies has apparently recently received an influx oflate type systems. There is an unusually largenumber of transition dwarfs; small galaxies with10ig. 7.— Distance moduli vs. Local Sheet ve-locities. Points with error bars: principal groups;crosses: individual galaxies in the field. Red: near-est Fornax; blue nearest NGC 1407.dE morphologies but with emission line or A starabsorption features. The NGC 1023 Group was imaged at CFHT overan extended region (Trentham & Tully 2009). Seethe distribution of galaxies on the sky with Fig-ure 9 and the runs of surface density and velocityin Figure 10. The statistics are poor with sucha small group but R t around NGC 1023 can beidentified at 350 kpc. Within this radius there are41 suspected members, 18 with known velocities.At a distance of 10.1 Mpc, R t = 350 kpc, σ p =133 ±
31 km s − , and the virial mass is 5 × M ⊙ .The survey region was extensive enough to doc-ument the ”Dressler effect” (Dressler 1980); theprominence of early types near the core and latetypes at large radii. Specifically, the transition canbe associated with the radius R t . Groups with mass below 5 × M ⊙ are sopoorly populated that they are hard to study at
212 210 208 206384042
Fig. 8.—
Top:
Distribution of galaxies in thevicinity of the NGC 5353/54 Group. Dark Blue:galaxies with velocities associated with the group.Cyan: foreground; green and red: background.Black crosses: unknown velocities. Dashed box:region of CFHT Megacam survey.
Bottom:
Runof surface density with radius. A discontinuity at1 . ◦ = 690 kpc is seen in the bottom panel. Acircle of this radius is superimposed on the toppanel.11 Fig. 9.— Distribution of galaxies in the vicinityof the NGC 1023 Group. The irregular box is theCFHT survey region. Red: early types; blue: latetypes; green: transition or uncertain types.
Fig. 10.—
Top:
Run of surface density with ra-dius.
Bottom:
Velocities as a function of radius.A discontinuity in density and transition to lowvelocity dispersion occurs at 2 ◦ = 350 kpc. A cir-cle of this radius is drawn around NGC 1023 inFig. 9.12arge distances, yet it is theoretically anticipatedand observed in our own vicinity that such groupsare common. Thanks to observations with HST,there is now a good understanding of the detaileddistribution of galaxies within 4 Mpc, with dis-tance measures with uncertainties less than 200kpc. For most of these systems there are veloci-ties accurate to ∼ − from HI observationswith radio telescopes. Within 4 Mpc at unob-scured Galactic latitudes there are four well knowncomplexes of galaxies: those of the Local Group,around Centaurus A, M81, and in the constella-tion of Sculptor. In addition, there are three as-sociations containing only dwarf galaxies. Theseassociations will be discussed in a later section. The Milky Way and M31 have colloquially beensaid to be members of the Local Group but it isnow understood that these two large galaxies liein separate halos within a common infall zone.Figure 11 illustrates the three-dimensional dis-tribution of the nearest galaxies. The massesof the two dominant halos can be probed bythe motions of gas, stars, and satellite galaxieswith greater detail than in more distant environ-ments. Recent estimate of the virial mass of theMilky Way range from 1 . × M ⊙ (Xue et al.2008) to 1 . × M ⊙ (Wilkinson & Evans1999). Recent estimates for M31 range from0 . × M ⊙ (Geehan et al. 2006) to 1 . × M ⊙ (Klypin et al. 2002), a slightly lowerrange than for the Milky Way even though M31is 40–50% more luminous. Recent observations ofthe proper motion of M31 help to restrict the rangeof plausible masses. The constraints have been re-viewed by van der Marel et al. (2012b) who findvirial masses of 1 . ± . × M ⊙ for M31 and1 . × M ⊙ for the Milky Way with more am-biguous uncertainty.The Milky Way and M31 second turnaroundhalos are discrete but the two galaxies lie within acommon infall zone approximated by the largercircles in Fig. 11. The infall is clearly seen inFigure 12. Velocities of galaxies in the LocalSheet rest frame are plotted as a function of three-dimensional distance from the nearer of the MilkyWay or M31. It is seen that galaxies within 280kpc of either of the dominant galaxies are usu-ally early type and disperse in velocity. Galaxiesat larger radii are usually late type and manifesta pattern in velocity characteristic of infall. The pattern suggests a zero velocity (first turnaround)surface at r t ∼
940 kpc. There are a couple ofsystems at intermediate radii that depart from theinfall pattern. NGC 6822 has probably taken awide swing around the Milky Way and Pegasushas probably passed once by M31.The discovery that many satellites of M31 areconfined to a thin plane with the kinematic sig-nature of rotation (Ibata et al. 2013; Conn et al.2013) adds an interesting twist to ideas aboutthe formation of the Local Group. Shaya & Tully(2013) suggest that almost all local satellites lie infour planes. They construct plausible orbits thatsuggest the satellites formed in strata as a wall wasbuilt around the evacuating Local Void. Massesare being increasingly constrained as proper mo-tions are measured.
The Centaurus A complex is dominated by the pe-culiar elliptical galaxy NGC 5128 = Cen A and thebeautiful spiral galaxy NGC 5236 = M83. Thereare now extensive HST observations of individualgalaxies throughout the region, with the resultthat good distances are known for most systemsfrom Tip of the Red Giant Branch (TRGB) lumi-nosities.The structure of the complex has been de-scribed by Karachentsev et al. (2002b, 2007).Figure 13 shows the distribution of galaxies onthe sky. It is seen that there are many earlytypes (Sa and earlier) around Cen A (SGL=160,SGB= −
5) and late types (later than Sa) aroundM83 (SGL=148, SGB=+1). The Cen A and M83families blend in a projection from our vantagepoint and in their velocities. We are remindedfrom the contours of extinction that the Cen Acomplex lies near the zone of avoidance and theremay be loss of information at SGL > ∼ . | b | < ◦ lie within the wedges centered on SGY= 0. MWM31Combined
Fig. 12.— Velocities of the nearest galaxies vs.the distance from the Milky Way (top) or M31(middle), whichever is nearer. The combined datais shown in the bottom panel. Open red: earlytypes; green star or cross: transition types; filledblue: late types. Galaxies within 280 kpc of oneof the two dominant galaxies have large velocitydispersions and are typically early types. Galax-ies at larger radii are predominantly late typesand have the velocity characteristics of first infallwithin ∼
940 kpc and cosmic expansion outsidethis radius.14ig. 13.— Projected distribution of galaxies in thevicinity of Cen A and M83 in super galactic coor-dinates. Early types identified by red squares andlate types identified by blue circles. Filled symbolsare within 1 Mpc of either Cen A or M83 in 3D.Crosses: uncertain distances. Obscuration con-tours are shown at factor 2 increments, with thedark contour at the level A B = 0 .
5. The Galacticequator lies at roughly SGL=180.
Cen AM83 N4945Cen A N5102N4945N5206 E383-87
Fig. 14.—
Top.
Distribution of galaxies in SGX vsSGY using distance information. The sample sep-arates into swarms around Cen A and M83 witha separation between centers of about 1.5 Mpc.
Bottom.
Distribution in SGY vs SGZ (edge-on tolocal structure) of all galaxies in the frame withmeasured distances within 3.8 Mpc in SGX. Sym-bol codes in the two panels have the same meaningas in Fig. 13. The Local Void extends above theCen A complex at positive SGZ.15hich are more informative but also have largererrors than projected distances. Within 600 kpcof Cen A there is a dispersion in 16 velocities of121 ±
30 km s − (unfortunately velocities are notknown for most of the dwarf spheroidals) and thevirial mass is 8 × M ⊙ . The dispersion ofvelocities for 9 galaxies around M83 is a smaller75 ±
25 km s − and the inferred virial mass is2 × M ⊙ . The systemic velocities of the two biggalaxies and of their groups are the same withinthe uncertainties. M83 and its companions arenear the zero velocity surface of the larger Cen Ahalo. The two currently distinct groups are on thebrink of falling together.Given the attention being received by the ap-parent adherence of satellites around M31 andthe Milky Way to thin planes (Ibata et al. 2013;Shaya & Tully 2013) it would be negligent to ig-nore the apparent occurrence of the phenomenonin the vicinity of Cen A seen in the lower panel ofFigure 14. TRGB distances have r.m.s. 5% uncer-tainties and every observed galaxy in the vicinityof Cen A is included in the figure (most of errorsin distance project into the SGX axis, normal tothe figure; almost none projects into SGZ). All thegalaxies with the possible exception of the mostdistant cases (NGC 5102 and ESO 383-87) appearto lie in two planes. Each plane has a long axisof about 600 kpc and a thickness of 100 kpc. Thethickness is reduced slightly from a vantage pointabout 5 ◦ above the SGX axis but already the pat-tern is striking in this view along the SGX cardinalaxis. In the case of the local planes, Shaya & Tullydrew attention to the probable role of evacuationfrom the Local Void in the development of layer-ing of structure. The proximity of the Local Voidis identified in the SGY − SGZ plot of Figure 14.The orientations of the two posited Cen A satelliteplanes suggest that they, like their local cousins,developed out of the evacuation of the void.
The M81 complex at 3.6 Mpc has traditionally in-cluded the region displayed in the top panel of Fig-ure 16. The nearest important neighbor is IC 342at a distance of 1.9 Mpc from M81. IC 342 ispart of the Maffei complex, off the right edge ofthe field shown in Fig. 16, in the direction to-ward the Galactic plane. Good distances are nowavailable for the major galaxies in the neighboringIC342/Maffei structure (Wu et al. 2014) but theregion is poorly studied because of obscuration so Fig. 15.— Velocities in the Local Sheet rest framewith 3D distance from Cen A. Red, early; blue,late and the major galaxies are given larger sym-bols. The majority of gas-deficient dwarfs do nothave known velocities so cannot be included in thisfigure. The dashed line at the velocity of Cen Aextends to 800 kpc, the domain of high velocitydispersion around Cen A.16ig. 16.— Galaxies in the plane of the sky inthe region of M81. A wide view is shown in thetop panel while the bottom panel zooms into thecentral region of the group. In the top panel, pre-viously known early types are in red, late typesare in blue, and members identified by Chiboucaset al. are in green. Contours of galactic extinc-tion are given in grey. The distorted rectangle isan outline of the CFHT imaging survey. In thebottom panel, early type galaxies that are new orpreviously known are colored red and orange re-spectively, likely tidal dwarfs near M81 are repre-sented by green triangles, while late type systemsare located by open squares and three newly iden-tified Blue Compact Dwarfs are labelled. Fig. 17.— Velocities with respect to the lumi-nosity weighted mean of M81-M82-NGC3077 as afunction of 3D distance. Open symbols: galax-ies within a projected radius of 370 kpc of M81.Galaxies at 3D radii 0.5 – 1.3 Mpc display the pat-tern expected of infall. The vertical dotted linesidentify the first and second turnaround radii, r t and r t .17ill not be given attention in this discussion.TRGB measurements based on HST observa-tions (Karachentsev et al. 2002a; Chiboucas et al.2013) provide a detailed three dimensional pictureof the M81 region. It is seen between Figures 16and 17 that there is a concentration of galaxiesaround M81 with a substantial dispersion in veloc-ities and there is a domain extending to ∼ . r t . Thetransition from negative to positive velocities rel-ative to the M81 group velocity centroid occurs ata radius of 1.41 Mpc and is taken to be the radiusof first turnaround, r t . The line-of-sight velocitydispersion from 23 galaxies within the core regionis 111 ±
23 km/s and the virial mass is 3 × M ⊙ . The Sculptor complex has NGC 253 at its core, 3.6Mpc from us. The three dimensional distributionof galaxies in the region are shown in the two pro-jections of Figure 18. Prior to the availability ofgood distances, NGC 55 and NGC 300 had beenconsidered part of an NGC 253 group. There is amore recent view that the complex is an extendedfilament (Jerjen et al. 1998; Karachentsev et al.2003).Rather, the excellent distances provided by theTRGB method show conclusively that NGC 55and NGC 300 are members of an entity that hasbeen called an association of dwarfs (Tully et al.2006). That association is quite distinct from theNGC 253 Group. The entity feels a force of attrac-tion from the combination of M31 and our Galaxythat is roughly three times the force from the his-torical Sculptor group. This association of dwarfswill be given attention in the next section.There is evidence for a collapsed region aroundNGC 253 but the numbers are very small. In Fig-ure 19 it is seen that two or three gas-rich galax-ies, including the intermediate-size NGC 247, arewithin the infall regime. The velocities of the in-falling galaxies provide an envelope for the velocitydispersion so it can be inferred that the velocitydispersion of satellites is at least ∼
70 km s − .The two satellites at slightly positive relative ve-locities in Fig. 19 that lie at 350 and 450 kpc fromNGC 253 are transition types, with the morpho-logical appearance of dE but detectable amounts of HI gas (Bouchard et al. 2005). These distancesof 400 ±
50 kpc from NGC 253 are rather large tobe within the second turnaround but the distancesare not in doubt because the sky projection com-ponents are dominant. The line-of-sight velocitydispersion of the four nearest galaxies to NGC 253is 47 km s − , obviously an uncertain number. Within the radius of 4 Mpc of our position thatincludes the halos containing Cen A, M81, andNGC 253 (and the obscured Maffei-IC342 com-plex) there are four or five entities that appearbound but include only small galaxies (Tully et al.2006). The characteristic radii of these entities of ∼
300 kpc is too large to suppose that they arerelaxed. In the main, it is suspected that theseassociations are still forming, with motions domi-nated by infall.In two of these nearby cases, the process ofcollapse seems sufficiently advanced and the num-bers of participants are sufficient to warrant a briefanalysis. One case involves the M B = − . ∼
100 kpcradius. The velocity dispersion for these fourgalaxies is σ p = 49 km s − and the virial massestimate is 3 × M ⊙ . Including all 8 galaxiesin the analysis, the bi-weight velocity dispersion is51 ±
18 km s − , the 3D gravitational radius dou-bles to 435 kpc, and the virial mass increases to7 × M ⊙ . The luminosity at B band is only1 . × L ⊙ so the implied mass-to-light ratio ishigh: M/L B ∼
230 to 470 M ⊙ /L ⊙ .18ig. 18.— Two supergalactic projections of galaxies in the Sculptor direction. The principal concentrationis around the dominant galaxy NGC 253. There is a secondary concentration associated with NGC 55. Theproximity to the Milky Way and M31 galaxies is seen.Fig. 19.— Velocities with respect to the dominant galaxy NGC 253. There is a hint of an infall regionwithin 800 kpc. 19 Fig. 20.— Two supergalactic projections of galaxies in the NGC 4214 association (larger red filled circles).The adjacent complexes include 3 other dwarf associations: NGC 3109 (blue triangles), 14+8 (green enclosingboxes), and ”dregs” (cyan stars). 20igure 20 shows the relationship between theNGC4214 association and other nearby galaxies.Color and special symbols draw attention to threeother smaller associations of dwarfs in the vicinitydiscussed by Tully et al. (2006). The NGC 3109association of 4 galaxies (14+12 group) is nearestthe Milky Way. Then quite near the NGC 4214group are four dwarf galaxies identified with the14+8 group and 4 more galaxies strung out over asufficient length that they are referred to as ‘dregs’in Tully et al. (2006).The other case worth attention involves fiveclosely associated galaxies and two outliers in-cluding NGC 55 ( M B = − .
8, type Sdm) andNGC 300 ( M B = − .
7, type Sd). The proximityof these galaxies to NGC 253 was shown in Fig. 18but M31 and our Galaxy have the greater gravi-tational influence. The velocity dispersion for the7 galaxies is 37 ±
14 km s − and virial mass es-timates, whether the inner 5 or the full 7, rangeover 3 − × M ⊙ .
3. Scaling Relations
As a prologue to the ensuing discussion, thereader is reminded that dimensions and masseshave been calculated from direct measurements ofdistances without recourse to systemic velocitiesand an assumption about the Hubble Constant.Of course, the distance scale is established by azero point set by nearby calibrators. The distancesare compatible with H = 75 km s − Mpc − (Sorce et al. 2012), so rescaling that would al-ter the choice of this parameter is monitored by h = H / R t and σ p would be di-rectly correlated. The relationship is seen in Fig-ure 21. The constant of proportionality is foundempirically to be 368 h ± − Mpc − .Two interesting relations are shown in Figures22 and 23. Each involves the virial mass calculatedaccording to M v = σ D r g G = απσ p R g G . (9)The parameter α that describes the nature of or-bits is given the value α = 2 . r g = ( π/ R g was defined in Eq. 1 so is ComaVirgoN5846N1407N5353/4N1023M81M31
Fig. 21.— Correlation between projected secondturnaround radius R t and the radial velocity dis-persion of galaxies within this radius σ p .calculated independently from r t = p / R t .What is being plotted in Fig. 22 is R t ∝ σ / p R / g .The derived correlation is R t = 0 . M / h − / Mpc (10)if the virial mass is in units of 10 M ⊙ . Thestandard deviation of the mean fit for the coef-ficient is 0 . r t = p / R t =0 . M / h − / Mpc. By comparison, the ra-dius r that encloses a mean density 200 timesthe critical density is r = 0 . M / h − / , so r t = 1 . r .The tight correlation implies the close correla-tions between projected dimensions R t = 1 . R g and statistically derived 3D dimensions r t =0 . r g . It follows that the correlation seen inFig. 23 would exist. In this plot the slope is fixedand the zero-point is determined by the relationsseen in Figs. 21 and 22. The mass–dispersion re-lation for groups obeys the law M v = 2 . × σ p h − M ⊙ (11)21ig. 22.— Correlation between the projected sec-ond turnaround radius R t and the virial mass cal-culated for galaxies within this radius M v .Fig. 23.— Correlation between the virial mass andthe velocity dispersion of galaxies within r t . where mass M v is in solar units and velocity dis-persion σ p is in km s − .
4. The Radius of First Turnaround
A measurement of the radius of first turnaround r t requires 3D information to distinguish thedecoupling of the infall zone from cosmic ex-pansion. Such information is currently availablein only a few cases. In the case of the tra-ditional Local Group, an initial discussion bySandage (1986) was carried forward most recentlyby Karachentsev et al. (2009). Graphical resultsdrawn from that reference are shown in Figure 24.The value of r t = 940 kpc displayed graphicallyin Figure 12 seems reasonably constrained to anuncertainty of 10% and encloses a mass of roughly3 × M ⊙ . However spherical symmetry is apoor approximation for the zero velocity surfaceof the Local Association because of the dumbbelldistribution of mass between M31 and the MilkyWay. Moreover, only 6 galaxies are well placed todefine the zero velocity surface. D LG , Mpc V L G , k m s − MW Fig. 24.— Velocities of nearby galaxies in the Lo-cal Group rest frame. The nearest galaxies have alarge velocity dispersion. Galaxies beyond 940 kpcare systematically redshifted. The solid curve is anempirical fit to galaxys beyond 900 kpc.The M81 Group offers a second opportunity tolocate a zero velocity surface. It is seen in Fig-ure 17 that r t ∼ . ∼ × M ⊙ . Then a third case is given bythe Virgo Cluster where ∼ × M ⊙ is enclosedwithin a zero velocity surface r t = 7 . ± . r t and themass within this radius: r t = 0 . M / h − / Mpc . (12)with a mean deviation of ± .
07 with three cases.This empirical determination can be comparedwith a simplification of the theoretical dependencyfor spherical collapse (Gary Mamon, private com-munication): r theory t = 0 . M / h − / (Ω Λ / . / Mpc . (13)
5. What is a Group?
It was asked in the introduction how ob-servers should identify groups. Most usefully, itshould approximate the definitions given by mod-elers. The parameter r t , the radius of secondturnaround, is very close to the parameter r ,the radius enclosing an overdensity of 200 timesclosure density, or r g , the gravitational radius. Amodeler identifies collapsed halos with dimensionsthat resemble r t . This is a dimension commonlyassociated with rich clusters. If it is accepted thatthe terms ‘group’ and ‘cluster’ are interchange-able, then the dimension an observer can use todefine collapsed structure is r t .By this measure, the Local Group is two groups,one around M31 and one around the Milky Way.Mass derivations result in similar estimates foreach of MW and M31 of 1 . × M ⊙ . Itwould follow that r t ∼
280 kpc for both groups.The pair are falling together, hence lie within acommon first turnaround surface enclosing ∼ × M ⊙ . Figure 11 illustrates the projected po-sitions of galaxies in the region and it is seen thatsatellites of early type are predominant within 280kpc of M31 and the Milky Way while satellites oflate type are predominant at larger radii. Fig-ure 12 shows radial velocities and it is seen thatsatellites within 280 kpc of one of the major galax-ies have large motions, positive and negative, whilethe outlying galaxies mostly display the character-istics of infall.The radius of first turnaround, r t can be a use-ful construct. It is proposed that entities definedby this radius be called ‘associations’. We live in the Milky Way Group in the Local Association. Ofcourse, since they are within a common infall en-velope, the Milky Way and M31 halos are destinedto merge. The current estimate is that first perias-tron is 3.9 Gyr in the future (van der Marel et al.2012a). It turns out that most other nearby struc-tures that have been called groups break up intomultiple parts. The M81 Association contains theM81 Group, the NGC 2403 Group, and a commoninfall region. The Centaurus Association containsthe Cen A Group and the M83 Group as majorcomponents. There is the Maffei–IC 342 Associa-tion.The ‘Main Disturber’ tidal index in the UpdatedNearby Galaxies Catalog (Karachentsev et al.2013) provides a description that approximatesthe definition of association given here. Makarov & Karachentsev(2011) have compiled a catalog of groups thatare reasonably consistent in scale with a secondturnaround definition. The confusion in the def-inition of the word ‘group’ is implicitly acknowl-edged in the study by Karachentsev (2005) of thenearby extragalactic structure. In that work, tra-ditional ‘groups’ are referred to as complexes orfilaments or clouds and the collapsed regions arecalled groups.
6. Dark Matter, Dark Energy, and LocalDynamics
The relationships between r t or r t and themasses internal to those radii are virtually inde-pendent of the dark energy content of the universe.However, because of the relative differences withrespect to the age of the universe between the on-set of collapse that is associated with r t and thecurrent age associated with r t , the ratio r t /r t has a dependence on the dark energy content.The equations 10 and 12 can be combined, afteradjustment for projection, to give r t /r t = 3 . ± . . (14)This ratio can be compared with a theoreti-cal prediction, never published but kindly pro-vided by Gary Mamon. The theoretical anal-ysis involves the virial radius in place of thesecond turnaround radius. The ratio of thetwo turnaround radii as a function of matterdensity are calculated by Mamon using alter-native approximations of the virial density by23itayama & Suto (1996) and Bryan & Norman(1998). Wojtak et al. (2005) provide a link be-tween virial and second turnaround radii. Theydefine the virial radius to enclose a region with101.9 times the critical density. They then deter-mine the following relation from simulated halosin the mass range 10 − M ⊙ : r v = 0 . M / h − / Mpc . (15)This formulation is very close to the 3D adjustedversion of Eq. 10 for the second turnaround radius.Hence r t ≃ . r v .Assuming the sum of matter and vacuum en-ergy densities add to the critical density, the ra-tio r t /r t ranges from 3.1 if matter density is ex-tremely low to 3.7 in the Einstein-de Sitter case.The observed ratio of 3.14 corresponds to the caseΩ matter = 0 .
15. Figure 25 illustrates the viabilityof alternative density parameters. The 68% prob-ability domain is constrained to Ω matter < . matter = 1 is only rejected with a sig-nificance of ∼ σ . We have here an interestingpotential test of the local influence of dark energybut the current uncertainties are very large.Another quantity of potential interest is thesurface of zero gravity, the surface around an over-dense region that is at the limit of what is des-tined to ever collapse onto the overdensity. Thissurface is well specified as a function of cosmolog-ical model in the case of spherical collapse. WithΩ Λ = 0 . r ZG ∼ . r t (Peirani & de Freitas Pacheco 2008). However itwas mentioned already with the discussion of r t that spherical collapse is generally a bad approx-imation because of the neghboring structures. Itmust be a very bad approximation at r ZG .
7. Evolved Groups, Spiral Groups, andDwarf Associations
Familiar groups can be roughly catagorizedby whether the majority of luminous ( M R < −
17) galaxies are elliptical/lenticular or spi-ral/irregular. Those dominated by early mor-phological types are considered to be dynamicallyevolved while those dominated by late types, areconsidered to be at an early stage of developmentthrough merging. The virial mass to blue lightratio, M v /L B , of groups is strongly correlatedwith both the virial mass and the morphological Fig. 25.— Measure of agreement between ob-served ratio of first and second turnaround radiiand expectation value as a function of matter den-sity in a universe with flat topology. The bestfit value of r t /r t = 3 .
14 implies Ω matter ∼ . σ the correspondence isΩ matter ∼ . L B = 3 . × M . exp − . /M . (16)Rather than being described by a continuouscurve, though, the distribution might be consid-ered to consist of three regimes. Groups domi-nated by early types typically have M v /L B valuesof several hundred in solar units. Groups domi-nated by late type systems, like the MW and M31groups, typically have M v /L B values below ornear 100. The prominence of young hot stars re-sults in a lot of light per unit mass. In the regimeof low mass groups of dwarfs M v /L B values canbe very large, suggesting inefficiency in convertingbaryons into stars.Fig. 26.— Virial mass vs. blue light for groups ofgalaxies in the Local Supercluster. Red: groupsdominated by early types; blue: groups in the ma-jority late types; green: associations consisting ofonly dwarf galaxies. The dashed line illustrates M/L B = 100. The curve is described by Eq. 16.Groups of both the early and late types werestudied during the CFHT Megacam imaging cam-paign. The completeness limit for membershipstudies was typically M R = −
11 or fainter. In Fig. 27.— Number of group members as a func-tion of the mass of parent halo. Red: only dwarfswith − < M R < −
11. Grey: only giants with M R < −
17. Solid line: correlation found with thedwarf sample: consistent with a constant numberof dwarfs per unit parent halo mass.an effort to study the properties and possible vari-ations in the luminosity function, galaxies wereseparated into two bins: ‘giants’ with M R < − − < M R < −
11. Small butsignificant variations in luminosity functions werefound. The variations could be characterized byvariations in the ratio of dwarfs to giants. Thegroups of predominantly early types have largerdwarf/giant ratios. An interesting result is foundif the number of giants or dwarfs is plotted againstthe group virial mass. The correlation is poor withgiants but very pronounced with dwarfs. The re-sult is shown in Figure 27. The straight line de-scribes the relation N d = 5 . M . ± . (17)where N d is the number of dwarfs with − < − M R < −
11 in the group. The slope is consis-tent within the errors with unity. The number ofdwarfs per unit halo mass is roughly constant. Acount of the number of dwarfs is a good measureof the mass of a group halo.25 . Summary
Halos can be observationally delineated by adensity drop and a transition between dispersedorbits and infall at the radii of second turnaround, r t . The scaling relations that are theoretically an-ticipated to exist over a wide range of halo massesare confirmed from observations of nearby col-lapsed regions. Specifically: • Over the mass range from the halo of M31to the Coma Cluster there is a correlation be-tween projected second turnaround radius R t and line-of-sight velocity dispersion σ p : σ p /R t =368 h km s − Mpc − . • Over the same mass range, there is thecorrelation between R t and the virial mass M measured in units of 10 M ⊙ : R t =0 . M / h − / Mpc. • These two correlations can be combined togive the alternative, but not independent, relation: M v /M ⊙ = 2 . × σ p h − .In a few nearby groups distance informa-tion is sufficient to constrain the radius of firstturnaround r t , the zero velocity surface be-tween infall and cosmic expansion. Three reason-ably studied collapse regions define the relation r t = 0 . M / h − / Mpc. The ratio of firstand second turnaround radii depends on the paceof the cosmic clock set by the matter-energy den-sity of the universe. At present the constraintsare not tight but a tentative measurement favorsΩ matter ∼ .
15 if the universe is topologically flat.Among the groups large and small that havebeen discussed in some detail, a couple of partic-ularly interesting details are worth highlighting.The NGC 1407 Group has drawn attention be-cause of the remarkably large blueshift of NGC1400. It is reported here that two additional mem-bers of the group, small dwarfs, now have mea-sured relative blueshifts that are almost as ex-treme. Then, given the attention that is beinggiven to the discovery that M31 satellites lie inthin planes, it is noteworthy to see a related be-havior in the galaxies around Centaurus A.Dwarf galaxies serve as markers of collapsedhalos because they are numerous. Dwarfs andgiants alike that are located within the radii ofsecond turnaround tend to be gas-poor ‘early’types while galaxies of all sizes outside the sec- ond turnaround tend to be gas-rich ‘late’ types.The transition between predominantly early andlate populations is another indicator of the radiusof second turnaround.Dwarf galaxies serve as markers for collapsedhalos in a way that is surprisingly quantitative. Ifthe number of dwarf galaxies with − < M R < −
11 are counted, then the number of these dwarfsin a halo depends linearly on the mass of the halo.Count the number of dwarfs and one has a measureof the mass of the halo.There are differences in the luminosity functionof galaxies with environment (Trentham & Tully2009). More dynamically evolved environmentshave somewhat steeper faint end slopes. However,it appears that the reason is not more dwarfs perunit mass in evolved environments but fewer inter-mediate luminosity systems in the vicinity of L ⋆ ,the luminosity that characterizes the exponentialcutoff from the faint end power law distribution. Itis suspected that intermediate luminosity systemsare being lost through mergers with the centraldominant galaxy. At faint luminosities, the pro-duction and depletion mechanisms are such thatthe number of dwarfs per unit halo mass remainsroughly constant.In a second paper in this series, the scaling re-lationships that have been defined in this studywill be applied to the compilation of a group cata-log of galaxies from the 2MASS redshift survey ofHuchra et al. (2012). Acknowledgements.
The research that has beendescribed has been realized with the contributionsof close collaborators Kristin Chiboucas, BradJacobs, Igor Karachentsev, Andi Mahdavi, EdShaya, and Neil Trentham. Special thanks to GaryMamon for enlightening conversations about in-fall timing and caustics (Gary, please publish thatpaper on infall timing). Ground-based observa-tions have been made with the Canada-France-Hawaii, Keck, and Subaru telescopes. Supportover the many years leading to this paper hasbeen provided by the US National Science Founda-tion and several awards from the Space TelescopeScience Institute in connection with observationswith Hubble Space Telescope.
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