A Catalog of Globular Cluster Systems: What Determines the Size of a Galaxy's Globular Cluster Population?
aa r X i v : . [ a s t r o - ph . GA ] J un Draft version June 11, 2013
Preprint typeset using L A TEX style emulateapj v. 5/2/11
A CATALOG OF GLOBULAR CLUSTER SYSTEMS: WHAT DETERMINES THE SIZE OF A GALAXY’SGLOBULAR CLUSTER POPULATION?
William E. Harris
Department of Physics & Astronomy, McMaster University, Hamilton ON L8S 4M1, Canada
Gretchen L. H. Harris
Department of Physics & Astronomy, University of Waterloo, Waterloo ON N2L 3G1, Canada
Matthew Alessi
Department of Physics & Astronomy, McMaster University, Hamilton ON L8S 4M1, Canada
Draft version June 11, 2013
ABSTRACTWe present a catalog of 422 galaxies with published measurements of their globular cluster (GC)populations. Of these, 248 are E galaxies, 93 are S0 galaxies, and 81 are spirals or irregulars. Amongvarious correlations of the total number of GCs with other global galaxy properties, we find that N GC correlates well though nonlinearly with the dynamical mass of the galaxy bulge M dyn = 4 σ e R e /G ,where σ e is the central velocity dispersion and R e the effective radius of the galaxy light profile. Wealso present updated versions of the GC specific frequency S N and specific mass S M versus host galaxyluminosity and baryonic mass. These graphs exhibit the previously known U-shape: highest S N or S M values occur for either dwarfs or supergiants, but in the midrange of galaxy size (10 − L ⊙ )the GC numbers fall along a well defined baseline value of S N ≃ S M = 0 .
1, similar among allgalaxy types. Along with other recent discussions, we suggest that this trend may represent the effectsof feedback, which systematically inhibited early star formation at either very low or very high galaxymass, but which had its minimum effect for intermediate masses. Our results strongly reinforce recentproposals that GC formation efficiency appears to be most nearly proportional to the galaxy halomass M halo . The mean “absolute” efficiency ratio for GC formation that we derive from the catalogdata is M GCS /M halo = 6 × − . We suggest that the galaxy-to-galaxy scatter around this meanvalue may arise in part because of differences in the relative timing of GC formation versus field-starformation. Finally, we find that an excellent empirical predictor of total GC population for galaxiesof all luminosities is N GC ∼ ( R e σ e ) . , a result consistent with Fundamental Plane scaling relations. Subject headings: galaxies: general – galaxies: star clusters – globular clusters: general INTRODUCTIONA globular cluster system (GCS) is the ensemble ofall such star clusters within a given galaxy. The historyof GCS studies in the astronomical literature can prop-erly be said to begin with Shapley’s (1918) work on theMilky Way GCS, which he used to make the first reliableestimate of the distance to the Galactic center. Nextpioneering steps were taken with reconnaisance of theM31 GCS (Hubble 1932; Kron & Mayall 1960; Vetesnik1962, among others) and the other Local Group galaxies(see Harris & Racine 1979; Harris 1991, for reviews ofthis early history). However, it was not until discoveryand measurement of the rich GC populations around theVirgo elliptical galaxies had begun (Baum 1955; Sandage1968; Racine 1968; Hanes 1977; Harris & Smith 1976;Strom et al. 1981) that GCS studies began to emerge asa distinct field.It is now realized that virtually all galaxies more lu-minous than ∼ × L ⊙ (that is, all but the tiniestdwarfs) contain old globular clusters, and that these mas-sive, compact star clusters represent a common thread [email protected]@[email protected] in the earliest star formation history in every type ofgalaxy. The first ‘catalog’ of GCSs (Harris & Racine1979) listed just 27 galaxies, all from either the Lo-cal Group or the Virgo cluster. By 1991 the num-ber had grown to 60 (Harris 1991) and a decade laterto 73 (Harris & Harris 2000), with the sample startingto include galaxies in a wider range of environments.Other compilations for different purposes were put to-gether by Brodie & Strader (2006), Peng et al. (2008),Spitler et al. (2008), and Georgiev et al. (2010).In the past decade, many new surveys of GCSs forgalaxies throughout the nearby universe have takenplace. The relevance of GCS properties to understand-ing galactic structure and early evolution is becomingincreasingly apparent, so the construction of a completenew catalog is well justified. A new list may reveal large-scale trends of GCS properties with galaxy type or en-vironment, and may also provide a springboard for de-signing new studies. Perhaps the most basic question,and one that dates back decades, is simply to ask whatdetermines the total population of GCs in a galaxy. Thetotal GC population size, N GC , must relate to the GCformation efficiency relative to the field-star populationas well as to the later dynamical evolution of the sys-tem. In this paper, we address these questions by using Harris Fig. 1.—
Distribution by distance of the 417 galaxies in thecatalog. For comparison, the Virgo cluster is at D = 16 Mpc andthe Coma cluster at D = 100 Mpc. a newly constructed GCS catalog to search for correla-tions of cluster population size with several other globalproperties of their host galaxies. THE DATA SAMPLE AND A GCS CATALOGWe have carried out an extensive literature search tofind published studies of galaxies that, at a minimum,give some useful, quantitative information for the totalnumber N GC of its globular clusters. Much of this mate-rial now comes from a few recent major surveys that havethe distinct advantage of being internally homogeneous,such as for the Virgo cluster galaxies (Peng et al. 2006),the Fornax cluster (Villegas et al. 2010), nearby dwarfgalaxies (Lotz et al. 2004; Georgiev et al. 2008, 2009),nearby E and S0 galaxies (Kundu & Whitmore 2001a,b;Larsen et al. 2001), and supergiant E galaxies (Blakeslee1997, 1999; Harris et al. 2006; Harris 2009). The Hub-ble Space Telescope (HST) cameras including WFPC2,ACS, and most recently WFC3 have provided a power-ful stimulus for the imaging and photometry that suchsurveys depend on.However, dozens of other GCS studies exist that arewidely scattered through the literature and we havesearched thoroughly for these as well. In total we haveextracted GCS data from 112 papers published up to De-cember 2012. These yield N GC measurements for 422 in-dividual galaxies, a dramatic increase over previous com-pilations.The distribution by distance for the galaxies in the cat-alog is shown in Figure 1. Here, the predominance of tar-gets in the Virgo-to-Fornax distance range (15 −
25 Mpc)is obvious, but the numbers of more remote galaxies aregradually increasing as new work goes on. In general,the HST cameras can reach GCSs reliably for D . D & ∼ harris/Databases.html.In addition to N GC , we include some selected obser-vational parameters describing the luminosities, masses,and scale sizes of the galaxies and that are available formost of the objects in the list. The full catalog containsthe following information: • Galaxy identification (from more than one catalogsource if appropriate). • Right ascension and declination (J2000). • Morphological type. • Foreground absorption A V , from NED(NASA/IPAC Extragalactic Database). • Distance D , primarily from the raw data in NED.For relatively nearby galaxies, wherever possiblewe adopt D − values measured from primary stan-dard candles based on resolved stellar popula-tions (Cepheids, red-giant-branch-tip stars, plan-etary nebulae luminosity functions, Mira stars, RRLyrae stars). Surface brightness fluctuations (SBF)measured from integrated light are also used as astandard candle. For each such galaxy we adoptthe average of the most recent individual measure-ments of those six primary methods (We emphasizethat our adopted values are not the averages givenin NED). For some slightly more distant galaxiesfor which these primary indicators are not avail-able, we use recent determinations from the Tully-Fisher relation as listed in NED. For still moredistant systems ( D &
30 Mpc) we use Hubble’slaw with H = 70 km s − Mpc − and with thegalaxy radial velocity corrected to the CMB refer-ence frame. • Absolute visual magnitude M TV , calculated fromthe distance modulus and the integrated magnitude V T if available from NED. In other cases where atotal V magnitude was unavailable we have used ablue magnitude B T and integrated ( B − V ) colortaken from the HyperLeda database. • Absolute near-infrared magnitude M TK , calculatedfrom the distance modulus and the integrated K magnitude from 2MASS. We use here the 2MASS K ( ext ) magnitude for each galaxy, a quantitywhich is available for 82% of the galaxies in ourcatalog. Although an alternate and perhaps prefer-able choice would be the frequently used K s band,this is unavailable for most of our galaxies. • Total number of globular clusters in the galaxy, N GC . • Stellar velocity dispersion σ e . This spectroscopicquantity is dominated by the bright inner part ofthe galaxy and in most cases represents the ve-locity dispersion of the bulge light. Where pos-sible we have taken the homogeneous σ e valueslobular Cluster Systems 3given by Gultekin et al. (2009), McElroy (1995),and McConnell et al. (2012). Otherwise, we use σ e as compiled in HyperLeda. In total, σ e measure-ments are available for 65% of the galaxies in thecatalog. • Effective radius R e enclosing half the total galaxylight, taken from NED or (secondarily) HyperLeda.Here, in the interests of the best possible combi-nation of homogeneity and completeness we useonly radii measured through optical photometry:primarily V whenever available, and secondarilyother nearby bands including g, r , or B . We donot use any values measured through infrared ornear-ultraviolet bands, since these give systemati-cally different R e from optical bandpasses. In total,optically based R e values are available for 81% ofthe galaxies. • Dynamical mass M dyn of the galaxy, calculatedfrom σ e and R e as described in Section 3 below. • Total stellar mass M GCS contained in the entireGC population of the galaxy, calculated as de-scribed below in Section 3.3. • Measured mass of the central supermassive blackhole (SMBH), a quantity of special interest al-though it is currently available for only 11% ofthe galaxies in our GCS list. SMBH data aretaken from Gultekin et al. (2009), McConnell et al.(2012), Graham (2008), and other sources listed inHarris & Harris (2011) and Harris et al. (2013).
Comments: (1) The only quantities available for all the galaxies inthe catalog are N GC , galaxy type, and luminosity M TV .In principle, a total magnitude obtained from a near-infrared band such as z, I , or K is a better photometricproxy for total stellar mass than are optical bands. Inter-nally homogeneous near-IR luminosities are available forsubsets of the data (for example, the Virgo and Fornaxcluster surveys; see Peng et al. 2008; Villegas et al. 2010,hereafter P08, V10), but at present we place most of ourreliance on absolute visual magnitudes because these areavailable for all of the targets, and allow us to comparevarious results readily with earlier work. In the discus-sion below, we also present some correlations with M K from the 2MASS K ( ext ) data, but these generally do notexhibit any smaller scatter than the ones using M TV .(2) The velocity dispersion σ e is a key quantity in manystudies of the “Fundamental Plane” (FP) of early-typegalaxies (e.g. Djorgovski & Davis 1987; Allanson et al.2009; Graves & Faber 2012, among many others). It is acritical element to dynamical estimates of galaxy mass(see below), and is a valuable indicator of the depthof the galaxy’s potential well (Loeb & Peebles 2003;Shankar et al. 2006). An additional advantage to includ-ing σ e here is that it appears to be stable with time forearly-type galaxies over a significant range of redshift, aswould be the case if large galaxies form their major corelight at high redshift and then evolve later by inside-outgrowth mainly through minor mergers (Bezanson et al.2009, 2011; Tiret et al. 2011; Patel et al. 2012). The scale radius R e , by contrast, is expected to grow withtime (e.g. Papovich et al. 2012; Shankar et al. 2013).(3) The population size N GC is only a simple first-order gauge of a GCS. Other GCS characteristics thatare of strong interest include the GCS radial profile , theGC luminosity distribution (GCLF), and the metallicitydistribution (MDF). The MDF in particular – in mostcases measured through broadband color indices – hasstimulated much ongoing discussion of the cluster for-mation process (e.g. Ashman & Zepf 1992; Larsen et al.2001; Peng et al. 2006; Harris 2009; Spitler et al. 2008;Mieske et al. 2010, among many others). Many of thesources listed in our bibliography discuss these othercharacteristics, but a comprehensive analysis of them ex-tends far beyond the goals of our present study. For eachgalaxy we have selected the studies that gave the best es-timates of the total GC population, not necessarily thebest analysis of the MDF or other characteristics.(4) It should be emphasized that the N GC values col-lected here are of greatly differing internal uncertaintyfrom one galaxy to the next and thus certainly do notmake up a homogeneous list. Ideally, N GC should be de-termined from imaging that is both deep enough to reachnearly to the faint limit of the GCLF, and also wide-fieldenough to cover the full radial extent of the GCS as wellas to determine the background contamination level ac-curately. These twin conditions are rarely met. Themost commonly used technique is to obtain GC photom-etry with a limiting magnitude near the “turnover” orpeak point of the GCLF and then fit a standard Gaus-sian GCLF shape to predict the total over all magni-tudes. If (as is usually the case) the field coverage doesnot sample the entire halo of the target galaxy, then someoutward extrapolation of the GCS radial profile beyondthe heavily populated inner regions is also needed to es-timate the total over all galactocentric radii. Good re-cent examples of the standard techniques can be found,for example, in Jord´an et al. (2007), P08, Young et al.(2012), and references cited there. In all cases the GCpopulation totals are simply the best attempts, with theavailable imaging data, to extrapolate over the full lu-minosity range and spatial range needed. Inspection ofthe data catalog will show that the quoted relative un-certainties ∆( N GC ) /N GC span a wide range: at best thepopulation total is known to ±
10% while at worst it maybe uncertain by as much as a factor of two. We returnto this point in the later discussion.(5) Earlier GCS lists were dominated by large E galax-ies with rich GC populations. The current catalog nowreduces many sampling biasses: it includes the completerange of galaxy environment, type, and luminosity, fromthe smallest dwarfs to the largest supergiants. Our cat-alog contains 248 ellipticals, 93 S0’s, and 81 spirals orirregulars. The smallest one in the list is the dSph KKS-55 at M TV = − . M TV ≃ −
24 and holdingup to 30,000 GCs each.(6) We note that for most galaxies in this catalog, wehave used the single literature source that gives the bestrecent estimate of the total GC population. This may notcorrespond to the best sources for other purposes, such asdiscussions of the MDF. In some cases we have averagedthe results from two or more sources that appear to give Harris
Fig. 2.— ‘Dynamical mass’ M dyn = 4 R e σ e /G plotted versustotal visual luminosity L V for the galaxies in the GCS catalog.Results are shown separately for the E/dE, S0, and spiral types.In each panel the diagonal line shows the best-fit linear solution tothe ellipticals, as described in the text. Fig. 3.— M dyn versus total visual luminosity L V for all galaxiesin the GCS catalog. The E/dE systems are plotted as open circles,S0 as red filled symbols, and S/Irr systems as blue crosses. Anarbitrary offset of +0.2 dex has been applied to (log M dyn ) for theS0 points, and +0.3 dex to the S/Irr points to bring them in linewith the E-galaxy solution (see text). The diagonal line shows thebest-fit linear solution to the ellipticals as in the previous figure. comparably good estimates of N GC . Lastly, in the cata-log we have chosen not to list any galaxies for which theestimated N GC was zero or negative. Such cases includea few very small dwarfs (Peng et al. 2008; Georgiev et al.2010) for which N GC ≤ CORRELATIONSUsing the entire database, we next explore some simplecorrelations of N GC with large-scale host galaxy proper-ties such as luminosity, dynamical mass, or scale size.The overall purpose is to use this new and larger datasetto search for reliable predictors of GC population size that can be calculated from the shortlist of simple struc-tural parameters that are available for most of the galax-ies. In doing so, we concentrate particularly on the ellip-ticals in the catalog, for two reasons: (a) they cover thelargest range in luminosity, from dwarf-spheroidal up tocD supergiants; and (b) they form the largest and mosthomogeneous subset of the GC measurements, makingup almost 60% of the entire catalog.In general we take pairs of parameters (x,y) in log/logspace and search for linear correlations of the normalform y = α + βx . Assume that we have a sample of n measured datapoints ( x i , y i ) with quoted measurementuncertainties ( σ xi , σ yi ). Our best-fit slope and zeropointare determined by minimizing the sum χ = n X i =1 ( y i − α − β ( x i − x )) ( σ yi + ǫ y ) + β ( σ xi + ǫ x ) (1)as in Tremaine et al. (2002) and particularly Novak et al.(2006). Here x is a suitable mean value over the thesample such that h x − x i ≃ α, β ).The parameters ( ǫ x , ǫ y ) are constants that representany additional variances in ( x, y ). These variances mightbe due to intrinsic (“cosmic”) scatter built in to the sam-ple population; or extra measurement uncertainties if thequoted ( σ x , σ y ) are underestimated; or a combinationof both effects. With only two pairs of measurements( x i , y i ) in the solution, and without any other externalconstraints, in general it will not be possible to solve in-dependently for both of ǫ x , ǫ y . Therefore in practice foreach solution described below, we set ǫ x = 0 and vary ǫ y until χ ν , the reduced χ per degree of freedom, equals 1.In the discussion below we refer to ǫ y determined fromthe solution as the residual dispersion for the dependentvariable y (see Novak et al. 2006).To set up several of the correlation solutions discussedlater, we also calculate (a) the total visual luminosity L V = 10 . M V ⊙ − M TV ) L V ⊙ with Solar M V ⊙ = 4 .
83; and(b) the dynamical mass, M dyn = 4 R e σ e G (2)following Wolf et al. (2010). Since the luminosity-weighted velocity dispersion is dominated by light fromwithin R e , and the dark-matter halo contributes a smallfraction of the mass within R e , M dyn is close to beingthe baryonic mass of the galactic bulge (e.g. Tiret et al.2011; Graves & Faber 2012). The dynamical mass canbe calculated for 61% of the galaxies in our catalog, i.e.the ones with measurements of both σ e and R e .As a preliminary step and a check of our procedures,we show in Figure 2 the direct correlation between M dyn and visual luminosity L V . The data are shown separatelyfor the E, S0, and spiral galaxies (note that no dwarf ir-regulars are present here, since they lack any “bulges”from which a velocity dispersion can be measured). It islobular Cluster Systems 5 Fig. 4.— M dyn plotted versus total K − band luminosity L K forthe galaxies in the GCS catalog. The E/dE systems are plottedas open circles, S0 as red filled symbols, and S/Irr systems as bluecrosses. The solid diagonal line shows the best-fit linear solutionto the entire sample, as described in the text, whereas the dashedline shows the two-part solution over the bright and faint rangesas in Table 2. worth emphasizing that L V and M dyn are observation-ally nearly independent measurements except insofar as R e relies on knowing the large-scale light profile of thegalaxy. For the E galaxies over their entire range, thebest-fit linear solution is listed in Table 2, along with theother correlations to be presented below. In the Table,the successive columns give (1) the pair of parameters(x,y) being fit, (2) the subsample of galaxy types usedin the solution, (3) the number of galaxies in the solu-tion, (4-6) the sample mean x , zeropoint α , and slope β , (7) the best-fit residual dispersion ǫ y , and (8) totalrms scatter σ y of the datapoints around the fitted so-lution. Throughout Table 2, the luminosities L V andmasses M dyn are in Solar units.The result M dyn ∼ L . V corresponds to the well knownsystematic increase in mass-to-light ratio with galaxysize derived from the Fundamental Plane (FP) of early-type galaxies (e.g. Faber et al. 1987; Jorgensen et al.1996; Cappellari et al. 2006; Allanson et al. 2009;Magoulas et al. 2012; Graves & Faber 2012). Quantita-tively, our E galaxy solution corresponds to( M/L ) V = 4 . L V / L V ⊙ ) . (3)The same slope also matches the S0 and spiral-galaxytrends (lower two panels of Fig. 2), although these diskgalaxies fall below the E line by 0.2-0.3 dex in mass (oralternately, they lie at higher luminosity for a given massby the same average factor). We use the same mass cal-culation formula (Eq. (2)) for all types, though it is notentirely clear that pure-spheroid (elliptical) systems anddisk systems should behave identically or should havethe same M/L . The measured scatter of ± .
27 dex rmsin log M dyn about the best-fit line is also encouraginglysmall, given that the luminosity and calculated mass arederived from a wide variety of observational sources for V T , σ e , and R e and are unavoidably a somewhat het-erogeneous sample. As will be seen below, we find very Fig. 5.—
Top panel:
Total number of globular clusters N GC plotted versus the visual luminosity of the host galaxy, for ellipticalgalaxies. Middle panel:
The same plot for S0 galaxies.
Lowerpanel:
The same plot for spiral and irregular galaxies. In all threepanels, the sloped line denotes a specific frequency S N ≡ similar scatters for most of our other correlations. InFigure 3, we show all galaxy types together where an off-set of 0.2 dex to the calculated mass has been applied tothe S0 systems to bring them in line with the ellipticals,and an offset of 0.3 dex to the S/Irr systems.In Figure 4 we show a similar correlation for the K − band luminosity versus M dyn . Plotting the threetypes of galaxies (E, S0, S/Irr) separately indicates littleor no zeropoint difference, so we show all three com-bined without offsets. Here we adopt (log L K /L K ⊙ )= 0 . . − M K ). The best-fit solution over all lu-minosities (Table 2) gives M dyn ∼ L . ± . K and hasa very similar scatter of ± .
26 dex. However, unlike L V (Fig. 3), the relation exhibits a noticeable nonlin-earity where the brighter galaxies follow a steeper slope.Splitting the data at M K = − . L K /L K ⊙ =10.3) gives the two additional solutions listed in Table2, where M dyn ∼ L . ± . K for fainter galaxies and M dyn ∼ L . ± . K for brighter ones.In summary, both V and K luminosities in our catalogcan act as similarly precise indicators of galaxy dynam-ical mass. Where necessary, in the discussion below wechoose to use L V because it is available for the entirecatalog of galaxies.3.1. Cluster Population and Galaxy Scale Parameters
We next plot total cluster population N GC versus theother listed scaling parameters: L V , σ e , R e , and M dyn .These are displayed in Figures 5 to 9, and selected best-fitsolutions are listed in Table 2. Comparison with the re-maining quantity in the list, the SMBH mass, is a specialtopic that has been the subject of discussion in severalrecent papers including Spitler & Forbes (2009) (here-after S09); Burkert & Tremaine (2010); Harris & Harris(2011); Snyder et al. (2011); Sadoun & Colin (2012);Rhode (2012); Harris et al. (2013), and will not be re-peated here.The most easily observable relation is between N GC Harris
Fig. 6.—
Top panel:
Total number of globular clusters N GC plotted versus the K − band luminosity of the host galaxy, for ellip-tical galaxies. Middle panel:
The same plot for S0 galaxies.
Lowerpanel:
The same plot for spiral and irregular galaxies. In all threepanels, the sloped line denotes a specific frequency S N ≡ and the galaxy V luminosity, which was also his-torically the first to be discussed in the literature(Jaschek 1957; Hanes 1977; Harris & Racine 1979;Harris & van den Bergh 1981). The data for all the sys-tems in the current catalog are shown in Figure 5. Ashas been found in all earlier discussions, N GC increasesvery roughly in direct proportion to host galaxy luminos-ity, but obvious systematic deviations occur at both thehigh- and low-luminosity ends of the scale, and betweenE/S0 systems and S/Irr ones.In Figure 6, we show the correlations between N GC andnear-infrared luminosity M K . The pattern is very muchthe same, and the scatter quite similar to the previousfigure. Since the N GC − M K graph appears to give muchthe same information, and M TV is available for a largersample of galaxies, we stick primarily with the use of thevisual-luminosity data in the following discussion.The connection between galaxy size and GC popula-tion is most often presented in terms of the specific fre-quency (Harris & van den Bergh 1981), S N ≡ N GC × . M TV +15) = (8 . × ) N GC L V /L V ⊙ . (4)In the K − band, if we adopt a typical color index ( V − K ) ≃ . S N ≡ N GC × . M TV +18 . = (4 . × ) N GC L K /L K ⊙ . (5)The trend of S N versus M TV is shown in Figure 10, forthe three subsets of data combined. The characteristicU-shaped distribution is the most prominent feature ofthe diagram: the intermediate-luminosity galaxies forma rather tight grouping in a “valley” around S N ≃ S N . This distribution wasapparent even in the first discussion of specific frequency Fig. 7.—
Left panels:
Total number of globular clusters N GC plotted versus the effective radius R e of the host galaxy. Elliptical,S0, and S/Irr galaxy types are plotted separately. Right panels:
Total number of globular clusters N GC plotted versus the bulgevelocity dispersion of the host galaxy. from a sample more than an order of magnitude smaller(see Fig. 4 of Harris & van den Bergh 1981). Two no-table recent versions with extensive discussions are givenby P08 and Georgiev et al. (2010) (hereafter G10), whichbuild on the newer surveys and particularly fill in thelower-luminosity range more extensively than before.In general the highest specific frequencies are found ei-ther in some E supergiants (particularly the cD or BCGgiants) or in dwarf spheroidals and nucleated dE,N galax-ies. S0 and disk galaxies have systematically lower S N than ellipticals, field E’s have lower S N than ones in richclusters of galaxies, and dE’s in denser environments fa-vor higher S N (e.g. Hanes 1977; Harris & van den Bergh1981; van den Bergh 1982, 2000; Durrell et al. 1996;Harris 2001; Brodie & Strader 2006, and P08).In most galaxies there are two clearly identifiable sub-sets of GCs that separate out by color or metallicity: theblue (metal-poor) population and the red (metal-rich)ones (e.g. Peng et al. 2006; Harris 2009; Mieske et al.2010). The ratio N(red)/N(blue) is dependent on galaxysize, with GC populations in lower-luminosity galax-ies of all types progressively more dominated by themetal-poor component (Peng et al. 2006). The blue GCsare likely to be the remnants of the very earliest star-forming stages of hierarchical merging, emerging out ofthe gas-rich and metal-poor protogalactic dwarfs (e.g.Burgarella et al. 2001; Moore et al. 2006, and P08), oradded later by accretion of low-mass metal-poor satel-lites. With photometric data of sufficient quality, it isalso possible to define S N or the GC mass fraction for redand blue types separately (e.g. Rhode et al. 2005, 2007;Brodie & Strader 2006; Spitler et al. 2008; Forte et al.2009). However, many of the galaxies in our catalog At the top end, the anomalously high − S N values seen inFig. 10 for four S0 or S galaxies are those for NGC 6041A, UGC3274, A2152-2, and IC 3651. All are distant and luminous systemsin rich clusters of galaxies, and close inspection of images suggeststhat they may simply be misclassified E/cD systems. lobular Cluster Systems 7 Fig. 8.—
Correlation of GC population size N GC versus thedynamical mass M dyn = 4 R e σ e /G . The data are shown separatelyfor ellipticals, S0’s, and spirals as in previous figures. In each panelthe s olid diagonal line shows the best-fit solution for the luminousE galaxies as discussed in the text, i.e. excluding the dwarfs. Inthe middle panel, the dashed line is the E-galaxy solution shifteddownward by 0.2 dex, while in the lower panel the dashed line isthe E line shifted downward by 0.3 dex (see text). Fig. 9.—
Correlation of GC population size N GC versus thedynamical mass M dyn = 4 R e · σ e /G . E galaxies are plotted asopen circles, S0’s as solid red circles, and spirals as blue crosses.The N − values for the S0 and spiral types have been normalized tothe E-galaxy level as described in the text. The s olid diagonal lineshows the best-fit solution for all galaxies with M dyn > M ⊙ . do not have sufficient photometric data to evaluate theblue/red ratios accurately, and we do not pursue thisquestion here. New photometric data aimed at obtaininghigh-quality blue/red population ratios for more galax-ies would be of great interest. In particular, it would beimportant to know how much of the scatter around themean S N relation at a given galaxy luminosity might bedue solely to differences in the relative number of blueGCs (and thus the efficiency of cluster formation at theearliest stages). The references cited above should be Fig. 10.—
Specific frequency S N versus the absolute visual mag-nitude M TV of the host galaxy. E and dE galaxies are plotted asopen circles, S0 systems as solid red circles, and spirals or irregularsas blue crosses. The horizontal line at bottom shows S N = 1. seen for more complete discussion.3.2. Other Correlations for N GC Going beyond specific frequency, we have exploredmore general correlations of N GC versus combinations ofscale size and velocity dispersion. We might, for example,hope to find choices which would more nearly linearizethe trend of N GC over the entire galaxy luminosity rangefrom dwarfs to supergiants.Neither R e nor σ e by itself is a good predictor of GCpopulation. As seen in Fig. 7, N GC versus those quan-tities exhibits quite a lot of scatter and behaves nonlin-early. The exception here is N ( R e ) for the spiral andirregular galaxies, which yields a roughly useful scalingin the cases where σ e is not available (see discussion be-low). N GC versus M dyn is plotted in Fig. 8 for the threegalaxy types, and in Fig. 9 for all galaxies combined.Similar trends of GCS numbers versus mass are also dis-cussed by P08, S09, and G10. The significant differencecompared with our work is that these previous studiesemployed photometrically determined masses (e.g. thecombination of a near-infrared luminosity and an as-sumed mass-to-light ratio), whereas we use M dyn whichstands independently of photometric indicators.For galaxy masses > M ⊙ , we find that N GC ∼ M . ± . dyn . That is, GCS population increases in almostexactly direct proportion to galaxy mass. For the smallergalaxies, the scaling is much shallower at N GC ∼ M . dyn and these also exhibit larger scatter (see particularlyFig. 9). We find as well that the S0-type galaxies lie be-low the ellipticals by ∆ log N GC ≃ − . if the same definition of M dyn is valid for disk galaxies andellipticals (cf. the caveats mentioned earlier), then diskgalaxies have fewer clusters per unit bulge mass than doellipticals, by factors of 1.5 to 2. This point is discussedmore extensively by G10. To plot up Fig. 9 we have ap- Harris Fig. 11.—
Correlation of GC population size N GC versus thebinding energy E b ∼ R e · σ e , as described in the text. In eachpanel the s olid diagonal line shows the best-fit solution for theluminous E galaxies, i.e. excluding the dwarfs. plied these offsets to the S and S0 types to bring themback to the E/dE line.In Figure 6, we show N GC now plotted against K − band infrared luminosity. In principle, if near-IR lu-minosity is a valid proxy for total stellar mass, then thisgraph should reveal the same basic trend as does Fig. 9.It does show the same trend, but the scatter is similar tothe correlations with L V and so is not additionally usefulfor the present purposes.In addition to M dyn ∼ R e σ e , another quantity usedoccasionally in the literature especially for pressure-supported systems such as star clusters, molecularclouds, or E galaxies, is the system’s binding energy E b ∼ M σ ∼ R e σ e (e.g. McLaughlin 2000; Hopkins et al.2007; Snyder et al. 2011). For completeness we showthe correlation of N GC and E b in Figure 11, where nu-merically we define E b = M dyn ( σ e / [200km s − ]) . Onceagain, the luminous galaxies (log E b > .
5) form a welldefined relation close to N GC ∼ E / b , with total scatterquite similar to the previous solution between N GC and M dyn (see Table 2). However, the dwarf galaxies standeven further off the mean line than before, so there ap-pears to be no additional advantage to using E b as apredictor of GC population.Going in the opposite direction to a smaller power of σ e has the numerical effect of reducing its importanceand bringing the dwarfs closer to the giant-galaxy line.We have explored a range of different empirical combi-nations and, as an example, we show the case for (log N GC ) against the direct product (log R e σ e ) in Figure12. This result comes close to giving a nearly linear cor-relation with encouragingly low scatter, over the entireluminosity range of galaxies from the smallest dwarfs tothe largest supergiants, a range of 5 orders of magnitudein mass. In performing the fit we have deleted the fivemost deviant points (three dwarfs, two giants), leaving N = 158 galaxies to determine the solution. In Fig. 12the E-galaxy solution is also shown superimposed on thedata for the 72 S0 and 19 spiral systems. Again, the E Fig. 12.—
GC population size N GC versus ( R e σ e ) as defined inthe text. The data are shown separately for ellipticals, S0’s, andspirals as in previous figures. In each panel the s olid diagonal lineshows the best-fit solution for the E galaxies, but now including b oth dwarfs and giants. In the second panel, the dashed line showsthe E-galaxy line shifted downward by 0.2 dex, while in the lowerpanel the dashed line shows the E-galaxy solution shifted down by0.3 dex. solution adequately matches the S0’s for a -0.2 dex shiftin log N GC , and matches the spirals for a -0.3 dex shift(shown as the dashed lines in the lower two panels).In brief, we find that the total globular cluster popula-tion of a galaxy is accurately predicted by the simple prod-uct of the galaxy’s effective radius R e and bulge velocitydispersion σ e . The specific relation for the E galaxies is N GC = (600 ± (cid:20) ( R e σ e / s) (cid:21) . ± . (6)The same relation can also be used for S0 and spiraltypes, with the zero-point shifts given above.This simple relation is not useful for late-type spiral orirregular galaxies where σ e is not defined or not measur-able. In those cases, a rough but still useful predictor of N GC appears to be the effective radius R e alone, as seenin Fig. 7 (lower left panel). For these types of galaxies,we find N GC = (38 ± R e / . ± .
53 dex in log N GC , significantly higherthan for the other relations presented above, and reflect-ing the intrinsically wide range of GC systems found instar-forming dwarf galaxies. Nevertheless, it should beuseful for giving first-order estimates if no other recourseis available.3.3. Specific Mass and Galaxy Scale Parameters
The total number of GCs in a galaxy is only a proxyfor a more physically relevant quantity, the total stellar mass M GCS contained in all the GCs. Ultimately, wewould like to know the typical fraction of baryonic massor total halo mass taken up by the GCs. Here we use the“specific mass” defined as a percentage of the previouslylobular Cluster Systems 9calculated dynamical mass of the host galaxy, S M = 100 M GCS M dyn . (8)This ratio should in principle be similar to the definition S M = 100 M GCS /M G⋆ used by P08, since as discussedabove, M dyn is nearly equal to the total bulge stellarmass with minor contributions from either dark matteror gas (except for major star-forming systems). By com-parison, G10 use S M = 100 M GCS / ( M ⋆ + M gas ), whichcan be significantly different for either gas-rich dwarfs orcD-type systems with massive amounts of hot halo gas.Perhaps more importantly however, our discussion of S M differs from those of P08, S09, or G10 in that they used photometrically estimated stellar masses M ⋆ versus ourdynamical masses.To define S M we need to add up the masses of theGCs in a given galaxy, or equivalently find the mean GCmass. For studies of GCSs such as the Virgo Clustersurvey (Peng et al.) or ones in very nearby galaxies, itis possible to obtain a nearly complete census of all theclusters in a given galaxy and explicitly add them up oneby one. For most of the galaxies in our catalog, however,we must adopt a more broad-brush procedure in hopes ofidentifying first-order trends. What helps considerablyis the empirical fact that the GCLF has a consistent,predictable shape across all galaxies, which convenientlyallows us to scale N GC to M GCS fairly straightforwardly.Usually the GCLF is represented by a simple Gaussianin number of GCs per unit magnitude interval, n ( M V ),with a characteristic peak µ V and standard deviation σ V .More detailed analysis suggests that a slightly asymmet-ric form such as the “evolved Schechter function” devel-oped by Jord´an et al. (2006, 2007) is a better match tothe data. However, this is only a minor concern becausethe total mass in the GCS is dominated by the clustersbrighter than the peak (turnover) point of the GCLF;the clusters fainter than the turnover make up typicallyonly ≃
20% of the total GCS mass. Thus the normalGaussian-type analytical approximation where σ V is de-termined by the bright half of the GCLF remains quiteuseful.A more important consideration is that µ V and σ V de-pend on galaxy luminosity, in the sense that the GCLFbecomes broader and brighter for bigger galaxies. Tointegrate over the GCLF, we follow the relations de-rived by Jord´an et al. (2006) and V10, adopting µ V = − . . M TV + 21 .
3) and σ V = 1 . − . M TV + 21 . Finally, to convert GC luminosity to mass, we use aconstant (
M/L ) V = 2 (McLaughlin & van der Marel2005). Figure 13 shows the resulting trend of mean in-dividual GC mass versus galaxy mass M dyn (where now h M GC i ≡ M GCS /N GC ). The residual scatter aroundthis relation is simply the visible result of the galaxy-to-galaxy differences in the calculated M dyn = f ( R e , σ e ) The exception is that we set a lower bound σ V ( min ) = 0 . σ/ ∆ m = − .
10 depends heavily on the more luminous galaxies,whereas for the dwarfs there is mainly a large scatter with no cleartrend. We note that S09 use a constant µ V and σ V to calculate M GCS . for a given galaxy luminosity. This graph would be adispersionless relation if, as in other papers, we had usedgalaxy luminosity to determine galaxy mass. The over-all trend is listed in Table 2 and shown in Figure 13and is well matched by a single power law, h M GC i =(2 . × ) M . dyn in Solar masses.The correlation solution for GCS mass versus galaxymass, for the E galaxies with L > L ⊙ (Table 2),yields M GCS ∼ M . ± . dyn , slightly but significantlysteeper than the N GC ∼ M . dyn dependence found ear-lier. The difference is a direct result of the second-ordertrend for mean GC mass to increase with host galaxysize. For the smaller E galaxies, the mean trend is M GCS ∼ M . ± . dyn , again quite similar to the depen-dence of N GC on M dyn .The plot of S M versus M dyn is shown in log/log formin Figure 14. The overall distribution is roughly simi-lar to that for S N (Fig. 10), though with less scatter ateither the high-luminosity or low-luminosity ends. Thisreduced scatter is partly a result of our use of dynamicalmasses rather than photometric masses for the galaxies(for example, two giant ellipticals may have the same lu-minosities, but if one of them is a cD-type or BCG, it willusually have a larger effective radius or central velocitydispersion and thus a higher dynamical mass).The practical penalty for using S M instead of S N isthat we cannot strictly include as many datapoints be-cause we need to have both R e and σ e to determine M dyn .We therefore supplement the dynamical data by addingin photometrically calculated masses from the knownconversion between L V and M dyn , from Eq. (3) and Ta-ble 2. These secondary masses are added for the galaxieswithout measured R e and σ e . As is evident in Fig. 14,the extra points are particularly valuable for the lower-luminosity galaxies. DISCUSSION4.1.
Population Scaling Relations
We can gain some more understanding of why the re-lation shown in Fig. 12 between cluster population and( R e σ e ) works by looking further at the scaling relationsamong galaxy mass, luminosity, size, and velocity disper-sion. For the giant ellipticals ( L > L ⊙ ), direct fits ofeach of R e and σ e versus luminosity give R e ∼ L . ± . and σ e ∼ L . ± . . Combining these then predicts( R e σ e ) ∼ L . ± . ∼ M . ± . dyn ∼ N . ± . GC , usingthe other correlations in Table 2 to translate from L to M dyn and then to N GC . Inverting the result then gives N GC ∼ ( R e σ e ) . ± . , which closely matches what weobtain from the direct solution in Fig. 12 and Table 2.The dwarf galaxies obey somewhat different scalingsamong size, dispersion, and luminosity, namely R e ∼ L . ± . and σ e ∼ L . ± . . Combining these leadsto ( R e σ e ) ∼ M . ± . dyn . However, this shallower trendis partly compensated by the shallower dependence ofcluster population on mass for the dwarfs (Table 2), N GC ∼ M . ± . ∼ ( R e σ e ) . ± . . The luminosityrange of the dwarf E’s is small enough that they can ac-commodate a relatively wide range of slopes, permittinga single linear relation across the entire range from dwarfspheroidals to supergiants to be a workable representa-0 Harris Fig. 13.—
Mean globular cluster mass h M GC i versus galaxymass M dyn . Here, the mean GC mass is defined as h M GC i = M GCS /N GC and the total mass in the globular cluster system iscalculated as described in Section 3.3. Symbols for the three galaxytypes (E, S0, S/Irr) are as in previous figures. The diagonal linegives the best-fit relation h M GC i ∼ M . dyn (Table 2). tion. Said differently, the GC population of a galaxy canbe seen as another outcome of the Fundamental Planefor early-type galaxies. Mass Fractions and Formation EfficiencyParameters
Discussion of the specific frequency and specific mass S N and S M quickly leads to the question of GC formationefficiency – the original reason for defining these ratios.A summary of the thinking during the early (pre-ΛCDM)literature can be found in Harris (2001). A more recentview with growing evidence is that the GCs, which arecompact stellar subsystems emerging from the densestand most massive sites of star formation, could be theobjects that form earliest in any starburst, followed bythe bulk of the field stars and the lower-mass star clustersthat soon dissolve into the field. First ideas along theselines were explored by Blakeslee (1997); McLaughlin(1999); Kavelaars (1999), and Harris & Harris (2002),pointing to the possibility of a universal GC formation ef-ficiency per unit baryonic mass including both stars andgas. If the later rounds of star formation after GC forma-tion are truncated or severely reduced by any combina-tion of external or internal quenching, then the resulting S N or S M observed long after the fact is a marker ofhow well the quenching worked. High − S N systems thuswould be field-star deficient (not “cluster-rich”) becausethe star formation was prevented from running to com-pletion. This interpretation has been developed furtherby P08, G10, and Spitler (2010); in particular, P08 esti-mate quantitatively that in dwarf galaxies the peak starformation epoch lagged the peak GC formation epoch by If we assume more generally that N GC ∼ R ae σ be , it can beshown from the scalings between R e , σ e , and L listed above thatany pair of exponents where a ≃ . b will work for both the giantsand dwarfs. However, the combination N ∼ ( Rσ ) . has the strongadvantage of simplicity and reproduces the actual data well. −
600 Myr.Different mechanisms for shutting down star formationwill operate at the opposite extremes of the galaxy massrange. For the dwarfs with their small potential wells, in-ternally driven feedback including starburst winds, pho-toionization, and ultimately SNe may eject a large frac-tion of the gas, while external quenching from tidal strip-ping of gas or external UV fields can also reduce the starformation efficiency (SFE). For the most massive galax-ies, AGN feedback and virial shock heating of infallinggas will lower the SFE (Dekel & Silk 1986; Dekel & Woo2003; Dekel & Birnboim 2006; Bower et al. 2006).The maximum SFE should then happen forintermediate-mass galaxies which are massive enoughto hold their star-forming gas against SN and starburstwinds, but in which AGN feedback or shock heatingare not intense enough to have much effect. Muchrecent literature has addressed this issue by tracing thechange in the ratio M ( halo ) /M ( baryon ) versus galaxysize, which displays the same U-shaped distributionthat we see in the S N and S M curves. To mentiononly two recent studies, Shankar et al. (2006) identifySFE(max) at a stellar mass M ⋆ ≃ . × M ⊙ or M ( halo ) ≃ M ⊙ , while from a combination of directstellar mass and halo mass observational determinationsLeauthaud et al. (2012) find the SFE maximum to beat M ⋆ ≃ × M ⊙ .Viewed in this light, the minimum of the S M or S N distributions becomes perhaps the most interesting re-gion of those diagrams. The baseline ratios S N ≃ S M ≃ . in the galaxies where starformation was globally the most efficient , what mass frac-tion went into the dense compact systems that we now see surviving as globular clusters. That is, these baselinevalues reveal what the “natural” GC formation efficiencyis when any disruption or quenching of star formation isat its least important. This present-day mass fraction is, however, only alower limit to the value at the time of formation, because(a) many low-mass or low-density star clusters are dis-rupted or dissolved over the subsequent Hubble time, and(b) even the surviving ones that started as the densest,most massive clumps have lost a large fraction of theirinitial mass through a combination of early rapid massloss (expulsion of residual gas, and SNe and stellar windsfrom massive stars) and later dynamical erosion. Thecurrent literature (Trenti et al. 2007; McLaughlin & Fall2008; Vesperini 2010) indicates that the individual globu-lar clusters that have survived to the present should havebeen ∼
10 times more massive when they were protoclus-ters than they are now. The conclusion we draw fromthese arguments is that the surviving GCs representedat least 1 percent of the star formation mass fraction inthe maximally efficient intermediate-mass galaxies. Ineither dwarf or supergiant galaxies, however, where S M may be an order of magnitude higher, the GCs we seetoday could – as protoclusters – have taken as much as10% of the gas that successfully formed stars. And thesemass fractions must only be lower limits to the amountsof star-forming gas that went into young star clusters,after accounting for the clusters that did not survive tothe present.The interpretation that M GCS is driven by the amountlobular Cluster Systems 11of gas mass initially present in the galaxy’s potential well(and not the gas mass that was actually consumed instar formation) then raises the possibility that M GCS should be more nearly proportional to M halo , or thetotal depth of the galaxy potential well. This direc-tion has been explored by Blakeslee (1997); McLaughlin(1999); Spitler et al. (2008), P08, S09, and G10. Follow-ing the notation of G10, we denote η ≡ M GCS /M halo .For the dwarf galaxies in particular (discussed at greaterlength by P08 and G10), our derivation that M GCS ∼ M . ± . dyn is in excellent agreement with the scaling M halo ∼ M . ⋆ obtained by Leauthaud et al. (2012) ifwe assume that M GCS ∝ M halo and also M ⋆ ≃ M dyn asabove.To derive the “absolute” GC formation efficiency pa-rameter η from our new sample of galaxies, we need toadopt a stellar-to-halo mass conversion relation M halo = f ( M ⋆ ), or else its inverse. By hypothesis we also sim-ply use M ⋆ ≃ M dyn and η = const as mentioned above.These steps then directly link M halo to M GCS and M ⋆ ,and the assumed value of η can be varied until a matchis achieved with our data. Our specific approach is tofix η by requiring the resulting S M vs. M ⋆ curve to passthrough the baseline S M ≃ . M ⋆ and M halo is givenby Behroozi et al. (2010) and Leauthaud et al. (2012),derived from a combination of methods for measuring M halo and M ⋆ over a wide range of luminosity regimes(see Eq. 24 from Behroozi or Eq. 13 from Leauthaud).Their empirical model shows a clear rise in the ratio M halo /M ⋆ at the low-mass and high-mass ends with aminimum at intermediate galaxies. However, their modelfunction and parameters give a curve for S M that is toosteep at each end to be satisfactory in detail for our pur-poses. A flexible and simpler conversion relation fromYang et al. (2008) and also used by S09 is M ⋆ = M ( M halo /M ) α + β (1 + M halo /M ) β . (9)To use this, we assume a value for M halo , which deter-mines M ⋆ (which by hypothesis equals our M dyn ). Fi-nally S M = 100 · M GCS /M ⋆ , which then defines a pointon Fig. 14. Repeating for a wide range of M halo thendefines a complete S M vs. M dyn curve.An illustrative fit of this model to the GCS data isshown as the solid curve in Fig. 14. This curve usesthe parameters η = 6 . × − , log M = 9.98, and log M = 10.7 along with exponents α = 0 . , β = 2 . At either the low-mass or high-mass end of the scale, the S M curve asymptotically approaches a simple power law.At the high-mass end, where ( M halo /M ) becomes large,it can quickly be shown that S M → const · M (1 − α ) /αdyn ,which for our fitted value α ≃ .
64 gives S M ∼ M . dyn .At the low-mass end where ( M halo /M ) is small, then S M → const · M − / ( α + β ) dyn , which for β = 2 .
88 gives The powerlaw-like slopes of the model curve at the high andlow mass ends are quite sensitive to the choices of ( α, β ), and thevalues we find to give a good fit are slightly different from the onesused by S09. Again, they used a different, simpler prescription for h M GC i and thus M GCS ; a smaller GCS dataset; and a differentprescription for computing galaxy masses.
Fig. 14.—
Specific mass S M = 100( M GCS /M dyn ) versus hostgalaxy mass. Solid dots are dynamical masses M dyn calculatedfrom ( R e , σ e ), while open circles are masses estimated from L V and Eq. 3, for galaxies without measurements of R e and σ e . Thesolid diagonal line at upper right shows the effect of changing M dyn by 1.0 dex (an increase in M dyn yields a proportionate decrease in S M ∼ M GCS /M dyn ). The horizontal line at S M = 0 . S M = 0 . S M ∼ M − . dyn .Our estimated η ≃ × − represents the absoluteefficiency of GC formation. By comparison, through dif-ferent combinations of methods for deriving the variousmasses and luminosities, G10 find a mean h η i ≃ × − ,while S09 find η ≃ × − . The agreement among thesediscussions is well within the scatter we can expect giventhe different assumptions for the definition of S M and themethods for finding galaxy luminosities, stellar masses,and dark-halo masses.We have suggested in the discussion above that thegalaxy mass where S M reaches a minimum representsthe level where star formation efficiency was the high-est. The S M interpolation curve in Fig. 14 reaches aminimum at M gal ≃ . × M ⊙ . Within the accuracypermitted by the scatter around the curve, this minimumpoint is strikingly similar to the galaxy-based estimatesof M gal ∼ × M ⊙ mentioned above for the point ofmaximum SFE, and emerges from quite a different lineof argument. Yet another piece of information in linewith these results is recent evidence (Spolaor et al. 2009,2010; Tortora et al. 2010) that the maximum metallicitygradient within galaxies occurs near M gal ∼ × M ⊙ ,as would be the case for galaxies that have had minimalinfluences from mergers and feedback during their pri-mary star-forming stages.Along with the GCS catalog itself, we view Fig. 14and the discussion above as the most important resultof this paper. The relative number of globular clustersin a galaxy as measured by either S N or S M differs con-siderably from one system to another, but still follows asystematic trend that can be matched by a single, con-2 Harrisstant ratio η = M GCS /M halo ≃ × − . Our resultsstrongly support recent work (S09, G10) that the glob-ular cluster system is a remarkably simple proxy for themost fundamental characteristic of a galaxy, namely itsdark-matter potential well.The hypothesis M GCS ∝ M halo can explain the basicshape of the S M distribution with its characteristic riseat extreme low and high luminosities, but it does notaddress the scatter that we see in any given mass range.Quantitatively this scatter is ± .
25 dex at any point onthe mean curve, or slightly less than a factor of two in S M . We should expect four generic sources of scatter: • Random measurement uncertainties in N GC andthus M GCS . The raw uncertainties or even the sta-tistical variance in N GC , as discussed above, maybe up to factors of 2 depending on the galaxy andare therefore probably the dominant source of theobserved scatter. • Random uncertainties in the quantities that de-termine M dyn , namely R e and σ e . These be-have differently from random scatter in N GC , since S M ∼ M GCS /M dyn , and thus S M and M dyn are notindependent quantities. Any error in M dyn wouldgenerate an equal, inverse change in S M and wouldshift points along a diagonal line in Fig. 14. Ifthis was, however, an important source of the ob-served scatter then it should be most visible at thehigh-mass end where the S M curve is nearly per-pendicular to that error line. In practice we seerather similar amounts of scatter over a wide rangeof masses, consistent with the expectation from theraw observations that R e and σ e are uncertain to .
10% (see Harris et al. 2013). • Intrinsic differences in GC formation efficiency be-tween galaxies of similar type and mass. These dif-ferences certainly exist (compare the classic well-studied cases of the Virgo giants M87 and M49,which have similar luminosities but GC popula-tions different by almost a factor of 3), but it isharder to make any general statements about theamount of such “cosmic scatter” at this stage. Alarge part of the scatter may be the result of envi-ronment: for example, P08 address this question indetail for dwarfs, and find evidence that dE galax-ies near dominant giants are more likely to havehigher − S N GCSs. By hypothesis these galaxiesmay have benefitted from being in deeper halo po-tential wells which increased massive star clusterformation. At the opposite end of the environmentscale, Cho et al. (2012) find that E galaxies in veryisolated environments have quite low specific fre-quencies in the range normally associated with spi-rals or S0’s. • The stochastic effects of different individual merg-ing and star-forming histories, which are hard torecover in full detail long after the fact. These mustalso play a role in generating galaxy-to-galaxy dif-ferences. Conversely, differences in dynamical GCdestruction interior to a galaxy should not be a ma-jor factor, since internal GC erosion rates should be similar for galaxies of the same type and luminos-ity . Going further into these intriguing questionsis beyond the scope of our paper.Lastly, although some dwarfs and supergiants standout as having exceptionally high S M , it is perhaps worthnoting as well that many dwarfs and giants have S M or S N values that are quite similar to those of theintermediate-luminosity galaxies in the “baseline” mid-dle range. The large size of our new catalog allows thisfeature of the distribution to stand out more clearly thanbefore. Quantitatively, fully 80% of the entire sample inFig. 14 falls within S M < . S M < .
2. The medianof the whole sample (less sensitive to outliers than themean) is at ˜ S M = 0 . S N = 3 and themedian is at ˜ S N = 2 . − S N GC system will result. Therefore wesuggest that the many dwarfs and giants with lower spe-cific frequencies could be ones in which the GC and field-star formation rates versus time were the same as in theintermediate galaxies.A next step in understandng the link between GC for-mation and their host galaxies will be to explore morethoroughly the relation with cluster metallicity (color):what does the correlation of S M versus galaxy mass looklike for the red and blue GC subgroups? Another majorquestion is the role of galaxy environment, and how muchof the scatter around the S M and S N relations is drivenby a galaxy’s location. Initial work on these questionshas started, but extensions to much bigger samples willbe valuable. SUMMARYWe summarize the results of our discussion as follows: • A new catalog of 422 galaxies with published mea-surements of their globular cluster systems is pre-sented along with a source bibliography. This list,based on a literature survey to the end of 2012,contains 248 ellipticals (dwarfs and giants), 93 S0’s,and 81 spirals and irregulars. • Total GC population N GC increases monotonicallywith either host galaxy luminosity or baryonicmass, but not in a simple linear way. In agreementwith other recent studies but now based on a largersample, we find that the GC specific frequency andspecific mass follow a U-shaped trend, with veryhigh S N at either very low or very high luminosity,but reaching a well defined mean value S N = 1 and S M = 0 . M dyn ∼ − × M ⊙ .This trend can be understood as the result of thedifferent kinds of feedback operating during galaxyformation: for the low-mass dwarfs, field-star for-mation is inhibited by radiative feedback, gas ejec-tion, and externally driven damping before it canlobular Cluster Systems 13run to completion; while for giants, early AGN ac-tivity and virial heating inhibit field-star formationafter the GCs have formed. At intermediate galaxymass, neither kind of feedback is as important andso these galaxies can form stars the most efficiently.Thus along with P08, G10 we identify these mini-mum S N , S M values as the baseline normal for starformation minimally damped by feedback or exter-nal quenching. • High- S N galaxies such as the extreme dwarfs or su-pergiants may be ones in which the GC formationepoch preceded the bulk of field-star formation andwas therefore less affected by feedback and quench-ing processes. • Previous recent studies including Peng et al.(2008); Spitler et al. (2008); Spitler & Forbes(2009); Georgiev et al. (2010) have explored theproposal that GC population size (or more impor-tantly, the mass M GCS ) is directly proportional tothe host galaxy halo mass M halo . Our work addssupport to this interpretation. We find that asingle constant ratio η ≡ M GCS /M halo = 6 × − is capable of reproducing the systematic trend ofspecific mass S M versus galaxy mass, over theentire range of galaxy sizes and masses. Thegalaxy-to-galaxy scatter anywhere around thisrelation is typically a factor of two. • We find that GC population size can also be ac-curately predicted by a simple product of galaxyeffective radius and velocity dispersion, as N GC ∼ ( R e σ e ) . . The residual scatter is ± .
32 dex, mak-ing it competitive with any other proposed correla-tion. We show that this relation can be roughlyunderstood from previously known FundamentalPlane scaling relations among galaxy luminosity,mass, and scale size.This work makes use of data products from the TwoMicron All-Sky Survey which is a joint project of theUniversity of Massachusetts and the Infrared Processingand Analysis Center/California Institute of Technology,funded by the National Aeronautics and Space Adminis-tration and the National Science Foundation. This workwas supported in part by the Natural Sciences and En-gineering Research Council of Canada through researchgrants to WEH, and by McMaster University throughpartial summer student salary to MA. GLHH wishes tothank ESO/Garching for a visiting scientist fellowship,where the first steps toward building this catalog werecarried out. We thank the anonymous referee for sugges-tions and comments that improved the presentation ofthis paper.
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TABLE 1Correlation Solutions (x, y) Galaxy Type N(sample) Mean Zeropoint Slope Residual RMS x α β Dispersion Scatter ǫ y σ y (log L V , log M dyn ) All Ellipticals 161 10.2 10.844 ± ± L K , log M dyn ) All 238 10.7 10.786 ± ± M K < − . ± ± M K > − . ± ± M dyn , log N GC ) Luminous E’s 139 11.2 2.924 ± ± ± ± E b , log N GC ) Luminous E’s 129 16.0 3.075 ± ± R e σ e , log N GC ) All Ellipticals 158 0.20 2.776 ± ± R e , log N GC ) S/Irr 60 0.4 1.582 ± ± M dyn , log M GCS ) Luminous E’s 125 11.4 8.625 ± ± ± ± L V , log R e ) Luminous E’s 136 10.5 0.648 ± ± ± ± L V , log σ e ) Luminous E’s 142 10.5 2.310 ± ± ± ± M dyn , log h M GC i ) All 242 10.7 5 . ± .
006 0 . ± ..