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# Globular Clusters in Virgo Ellipticals: Unexpected Results for Giants, Dwarfs, and Nuclei from ACS Imaging Jay Strader, Jean P. Brodie, Lee Spitler, Michael A. Beasley UCO/Lick Observatory, University of California, Santa Cruz, CA 95064 strader@ucolick.org, brodie@ucolick.org, lees@ucolick.org, mbeasley@ucolick.org ###### Abstract We have analyzed archival _Hubble Space Telescope_/Advanced Camera for Surveys images in \(g\) and \(z\) of the globular cluster (GC) systems of 53 ellipticals in the Virgo Cluster, spanning massive galaxies to dwarf ellipticals (dEs). Several new results emerged: (i) In the giant ellipticals (gEs) M87 and NGC 4649, there is a correlation between luminosity and color for _individual_ metal-poor GCs, such that more massive GCs are more metal-rich. A plausible interpretation of this result is self-enrichment, and may suggest that these GCs once possessed dark matter halos. (ii) In some gEs (most notably M87), there is an "interloping" subpopulation of GCs with intermediate colors (1.0 \(<g-z<\) 1.25) and a narrow magnitude range (0.5 mag) near the turnover of the GC luminosity function. These GCs look otherwise identical to the classic metal-poor and metal-rich GC subpopulations. (iii) The dispersion in color is nearly twice as large for the metal-rich GCs than the metal-poor GCs. However, there is evidence for a nonlinear relation between \(g-z\) and metallicity, and the dispersion in metallicity may be the same for both subpopulations. (iv) Very luminous, intermediate-color GCs are common in gEs. These objects may be remnants of many stripped dwarfs, analogues of \(\omega\) Cen in the Galaxy. (v) There is a continuity of GC system colors from gEs to some dEs: in particular, many dEs have metal-rich GC subpopulations. We also confirm the GC color-galaxy luminosity relations found previously for both metal-poor and metal-rich GC subpopulations. (vi) There are large differences in GC specific frequency among dEs, independent of the presence of a nucleus and the fraction of metal-rich GCs. Over \(-15<M_{B}<-18\), there is little correlation between specific frequency and \(M_{B}\) (in contrast to previous studies). But we do find evidence for two separate \(S_{N}\) classes of dEs: those with \(B\)-band \(S_{N}\sim 2\), and dEs with populous GC systems that have \(S_{N}\) ranging from \(\sim 5-20\) with median \(S_{N}\sim 10\). Together, these points suggest multiple formation channels for dEs in the Virgo Cluster. (vii) The peak of the GC luminosity

function (GCLF) is the same for both gEs and dEs. This is contrary to expectations of dynamical friction on massive GCs, unless the primordial GCLF varies between gEs and dEs. Among gEs the GCLF turnover varies by a surprising small 0.05 mag, an encouraging result for its use as an accurate standard candle. (viii) dE,Ns appear bimodal in their nuclear properties: there are small bright red nuclei consistent with formation by dynamical friction of GCs, and larger faint blue nuclei which appear to have formed by a dissipative process with little contribution from GCs. The role of dynamical evolution in shaping the present-day properties of dE GC systems and their nuclei remains ambiguous. globular clusters: general -- galaxies: star clusters -- galaxies: formation ## 1 Introduction It is increasingly apparent that globular clusters (GCs) offer important constraints on the star formation and assembly histories of galaxies. Recent spectroscopic studies of GCs in massive early-type galaxies (e.g., Strader _et al._ 2005) indicate that the bulk of star formation occurred at relatively high redshift (\(z\gtrsim 2\)) in high density environments (as environmental density and galaxy mass decrease, the fraction of younger GCs may increase; see Puzia _et al._ 2005). These findings allow the age-metallicity degeneracy to be broken and lead to the conclusion that the bimodal color distributions seen in most nearby luminous galaxies are due primarily to two old GC subpopulations: metal-poor (blue) and metal-rich (red). The metallicities of these peaks correlate with host galaxy luminosity (Larsen _et al._ 2001, Strader, Brodie, & Forbes 2004; see also Lotz, Miller, & Ferguson 2004 for dwarfs). Most recent photometric studies of GC systems in ellipticals have used the Wide Field and Planetary Camera 2 (WFPC2) on the Hubble Space Telescope (HST). Compared to ground-based imaging, this strategy gains photometric accuracy and minimizes contamination at the expense of small spatial coverage. Among the larger HST studies of early-type galaxies utilizing deep imaging are Larsen _et al._ (2001) and Kundu & Whitmore (2001). These studies found bimodality in many of their sample galaxies (extending down to low-luminosity ellipticals) and a nearly uniform log-normal GC luminosity function (GCLF) with a peak at \(M_{V}\sim-7.4\). However, the GC systems of dwarf ellipticals (dEs) are more poorly understood. The large HST surveys to date (primarily of Virgo and Fornax) have been limited to relatively shallow snapshot imaging; this precluded the study of color and luminosity distributions in detail. Among the suggestions of this initial work are a correlation of increasing specific frequency (\(S_{N}\)) with decreasing galaxy luminosity, a dichotomy in

the GC systems of nucleated and non-nucleated dEs (dE,N and dE,noN, respectively), and the difficulty of making dE nuclei as observed through dynamical friction of GCs (Miller _et al._ 1998, Lotz _et al._ 2001). Data from the Advanced Camera for Surveys (ACS) Virgo Cluster Survey (HST GO 9401, P. I. Cote) offers an important step forward in understanding the detailed properties of the GC systems of ellipticals over a wide range in galaxy mass. The use of F475W and F850LP filters (henceforth called \(g\) and \(z\) for convenience, though the filters do not precisely match the Sloan ones) allows a much wider spectral baseline for metallicity separation than \(V\) and \(I\). Though only a single orbit is used per galaxy, the increased sensitivity of ACS (compared to WFPC2) allows one to reach \(\sim 3\) mag beyond the turnover of the GCLF, encompassing most of the GCs in a given galaxy. More accurate photometry for the brighter GCs is also possible. Finally, the field of view of ACS is twice that of WFPC2. Together, these attributes allow a study of the color and luminosity distribution of GCs in a large sample of galaxies in much more detail than previously possible. In what follows, elliptical (unabbreviated) refers to all galaxies in our sample and denotes no specific luminosity. The three brightest galaxies are described as giant ellipticals (gEs); these have \(M_{B}\leq-21.4\). Galaxies of intermediate or high luminosity are called Es. Faint galaxies with exponential surface brightness profiles are dEs; the transition from E to dE occurs traditionally at \(M_{B}\sim-18\) (Kormendy 1985). We include two galaxies with \(M_{B}=-18.1\) (VCC 1422 and VCC 1261) under this heading, since these galaxies have nuclei similar to those commonly found among dEs. The E/dE classifications have been taken from the literature and we do not perform independent surface photometry in this paper (though we note the increasing debate in the literature about whether this dichotomy is real, e.g., Graham & Guzman 2003). We have updated the nucleation status of a dE if appropriate; the vast majority of the dEs in our study have nuclei. ## 2 Data Reduction and Analysis All data were taken as part of the ACS Virgo Cluster Survey (Cote _et al._ 2004); this survey includes both ellipticals and S0s. We used all galaxies classified as ellipticals, excepting a few dwarfs quite close to luminous Es whose GC systems could not be isolated. This left a final sample of 53 galaxies. Images were first processed through the standard ACS pipeline. _Multidrizzle_ was utilized for image combining and cosmic ray rejection. GC candidates were selected as matched-filter detections on 20 \(\times\) 20 pixel median-subtracted images. Using DAOPHOT II (Stetson 1993), aperture photometry was performed in a 5-pixel aperture and adjusted to a 10-pixel aperture using corrections of \(-0.09\) in \(g\) and \(-0.15\)

in \(z\). These are median corrections derived from bright objects in the five most luminous galaxies in the Virgo Cluster Survey: VCC 1226, VCC 1316, VCC 1978, VCC 881, and VCC 798. These 10-pixel magnitudes were then corrected to a nominal infinite aperture using values of \(-0.10\) in \(g\) and \(-0.12\) in \(z\) (Sirianni _et al._ 2005; this paper describes the photometric calibration of ACS). Finally, the magnitudes were transformed to the AB system using zeropoints from Sirianni _et al._ (26.068 and 24.862 for \(g\) and \(z\), respectively), and corrected for Galactic reddening using the maps of Schlegel Finkbeiner, & Davis (1998). Most GCs at the distance of Virgo are well-resolved in ACS imaging. Half-light radii (\(r_{h}\)) for GC candidates were measured on \(g\) images (since the \(g\) PSF is more centrally concentrated) using the _ishape_ routine (Larsen 1999). For each object, King models with fixed \(c=30\) (for \(c=r_{tidal}/r_{core}\)) and varying \(r_{h}\) were convolved with a distant-dependent empirical PSF derived from bright isolated stars in the images to find the best-fit \(r_{h}\). This \(c\) is typical of non core-collapsed GCs in the Milky Way (Trager, King, & Djorgovski 1995). We experimented with allowing \(c\) to vary, but it was poorly constrained for most GCs. However, the adopted \(c\) in _ishape_ has little effect on the derived \(r_{h}\) (Larsen 1999). To convert these measured sizes into physical units, galaxy distance estimates are required. We used those derived from surface brightness fluctuation measurements in the literature when possible: these were available from Tonry _et al._ (2001) for the bright galaxies and from Jerjen _et al._ (2004) for several dEs. For the remainder of the galaxies we used a fixed distance of 17 Mpc, which is the mean of the ellipticals in Tonry _et al._ Due to the depth of the images (\(z\gtrsim 25\)), some of the fields suffer significant contamination from foreground stars and especially background galaxies. Using the gEs and several of the more populous dEs as fiducials, we chose the following structural cuts to reduce interlopers: 0.55 \(<\) sharp \(<\) 0.9, \(-0.5<\) round \(<0.5\), and \(1<r_{h}\) (pc) \(<13\), where the sharp and round parameters are from DAOPHOT. A large upper limit for \(r_{h}\) is used since the size measurements skew systematically larger for fainter GCs. We further applied a color cut of \(0.5<g-z<2.0\) (\(>0.3\) mag to each blue and red of the limiting metallicities expected for old GCs; Jordan _et al._ 2004) and an error limit \(<\) 0.15 mag. In practice, this magnitude limit excluded most GCs within the innermost few arcsec of the brightest galaxies (whose GC systems are quite populous). Finally, we visually inspected all GC candidates, and excluded those which were obviously background galaxies. Our criteria are illustrated visually in Figure 1 for the bright dE VCC 1087, which displays a good mix of actual GCs and contaminants. These cuts remove nearly all foreground stars. However, compact galaxies (or compact star-forming regions within larger galaxies) with the appropriate colors can masquerade as GCs. In some images, clusters of galaxies are clearly visible. The increasing numbers of

background sources at \(z\gtrsim 23\), combined with the difficulty of accurate size measurements below this magnitude, makes efficient rejection of contaminants challenging. For gEs this is a minimal problem, due to the large number of GCs within the ACS field of view (hundreds to \(\sim 1700\) for M87). But with dEs and even low-luminosity Es, contaminants can represent a large fraction of the GC candidates. Due to these concerns, we chose only to use GCs brighter than \(z=23.5\) to study the colors and total numbers of GCs for the remainder of the paper. However, we used minimal cuts to study GC luminosity functions; this is described in more detail in SS4. Basic data about the 53 galaxies in our sample are given in Table 1, along with GC system information as discussed below. ## 3 Color Distributions ### Massive Ellipticals In Figures 2 and 3 we show the color-magnitude diagrams (CMDs) for the three most luminous galaxies in our sample: NGC 4472 (\(M_{B}=-21.9\)), M87 (\(M_{B}=-21.5\)), and NGC 4649 (\(M_{B}=-21.4\)). Figure 4 is a plot of magnitude vs. photometric error in \(g\) and \(z\) for M87. The CMDs in Figures 2 and 3 contain considerable structure only apparent because of the large number of GCs; we have chosen to discuss them in some detail. All three gEs clearly show the bimodality typical of massive galaxies, with blue and red peaks of \(g-z\sim 0.9\) and \(\sim 1.4\), respectively. This separation is twice as large as is typical of studies of GC systems in \(V-I\) (e.g., Larsen _et al._ 2001; Kundu & Whitmore 2001), due to the larger metallicity sensitivity of the \(g-z\) baseline. However, a new result is that the red peak is clearly broader than the blue peak; at bright magnitudes (\(z<22\)) there is little photometric error so this must be due to real color differences. To gauge the size of this effect, we fit a heteroscedastic normal mixture model to the M87 colors in the range \(21<z<22\). Subtracting a median photometric error of 0.02 mag in quadrature, we find \(\sigma_{blue}\sim 0.07\) and \(\sigma_{red}\sim 0.14\). These \(\sigma\) values may be overestimates because of the presence of contaminants in the tails of the color distributions, but provide first-order estimates for investigation. Given the lack of evidence for significant age differences among bright GCs in massive early-type galaxies (Strader _et al._ 2005), it is reasonable to attribute the dispersion in \(g-z\) entirely to metallicity. To convert these dispersions into metallicity, we must find a relation between [M/H] and \(g-z\). Jordan _et al._ (2004) used Bruzual & Charlot (2003) models to find a linear relationship in the range \(-2.3\leq\) [M/H] \(\leq+0.4\), however, the relation may be nonlinear for

low metallicities. We fit a quadratic relation for [M/H] and \(g-z\) using Maraston (2005) model predictions for \(g-z\) for four metallicities (\(-2.25,-1.35,-0.33,0\)) and Bruzual & Charlot (2003) predictions at five metallicities (\(-2.3,-1.7,-0.7,-0.4,0\)). Both sets of models assume a 13 Gyr stellar population and a Salpeter initial mass function. The resulting fit is: [M/H] \(=-8.088+9.081(g-z)-2.524(g-z)^{2}\). Using this relation, the blue and red GC dispersions correspond to \(1\sigma\) metallicity ranges of (\(-2.0\), \(-1.4\)) and (\(-0.7\), \(-0.1\)), respectively. Thus, despite the wider color range of the red GCs, their logarithmic metallicity range appears no wider than that of the blue GCs due to the nonlinear relationship between \(g-z\) and metallicity. A caveat is that this conclusion depends critically upon our assumed relation, which is likely to be most uncertain in the metal-poor regime where the stellar libraries of the models have few stars. In M87, there is a clear enhancement of GCs at \(z\sim 22.5\) with _intermediate_ colors, giving the CMD the appearance of a "cosmic H". This is illustrated more clearly in Figure 5, which shows color histograms of GCs in the regions \(22.2<z<22.7\) and \(22.8<z<23.3\), just below. Such a subpopulation of "H" clusters is also present, albeit less clearly and slightly fainter, in NGC 4472. Since this is near the turnover of the GCLF (with the largest number of GCs per magnitude bin), it is difficult to acertain whether the enhancement is present at all colors or only in a narrow range. However, this subpopulation appears normal in all other respects. Defining a fiducial sample as lying in the range: \(1.0<g-z<1.25\) and \(22.2<z<22.7\), the sizes and radial distribution of these GCs lie between those of the blue and red GCs, though perhaps more similar to the blue ones. Visually they are indistinguishable from GCs of similar luminosity. With current data we cannot say how common these "H" GCs are in massive ellipticals, though their presence in NGC 4472 suggests that in the Virgo Cluster the phenomenon is not limited to M87. Here only 34 GCs fall into the limits defined above (though this is unlikely to define a complete sample); if this subpopulation scales with GC system richness, 10 or fewer might be expected in other luminous galaxies, rendering their detection difficult. At \(z\sim 22.5\) spectroscopy of these GCs is feasible (though difficult), and could help establish whether their intermediate colors are due principally to metallicity or age, and whether they have kinematics distinct from the blue or red GCs. There may be a tail of these objects that extend to brighter magnitudes, but it is difficult to tell whether these are just outlying GCs in the normal blue or red subpopulations. Also of interest is a group of anomalously bright GCs (\(z\lesssim 20\)), which have a wide range in color (\(0.9<g-z<1.5\)) and in some galaxies are separated from the bulk of the GC system by 0.1 mag or more. In M87, these GCs are on average \(15\%\) larger (with mean \(r_{h}=2.7\) pc) than GCs in the rest of the system, and have median galactocentric distances \(\sim 10\%\) smaller (\(\sim 5\) kpc) than the GC system as a whole. The dispersions in these properties appear consistent with the GC system as a whole, but with few bright

clusters this is difficult to constrain. Some of these luminous GCs are likely in the tail of the normal blue and red subpopulations, but given the wide range in colors (including many with intermediate colors), small galactocentric radii, and the larger-than-average sizes, a portion may also be the stripped nuclei of dwarf galaxies--analogues of \(\omega\) Cen in the Galaxy (Majewski _et al._ 2000). The color distribution of dE nuclei in this sample (see below) peaks at \(g-z\sim 1.0-1.1\), consistent with the blue end of the intermediate-color objects. The surface brightness profiles of these objects resemble those of other GCs and do not have the exponential profiles seen in some ultra-compact dwarf galaxies (de Propris _et al._ 2005), though we note our size selection criterion for GCs would exclude some Virgo UCDs (Hasegan _et al._ 2005). Similar bright, intermediate-color GCs have also been found in the NGC 1399 (Dirsch _et al._ 2003), NGC 4636 (Dirsch, Schuberth, & Richtler 2005), and NGC 1407 (Cenarro _et al._ 2005); they appear to be a common feature of gEs. #### 3.1.1 The Blue Tilt A feature present in the CMDs of M87 and NGC 4649 is a _tilt_ of the color distribution of blue GCs, in the sense that the mean color of the blue GCs becomes redder with increasing luminosity. No such trend is apparent for the red GCs. A precise measure of this observation is challenging; due to the multiple subpopulations and "H" GCs, a direct linear fit is not viable. Instead, we divided the M87 GC candidates into four 0.5 mag bins in the range \(20<z<22\) and one 0.4 mag bin (\(22.8<z<23.2\), avoiding the "H" GCs). To each of these bins we fit a heteroscedastic normal mixture model, and then fit a weighted linear model to the resulting blue peaks. This model is \(g-z=-0.043\,z+1.848\); the slope is \(4\sigma\) significant. A fit to the corresponding red peaks is consistent with a slope of zero. These fits, as well as the binned values, are overplotted on the M87 CMD in Figure 2. Including a bin with the "H" GCs (\(22<z<22.8\)) gives a slope which is (unsurprisingly) slightly more shallow (\(-0.037\)) but still significant. NGC 4649 has fewer GCs than M87 and appears to have no "H" GCs, so for this galaxy we fit three 1.0 mag bins in the range \(20<z<23\). The resulting blue GC model is \(g-z=-0.040\,z+1.817\), which agrees very well with that of M87, and there is again no significant evidence for a nonzero red GC slope. The smoothness of the change argues against stochastic stellar population changes (e.g., horizontal branch stars, blue stragglers) as the cause of the trend. If due to age, its size--\(\sim 0.12-0.13\) mag in \(g-z\) over \(\sim 3\) mag in \(z\)--would require an unlikely age spread of \(\sim 7-8\) Gyr at low metallicity using Maraston (2005) models. If due to metallicity, the color-metallicity relation derived above indicates the trend corresponds to a mean slope of \(\sim 0.15-0.2\) dex/mag. For blue GCs in these galaxies, _metallicity correlates with mass_.

Interpretations of this surprising finding are discussed below; first we consider whether a bias in observation or analysis might be the cause. Given that the correlation extends over a large range in GC luminosity, and is not seen for red GCs, selection bias (choosing redder GCs at bright magnitudes and bluer GCs at faint magnitudes) seems unlikely to be a factor. There is no significant correlation between GC luminosity and galactocentric radius, ruling out a radial variation in any quantity as a cause (e.g., dust). Together these facts also suggest that a systematic photometric error cannot be blamed. To physically produce the observed correlation, either more massive GCs must have formed from more enriched gas, or individual GCs must have self-enriched. In the former picture, we could imagine blue GCs forming in proto-dwarf galaxies with varying metal enrichment. The essential problem is that there is no evidence that the GCLF varies strongly among dEs, as we would need the most metal-rich dEs to have few or no low-mass GCs to preserve the relation. GC self-enrichment might explain the correlation, as more massive GCs could retain a larger fraction of supernovae (SNe) ejecta. The self-enrichment of GCs has been studied in some detail as a possible origin to the chemical inhomogeneities observed among stars in Galactic GCs (e.g., Smith 1987). Early works (e.g., Dopita & Smith 1986) argued that only the most massive GCs could retain enough gas to self-enrich, but this depends critically on the assumed initial metal abundance of the proto-GC cloud and on the details of the cooling curve. Morgan & Lake (1989) found that a more accurate cooling curve reduced the critical mass to \(\geq 10^{5}M_{\odot}\) in a "supershell" model, as suggested by Cayrel (1986). In the model of Parmentier _et al._ (1999), proto-GC clouds are confined by a hot protogalactic medium, and this model in fact predicts an _inverse_ GC mass-metallicity relation, in which the most massive GCs are the most metal-poor (Parmentier & Gilmore 2001). Clearly a wide range of models exist, and it is possible that with the appropriate initial conditions and physical mechanism a self-enrichment model of this sort can be made to work. Another possibility is that the blue GCs formed inside individual dark matter (DM) halos. This scenario was first proposed by Peebles (1984), but fell into disfavor (Moore 1996) after studies of Galactic GCs found low mass-to-light ratios (Pryor _et al._ 1989) and tidal tails were observed around several GCs (e.g., Pal 5; Odenkirchen _et al._ 2003). Recently, Bromm & Clarke (2002) and Mashchenko & Sills (2005a,b) have used numerical simulations to argue that GCs with primordial DM halos could lose the bulk of the DM through either violent relaxation at early times or subsequent tidal stripping. If true, then a present-day lack of DM does not necessarily imply that GCs never had DM halos. It seems qualitatively plausible to produce the correlation in this context, but whether it could be sustained in

detail requires additional simulation. Any such model need also be compared to the rather stringent set of other observations of blue GCs (some of which are not usually considered), including the lack of GC mass-radius and metallicity-galactocentric radius relations and the presence of a _global_ correlation between the mean metallicity of blue GCs and parent galaxy mass. In addition, since the Galaxy itself (and perhaps NGC 4472) show no obvious blue GC mass-metallicity relationship, variations among galaxies are needed. It is also important to explain why the red GCs do not show such a relation. If the mass-metallicity relation was in terms of _absolute_ metallicity, then a small increase in metallicity (\(0.01-0.02Z_{\odot}\)) could be visible among the blue GCs but not among the red GCs. Even if no weak relation exits, one cannot rule out a metallicity-dependent process that results in a relation only for the blue GCs even if both subpopulations formed the same way. Many other physical properties of the blue and red GCs are similar enough (e.g., GC mass functions, sizes) that it may be challenging to invoke completely different formation mechanisms. Some of the similarities could be due to post-formation dynamical destruction of low-mass or diffuse GCs, which might act to erase initial variations in some GC system properties. No consensus exists in the literature on the effectiveness of GC destruction in shaping the present-day GC mass function (e.g., Vesperini 2001; Fall & Zhang 2001). ### Subpopulation Colors and Numbers The GC color distributions were modeled using the Bayesian program Nmix (Richardson & Green 1997), which fits normal heteroscedastic mixture models. The number of subpopulations is a free parameter (ranging to 10). For nearly all of the bright galaxies, two subpopulations were preferred; in no such galaxy was there a strong preference for a uni- or trimodal color distribution. Thus we report bimodal fits for these galaxies. While many of the dE color distributions visually appear bimodal, they generally had too few GCs to constrain the number of subpopulations with this algorithm. We adopted the following solution: we fit one peak to galaxies which only had GCs with \(g-z<1\); to the remaining galaxies we fit two peaks. A histogram of the dE GC colors (Figure 6) shows bimodality, which suggests that this approach is reasonable. Of course, some of the dEs have few GCs, so the peak locations may be quite uncertain. Linear relationships have previously been reported between parent galaxy luminosity and the mean colors (peak/mode of a Gaussian fit) of the red (Larsen _et al._ 2001; Forbes & Forte 2001) and blue (Strader, Forbes, & Brodie 2004; Larsen _et al._ 2001; Lotz, Miller & Ferguson 2004) subpopulations. Except for the massive Es, the individual subpopulations in this study have few GCs. Thus, errors on peak measurements are significant for most

galaxies. Nevertheless, Figure 7 shows that clear linear relationships are present for both the blue and red GCs over the full \(\sim 6-7\) mag range in parent galaxy luminosity. These weighted relations are \(g-z=-0.014\,M_{B}+0.642\) and \(g-z=-0.053\,M_{B}+0.225\) for the blue and red GCs, respectively. The plotted error bars are standard errors of the mean. The fits exclude the gE NGC 4365, whose anomalous GC system has been discussed in detail elsewhere (Larsen, Brodie, & Strader 2005, Brodie _et al._ 2005, Larsen _et al._ 2003, Puzia _et al._ 2002). For the red GCs there is a hint that the slope may flatten out for the faintest galaxies, but a runs test on the red residuals gave \(p=0.33\) (and \(p=0.45\) for the blue GCs), suggesting reasonable model fits. Many possible systematic errors could affect the faintest galaxies, e.g., the larger effects of contamination, and the uncertainty in the distances to individual galaxies, which could change their \(M_{B}\) by \(\sim 0.2-0.3\) mag. Thus, one cannot conclude that the GC color-galaxy luminosity relations are well-constrained at the faint end of our sample. However, they are consistent with extrapolations from brighter galaxies. Our results are also consistent with previous slope measurements: Larsen _et al._ (2001) and Strader _et al._ (2004) found that the \(V-I\) red:blue ratio of slopes is \(\sim 2\), while we find \(\sim 3.7\) in \(g-z\). This is consistent with \(g-z\propto 2\,(V-I)\), a rough initial estimate of color conversion (Brodie _et al._ 2005). There is at least one ongoing program to study Galactic GCs in the Sloan filter set which should improve this (and similar) conversions considerably. It does appear that the residuals of the blue and red peak values are correlated; this is probably unavoidable when fitting heteroscedastic mixture models to populations which are not well-separated. Since the red and blue GC subpopulations clearly have different dispersions where this can be tested in detail, fitting homoscedastic models does not make sense. We experimented with fitting two-component models with the variances fixed to the mean value for the brightest galaxies, for which the large number of GCs (at least partially) breaks the degeneracy between peak location and dispersion. The slopes of the resulting blue and red relations are similar to those found using the above approach: \(-0.012\) and \(-0.057\), respectively. However, the blue peak values for many of the galaxies appear to be artificially high--this may be because galaxies less massive than gEs have smaller intra-subpopulation metallicity spreads. Thus we have chosen to leave the original fits as our final values. The very existence of red GCs in faint galaxies with \(M_{B}\sim-15\) to \(-16\) is an interesting and somewhat unexpected result. In massive early-type galaxies and many spiral bulges, the number of red GCs normalized to spheroid luminosity is approximately constant (Forbes, Brodie, & Larsen 2001). This suggests that red GCs formed along with the spheroidal field stars at \(\sim\) constant efficiency. However, many properties of dEs (e.g., surface brightness profiles, M/L ratios, spatial/velocity distribution, stellar populations) suggest that their formation mechanism was different from massive Es (e.g., Kormendy 1985, though see Graham & Guzman 2003 for a different view). A continuity of red GC properties between Es and at

least some of the dEs in our sample could imply either that their formation mechanisms were more similar than expected or that red GCs are formed by a self-regulating, local process that can occur in a variety of contexts. The mixture modeling also returns the number of GCs in each subpopulation. In Figure 8 we plot the fraction of blue GCs vs. galaxy luminosity. There is a general trend (with a large scatter) for an increasing proportion of blue GCs with decreasing galaxy luminosity. The average fraction of blue GCs for the dEs is \(\sim 0.7\), with many galaxies having no red GCs. The fraction asymptotes to \(\sim 0.4-0.5\) for the gEs, but since these data cover only the central, more red-dominated part of their GC systems, the global fraction is likely higher (e.g., in a wide-field study of three gEs by Rhode & Zepf 2004, the blue GC fraction ranged from 0.6-0.7.). The GC systems of the dEs fall entirely within the ACS field of view (see discussion below), so their measured blue GC fractions are global. These results show clearly that the classic correlation between GC metallicity/color and galaxy luminosity (Brodie & Huchra 1991) is a combination of two effects: the decreasing ratio of blue to red GCs with increasing galaxy luminosity, and, more importantly, the GC color-galaxy luminosity relations which exist for _both_ subpopulations. There does not seem to be a single "primordial" GC color-galaxy luminosity relation, as assumed in the accretion scenario for GC bimodality (e.g., Cote, Marzke, & West 2002) ## 4 Luminosity Functions and Nuclei Many previous works have found (e.g., Harris 1991, Secker 1992, Kundu & Whitmore 2001, Larsen _et al._ 2001) that the GCLF in massive galaxies is well-fit by a Gaussian or \(t_{5}\) distribution with similar properties among well-studied galaxies: \(M_{V}\sim-7.4\), \(\sigma_{V}\sim 1.3\). However, the shape among dwarf galaxies is poorly known. Individual galaxies have too few GCs for a robust fit, and thus a composite GCLF of many dwarfs is necessary. Using ground-based imaging, Durrell _et al._ (1996) found that the turnover of the summed GCLF of 11 dEs in Virgo was \(\sim 0.4\pm 0.3\) mag fainter the M87 turnover. Lotz _et al._ (2001) presented HST/WFPC2 snapshot imaging of 51 dEs in Virgo and Fornax; the summed GCLF in Lotz _et al._ is not discussed, but a conference proceeding using the same data (Miller 2002) suggests \(M_{V}\sim-7.4\) and \(-7.3\) for Virgo and Fornax, respectively. We study the dE GCLF through comparison to gEs (VCC 1316-M87, VCC 1226-NGC 4472, and VCC 1978-NGC 4649) previously found to have "normal" GCLFs (Larsen _et al._ 2001). For dwarfs we constructed a summed GCLF of all 37 dEs (\(M_{B}<-18.2\) in our sample). Since the individual distance moduli of the dEs are unknown, this GCLF could

have additional scatter because of the range of galaxy distances; we find below that this appears to be quite a small effect. The individual GC systems of dEs are quite concentrated: most GCs are within \(30-40\arcsec\), beyond which the background contamination rises sharply (size measurements are unreliable at these faint magnitudes). Thus for the dEs we selected only those GCs within \(30\arcsec\) of the center of the galaxy. In the previous sections we used photometric and structural cuts to reduce the number of background galaxies and foreground stars interloping in our GC samples. However, these same cuts cannot be directly used to fit GCLFs, since these measurements become increasingly inaccurate for faint GCs (which may then be incorrectly removed). For example, applying only the color cut (\(0.5<g-z<2.0\)) vs. the full selection criteria to M87 changes the number of GCs with \(z<23.5\) by 16% (1444 vs. 1210) and shifts the peak of the GCLF by \(\sim 0.4\) mag, which is quite significant. We directly fit \(t_{5}\) distributions to the three gE GCLFs using using the code of Secker & Harris (1993), which uses maximum-likelihood fitting and incorporates photometric errors and incompleteness (Gaussian fits gave similar results within the errors; note that \(\sigma_{gauss}=1.29\,\sigma_{t5}\)). These GCLFs only have a color cut applied: \(0.5<g-z<2.0\) for M87 and NGC 4472 and \(0.85<g-z<2.0\) for NGC 4649 (since star-forming regions in the nearby spiral NGC 4647 strongly contaminate the blue part of the CMD). The results are given in Table 2. We fit both total populations and blue and red GCs separately, using the color cuts given in the table. Similar to what is commonly seen in \(V\)-band GCLFs (e.g., Larsen _et al._ 2001), the \(g\) turnovers are \(\sim 0.3-0.4\) mag brighter for the blue GCs than the red GCs. This is predicted for equal-mass/age turnovers separated by \(\sim 1\) dex in metallicity (Ashman, Conti & Zepf 1995). However, in \(z\) the blue GCs are only slightly brighter than the red GCs (the mean difference is negligible, but the blue GCs are brighter in M87 and NGC 4472 and fainter in NGC 4649, which may be biased slightly faint because of contamination). This difference between \(g\) and \(z\) is qualitatively consistent with stellar population models: Maraston (2005) models predict that equal-mass 13 Gyr GCs with [M/H] = \(-1.35\) and \(-0.33\) will have \(\Delta g=0.6\) and \(\Delta z=0.2\); the \(z\) difference is one-third of the \(g\) difference. This effect is probably due to the greater sensitivity of \(g\) to the turnoff region and the larger number of metal lines in the blue. The errors for the blue GCLF parameters may be slightly larger than formally stated because of the presence of the "H" GCs discussed previously. Variations in this feature and in the blue tilt among galaxies could represent a fundamental limitation to the accuracy to using _only_ the blue GCLF turnover as a standard candle (see, e.g., Kissler-Patig 2000). However, the total peak locations themselves are quite constant, with a range of only 0.03 mag in \(g\) and 0.05 mag in \(z\). At least for the well-populated old GC systems of gEs, the GCLF turnover appears to be an accurate distance indicator whose primary limitation is accurate photometry.

Unfortunately, we cannot directly fit the composite dE GCLF as for the gEs--it is far too contaminated. Instead, we must first correct for background objects. As noted above, the GC systems of the dEs are very centrally concentrated. Thus we can use the outer regions of the dE images as a fiducial background. We defined a background sample in the radial range \(1.1-1.25\arcmin\) (corresponding to 5.4-6.2 kpc at a distance of 17 Mpc), then subtracted the resulting \(z\) GCLF in 0.1 mag bins from the central \(30\arcsec\) sample with the appropriate areal correction factor. We confined the fit to candidates with \(0.7<g-z<1.25\) because of the small color range of the dEs. The resulting fit (performed as above) gave a turnover of \(M_{z}=-8.14\pm 0.14\), compared to the weighted mean of \(M_{z}=-8.19\) for the gEs. These are consistent. Since the \(z\) GCLFs are being used, the overall color differences between the GC systems should have little effect on the peak locations. The dispersion of the dEs (\(\sigma_{z}=0.74\)) is less than that of the gEs (\(\sigma_{z}=1.03\)); this is probably partially due to the smaller color range of the dEs. It also indicates that the range of galaxy distances is probably not significant, consistent with the projected appearance of many of the dEs near the Virgo cluster core (Binggeli _et al._ 1987). That the peak of the GCLF appears to be the same for both gEs and dEs is perhaps puzzling. Even if both galaxy types had similar primordial GCLFs, analytic calculations and numerical simulations of GC evolution in the tidal field of the dEs suggest that dynamical friction should destroy high-mass GCs within several Gyr (Oh & Lin 2000, Lotz _et al._ 2001). Tidal shocks (which tend to destroy low-mass GCs) are more significant in massive galaxies, so it is unlikely that the loss of high-mass GCs in dEs could be balanced by a corresponding amount of low-mass GC destruction to preserve a constant turnover. _ab initio_ variations in the GCLF between dEs and gEs are another possibility, but cannot be constrained at present. If we see no evidence in the dE GCLF turnover for dynamical friction, does this provide significant evidence against this popular scenario for forming dE nuclei? In order study the relationship between the nuclei and GCs of our sample galaxies, we performed photometry and size measurements on all of the dE nuclei. The procedures were identical to those described above for the GCs, except that aperture corrections were derived through analytic integration of the derived King profile, since many of the nuclei were large compared to typical GC sizes. The properties of the nuclei are listed in Table 3. In Figure 9 we plot parent galaxy \(M_{B}\) vs. \(M_{z}\) of the nucleus. Symbol size is proportional to nuclear size. The overplotted dashed line represents Monte Carlo simulations of nuclear formation through dynamical friction of GCs from Lotz _et al._(2001). These assume 5 Gyr of orbital decay (implicitly assuming the GCs are \(\sim 5\) Gyr old); we have converted their \(M_{V}\) to \(M_{z}\) using stellar populations models of Maraston (2005). Lotz _et al._ found that the expected

dynamical friction was inconsistent with observations, since the observed nuclei were fainter than predicted by their simulations. They argued that some additional process was needed to oppose dynamical friction (e.g., tidal torques). At first glance, this is supported by Figure 9--for galaxies in the luminosity range \(-15<M_{B}<-17\), there is a \(\sim 3\) mag range in nuclear luminosity. But most of the bright nuclei are quite small, while those much fainter than predicted from the simulations are generally larger (by factors of 2-4). In addition, the small nuclei tend to be redder than the large ones; some of the large faint nuclei are blue enough that they require some recent star formation. While not a one-to-one relation, it seems plausible that these luminosity and size distributions reflect two different channels of nuclear formation: small bright red nuclei are formed by dynamical friction, while large faint blue nuclei are formed by a dissipative process with little or no contribution from GCs. The red nuclei have typical colors of \(g-z\sim 1.1\), consistent with those of the red GCs in dEs. This implies that red GCs (rather than blue GCs) would need to be the primary "fuel" for nuclei built by dynamical friction, at least among the brigher dEs. The formation of blue nuclei could happen as the dE progenitor entered the cluster, or during tidal interactions that drive gas to the center of the galaxy (i.e., harassment, Mayer _et al._ 2001). Present simulations do not resolve the central parts of the galaxy with adequate resolution to estimate the size of a central high-density component, but presumably this will be possible in the future. It is unclear how to square the hypothesis that some nuclei are built by dynamical friction with the GCLF results from above. Perhaps the small \(\sigma_{z}\) of the composite dE (compared to the gEs) is additional evidence of GC destruction. Our background correction for the dE GCLF is potential source of systematic error in the turnover determination. Deeper ACS photometry for several of the brighter Virgo dEs could be quite helpful, since it could give both better rejection of background sources and more accurate SBF distances to the individual dEs. ## 5 Specific Frequencies For the Es in our sample, the ACS FOV covers only a fraction of their GC system, and no robust conclusions about their total number of GCs can be drawn without uncertain extrapolation. The GC systems of fainter Es and dEs are less extended and fall mostly or entirely within ACS pointings. As previously discussed, \(S_{N}\) is often used as a measure of the richness of a GC system. In fact, Harris & van den Bergh (1981) originally defined \(S_{N}\) in terms of the _bright_ GCs in a galaxy: the total number was taken as the number brighter than the turnover doubled. The justification for this procedure was that the faint end of the

GCLF was often ill-defined, with falling completeness and rising contamination. In addition, for a typical log-normal GCLF centered at \(M_{V}=-7.4\sim 3\times 10^{5}M_{\odot}\), 90% or more of the total GC system mass is in the bright half of the LF. Many of the dEs in this study have tens or fewer detected GCs, making accurate measurements of the shape of the LF impossible. Thus counting only those GCs brighter than the turnover is unwise; small variations in the peak of the GCLF among galaxies (either due to intrinsic differences or the unknown galaxy distances) could cause a large fraction of GC candidates to be included or excluded. As discussed in SS2, we chose \(z=23.5\) as the magnitude limit for our study. This is \(\sim 0.5\) mag beyond the turnover of the composite dE GCLF discussed in the previous section (\(M_{z}\sim-8.1\), \(\sigma_{z}\sim 0.7\)) at a nominal Virgo Cluster distance of 17 Mpc. For such galaxies our magnitude cutoff corresponds to \(\sim 75\)% completeness. Thus, to get total GC populations for the dEs, we divided the number of GCs brighter than \(z=23.5\) by this factor (for the few dEs with individual distance moduli, we integrated the LF to find the appropriate correction factor). Finally, as discussed in SS4, the photometric and structural cuts used to reduce contaminants also remove real GCs. Using M87 as a standard, 16.2% of real GCs with \(z<23.5\) were falsely removed; we add back this statistical correction to produce our final estimate of the total GC population. Due to the small radial extent of the dE GC systems (primarily within a projected galactocentric radius of \(\sim 30-40\arcsec\)) no correction for spatial coverage is applied. We calculated \(B\)-band \(S_{N}\) using the absolute magnitudes in Table 1. Unless otherwise noted, all \(S_{N}\) refer to \(B\)-band values. These \(S_{N}\) can be converted to the standard \(V\)-band \(S_{N}\) by dividing our values by a factor \(10^{0.4(B-V)}\); a typical dE has \(B-V=0.8\), which corresponds to a conversation factor \(\sim 2.1\). \(S_{N}\) is plotted against \(M_{B}\) in Figure 10. In this figure the dE,N galaxies are filled circles and the dE,noN galaxies are empty circles. The symbol size is proportional to the fraction of blue GCs. Miller _et al._ (1998) found that dE,noNs had, on average, lower \(S_{N}\) than dE,Ns, and that for both classes of dE there was an inverse correlation between \(S_{N}\) and galaxy luminosity. We do not see any substantial difference between dE,noN and dE,N galaxies. There does appear to be a weak correlation between \(S_{N}\) and \(M_{B}\) in our data (with quite large \(S_{N}\) for some of the faintest galaxies), but there is a large spread in \(S_{N}\) for most luminosities. The \(S_{N}\) does not appear to depend strongly on the fraction of blue GCs, so differing mixes of subpopulations cannot be responsible for the \(S_{N}\) variations. Thus the observed values correspond to total blue GC subpopulations which vary by a factor of up to \(\sim 10\) at a given luminosity. There appear to be two sets of galaxies: one group with \(S_{N}\sim 2\), the other group ranging from \(S_{N}\sim 5-20\) and a slight enhancement at \(S_{N}\sim 10\). We only have one galaxy in common with Miller _et al._ (VCC 9), but our \(S_{N}\) for this single galaxy is consistent with theirs within the

errors. We note again that background contamination remains as a systematic uncertainty for total GC populations. It is tempting to argue that these groups represent different formation channels for dEs, e.g., the fading/quenching of dIrrs, harassment of low-mass spirals, or simply a continuation of the E sequence to fainter magnitudes. In some scenarios for blue GC formation (e.g., Strader _et al._ 2005; Rhode _et al._ 2005), galaxies with roughly similar masses in a given environment would be expected to have similar numbers of blue GCs per unit mass of their DM halos. Variations in \(S_{N}\) at fixed luminosity would then be due to differences in the efficiency of converting baryons to stars--either early in the galaxy's history (due to feedback) or later, due to galaxy transformation processes as described above. Two pieces of observational evidence should help to differentiate among the possibilities. First, GC kinematics can be used directly to constrain dE mass-to-light ratios (e.g., Beasley _et al._ 2005). It will be interesting to see whether GC kinematics are connected to the apparent dichotomy of rotating vs. non-rotating dEs (Pedraz _et al._ 2002, Geha _et al._ 2002, 2003, van Zee _et al._ 2004). Second, more detailed stellar population studies of dEs are needed, especially any method which (unlike integrated light spectroscopy) can break the burst strength-age degeneracy. A rather ad hoc alternative is that the efficiency of blue GC formation varies substantially among Virgo dEs, but this cannot be constrained at present. It is interesting that the fraction of blue GCs appears unrelated both to the nucleation of the dE and the \(S_{N}\) variations; this suggests that whatever process leads to red GC formation in dEs is independent of these other factors. ## 6 Summary We have presented a detailed analysis of the GC color and luminosity distributions of several gEs and of the colors, specific frequencies, luminosity functions, and nuclei of a large sample of dEs. The most interesting feature in the gEs M87 and NGC 4649 is a correlation between mass and metallicity for individual blue GCs. Self-enrichment is a plausible interpretation of this observation, and could suggest that these GCs once possessed dark matter halos (which may have been subsequently stripped). Among the other new features observed are very luminous (\(z\gtrsim 20\)) GCs with intermediate to red colors. These objects are slightly larger than typical GCs and may be remnants of stripped dwarf galaxies. Next, we see an intermediate-color group of GCs which lies near the GCLF turnover and in the gap between the blue and red GCs. Also, the color spread among the red GCs is nearly twice that of the blue GCs, but because the relation between \(g-z\) and metallicity appears to be nonlinear, the \(1\sigma\) dispersion in metallicity (\(\sim 0.6\) dex) may be the same for

UCO/Lick Observatory, University of California, Santa Cruz, CA 95064 strader@ucolick.org, brodie@ucolick.org, lees@ucolick.org, mbeasley@ucolick.org ###### Abstract We have analyzed archival _Hubble Space Telescope_/Advanced Camera for Surveys images in \(g\) and \(z\) of the globular cluster (GC) systems of 53 ellipticals in the Virgo Cluster, spanning massive galaxies to dwarf ellipticals (dEs). Several new results emerged: (i) In the giant ellipticals (gEs) M87 and NGC 4649, there is a correlation between luminosity and color for _individual_ metal-poor GCs, such that more massive GCs are more metal-rich. A plausible interpretation of this result is self-enrichment, and may suggest that these GCs once possessed dark matter halos. (ii) In some gEs (most notably M87), there is an "interloping" subpopulation of GCs with intermediate colors (1.0 \(<g-z<\) 1.25) and a narrow magnitude range (0.5 mag) near the turnover of the GC luminosity function. These GCs look otherwise identical to the classic metal-poor and metal-rich GC subpopulations. (iii) The dispersion in color is nearly twice as large for the metal-rich GCs than the metal-poor GCs. However, there is evidence for a nonlinear relation between \(g-z\) and metallicity, and the dispersion in metallicity may be the same for both subpopulations. (iv) Very luminous, intermediate-color GCs are common in gEs. These objects may be remnants of many stripped dwarfs, analogues of \(\omega\) Cen in the Galaxy. (v) There is a continuity of GC system colors from gEs to some dEs: in particular, many dEs have metal-rich GC subpopulations. We also confirm the GC color-galaxy luminosity relations found previously for both metal-poor and metal-rich GC subpopulations. (vi) There are large differences in GC specific frequency among dEs, independent of the presence of a nucleus and the fraction of metal-rich GCs. Over \(-15<M_{B}<-18\), there is little correlation between specific frequency and \(M_{B}\) (in contrast to previous studies). But we do find evidence for two separate \(S_{N}\) classes of dEs: those with \(B\)-band \(S_{N}\sim 2\), and dEs with populous GC systems that have \(S_{N}\) ranging from \(\sim 5-20\) with median \(S_{N}\sim 10\). Together, these points suggest multiple formation channels for dEs in the Virgo Cluster. (vii) The peak of the GC luminosity function (GCLF) is the same for both gEs and dEs. This is contrary to expectations of dynamical friction on massive GCs, unless the primordial GCLF varies between gEs and dEs. Among gEs the GCLF turnover varies by a surprising small 0.05 mag, an encouraging result for its use as an accurate standard candle. (viii) dE,Ns appear bimodal in their nuclear properties: there are small bright red nuclei consistent with formation by dynamical friction of GCs, and larger faint blue nuclei which appear to have formed by a dissipative process with little contribution from GCs. The role of dynamical evolution in shaping the present-day properties of dE GC systems and their nuclei remains ambiguous. globular clusters: general -- galaxies: star clusters -- galaxies: formation ## 1 Introduction It is increasingly apparent that globular clusters (GCs) offer important constraints on the star formation and assembly histories of galaxies. Recent spectroscopic studies of GCs in massive early-type galaxies (e.g., Strader _et al._ 2005) indicate that the bulk of star formation occurred at relatively high redshift (\(z\gtrsim 2\)) in high density environments (as environmental density and galaxy mass decrease, the fraction of younger GCs may increase; see Puzia _et al._ 2005). These findings allow the age-metallicity degeneracy to be broken and lead to the conclusion that the bimodal color distributions seen in most nearby luminous galaxies are due primarily to two old GC subpopulations: metal-poor (blue) and metal-rich (red). The metallicities of these peaks correlate with host galaxy luminosity (Larsen _et al._ 2001, Strader, Brodie, & Forbes 2004; see also Lotz, Miller, & Ferguson 2004 for dwarfs). Most recent photometric studies of GC systems in ellipticals have used the Wide Field and Planetary Camera 2 (WFPC2) on the Hubble Space Telescope (HST). Compared to ground-based imaging, this strategy gains photometric accuracy and minimizes contamination at the expense of small spatial coverage. Among the larger HST studies of early-type galaxies utilizing deep imaging are Larsen _et al._ (2001) and Kundu & Whitmore (2001). These studies found bimodality in many of their sample galaxies (extending down to low-luminosity ellipticals) and a nearly uniform log-normal GC luminosity function (GCLF) with a peak at \(M_{V}\sim-7.4\). However, the GC systems of dwarf ellipticals (dEs) are more poorly understood. The large HST surveys to date (primarily of Virgo and Fornax) have been limited to relatively shallow snapshot imaging; this precluded the study of color and luminosity distributions in detail. Among the suggestions of this initial work are a correlation of increasing specific frequency (\(S_{N}\)) with decreasing galaxy luminosity, a dichotomy in the GC systems of nucleated and non-nucleated dEs (dE,N and dE,noN, respectively), and the difficulty of making dE nuclei as observed through dynamical friction of GCs (Miller _et al._ 1998, Lotz _et al._ 2001). Data from the Advanced Camera for Surveys (ACS) Virgo Cluster Survey (HST GO 9401, P. I. Cote) offers an important step forward in understanding the detailed properties of the GC systems of ellipticals over a wide range in galaxy mass. The use of F475W and F850LP filters (henceforth called \(g\) and \(z\) for convenience, though the filters do not precisely match the Sloan ones) allows a much wider spectral baseline for metallicity separation than \(V\) and \(I\). Though only a single orbit is used per galaxy, the increased sensitivity of ACS (compared to WFPC2) allows one to reach \(\sim 3\) mag beyond the turnover of the GCLF, encompassing most of the GCs in a given galaxy. More accurate photometry for the brighter GCs is also possible. Finally, the field of view of ACS is twice that of WFPC2. Together, these attributes allow a study of the color and luminosity distribution of GCs in a large sample of galaxies in much more detail than previously possible. In what follows, elliptical (unabbreviated) refers to all galaxies in our sample and denotes no specific luminosity. The three brightest galaxies are described as giant ellipticals (gEs); these have \(M_{B}\leq-21.4\). Galaxies of intermediate or high luminosity are called Es. Faint galaxies with exponential surface brightness profiles are dEs; the transition from E to dE occurs traditionally at \(M_{B}\sim-18\) (Kormendy 1985). We include two galaxies with \(M_{B}=-18.1\) (VCC 1422 and VCC 1261) under this heading, since these galaxies have nuclei similar to those commonly found among dEs. The E/dE classifications have been taken from the literature and we do not perform independent surface photometry in this paper (though we note the increasing debate in the literature about whether this dichotomy is real, e.g., Graham & Guzman 2003). We have updated the nucleation status of a dE if appropriate; the vast majority of the dEs in our study have nuclei. ## 2 Data Reduction and Analysis All data were taken as part of the ACS Virgo Cluster Survey (Cote _et al._ 2004); this survey includes both ellipticals and S0s. We used all galaxies classified as ellipticals, excepting a few dwarfs quite close to luminous Es whose GC systems could not be isolated. This left a final sample of 53 galaxies. Images were first processed through the standard ACS pipeline. _Multidrizzle_ was utilized for image combining and cosmic ray rejection. GC candidates were selected as matched-filter detections on 20 \(\times\) 20 pixel median-subtracted images. Using DAOPHOT II (Stetson 1993), aperture photometry was performed in a 5-pixel aperture and adjusted to a 10-pixel aperture using corrections of \(-0.09\) in \(g\) and \(-0.15\) in \(z\). These are median corrections derived from bright objects in the five most luminous galaxies in the Virgo Cluster Survey: VCC 1226, VCC 1316, VCC 1978, VCC 881, and VCC 798. These 10-pixel magnitudes were then corrected to a nominal infinite aperture using values of \(-0.10\) in \(g\) and \(-0.12\) in \(z\) (Sirianni _et al._ 2005; this paper describes the photometric calibration of ACS). Finally, the magnitudes were transformed to the AB system using zeropoints from Sirianni _et al._ (26.068 and 24.862 for \(g\) and \(z\), respectively), and corrected for Galactic reddening using the maps of Schlegel Finkbeiner, & Davis (1998). Most GCs at the distance of Virgo are well-resolved in ACS imaging. Half-light radii (\(r_{h}\)) for GC candidates were measured on \(g\) images (since the \(g\) PSF is more centrally concentrated) using the _ishape_ routine (Larsen 1999). For each object, King models with fixed \(c=30\) (for \(c=r_{tidal}/r_{core}\)) and varying \(r_{h}\) were convolved with a distant-dependent empirical PSF derived from bright isolated stars in the images to find the best-fit \(r_{h}\). This \(c\) is typical of non core-collapsed GCs in the Milky Way (Trager, King, & Djorgovski 1995). We experimented with allowing \(c\) to vary, but it was poorly constrained for most GCs. However, the adopted \(c\) in _ishape_ has little effect on the derived \(r_{h}\) (Larsen 1999). To convert these measured sizes into physical units, galaxy distance estimates are required. We used those derived from surface brightness fluctuation measurements in the literature when possible: these were available from Tonry _et al._ (2001) for the bright galaxies and from Jerjen _et al._ (2004) for several dEs. For the remainder of the galaxies we used a fixed distance of 17 Mpc, which is the mean of the ellipticals in Tonry _et al._ Due to the depth of the images (\(z\gtrsim 25\)), some of the fields suffer significant contamination from foreground stars and especially background galaxies. Using the gEs and several of the more populous dEs as fiducials, we chose the following structural cuts to reduce interlopers: 0.55 \(<\) sharp \(<\) 0.9, \(-0.5<\) round \(<0.5\), and \(1<r_{h}\) (pc) \(<13\), where the sharp and round parameters are from DAOPHOT. A large upper limit for \(r_{h}\) is used since the size measurements skew systematically larger for fainter GCs. We further applied a color cut of \(0.5<g-z<2.0\) (\(>0.3\) mag to each blue and red of the limiting metallicities expected for old GCs; Jordan _et al._ 2004) and an error limit \(<\) 0.15 mag. In practice, this magnitude limit excluded most GCs within the innermost few arcsec of the brightest galaxies (whose GC systems are quite populous). Finally, we visually inspected all GC candidates, and excluded those which were obviously background galaxies. Our criteria are illustrated visually in Figure 1 for the bright dE VCC 1087, which displays a good mix of actual GCs and contaminants. These cuts remove nearly all foreground stars. However, compact galaxies (or compact star-forming regions within larger galaxies) with the appropriate colors can masquerade as GCs. In some images, clusters of galaxies are clearly visible. The increasing numbers of background sources at \(z\gtrsim 23\), combined with the difficulty of accurate size measurements below this magnitude, makes efficient rejection of contaminants challenging. For gEs this is a minimal problem, due to the large number of GCs within the ACS field of view (hundreds to \(\sim 1700\) for M87). But with dEs and even low-luminosity Es, contaminants can represent a large fraction of the GC candidates. Due to these concerns, we chose only to use GCs brighter than \(z=23.5\) to study the colors and total numbers of GCs for the remainder of the paper. However, we used minimal cuts to study GC luminosity functions; this is described in more detail in SS4. Basic data about the 53 galaxies in our sample are given in Table 1, along with GC system information as discussed below. ## 3 Color Distributions ### Massive Ellipticals In Figures 2 and 3 we show the color-magnitude diagrams (CMDs) for the three most luminous galaxies in our sample: NGC 4472 (\(M_{B}=-21.9\)), M87 (\(M_{B}=-21.5\)), and NGC 4649 (\(M_{B}=-21.4\)). Figure 4 is a plot of magnitude vs. photometric error in \(g\) and \(z\) for M87. The CMDs in Figures 2 and 3 contain considerable structure only apparent because of the large number of GCs; we have chosen to discuss them in some detail. All three gEs clearly show the bimodality typical of massive galaxies, with blue and red peaks of \(g-z\sim 0.9\) and \(\sim 1.4\), respectively. This separation is twice as large as is typical of studies of GC systems in \(V-I\) (e.g., Larsen _et al._ 2001; Kundu & Whitmore 2001), due to the larger metallicity sensitivity of the \(g-z\) baseline. However, a new result is that the red peak is clearly broader than the blue peak; at bright magnitudes (\(z<22\)) there is little photometric error so this must be due to real color differences. To gauge the size of this effect, we fit a heteroscedastic normal mixture model to the M87 colors in the range \(21<z<22\). Subtracting a median photometric error of 0.02 mag in quadrature, we find \(\sigma_{blue}\sim 0.07\) and \(\sigma_{red}\sim 0.14\). These \(\sigma\) values may be overestimates because of the presence of contaminants in the tails of the color distributions, but provide first-order estimates for investigation. Given the lack of evidence for significant age differences among bright GCs in massive early-type galaxies (Strader _et al._ 2005), it is reasonable to attribute the dispersion in \(g-z\) entirely to metallicity. To convert these dispersions into metallicity, we must find a relation between [M/H] and \(g-z\). Jordan _et al._ (2004) used Bruzual & Charlot (2003) models to find a linear relationship in the range \(-2.3\leq\) [M/H] \(\leq+0.4\), however, the relation may be nonlinear for low metallicities. We fit a quadratic relation for [M/H] and \(g-z\) using Maraston (2005) model predictions for \(g-z\) for four metallicities (\(-2.25,-1.35,-0.33,0\)) and Bruzual & Charlot (2003) predictions at five metallicities (\(-2.3,-1.7,-0.7,-0.4,0\)). Both sets of models assume a 13 Gyr stellar population and a Salpeter initial mass function. The resulting fit is: [M/H] \(=-8.088+9.081(g-z)-2.524(g-z)^{2}\). Using this relation, the blue and red GC dispersions correspond to \(1\sigma\) metallicity ranges of (\(-2.0\), \(-1.4\)) and (\(-0.7\), \(-0.1\)), respectively. Thus, despite the wider color range of the red GCs, their logarithmic metallicity range appears no wider than that of the blue GCs due to the nonlinear relationship between \(g-z\) and metallicity. A caveat is that this conclusion depends critically upon our assumed relation, which is likely to be most uncertain in the metal-poor regime where the stellar libraries of the models have few stars. In M87, there is a clear enhancement of GCs at \(z\sim 22.5\) with _intermediate_ colors, giving the CMD the appearance of a "cosmic H". This is illustrated more clearly in Figure 5, which shows color histograms of GCs in the regions \(22.2<z<22.7\) and \(22.8<z<23.3\), just below. Such a subpopulation of "H" clusters is also present, albeit less clearly and slightly fainter, in NGC 4472. Since this is near the turnover of the GCLF (with the largest number of GCs per magnitude bin), it is difficult to acertain whether the enhancement is present at all colors or only in a narrow range. However, this subpopulation appears normal in all other respects. Defining a fiducial sample as lying in the range: \(1.0<g-z<1.25\) and \(22.2<z<22.7\), the sizes and radial distribution of these GCs lie between those of the blue and red GCs, though perhaps more similar to the blue ones. Visually they are indistinguishable from GCs of similar luminosity. With current data we cannot say how common these "H" GCs are in massive ellipticals, though their presence in NGC 4472 suggests that in the Virgo Cluster the phenomenon is not limited to M87. Here only 34 GCs fall into the limits defined above (though this is unlikely to define a complete sample); if this subpopulation scales with GC system richness, 10 or fewer might be expected in other luminous galaxies, rendering their detection difficult. At \(z\sim 22.5\) spectroscopy of these GCs is feasible (though difficult), and could help establish whether their intermediate colors are due principally to metallicity or age, and whether they have kinematics distinct from the blue or red GCs. There may be a tail of these objects that extend to brighter magnitudes, but it is difficult to tell whether these are just outlying GCs in the normal blue or red subpopulations. Also of interest is a group of anomalously bright GCs (\(z\lesssim 20\)), which have a wide range in color (\(0.9<g-z<1.5\)) and in some galaxies are separated from the bulk of the GC system by 0.1 mag or more. In M87, these GCs are on average \(15\%\) larger (with mean \(r_{h}=2.7\) pc) than GCs in the rest of the system, and have median galactocentric distances \(\sim 10\%\) smaller (\(\sim 5\) kpc) than the GC system as a whole. The dispersions in these properties appear consistent with the GC system as a whole, but with few bright clusters this is difficult to constrain. Some of these luminous GCs are likely in the tail of the normal blue and red subpopulations, but given the wide range in colors (including many with intermediate colors), small galactocentric radii, and the larger-than-average sizes, a portion may also be the stripped nuclei of dwarf galaxies--analogues of \(\omega\) Cen in the Galaxy (Majewski _et al._ 2000). The color distribution of dE nuclei in this sample (see below) peaks at \(g-z\sim 1.0-1.1\), consistent with the blue end of the intermediate-color objects. The surface brightness profiles of these objects resemble those of other GCs and do not have the exponential profiles seen in some ultra-compact dwarf galaxies (de Propris _et al._ 2005), though we note our size selection criterion for GCs would exclude some Virgo UCDs (Hasegan _et al._ 2005). Similar bright, intermediate-color GCs have also been found in the NGC 1399 (Dirsch _et al._ 2003), NGC 4636 (Dirsch, Schuberth, & Richtler 2005), and NGC 1407 (Cenarro _et al._ 2005); they appear to be a common feature of gEs. #### 3.1.1 The Blue Tilt A feature present in the CMDs of M87 and NGC 4649 is a _tilt_ of the color distribution of blue GCs, in the sense that the mean color of the blue GCs becomes redder with increasing luminosity. No such trend is apparent for the red GCs. A precise measure of this observation is challenging; due to the multiple subpopulations and "H" GCs, a direct linear fit is not viable. Instead, we divided the M87 GC candidates into four 0.5 mag bins in the range \(20<z<22\) and one 0.4 mag bin (\(22.8<z<23.2\), avoiding the "H" GCs). To each of these bins we fit a heteroscedastic normal mixture model, and then fit a weighted linear model to the resulting blue peaks. This model is \(g-z=-0.043\,z+1.848\); the slope is \(4\sigma\) significant. A fit to the corresponding red peaks is consistent with a slope of zero. These fits, as well as the binned values, are overplotted on the M87 CMD in Figure 2. Including a bin with the "H" GCs (\(22<z<22.8\)) gives a slope which is (unsurprisingly) slightly more shallow (\(-0.037\)) but still significant. NGC 4649 has fewer GCs than M87 and appears to have no "H" GCs, so for this galaxy we fit three 1.0 mag bins in the range \(20<z<23\). The resulting blue GC model is \(g-z=-0.040\,z+1.817\), which agrees very well with that of M87, and there is again no significant evidence for a nonzero red GC slope. The smoothness of the change argues against stochastic stellar population changes (e.g., horizontal branch stars, blue stragglers) as the cause of the trend. If due to age, its size--\(\sim 0.12-0.13\) mag in \(g-z\) over \(\sim 3\) mag in \(z\)--would require an unlikely age spread of \(\sim 7-8\) Gyr at low metallicity using Maraston (2005) models. If due to metallicity, the color-metallicity relation derived above indicates the trend corresponds to a mean slope of \(\sim 0.15-0.2\) dex/mag. For blue GCs in these galaxies, _metallicity correlates with mass_. Interpretations of this surprising finding are discussed below; first we consider whether a bias in observation or analysis might be the cause. Given that the correlation extends over a large range in GC luminosity, and is not seen for red GCs, selection bias (choosing redder GCs at bright magnitudes and bluer GCs at faint magnitudes) seems unlikely to be a factor. There is no significant correlation between GC luminosity and galactocentric radius, ruling out a radial variation in any quantity as a cause (e.g., dust). Together these facts also suggest that a systematic photometric error cannot be blamed. To physically produce the observed correlation, either more massive GCs must have formed from more enriched gas, or individual GCs must have self-enriched. In the former picture, we could imagine blue GCs forming in proto-dwarf galaxies with varying metal enrichment. The essential problem is that there is no evidence that the GCLF varies strongly among dEs, as we would need the most metal-rich dEs to have few or no low-mass GCs to preserve the relation. GC self-enrichment might explain the correlation, as more massive GCs could retain a larger fraction of supernovae (SNe) ejecta. The self-enrichment of GCs has been studied in some detail as a possible origin to the chemical inhomogeneities observed among stars in Galactic GCs (e.g., Smith 1987). Early works (e.g., Dopita & Smith 1986) argued that only the most massive GCs could retain enough gas to self-enrich, but this depends critically on the assumed initial metal abundance of the proto-GC cloud and on the details of the cooling curve. Morgan & Lake (1989) found that a more accurate cooling curve reduced the critical mass to \(\geq 10^{5}M_{\odot}\) in a "supershell" model, as suggested by Cayrel (1986). In the model of Parmentier _et al._ (1999), proto-GC clouds are confined by a hot protogalactic medium, and this model in fact predicts an _inverse_ GC mass-metallicity relation, in which the most massive GCs are the most metal-poor (Parmentier & Gilmore 2001). Clearly a wide range of models exist, and it is possible that with the appropriate initial conditions and physical mechanism a self-enrichment model of this sort can be made to work. Another possibility is that the blue GCs formed inside individual dark matter (DM) halos. This scenario was first proposed by Peebles (1984), but fell into disfavor (Moore 1996) after studies of Galactic GCs found low mass-to-light ratios (Pryor _et al._ 1989) and tidal tails were observed around several GCs (e.g., Pal 5; Odenkirchen _et al._ 2003). Recently, Bromm & Clarke (2002) and Mashchenko & Sills (2005a,b) have used numerical simulations to argue that GCs with primordial DM halos could lose the bulk of the DM through either violent relaxation at early times or subsequent tidal stripping. If true, then a present-day lack of DM does not necessarily imply that GCs never had DM halos. It seems qualitatively plausible to produce the correlation in this context, but whether it could be sustained in detail requires additional simulation. Any such model need also be compared to the rather stringent set of other observations of blue GCs (some of which are not usually considered), including the lack of GC mass-radius and metallicity-galactocentric radius relations and the presence of a _global_ correlation between the mean metallicity of blue GCs and parent galaxy mass. In addition, since the Galaxy itself (and perhaps NGC 4472) show no obvious blue GC mass-metallicity relationship, variations among galaxies are needed. It is also important to explain why the red GCs do not show such a relation. If the mass-metallicity relation was in terms of _absolute_ metallicity, then a small increase in metallicity (\(0.01-0.02Z_{\odot}\)) could be visible among the blue GCs but not among the red GCs. Even if no weak relation exits, one cannot rule out a metallicity-dependent process that results in a relation only for the blue GCs even if both subpopulations formed the same way. Many other physical properties of the blue and red GCs are similar enough (e.g., GC mass functions, sizes) that it may be challenging to invoke completely different formation mechanisms. Some of the similarities could be due to post-formation dynamical destruction of low-mass or diffuse GCs, which might act to erase initial variations in some GC system properties. No consensus exists in the literature on the effectiveness of GC destruction in shaping the present-day GC mass function (e.g., Vesperini 2001; Fall & Zhang 2001). ### Subpopulation Colors and Numbers The GC color distributions were modeled using the Bayesian program Nmix (Richardson & Green 1997), which fits normal heteroscedastic mixture models. The number of subpopulations is a free parameter (ranging to 10). For nearly all of the bright galaxies, two subpopulations were preferred; in no such galaxy was there a strong preference for a uni- or trimodal color distribution. Thus we report bimodal fits for these galaxies. While many of the dE color distributions visually appear bimodal, they generally had too few GCs to constrain the number of subpopulations with this algorithm. We adopted the following solution: we fit one peak to galaxies which only had GCs with \(g-z<1\); to the remaining galaxies we fit two peaks. A histogram of the dE GC colors (Figure 6) shows bimodality, which suggests that this approach is reasonable. Of course, some of the dEs have few GCs, so the peak locations may be quite uncertain. Linear relationships have previously been reported between parent galaxy luminosity and the mean colors (peak/mode of a Gaussian fit) of the red (Larsen _et al._ 2001; Forbes & Forte 2001) and blue (Strader, Forbes, & Brodie 2004; Larsen _et al._ 2001; Lotz, Miller & Ferguson 2004) subpopulations. Except for the massive Es, the individual subpopulations in this study have few GCs. Thus, errors on peak measurements are significant for most galaxies. Nevertheless, Figure 7 shows that clear linear relationships are present for both the blue and red GCs over the full \(\sim 6-7\) mag range in parent galaxy luminosity. These weighted relations are \(g-z=-0.014\,M_{B}+0.642\) and \(g-z=-0.053\,M_{B}+0.225\) for the blue and red GCs, respectively. The plotted error bars are standard errors of the mean. The fits exclude the gE NGC 4365, whose anomalous GC system has been discussed in detail elsewhere (Larsen, Brodie, & Strader 2005, Brodie _et al._ 2005, Larsen _et al._ 2003, Puzia _et al._ 2002). For the red GCs there is a hint that the slope may flatten out for the faintest galaxies, but a runs test on the red residuals gave \(p=0.33\) (and \(p=0.45\) for the blue GCs), suggesting reasonable model fits. Many possible systematic errors could affect the faintest galaxies, e.g., the larger effects of contamination, and the uncertainty in the distances to individual galaxies, which could change their \(M_{B}\) by \(\sim 0.2-0.3\) mag. Thus, one cannot conclude that the GC color-galaxy luminosity relations are well-constrained at the faint end of our sample. However, they are consistent with extrapolations from brighter galaxies. Our results are also consistent with previous slope measurements: Larsen _et al._ (2001) and Strader _et al._ (2004) found that the \(V-I\) red:blue ratio of slopes is \(\sim 2\), while we find \(\sim 3.7\) in \(g-z\). This is consistent with \(g-z\propto 2\,(V-I)\), a rough initial estimate of color conversion (Brodie _et al._ 2005). There is at least one ongoing program to study Galactic GCs in the Sloan filter set which should improve this (and similar) conversions considerably. It does appear that the residuals of the blue and red peak values are correlated; this is probably unavoidable when fitting heteroscedastic mixture models to populations which are not well-separated. Since the red and blue GC subpopulations clearly have different dispersions where this can be tested in detail, fitting homoscedastic models does not make sense. We experimented with fitting two-component models with the variances fixed to the mean value for the brightest galaxies, for which the large number of GCs (at least partially) breaks the degeneracy between peak location and dispersion. The slopes of the resulting blue and red relations are similar to those found using the above approach: \(-0.012\) and \(-0.057\), respectively. However, the blue peak values for many of the galaxies appear to be artificially high--this may be because galaxies less massive than gEs have smaller intra-subpopulation metallicity spreads. Thus we have chosen to leave the original fits as our final values. The very existence of red GCs in faint galaxies with \(M_{B}\sim-15\) to \(-16\) is an interesting and somewhat unexpected result. In massive early-type galaxies and many spiral bulges, the number of red GCs normalized to spheroid luminosity is approximately constant (Forbes, Brodie, & Larsen 2001). This suggests that red GCs formed along with the spheroidal field stars at \(\sim\) constant efficiency. However, many properties of dEs (e.g., surface brightness profiles, M/L ratios, spatial/velocity distribution, stellar populations) suggest that their formation mechanism was different from massive Es (e.g., Kormendy 1985, though see Graham & Guzman 2003 for a different view). A continuity of red GC properties between Es and at least some of the dEs in our sample could imply either that their formation mechanisms were more similar than expected or that red GCs are formed by a self-regulating, local process that can occur in a variety of contexts. The mixture modeling also returns the number of GCs in each subpopulation. In Figure 8 we plot the fraction of blue GCs vs. galaxy luminosity. There is a general trend (with a large scatter) for an increasing proportion of blue GCs with decreasing galaxy luminosity. The average fraction of blue GCs for the dEs is \(\sim 0.7\), with many galaxies having no red GCs. The fraction asymptotes to \(\sim 0.4-0.5\) for the gEs, but since these data cover only the central, more red-dominated part of their GC systems, the global fraction is likely higher (e.g., in a wide-field study of three gEs by Rhode & Zepf 2004, the blue GC fraction ranged from 0.6-0.7.). The GC systems of the dEs fall entirely within the ACS field of view (see discussion below), so their measured blue GC fractions are global. These results show clearly that the classic correlation between GC metallicity/color and galaxy luminosity (Brodie & Huchra 1991) is a combination of two effects: the decreasing ratio of blue to red GCs with increasing galaxy luminosity, and, more importantly, the GC color-galaxy luminosity relations which exist for _both_ subpopulations. There does not seem to be a single "primordial" GC color-galaxy luminosity relation, as assumed in the accretion scenario for GC bimodality (e.g., Cote, Marzke, & West 2002) ## 4 Luminosity Functions and Nuclei Many previous works have found (e.g., Harris 1991, Secker 1992, Kundu & Whitmore 2001, Larsen _et al._ 2001) that the GCLF in massive galaxies is well-fit by a Gaussian or \(t_{5}\) distribution with similar properties among well-studied galaxies: \(M_{V}\sim-7.4\), \(\sigma_{V}\sim 1.3\). However, the shape among dwarf galaxies is poorly known. Individual galaxies have too few GCs for a robust fit, and thus a composite GCLF of many dwarfs is necessary. Using ground-based imaging, Durrell _et al._ (1996) found that the turnover of the summed GCLF of 11 dEs in Virgo was \(\sim 0.4\pm 0.3\) mag fainter the M87 turnover. Lotz _et al._ (2001) presented HST/WFPC2 snapshot imaging of 51 dEs in Virgo and Fornax; the summed GCLF in Lotz _et al._ is not discussed, but a conference proceeding using the same data (Miller 2002) suggests \(M_{V}\sim-7.4\) and \(-7.3\) for Virgo and Fornax, respectively. We study the dE GCLF through comparison to gEs (VCC 1316-M87, VCC 1226-NGC 4472, and VCC 1978-NGC 4649) previously found to have "normal" GCLFs (Larsen _et al._ 2001). For dwarfs we constructed a summed GCLF of all 37 dEs (\(M_{B}<-18.2\) in our sample). Since the individual distance moduli of the dEs are unknown, this GCLF could have additional scatter because of the range of galaxy distances; we find below that this appears to be quite a small effect. The individual GC systems of dEs are quite concentrated: most GCs are within \(30-40\arcsec\), beyond which the background contamination rises sharply (size measurements are unreliable at these faint magnitudes). Thus for the dEs we selected only those GCs within \(30\arcsec\) of the center of the galaxy. In the previous sections we used photometric and structural cuts to reduce the number of background galaxies and foreground stars interloping in our GC samples. However, these same cuts cannot be directly used to fit GCLFs, since these measurements become increasingly inaccurate for faint GCs (which may then be incorrectly removed). For example, applying only the color cut (\(0.5<g-z<2.0\)) vs. the full selection criteria to M87 changes the number of GCs with \(z<23.5\) by 16% (1444 vs. 1210) and shifts the peak of the GCLF by \(\sim 0.4\) mag, which is quite significant. We directly fit \(t_{5}\) distributions to the three gE GCLFs using using the code of Secker & Harris (1993), which uses maximum-likelihood fitting and incorporates photometric errors and incompleteness (Gaussian fits gave similar results within the errors; note that \(\sigma_{gauss}=1.29\,\sigma_{t5}\)). These GCLFs only have a color cut applied: \(0.5<g-z<2.0\) for M87 and NGC 4472 and \(0.85<g-z<2.0\) for NGC 4649 (since star-forming regions in the nearby spiral NGC 4647 strongly contaminate the blue part of the CMD). The results are given in Table 2. We fit both total populations and blue and red GCs separately, using the color cuts given in the table. Similar to what is commonly seen in \(V\)-band GCLFs (e.g., Larsen _et al._ 2001), the \(g\) turnovers are \(\sim 0.3-0.4\) mag brighter for the blue GCs than the red GCs. This is predicted for equal-mass/age turnovers separated by \(\sim 1\) dex in metallicity (Ashman, Conti & Zepf 1995). However, in \(z\) the blue GCs are only slightly brighter than the red GCs (the mean difference is negligible, but the blue GCs are brighter in M87 and NGC 4472 and fainter in NGC 4649, which may be biased slightly faint because of contamination). This difference between \(g\) and \(z\) is qualitatively consistent with stellar population models: Maraston (2005) models predict that equal-mass 13 Gyr GCs with [M/H] = \(-1.35\) and \(-0.33\) will have \(\Delta g=0.6\) and \(\Delta z=0.2\); the \(z\) difference is one-third of the \(g\) difference. This effect is probably due to the greater sensitivity of \(g\) to the turnoff region and the larger number of metal lines in the blue. The errors for the blue GCLF parameters may be slightly larger than formally stated because of the presence of the "H" GCs discussed previously. Variations in this feature and in the blue tilt among galaxies could represent a fundamental limitation to the accuracy to using _only_ the blue GCLF turnover as a standard candle (see, e.g., Kissler-Patig 2000). However, the total peak locations themselves are quite constant, with a range of only 0.03 mag in \(g\) and 0.05 mag in \(z\). At least for the well-populated old GC systems of gEs, the GCLF turnover appears to be an accurate distance indicator whose primary limitation is accurate photometry. Unfortunately, we cannot directly fit the composite dE GCLF as for the gEs--it is far too contaminated. Instead, we must first correct for background objects. As noted above, the GC systems of the dEs are very centrally concentrated. Thus we can use the outer regions of the dE images as a fiducial background. We defined a background sample in the radial range \(1.1-1.25\arcmin\) (corresponding to 5.4-6.2 kpc at a distance of 17 Mpc), then subtracted the resulting \(z\) GCLF in 0.1 mag bins from the central \(30\arcsec\) sample with the appropriate areal correction factor. We confined the fit to candidates with \(0.7<g-z<1.25\) because of the small color range of the dEs. The resulting fit (performed as above) gave a turnover of \(M_{z}=-8.14\pm 0.14\), compared to the weighted mean of \(M_{z}=-8.19\) for the gEs. These are consistent. Since the \(z\) GCLFs are being used, the overall color differences between the GC systems should have little effect on the peak locations. The dispersion of the dEs (\(\sigma_{z}=0.74\)) is less than that of the gEs (\(\sigma_{z}=1.03\)); this is probably partially due to the smaller color range of the dEs. It also indicates that the range of galaxy distances is probably not significant, consistent with the projected appearance of many of the dEs near the Virgo cluster core (Binggeli _et al._ 1987). That the peak of the GCLF appears to be the same for both gEs and dEs is perhaps puzzling. Even if both galaxy types had similar primordial GCLFs, analytic calculations and numerical simulations of GC evolution in the tidal field of the dEs suggest that dynamical friction should destroy high-mass GCs within several Gyr (Oh & Lin 2000, Lotz _et al._ 2001). Tidal shocks (which tend to destroy low-mass GCs) are more significant in massive galaxies, so it is unlikely that the loss of high-mass GCs in dEs could be balanced by a corresponding amount of low-mass GC destruction to preserve a constant turnover. _ab initio_ variations in the GCLF between dEs and gEs are another possibility, but cannot be constrained at present. If we see no evidence in the dE GCLF turnover for dynamical friction, does this provide significant evidence against this popular scenario for forming dE nuclei? In order study the relationship between the nuclei and GCs of our sample galaxies, we performed photometry and size measurements on all of the dE nuclei. The procedures were identical to those described above for the GCs, except that aperture corrections were derived through analytic integration of the derived King profile, since many of the nuclei were large compared to typical GC sizes. The properties of the nuclei are listed in Table 3. In Figure 9 we plot parent galaxy \(M_{B}\) vs. \(M_{z}\) of the nucleus. Symbol size is proportional to nuclear size. The overplotted dashed line represents Monte Carlo simulations of nuclear formation through dynamical friction of GCs from Lotz _et al._(2001). These assume 5 Gyr of orbital decay (implicitly assuming the GCs are \(\sim 5\) Gyr old); we have converted their \(M_{V}\) to \(M_{z}\) using stellar populations models of Maraston (2005). Lotz _et al._ found that the expected dynamical friction was inconsistent with observations, since the observed nuclei were fainter than predicted by their simulations. They argued that some additional process was needed to oppose dynamical friction (e.g., tidal torques). At first glance, this is supported by Figure 9--for galaxies in the luminosity range \(-15<M_{B}<-17\), there is a \(\sim 3\) mag range in nuclear luminosity. But most of the bright nuclei are quite small, while those much fainter than predicted from the simulations are generally larger (by factors of 2-4). In addition, the small nuclei tend to be redder than the large ones; some of the large faint nuclei are blue enough that they require some recent star formation. While not a one-to-one relation, it seems plausible that these luminosity and size distributions reflect two different channels of nuclear formation: small bright red nuclei are formed by dynamical friction, while large faint blue nuclei are formed by a dissipative process with little or no contribution from GCs. The red nuclei have typical colors of \(g-z\sim 1.1\), consistent with those of the red GCs in dEs. This implies that red GCs (rather than blue GCs) would need to be the primary "fuel" for nuclei built by dynamical friction, at least among the brigher dEs. The formation of blue nuclei could happen as the dE progenitor entered the cluster, or during tidal interactions that drive gas to the center of the galaxy (i.e., harassment, Mayer _et al._ 2001). Present simulations do not resolve the central parts of the galaxy with adequate resolution to estimate the size of a central high-density component, but presumably this will be possible in the future. It is unclear how to square the hypothesis that some nuclei are built by dynamical friction with the GCLF results from above. Perhaps the small \(\sigma_{z}\) of the composite dE (compared to the gEs) is additional evidence of GC destruction. Our background correction for the dE GCLF is potential source of systematic error in the turnover determination. Deeper ACS photometry for several of the brighter Virgo dEs could be quite helpful, since it could give both better rejection of background sources and more accurate SBF distances to the individual dEs. ## 5 Specific Frequencies For the Es in our sample, the ACS FOV covers only a fraction of their GC system, and no robust conclusions about their total number of GCs can be drawn without uncertain extrapolation. The GC systems of fainter Es and dEs are less extended and fall mostly or entirely within ACS pointings. As previously discussed, \(S_{N}\) is often used as a measure of the richness of a GC system. In fact, Harris & van den Bergh (1981) originally defined \(S_{N}\) in terms of the _bright_ GCs in a galaxy: the total number was taken as the number brighter than the turnover doubled. The justification for this procedure was that the faint end of the GCLF was often ill-defined, with falling completeness and rising contamination. In addition, for a typical log-normal GCLF centered at \(M_{V}=-7.4\sim 3\times 10^{5}M_{\odot}\), 90% or more of the total GC system mass is in the bright half of the LF. Many of the dEs in this study have tens or fewer detected GCs, making accurate measurements of the shape of the LF impossible. Thus counting only those GCs brighter than the turnover is unwise; small variations in the peak of the GCLF among galaxies (either due to intrinsic differences or the unknown galaxy distances) could cause a large fraction of GC candidates to be included or excluded. As discussed in SS2, we chose \(z=23.5\) as the magnitude limit for our study. This is \(\sim 0.5\) mag beyond the turnover of the composite dE GCLF discussed in the previous section (\(M_{z}\sim-8.1\), \(\sigma_{z}\sim 0.7\)) at a nominal Virgo Cluster distance of 17 Mpc. For such galaxies our magnitude cutoff corresponds to \(\sim 75\)% completeness. Thus, to get total GC populations for the dEs, we divided the number of GCs brighter than \(z=23.5\) by this factor (for the few dEs with individual distance moduli, we integrated the LF to find the appropriate correction factor). Finally, as discussed in SS4, the photometric and structural cuts used to reduce contaminants also remove real GCs. Using M87 as a standard, 16.2% of real GCs with \(z<23.5\) were falsely removed; we add back this statistical correction to produce our final estimate of the total GC population. Due to the small radial extent of the dE GC systems (primarily within a projected galactocentric radius of \(\sim 30-40\arcsec\)) no correction for spatial coverage is applied. We calculated \(B\)-band \(S_{N}\) using the absolute magnitudes in Table 1. Unless otherwise noted, all \(S_{N}\) refer to \(B\)-band values. These \(S_{N}\) can be converted to the standard \(V\)-band \(S_{N}\) by dividing our values by a factor \(10^{0.4(B-V)}\); a typical dE has \(B-V=0.8\), which corresponds to a conversation factor \(\sim 2.1\). \(S_{N}\) is plotted against \(M_{B}\) in Figure 10. In this figure the dE,N galaxies are filled circles and the dE,noN galaxies are empty circles. The symbol size is proportional to the fraction of blue GCs. Miller _et al._ (1998) found that dE,noNs had, on average, lower \(S_{N}\) than dE,Ns, and that for both classes of dE there was an inverse correlation between \(S_{N}\) and galaxy luminosity. We do not see any substantial difference between dE,noN and dE,N galaxies. There does appear to be a weak correlation between \(S_{N}\) and \(M_{B}\) in our data (with quite large \(S_{N}\) for some of the faintest galaxies), but there is a large spread in \(S_{N}\) for most luminosities. The \(S_{N}\) does not appear to depend strongly on the fraction of blue GCs, so differing mixes of subpopulations cannot be responsible for the \(S_{N}\) variations. Thus the observed values correspond to total blue GC subpopulations which vary by a factor of up to \(\sim 10\) at a given luminosity. There appear to be two sets of galaxies: one group with \(S_{N}\sim 2\), the other group ranging from \(S_{N}\sim 5-20\) and a slight enhancement at \(S_{N}\sim 10\). We only have one galaxy in common with Miller _et al._ (VCC 9), but our \(S_{N}\) for this single galaxy is consistent with theirs within the errors. We note again that background contamination remains as a systematic uncertainty for total GC populations. It is tempting to argue that these groups represent different formation channels for dEs, e.g., the fading/quenching of dIrrs, harassment of low-mass spirals, or simply a continuation of the E sequence to fainter magnitudes. In some scenarios for blue GC formation (e.g., Strader _et al._ 2005; Rhode _et al._ 2005), galaxies with roughly similar masses in a given environment would be expected to have similar numbers of blue GCs per unit mass of their DM halos. Variations in \(S_{N}\) at fixed luminosity would then be due to differences in the efficiency of converting baryons to stars--either early in the galaxy's history (due to feedback) or later, due to galaxy transformation processes as described above. Two pieces of observational evidence should help to differentiate among the possibilities. First, GC kinematics can be used directly to constrain dE mass-to-light ratios (e.g., Beasley _et al._ 2005). It will be interesting to see whether GC kinematics are connected to the apparent dichotomy of rotating vs. non-rotating dEs (Pedraz _et al._ 2002, Geha _et al._ 2002, 2003, van Zee _et al._ 2004). Second, more detailed stellar population studies of dEs are needed, especially any method which (unlike integrated light spectroscopy) can break the burst strength-age degeneracy. A rather ad hoc alternative is that the efficiency of blue GC formation varies substantially among Virgo dEs, but this cannot be constrained at present. It is interesting that the fraction of blue GCs appears unrelated both to the nucleation of the dE and the \(S_{N}\) variations; this suggests that whatever process leads to red GC formation in dEs is independent of these other factors. ## 6 Summary We have presented a detailed analysis of the GC color and luminosity distributions of several gEs and of the colors, specific frequencies, luminosity functions, and nuclei of a large sample of dEs. The most interesting feature in the gEs M87 and NGC 4649 is a correlation between mass and metallicity for individual blue GCs. Self-enrichment is a plausible interpretation of this observation, and could suggest that these GCs once possessed dark matter halos (which may have been subsequently stripped). Among the other new features observed are very luminous (\(z\gtrsim 20\)) GCs with intermediate to red colors. These objects are slightly larger than typical GCs and may be remnants of stripped dwarf galaxies. Next, we see an intermediate-color group of GCs which lies near the GCLF turnover and in the gap between the blue and red GCs. Also, the color spread among the red GCs is nearly twice that of the blue GCs, but because the relation between \(g-z\) and metallicity appears to be nonlinear, the \(1\sigma\) dispersion in metallicity (\(\sim 0.6\) dex) may be the same for both subpopulations. The peak of the GCLF is the same in the gEs and a composite dE, modulo uncertainties in background subtraction for the dEs. This observation may be difficult to square with theoretical expectations that dynamical friction should deplete massive GCs in less than a Hubble time, and with the properties of dE nuclei. There appear to be two classes of nuclei: small bright red nuclei consistent with formation by dynamical friction of GCs, and larger faint blue nuclei which appear to have formed by a dissipative process with little contribution from GCs. Though dominated by blue GCs, many dEs appear to have bimodal color distributions, with significant red GC subpopulations. The colors of these GCs form a continuity with those of more massive galaxies; both the mean blue and red GC colors of dEs appear consistent with extrapolations of the GC color-galaxy luminosity relations for luminous ellipticals. We confirm these relations for both blue and red GC subpopulations. While previous works found an inverse correlation between dE \(S_{N}\) and galaxy luminosity, we find little support for such a relation. 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# Keck/HIRES Spectroscopy of Four Candidate Solar Twins Jeremy R. King Department of Physics and Astronomy, 118 Kinard Laboratory, Clemson University, Clemson, SC 29634-0978 jking2@ces.clemson.edu Ann M. Boesgaard1 Footnote 1: affiliation: Visiting Astronomer, W.M. Keck Observatory, jointly operated by the California Institute of Technology and the University of California. Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, HI 96822 boes@ifa.hawaii.edu Simon C. Schuler Department of Physics and Astronomy, 118 Kinard Laboratory, Clemson University, Clemson, SC 29634-0978 sschule@ces.clemson.edu ###### Abstract We use high S/N, high-resolution Keck/HIRES spectroscopy of 4 solar twin candidates (HIP 71813, 76114, 77718, 78399) pulled from our _Hipparcos_-based Ca II H & K survey to carry out parameter and abundance analyses of these objects. Our spectroscopic \(T_{\rm eff}\) estimates are some \({\sim}100\) K hotter than the photometric scale of the recent Geneva-Copenhagen survey; several lines of evidence suggest the photometric temperatures are too cool at solar \(T_{\rm eff}\). At the same time, our abundances for the 3 solar twin candidates included in the Geneva-Copenhagen survey are in outstanding agreement with the photometric metallicities; there is no sign of the anomalously low photometric metallicities derived for some late-G UMa group and Hyades dwarfs. A first radial velocity determination is made for HIP 78399, and \(UVW\) kinematics derived for all stars. HIP 71813 appears to be a kinematic member of the Wolf 630 moving group (a structure apparently reidentified in a recent analysis of late-type _Hipparcos_ stars), but its

metallicity is 0.1 dex higher than the most recent estimate of this group's metallicity. While certainly "solar-type" stars, HIP 76114 and 77718 are a few percent less massive, significantly older, and metal-poor compared to the Sun; they are neither good solar twin candidates nor solar analogs providing a look at the Sun at some other point in its evolution. HIP 71813 appears to be an excellent solar analog of age \({\sim}8\) Gyr. Our results for HIP 78399 suggest the promise of this star as a solar twin may be equivalent to the "closest ever solar twin" HR 6060; follow up study of this star is encouraged. stars: abundances -- stars: activity -- stars: atmospheres -- stars: evolution -- stars: fundamental parameters -- stars: late-type ## 1 Introduction The deliberate search for and study of solar analogs has been ongoing for nearly 30 years, initiating with the seminal early works of Hardorp (e.g., Hardorp 1978). Cayrel de Strobel (1996) gives an authoritative review of this early history, many photometric and spectroscopic results, and the astrophysical motivations for studying solar analogs. As of a decade ago, these motivations were of a strong fundamental and utilitarian nature, seeking answers to such questions as: (a) what is the solar color? b) how well do photometric indices predict spectroscopic properties? c) how robust are spectral types at describing or predicting the totality of a stellar spectrum? d) are there other stars that can be used as exact photometric and/or spectroscopic proxies for the Sun in the course of astrophysical research programs? While these important questions remain incompletely answered and of great interest, the study of solar analogs and search for solar twins has taken on renewed importance. Much of this has been driven by the detection of planetary companions around solar-type stars; the impact of these detections on solar analog research was foreshadowed with great prescience by Cayrel de Strobel (1996). Precision radial velocity searches for exoplanets are most robust when applied to slowly rotating and inactive stars; solar analogs are thus fruitful targets-metal-rich ones apparently even more fruitful (Fischer & Valenti, 2005). The appeal in searching for elusive terrestrial exoplanets around solar analogs remains a natural one given the existence of our own solar system. Solar analogs of various age also provide a mechanism to examine the past and future evolution of the Sun without significant or total recourse to stellar models. Such efforts looking at the sun in time (Ribas et al., 2005) now appear to be critical complements to studying

the evolution of planets and life surrounding solar-type stars. For example, it has been suggested that solar-type stars may be subject to highly energetic superflare outbursts, perhaps induced by orbiting planets, that would have dramatic effects on atmospheres surrounding and lifeforms inhabiting orbiting planets (Rubenstein & Schaefer, 2000; Schaefer, King & Deliyannis, 2000). It also seems clear that the nominal non-stochastic gradual evolution of solar-type chromospheres has important implications for a diversity of planetary physics (in our own solar system and others): the structure and chemistry of planetary atmospheres, the water budget on Mars, and even the evolution of planetary surfaces (Ribas et al., 2005); such issues are critical ones to understand in the development and evolution of life. The utilitarian importance of studying solar analogs has also persisted. For example, there should be little argument that differential spectroscopic analyses performed relative to the Sun are most reliable when applied to stars like the Sun- early G dwarfs. Happily, such objects can be found in a large variety of stellar populations having an extreme range of metallicity and age. The development of large aperture telescopes and improved instrumentation such as multi-object spectrographs and wide field imagers over the next decade or so mean that the stellar astronomy community is poised to undertake abundance surveys of tens or hundreds of thousands of Galactic stars. Critical questions confronting such ambitious but inevitable initiatives include: a) how reliable are photometric metallicities? b) can low-resolution spectroscopy yield results as robust as those from high-resolution spectroscopy? c) will automated spectroscopic analyses needed to handle such large datasets yield reliable results? All these questions can be addressed well by comparison with the results of high-resolution spectroscopy of solar analogs. Despite the importance of carrying out high-resolution spectroscopic analyses of solar analogs, efforts at doing so have been deliberate in pace. Recent exceptions to this include the solar analog studies of Gaidos, Henry & Henry (2000) and Soubiran & Triaud (2004). Here, we present the first results from a small contribution aimed at remedying this pace of study. Using the results of Dr. D. Soderblom's recent chromospheric Ca II H & K survey of nearby (\(d{\leq}60\) pc) late-F to early-K dwarfs in the _Hipparcos_ catalog, we have selected a sample of poorly-studied solar twin candidates having \(0.63{\leq}(B-V){\leq}0.66\), Ca II chromospheric fluxes within a few tenths of a dex of the mean solar value, and \(M_{V}\) within a few tenths of a magnitude of the solar value; there are roughly 150 such objects accessible from the northern hemisphere. These objects have been or are being observed as time allows during other observing programs. Here, we present echelle spectroscopy of 4 candidates obtained with Keck/HIRES. The objects are HIP 71813, 76114, 77718, and 78399. ## 2

Data and Analysis ### Observations and Reductions Our 4 solar twin candidates were observed on UT July 8 2004 using the Keck I 10-m, its HIRES echelle spectrograph, and a Tektronix \(2048{\times}2048\) CCD detector. The chosen slit width and cross-disperser setting yielded spectra from 4475 to 6900 A at a resolution of \(R{\sim}45,000\). Exposure times ranged from 3 to \({\sim}6\) minutes, achieving per pixel S/N in the continuum near 6707 A of \({\sim}400\). A log of the observations containing cross-identifications is presented in Table 1. Standard reductions were carried out including debiasing, flat-fielding, order identification/tracing/extraction, and wavelength calibration (via solutions calculated for an internal Th-Ar lamp). The H\(\alpha\) and H\(\beta\) features are located at the blue edge of their respective orders; the lack of surrounding wavelength coverage with which to accomplish continuum normalization thus prevented us from using Balmer profile fitting to independently determine \(T_{\rm eff}\). Samples of the spectra in the \({\lambda}6707\) Li I region can be found later in Figures 3 and 4. + Footnote †: margin: Fig. 5 ### Parameters and Abundances Clean, "case a" Fe I and Fe II lines from the list of Thevenin (1990) were selected for measurement in our 4 solar twin candidate spectra and a similarly high S/N and \(R{\sim}45,000\) Keck/HIRES lunar spectrum (described in King et al. (1997)) used as a solar proxy spectrum. Equivalent widths were measured using the profile fitting routines in the 1-d spectrum analysis software package SPECTRE(Fitzpatrick & Sneden, 1987). Line strengths of all the features measured in each star and our solar proxy spectrum can be found in Table 2. Abundances were derived from the equivalent widths using the 2002 version of the LTE analysis package MOOG and Kurucz model atmospheres interpolated from ATLAS9 grids. Oscillator strengths were taken from Thevenin (1990); the accuracy of these is irrelevant inasmuch as normalized abundances [x/H] were formed on a line-by-line basis using solar abundances derived in the same manner. The solar model atmosphere was characterized by \(T_{\rm eff}=5777\) K, log \(g=4.44\), a metallicity of [m/H]=0., and a microturbulent velocity of \({\xi}=1.25\); the latter is intermediate to the values of \({\xi}\) from the calibrations of Edvardsson et al. (1993) and Allende Prieto et al. (2004). An enhancement factor of 2.2 was applied to the van der Waals broadening coefficients for all lines. + Footnote †: margin: Fig. 5 Stellar parameters were determined as part of the Fe analysis in the usual fashion. \(T_{\rm eff}\) and \({\xi}\) were determined by requiring zero correlation coefficient between the _solar normalized_ abundances (i.e., [Fe/H]; again, accomplished on a line-by-line ba

sis) and the lower excitation potential and reduced equivalent width, respectively. This approach leads to unique solutions when there is no underlying correlation between excitation potential and reduced equivalent width. We show in Figure 1 that there is no such underlying correlation in our Fe I sample. Figure 2 displays the Fe I-based line-by-line [Fe/H] values versus both lower excitation potential (top) and reduced equivalent width (bottom) using our final model atmosphere parameters for the case of HIP 76114; the linear correlation coefficients in both planes are \({\sim}0.00\). Our abundance analysis is thus a purely differential one, and the derived parameters do not depend on the rigorous accuracy of the \(gf\) values. The 1\(\sigma\) level uncertainties in \(T_{\rm eff}\) and \({\xi}\) were determined by finding the values of these parameters where the respective correlation coefficients became significant at the 1\({\sigma}\) confidence level. Gravity estimates were made via ionization balance of Fe. The error estimates for log \(g\) include uncertainties in both [Fe I/H] and [Fe II/H] due to measurement uncertainty, \(T_{\rm eff}\) errors, and \({\xi}\) errors. The final parameters and their uncertainties can be found in the summary of results in Table 4. ++ Footnote †: margin: Fig. 5 Footnote †: margin: Fig. 5 Abundances of Al, Ca, Ti, and Ni were derived in a similar fashion using the line data in Table 2 and model atmospheres characterized by the parameters determined from the Fe data. Abundances of a given species were normalized on a line-by-line basis using the values derived from the solar spectrum, and then averaged together. Typical errors in the mean are only 0.01-0.02 dex, indicative of the quality of the data. The sensitivity of the derived abundances to arbitrarily selected fiducial variations in the stellar parameters (\({\pm}100\) K in \(T_{\rm eff}\); \({\pm}0.2\) dex in log \(g\); and \({\pm}0.2\) km s\({}^{-1}\) in microturbulence) are provided for each element in Table 3. Coupling these with the parameter uncertainties and the statistical uncertainties in the mean yielded total uncertainties in the abundance ratio of each element. The mean abundances and the \(1{\sigma}\) uncertainties are given in Table 4. ++ Footnote †: margin: Fig. 5 Footnote †: margin: Fig. 5 ### Oxygen Abundances O abundances were derived from the measured equivalent widths of the \({\lambda}6300\) [O I] feature (Table 2) using the blends package in MOOG to account for contamination by a Ni I feature at 6300.34. Isotopic components (Johansson et al., 2003) of Ni were taken into account with the \(gf\) values taken from Bensby, Feltzing & Lundstrom (2004); the [O I] \(gf\) value (-9.717) is taken from Allende Prieto, Lambert, & Asplund (2001). The assumed Ni abundances were taken as [Ni/H]=0.00, -0.04, -0.16, and -0.01 for HIP 71813, 76114, 77718, and 78399 respectively. Abundances are given in Table 4. Uncertainties in [O/H] are dominated by those in the equivalent widths (\(0.5\) mA) measurements of the stars and the Sun, and that in log \(g\) (\(0.12\) dex). These uncertainties from these 3 sources were added in

quadrature to yield the total uncertainties associated with the [O/H] values given in Table 4. ### Lithium Abundances Li abundances were derived from the \({\lambda}6707\) Li I resonance features via spectrum synthesis. Utilizing the derived parameters, synthetic spectra of varying Li abundance were created in MOOG using the line list from King et al. (1997). No contribution from \({}^{6}\)Li was assumed, a reasonable assumption given that the Li abundances in our objects are well-below meteoritic (log \(N\)(Li)=3.31; \({}^{6}\)Li/\({}^{7}\)Li\(=0.08\)). Smoothing was carried out by convolving the synthetic spectra with Gaussians having FWHM values measured from clean, weak lines measured in our spectra. Comparison of the syntheses (solid lines) and the Keck/HIRES spectra in the \({\lambda}6707\) region are shown in Figures 3 and 4. Total uncertainties include those due to uncertainties in the \(T_{\rm eff}\) value (Table 3) and in the fit itself. The Li results are listed in Table 4. ++ Footnote †: margin: Fig. 5 Footnote †: margin: Fig. 5 ### Rotational Velocity and Chromospheric Emission The same FWHM values measured for each star and used to smooth the syntheses were assumed to be the quadrature sum of components due to spectrograph resolution and (twice the projected) rotational velocity. The resulting \(v\) sin \(i\) values are listed in Table 4. Inasmuch as we assume no contribution from macroturbulent broadening mechanisms, we present these estimates as upper limits to the projected rotational velocity. The Ca II H&K chromospheric emission indices of our objects are listed in Table 4 and come from the low-resolution (\(R{\sim}2000\)) KPNO coude' feed-based survey of D. Soderblom. ### Masses and Ages Masses and ages of the Sun and our four solar twin candidates were estimated by placing them in the \(M_{V}\) versus \(T_{\rm eff}\) plane using our temperature estimates and the Hipparcos-based absolute visual magnitudes. Comparison of these positions with isochrones and sequences of constant mass taken from appropriate metallicity Yonsei-Yale Isochrones (Yi, Kim & Demarque, 2003) (as updated by Demarque et al. (2004)) yielded the mass and age estimates in Table 4. The uncertainties in mass and age are calculated assuming the influence of uncertainties in our \(T_{\rm eff}\) and \(M_{V}\) values; including the uncertainty in our metallicity esti

mates (\({\sigma}{\sim}0.04\) dex) has a negligible effect on the uncertainty of our estimated masses, but would contribute an additional 0.4 Gyr uncertainty in the age estimates. The HR diagrams containing our objects and these isochrones are shown in Figure 5. + Footnote †: margin: Fig. 5 ## 3 Results and Discussion ### Comparison with Previous Results HIP 71813 is included in the recent Geneva-Copenhagen solar neighborhood survey of Nordstrom et al. (2004). Their photometric metallicity determination of [Fe/H]\(=+0.01\) is in outstanding agreement with our Al, Ca, Ti, Fe, and Ni abundances, which range from \(-0.02\) to \(+0.02\). Their photometric \(T_{\rm eff}\) estimate of 5662 K is some 90 K lower than our spectroscopic value. If the solar color, \((B-V)_{\odot}=0.642\), adopted in Table 4 is to be believed, then our \(T_{\rm eff}\) value would seem to be more consistent with the nearly indistinguishable \((B-V)\) index (0.644) of HIP 71813. HIP 76114 is also included in the Geneva-Copenhagen survey. The Nordstrom et al. (2004) photometric metallicity of [Fe/H]\(=-0.05\) is also in outstanding agreement with our Al, Ca, Ti, Fe, and Ni abundances, which range from \(-0.06\) to \(-0.02\). The photometric \(T_{\rm eff}\) difference between HIP 71813 and 76114 (former minus latter) of 52 K is in excellent agreement with our spectroscopic difference of 40 K. HIP 77718 has a photometric metallicity, [Fe/H]\(=-0.19\) from the Nordstrom et al. (2004) solar neighborhood survey that is in good agreement with our Al, Ti, Fe, and Ni determinations, which range from \(-0.15\) to \(-0.22\); our [Ca/H] abundance of \(-0.09\) appears only mildly anomalous in comparison. The HIP 77718 minus 71813 photometric \(T_{\rm eff}\) difference of 92 K is in outstanding agreement with the 90 K spectroscopic difference. Gray et al. (2003) have determined the parameters and overall abundance of HIP 77718 via the analysis of low resolution blue spectra as part of their NStar survey. The independent spectroscopic \(T_{\rm eff}\) estimate, made via different comparisons of different spectral features in a different part of the spectrum, of 5859 K is only 19 K larger than our own and 105 K larger than the photometric value. The Gray et al. (2003) metallicity of [m/H]\(=-0.15\) is indistinguishable from our own result. HIP 78399 has not been subjected to any published abundance or high-resolution spectroscopic analysis that we are aware of. Accordingly, it lacks a radial velocity determination. We remedied this by determining a radial velocity relative to HIP 76114 via cross-correlation of the spectra in the 6160-6173 A range. We assumed the precision radial velocity of \(-35.7\) km/s from Nidever et al. (2002) for HIP 76114. Cross-correlation of the telluric B-Band

spectra in the 6880 A region revealed a 9.9 km/s offset between the spectra. While larger than anticipated, this intra-night drift was confirmed by comparison of telluric water vapor features in the 6300 A region. Accounting for this drift and the appropriate relative heliocentric corrections, we find a radial velocity of \(-24.7{\pm}0.7\) km/s for HIP 78399. ### HIP 71813 and the Wolf 630 Moving Group Eggen (1969) included HIP71813 as a member of the Wolf 630 moving group. Membership in this putative kinematic population was defined by Eggen in a vast series papers as traced in the work of Mcdonald & Hearnshaw (1983). Regardless of one's view on the reality of these kinematic assemblages, it is likely that the recent passing of O. Eggen has meant that a wealth of modern data (in particular _Hipparcos_ parallaxes and precision radial velocities) has not yet been brought to bear on the reality, properties, and detailed membership of the Wolf 630 group. A notable exception is the work of Skuljan, Hearnshaw & Cottrell (1999), who find a clustering of late-type stars at \((U,V)=(+20,-30)\) km s\({}^{-1}\) that is absent in the kinematic phase space of their early-type stellar sample; this is highly suggestive of an old moving group at Eggen's suggested position of the Wolf 630 group in the Bottlinger (\(U\),\(V\)) diagram. The salient characteristics identified by Eggen for the Wolf 630 group are a) a kinematically old disk population, b) a characteristic Galactic rotational velocity of \(V=-33\) km/s, and c) a color-luminosity array similar to M 67. It is beyond the scope of this paper to revisit or refine characteristics of the Wolf 630 group. However, several notes can be made. First, our 8 Gyr age estimate for HIP 71813 is certainly consistent with an old disk object. Second, if the estimate of Taylor (2000) of [Fe/H]\(=-0.12\) for the Wolf 630 group metallicity is accurate, then HIP 71813 would not seem to be a member. Third, using _Hipparcos_ parallaxes and proper motions, and modern radial velocity determinations (Nordstrom et al., 2004; Tinney & Reid, 1998), the \(UVW\) kinematics of HIP 71813 can be compared with those of Wolf 629, a Wolf 630 group defining member according to Eggen. The heliocentric Galactic velocities of all our objects are listed in Table 4. The (U,V)=(+21.3\({\pm}1.5\),-36.3\({\pm}1.3\)) results for HIP 71813 are in excellent agreement with those for Wolf 629 (+21.0\({\pm}1.3\),-33.4\({\pm}1.0\)), and consistent with the canonical Wolf 630 group values (26, -33) given by Eggen (1969). None of our other candidate solar twins has kinematics, which are listed in Table 4, consistent with those of the Wolf 630 group.

### Solar Twin Status Evaluation _HIP 71813_. The \(T_{\rm eff}\) value, light metal-abundances, and chromospheric Ca II emission of HIP 71813 are indistinguishable from solar values. The Li abundance, however, appears to be depleted by a factor of \({\geq}2\) compared to the Sun. More importantly, however, the star appears significantly more evolved than the sun. The \(M_{V}\) and log \(g\) values are significantly lower than the solar values, and our estimated age is a factor of 2 older than the Sun's. While clearly an inappropriate solar twin candidate, the star would appear to be an excellent solar analog of significantly older age. _HIP 76114_. HIP 76114 is marginally cooler than the Sun, \({\Delta}T_{\rm eff}=-67\) K. While any of the light element abundances alone are indistinguishable from solar, taken together they suggest a metallicity some 0.04 dex lower than solar; this is confirmed by the photometric metallicity of Nordstrom et al. (2004). The Ca II emission and Li abundance is solar within the uncertainties, but the star appears marginally evolved relative to the Sun as indicated by its slightly lower \(M_{V}\) and log \(g\) values; table 4 suggests that HIP 76114 is \({\geq}1.5\) Gyr older than the Sun. This object may be a suitable solar analog of slightly older age, albeit of likely slightly lower metallicity, that can be included in studies looking at solar evolution. _HIP 77718_. While the Ca II chromospheric emission and age determination of HIP 77718 are observationally indistinguishable from the Sun, our analysis indicates this star is clearly warmer (\({\Delta}T_{\rm eff}=63\) K) and some 0.16 dex metal-poor relative to solar; both the warmer temperature and slightly metal-poor nature are independently confirmed by the spectroscopic analysis of Gray et al. (2003). The Li abundance is some 20 times higher than solar. This difference may be related to reduced PMS Li depletion due to lower metallicity or reduced main-sequence depletion due to a younger age; our observations can not distinguish between these possibilities. Regardless, this star is not a good solar twin candidate, nor an optimal metal-poor or younger solar analog. _HIP 78399_. The poorly-studied HIP 78399 appears to hold great promise as a solar twin candidate. Its \(T_{\rm eff}\), luminosity, mass, age, light metal abundances, and rotational velocity are all indistinguishable from solar values. The only marked difference seen is the Li abundance, which is a factor of \({\sim}6\) larger than the solar photospheric abundance. While the evolution of Li depletion in solar-type stars is a complex and still incompletely understood process subject to vigorous investigation, this difference may suggest a slightly younger age for HIP 78399, which is allowed by our age determination and may be consistent with a slightly larger Ca II chromospheric flux. Currently, the "closest ever solar twin" title belongs to HR 6060 (Porto De Mello & Da Silva, 1997). Several spectroscopic analyses of this star have been carried out (Luck & Heiter

, 2005; Allende Prieto et al., 2004; Gray et al., 2003; Porto De Mello & Da Silva, 1997). \(T_{\rm eff}\) estimates range from 5693 to 5835 K, and [Fe/H] estimates from -0.06 to +0.05; the precision \(T_{\rm eff}\) analysis using line ratios (Gray, 1995) indicates a \(T_{\rm eff}\) difference with respect to the Sun of 17 K. The Ca II H&K emission index (-5.00, Gray 1995) and rotational velocity (\({\leq}3\) km/s, De Mello & Da Silva 1997) are indistinguishable from solar values. The _Hipparcos_-based absolute magnitude strongly suggests the mass and age of HR 6060 are virtually identical to the Sun's (Porto De Mello & Da Silva, 1997). Just as for HIP 78399, the only glaring outlying parameter is Li abundance, which is a factor of \({\sim}4\) larger than the solar photospheric Li abundance (Stephens, 1997). The work of Jones, Fischer & Soderblom (1999) on 1 \(M_{\odot}\) stars in the solar-age and -abundance cluster M67 suggests that we can expect such objects to exhibit a \({\sim}1\) dex range in Li; thus, the Sun may not be an especially good Li "standard". Based on our analysis, we believe there is a case to be made that HIP 78399 share the stage with HR 6060 as the closest ever solar twin. For those engaged in studies of solar twins or the Sun in time, HIP 78399 is certainly worthy of closer follow-up study. Particularly valuable would be: a) refining its T\({}_{\rm eff}\) and luminosity estimates relative to the Sun via Balmer line profile fitting, analysis of line ratios, etc. b) analysis of the \({\lambda}7774\) O I lines to confirm whether its O abundance is truly subsolar, c) performing an independent check on its relative age via the [Th/Nd] ratio (Morell, Kallander, & Butcher, 1992), and d) determining a \({}^{9}\)Be abundance, which is more immune to the effects of stellar depletion and also contains embedded information about the "personal" integrated Galactic cosmic-ray history of matter comprised by the star. ## 4 Summary We have carried out high S/N high-resolution Keck/HIRES spectroscopy of four candidate solar twins drawn from a _Hipparcos_-defined Ca II H&K survey. Parameters, abundances, masses, ages, and kinematics have been derived in a differential fine analysis. Comparisons suggests that the _relative_ photometric \(T_{\rm eff}\) values of Nordstrom et al. (2004) and our spectroscopic temperatures are indistinguishably robust; however, the photometric \(T_{\rm eff}\) values are typically 100 K cooler. There are several lines of evidence that suggest the photometric scale is misanchored (at least near solar \(T_{\rm eff}\)). First, if the solar color of Cayrel de Strobel (1996) is nearly correct, then our spectroscopic \(T_{\rm eff}\) values are in outstanding accord with the colors of HIP 71813 and 78399. Second, the independent analysis of HIP 77718 by Gray et al. (2003) using different spectral features in the blue yields a spectroscopic \(T_{\rm eff}\) in outstanding agreement with our own. Third, the Nordstrom et al. (2004) photometric \(T_{\rm eff}\) estimate for the "closest ever solar twin" HR 6060 is 5688 K, some \(100\) K lower than the precision

\(T_{\rm eff}\) estimate of Gray (1995). At the same time, our light metal-abundances are in excellent agreement with the photometric metallicity estimates for the 3 of our objects in Nordstrom et al. (2004), differing by no more than a few hundredths of a dex. There is no sign of the abnormally low photometric metallicity values seen for some very cool Pop I dwarfs in the Hyades and UMa group as noted by King & Schuler (2005). As these authors note, anomalous photometric estimates may be restricted to late G dwarfs. Our spectroscopic metallicity for HIP 77718 is in nearly exact agreement with that derived from low-resolution blue spectra by Gray et al. (2003). We present the first abundances and radial velocity estimate for HIP 78399. Using the radial velocities and _Hipparcos_ proper motions and parallaxes, we derive the \(UVW\) kinematics of our four solar twin candidates. The position of HIP 71813 in the \((U,V)\) plane is consistent with membership in Eggen's Wolf 630 moving group, a kinematic structure of late-type _Hipparcos_ stars apparently verified by Skuljan, Hearnshaw & Cottrell (1999). Our metallicity for HIP 71813, [Fe/H]\(=-0.02\), is 0.1 dex higher than the Wolf 630 estimate of Taylor (2000), however. Revisiting the characteristic metallicity via identification of assured Wolf 630 group members using _Hipparcos_ data and new precision radial velocities, and follow-up high resolution spectroscopy to determine abundances would be of great value. HIP 77718 is \({\sim}70\) K warmer than the Sun, significantly more metal-poor ([m/H]\({\sim}-0.16\)), significantly more Li-rich (log \(N\)(Li)\({\sim}2.3\)) and a few percent lass massive than the Sun; we deem it neither a suitable solar twin nor solar analog to trace the evolution of the Sun. The light-metal and Li abundances of HIP 76114 are much closer to solar. However, HIP 76114 does appear to be slightly metal-poor ([m/H]\(=-0.04\)), cooler \({\Delta}T_{\rm eff}=67\) K, older \({\Delta}{\tau}{\geq}3\) Gyr, and a few percent less massive compared to the Sun. HIP 71813 appears to be an excellent solar analog of solar abundance, mass, and \(T_{\rm eff}\), but advanced age-\(M_{V}=4.45\) and \({\tau}{\sim}8\) Gyr; the more evolved state of this star is likely reflected in the subsolar upper limit to its Li abundance. Finally, our first ever analysis of HIP 78399 suggests this object may be a solar twin candidate of quality comparable to the "closest ever solar twin" HR 6060 (Porto De Mello & Da Silva, 1997). The \(T_{\rm eff}\), mass, age, and light metal abundances of this object are indistinguishable from solar given the uncertainties. The only obvious difference is that which characterizes HR 6060 as well- a Li abundance a factor of a few larger than the solar photospheric value. This object merits additional study as a solar twin to refine its parameters; of particular interest will be confirming our subsolar O abundance derived from the very weak \({\lambda}6300\) [O I] feature. We are indebted to Dr. David Soderblom for the use of his nearby star activity catalog from which our objects were selected and for his valuable comments on the manuscript. It is a

# Initial Populations of Black Holes in Star Clusters Krzysztof Belczynski12 Aleksander Sadowski3 , Frederic A. Rasio4 , Tomasz Bulik 5 Footnote 1: footnotetext: Statistics is much better for single stars than binaries; and even with only \(2\times 10^{5}\) single stars we obtain usually thousands, and minimum several hundred, BHs. For example see Tables 2-5 Footnote 2: The number of single BHs formed out of binary systems may be inferred by comparing the numbers of binary BHs with the single BHs listed under “binary disruption” and “binary mergers” in Tables 2–5, 7 and 8 Footnote 3: affiliationmark: Footnote 4: affiliationmark: Footnote 5: affiliationmark: \({}^{1}\) New Mexico State University, Dept of Astronomy, 1320 Fregner Mall, Las Cruces, NM 88003 \({}^{2}\) Tombaugh Fellow \({}^{3}\) Astronomical Observatory, Warsaw University, Al. Ujazdowskie 4, 00-478, Warsaw, Poland \({}^{4}\) Northwestern University, Dept of Physics and Astronomy, 2145 Sheridan Rd, Evanston, IL 60208 \({}^{5}\) Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warszawa, Poland kbelczyn@nmsu.edu, oleks@camk.edu.pl, rasio@northwestern.edu, bulik@camk.edu.pl ###### Abstract Using an updated population synthesis code we study the formation and evolution of black holes (BHs) in young star clusters following a massive starburst. This study continues and improves on the initial work described by Belczynski, Sadowski & Rasio (2004). In our new calculations we account for the possible ejections of BHs and their progenitors from clusters because of natal kicks imparted by supernovae and recoil following binary disruptions. The results indicate that the properties of both retained BHs in clusters and ejected BHs (forming a field population) depend sensitively on the depth of the cluster potential. In particular, most BHs ejected from binaries are also ejected from clusters with central escape speeds \(V_{\rm esc}\lesssim 100\,{\rm km}\,{\rm s}^{-1}\). Conversely, most BHs remaining in binaries are retained by clusters with \(V_{\rm esc}\gtrsim 50\,{\rm km}\,{\rm s}^{-1}\). BHs from single star evolution are also affected significantly: about half of the BHs originating from primordial single stars are ejected from clusters with \(V_{\rm esc}\lesssim 50\,{\rm km}\,{\rm s}^{-1}\). Our results lay a foundation for theoretical studies of the formation of BH X-ray binaries in and around star clusters, including possible "ultra-luminous" sources, as well as merging BH-BH binaries detectable with future gravitational-wave observatories. Subject headings: binaries: close -- black hole physics -- gravitational waves -- stars: evolution ## 1. INTRODUCTION ### Black Holes in Star Clusters Theoretical arguments and many observations suggest that BHs should form in significant numbers in star clusters. Simple assumptions about the stellar initial mass function (IMF) and stellar evolution indicate that out of \(N\) stars formed initially, \(\sim 10^{-4}-10^{-3}\,N\) should produce BHs as remnants after \(\sim 20\,\)Myr. Thus any star cluster containing initially more than \(\sim 10^{4}\) stars should contain at least some BHs; large super star clusters and globular clusters should have formed many hundreds of BHs initially, and even larger systems such as galactic nuclei may contain many thousands to tens of thousands. Not surprisingly, observations are most sensitive to (and have provided constraints mainly on) the most massive BHs that may be present in the cores of very dense clusters (van der Marel 2004). For example, recent observations and dynamical modeling of the globular clusters M15 and G1 indicate the presence of a central BH with a mass \(\sim 10^{3}-10^{4}\,M_{\odot}\) (Gerssen et al. 2002, 2003; Gebhardt et al. 2002, 2005). However, direct \(N\)-body simulations by Baumgardt et al. (2003a,b) suggest that the observations of M15 and G1, and, in general, the properties of all _core-collapsed_ clusters, could be explained equally well by the presence of many compact remnants (heavy white dwarfs, neutron stars, or \(\sim 3-15{\,M_{\odot}}\) BHs) near the center without a massive BH (cf. van der Marel 2004; Gebhardt et al. 2005). On the other hand, \(N\)-body simulations also suggest that many _non-core-collapsed_ clusters (representing about 80% of globular clusters in the Milky Way) could contain central massive BH (Baumgardt et al. 2004, 2005). In any case, when the correlation between central BH mass and bulge mass in galaxies (e.g., Haring & Rix 2004) is extrapolated to smaller stellar systems like globular clusters, the inferred BH masses are indeed \(\sim 10^{3}-10^{4}\,M_{\odot}\). These are much larger than a canonical \(\sim 10\,M_{\odot}\) stellar-mass BH (see, however, SS3.1.6), but much smaller than the \(\sim 10^{6}-10^{9}\,M_{\odot}\) of supermassive BHs. Hence, these objects are often called _intermediate-mass black holes_ (IMBHs; see, e.g., Miller & Colbert 2004). Further observational evidence for IMBHs in dense star clusters comes from many recent _Chandra_ and XMM-_Newton_ observations of "ultra-luminous" X-ray sources (ULXs), which are often (although not always) clearly associated with young star clusters and whose high X-ray luminosities in many cases suggest a compact object mass of at least \(\sim 10^{2}\,M_{\odot}\) (Cropper et al. 2004; Ebisuzaki et al. 2001; Kaaret et al. 2001; Miller et al. 2003). In many cases, however, beamed emission by an accreting stellar-mass BH may provide an alternative explanation (King et al. 2001; King 2004; Zezas & Fabbiano 2002). One natural path to the formation of a massive object at the center of any young stellar system with a high enough density is through runaway collisions and mergers of massive stars following gravothermal contraction and core collapse (Ebisuzaki et al. 2001; Portegies Zwart & McMillan 2002; Gurkan, Freitag, & Rasio 2004). These runaways occur when massive stars can drive core collapse _before they evolve_. Alternatively, if the most massive stars in the cluster are allowed to evolve and produce supernovae, the gravothermal contraction of the cluster will be reversed by the sudden mass loss, and many stellar-mass BHs will be formed. The final fate of a cluster with a significant component of stellar-mass BHs remains highly uncertain. This is because realistic dynamical simulations for such clusters (containing a large number of BHs _and_ ordinary stars with

a realistic mass spectrum) have yet to be performed. For old and relatively small systems (such as small globular clusters), complete evaporation is likely (with essentially all the stellar-mass BHs ejected from the cluster through three-body and four-body interactions in the dense core). This is expected theoretically on the basis of simple qualitative arguments based on Spitzer's "mass-segregation instability" applied to BHs (Kulkarni et al. 1993; Sigurdsson & Hernquist 1993; Watters et al. 2000) and has been demonstrated by dynamical simulations (Portegies Zwart & McMillan 2000; O'Leary et al. 2005). However, it has been suggested that, if stellar-mass BHs are formed with a relatively broad mass spectrum (a likely outcome for stars of very low metallicity; see Heger et al. 2003), the most massive BH could resist ejection, even from a cluster with low escape velocity. These more massive BHs could then grow by repeatedly forming binaries (through exchange interactions) with other BHs and merging with their companions (Miller & Hamilton 2002; Gultekin, Miller, & Hamilton 2004). However, as most interactions will probably result in the ejection of one of the lighter BHs, it is unclear whether any object could grow substantially through this mechanism before running out of companions to merge with. A single stellar-mass BH remaining at the center of a globular cluster is very unlikely to become detectable as an X-ray binary (Kalogera, King, & Rasio 2004). In addition to its obvious relevance to X-ray astronomy, the dynamics of BHs in clusters also plays an important role in the theoretical modeling of gravitational-wave (GW) sources and the development of data analysis and detection strategies for these sources. In particular, the growth of a massive BH by repeated mergers of stellar-mass BHs spiraling into an IMBH at the center of a dense star cluster may provide an important source of low-frequency GWs for LISA, the Laser-Interferometer Space Antenna (Miller 2002; Will 2004). Similarly, dynamical hardening and ejections of binaries from dense clusters of stellar-mass BHs could lead to greatly enhanced rates of BH-BH mergers detectable by LIGO and other ground-based interferometers (Portegies Zwart & McMillan 2000; O'Leary et al. 2005). A crucial starting point for any detailed study of BHs in clusters is an accurate description of the initial BH population. Here, "initial" means on a timescale short compared to the later dynamical evolution timescale. Indeed most \(N\)-body simulations of star cluster dynamics never attempt to model the brief, initial phase of rapid massive star evolution leading to BH formation. The goal of our work here is to provide the most up-to-date and detailed description of these initial BH populations. This means that we must compute the evolution of a large number of massive stars, including a large fraction of binaries, all the way to BH formation, i.e., on a timescale \(\sim 10-100\,\)Myr, taking into account a variety of possible cluster environments. ### Previous Work In a previous study (Belczynski et al. 2004; hereafter Paper I) we studied young populations of BHs formed in a massive starburst, without explicitly taking into account that most stars are formed in clusters. For many representative models we computed the numbers of BHs, both single and in various types of binaries, at various ages, as well as the physical properties of different systems (e.g., binary period and BH mass distributions). We also discussed in detail the evolutionary channels responsible for these properties. In this follow-up study, we consider the possible ejection of these BHs from star clusters with different escape speeds, taking into account the recoil imparted by supernovae (SNe) and binary disruptions. During SNe, mass loss and any asymmetry in the explosion (e.g., in neutrino emission) can impart large extra speeds to newly formed compact objects. If a compact object is formed in a binary system, the binary may either _(i)_ survive the explosion, but its orbital parameters are changed and the system (center-of-mass) speed changes, or _(ii)_ the binary is disrupted and the newly formed compact object and its companion fly apart on separate trajectories. The secondary star in a binary may later undergo a SN explosion as well, provided that it is massive enough. The effects of this second explosion are equally important in determining the final characteristics of compact objects. In Paper I we included the effects of SNe, both natal kicks and mass loss, on the formation and evolution of BHs (single and in binaries), but we did not keep track of which BHs and binaries would be retained in their parent cluster. Starbursts form most of their stars in dense clusters with a broad range of masses and central potentials (and hence escape speeds; see, e.g., Elmegreen et al. 2002; McCrady et al. 2003; Melo et al. 2005). Smaller clusters of \(\sim 10^{4}\,M_{\odot}\) (open clusters or "young populous clusters," such as the Arches and Quintuplet clusters in our Galactic center) could have escape speeds as low as \(V_{\rm esc}\lesssim 10\,{\rm km}\,{\rm s}^{-1}\) while the largest "super star clusters" with much deeper potential wells could have \(V_{\rm esc}\gtrsim 100\,{\rm km}\,{\rm s}^{-1}\). On the other hand the natal kick velocities could be relatively high, \(\sim 100-500\,{\rm km}\,{\rm s}^{-1}\) for low-mass BHs, so that a large fraction of BHs might leave the cluster early in the evolution. Here we repeat our study of young BH populations taking into account ejections from star clusters. We perform our calculations with a slightly updated version of our population synthesis code StarTrack (SS2) and we present results for both the retained cluster BH populations and the ejected BHs, which will eventually become part of the field BH population surrounding the surviving clusters. Our models and assumptions are discussed in SS2, with particular emphasis on the updates since Paper I. In SS3 we present our new results and in SS4 we provide a summary and discussion. ## 2. MODEL DESCRIPTION AND ASSUMPTIONS ### Population Synthesis Code Our investigation is based on a standard population synthesis method. We use the StarTrack code (Belczynski, Kalogera & Bulik 2002, hereafter BKB02), which has been revised and improved significantly over the past few years (Belczynski et al. 2006). Our calculations do not include any treatment of dynamical interactions (collisions) between binaries and single stars or other binaries1. In particular, the star clusters we consider are assumed to have avoided the 'runaway collision instability" that

can drive rapid collisions and mergers of massive main-sequence stars during an early episode of cluster core collapse (Freitag et al. 2006a,b). Instead, our results can provide highly realistic initial conditions for dynamical simulations of dense star clusters in which the early phase of massive star evolution proceeded 'normally," without significant influence from cluster dynamics. Footnote 1: footnotetext: Statistics is much better for single stars than binaries; and even with only \(2\times 10^{5}\) single stars we obtain usually thousands, and minimum several hundred, BHs. For example see Tables 2-5 All stars are evolved based on the metallicity- and wind-mass-loss-dependent models of Hurley, Pols & Tout (2002), with a few improvements described in BKB02. The main code parameters we use correspond to the standard model presented in SS2 of BKB02 and are also described in Paper I. Each star, either single or a binary component, is placed initially on the zero-age main sequence (ZAMS) and then evolved through a sequence of distinct phases: main sequence (MS), Hertzsprung Gap (HG), red giant branch (RG), core He burning (CHeB), asymptotic giant branch (AGB); if a star gets stripped of its H-rich envelope, either through wind mass loss or Roche lobe overflow (RLOF) it becomes a naked helium star (He). The nuclear evolution leads ultimately to the formation of a compact object. Depending on the pre-collapse mass and initial composition this may be a white dwarf (WD), a neutron star (NS) or a BH. The population synthesis code allows us to study the evolution of both single and binary stars. Binary star components are evolved as single stars while no interactions are taking place. We model the following processes, which can alter the binary orbit and subsequent evolution of the components: tidal interactions, magnetic braking, gravitational radiation, and angular momentum changes due to mass loss. Binary components may interact through mass transfer and accretion phases. We take into account various modes of mass transfer: wind accretion and RLOF; conservative and non-conservative; stable or dynamically unstable (leading to common-envelope evolution). The mass transfer rates are calculated from the specific binary configurations and physical properties (masses, evolutionary stages, etc.) of the stars involved. Binary components may loose or gain mass, while the binary orbit may either expand or shrink in response. Moreover, we allow for binary mergers driven by orbital decay. In this study, we evolve binary merger products assuming that they restart on the ZAMS. An exception is made when a BH takes part in the merger, in which case we assume the remnant object to be a BH again. The mass of the merger product is assumed equal to the total parent binary mass for unevolved and compact remnant components; however we assume complete envelope mass loss from any evolved star (HG, RG, CHeB, or evolved He star) involved in a merger. A few additions and updates to StarTrack since Paper I are worth mentioning here (see Belczynski et al. 2006 for more details). System velocities are now tracked for all stars (single and binaries) after SNe (see SS2.3). The new magnetic braking law of Ivanova & Taam (2003) has been adopted, although this has minimal impact on our results for BHs. Two new types of WDs have been introduced: hydrogen and "hybrid" (these are possible BH donors in binaries). An improved criterion is adopted for CHeB stars to discriminate between those with convective (\(M<7{\,M_{\odot}}\)) and radiative (\(M\geq 7{\,M_{\odot}}\)) envelopes; this affects the stellar response to mass loss. We have also added a new tidal term for RLOF rate calculations. Some minor problems in the calculations of Paper I were also identified and are corrected in this study. The evolution of a small fraction of BH RLOF systems with donors at the end of the RG stage was terminated when the donor contracted and detached after entering a CHeB phase. However, the donor may restart RLOF during expansion on the AGB, which is now properly accounted for. Another small fraction of systems, evolving through the rapid RLOF phase with HG donors, were previously classified as mergers and subsequently evolved as single stars (merger products). However, the RLOF at that stage may be dynamically stable and in some cases a binary system may survive and continue its evolution, which is now also properly taken into account. None of these corrections affect the results of Paper I significantly. ### Black Hole Formation Black holes originate from the most massive stars. The formation time is calculated for each star using the stellar models of Hurley et al. (2000) and Woosley (1986). For intermediate-mass stars the FeNi core collapses and forms a hot proto-NS or a low-mass BH. Part of the envelope falls back onto the central object while the rest is assumed to be ejected in a SN explosion. We use the results of Fryer (1999) and Fryer & Kalogera (2001) to determine how much matter is ejected. In general, for the highest masses (\(>30\,M_{\odot}\) for low-metallicity models) total fall-back is expected, with no accompanying SN explosion. Motivated by the large observed velocities of radio pulsars we assume significant asymmetries in SN explosions. Here we adopt the kick velocity distribution of Arzoumanian, Cordes & Chernoff (2002), taking into consideration more recent observations (e.g., White & Van Paradijs 1996; Mirabel & Rodrigues 2003). NSs receive full kicks drawn from the bimodal distribution of Arzoumanian et al. (2002). Many BHs form through partial fall back of material initially expelled in a SN explosion, but then accreted back onto the central BH. For these the kick velocity is lowered proportionally to the mass of accreted material (for details see BKB02). For the most massive stars, the BH forms silently through a direct collapse without accompanying SN explosion, and in this case we assume _no_ BH natal kick. The mass loss and kick velocity together determine whether a binary hosting the BH progenitor is disrupted by the SN explosion. Our calculated initial-to-final mass relation for various metallicities is discussed in detail in Paper I, where it is also demonstrated that (within our BH kick model) for solar metallicity many BHs are formed with lowered kicks through fall back. This occurs for single stars with initial masses in the range \(20-42M_{\odot}\) and \(50-70M_{\odot}\). For metallicity \(Z=0.001\), BHs receive a kick in the narrower ranges \(18-25M_{\odot}\) and \(39-54M_{\odot}\), while for \(Z=0.0001\) only BHs formed from stars of \(18-24M_{\odot}\) receive kicks, with others forming silently. ### Spatial Velocities All stars, single and binaries, are assumed to have zero initial velocities. This means we are neglecting their orbital speeds within the cluster. Indeed, for a variety of reasons (e.g., relaxation toward energy equipartition, formation near the cluster center), massive stars (BH progenitors) are expected to have lower velocity dispersions

given system. Masses of single stars and binary primaries (more massive components) are drawn from the three-component, power-law IMF of Kroupa, Tout, & Gilmore (1993) (see also Kroupa & Weidner 2003) with slope \(\alpha_{1}=-1.3\) within the initial mass range \(0.08-0.5{\,M_{\odot}}\), \(\alpha_{2}=-2.2\) for stars within \(0.5-1.0{\,M_{\odot}}\), and \(\alpha_{3}=-2.35\) within \(1.0\,M_{\odot}-M_{\rm max}\). The binary secondary masses are generated from an assumed flat mass ratio distribution (\(q=M_{\rm a}/M_{\rm b}\); \(M_{\rm a},\ M_{\rm b}\) denoting the mass of the primary and secondary, respectively). The mass ratio is drawn from the interval \(q_{\rm min}\) to 1, where \(q_{\rm min}=0.08{\,M_{\odot}}/M_{\rm a}\), ensuring that the mass of the secondary does not fall below the hydrogen burning limit. The only exception is model B, in which both the primary and the secondary masses are sampled independently from the assumed IMF (i.e., the component masses are not correlated). This IMF is easily integrated to find the total mass contained in single and binary stars for any adopted \(\alpha_{1},\alpha_{2},\alpha_{3}\) values. The particular choice of low-mass slope of the IMF (\(\alpha_{1},\alpha_{2}\)) does not change our results, since low-mass stars do not contribute to the BH populations. However, as most of the initial stellar mass is contained in low-mass stars, a small change in the IMF slope at the low-mass end can significantly change the mass normalization. In our simulations, we do not evolve all the single stars and binaries described above since the low-mass stars cannot form BHs. Out of the total population described above we evolve only the single stars with masses higher than \(4{\,M_{\odot}}\) and the binaries with primaries more massive than \(4{\,M_{\odot}}\) (no constraint is placed on the mass of the secondary, except that it must be above \(0.08{\,M_{\odot}}\)). All models were calculated with \(10^{6}\) massive primordial binaries. We also evolved \(2\times 10^{5}\) massive single stars but then scaled up our results to represent \(10^{6}\) single stars1. The mass evolved in single stars and binaries was then calculated and, by extrapolation of the IMF (down to hydrogen burning limit), the total initial cluster mass was determined for each model simulation. Footnote 1: footnotetext: Statistics is much better for single stars than binaries; and even with only \(2\times 10^{5}\) single stars we obtain usually thousands, and minimum several hundred, BHs. For example see Tables 2-5 In the discussion of our results we assume an initial (primordial) binary fraction of \(f_{\rm bin}=50\%\), unless stated otherwise (i.e., tables and figures usually assume equal numbers of single stars and binaries initially, with 2/3 of stars in binaries). However, our results can easily be generalized to other primordial binary fractions \(f_{\rm{bin}}\) by simply weighing differently the numbers obtained for single stars and for binaries. Our assumed distribution of initial binary separations follows Abt (1983). Specifically, we take a flat distribution in \(\log a\), so that the probability density \(\Gamma(a)\propto{1\over a}\). This is applied between a minimum value, such that the primary's initial radius (on the zero-age main sequence) is half the radius of its Roche lobe, and a maximum value of \(10^{5}\,{\rm R}_{\odot}\). We also adopt a standard thermal eccentricity distribution for initial binaries, \(\Xi(e)=2e\), in the range \(e=0-1\) (e.g., Heggie 1975; Duquennoy & Mayor 1991). #### 2.4.4 Cluster Properties The only cluster parameter that enters directly in our simulations is the escape speed \(V_{\rm esc}\) from the cluster core. All single and binary BHs are assumed immediately ejected from the cluster if they acquire a speed exceeding \(V_{\rm esc}\). We do not take into account ejections from the cluster halo (where the escape speed would be lower) as all BHs and their progenitors are expected to be concentrated near the cluster center. In Tables 2, 3, 4, 5 we present results of simulations for our standard model corresponding to four different values of the escape speed: \(V_{\rm esc}=10,~{}50,~{}100,~{}300{\,{\rm km}\,{\rm s}^{-1}}\). For any assumed cluster model the escape speed can be related to the total mass \(M_{\rm cl}\) and half-mass radius \(R_{\rm h}\): \[V_{\rm esc}=f_{\rm cls}\,\left(\frac{M_{\rm cl}}{10^{6}{\,M_{\odot}}}\right)^{ 1/2}\,\left(\frac{R_{\rm h}}{1\,{\rm pc}}\right)^{-1/2}.\] (18) For example, for a simple Plummer sphere we have \(f_{\rm cls}=106{\,{\rm km}\,{\rm s}^{-1}}\), while for King models with dimensionless central potentials \(W_{0}=3,~{}5,~{}7,~{}9\) and 11, the values are \(f_{\rm cls}=105.2\), 108.5, 119.3, 157.7, and \(184.0{\,{\rm km}\,{\rm s}^{-1}}\), respectively. For our four considered values of the escape speed, \(V_{\rm esc}=10\), 50, 100, and \(300{\,{\rm km}\,{\rm s}^{-1}}\), in a \(W_{0}=3\) King model with \(R_{\rm h}=1\,\)pc (typical for a variety of star clusters), the corresponding cluster masses are \(M_{\rm cl}=0.009\), 0.226, 0.904, and \(8.132\ \times 10^{6}{\,M_{\odot}}\), respectively. In each table we present the properties of BH populations at five different cluster ages: 8.7, 11.0, 15.8, 41.7 and \(103.8\,\)Myr. These correspond to MS turnoff masses of 25, 20, 15, 8 and \(5\,M_{\odot}\), respectively. The tables include information on both the BHs retained in the clusters (with velocities \(<V_{\rm esc}\)) and those ejected from clusters. ## 3. RESULTS ### Standard Reference Model #### 3.1.1 Black Hole Spatial Velocities In Figure 2 we show distributions of spatial velocities for _all_ single and binary BHs shortly after the initial starburst (at \(8.7\,\)Myr). The distribution shows a rather broad peak around \(\sim 30-300{\,{\rm km}\,{\rm s}^{-1}}\), but also includes a large fraction (\(\sim\) 2/3) of BHs formed with no kick. The peak originates from a mixture of low-velocity binary BHs and high-velocity single BHs. The no-kick single and binary BHs originate from the most massive stars, which have formed BHs silently and without a kick. All the no-kick systems (with zero velocity assumed) were placed on the extreme left side of all distributions in Figure 2 to show their contribution in relation to other non-zero velocity systems (the bin area is chosen so as to represent their actual number, although the placement of the bin along the velocity axis is arbitrary). Binary stars hosting BHs survive only if the natal kicks they received were relatively small, since high-magnitude kicks tend to disrupt the systems. We see (middle panel of Fig. 2) that most BH binaries have spatial velocities around \(50{\,{\rm km}\,{\rm s}^{-1}}\), which originate from the low-velocity side of the bimodal Arzoumanian et al. (2002) distribution. Single BHs originating from single stars follow closely the bimodal distribution of natal kicks, but the final BH velocities are slightly lower because of fall-back and direct BH formation (see SS2.3). The low- and high-velocity single BHs have speeds around \(50{\,{\rm km}\,{\rm s}^{-1}}\) and \(250{\,{\rm km}\,{\rm s}^{-1}}\), respectively. Single BHs originating from binary disruptions gain high speeds (\(\sim 100-400{\,{\rm km}\,{\rm s}^{-1}}\)), since binaries

are disrupted when a high-magnitude kick occurs. Finally, the single BHs formed through binary mergers have the lowest (nonzero) velocities (\(\sim 10-100{\,{\rm km}\,{\rm s}^{-1}}\)), since they are the most massive BHs and therefore most affected by fall-back. In Figure 3 we show the velocity distributions at a later time (\(103.8\,\)Myr) when essentially all BHs have formed, and no more SNe explosions are expected, so the velocity distribution is no longer evolving (the MS turnoff mass for that time is down to \(5{\,M_{\odot}}\)). The velocities have now shifted to somewhat higher values (with a single peak at \(\simeq 200{\,{\rm km}\,{\rm s}^{-1}}\) for non-zero velocity BHs), while the relative contribution of no-kick systems drops to around 1/3. At this later time the population is more dominated by single BHs. Most of the non-zero velocity single BHs come from binary disruptions (see middle panel of Fig. 3) and therefore they have received larger kicks, shifting the overall distribution toward slightly higher velocities. Also, at later times, lower-mass BH progenitors go through SN explosions, and they receive on average larger kicks (since for lower masses there is less fall back). We note that most of the non-zero-velocity BHs gain speeds of \(50-200{\,{\rm km}\,{\rm s}^{-1}}\) in SNe explosions. Depending on the properties of a given cluster they may be ejected or retained, and will either populate the field or undergo subsequent dynamical evolution in the cluster. We now discuss separately the properties of the retained and ejected BH populations. #### 3.1.2 Properties of Retained (Cluster) BH Populations Retained BHs in clusters could be found either in binaries or as single objects. Binary BHs are found with different types of companion stars, while single BHs may have formed through various channels, which we also list in Tables 2 - 5. Shortly after the starburst the most frequent BH companions are massive MS stars, but, as the population evolves, these massive MS companions finish their lives and form additional BHs. Double BH-BH systems begin to dominate the binary BH population after about \(15\,\)Myr. At later times less massive stars evolve off the MS and start contributing to the sub-population of BHs with evolved companions (CHeB stars being the dominant companion type, with a relatively long lifetime in that phase) or other remnants as companions (WDs and NSs). Once the majority of stars massive enough to make BHs end their lives (around \(10-15\,\)Myr) we observe a general decrease in the total number of BHs in binaries. The number of BHs in binaries is depleted through the disruptive effects of SNe and binary mergers (e.g., during CE phases). Both processes enhance the single BH population. This single population is dominated by the BHs formed from primordial single stars (assuming \(f_{\rm bin}=50\%\)). The formation along this single star channel stops early on when all single massive stars have finished evolution and formed BHs (at \(\simeq 10-15\,\)Myr). In contrast, the contribution of single BHs from binary disruptions and mergers is increasing with time, but eventually it also saturates (at \(\simeq 50-100\,\)Myr), since there are fewer potential BH progenitor binaries as the massive stars die off. In general, the single BHs are much more numerous in young cluster environments than binary BHs. At early times (\(\simeq 10\,\)Myr) they dominate by a factor \(2-4\), but later the ratio of single to binary BHs increases to almost 10 (after \(\sim 100\,\)Myr), as many binaries merge or are disrupted (adding to the single population). #### 3.1.3 Properties of Ejected (Field) BH Populations Tables 2 - 5 show also the properties of BHs ejected from their parent clusters, assuming different escape speeds. Significant fractions (\(\gtrsim 0.4\)) of single and binary BHs are likely to be ejected from any cluster with escape speed \(V_{\rm esc}\lesssim 100{\,{\rm km}\,{\rm s}^{-1}}\). In general, single BHs are more prone to ejection since they gain larger speeds in SNe explosions (compared to heavier binaries). Early on the number of ejected BHs increases with time as new BHs of lower mass (and hence receiving larger kicks) are being formed. At later times (after \(\simeq 15\,\)Myr), the number of fast BHs remains basically unchanged. Ejected binaries consist mostly of BH-MS and BH-BH pairs in comparable numbers. Rare BH-NS binaries are ejected more easily than other types since they experience two kicks. Single ejected BHs consist mostly of BHs originating from single stars which have received large kicks and from the components of a disrupted binary (the involved kicks were rather large to allow for disruption). #### 3.1.4 Dependence on Cluster Escape Velocity and Initial Binary Fraction In Table 6 we list fractions of retained BHs at \(103.8\,\)Myr after the starburst. The results are presented for initial cluster binary fractions of \(f_{\rm bin}=0,\ 50,\ 100\%\), and can be linearly interpolated for the desired \(f_{\rm bin}\). For our standard model the results are shown for the four considered escape velocities. For an initial cluster binary fraction \(f_{\rm bin}=50\%\) we find that the retained fraction can vary from \(\sim 0.4\) for low escape velocities (\(V_{\rm esc}=10{\,{\rm km}\,{\rm s}^{-1}}\)) to \(\sim 0.9\) for high velocities (\(V_{\rm esc}=300{\,{\rm km}\,{\rm s}^{-1}}\)). For escape velocities typical of globular clusters or super star clusters (\(V_{\rm esc}\sim 50{\,{\rm km}\,{\rm s}^{-1}}\)), retained and ejected fractions are about equal. The retained fractions for various types of systems are plotted as a function of \(V_{\rm esc}\) in Figure 4. All curves are normalized to total number of BHs, both single and binaries. Results are listed in Table 6 for different binary fractions. In particular, these can be used to study the limiting cases of pure binary populations (\(f_{\rm bin}=100\%\)) and pure single star populations (\(f_{\rm bin}=0\%\)). Note that even an initial population with all massive stars in binaries will form many single BHs through binary disruptions and binary mergers. We also note the decrease of the retained fraction with increasing initial binary fraction. Clusters containing more binaries tend to lose relatively more BHs through binary disruptions in SNe compared to single star populations. #### 3.1.5 Orbital Periods of Black Hole Binaries Figure 5 presents the period distribution of BH binaries for our standard model (for the characteristic escape velocity \(V_{\rm esc}=50{\,{\rm km}\,{\rm s}^{-1}}\)). We show separately populations retained and ejected from a cluster. The distributions for different values of the escape velocity are similar. In Paper I we obtained a double-peaked period distribution for BHs in field populations: tighter binaries were found around \(P_{\rm orb}\sim 10\,\)d while wider systems peaked around \(P_{\rm orb}\sim 10^{5}\,\)d. The shape of this distribution comes

from the property that tighter BH progenitor systems experienced at least one RLOF/CE episode leading to orbital decay, while wider systems never interacted and stayed close to their initial periods. The two peaks are clearly separated with a demarcation period \(P_{\rm s}\sim 10^{3}\,\)d. It is easily seen here that slow and fast BH populations add up to the original double-peaked distribution of Paper I. Only the shortest-period and hence most tightly bound systems (\(P_{\rm orb}<P_{\rm s}\)) survive SN explosions and they form a population of fast, short-period BH binaries (see bottom panel of Fig. 5). In contrast, systems retained in clusters have again a double-peaked orbital period distribution. The slowest systems have rather large periods (\(P_{\rm orb}\sim 10^{5}\,\)d) and they will likely get disrupted through dynamical interactions in the dense cluster core. The short-period cluster binaries (\(P_{\rm orb}\sim 10-100\,\)d) are much less numerous, since most of the short-period systems gained high post-SN velocities and contributed to the ejected population. Compared to Paper I we note that the inclusion of ejections further depletes the cluster _hard_ binary BH population. Only about 1/3 of systems are found with periods below \(P_{\rm s}\), half of which are retained within a cluster with \(V_{\rm esc}=50{\,{\rm km}\,{\rm s}^{-1}}\). For a cluster with \(V_{\rm esc}=100{\,{\rm km}\,{\rm s}^{-1}}\) about 80% of the short-period systems are retained. #### 3.1.6 Black Hole Masses Black hole mass distributions are presented in Figures 6, 7 for \(V_{\rm esc}=50{\,{\rm km}\,{\rm s}^{-1}}\). With few exceptions the models for different escape velocity values are very similar. The retained and ejected populations are shown in separate panels. The retained populations of BHs shown in the top panel of Figure 6 have a characteristic triple-peaked mass distribution: a first peak at \(M_{\rm BH}\sim 6-8{\,M_{\odot}}\), a second one at \(M_{\rm BH}\sim 10-16{\,M_{\odot}}\), and third at \(M_{\rm BH}\sim 22-26{\,M_{\odot}}\); beyond this it steeply falls off with increasing mass. The shape of the distribution is determined by the combination of IMF and initial-to-final mass relation for single BHs (presented and discussed in detail in Paper I): the most massive stars (\(\geq 50{\,M_{\odot}}\)) form BHs with masses in the range \(\sim 10-16{\,M_{\odot}}\); stars within an initial mass range \(25-35{\,M_{\odot}}\) form BHs of mass \(\sim 25{\,M_{\odot}}\); stars of initially \(40-50{\,M_{\odot}}\) tend to form \(7{\,M_{\odot}}\) BHs. Both single and binary BHs contribute significantly to the second and third peaks. However, only single stars are responsible for a first narrow peak corresponding to a pile up of BHs in the initial-to-final mass relation around \(6-8{\,M_{\odot}}\). This characteristic feature is a result of a very sharp transition in single star evolution, from H-rich to naked helium stars, which is caused by wind mass loss and the more effective envelope removal for single stars above a certain initial mass. In binary stars, removal of the envelope can happen not only through stellar winds but also through RLOF, and so it is allowed for the entire mass range and the first peak is washed out. The ejected populations, shown in the bottom panel of Figure 6, are dominated by single BHs (due to their high average speeds) with masses \(\sim 3-30{\,M_{\odot}}\). The distributions have one sharp peak at \(M_{\rm BH}\sim 6-8{\,M_{\odot}}\), corresponding to the first low-mass peak in the distribution for retained populations. The high-mass BHs are very rare in the ejected populations since the kick magnitudes decrease with increasing BH mass (because of significant fall-back or direct BH formation at the high-mass end). BHs in binary systems reach a maximum mass of about \(30{\,M_{\odot}}\) for both cluster and ejected populations. Most single BHs have masses below \(30{\,M_{\odot}}\). However, the tail of the single BH mass distribution extends to \(\sim 50{\,M_{\odot}}\) for ejected populations and to about \(80{\,M_{\odot}}\) for cluster populations. This is shown in Figure 7 (note a change of vertical scale as compared to Figure 6). The highest-mass BHs are always retained in the clusters and they are formed through binary mergers. These mergers are the result of early CE evolution of massive binaries. The most common merger types are MS-MS, HG-MS and BH-HG mergers. During mergers involving HG stars we assume that the envelope of the HG star is lost, while the BH/MS star and the compact core of the HG star merge to form a new, more massive object. The merger product is then evolved and it may eventually form a single BH. Even with significant mass loss through stellar winds and during the merger process, a small fraction of BHs reach very high masses, up to about \(80{\,M_{\odot}}\). With a less conservative assumption, allowing some fraction of the HG star envelope to be accreted onto the companion in a merger, the maximum BH mass could then reach even higher values \(\gtrsim 100{\,M_{\odot}}\). In Figure 8 we show the results of a calculation with the merger product's mass always assumed equal to the total binary mass. Although the amount of mass loss in a merger is rather uncertain, the two models above (with and without mass loss) indicate that binary star evolution could lead to the formation of single \(\sim 100{\,M_{\odot}}\) BHs. These most massive BHs form very early in the evolution of a cluster (first \(\sim 5-10\,\)Myr) since they originate from the most massive and rapidly evolving stars. These BHs are retained in clusters (direct/silent BH formation with no associated natal kick) and they may act as potential seeds for building up intermediate-mass BHs through dynamical interactions during the subsequent cluster evolution (Miller & Hamilton 2002; O'Leary et al. 2005). ### Parameter Study #### 3.2.1 Black Hole Spatial Velocities For most alternative models the velocity distributions are similar to those found in the reference model (see Fig. 2 and 3). These distributions are generally characterized by the same wide, high-velocity peak (tens to hundreds of \({\,{\rm km}\,{\rm s}^{-1}}\)) and a rather large population of zero-kick BHs. In particular, for models B, F, G2, J and H, the distributions are almost identical to those of the reference model at all times. For models D, G1, and I the distributions show slight differences. With lowered CE efficiency (model D) it is found that there are fewer fast binary BHs, and most surviving binaries do not gain higher velocities at early times. Basically, many tight binaries that survived SN explosions in the reference model have now merged in a first CE phase, even before the first SN explosion occurred. In model G1, in which we consider only primordial stars up to \(M_{\rm max}=50\,{\,M_{\odot}}\), the population of massive BHs formed through direct collapse (with no kick) is significantly reduced. This results in a velocity distribution similar to that of the reference model for non-zero velocity systems, but with a much lower number of zero-kick BHs.

The model I distribution is slightly different, especially at early times when most BHs form with no kick, since in this model we consider only the most massive BHs formed mainly through direct collapse. A few models show more significant differences. Different metallicities lead to changes in BH velocities, especially at early times. For very low metallicity (model C1) almost all BHs are formed with no kick, while for high, solar-like metallicity (model C2) most BHs have non-zero velocities in a wide range (\(\sim 10-1000{\,{\rm km}\,{\rm s}^{-1}}\)). Metallicity strongly affects the wind mass loss rates, which are most important for the evolution of the most massive stars (i.e., at early times). In particular, for low-\(Z\) values, the wind mass loss rates are smaller (hence more high-mass pre-SN stars and direct collapses), while for high \(Z\) the winds are very effective in removing mass from BH progenitors (hence smaller mass pre-SN stars, and more fall-back BH formation). The most significant difference is found in model E, where we allow for full BH kicks. All BHs are formed with rather high (\(\geq 100{\,{\rm km}\,{\rm s}^{-1}}\)) velocities. The distribution, shown in Figures 15 and 16, is double-peaked both for early and late times. The single stars dominate the population, forming the low- (\(\sim 100{\,{\rm km}\,{\rm s}^{-1}}\)) and high-velocity component (\(\sim 500{\,{\rm km}\,{\rm s}^{-1}}\)), a direct result of the adopted bimodal natal kick velocity distribution. Binary stars are found at lower velocities (\(\sim 100{\,{\rm km}\,{\rm s}^{-1}}\)) but they are only a minor contributor to the overall BH population since most of them are now disrupted at the first SN explosion. #### 3.2.2 Properties of Retained (Cluster) BH Populations In Table 7 we present the properties of cluster BH populations \(11\,\)Myr after the starburst. Results for the various models may be easily compared with our reference model. Binary BHs for different model assumptions are still in general dominated by BH-MS and BH-BH binaries. These systems appear in comparable numbers in most models. Only for models B and E do we find a smaller contribution of BH-BH binaries (\(\sim\) 5% and almost zero for models E and B, respectively). In model B the independent choice of masses produces systems with extreme mass ratios, so that massive primordial binaries with two BH progenitors are very rare. Obviously for model E, in which the two BHs receive full kicks, the BH-BH binary formation is strongly suppressed by binary disruptions. The highest number of binaries containing BHs is found in our model with the lowest tested metallicity (C1). For low metallicities BHs form preferentially with high masses (low wind mass loss rates) through direct collapse with no kick. In contrast model E, assuming full BH kicks, results in the lowest number of BH binaries. Many models (D, G2, H, I, J) result in very similar contents to our reference model. It is worth noting in particular that the CE treatment (either lowered efficiency in model D, or different prescription in model J) does not appear to play a significant role in determining cluster initial binary BH populations. For all models the single BHs dominate the population even at very early times (as early as \(11\,\)Myr). Single BHs originate predominantly from primordial single stars, with smaller contributions from disrupted binaries and binary mergers. The basic general trends seen in our reference model are preserved in other models. Also most models (B, C1, D, G2, H, I, J) form similar numbers of single BHs as our reference model. It is found, as in the binary populations, that the highest number of single BHs is seen in our model with lowest metallicity (C1), while the model with full BH kicks (E) generates the lowest number of single BHs retained in a cluster. At \(103.5\,\)Myr (see Table 8), when no more BHs are being formed, single BHs strongly dominate (by about an order of magnitude) over binary BHs. Single BHs still originate mostly from primordial single stars, but there is an increased contribution from binary mergers and disruptions. The binary population remains dominated by BH-BH and BH-MS systems in most models, but with an increased contribution from other evolved systems (BH-WD and BH-NS) compared to earlier times. Note that only in model E does the number of systems other than BH-BH and BH-MS end up dominating the binary population. #### 3.2.3 Properties of Ejected (Field) BH Populations In Tables 7 and 8 we also characterize the populations of ejected (field) BHs for various models. Results for both times are comparable for binary BHs, but with significantly more single BHs being ejected at later times. The BH-MS and BH-BH binaries, which dominate the total populations, are also found to be most effectively ejected from clusters. However, BH-NS systems, receiving two natal kicks, are also found to be easily ejected. Indeed, in many models (C1, C2, E, F, G1, G2, H, J), they constitute a significant fraction of ejected systems. Contrary to our intuitive expectation, evolution with the full BH kicks (model E) does not generate a particularly large population of fast BH binaries. In fact, the ejected population is smaller than in the reference model. Higher kicks are much more effective in binary disruption than in binary ejection. The numbers of fast single BHs are comparable in most models (A, B, D, G1, G2, H, J), with the ejected populations usually consisting equally of BHs coming from binary disruptions and primordial single stars, with a smaller contribution from merger BHs. For models with massive BHs (C1 and I) which receive small kicks there are fewer single BHs in the ejected population (by a factor \(\sim 2\)). On the other hand, for the model with full BH kicks (E), the ejected single BH population is larger (by a factor of \(\sim 3\)) compared to the reference model. #### 3.2.4 Dependence on Cluster Escape Velocity and Initial Binary Fraction Retained fractions for different evolutionary models follow in general the same trends as in our reference model, i.e., retained fractions decrease with increasing initial binary fraction. The exception to that trend is for models with full BH kicks (E), increased metallicity (C2) or uncorrelated binary component masses (B). Also, independent of the escape velocity, it is found that at least \(\sim 40\%\) of BHs are retained simply because of no-kick BHs (for an initial binary fraction of 50%), with the obvious exception of the model with full BH kicks (E). In particular, for models C1, D, F, G1, G2, H, I, and J, the dependence of the retained fraction on \(V_{\rm esc}\) is very similar to that seen in the reference model (see Fig. 4). In model B, th secondary mass is on average very small compared to the BH mass (due to our choice of initial conditions for this model). Therefore, BHs in binary systems