# A Photometric Survey of Globular Cluster Systems in Brightest Cluster
Galaxies

William E. Harris<EMAIL_ADDRESS>Department of Physics and
Astronomy, McMaster University, Hamilton, ON L8S 4M1, Canada

###### Abstract

Hubble Space Telescope imaging for 26 giant early-type galaxies, all drawn
from the MAST archive, is used to carry out photometry of their surrounding
globular cluster (GC) systems. Most of these targets are Brightest Cluster
Galaxies (BCGs) and their distances range from 24 to 210 Mpc. The catalogs of
photometry, completed with DOLPHOT, are publicly available. The GC color
indices are converted to [Fe/H] through a combination of 12-Gyr SSP (Single
Stellar Population) models and direct spectroscopic calibration of the
fiducial color index (F475W-F850LP). All the resulting metallicity
distribution functions (MDFs) can be accurately matched by bimodal Gaussian
functions. The GC systems in all the galaxies also exhibit shallow metallicity
gradients with projected galactocentric distance that average $Z\sim
R_{gc}^{-0.3}$. Several parameters of the MDFs including the means,
dispersions, and blue/red fractions are summarized. Perhaps the most
interesting new result is the trend of blue/red GC fraction with galaxy mass,
which connects with predictions from recent simulations of GC formation within
hierarchical assembly of large galaxies. The observed trend reveals two major
transition stages: for low-mass galaxies, the metal-rich (red) GC fraction
$f(red)$ increases steadily with galaxy mass, until halo mass $M_{h}\simeq
3\times 10^{12}M_{\odot}$. Above this point, more than half the metal-poor
(blue) GCs come from accreted satellites and $f(red)$ starts declining. But
above a still higher transition point near $M_{h}\simeq 10^{14}M_{\odot}$, the
data hint that $f(red)$ may start to increase again because the metal-rich GCs
also become dominated by accreted systems.

galaxy mass, globular clusters

††journal: ApJ††software: DOLPHOT (Dolphin, 2000), IRAF (Tody, 1986, 1993),
Daophot (Stetson, 1987), ISHAPE (Larsen, 1999), Astrodrizzle (Hack et al.,
2012)

##  1 Introduction

Globular clusters (GCs), the old massive star clusters formed in the early
stages of galaxy assembly, are routinely found in all galaxies except the
smallest dwarfs (Harris, 2010; Forbes et al., 2018a; Beasley, 2020; Eadie et
al., 2022). Thanks to the rapid development of theoretical models and
simulations in recent years, a basic framework has now been established that
follows GC formation in the gas-rich halos that build up during hierarchical
assembly of galaxies (among others, see Côté et al., 1998; Kravtsov & Gnedin,
2005; Muratov & Gnedin, 2010; Kruijssen, 2012; Tonini, 2013; Boylan-Kolchin,
2017; Pfeffer et al., 2018; El-Badry et al., 2019; Choksi et al., 2018; Choksi
& Gnedin, 2019; Kruijssen et al., 2019a; Reina-Campos et al., 2019; Halbesma
et al., 2020).

Confronting the impressive state of current theory with observations requires
high quality data. Challenging observational questions about GC populations
remain at both the low-mass and high-mass ends of the galaxy mass spectrum.
Much recent interest has focussed on the lowest-mass galaxies and their
surprising diversity of GC populations (e.g. Forbes et al., 2018b; Lim et al.,
2018; Amorisco et al., 2018; Saifollahi et al., 2021; Trujillo-Gomez et al.,
2021; Eadie et al., 2022; Danieli et al., 2022; Carlsten et al., 2022; van
Dokkum et al., 2022). It is not yet clear, for example, if the near-linear
proportionality between GC system mass and total galaxy mass that has been
established from large galaxies (Blakeslee, 1997; Spitler & Forbes, 2009;
Hudson et al., 2014; Harris et al., 2015, 2017a; Burkert & Forbes, 2020)
extends downward into the dwarf regime even in the presence of much scatter
(Forbes et al., 2018b; Prole et al., 2019; Doppel et al., 2021; Eadie et al.,
2022; Zaritsky, 2022).

But there is also a need for improved observational data aimed at the GC
systems in the rare highest-mass galaxies. These have the most complex history
of assembly through mergers and accretions that happened intensely at high
redshift but continue right up to (and beyond) the present day. Traces of that
history have been left behind in their GC systems including the GC mass
distribution, their radial distribution throughout the halo, and their
metallicity distribution including the proportions of ‘blue’ (metal-poor) and
‘red’ (metal-rich) clusters. At the very highest galaxy masses, it is also not
yet clear if the $M_{halo}-M_{GCS}$ relation is genuinely linear or displays
curvature (e.g. Harris et al., 2017a; Boylan-Kolchin, 2017; El-Badry et al.,
2019; Choksi & Gnedin, 2019).

The Hubble Space Telescope (HST) MAST archive has been an extraordinarily
valuable resource for work of this type. Many giant galaxies lie in the
distance range $\lesssim 200$ Mpc within which GC populations are readily
imaged with reasonable exposure times with the ACS and WFC3 cameras.
Photometry of the GC systems for many such targets has been published in
previous HST programs (Harris et al., 2006; Harris, 2009a; Harris et al.,
2014, 2016, 2017b), but more can be found in the Archive. These are not all in
the same color indices, so all the photometry needs to be put on an internally
homogeneous basis in the process of data analysis.

The purpose of the present study is to provide more of the necessary data for
an investigation of the largest galaxies. The MAST Archive was searched for
target galaxies that were at distances $\gtrsim 25$ Mpc (guaranteeing that
their GCs would be near-starlike in morphology and permitting stellar
photometry codes to be used, as described below); with imaging in at least two
optical/near-IR filters; and with exposure times deep enough to cover at least
the bright half of the standard GC luminosity function with $SNR\gtrsim 5$
(effectively, one or more orbits per filter). This search yielded the 26
galaxies listed in Table 1, all imaged with the ACS Wide Field Camera. Most of
these are BCGs (Brightest Cluster Galaxies), but some are in smaller galaxy
groups.111There are roughly 20 more in a similar distance range imaged with
WFC3/UVIS, and these will be the subject of an upcoming study. In this paper,
the photometry of the GC populations around these galaxies is presented along
with a first look at their metallicity distributions in particular.

Organization of the paper is as follows: Section 2 describes details of the
photometric measurement and completeness tests, and Section 3 the resulting
color-magnitude diagrams for the GCs in each galaxy and the color
distributions. Section 4 describes tests of the effect of any marginal
resolution of the GCs on their measured colors. Section 5 lays out a
preliminary conversion of the different color indices into [Fe/H]
metallicities, derived from a combination of SSP models and observational
spectroscopic calibration, while Section 6 shows the metallicity distributions
for all galaxies now put onto a common system. A brief analysis of the
distribution shapes and parameters is shown, with perhaps the most intriguing
result showing up in the correlation of the relative numbers of blue vs. red
GCs as a function of galaxy mass. Section 7 ends with a summary of the
findings and prospects for future work.

Table 1: List of Program Galaxies

Galaxy | Environment | $d$ | $M_{V}^{T}$ | $A_{V}$ | $R_{e}$ | HST | Blue Filter | Red Filter
---|---|---|---|---|---|---|---|---
|  | (Mpc) | (mag) | (mag) | (kpc) | Program | (sec) | (sec)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
NGC1129 | AWM7 | 71 | $-22.3:$ | 0.309 | 24.2 | 13698 | F435W (2416s) | F606W (2288s)
NGC1132 | N1132 Group | 101 | $-22.87$ | 0.176 | 16.5 | 10558 | F475W (7800s) | F850LP (9630s)
NGC1272 | Perseus/Abell 426 | 74 | $-23.09$ | 0.176 | 20.5 | 10201 | F555W (2368s) | F814W (2260s)
NGC1275 | Perseus/Abell 426 | 74 | $-22.74$ | 0.447 | 6.1 | 15235 | F475W (2436s) | F814W (2325s)
NGC1278 | Perseus/Abell 426 | 74 | $-21.46$ | 0.452 | 8.8 | 15235 | F475W (2577s) | F814W (2429s)
NGC1407 | Eridanus/N1407 | 23 | $-22.04$ | 0.187 | 7.8 | 9427 | F435W (1500s) | F814W (680s)
NGC3258 | Antlia/Abell S0636 | 45 | $-21.95$ | 0.221 | 6.5 | 9427 | F435W (5360s) | F814W (2280s)
NGC3268 | Antlia/Abell S0636 | 45 | $-21.95$ | 0.281 | 7.8 | 9427 | F435W (5360s) | F814W (2280s)
NGC3348 | CfA69 | 42 | $-22.20$ | 0.204 | 5.3 | 9427 | F435W (7200s) | F814W (2120s)
NGC4696 | Cen30/Abell 3526 | 46 | $-23.4:$ | 0.306 | 11: | 9427 | F435W (5440s) | F814W (2320s)
NGC4874 | Coma/Abell 1656 | 103 | $-23.37$ | 0.025 | 20.1 | 10861,11711 | F475W (5071s) | F814W (11825s)
NGC4889 | Coma/Abell 1656 | 103 | $-23.69$ | 0.025 | 15.0 | 11711,14361 | F475W (7138s) | F814W (11098s)
NGC5322 | CfA122 | 27 | $-22.00$ | 0.038 | 4.4 | 9427 | F435W (3390s) | F814W (820s)
NGC5557 | CfA141 | 38 | $-22.33$ | 0.016 | 5.5 | 9427 | F435W (5260s) | F814W (2400s)
NGC6166 | Abell 2199 | 130 | $-23.65$ | 0.031 | 30.0 | 12238 | F475W (5370s) | F814W (4885s)
NGC7049 | N7049 Group | 30 | $-21.84$ | 0.153 | 3.6 | 9427 | F435W (3480s) | F814W (1200s)
NGC7626 | Pegasus/LGC473 | 44 | $-22.06$ | 0.197 | 8.2 | 9427 | F435W (7720s) | F814W (2600s)
NGC7720 | Abell 2634 | 130 | $-23.39$ | 0.194 | 19.4 | 12238 | F475W (5282s) | F814W (5278s)
IC4051 | Coma/Abell 1656 | 103 | $-21.68$ | 0.025 | 4.7 | 12918 | F475W (2569s) | F814W (1310s)
UGC9799 | Abell 2052 | 154 | $-23.2:$ | 0.102 | 14: | 12238 | F475W (7977s) | F814W (5253s)
UGC10143 | Abell 2147 | 151 | $-22.9:$ | 0.086 | 23.0 | 12238 | F475W (10726s) | F814W (5262s)
ESO306-G017 | Abell S0540 | 154 | $-23.63$ | 0.090 | 18: | 10558 | F475W (7059s) | F850LP (8574s)
ESO325-G004 | Abell S0740 | 149 | $-23.10$ | 0.166 | 11: | 10710,10429 | F475W (5901s) | F814W (18078s)
ESO383-G076 | Abell 3571 | 171 | $-24.25$ | 0.149 | 16: | 10429,12238 | F475W (10830s) | F814W (21081s)
ESO444-G046 | Abell 3558 | 210 | $-23.81$ | 0.137 | 17: | 10429,12238 | F475W (20282s) | F814W (35426s)
ESO509-G008 | Abell 1736 | 153 | $-23.04$ | 0.144 | 9: | 10429,12238 | F475W (10758s) | F814W (17310s)

  * •

Key to columns: (1) Galaxy identification; (2) Group or cluster membership;
(3) Distance $d$, calculated as $d=cz/H_{0}$ where $cz$ is the group redshift
relative to the CBR frame and $H_{0}=70$ km s-1 Mpc-1; (4) Integrated $V-$band
luminosity; (5) Foreground extinction in $V$; (6) Effective radius of the
galaxy light profile in the $V-$band; (7) ID number(s) of the original HST GO
program(s) from which the imaging data are taken; (8,9) the filters and
exposure times of the raw observations. Unless otherwise noted all images are
taken with the ACS/WFC camera. Values for $M_{V}^{T}$, $A_{V}$, and $R_{e}$
are drawn from NED.

##  2 Measurement Procedures

Photometry was carried out with DOLPHOT (Dolphin, 2000), a widely used package
particularly designed for use with the HST ACS and WFC3 cameras. DOLPHOT
starts by using a deep ‘reference image’ for the target field for the purpose
of object detection and location: the candidate stars found from the reference
image are then measured on every individual _*.flc_ exposure and the results
averaged to give final magnitudes. For the galaxy fields studied here, this
deep image was constructed with _astrodrizzle_ in most cases from the summed
exposures in the redder filter (usually F814W or F850LP).

DOLPHOT contains a large number of adjustable parameters for object detection,
aperture photometry, PSF fitting, and other steps in the process (see, e.g.
Dalcanton et al., 2012; Williams et al., 2014; Cohen et al., 2020, for much
more extensive discussion). For all the galaxies studied here, many of these
parameter choices are not critically sensitive since the fields are all quite
uncrowded in any absolute sense (even though the galaxy may be surrounded by
many thousands of its GCs, this still leaves typically $10^{2-3}$ resolution
elements per object). A list of some of the key parameters adopted here is
given in Table 2. These same parametric values were used for every field to
maximize homogeneity.

Throughout this study, the magnitudes are in the natural ACS/WFC filter system
and on the Vegamag magnitude scale.

Figure 1: Use of DOLPHOT selection parameters _sharp_ and _chi_ for culling of nonstellar objects. Black symbols are real objects on the field and superimposed blue symbols are artificial stars injected into the image. Red lines mark the exclusion boundaries outside of which measured objects were rejected as distinguishably nonstellar. Figure 2: Artificial-star test with DOLPHOT for the NGC 4874 field. The CMD for the input population of fake stars is shown in the left panel (note here the y-axis is the blue filter), while the CMD for the recovered stars before any culling by SNR, _chi, sharp_ is in the middle panel. The remaining objects after all culling steps are shown in the right panel. The 50% detection completeness level is shown as the dashed red line. Table 2: Selected DOLPHOT Parameters Parameter | Adopted Value
---|---
RAper | 6.0 px
RSky | 15, 25 px
RPSF | 15 px
aprad | 15
apsky | 25, 30 px
SigFind | 3.0
FPSF | Lorentz
FitSky | 2
PSFPhot | 1
SigPSF | 5
PSFStep | 0.25 px

Once the DOLPHOT run was finished, the list of measured objects was rigorously
culled to remove nonstellar or other unwanted objects. In this section, the
NGC 4874 field is used as a template example to describe the details of this
procedure.

In all target fields, the overwhelming source of field contamination consists
of small, faint background galaxies, whereas the GCs that are the objects of
this study are unresolved or near-starlike. In the first round of culling,
objects classified as $n_{type}>1$ (i.e., not ‘bright’ stars) were rejected,
as were those with magnitudes $m=99.999$ in either filter, or those with
$SNR<4.0$. After these initial removals, the parameters _sharp, chi_
describing the goodness of fit to the PSF were used as shown in Figure 1. For
_sharp_ , any objects lying outside the curved boundary lines shown in the
figure were rejected. A simple quadratic curve was used for the boundary lines
to reflect the gradually increasing random dispersion in the _sharp_ index
towards fainter magnitude. The artificial-star population (see below) was also
used as a check that the limits were not excluding true starlike sources. For
_chi_ , a boundary line was set at a constant value and any objects falling
above the line were rejected. In practice, _sharp, chi_ , and the photometric
error are well correlated (that is, objects with high chi-values usually have
high sharp values and high photometric uncertainties as well). As can be seen
in Fig. 1, the measured objects remaining after the first basic culling by
$n_{type}$, SNR, and magnitude are already dominated by the starlike sources
that include the GC candidates, allowing the exclusion boundaries for _chi,
sharp_ to be placed conservatively.

Figure 3: _Upper panel:_ Photometric measurement uncertainty versus magnitude,
determined from the artificial-star tests. The interpolation lines follow the
equation given in the text. _Lower panel:_ Photometric detection completeness
curves versus magnitude as determined from artificial-star tests. This example
is for NGC 4874. The sigmoid curves fitted to each of the two filters follow
the equation given in the text.

Figure 4: _Left panel:_ Detection completeness as a function of F814W
magnitude, now plotted in five radial zones around NGC 4874. The _heavy solid
curve_ shows the solution for all zones combined. For comparison, the
effective radius of the galaxy light profile is $r_{e}=40^{\prime\prime}$ for
NGC 4874. _Right panel:_ The same material for NGC 5322, a less luminous and
more centrally concentrated galaxy. In this case, $r_{e}=33^{\prime\prime}$.
Figure 5: Color-magnitude diagram for the NGC 4874 field. _Left panel:_ The
CMD from all the DOLPHOT measurements before any culling steps. _Middle
panel:_ The CMD after removal of objects with $ntype>1$ (nonstellar),
magnitude = 99.999 in either filter, and $SNR<4$. _Right panel:_ The CMD after
final additional culling by _chi_ and _sharp_. In all panels, the red dashed
line indicates the 50% detection completeness level, while the blue lines mark
out the objects used to define the color and metallicity histograms for the
globular cluster population.

Photometric measurement uncertainty (rms scatter) $\sigma(m)$ and completeness
of detection $f(m)$ were determined from artificial-star tests run with the
DOLPHOT tools. Typically 10000-20000 fake stars per galaxy were used, covering
the magnitude and color ranges of the real data. DOLPHOT adds and measures
these one at a time so the degree of crowding is never affected. Figure 2
shows the color-magnitude diagram (CMD) for the population of fake stars
inserted into the NGC 4874 image (left panel), the CMD for the ones that were
recovered and measured before any culling (middle panel), and finally the ones
remaining after all the same culling steps that were applied to the real data
(right panel).

The measurement uncertainty $\sigma(m)$ and the detection completeness $f(m)$
were also determined from these artificial-star tests. A convenient and simple
function that provides a good approximation to the increase of $\sigma$ with
magnitude is

$\sigma(m)=\beta_{1}e^{\beta_{2}(m-\beta_{3})}.$ (1)

while the falloff of completeness $f$ with magnitude can be described by a
sigmoid curve (Harris et al., 2016)

$f(m)=1/(1+e^{\alpha(m-m_{0})})$ (2)

where $m_{0}$ is the magnitude at which completeness is 50% and $\alpha$
measures the steepness of decline of the $f-$curve. Both of these trends are
shown in Figure 3 for our template case of NGC 4874: the particular values for
the various parameters ($\beta_{1},\beta_{2},\beta_{3},\alpha,m_{0}$) are
listed for all the fields in Table 3. These differ from field to field
depending on exposure times and filters, but the same shapes for the fitting
formulae hold. The best-fit estimates for $\beta_{2}$ are consistently near
$0.6\pm 0.1$.

Completeness of detection will at some level depend on galactocentric
location, since objects become progressively harder to detect against the
higher background light and sky noise in the inner regions of the galaxy.
However, a feature of the giant galaxies studied here that mitigates this
dependence is that they are much more diffuse in structure than smaller
galaxies, along with much lower central surface brightnesses (e.g. Kormendy et
al., 2009). In addition, the great majority of the GCs measured in this study
are drawn from the mid- to outer-halo regions that are well outside the
$\lesssim 1R_{e}$ central region where the bulge light is brightest. Figure 4
shows the completeness curves obtained from the artificial-star runs in NGC
4874 within five different radial zones, and also for NGC 5322, a less
luminous galaxy with smaller effective radius, in four radial zones. Though
there is a trend for the 50% completeness level $m_{0}$ to become fainter with
increasing radius, the trend is small, and as will be seen below it results in
no important effects on the final GC color distributions.

The measurements after various stages of culling are shown in the CMDs of
Figure 5, which shows dereddened colors and magnitudes. In the left panel, all
measured objects found by DOLPHOT are shown: the GC red and blue vertical
sequences are already quite prominent, but much faint contamination is
present. The objects remaining after rejection by $n_{type}$ and SNR are shown
in the middle panel, and the ones remaining after final culling by _chi,
sharp_ are shown in the right panel. The final step effectively removes only
some very red or blue objects well away from the GC sequences, plus a larger
number of faint blue objects that are below the completeness limit. With a
single exception (namely NGC 1275; see below), no young population of massive
star clusters that would appear on the luminous blue side of the CMD is
evident in any of the target galaxies.

Figure 6: Distribution of GC candidates brighter than F814W=26.0 around NGC
4874. Holes in the distribution indicate satellite galaxies removed from the
data list.

The spatial distribution of GC candidates with $F814W<26$ across the NGC 4874
field is shown in Figure 6. The GC population now totally dominates the
numbers of objects brighter than this level.

Final lists of the photometry for each galaxy include (a) object positions (x,
y on the reference image, and J2000 RA and Dec); (b) magnitude and uncertainty
in each filter; and (c) DOLPHOT output parameters including $\chi$, SNR,
sharpness, roundness, and crowdedness in each filter. An example is shown in
Table 4 and all data, along with the reference images, can be obtained at the
DOIs listed in the Acknowledgements below, or from the author’s website at
_https://physics.mcmaster.ca/Fac_Harris/Supergiants.html_.

##  3 Color-Magnitude Diagrams and the Color Distribution Functions

The resulting CMDs are shown in Figures 7, 8, and 9. The 26 program galaxies
are grouped by color index for easier intercomparison. Galaxies measured in
the two indices $(F475W-F814W)$ and $(F435W-F814W)$ account for 22 of the 26
galaxies in the program. In most cases, the familiar blue and red sequences
are fairly clearly visible, and once the colors are dereddened, these
sequences fall at very similar colors from one galaxy to another. In Figure
10, the top two panels show the combined CMD for all 13 galaxies that were
measured in $(F475W,F814W)$ in the form of absolute magnitude versus intrinsic
color. The same data are shown in the right-hand panel in Hess-diagram form as
a smoothed contour plot. The lower pair of panels shows the same for all 9
galaxies measured in $(F435W,F814W)$. The galaxies in the upper panels are on
average the most luminous ones in the whole list, and not surprisingly with
the largest GC populations.

One notable feature of Fig. 10 that is also visible in several of the
individual CMDs is the upward extension of the red, more metal-rich sequence
beyond $M_{I}\lesssim-12$. For a mass-to-light ratio of 2, this range
corresponds to $M\gtrsim 5\times 10^{6}M_{\odot}$ and certainly reaches well
into the luminosity range of the Ultra-Compact Dwarfs (UCDs). This regime
includes the “superluminous” GCs discussed in Harris et al. (2014), who
demonstrate that there are systematically more of these objects than would be
expected from a normal statistical extension of the standard lognormal GC
luminosity function. Notably, in the lower panels of Fig. 10, which include
less luminous galaxies, no such red-sequence extension is seen. Further
discussion and interpretation of these points is given in Harris et al.
(2014).

Table 3: Photometric Parameters

Galaxy |  $m_{0}$(blue) |  $\alpha$(blue) |  $m_{0}$(red) |  $\alpha$(red) |  $\beta_{1},\beta_{2},\beta_{3}$(blue) |  $\beta_{1},\beta_{2},\beta_{3}$(red) | Color Range | $m(lim)$
---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
| F475W | F814W | F475W | F814W | (F475W-F814W)0 | F814W0
NGC1275 | 27.72 | 1.90 | 26.45 | 1.57 | 0.15, 0.68, 27.0 | 0.05, 0.62, 24.5 | 1.0 - 2.2 | 25.0
NGC1278 | 27.83 | 2.40 | 26.33 | 1.60 | 0.09, 0.61, 27.0 | 0.04, 0.59, 24.5 | 1.0 - 2.2 | 25.0
NGC4874 | 28.37 | 3.38 | 26.33 | 1.39 | 0.11, 0.58, 27.5 | 0.05, 0.62, 25.5 | 1.0 - 2.2 | 26.0
NGC4889 | 28.49 | 3.53 | 26.46 | 1.28 | 0.08, 0.61, 27.0 | 0.04, 0.64, 25.5 | 1.0 - 2.2 | 26.0
NGC6166 | 28.64 | 2.61 | 26.65 | 1.42 | 0.10, 0.68, 27.5 | 0.04, 0.65, 25.0 | 1.0 - 2.2 | 26.0
NGC7720 | 28.86 | 9.62 | 27.07 | 2.78 | 0.09, 0.66, 27.5 | 0.04, 0.68, 25.0 | 1.0 - 2.2 | 26.5
UGC9799 | 28.90 | 3.14 | 27.30 | 1.36 | 0.13, 0.66, 28.0 | 0.05, 0.653, 25.5 | 1.0 - 2.2 | 26.5
UGC10143 | 29.09 | 3.16 | 27.54 | 1.66 | 0.11, 0.69, 28.0 | 0.05, 0.63, 25.5 | 1.0 - 2.2 | 26.5
ESO325-G004 | 28.22 | 1.67 | 26.70 | 1.03 | 0.14, 0.67, 28.0 | 0.04, 0.62, 25.5 | 1.0 - 2.2. | 26.0
ESO383-G076 | 28.71 | 1.96 | 26.69 | 0.94 | 0.13, 0.65, 28.0 | 0.05, 0.56, 26.0 | 1.0 - 2.2 | 26.0
ESO444-G046 | 29.36 | 3.40 | 27.71 | 1.39 | 0.10, 0.65, 28.0 | 0.04, 0.61, 25.5 | 1.0 - 2.2 | 27.0
ESO509-G008 | 28.62 | 2.04 | 27.10 | 1.51 | 0.10, 0.64, 28.0 | 0.03, 0.67, 25.5 | 1.0 - 2.2 | 26.5
IC4051 | 28.02 | 3.26 | 26.03 | 1.66 | 0.15, 0.59, 28.0 | 0.08, 0.59, 25.5 | 1.0 - 2.2. | 25.5
| F435W | F814W | F435W | F814W | (F435W-F814W)0 | F814W0
NGC1407 | 27.14 | 3.04 | 25.11 | 1.57 | 0.08, 0.59, 26.0 | 0.10, 0.58, 25.0 | 1.15 - 2.70 | 24.4
NGC3258 | 27.87 | 2.25 | 26.37 | 2.20 | 0.05, 0.69, 26.0 | 0.06, 0.66, 25.0 | 1.15 - 2.70 | 25.5
NGC3268 | 27.93 | 2.58 | 26.40 | 2.69 | 0.04, 0.62, 26.0 | 0.06, 0.60, 25.0 | 1.15 - 2.70 | 25.5
NGC3348 | 28.07 | 3.27 | 26.54 | 2.49 | 0.04, 0.62, 26.0 | 0.04, 0.61, 25.0 | 1.15 - 2.70 | 26.0
NGC4696 | 27.96 | 2.64 | 26.44 | 2.30 | 0.04, 0.72, 26.0 | 0.06, 0.69, 25.0 | 1.15 - 2.70 | 25.5
NGC5322 | 27.62 | 2.29 | 26.09 | 2.47 | 0.05, 0.62, 26.0 | 0.06, 0.53, 25.0 | 1.15 - 2.70 | 25.5
NGC5557 | 28.10 | 2.55 | 26.63 | 2.34 | 0.04, 0.60, 26.0 | 0.05, 0.57, 25.0 | 1.15 - 2.70 | 26.0
NGC7049 | 27.38 | 3.61 | 25.84 | 2.00 | 0.09, 0.73, 26.0 | 0.09, 0.64, 25.0 | 1.15 - 2.70 | 25.2
NGC7626 | 28.16 | 2.37 | 26.67 | 2.15 | 0.07, 0.64, 27.0 | 0.06, 0.61, 25.0 | 1.15 - 2.70 | 26.0
| F475W | F850LP | F475W | F850LP | (F475W- F850LP)0 | F850LP0
NGC1132 | 28.58 | 2.17 | 26.17 | 1.11 | 0.06, 0.61, 27.0 | 0.07, 0.57, 25.5 | 1.0 - 2.4 | 26.0
ESO306-G017 | 28.37 | 1.59 | 26.63 | 2.02 | 0.12, 0.65, 28.0 | 0.07, 0.59, 25.5 | 1.0 - 2.4 | 26.0
| F435W | F606W | F435W | F606W | (F435W-F606W)0 | F606W0
NGC1129 | 27.80 | 2.84 | 26.23 | 1.43 | 0.07, 0.64, 26.0 | 0.03, 0.65, 25.0 | 0.7 - 1.8 | 26.0
| F555W | F814W | F555W | F814W | (F555W-F814W)0 | F814W0
NGC1272 | 27.78 | 5.73 | 26.29 | 1.96 | 0.22, 0.63, 28.0 | 0.07, 0.58, 25.5 | 0.8 - 1.8 | 25.5

Key to columns: (1) Galaxy ID; (2,3) Completeness curve parameters for the
blue filter (Eq.2); (4,5) Completeness curve parameters for the red filter;
(6) Measurement uncertainty parameters for the blue filter (Eq.1): (7)
Measurement uncertainty parameters for the red filter; (8) Selection range of
dereddened colors including GC candidates; (9) Faint limit of intrinsic (red)
magnitude for selection of GCs. The GC candidates selected for the CDF and MDF
are the objects in the color-magnitude diagram brighter than $m(lim)$ and
within the given color range.

Table 4: DOLPHOT Photometry for NGC 4874 Field

RA | Dec | x(px) | y(px) | F475W | $\pm$ | F814W | $\pm$
---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
194.8888158 | 27.9473826 | 3004.360 | 4224.360 | 24.857 | 0.010 | 20.914 | 0.002
194.8728227 | 27.9648258 | 988.080 | 4099.310 | 22.827 | 0.014 | 21.222 | 0.002
194.9314789 | 27.962022 | 4326.600 | 834.530 | 25.655 | 0.021 | 21.814 | 0.002
194.8990624 | 27.9774308 | 1562.750 | 1795.640 | 23.663 | 0.006 | 21.968 | 0.002
194.8854043 | 27.9577357 | 2134.520 | 3794.070 | 23.668 | 0.007 | 22.008 | 0.003
194.8747698 | 27.9706598 | 705.980 | 3630.200 | 23.928 | 0.007 | 22.191 | 0.003
194.9120388 | 27.9656600 | 3040.670 | 1751.930 | 23.814 | 0.007 | 22.218 | 0.003

  * •

Note: The partial table listed here illustrates the form and content of the
complete data. Final photometry for all the galaxies can be accessed through
the author’s website or through the DOIs listed in the Acknowledgements below.
The final lists accessible on-line also include other DOLPHOT parameters (chi,
SNR, sharpness, roundness, crowdedness).

Figure 7: Color-magnitude diagrams for the GC populations around the 13
galaxies measured in the (F475W, F814W) filters. Data are plotted as absolute
magnitude $M_{814}$ versus intrinsic color for easier comparison with the next
two figures. The dashed red lines show the 50% completeness limits as
determined from artificial-star tests. Figure 8: Color-magnitude diagrams for
the GC populations around the nine galaxies measured in the (F435W, F814W)
filters. Figure 9: Color-magnitude diagrams for the GC populations around four
galaxies measured in other filter combinations.  Figure 10: _Upper panels:_
CMD for all 13 galaxies measured in (F475W, F814W) combined, along with its
Hess-diagram contour version. _Lower panels:_ Combined CMD for all 9 galaxies
measured in (F435W, F814W). For the Hess diagrams, contour levels are
logarithmic.

The distribution of GC colors (the CDF) is defined from the objects within the
intrinsic (dereddened) color range as given in column (8) of Table 3, and
brighter than $m(lim)$ as given in column (9). The magnitude limit $m(lim)$ is
conservatively brighter (by about 0.5 mag) than the 50% completeness limit to
ensure that the CDFs will not be biased by incompleteness. In the template
example of NGC 4874, this final selection box is marked out with the blue
dashed lines in Fig. 5.

The CDFs can, however, be explicitly corrected for incompleteness and the
effect of doing this is illustrated in Figure 11. Here, the CDF from the raw
counts is shown as the magenta histogram, while the completeness-corrected
version is shown as the black histogram. In the completeness-corrected
version, each object is counted as $(1/f)$ objects where $f$ is the
completeness fraction defined by Eq. 2. In addition, the completeness
correction takes into account the projected galactocentric location of each
object, since the fitted values of $(m_{0},\alpha)$ in Eq. 2 depend slightly
on radius as shown in Fig. 4.

In Fig. 11 both curves have been normalized to the same total population to
highlight any changes in the shape of the distribution. The _total_ equivalent
number of objects $n_{corr}=\sum(1/f_{i})$ in each color bin is
correspondingly larger than the raw counts typically by about 20%, but the two
_fractional_ distributions across all the bins (as shown in Fig. 11) are very
similar to within their statistical uncertainties. This robust feature of the
CDFs is partly due to a conservative magnitude cutoff as noted above, but also
due to the empirical feature that the radial metallicity gradients of the GC
systems in these giant galaxies are quite shallow (discussed below). Tihat is,
to first approximation the shape of the CDF is insensitive to the chosen
radial range, at least within the limits of the current data.

The CDFs for all the program galaxies are displayed as histograms in Figures
12, 13, and 14. The numbers of GCs in 0.05-magnitude bins in color, along with
their Poisson errorbars ($N\pm N^{1/2}$), are plotted versus dereddened color.
The overall shapes of these CDFs are familiar from many previous photometric
studies of GC systems in large galaxies (Geisler et al., 1996; Larsen et al.,
2001; Rhode & Zepf, 2004; Peng et al., 2006; Bassino et al., 2006; Harris,
2009b; Cho et al., 2012; Kim et al., 2013; Fensch et al., 2014; Escudero et
al., 2015; Harris et al., 2017b, among numerous others): a relatively high,
narrow peak on the blue side with a broader, lower distribution on the red
side.

All the CDFs have been fit with a bimodal-Gaussian function with the GMM code
(Muratov & Gnedin, 2010). The blue and red subcomponents from the solutions
are shown as the dotted lines in the figures. In general, the bimodal-Gaussian
assumption provides a good match to all the galaxies here, and perhaps better
the larger the GC population becomes. The parameters derived from the
solutions are listed in Table 5, and include the centroids of the blue and red
modes ($\mu_{1},\mu_{2}$); the standard deviations of the two modes
($\sigma_{1},\sigma_{2}$); the fraction of the total population taken up by
the blue mode ($f_{1}$); and two other parameters ($DD,H_{12}$) that relate to
the shape of the CDF. $DD$ (Muratov & Gnedin, 2010) measures the separation of
the two modes relative to their intrinsic widths,

$DD={(\mu_{2}-\mu_{1})\over([\sigma_{1}^{2}+\sigma_{2}^{2}]/2)^{1/2}}.$ (3)

The observed $DD$ values range from a low of 1.34 (UGC 10143) to a high of
3.24 (NGC 1407). For clearly separated modes $DD\gtrsim 2$ is normally
expected, but the sample sizes of GCs here are so large that even for the
galaxies in the list with $DD<2$ a unimodal Gaussian solution is strongly
rejected by GMM (at 99% confidence and higher). The other parameter, the
_height ratio_ $H_{12}$, is defined as the maximum height (amplitude) of the
blue component divided by the maximum height of the red component and can be
expressed in terms of the Gaussian fit parameters as

$H_{12}={\sigma_{2}\over\sigma_{1}}\cdot{f_{1}\over(1-f_{1})}-1.$ (4)

$H_{12}$ is zero where the modes are of equal height, positive where the blue
peak is higher, and negative where the red peak is higher.

Figure 11: Color distribution function for the NGC 4874 system. The magenta
dots, errorbars, and lines show the raw CDF for the color range
$1.0<(F475W-F814W)_{0}<2.2$ and magnitude range $F814W_{0}<26.0$, plotted as
the fraction of the total per 0.05-mag bin. The black histogram shows the CDF
corrected for photometric completeness, including the dependence of
completeness on projected galactocentric radius; see text. Figure 12: Color
distribution functions (CDFs) for the galaxies measured in (F475W, F814W).
Bimodal-Gaussian fits are shown as the solid lines, with the blue and red
subcomponents as the dashed lines. Figure 13: Color distribution functions for
the galaxies measured in (F435W, F814W). Figure 14: Color distribution
functions for the galaxies measured in other filter combinations.

##  4 Effects of GC Intrinsic Sizes

Most of the galaxies analyzed in this paper are at $d\gtrsim 50$ Mpc, for
which the intrinsic scale sizes of their GCs (with typical half-mass radii
$r_{h}\sim 3$ pc) are small enough that they are effectively unresolved.
Stellar photometry codes like DOLPHOT (Dolphin, 2000) or Daophot (Stetson,
1987) are therefore very well suited to measuring them. However, a few of the
galaxies have $d\lesssim 40$ Mpc and for these, a significant fraction of
their GCs will be marginally nonstellar (Harris, 2009b). For such objects,
photometry based on PSF fitting will not be ideal. For the still nearer Virgo
galaxies ($d\simeq 16$ Mpc) and a few other systems not studied here, the GCs
are partially resolved well enough that measurements of their half-light radii
are interesting in their own right (Jordán et al., 2005; Harris et al., 2010).
For the galaxies in the study by Harris (2009b), which are in the distance
range $d\simeq 25-50$ Mpc, aperture photometry and a curve of growth technique
were used to obtain the final GC magnitudes and colors, rather than PSF
fitting. However, the initial survey of those same galaxies (Harris et al.,
2006) employed PSF fitting, with scarcely different results.

###  4.1 ISHAPE versus DOLPHOT

Here, a method similar to that of Jordán et al. (2005) is used to test the
validity of the photometry, and especially the color indices on which the GC
metallicity estimates are ultimately based (see below). This test is done on
the nearest galaxy in the list, NGC 1407 at $d=23$ Mpc. At this distance, a
standard GC radius of 3 pc corresponds to an angular radius of
$0.027^{\prime\prime}$, about half an ACS pixel or 1/4 of the stellar fwhm.

A fitting routine designed for exactly this task is ISHAPE (Larsen, 1999). For
each candidate GC in the photometry list, a King-model profile (King, 1962) is
convolved with the PSF for the image, and the assumed profile width (fwhm) is
adjusted till an optimum fit is achieved. An “average” KING30 profile was
assumed ($r_{t}/r_{c}=30$). For the PSF, following standard usage of ISHAPE,
_iraf/daophot_ was used to do photometry on the *.drc images in each filter,
along with its routines _phot/pstselect/psf_ to produce empirical PSFs from 34
bright, uncrowded stars across the frame in both filters. ISHAPE was then run
on the entire list of measured objects in the CMD for NGC 1407, again in both
filters. The ISHAPE parameter FITRAD was assumed to be 5 px, or typically 10
times larger than the GC half-light radii. A test run with FITRAD = 7 px was
also made, which produced no systematic differences compared with the 5-px
run.

The output parameters from ISHAPE include not just the fwhm and ellipticity of
the King-model profile, but also the total fluxes, which can then be converted
to total magnitudes and colors, and finally compared directly with the DOLPHOT
results. These comparisons are shown in Figure 15. The upper panel shows the
difference in F814W magnitude between the two codes (ISHAPE $-$ DOLPHOT). Not
surprisingly, there is a nonzero offset in the sense that DOLPHOT measures
these marginally resolved GCs slightly brighter than does ISHAPE.222The
distribution of points along short diagonal lines in the figure is the result
of some quantization of the fluxes within ISHAPE. The mean difference is
$\langle\Delta(F814W)\rangle=+0.077\pm 0.003$ with an rms scatter of 0.087
mag.

The much more important test is in the comparison of color indices, shown in
the lower panel of Fig. 15. Here, no systematic offset is seen; the mean is
$(-0.004\pm 0.004)$ with rms scatter of 0.14 mag. Clearly, net magnitude
offsets are present in both filters, but they are nearly equal and yield color
indices that are closely similar in both codes. These tests support the use of
the measured color indices from DOLPHOT for derivation of the CDF and MDF.

Figure 15: Comparison of photometry from DOLPHOT and ISHAPE, for the GCs in
NGC 1407, the closest galaxy in the list. _Upper panel:_ Difference between
the F814W magnitudes measured by the two codes, in the sense (ISHAPE $-$
DOLPHOT). _Lower panel:_ Difference between the two color indices
(F435W-F814W). Figure 16: Predicted HST/ACS color indices calculated from the
Parsec SSP models, plotted versus metallicity [m/H]. Each dot shows one
synthetic model GC for $10^{5}M_{\odot}$ and a 12 Gy age. In each panel the
solid line shows the best-fit quadratic interpolation relation given in the
upper half of Table 6.

3100

Figure 17: Color index $(F475W-F850LP)_{0}$ versus metallicity, from six
recent literature sources. The conversion relation adopted here is shown as
the heavy solid line as given in Eq. 5. For comparison, the black dotted line
indicates the predicted relation from the Parsec SSP models without
adjustment.

###  4.2 Notes on Individual Galaxies

_NGC 1129:_ The galaxy is located near the upper right corner of the ACS field
at (3600,3160) px. Its GC system is relatively compact and a radial falloff of
the GC numbers is clearly seen, but numerous objects with GC-like colors and
magnitudes also populate the lower half of the field ($y<2000$ px) that appear
to be contamination from companion galaxies or possibly an intergalactic
population. For the present study, only the data with $y>2000$ px are used.
This galaxy needs additional investigation to clarify the contamination issue
as well as the puzzling result for the transformation of color to metallicity
(see discussion below).

_NGC 1272:_ A color-magnitude array for the GC population was previously
measured by Penny et al. (2012).

_NGC 1275:_ This active galaxy has a large population of young star clusters
in its inner regions that has attracted much previous interest. For HST-based
photometry of the young cluster population, see, for example, Holtzman et al.
(1992); Carlson et al. (1998); Canning et al. (2014); Lim et al. (2020, 2022).

_IC 4051:_ GC photometry in this Coma giant ETG drawn from HST/WFPC2 imaging
was presented in Baum et al. (1997) and Woodworth & Harris (2000). This galaxy
and its GC system have an unusually compact spatial central concentration.

_NGC 4874, NGC 4889:_ These are the two centrally dominant giants in the Coma
cluster. Previous HST photometry for their GC systems was presented in Harris
et al. (2009) with WFPC2 data, and in Peng et al. (2011); Harris et al.
(2017b) with the same ACS data used in the present study.

_NGC 7720:_ This galaxy has a double core, with two near-equal nuclei
separated by just $12^{\prime\prime}$ (7.5 kpc); see Laine et al. (2003).
Somewhat of necessity, the GC population is treated here as if it belongs to
one combined galaxy.

##  5 Transformations to Metallicity

The major results of this study consist of the photometric database and the
resulting CDFs, as laid out in the previous sections. Following on from that,
a brief discussion at the metallicity distribution functions (MDFs) is given.
For reasons to be discussed below, the MDFs and their resulting parametric
descriptions, at this stage, should be viewed as only a first look.

Because the CMDs for the program galaxies are in several different filter
pairs, the CDFs and their bimodal-Gaussian fit parameters cannot be directly
intercompared. A necessary first step could be to transform them all into a
common filter system, for example through the use of SSP models or published
empirical transformations between the various filters. Though observationally
based transformations would normally be preferred, two practical issues that
make such a step uncertain are that (a) published empirical conversions
(Sirianni et al., 2005) were derived for single stars and not the combined
light of star clusters; and (b) there are no calibrating galaxies available
whose GC systems have been measured in all the five different indices in this
study; hardly any that have even been measured in more than two filters.

However, a bigger issue is that the CDFs are not what we ultimately want. What
is really wanted are the MDFs that the CDFs represent, and constructing MDFs
requires confronting the well known problem of transforming color to
metallicity. All contemporary SSP models show that these transformations are
nonlinear to various degrees in optical/NIR bands (e.g. Yoon et al., 2006,
2011; Peng et al., 2006; Cantiello & Blakeslee, 2007; Cantiello et al., 2014;
Usher et al., 2012). The physical cause of the nonlinear dependence of the
color-to-metallicity relation (CMR) is simply that the integrated color of a
GC is dominated by the light from its giant and subgiant stars, and the color
of the red-giant branch becomes more sensitive to metallicity as [Fe/H]
increases. That is, the isochrone lines shift redward progressively faster
with increasing metallicity.

Theoretical modelling exercises (e.g. Yoon et al., 2006, 2011; Cantiello &
Blakeslee, 2007; Cantiello et al., 2014) show that with appropriately
constructed (very) nonlinear CMRs, a bimodal CDF can result from a flat or
unimodal MDF. (Contrarily, an intrinsically bimodal MDF can be converted to a
flat or unimodal CDF with a suitably nonlinear transformation.) Notably,
however, MDFs for the GC systems in several nearby large galaxies have been
constructed directly from spectroscopic samples (Peng et al., 2006; Woodley et
al., 2010; Alves-Brito et al., 2011; Usher et al., 2012; Brodie et al., 2012;
Harris et al., 2016; Caldwell & Romanowsky, 2016; Villaume et al., 2019;
Fahrion et al., 2020). These spectroscopic MDFs bypass the CMR issue, and
importantly, they usually show bimodality.

Among the five color indices used here, one of them stands out above the
others because several studies exist that connect it directly, and
empirically, with metallicity. This is (F475W-F850LP) (equivalent to $(g-z)$
in the AB magnitude system). For this index, CMRs built from spectroscopic
metallicities in either [Fe/H] or [m/H] have been published by Peng et al.
(2006); Sinnott et al. (2010); Blakeslee et al. (2010); Usher et al. (2012);
Vanderbeke et al. (2014); Harris et al. (2017b); Villaume et al. (2019);
Fahrion et al. (2020). In some of these studies a two-part piecewise linear
correlation is assumed, and in other cases a continuous quadratic or other
nonlinear shape is assumed, but no single definitive version can be said to
exist as yet. All these relations are limited by the observational scatter in
the measured color indices and spectra, as well as by the relatively small
samples of GCs being used, which may include only some dozens of clusters.
These issues of scatter and sample size have led to simple linear or
piecewise-linear CMRs being used in the past for many broadband color indices
(e.g. Barmby et al., 2000; Harris et al., 2006; Peng et al., 2006; Vanderbeke
et al., 2014; Fahrion et al., 2020).

To go a step further, the approach used here is to define a hybrid combination
of SSP models and empirical data. First, a baseline CMR between the fiducial
color index (F475W-F850LP)0 and metallicity [Fe/H] is defined; and then
second, SSP models are used to convert every other measured color index into
(F475W-F850LP)0 and thus into metallicity.

For the purposes of this preliminary investigation, the widely used
ParsecV1.2S stellar models (Marigo et al., 2017) are adopted here333accessible
at http://stev.oapd.inaf.it/cgi-bin/cmd. These SSPs conveniently allow model
GCs to be produced at any given metallicity, age, and cluster mass. The input
parameters adopted include the OBC bolometric corrections and a Kroupa
canonical two-part powerlaw IMF. Each model GC was computed for a total mass
of $10^{5}M_{\odot}$, no internal spread in metallicity, and an age of 12 Gyr
(but see below for the effects of age differences). The total luminosity of
each model GC was calculated in the standard set of HST ACS broadband filters,
from which all the color indices used in this study could be constructed. The
individual model GCs covered the full available range of metallicities from
[m/H] $=-2.2$ to +0.4 in steps of 0.1 dex. The resulting plots of the
predicted color indices versus [m/H], without any adjustment (see below), are
shown in Figure 16. The interpolation curves shown there are simple quadratic
functions, with coefficients and fitting uncertainties as listed in the upper
half of Table 6. The residual scatter of points around the best-fit lines is
the result of stochastic differences in populating each synthetic cluster with
a finite number of stars. Although each cluster contains $\sim 10^{5}$ stars,
the total light is dominated by the red giants, and only some dozens of those
are present in each model cluster. The curvature of each line shows explicitly
how the sensitivity of color to metallicity increases steadily with increasing
metallicity.

These models are next tied to spectroscopically based transformations of
(F475W-F850LP) to metallicity [Fe/H]. The relations from six recent studies as
mentioned above are shown in Figure 17, including Peng et al. (2006); Sinnott
et al. (2010); Usher et al. (2012); Harris et al. (2017b); Villaume et al.
(2019), and Fahrion et al. (2020). Where necessary, $(g-z)$ colors in the AB
magnitude system have been converted to the Vegamag scale with
$(F475W-F850LP)=(g-z)+0.636$ (Sirianni et al., 2005). Although all these
studies show rather similar conversions, some use [m/H] for metallicity (Usher
et al., 2012; Fahrion et al., 2020) and the others use [Fe/H] (Peng et al.,
2006; Sinnott et al., 2010; Harris et al., 2017b; Villaume et al., 2019). In
this study, conversions will be made to [Fe/H].

As noted above, the observational samples of GCs are small enough and show
enough cluster-to-cluster scatter that a definitive transformation is
difficult to define. For the purposes of the present study, the Parsec SSPs
are used to define the _shape_ of the CMR, but the observational data are used
to fix the _zeropoint_. The SSPs themselves give scaled-Solar [m/H]
abundances. The adopted result is shown in Fig. 17 as the heavy solid line and
is given by the quadratic curve

$(F475W-F850LP)_{0}=(2.158\pm 0.011)+\\\ (0.5708\pm 0.0263)[Fe/H]+(0.1003\pm
0.0128)[Fe/H]^{2}\,.$ (5)

Notably, however, the raw conversion relation from the SSPs without any
adjustment already falls within the spread of the observational relations and,
in the end, was shifted by only $-0.12$ dex in metallicity to yield the
transformation in Eq. 5. Given the spread of the observational calibrations,
to within $\sim\pm 0.1$ dex the final placement of this curve remains a matter
of judgment, and the final choice was simply to fit the ensemble of
calibrations that use [Fe/H] rather than [m/H].

The second necessary step is the conversion of any of the other color indices
into (F475W-F850LP). The curves shown in Fig. 16 are used for this purpose.
The ratio of any pair of indices is nearly constant with metallicity, and
these ratios are listed in the last column of Table 6 (upper half). That is,
the (F475W-F850LP) index is multiplied by the ratio listed in the table to
give each of the other indices (for example, (F475W-F814W) = 0.884
(F475W-F850LP)). Combining these ratios with Eq. 5 gives the final set of
conversions between each color index and [Fe/H] listed in the lower half of
Table 6.444Again, the coefficients listed in the upper half of the table give
the color index as a function of [m/H] directly from the SSP models. The
coefficients in the lower half of the table give color index as a function of
[Fe/H] after the empirical calibration is done. To convert color into
metallicity, these quadratic relations are numerically inverted.

The procedure for deriving metallicities used here bears some resemblance to
the multiparameter fitting process of de Meulenaer et al. (2017) for the star
clusters in M31. They employ the same Marigo et al. (2017) SSP models to
derive age, mass, metallicity, and foreground extinction, all of which span
wide ranges for the M31 cluster population as a whole. They find that the
integrated photometric colors from the ACS and WFC3 cameras yield
metallicities and ages in reasonable agreement with direct spectroscopy,
though residual nonlinear trends remain that resemble those found here. The
photometry for the giant ETGs studied here represents a much simpler problem,
since no young clusters or differential extinction are present.

The hybrid procedure used here to define the CMR is clearly far from ideal. In
particular, the transformations of the different color indices into the
fiducial index (F475W-F850LP) rely heavily on the SSP models, as does the
adopted shape for the final CMR. A significant and very helpful step would be
to measure the GCs in the entire set of color indices for at least one of
these giant ETGs and preferably a selected set of them. Reliance on the models
would be very much reduced as a result.

A related and more fundamental issue is in the nature of the CMR itself. The
implicit assumption made here, and in most other studies, is that GCs are
similar enough objects in different galaxies that a single CMR can be applied
across all of them. The possibility that GCs in different galaxies might
follow different CMRs is discussed by, e.g., Usher et al. (2012); Powalka et
al. (2016) and Villaume et al. (2019). Although the galaxies studied here are
all of similar type (giant ETGs), perhaps the most obvious single concern is
the expected range of GC ages, which will differ from one galaxy to another
depending on their specific history of hierarchical growth. In the present
discussion, the blanket assumption of a 12-Gyr age is made, but a test of the
sensitivity of the predicted MDF to the assumed GC age is shown in the next
section.

Nevertheless, some useful consistency checks of the procedure can be done
through the available integrated photometry for the Milky Way GCs. One example
is shown in Figure 18. Bellini et al. (2015) provide integrated, dereddened
colors for a subset of the Milky Way GCs measured directly in the ACS F606W,
F814W filters. These colors can then be plotted versus their [Fe/H] values
(Harris, 1996, 2010 edition). The predicted relation as given in Table 6
matches the data well within the scatter.

A less direct but useful check can be made from the $BVI$ integrated colors of
the Milky Way GCs listed in Harris (1996). From the conversion equations given
in Sirianni et al. (2005), their dereddened $(B-I)_{0}$ values were
transformed into the HST/ACS color indices (F475W-F814W) and (F435W-F814W);
$(B-V)_{0}$ was transformed into (F435W-F606W); and $(V-I)_{0}$ was
transformed into (F555W-F814W). These four indices are plotted versus [Fe/H]
in Figure 19. The adopted transformations in Table 6 are drawn in for
comparison. Except for (F435W-F606W), where the Milky Way GC data fall below
the line by $\sim 0.1$ mag, the conversion lines and the datapoints agree
reasonably well, even considering the fact that the Sirianni conversion
equations are defined for single stars rather than composite stellar systems.

Table 5: Bimodal Gaussian Parameters for the CDFs Galaxy | $N$ | $\mu_{1}$ | $\mu_{2}$ | $\sigma_{1}$ | $\sigma_{2}$ | $f_{1}$ | DD | $H_{12}$
---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
$(F475W-F814W)_{0}$ |
NGC 1275 | 2704 | 1.336 (0.009) | 1.621 (0.018) | 0.102 (0.010) | 0.204 (0.008) | 0.311 (0.049) | 1.76 (0.11) | -0.10
NGC 1278 | 1107 | 1.368 (0.009) | 1.606 (0.021) | 0.084 (0.008) | 0.199 (0.008) | 0.390 (0.058) | 1.55 (0.14) | 0.51
NGC 4874 | 4632 | 1.330 (0.005) | 1.607 (0.011) | 0.082 (0.005) | 0.192 (0.005) | 0.305 (0.029) | 1.77 (0.09) | 0.03
NGC 4889 | 3054 | 1.333 (0.005) | 1.619 (0.013) | 0.086 (0.006) | 0.198 (0.007) | 0.327 (0.034) | 1.87 (0.11) | 0.12
NGC 6166 | 3162 | 1.363 (0.010) | 1.673 (0.023) | 0.098 (0.009) | 0.179 (0.011) | 0.368 (0.054) | 2.15 (0.16) | 0.06
NGC 7720 | 4242 | 1.366 (0.008) | 1.684 (0.015) | 0.135 (0.005) | 0.198 (0.006) | 0.439 (0.034) | 1.88 (0.09) | 0.15
UGC 9799 | 4094 | 1.318 (0.005) | 1.623 (0.013) | 0.106 (0.007) | 0.201 (0.006) | 0.332 (0.033) | 1.89 (0.09) | -0.06
UGC 10143 | 3623 | 1.385 (0.013) | 1.617 (0.035) | 0.121 (0.013) | 0.212 (0.009) | 0.443 (0.085) | 1.34 (0.17) | 0.39
ESO325-G004 | 941 | 1.399 (0.014) | 1.663 (0.022) | 0.075 (0.014) | 0.219 (0.008) | 0.242 (0.059) | 1.61 (0.13) | -0.07
ESO383-G076 | 2906 | 1.312 (0.005) | 1.575 (0.008) | 0.074 (0.006) | 0.192 (0.004) | 0.254 (0.023) | 1.81 (0.07) | -0.12
ESO444-G046 | 7121 | 1.331 (0.007) | 1.626 (0.008) | 0.117 (0.004) | 0.211 (0.002) | 0.386 (0.024) | 1.73 (0.04) | 0.13
ESO509-G008 | 995 | 1.292 (0.008) | 1.582 (0.024) | 0.101 (0.008) | 0.205 (0.010) | 0.497 (0.054) | 1.79 (0.19) | 1.01
IC 4051 | 975 | 1.338 (0.012) | 1.626 (0.019) | 0.103 (0.007) | 0.189 (0.010) | 0.343 (0.049) | 1.90 (0.18) | -0.04
$(F435W-F814W)_{0}$ |
NGC 1407 | 1029 | 1.605 (0.014) | 2.097 (0.011) | 0.131 (0.010) | 0.170 (0.008) | 0.353 (0.025) | 3.24 (0.12) | -0.29
NGC 3258 | 1880 | 1.515 (0.006) | 1.941 (0.017) | 0.114 (0.004) | 0.224 (0.011) | 0.438 (0.027) | 2.39 (0.15) | 0.53
NGC 3268 | 1568 | 1.517 (0.009) | 1.904 (0.031) | 0.098 (0.013) | 0.240 (0.019) | 0.307 (0.057) | 2.11 (0.26) | 0.09
NGC 3348 | 1056 | 1.578 (0.140) | 2.098 (0.015) | 0.140 (0.011) | 0.197 )0.008) | 0.341 (0.028) | 3.04 (0.11) | -0.27
NGC 4696 | 2170 | 1.611 (0.009) | 2.058 (0.026) | 0.127 (0.009) | 0.205 (0.017) | 0.439 (0.048) | 2.67 (0.21) | 0.26
NGC 5322 | 589 | 1.609 (0.018) | 2.030 (0.058) | 0.150 (0.037) | 0.215 (0.034) | 0.399 (0.118) | 2.27 (0.38) | -0.05
NGC 5557 | 699 | 1.581 (0.015) | 2.078 (0.017) | 0.143 (0.011) | 0.196 (0.014) | 0.424 (0.035) | 2.90 (0.15) | 0.01
NGC 7049 | 356 | 1.669 (0.034) | 2.165 (0.059) | 0.159 (0.029) | 0.180 (0.033) | 0.521 (0.097) | 2.92 (0.36) | 0.23
NGC 7626 | 1611 | 1.593 (0.012) | 2.062 (0.015) | 0.129 (0.007) | 0.185 (0.009) | 0.426 (0.025) | 2.95 (0.11) | 0.06
$(F475W-F850LP)_{0}$ |
NGC 1132 | 1972 | 1.513 (0.009) | 1.931 (0.021) | 0.136 (0.008) | 0.182 (0.010) | 0.570 (0.037) | 2.61 (0.15) | 0.77
ESO306-G017 | 2591 | 1.573 (0.014) | 1.962 (0.021) | 0.142 (0.008) | 0.172 (0.010) | 0.512 (0.044) | 2.47 (0.12) | 0.27
$(F555W-F814W)_{0}$ |
NGC 1272 | 2201 | 0.991 (0.007) | 1.215 (0.008) | 0.079 (0.005) | 0.165 (0.004) | 0.391 (0.030) | 1.73 (0.07) | 0.34
$(F435W-F606W)_{0}$ |
NGC 1129 | 1035 | 1.021 (0.024) | 1.315 (0.034) | 0.122 (0.013) | 0.187 (0.013) | 0.374 (0.092) | 1.86 (0.17) | -0.08
Figure 18: Color versus metallicity [Fe/H] for the ACS color index $(F606W-F814W)$. The intrinsic, dereddened colors for Milky Way globular clusters measured by Bellini et al. (2015) are shown as the solid dots. The _solid line_ is the adopted relation between [Fe/H] and $(F606W-F814W)_{0}$ as given in Table 6.  Figure 19: Color versus metallicity [Fe/H] for the additional color indices used in the present study. In each case the _solid line_ shows the final relation between [Fe/H] and the given color index as listed in Table 6. Data points (solid dots) are dereddened color indices for Milky Way GCs, transformed from $BVI$ into the Vegamag HST/ACS filter system. Table 6: Color/Metallicity Transformations for the ACS/WFC filters Color Index | $a_{0}$ | $a_{1}$ | $a_{2}$ | Conversion Ratio
---|---|---|---|---
Predicted Transformations From SSP Models: color
$=a_{0}+a_{1}[m/H]+a_{2}[m/H]^{2}$
(F475W-F850LP) | 2.091 (0.011) | 0.5468 (0.0263) | 0.1003 (0.0128) | 1.000
(F475W-F814W) | 1.833 (0.009) | 0.4516 (0.0203) | 0.0780 (0.0099) | 0.884
(F435W-F814W) | 2.262 (0.010) | 0.5775 (0.0230) | 0.0947 (0.0112) | 1.077
(F435W-F606W) | 1.308 (0.004) | 0.3261 (0.0105) | 0.0436 (0.0051) | 0.617
(F555W-F814W) | 1.300 (0.007) | 0.3283 (0.0174) | 0.0620 (0.0085) | 0.629
(F606W-F814W) | 0.955 (0.006) | 0.2514 (0.0148) | 0.0511 (0.0072) | 0.461
Final Calibrated Transformations: color $=a_{0}+a_{1}[Fe/H]+a_{2}[Fe/H]^{2}$
(F475W-F850LP) | 2.158 (0.011) | 0.5708 (0.0263) | 0.1003 (0.0128) |
(F475W-F814W) | 1.908 (0.009) | 0.5050 (0.0203) | 0.0890 (0.0099) |
(F435W-F814W) | 2.324 (0.010) | 0.6148 (0.0230) | 0.1080 (0.0112) |
(F435W-F606W) | 1.331 (0.004) | 0.3520 (0.0105) | 0.0618 (0.0051) |
(F555W-F814W) | 1.357 (0.007) | 0.3589 (0.0174) | 0.0630 (0.0085) |
(F606W-F814W) | 0.994 (0.006) | 0.2629 (0.0148) | 0.0462 (0.0072) |
Table 7: Metallicity Gradient Parameters Galaxy | Intercept $a$ | Slope $b$ |  $(R/R_{e})$ Range
---|---|---|---
(1) | (2) | (3) | (4)
NGC1275 |  $-0.816$ (0.025) |  $-0.215$ (0.041) | $0.36-8.73$
NGC1278 |  $-0.924$ (0.023) |  $-0.298$ (0.054) | $0.20-6.30$
NGC4874 |  $-0.942$ (0.011) |  $-0.273$ (0.039) | $0.30-3.85$
NGC4889 |  $-0.879$ (0.016) |  $-0.348$ (0.041) | $0.40-5.89$
NGC6166 |  $-0.882$ (0.011) |  $-0.402$ (0.044) | $0.32-3.44$
NGC7720 |  $-0.822$ (0.012) |  $-0.319$ (0.034) | $0.32-6.15$
IC4051 |  $-0.520$ (0.041) |  $-0.657$ (0.057) | $1.32-22.5$
UGC9799 |  $-0.766$ (0.018) |  $-0.486$ (0.037) | $0.53-7.55$
UGC10143 |  $-0.939$ (0.012) |  $-0.251$ (0.033) | $0.15-5.62$
ESO325-G004 |  $-0.651$ (0.031) |  $-0.260$ (0.060) | $0.40-10.2$
ESO383-G076 |  $-0.956$ (0.022) |  $-0.143$ (0.045) | $0.60-8.95$
ESO444-G046 |  $-0.951$ (0.015) |  $-0.075$ (0.027) | $0.52-10.7$
ESO509-G008 |  $-0.987$ (0.062) |  $-0.352$ (0.093) | $1.51-14.1$
NGC1407 |  $-0.778$ (0.021) |  $-0.303$ (0.078) | $0.21-2.96$
NGC3258 |  $-1.088$ (0.024) |  $-0.280$ (0.063) | $0.23-4.97$
NGC3268 |  $-1.072$ (0.026) |  $-0.188$ (0.084) | $0.41-4.27$
NGC3348 |  $-0.786$ (0.030) |  $-0.033$ (0.082) | $0.55-8.09$
NGC4696 |  $-0.913$ (0.016) |  $-0.266$ (0.050) | $0.13-4.41$
NGC5322 |  $-0.910$ (0.033) |  $-0.117$ (0.099) | $0.46-6.21$
NGC5557 |  $-0.838$ (0.039) |  $-0.357$ (0.111) | $0.67-6.73$
NGC7049 |  $-0.697$ (0.070) |  $-0.402$ (0.072) | $0.38-3.99$
NGC7626 |  $-0.891$ (0.019) |  $-0.421$ (0.123) | $0.32-2.74$
NGC1132 |  $-0.971$ (0.019) |  $-0.344$ (0.062) | $0.31-4.59$
ESO306-G017 |  $-0.736$ (0.021) |  $-0.404$ (0.054) | $0.62-6.80$
NGC1129 |  $-0.474$ (0.030) |  $+0.005$ (0.095) | $0.2-2.5$
NGC1272 |  $-0.786$ (0.014) |  $-0.638$ (0.057) | $0.26-3.21$

Table 8: Bimodal Gaussian Parameters for the MDFs Galaxy | $N$ | $\mu_{1}$ | $\mu_{2}$ | $\sigma_{1}$ | $\sigma_{2}$ | $f_{1}$ | DD | $H_{12}$ | [Fe/H]mid
---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
NGC 1275 | 2597 | -1.305 (0.052) | -0.407 (0.035) | 0.506 (0.022) | 0.355 (0.015) | 0.599 (0.047) | 2.05 (0.11) | 0.05 | -0.77
NGC 1278 | 1083 | -1.262 (0.044) | -0.365 (0.051) | 0.451 (0.026) | 0.314 (0.026) | 0.722 (0.052) | 2.31 (0.14) | 0.81 | -0.63
NGC 4874 | 4512 | -1.408 (0.028) | -0.467 (0.020) | 0.446 (0.014) | 0.345 (0.010) | 0.546 (0.025) | 2.36 (0.08) | -0.07 | -0.89
NGC 4889 | 2957 | -1.408 (0.034) | -0.442 (0.026) | 0.434 (0.019) | 0.350 (0.013) | 0.546 (0.030) | 2.45 (0.09) | -0.03 | -0.88
NGC 6166 | 3110 | -1.209 (0.034) | -0.311 (0.021) | 0.505 (0.017) | 0.297 (0.012) | 0.651 (0.030) | 2.17 (0.09) | 0.10 | -0.63
NGC 7720 | 4031 | -1.178 (0.027) | -0.296 (0.026) | 0.549 (0.012) | 0.345 (0.012) | 0.677 (0.027) | 1.92 (0.07) | 0.32 | -0.57
UGC 9799 | 3881 | -1.426 (0.035) | -0.459 (0.025) | 0.473 (0.017) | 0.375 (0.013) | 0.518 (0.029) | 2.27 (0.08) | -0.15 | -0.91
UGC 10143 | 3124 | -1.156 (0.048) | -0.323 (0.066) | 0.518 (0.019) | 0.346 (0.026) | 0.792 (0.059) | 1.89 (0.11) | 1.54 | -0.43
ESO325-G004 | 926 | -1.095 (0.073) | -0.209 (0.068) | 0.436 (0.037) | 0.318 (0.027) | 0.625 (0.076) | 2.33 (0.16) | 0.22 | -0.55
ESO383-G076 | 2806 | -1.510 (0.040) | -0.575 (0.027) | 0.419 (0.020) | 0.368 (0.015) | 0.472 (0.033) | 2.37 (0.10) | -0.21 | -1.05
ESO444-G046 | 6703 | -1.447 (0.051) | -0.496 (0.044) | 0.479 (0.019) | 0.422 (0.017) | 0.514 (0.048) | 2.11 (0.06) | -0.07 | -0.96
ESO509-G008 | 899 | -1.590 (0.053) | -0.543 (0.063) | 0.448 (0.027) | 0.374 (0.033) | 0.634 (0.052) | 2.54 (0.14) | 0.45 | -0.96
IC 4051 | 928 | -1.323 (0.058) | -0.438 (0.042) | 0.460 (0.030) | 0.332 (0.019) | 0.573 (0.053) | 2.21 (0.17) | -0.03 | -0.81
NGC 1407 | 979 | -1.431 (0.066) | -0.357 (0.019) | 0.505 (0.031) | 0.278 (0.013) | 0.373 (0.029) | 2.63 (0.22) | -0.67 | -0.87
NGC 3258 | 1633 | -1.791 (0.031) | -0.622 (0.027) | 0.401 (0.015) | 0.376 (0.017) | 0.461 (0.023) | 3.01 (0.10) | -0.20 | -1.22
NGC 3268 | 1418 | -1.790 (0.042) | -0.645 (0.029) | 0.388 (0.020) | 0.372 (0.018) | 0.406 (0.028) | 3.01 (0.10) | -0.34 | -1.26
NGC 3348 | 985 | -1.536 (0.084) | -0.362 (0.035) | 0.465 (0.042) | 0.326 (0.021) | 0.370 (0.044) | 2.92 (0.26) | -0.59 | -0.96
NGC 4696 | 2046 | -1.465 (0.027) | -0.399 (0.018) | 0.435 (0.017) | 0.312 (0.012) | 0.514 (0.020) | 2.81 (0.10) | -0.24 | -0.88
NGC 5322 | 547 | -1.474 (0.073) | -0.462 (0.046) | 0.408 (0.041) | 0.328 (0.027) | 0.457 (0.057) | 2.74 (0.23) | -0.32 | -0.96
NGC 5557 | 649 | -1.526 (0.098) | -0.379 (0.042) | 0.496 (0.050) | 0.293 (0.037) | 0.485 (0.060) | 2.81 (0.30) | -0.44 | -0.88
NGC 7049 | 330 | -1.311 (0.135) | -0.259 (0.103) | 0.420 (0.070) | 0.270 (0.059) | 0.555 (0.102) | 2.94 (0.30) | -0.20 | -0.69
NGC 7626 | 1510 | -1.472 (0.042) | -0.390 (0.023) | 0.492 (0.021) | 0.286 (0.014) | 0.514 (0.027) | 2.69 (0.13) | -0.38 | -0.85
NGC 1132 | 1852 | -1.418 (0.036) | -0.357 (0.031) | 0.475 (0.022) | 0.314 (0.015) | 0.637 (0.031) | 2.64 (0.10) | 0.16 | -0.76
ESO306-G017 | 2521 | -1.162 (0.027) | -0.274 (0.024) | 0.532 (0.012) | 0.278 (0.013) | 0.674 (0.028) | 2.09 (0.07) | 0.08 | -0.56
NGC 1272 | 2151 | -1.200 (0.069) | -0.275 (0.068) | 0.499 (0.028) | 0.432 (0.027) | 0.588 (0.069) | 1.98 (0.11) | 0.24 | -0.65
NGC 1129 | 997 | -0.742 (0.068) | 0.167 (0.059) | 0.614 (0.025) | 0.373 (0.028) | 0.651 (0.062) | 1.89 (0.14) | 0.13 | -0.14
Mean |  | -1.377 (0.044) | -0.404 (0.025) | 0.465 (0.009) | 0.336 (0.008) | 0.556 (0.021) | 2.45 (0.07) | 0.00 (0.09) | -0.823 (0.040)

Table 9: Additional Global Parameters for the MDFs Galaxy |  $\langle$[Fe/H]$\rangle$ |  $\sigma$[Fe/H] | Median | Skewness | Kurtosis | IQR | IDR
---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
NGC 1275 | -0.945 (0.012) | 0.631 (0.009) | -0.908 (0.020) | -0.200 (0.048) | -0.610 (0.096) | 0.956 (0.019) | 1.649 (0.024)
NGC 1278 | -1.013 (0.018) | 0.580 (0.012) | -1.046 (0.030) | 0.038 (0.074) | -0.577 (0.149) | 0.873 (0.028) | 1.513 (0.042)
NGC 4874 | -0.981 (0.009) | 0.618 (0.007) | -0.949 (0.015) | -0.155 (0.036) | -0.703 (0.073) | 0.959 (0.013) | 1.616 (0.020)
NGC 4889 | -0.970 (0.011) | 0.625 (0.008) | -0.952 (0.022) | -0.107 (0.045) | -0.758 (0.090) | 0.981 (0.017) | 1.652 (0.024)
NGC 6166 | -0.896 (0.011) | 0.616 (0.008) | -0.856 (0.018) | -0.224 (0.044) | -0.672 (0.088) | 0.968 (0.017) | 1.611 (0.024)
NGC 7720 | -0.893 (0.010) | 0.642 (0.007) | -0.873 (0.016) | -0.206 (0.039) | -0.558 (0.077) | 0.961 (0.015) | 1.677 (0.026)
UGC9799 | -0.961 (0.010) | 0.646 (0.007) | -0.919 (0.014) | -0.192 (0.039) | -0.648 (0.079) | 0.994 (0.015) | 1.667 (0.019)
UGC10143 | -0.983 (0.011) | 0.593 (0.008) | -0.991 (0.013) | -0.033 (0.044) | -0.442 (0.088) | 0.849 (0.017) | 1.571 (0.026)
ESO325-G004 | -0.763 (0.019) | 0.584 (0.014) | -0.757 (0.025) | -0.075 (0.080) | -0.725 (0.161) | 0.926 (0.030) | 1.524 (0.044)
ESO383-G076 | -1.016 (0.012) | 0.610 (0.008) | -0.967 (0.014) | -0.174 (0.046) | -0.675 (0.092) | 0.949 (0.020) | 1.597 (0.025)
ESO444-G046 | -0.985 (0.008) | 0.656 (0.006) | -0.958 (0.014) | -0.108 (0.030) | -0.647 (0.060) | 0.976 (0.013) | 1.741 (0.016)
ESO509-G008 | -1.207 (0.022) | 0.659 (0.016) | -1.246 (0.033) | 0.102 (0.082) | -0.761 (0.163) | 1.023 (0.033) | 1.740 (0.043)
IC 4051 | -0.946 (0.020) | 0.600 (0.014) | -0.926 (0.038) | -0.192 (0.080) | -0.649 (0.160) | 0.913 (0.027) | 1.577 (0.048)
NGC 1407 | -0.758 (0.021) | 0.643 (0.015) | -0.555 (0.019) | -0.772 (0.078) | -0.308 (0.156) | 0.926 (0.040) | 1.669 (0.041)
NGC 3258 | -1.161 (0.017) | 0.700 (0.012) | -1.083 (0.037) | -0.133 (0.061) | -0.988 (0.121) | 1.151 (0.020) | 1.845 (0.038)
NGC 3268 | -1.111 (0.018) | 0.678 (0.013) | -1.012 (0.030) | -0.246 (0.065) | -0.925 (0.130) | 1.129 (0.023) | 1.796 (0.039)
NGC 3348 | -0.796 (0.022) | 0.685 (0.015) | -0.612 (0.031) | -0.583 (0.078) | -0.655 (0.156) | 1.030 (0.056) | 1.841 (0.039)
NGC 4696 | -0.947 (0.014) | 0.655 (0.010) | -0.879 (0.031) | -0.228 (0.054) | -0.893 (0.108) | 1.080 (0.021) | 1.667 (0.025)
NGC 5322 | -0.925 (0.027) | 0.624 (0.019) | -0.851 (0.052) | -0.272 (0.104) | -0.811 (0.209) | 1.003 (0.038) | 1.643 (0.043)
NGC 5557 | -0.936 (0.028) | 0.702 (0.019) | -0.787 (0.046) | -0.411 (0.096) | -0.904 (0.192) | 1.165 (0.041) | 1.817 (0.050)
NGC 7049 | -0.842 (0.035) | 0.639 (0.025) | -0.801 (0.052) | -0.189 (0.134) | -1.148 (0.268) | 1.110 (0.042) | 1.700 (0.056)
NGC 7626 | -0.946 (0.017) | 0.676 (0.012) | -0.814 (0.033) | -0.380 (0.063) | -0.892 (0.126) | 1.100 (0.025) | 1.796 (0.026)
NGC 1132 | -1.033 (0.015) | 0.664 (0.011) | -1.042 (0.029) | -0.061 (0.057) | -0.852 (0.114) | 1.061 (0.020) | 1.746 (0.029)
ESO306-G017 | -0.872 (0.012) | 0.624 (0.009) | -0.836 (0.021) | -0.293 (0.049) | -0.672 (0.097) | 0.966 (0.019) | 1.639 (0.033)
NGC 1272 | -0.819 (0.014) | 0.656 (0.010) | -0.812 (0.017) | -0.029 (0.053) | -0.504 (0.106) | 0.977 (0.019) | 1.698 (0.030)
NGC 1129 | -0.457 (0.022) | 0.710 (0.016) | -0.414 (0.032) | -0.284 (0.077) | -0.530 (0.155) | 1.047 (0.035) | 1.866 (0.052)
Mean | -0.971 (0.033) | 0.641 (0.010) | -0.913 (0.049) | -0.226 (0.067) | -0.697 (0.057) | 1.000 (0.029) | 1.678 (0.027)
Figure 20: Metallicity distributions (MDFs) for the 26 galaxies in this study,
displayed in order of increasing luminosity, left to right along each row and
then down to the next row. Each panel shows the MDF transformed from the
corresponding CDF shown in the previous figures. In each case the best-fit
bimodal Gaussian curve is shown as the solid line along with its blue and red
components (dashed lines). Figure 21: _Upper panel:_ Color distribution
function (CDF) for the globular clusters in NGC 4874 within the selected box
shown in the CMD of Fig. 5. The data binned in 0.05-mag steps are plotted as
the histogram, while the best-fit bimodal Gaussian curve is superimposed as
the solid line. The blue and red Gaussian subcomponents are plotted as the
dashed lines. _Lower panel:_ Metallicity distribution function (MDF) after
transformation of the color indices to [Fe/H] as described in the text. The
bimodal-Gaussian fit and its two components are shown as well. Figure 22: Blue
(metal-poor) fraction of GCs from the MDFs plotted versus the same quantity
from the CDFs. Removing the nonlinearity in the color indices shifts the
proportions of blue and red clusters estimated by the Gaussian Mixture
Modelling fits, such that the blue component in the MDF is broader and more
populated than in the CDF.  Figure 23: The calculated MDF for NGC 4874, for
five different assumed ages of the GC population running from 9 to 13 Gy. All
the MDFs are generated from the same observed CDF in $(F475W-F814W)_{0}$. A
smaller assumed age requires a higher metallicity to generate the same
observed CDF.  Figure 24: Metallicity versus galactocentric radius for all 26
systems. The radius of each object is in units of the effective radius $R_{e}$
of the galaxy light profile. Solid curves show the least-squares fit to the
distributions as described in the text. Figure 25: The powerlaw exponent $b$
defining the GC metallicity gradient for each galaxy is plotted versus galaxy
luminosity. The sample mean at $\langle b\rangle=-0.3$ is drawn in as the
dashed line.  Figure 26: Parameters characterizing the MDFs, plotted versus
galaxy stellar mass. Successive panels show: (1) Mean metallicity, (2)
[Fe/H]mid, the metallicity at which the blue and red modes cross; (3) Median
metallicity; (4) Three measures of the global spread of the MDF including the
standard deviation $\sigma$ (blue points), InterQuartile Range (magenta), and
InterDecile Range (green); (5) the mean metallicities $\mu_{1},\mu_{2}$ of the
blue and red modes from the bimodal-Gaussian fit; (6) the standard deviations
$\sigma_{1},\sigma_{2}$ of the two modes; (7) Peak separation parameter DD;
(8) mode height ratio $H_{12}$; (9) skewness of the MDF; and (10) kurtosis of
the MDF.  Figure 27: The peak-separation parameter DD (upper panel) and blue-
GC fraction (lower panel) are plotted versus [Fe/H]mid, the metallicity at
which the blue and red mode Gaussian curves are equal.  Figure 28: Blue-GC
fraction $f_{1}=N(blue)/N(tot)$ estimated from the bimodal-Gaussian MDF
fitting, plotted versus galaxy stellar mass $M_{*}$. Here the samples are
restricted to the radial range $0.5-5R_{e}$ for each galaxy to facilitate
comparison. Figure 29: Red-GC fraction $f_{2}=N(red)/N(tot)$ plotted versus
galaxy halo mass as described in the text. The giant ETGs studied here are
plotted as large blue circles, while the other symbols are galaxies from
Virgo, Fornax, and the Local Group that populate the lower-mass range of the
diagram. The solid line shows the trend of the running weighted mean,
calculated with nine objects per bin.  Figure 30: _Upper panel:_ Median [Fe/H]
for the GC populations. plotted versus halo mass $M_{h}$. Symbols are as in
the previous figure. _Lower panel:_ The standard deviation $\sigma_{[Fe/H]}$
of the metallicity distribution, plotted versus halo mass $M_{h}$.

##  6 The Metallicity Distributions

For each of the 26 galaxies, the dereddened color of every individual GC
within the magnitude and color limits specified above was converted to [Fe/H]
by inversion of the quadratic relations in Table 6. From these, the MDFs were
generated. The final MDFs for all 26 galaxies studied here are shown in Figure
20, where now every galaxy has been put onto a common system for direct
comparison. Here the galaxies are now shown in order of increasing luminosity.

The CDF and resulting MDF for our template object, NGC 4874, are shown for
direct comparison in Figure 21. The difference is striking: both are clearly
bimodal in form and a bimodal-Gaussian model fits both distributions equally
well, but the numbers of clusters allocated to the ‘red’ and ‘blue’ subgroups
are almost the mirror image of each other. At lower metallicities, the
shallower dependence of color on metallicity turns the broad blue component of
the MDF into a narrow blue component in the CDF.

Another way to display the difference between the CDF and MDF is in Figure 22.
Here, the fraction $f_{1}$ of the total GC population assigned by the GMM fits
to the metal-poor component in the MDF is plotted versus the same component in
the CDF. In all cases, removal of the nonlinearity in the color indices shifts
the best-fit result to a higher blue fraction in the MDF. It it worth noting
that the (F435W-F814W) index (analogous to B-I) is closest to giving 1:1
agreement: among the color indices used here, it has the widest range in color
and is close to being linearly proportional to metallicity especially for
[Fe/H] $>-1.5$. In the consistency test shown in Fig. 19, it also shows the
closest agreement with the colors of the Milky Way GCs.

As seen in Fig. 20, many of the individual MDFs are unimodal in the strict
sense that there is no deep inflection point at intermediate [Fe/H] dividing
the components. The same is true for the CDFs described above. Nevertheless, a
single symmetric function does not match any of the observed systems, whereas
the bimodal-Gaussian model proves to fit the MDFs just as well as it did for
the CDFs. But the two modes for these massive galaxies are much more heavily
overlapped than they are in less massive galaxies (Peng et al., 2006). As
pointed out by Choksi & Gnedin (2019), these broad MDFs are more of a
continuum and the two-component division is to some extent an arbitrary
procedure given that GCs are forming and being accreted almost continuously in
the growth history of a giant galaxy.

###  6.1 Age Effects

To test the sensitivity of the MDFs to the assumed GC age, SSP models were
generated as described above for ages of 9 to 13 Gyr in 1-Gyr steps, and the
CMR recalculated. The resulting MDFs for the test case of NGC 4874 are shown
in Figure 23. If the assumed age is lower, then the GC metallicity needs to be
higher in order to yield the same observed CDF. In this old age range,
however, the differences are second-order: going from 9 to 13 Gyr, the MDF
shifts by $\sim 0.2$ dex, but the overall bimodal MDF shape stays much the
same.

In a further stage of modelling, an age/metallicity relation could be built
in, such as is observed in the Milky Way (Dotter et al., 2011; Leaman et al.,
2013; Usher et al., 2019), along with some level of age spread at any
metallicity. More extensive modelling of this type will be the subject of
future work.

###  6.2 Metallicity Gradients?

At this stage, no selection of GCs has been made by galactocentric radius. In
practice this choice has little effect on the MDFs because the radial
metallicity gradients in these giant galaxies are shallow. Figure 24 shows the
distribution of [Fe/H] versus radius $R$ normalized to the effective radius
$R_{e}$ of the optical light profile. In each case a simple least-squares fit
was done with the form

${\rm[Fe/H]}=a+b~{}{\rm log}(R/R_{e})$ (6)

with the solutions shown as the solid lines in the panels, and as listed in
Table 7.

Metallicity gradients are an outcome of the star formation and merger
histories of galaxies. Because of their high luminosity and easy detectability
at all radii, GCs provide a way to trace gradients to much larger radii than
is possible with integrated light. Systematic radial changes in mean GC
metallicity can arise from intrinsic gradients in the red and blue
subpopulations, or as a ‘population gradient’ due to the radial change in the
fraction of red and blue clusters, or both (Geisler et al., 1996; Harris,
2009a; Liu et al., 2011; Forbes & Remus, 2018). Either _in situ_ star
formation or later accretion dominated by low-metallicity dwarfs can leave
behind gradients, along with considerable galaxy-to-galaxy differences in the
outcomes.

As is seen in Fig. 24, shallow gradients in mean metallicity are present for
most of the galaxies in the present sample, though the stronger impression is
that of very large scatter of the individual GC metallicities at all radii.
The mean value for the powerlaw exponent across the entire sample is $\langle
b\rangle=-0.301\pm 0.031$ with an rms scatter of $\pm 0.16$ and no clear trend
with galaxy luminosity. Figure 25 shows the individual results plotted versus
galaxy luminosity. In other words, the mean GC metallicity in these large
galaxies scales as $Z\sim R^{-0.3}$ and with significant individual variation.

Liu et al. (2011) give an extensive discussion of the color and metallicity
gradients of the GCSs in 76 Virgo and Fornax Survey member galaxies. They find
that the mean color gradient is $\Delta(g-z)/\Delta{\rm log}R=-0.112$ with rms
scatter $\pm 0.077$, which from the Blakeslee et al. (2010) transformation of
$(g-z)$ to [Fe/H] (a quartic polynomial form) converts to a mean exponent
$\langle b\rangle=-0.387\pm 0.034$ with rms scatter of $\pm 0.28$. The
conversion relation used here for (F475W-F850LP), however, would give a mean
$\langle b\rangle\simeq-0.3\pm 0.02$ with rms scatter $\pm 0.21$, highly
consistent with the results here. Almost all of the Virgo and Fornax members
are less luminous than the giants studied here, extending down into the dwarf
regime as low as $M_{V}\sim-16$. There is some indication that for the
smallest dwarfs in their sample the mean gradients become shallower or even
absent. Gradients in the range $b\sim-0.3$ are also found in the SLUGGS sample
of galaxies (Pastorello et al., 2015).

Forbes & Remus (2018) give results for a collated sample of large galaxies and
find somewhat smaller slopes in the range $b\simeq-0.1$ to $-0.2$, though
these are obtained with a different (linear) transformation from colors. GC
samples built on purely spectroscopic data are more rare (e.g. Woodley et al.,
2010; Pota et al., 2015; Pastorello et al., 2015; Caldwell & Romanowsky, 2016;
Villaume et al., 2019; Ko et al., 2022) but have generally shown negative
gradients in line with the photometrically based ones shown here. Future work
will discuss more comprehensively the gradients associated with the blue and
red subpopulations, and links with contemporary formation models (cf. the
references cited above).

###  6.3 MDF Parameters

Just as for the CDFs, bimodal Gaussian fits were made to all of the MDFs in
Fig. 20, with the quantitative results listed in Table 8. In addition to the
peak [Fe/H] values $(\mu_{1},\mu_{2})$ and dispersions
$(\sigma_{1},\sigma_{2})$ of the two subcomponents, the table also includes
the fraction $f_{1}$ (number of blue GCs relative to the total), the mode
separation parameter $DD$, the mode height ratio $H_{12}$, and finally the
metallicity [Fe/H]mid at which the blue and red modes are equal (i.e. the
metallicity at which the two Gaussian curves cross). The sample means of each
quantity are listed at the end of the table.

Table 9 includes a further list of other parameters characterizing the MDFs
that do not depend on any particular assumption about the intrinsic shape of
the distribution. These include the global mean metallicity
$\langle$[Fe/H]$\rangle$; the global standard deviation $\sigma$[Fe/H]; the
median metallicity; the skewness and kurtosis of the MDF555The definition of
kurtosis used here is the “excess kurtosis”, which equals zero for a Gaussian
distribution.; and the InterQuartile and InterDecile Ranges (IQR, IDR).

The correlations of most of these parameters versus galaxy stellar mass are
shown in Figure 26. The stellar mass is calculated as

$M_{*}=(M/L)_{V}\cdot 10^{0.4(4.83-M_{V}^{T})}$ (7)

where the mass-to-light ratio is adopted as $(M/L)_{V}=3.5$ from the
calibrations of Bell et al. (2003), assuming a Chabrier IMF and a mean
intrinsic color $(B-V)_{0}=0.9$ for these giant ETGs.

In Fig. 26, many of the parameters are remarkably uniform across a factor of
about 20 in mass range. These near-constant parameters include the mean,
median, and mid-[Fe/H] metallicities, the central locations $\mu_{1},\mu_{2}$
of the Gaussian components and their dispersions $\sigma_{1},\sigma_{2}$, and
the measures of the total spread of the MDF ($\sigma$, IQR, and IDR). The mode
dispersions show systematic increases with mass at a barely significant level:
$\Delta\sigma_{1}/\Delta{\rm log}M_{*}=0.035\pm 0.028$, and
$\Delta\sigma_{2}/\Delta{\rm log}M_{*}=0.043\pm 0.027$.

For comparison, for lower-mass galaxies in Virgo Peng et al. (2006) found that
the mean metallicities of the blue and red components increase as $Z_{1}\sim
M_{*}^{0.16},Z_{2}\sim M_{*}^{0.26}$ (albeit derived from a slightly different
CMR as discussed above). The results here for the more massive ETGs suggest
that this scaling does not continue upward, instead settling to a more uniform
level as expected from recent simulations such as Choksi & Gnedin (2019,
particularly their Figure 4) and El-Badry et al. (2019).

The three measures of MDF total width ($\sigma$, IQR, IDR) are extremely well
correlated with each other and essentially equivalent. Quantitatively, there
is also little difference in their ratios of mean value to rms scatter. The
one practical disadvantage of the IDR (or even the IQR) is simply it they will
become undefined for dwarf galaxies with very small GC numbers.

There is one obvious outlier in the graphs of mean metallicity and
$\mu_{1},\mu_{2}$, which is NGC 1129. Its transformed [Fe/H] values are more
metal-rich than expected by a surprising 0.6 dex, which is equivalent to the
colors being too red by more than 0.2 mag. NGC 1129 is the only CDF measured
in $(F435W-F606W)$, for which the predicted CMR was seen in Fig. 19 to be
significantly offset relative to the Milky Way GCs by about the same amount.
The reason for this discrepancy is not clear at present and will require
further investigation, and ultimately probably remeasurement in different
filters.

Theoretical work predicting the MDFs for GC systems has been quite limited up
until recent years but now growing. The common bimodal shape of the
_metallicity_ distribution (as opposed to the color distribution) in large
galaxies has been shown by several authors to be a natural and frequent
outcome of hierarchical growth (Côté et al., 1998; Muratov & Gnedin, 2010;
Tonini, 2013; Choksi et al., 2018; Choksi & Gnedin, 2019; El-Badry et al.,
2019; Kruijssen et al., 2019a; Halbesma et al., 2020). To first order, high-
metallicity GCs are formed _in situ_ in the massive potential well of the main
progenitor branch while low-metallicity GCs originate _ex situ_ within the
many metal-poor dwarfs that are accreted throughout its growth history
(Beasley, 2020; Choksi et al., 2018; Forbes et al., 2018a; Forbes & Remus,
2018; El-Badry et al., 2019). This is not an exclusive division, though, since
blue GCs also form early along the main progenitor branch within the small
potential wells that populate the base of the merger tree, and red GCs can be
accreted later any time a merger with another large galaxy takes place (Choksi
et al., 2018).

One of the simplest tests of recent theoretical predictions is the mean or
median metallicity of the entire GC system. Observationally (Table 9) the data
give a mean $\langle$Fe/H$\rangle=-0.97$, and median
$\overline{{\rm{[Fe/H]}}}=-0.91$. A common theme of the models at present
seems to be overestimation of the mean, though for differing reasons. Pfeffer
et al. (2018) predict $\overline{{\rm{[Fe/H]}}}$ from the E-MOSAICS
simulations almost 0.5 dex higher than the observations give for large
galaxies, which they ascribe to insufficient disruption of small disk clusters
that have high metallicities. Halbesma et al. (2020) show MDFs for their Milky
Way and M31 analogs derived from the Auriga simulations that show much
individual variation but are often too metal-rich, suggested to be a result of
the way the star particles are assigned to GCs or field stars. The models from
Choksi et al. (2018) and Choksi & Gnedin (2019) built on the Illustris
simulations (Pillepich et al., 2018) come close at present to matching the
data: they show that when the full effect of merging and accretion is
accounted for, the mean metallicity of the GC system increases steadily with
galaxy mass up to $M_{*}\sim 10^{11}M_{\odot}$ and then roughly levels off at
$\langle$Fe/H$\rangle\simeq-0.8$, about $0.1-0.2$ dex higher than the
observations in Fig. 20 suggest. Their definition of GC metallicity, however,
uses a theoretical prescription for the mass-metallicity relation between halo
mass and its star-forming gas, and so it is not clear if this small offset is
important.

The Choksi et al. models also predict that the overall dispersion
$\sigma_{\rm{[Fe/H]}}$ shows a shallow increase with galaxy mass but again
levels off at $\simeq 0.5-0.6$ dex at high mass, whereas the observations in
Table 9 give $\sigma_{\rm{[Fe/H]}}=0.64\pm 0.01$. As noted above, their
predicted metallicities for the red and blue modes considered separately as
functions of present-day galaxy mass are also within $\sim 0.1$ dex of the
observations.

Finally, two other correlations between MDF parameters are shown in Figure 27.
These plot the mode separation parameter DD and the blue fraction $f_{1}$
versus [Fe/H]mid, the metallicity at which the blue and red Gaussian
components give equal probabilities. Both show well defined trends that arise
from the MDF shape. If $f_{1}$ is low, the fact that the component dispersions
$\sigma_{1},\sigma_{2}$ stay almost constant means that as [Fe/H]mid shifts to
lower metallicity, the sharper red mode stands out more clearly, and the mode
separation DD increases. Consequently, the height ratio $H_{12}$ also
increases, as can be seen from inspection of Table 8.

###  6.4 Trends Over a Wider Mass Range

One comparison with theory that has interesting new potential for constraining
models (Choksi & Gnedin, 2019; El-Badry et al., 2019) is the correlation shown
in Figure 28, the metal-poor GC fraction $f_{1}$ versus galaxy mass $M_{*}$.
An advantage of focussing on $f_{1}$ or $f_{2}=(1-f_{1})$ is that it should be
relatively insensitive to the zeropoint of the metallicity scale from
different color indices, as long as the CMR transformation has successfully
removed the nonlinearity in the colors.666The case of NGC 1129 discussed above
provides some evidence for this. It is an obvious outlier in the graphs of
$\mu_{1},\mu_{2}$ and mean metallicity, but for $f_{1}$ and the metallicity
dispersions it fits well with the distributions of the other galaxies. In Fig.
28, to minimize any biases from sampling different parts of the halo in
different galaxies, $f_{1}$ has been recalculated within the restricted radial
range of $(0.5R_{e}-5R_{e})$ rather than the entire range of the data (though
in practice, constraining the radial range this way has very little effect
given the shallowness of the metallicity gradients in all the systems).

This correlation hints at some intriguing behavior with increasing galaxy mass
that is anticipated from current simulations of hierarchical growth. However,
the mass range shown is not much more than a factor of 10, and the numbers of
datapoints are still limited. To gain a broader look at the trend of red/blue
fractions over a wider mass range, the BCG sample here has been supplemented
by GC metallicity data for smaller galaxies from the Virgo and Fornax ACS
imaging surveys (Peng et al., 2006; Villegas et al., 2010). From the
photometric catalogs for both these programs (Jordán et al., 2009, 2015), the
$(g-z)$ colors for the individual GCs in each galaxy were extracted, the de-
reddened indices converted to (F475W-F850LP) = $(g-z)_{0}+0.636$, and then
into metallicity through Eq. 5. As in Liu et al. (2019), only GC candidates
with probability $p\geq 0.5$ were adopted to reduce possible sample
contamination. Where the total numbers of GCs were large enough, GMM bimodal-
Gaussian fits were then done for each Virgo or Fornax galaxy to extract the
same MDF parameters as described above for the BCGs. For the smaller dwarfs
where the numbers of clusters were too low to permit a bimodal-Gaussian fit,
the list of GCs was simply divided at [Fe/H] = -1 with lower-metallicity ones
counted as ‘blue’ and higher-metallicity ones as ‘red’.

Finally, to extend the galaxy mass range down to the lowest possible levels,
the Local Group dwarfs containing GCs were added, with spectroscopic and
photometric data from a range of individual sources (Forbes et al., 2018b;
Forbes, 2020; Colucci et al., 2011; Piatti et al., 2018; Law & Majewski, 2010;
Massari et al., 2019; Dalessandro et al., 2016; Eadie et al., 2022; Kruijssen
et al., 2019b; Da Costa & Mould, 1988; de Boer & Fraser, 2016; Pace et al.,
2021; Cole et al., 2017; Georgiev et al., 2010; Larsen et al., 2014;
Veljanoski et al., 2013, 2015; Beasley et al., 2019; Wang et al., 2019;
Caldwell et al., 2017; Crnojević et al., 2016; Cusano et al., 2016). For these
dwarfs the numbers of clusters are too small for GMM solutions, so again the
samples were simply divided at [Fe/H] = -1 as above.

The combined sample is plotted in Figure 29. Here, in order to compare more
easily with relevant theory (Choksi & Gnedin, 2019; El-Badry et al., 2019) the
numbers have been recast as $f_{2}$ (the red-GC fraction) versus galaxy halo
mass $M_{h}$. Conversion of $M_{*}$ to $M_{h}$ has been done through the SHMR
(stellar-to-halo mass ratio) prescription in Hudson et al. (2015), which uses
weak lensing as the basis of calibration.

Fig. 29, despite the internal scatter, reveals a complex and non-monotonic
overall trend. For the range between $M_{h}\simeq 10^{10}M_{\odot}$ and
$3\times 10^{12}M_{\odot}$, the red fraction scales roughly as $f_{2}\simeq
0.25\cdot{\rm log}(M_{h}/{10^{10}M_{\odot})}$. At higher mass, $f_{2}$
reverses and declines again, reaching a local minimum of $f_{2}(min)\simeq
0.4$ at $M_{h}\sim 10^{14}M_{\odot}$. Past that point, however, $f_{2}$ begins
to rise again, reaching $\sim 0.6$ for the very most massive BCGs in the
observed sample. The solid line shows the mean trend of $f_{2}$ versus galaxy
mass, calculated as a weighted mean with nine consecutive points per bin.

In the dwarf regime, very large scatter is present near $10^{11}M_{\odot}$.
Some of this spread is due to small-number statistical scatter, but this is
also the low-luminosity and low-N range where the relative effects of field
contamination would be worst (see the discussions in Jordán et al., 2009,
2015; Liu et al., 2019), since for the Virgo and Fornax dwarfs the numbers of
GCs are determined only after subtraction of local number densities of objects
that match GCs in size and morphology. For the still smaller Local Group
members the same issue does not come up, since their GCs can be individually
identified.

The trend of $f_{2}$ seen here is expected from current understanding of
hierarchical growth. Figures 6 and 7 of Choksi & Gnedin (2019) and Figure 9 of
El-Badry et al. (2019) indicate the predicted trends. At first, starting from
the smallest dwarfs, $f_{2}$ increases with increasing halo mass because more
massive halos contain more enriched gas, and the mass fraction accreted from
satellites stays relatively low. For these smaller galaxies, their GC
population is dominated by the _in situ_ component. But at higher mass the
accreted mass fraction (mostly from metal-poor dwarfs) rises strongly, and
eventually accreted GCs dominate the _blue_ GC population, forcing the red
fraction $f_{2}$ to decrease again. Then, at still higher mass, accreted GCs
also start to dominate the _red_ GC population because of mergers with other
large galaxies that bring in metal-rich GCs. This effect slows the decrease of
$f_{2}$ or even drives it back upward again depending on the details of the
merger history and the total numbers of clusters involved.

More quantitatively, from the simulations the first reversal point at $\sim
3\times 10^{12}M_{\odot}$ is where 50% of the metal-poor GC population is from
accreted satellites, whereas the second reversal point at $\sim 5\times
10^{13}M_{\odot}$ is where 50% of the metal-rich GCs are from accreted systems
(Choksi & Gnedin, 2019). These two transition masses are quite close to what
is seen in Fig. 29. Different individual merger histories will generate
scatter in the overall trend, as is evident in the data. But an additional
quantitative point of agreement with the predictions of both Choksi & Gnedin
(2019) and El-Badry et al. (2019) is that $f_{2}$ reaches a maximum near
$\simeq 0.6$ across the entire mass range. Interestingly, this maximum is
reached twice: once at a few $\times 10^{12}M_{\odot}$ and again beyond
$10^{15}M_{\odot}.$

In brief, the trend of red/blue GC fraction (Fig. 29) allows us to identify
three important mass transition points ($M_{0},M_{1},M_{2}$) that connect the
observed data with current theory:

  1. 1.

$M_{0}\simeq 10^{10}M_{\odot}$, the galaxy halo mass below which no galaxies
have metal-rich GCs ($f_{2}\rightarrow 0$). Dwarf galaxies below this limit
may have produced some star clusters, but most of their clusters would have
been too low-mass to survive to the present day, and the small initial gas
mass in the halo was disrupted after feedback from SNe and radiative heating
(cf. the discussions of Kruijssen, 2019; Eadie et al., 2022).

  2. 2.

$M_{1}\simeq 3\times 10^{12}M_{\odot}$, the halo mass at which accreted GCs
start to dominate the metal-poor GC component.

  3. 3.

$M_{2}\simeq 10^{14}M_{\odot}$, the halo mass at which accreted GCs also start
to dominate the metal-rich GC component. $M_{1}$ and $M_{2}$ both mark
reversal points in the proportions of red and blue GCs.

Figure 30 (upper panel) shows the trend of the observed median metallicity for
the same combined sample of galaxies. For smaller galaxies, the median rises
steadily with the increasing presence of red GCs, but levels off past
$M_{h}\sim 10^{13}M_{\odot}$ as the total GC population becomes dominated by
accreted systems with a wide mixture of both metal-rich and metal-poor
clusters. Again, this behavior of the mean or median [Fe/H] matches the
predictions of the current suites of simulations cited above.

Lastly, Figure 30 (lower panel) shows the rms spread (standard deviation) of
the MDF considered as a whole, plotted versus galaxy mass. This is seen to
stay nearly constant at $\sigma\simeq 0.6$ dex for $M_{h}>10^{11}M_{\odot}$,
scattering to lower values only for some of the lowest-luminosity systems that
contain only metal-poor GCs. The models discussed in Choksi & Gnedin (2019,
see their Figure 5) predict a $\sigma$ versus mass trend starting near 0.4 dex
at $M_{h}=10^{11}M_{\odot}$ and then gradually rising to $\simeq 0.6$ at
higher mass. Thus in this case, the simulations underestimate the
observational spread.

##  7 Overview and Conclusions

A summary of the main results of this study is as follows:

  1. 1.

From the HST MAST archive, data for 26 giant ETGs with large globular cluster
populations and imaged with the ACS/WFC camera have been reduced homogeneously
with the DOLPHOT photometry code. All of these systems are distant enough that
their GCs appear near-starlike, and are imaged with long enough exposure times
to cover at least the bright half of the GC luminosity distribution.

  2. 2.

From the photometry, the color distribution function (CDF) of the GC
population is generated for each galaxy. A total of 13 galaxies are measured
in (F475W-F814W), nine in (F435W-F814W), two in (F475W-F850LP), and one each
in (F435W-F606W) and (F555W-F814W). All the photometry is publicly available.

  3. 3.

Bimodal-Gaussian fits prove to be accurate descriptions of every CDF. The only
system in this set of galaxies showing evidence of a very young cluster
population and recent star formation is the Perseus giant NGC 1275, well known
to be an active giant with large amounts of gas. The final CDFs are shown to
be insensitive to corrections for galactocentric radius, photometric
completeness, and (for the closest targets) marginal nonstellar morphology of
the GCs.

  4. 4.

For the most luminous galaxies, the red (metal-rich) GC sequence extends
upward to distinctly higher luminosities than does the blue sequence, reaching
predicted masses well above $5\times 10^{6}M_{\odot}$ and into the UCD regime.

  5. 5.

The measured GC colors are converted to metallicity [Fe/H] through a through a
simple quadratic color-to-metallicity relation, built on a combination of
$12-$Gyr SSP models and empirical spectroscopic calibrations of the index
(F475W-F850LP) that are in the previous literature. From the CMRs for every
color index, metallicity distribution functions (MDFs) are generated on a
common scale. Among the six optical/near-IR color indices tested here, the
most satisfactory one for these purposes appears to be (F435W-F814W) for its
dynamic range in color.

  6. 6.

All the program galaxies exhibit shallow metallicity gradients versus
projected galactocentric distance: on average, heavy-element abundance scales
as $Z\sim R^{-0.3}$ though with significant individual variation. This result
agrees well with typical gradients found in a wide range of less luminous
galaxies.

  7. 7.

Bimodal-Gaussian fits also prove to accurately match all the MDFs as well,
though with the important difference that the blue (metal-poor) mode is much
broader and has a higher proportion of the GC totals in the MDF than in the
CDF. Age differences in the range $9-13$ Gyr will make second-order
differences in the [Fe/H] scale, in the sense that a younger assumed age needs
slightly higher metallicity to yield the same color.

  8. 8.

Several parameters of the MDFs are calculated including the mean and
dispersion of the blue and red modes, the fraction of the total in each mode,
and several indicators of the total mean, range, and shape of the MDF as a
whole. The intrinsic dispersions of the blue and red modes show shallow
systematic increases with galaxy stellar mass $M_{*}$, but most of the other
parameters remain remarkably uniform.

  9. 9.

A plot of blue/red GC fraction versus galaxy mass shows interestingly complex
behavior that has the potential to put new constraints on current theoretical
predictions of hierarchical assembly of these large galaxies. For
$M_{h}\gtrsim 3\times 10^{12}M_{\odot}$, the blue GC component is expected to
become dominated by accreted systems, driving a decrease in the red-GC
fraction $f_{2}$; but at still higher mass above $\sim 10^{14}M_{\odot}$ the
red GCs also become dominated by accreted systems, causing $f_{2}$ to increase
again. This double reversal in the trend of red/blue fraction versus galaxy
mass is now tentatively seen across the full mass range of observed galaxies,
with the reversal points in approximate agreement with recent theory.

One obvious limitation of the discussion here is the assumption of a single
age (12 Gyr) for all GCs in the conversion of color to metallicity. In a next
stage of analysis, following what is known about the Milky Way GCs and other
nearby galaxies, sample age/metallicity relations can be built in along with
some intrinsic scatter. In addition, different SSP models should be used to
test the consistency and reliability of the color transformations used here.

On the observational side, new photometry of the GCs in one or more of these
giant galaxies should be carried out that would measure the GCs in the entire
set of color indices, not just one or two. Reliance on the models to define
the conversions of one color index to another would be drastically reduced,
increasing the confidence in the eventual metallicity distributions.

Much else can be done with the current dataset in other directions. The radial
distributions of the GCs and the GC luminosity distributions can be measured,
leading to total GC population estimates, specific frequencies, and a stronger
definition of the correlation of total GC system mass versus galaxy halo mass
at its high-mass end. The shape of the MDF can also be traced as a function of
GC luminosity (mass) and projected location in the halo. These directions will
be the subject of followup work.

## Acknowledgements

The complete set of reduced photometry, along with the reference images that
define the local xy coordinate systems for each galaxy field, can be obtained
at this DOI: https://doi.org/10.5281/zenodo.7342871 (catalog
10.5281/zenodo.7342871).

The imaging data used in this paper were obtained from the Mikulski Archive
for Space Telescopes (MAST) at the Space Telescopes Science Institute. The
spcific observations analyzed can be accessed via these DOIs:
https://doi.org/10.17909/8c69-1q86 (catalog 10.17909/8c69-1q86),
https://doi.org/10.17909/36nb-z070 (catalog 10.17909/36nb-z070),
https://doi.org/10.17909/pvn9-ya02 (catalog 10.17909/pvn9-ya02),
https://doi.org/10.17909/64m6-1c38 (catalog 10.17909/64m6-1c38),
https://doi.org/10.17909/bk3t-td27 (catalog 10.17909/bk3t-td27),
https://doi.org/10.17909/3x7e-fj90 (catalog 10.17909/3x7e-fj90),
https://doi.org/10.17909/qssg-zb94 (catalog 10.17909/qssg-zb94),
https://doi.org/10.17909/qafy-dv12 (catalog 10.17909/qafy-dv12),
https://doi.org/10.17909/fk8b-jv82 (catalog 10.17909/fk8b-jv82).

The author acknowledges the financial support of NSERC (Natural Sciences and
Engineering Research Council of Canada). For very helpful practical advice
during early stages of this work, I am grateful to Pat Durrell, Doug Welch,
Guillaume Hewitt, and Andy Dolphin. HST (ACS)

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