Zero-Age Horizontal Branch Models for -2.5<= [Fe/H]<= -0.5 and Their Implications for the Apparent Distance Moduli of Globular Clusters

Grids of zero-age horizontal branch (ZAHB) models are presented, along with a suitable interpolation code, for -2.5<= [Fe/H]<= -0.5, in steps of 0.2 dex, assuming Y = 0.25 and 0.29, [O/Fe] = +0.4 and +0.6, and [m/Fe] = 0.4 for all of the other alpha elements. The HB populations of 37 globular clusters (GCs) are fitted to these ZAHBs to derive their apparent distance moduli, (m-M)_V. With few exceptions, the best estimates of their reddenings from dust maps are adopted. The distance moduli are constrained using the prediction that (M_F606W-M_F814W)_0 colours of metal-poor, main-sequence stars at M_F606W>~ 5.0 have very little sensitivity to [Fe/H]. Intrinsic (M_F336W-M_F606W)_0 colours of blue HB stars, which provide valuable connections between GCs with exclusively blue HBs and other clusters of similar metallicity that also have red HB components, limit the uncertainties of relative (m-M)_V values to within +/- 0.03-0.04 mag. The ZAHB-based distances agree quite well with the distances derived by Baumgardt&Vasiliev (2021, MNRAS, 505, 5957). Their implications for GC ages are briefly discussed. Stellar rotation and mass loss appear to be more important than helium abundance variations in explaining the colour-magnitude diagrams of second-parameter GCs (those with anomalously very blue HBs for their metallicities).


INTRODUCTION
A zero-age horizontal branch (ZAHB) typically consists of a few dozen stellar models that predict the luminosities and effective temperatures (T eff s) of stars that are just beginning their core He-burning phases of evolution.Such models have the same helium core mass and chemical abundance profiles above the core, as derived from an appropriate red-giant precursor (see, e.g., Piersanti et al. 2004, Serenelli & Weiss 2005, VandenBerg et al. 2016), but a range of envelope, and hence total, masses to take into account star-to-star variations in mass loss along the upper redgiant branch (RGB) during the preceding evolution.Fortunately, ZAHBs that are applicable to old stellar populations are nearly ageindependent, as shown most recently by Denissenkov et al. (2017, hereafter DVKF17) and Pietrinferni et al. (2021).Consequently, it is necessary to compute only a single evolutionary track, for a mass that has a predicted lifetime of 12 Gyr or so, to the tip of the RGB in order to obtain the information that is needed to generate a ZAHB for the assumed initial abundances of helium and the metals.In effect, ZAHBs are decoupled from isochrones; i.e., the same ZAHB ⋆ E-mail: vandenbe@uvic.ca can be fitted to the HB population of a given globular cluster (GC) irrespective of the age of the isochrone that provides the best fit to the turnoff (TO) observations.Because ZAHB sequences are nearly horizontal at colours that are typical of upper main-sequence (MS) and TO stars in optical colour-magnitude diagrams (CMDs) of GCs, they have often been used to derive the apparent distance moduli of these systems, particularly when they contain red HB populations (e.g., VandenBerg 2000;Recio-Blanco et al. 2005;VandenBerg et al. 2013, hereafter VBLC13).The fitting of observed HBs to ZAHB models has the advantage of being insensitive to T eff and reddening uncertainties, in addition to having a relatively weak dependence on metallicity.Most stellar evolutionary computations over the years have found that the slope of the M V (HB) versus [Fe/H] relation in the vicinity of the instability strip is ≈ 0.18-0.20 (Lee et al. 1990;Cassisi et al. 1999, and references therein;VandenBerg et al. 2000), which means that a 0.1 dex error in the adopted metallicity results in an error of < ∼ 0.02 mag in the derived value of (m − M) V .Provided that the ZAHB models are trustworthy, ages that are based on the difference in magnitude between the ZAHB and the TO, when evaluated using the methods described by VBLC13, should be especially reliable.(This also requires, of course, that the stel-lar models are generated for best estimates of the chemical abundances.) The main difficulty with this approach is observational, insofar as the flat portions of ZAHBs are not populated in a large number of Galactic GCs, many of which have metallicities in the range −2.0 < ∼ [Fe/H] < ∼ −1.5.When the core He-burning stars are distributed along steep, blue tails (in optical CMDs), small errors in the predicted colours of the stellar models or in the adopted reddenings can easily alter the distance moduli that are obtained from fits of the HB populations to ZAHB models by as much as 0.1-0.2mag.However, as shown below, it is possible to do much better than that if other considerations are taken into account -notably, the near independence of the location of the MS fiducials of the most metal-poor GCs on the [(M F606W − M F814W ) 0 , M F606W ]diagram and similar CMDs.In fact, it turns out that ultraviolet photometry from the Hubble Space Telescope (HST) UV Legacy Survey (Piotto et al. 2015, Nardiello et al. 2018, hereafter NLP18) can also be used to place tight constraints on the relative reddenings and distance moduli of GCs that have similar metallicities but very different HB morphologies (such as M 3 and M 13).
Although the discovery of He abundance variations within GCs (Piotto et al. 2005, Milone et al. 2012) has made the modeling of the entire distributions of their HB stars on various CMDs quite challenging (see, e.g., DVKF17), the presence of multiple stellar populations should not affect ZAHB-based determinations of (m − M) V .Based on their survey of 57 GCs, Milone et al. (2018) concluded that all of the clusters in their sample contain (at least) two chemically distinct stellar populations, which they refer to as 1G and 2G (i.e., first and second generation), and that all 2G stars are helium enhanced with respect to those classified as 1G, with enhancements that apparently vary from ∆ Y < ∼ 0.01 to > ∼ 0.10, where Y is the mass-fraction abundance of helium.As a consequence of having higher Y, 2G HB stars near their "zero-age" CMD locations can be expected to be brighter, at a given colour, than their 1G counterparts.Thus, the faintest of the reddest HB stars should, presumably, be well represented by ZAHB models for initial helium abundances that are close to the primordial value (Y P ≈ 0.247; see, e.g., Pitrou et al. 2018, Planck Collaboration 2020). 1his expectation is supported by the direct spectroscopic determination of the He abundance in NGC 6752 by Villanova et al. (2009), who obtained a mean value of Y = 0.245 ± 0.012 for 7 HB stars that hot enough (T eff > 8500 K) to have measureable photospheric helium lines, but not so hot (T eff < 11, 500 K) that the surface chemistry is modified by gravitational settling.Even though Villanova et al. (2012) obtained a significantly larger value of Y (0.29 ± 0.01) from a similar study of the blue HB stars in M 4, they argued that these stars are likely to be helium-enhanced, and therefore members of the 2G population of M 4, because they are brighter than the faintest red HB stars in this cluster.In fact, the downward tilt of the M 4 HB, in the direction from blue to red, resembles that seen in NGC 6362 (Brocato et al. 1999), which was the subject of a recent study by VandenBerg & Denissenkov (2018).The latter concluded from their fits of ZAHB models to the non-variable HB stars on the (B − V) 0 , M V diagram, from an analysis of the observed periods of member RR Lyrae variables, and from full simulations of the cluster HB population, that its blue HB stars have higher He abundances by as much as ∆ Y ≈ 0.03 than those on the red side of the instability strip.Hence, the available evidence indicates that ZAHBs for Y ≈ Y P should be relevant to at least some fraction of the reddest HB stars in all GCs, even those that have exclusively blue HB populations, such as NGC 6752.
This investigation has been carried out to (i) provide new ZAHB models to complement several of the grids of tracks for earlier evolutionary phases that were recently made available by VandenBerg et al. (2022a) and VandenBerg (2023), and (ii) examine the implications of such models for the apparent distance moduli of GCs.The computation of the ZAHBs and the methods used to determine the (m − M) V values of interest are described in § 2 and § 3, respectively.The distance moduli of 37 GCs with metallicities in the range −2.4 < ∼ [Fe/H] < ∼ −0.6, supported by intercomparisons of the cluster HB populations on UV-optical CMDs, are derived in § 4. The possible role of stellar rotation and mass loss in explaining the CMDs of second-parameter clusters is discussed in § 5.A summary of the main results of this study is provided in § 6, which also compares the derived distance moduli with those obtained by VBLC13 and by Baumgardt & Vasiliev (2021), who concluded from a thorough analysis of literature values, supplemented by their own findings that utilized Gaia and various HST observations, that their distance determinations should be accurate to within a few percent for most clusters.

THE COMPUTATION OF THE ZAHB SEQUENCES
The ZAHB models presented in this investigation were computed using the stellar structure and evolution program described by VandenBerg (2023, and references therein), who provides complementary evolutionary tracks for the MS and RGB phases, along with the means to produce isochrones from them.A brief summary of the physics that has been incorporated in the Victoria code since 2012, when it was last revised (see VandenBerg et al. 2012), is provided in Table 1.Of the physics components listed therein, diffusion likely has the greatest impact on ZAHB models because, in comparison with the predictions of non-diffusive computations, it results in lower surface He abundances in red giants after the first dredge-up by ∆ Y ∼ 0.015, with some dependence on mass and metallicity, leading to fainter ZAHBs by ∼ 0.05 mag (see, e.g., Proffitt & VandenBerg 1991).The Victoria code does not follow the settling of the metals (for some discussion of this point, see VandenBerg et al. 2014, their § 2) or radiative accelerations, though it includes an ad hoc treatment of extra mixing below surface convection zones, when they exist, to limit the efficiency of diffusion in order to satisfy various empirical constraints.All of this is thoroughly discussed by VandenBerg et al. (2012, see their § 3) and need not be repeated here.However, it is worth mentioning that evolutionary calculations which employ much more sophisticated treatments of diffusive processes (notably by Michaud et al. 2007) yield ZAHB luminosities that differ by ∆ M bol ≈ 0.038 mag compared with those that neglect this physics (also see Michaud et al. 2010).This is less than the determination from Victoria models by only ∼ 0.02 mag.
The adoption of alternative equation-of-state formulations (e.g., Rogers et al. 1996), nuclear reactions (e.g., the latest JINA Reaclib database; Cyburt et al. 2010), and radiative opacities (e.g., Opacity Project data; Badnell et al. 2005) can be expected to have some impact on the predicted properties of ZAHB models, just as Cassisi et al. (2021) have found in their study of the uncertainties of conductive opacities in moderately to strongly degenerate a Opacities for stellar interior conditions were generated via the web site https://opalopacity.llnl.gov;complementary data for the outer layers of stars were computed using the code described by Ferguson et al.  b For reactions not explicitly identified, rates from the NACRE compilation (Angulo et al. 1999) were adopted.
gases.Nevertheless, the differences in the resultant stellar models are likely to be relatively small given that the basic physics of lower mass stars appears to be quite well understood.This is suggested, e.g., by the fact that the physics in Table 1 leads to HB-precursor, RGB-tip models with luminosities that are in excellent agreement with empirical constraints (see VandenBerg 2023), and by comparisons of stellar models based on completely independent codes (see below).
The top panel of Figure 1 illustrates one of the four grids of ZAHB models that was computed for this study.It was generated for metallicities ranging from [Fe/H] = −2.5 to −0.5, in steps of 0.2 dex, assuming Y = 0.250 and [O/Fe] = +0.6, with [m/Fe] = +0.4 for all of the other α elements.This mixture of the metals is the same as the one that was given the name "a4xO_p2" by VandenBerg et al. (2022a).An otherwise identical set of ZAHB models was computed on the assumption of Y = 0.290, while the two remaining grids adopted the same values of [Fe/H] and Y but the "a4xO_p0" mix, which has [O/Fe] = [α/Fe] = +0.4.With the aid of a suitable interpolation code (see below), it is possible to obtain individual ZAHB loci for any values of [Fe/H], Y, and [O/Fe] within the aforementioned ranges.
To generate a sequence of ZAHB models for an adopted initial chemical composition, a track was computed from a fully convective structure on the Hayashi line to the tip of the RGB assuming a mass for which the predicted age at the onset of the He flash is ≈ 12.5 Gyr.(Minor variations in the adopted mass do not affect the resultant ZAHB; see DVKF17, Pietrinferni et al. 2021.)The chemical abundance profiles from this RGB precursor were then inserted into a previously converged ZAHB model and the mass-fraction abundance of 12 C in the core was increased by 0.04 to take into account the production of carbon that is predicted to occur during the flash (e.g., Sweigart 1987).The resultant structure was then relaxed over 25 short timesteps until the age of the model reached 2.5 Myr.The same procedure was followed in generating the next model in the sequence except that the mass of the envelope was reduced by a small amount before that model was relaxed to its ZAHB location on the H-R diagram.This process continued until the lowest mass model in the sequence had a predicted temperature that exceeded log T eff = 4.2.Several ZAHB models with masses above that of the reference RGB precursor were also computed so that all ZAHBs include models with masses 0.95M ⊙ .Having all of the ZAHBs run from a common T eff at the hot end to a common mass at the cool end simplified the writing of a suitable interpolation program.Each ZAHB typically consists of several dozen stellar models.
Fig. 1b shows that ZAHB models from the latest BaSTI computations (Pietrinferni et al. 2021) are in excellent agreement with those presented in this paper despite minor differences in the reference solar abundances and in the physics ingredients (notably the treatment of diffusion) that have been incorporated into the respective evolutionary codes.Even the predicted luminosities and effective temperatures of ZAHB models for the same mass are almost identical.For instance, the ZAHB sequences for [Fe/H] = −1.05are both terminated at their cool ends by nearly coincident 0.85M ⊙ models, whereas the highest masses considered in the ZAHBs for [Fe/H] = −2.2 are just slightly offset from each other, but in the expected given that they have masses of 0.80M ⊙ (BaSTI) and 0.797M ⊙ (this study).It may be recalled (see VandenBerg 2023) that the turnoff luminosity versus age relations predicted by BaSTI and Victoria-Regina isochrones are also essentially indistinguishable when both assume the same metallicity and C+N+O abundance -though they have somewhat different capabilities insofar as the fitting of the shapes of observed CMDs is concerned, which is probably due mostly to differences in the adopted diffusion physics and bolometric corrections (BCs).
VandenBerg et al. ( 2016) have already demonstrated that the ZAHBs (and isochrones) produced by the Victoria and MESA (Paxton et al. 2011) codes are in equally fine agreement when close to the same physics is adopted.This particular comparison provides an important test of the reliability of the numerical procedures that are used in this study (and by Pietrinferni et al. 2021) to generate ZAHB models because MESA follows the evolution of lower mass stars through the He flash, thereby producing a continuous track between the tip of the RGB and the ZAHB.Once the most massive ZAHB model has been created in this way, those for lower masses are obtained by removing mass from the initial model in small increments, and then relaxing the resultant structures over many short timesteps.Such good agreement of the results from completely independent stellar evolution programs is truly very satisfying.
As the interpolation of the ZAHB grids is reasonably straightforward, just a few comments on the adopted procedures should suffice.At log T eff values above some minimum that is common to all of the ZAHBs (e.g., over the range in temperature between the two vertical dashed lines in Fig. 1a), interpolations are carried out at constant values of log T eff , in steps of 0.005 dex, using [Fe/H] as the independent variable, to obtain the masses and luminosities at the adopted temperatures, for any metallicity of interest within the range −2.5 [Fe/H] −0.5.The parts of the ZAHBs that extend to cooler temperatures (e.g., to the right of the dashed line at log T eff = 3.8) are divided into 20 equidistant segments, where each increment in "distance" is Once the original ZAHBs in each grid have been interpolated to give the masses, luminosities, and temperatures at the endpoints of these segments, these properties are interpolated using Akima (1970) splines to yield their values at the selected metallicity.The result of this process will be four ZAHBs for the same [Fe/H] value, but for the two grid values of Y and the two grid values of [O/Fe].Linear interpolations are then used, either at constant log T eff values over the temperature range that is common to the ZAHBs for different Y and the same oxygen abundances, or at constant mass at cooler temperatures, to obtain the ZAHBs for the He abundance of interest.In the final step, the latter are similarly interpolated, using [O/Fe] as the independent variable instead of Y, to obtain the ZAHB for the desired chemical abundances ([Fe/H], Y, and [O/Fe]).
The solid curves in Figure 2 represent the same ZAHBs from Fig. 1a for [Fe/H] = −2.5, −1.7, and −0.9 after they have been transposed to the (M F606W − M F814W , M F606W )-diagram using the BCs given by VandenBerg et al. (2022a), which differ only slightly from those supplied by Casagrande & VandenBerg (2014) for filters at optical and longer wavelengths.(In the case of ultraviolet filters, the 2022 BCs differ significantly from, and improve upon, earlier transformations; see VandenBerg et al. 2022b.)To illustrate the effects of varying the metal abundance, as well as the amount of mass loss that would have occurred prior to reaching the ZAHB, several points have been identified along each curve.The reddest of these points give the predicted ZAHB locations of 12.5 Gyr stars if no mass loss is assumed to occur during the preceding evolution.Thus, for instance, HB stars in 12.5 Gyr GCs with [Fe/H] = −2.5 should all have intrinsic M F606W − M F814W colors < ∼ 0.19, whereas similar HBs in coeval clusters with [Fe/H] = −1.7 could be as red as (M F606W − M F814W ) 0 = 0.60.
The consequences of mass loss, in increments of 0.04M ⊙ , are indicated by successive bluer points along each ZAHB, in the direction from right to left.If, e.g., red giants in a 12.5 Gyr GC with [Fe/H] = −2.5 were to lose between 0.0 and 0.16M ⊙ during their RGB evolution, they would be expected to populate the ZAHB over the ranges in magnitude and colour between the brightest and faintest filled circles in black.Moreover, 12.5 Gyr clusters with [Fe/H] = −1.7 and −0.9 are not predicted to have any blue HB stars unless they lose > ∼ 0.16M ⊙ or > ∼ 0.28M ⊙ , respectively, before arriving at their respective ZAHB locations.It is also apparent from the density of points along higher metallicity ZAHBs that, once the mass loss has exceeded some threshold, further incremental increases result in large separations of the respective ZAHB models (see, e.g., the results for [Fe/H] = −0.9).Thus, most of the HB stars in metal-rich clusters will necessarily be concentrated in red clumps, while the HBs of intermediate-metal-poor GCs can be expected to span fairly broad colour ranges.The plots of cluster HBs that are presented in § 4 leave little doubt as to the overriding importance of metallicity and star-to-star mass-loss variations in determining their distributions as a function of colour -something that has been known since the pioneering work by Rood (1973).
It has long been recognized, as well, that helium, [CNO/Fe], and age may play important roles in determining the morphologies of observed HBs (e.g., Rood 1973, Lee et al. 1990, and the review by Catelan 2009, and references therein).Higher Y, which has the effect of increasing the luminosities of HB stars and shifting their distributions to bluer colours, as shown in Fig. 2a, could well be primarily responsible (see, e.g., Caloi & D'Antona 2005, DVKF17) for the extended blue HB tails that are characteristic of many GCs; see Brown et al. (2016) for plots of the HB populations of the GCs that were included in the HST UV Legacy Survey carried out by Piotto et al. (2015).However, this complication should not be a concern for the determination of GC distances from ZAHB models, provided that a significant fraction of the cluster stars have close to the primordial He abundance; recall the discussion of this point in § 1.The bottom panel of Fig. 2 shows that bluer HBs can also be produced in GCs if they have reduced O, or more generally, lower C+N+O abundances, but such variations clearly have only minimal effects on the ZAHBs themselves.As already mentioned, ZAHBs are age-independent, though higher ages imply lower TO masses and therefore somewhat bluer HB distibutions.
An important feature of CMDs that are constructed from optical and longer wavelength observations is that the locations of the blue tails of ZAHBs (at, e.g., M F606W − M F814W < ∼ 0.05 in Fig. 2) are only weakly dependent on [Fe/H], Y, and [O/Fe], especially at very low metallicities ([Fe/H] < ∼ −1.7).As a result, fits of very blue HB populations to ZAHB loci depend almost entirely on their apparent distance moduli and reddenings.However, because ZAHBs are so steeply sloped at blue colours, even small errors in the reddenings will translate to large errors in the inferred distance moduli unless other constraints are invoked.Such a constraint is suggested by Figure 3, which shows that the upper-MS portions of isochrones, when plotted on the same CMDs that are constructed from F606W, F814W observations, have very little dependence on the assumed chemical abundances; at M F606W > 5.0, in particular, the isochrones for [Fe/H] < −1.9 are indistinguishable, while the colour separations of iscochrones for higher metallicities that differ by 0.2 dex amount to only a few thousandths of a magnitude.
The differences in the predicted M F606W − M F814W colours at M F606W = 5.5 are listed in Table 2 for isochrones which are very similar to those that appear in Fig. 3, except for the assumption of a slight variation of Y with [Fe/H], as noted therein.According to these results, the median MS fiducial sequences of GCs with [Fe/H] values between −2.5 and −2.3 (and the same values of [α/Fe]) are expected to have the same colour to within 0.0017 mag.Even in the case of clusters with −1.7 [Fe/H] −1.5, the median (M F606W − M F814W ) 0 colour at M F606W ≈ 5.5 should not differ by more than 0.0057 mag.In fact, cluster-to-cluster variations  Milone et al. 2018, their Table 4).At ∼ 1.5 mag below the TO, the CMD locations of isochrones are obviously independent of age.
Despite sharing many attributes (steep slopes, little or no sensitivity to age or metallicity), there is one crucial difference between the blue tails of ZAHBs and the portions of isochrones that are relevant to upper-MS stars; namely, the blue tails of ZAHBs have negative slopes while isochrones have positive slopes.Consequently, if a higher reddening were adopted, the fitting of an observed HB to the same ZAHB would yield a smaller distance modulus, while the fitting (at M F606W > 5.0) of the MS stars in the same GC to the MS fiducial sequences of other GCs with similar metallicities would result in a larger distance modulus.In principle, therefore, consistent fits of both the HB and MS populations in GCs are possible only for certain specific values of E(B − V) and (m − M) V .
The main point to be drawn from Fig. 3 is that, when the HB populations of clusters of very low metallicities are fitted to ZAHB loci, the inferred reddenings and distance moduli must be such that their MS fiducial sequences superimpose one another.Importantly, the approach used here to determine these properties have very little dependence on the cluster metallicities and their uncertainties, and no dependence whatsoever on the cluster ages.The next section describes in more detail the methods that have been used to determine the apparent distance moduli of GCs.

CAVEATS AND METHODOLOGY
When VBLC13 compared their ZAHBs, which are very similar to those computed for this study, with the HST Advanced Camera for Surveys (ACS) photometry obtained by Sarajedini et al. (2007, hereafter SBC07), they found that the models matched the lower bounds of the observed distributions of HB stars quite well when reddenings from the Schlegel et al. (1998, hereafter SFD98) dust maps and metallicities given by Carretta et al. (2009, hereafter CBG09) were adopted.In particular, there were no obvious discrepancies between the predicted and observed colours along the bluest HBs.However, it was necessary to apply a blueward ad-justment to the isochrones that were fitted to the upper MS and TO observations by δ(M F606W − M F814W ) ∼ 0.015-0.03mag in order to reproduce the TO colours.Although the main cause of such discrepancies was not known at the time, the new reduction and calibration of the same data by NLP18 has largely eliminated this problem (see VandenBerg et al. 2022b).
Since this recalibration also impacts the colours of HB stars, ZAHB models are no longer able to reproduce their CMD locations quite as well.To obtain fits of observed HB populations to ZAHB loci that are similar to those reported by VBLC13, it is necessary to adjust the predicted M F606W − M F814W colours by ≈ +0.012 mag.Unfortunately, this offset varies from cluster to cluster by a few to several thousandths of a magnitude, even when considering GCs that have the same metallicities to within 0.1 dex.Similar differences exist between the observed photometry of upper-MS stars given by SBC07, on the one hand, and by NLP18, on the other; e.g., the median TO colours of NGC 5053, M 92, and M 68 that are derived from the 2007 and 2018 investigations differ by 0.013, 0.015, and 0.025 mag, respectively.Although such differences are within the photometric uncertainties tabulated by NLP18, they present a major problem for any study, such as the present one, that requires highly homogeneous photometry.It is clear from the discussion of Figs. 2 and 3 and Table 2 that differences in the observed (M F606W − M F814W ) 0 colours as small as ∼ 0.002 mag matter; i.e., the adopted photometry must be exceedingly accurate in a differential sense.
Considerable time and effort was expended in examining both the original ACS F606W, F814W observations provided by SBC07 and the recalibrated versions of the same data given by NLP18.For most GCs, the HB photometry from the latter source could be fitted quite well to the ZAHB models (aside from the small zeropoint offset mentioned above) and it was usually, but not always, possible to obtain satisfactory superpositions of the MS fiducial sequences of clusters of similar [Fe/H] on the assumption of the best estimates of the foregound reddenings.However, satisfactory consistency could not be found for several clusters (including, e.g., M 68, M 2, NGC 6752).Because no such difficulties were encountered in similar analyses of the original ACS photometry by SBC07, the decision was made to adopt these data in this investigation.Interestingly, one consequence of this choice is that good fits of the observed HB populations to ZAHB models could be obtained on the assumption of the reddenings from dust maps, or very close to them.Both of these findings indicated a clear preference for the SBC07 observations, which appear to be very homogeneous. 2n this investigation, GCs have been separated into a number of different metallicity bins.For those clusters with the bluest HBs, initial estimates of (m − M) V were obtained by fitting their HB populations to suitable ZAHB models, on the assumption of reddenings from dust maps.As reported by Schlafly & Finkbeiner (2011, hereafter SF11), their reddening maps yield E(B − V) values that are ≈ 86% of those given by the earlier SFD98 maps.However, roughly half of the reduction can be attributed to differences in the spectral energy distributions of the stars that are used to derive the extinction law.As a result, the current best estimate of the nominal reddening of a given cluster -i.e., the E(B − V) value that applies to early-type stars, which is the usual convention for the majority of reddening determinations, including those by SFD98 -will be close to , which is equivalent to the average E(B − V) value from the two reddening maps. 3With this choice, the actual colour excesses that apply to the upper-MS populations in GCs can be calculated using the values of R η given by Casagrande & VandenBerg (2014, their Table A1), where R η = A η /E(B − V) and A η is the extinction in the η bandpass.Hence, for filters η and ζ, Because the reddening produced by a given amount of dust is larger for stars of earlier spectral types, it is important to use appropriate values of R η when dealing with blue HBs.Unfortunately, as already noted, the BCs provided by Casagrande & VandenBerg (2014) to take reddenings up to E(B − V) = 0.70 into account, are limited to stars with T eff 8000 K.However, Figure 4 shows that the dependencies of R F606W , R F814W , and R F606W − R F814W on T eff are sufficiently weak and well defined that the relations which are obtained from ZAHB models with T eff < 8000 K can be reliably extrapolated to higher temperatures -specifically to T eff = 10, 000 K, which is close to the temperature at which ZAHB loci on optical CMDs make a transition from steep blue tails to nearly horizontal morphologies.Based on the results shown in Fig. 4, the extinctions and the reddenings of observed HB stars located near the tops of blue tails have been calculated assuming R F606W = 2.977 and R F814W = 1.895.The robustness of these determinations is indicated by the fact that the differences in the extrapolated values of R F606W and R F814W are in excellent agreement with the extrapolated values of R F606W − R F814W at T eff > 8000 K. (Since F336W, F438W observations also play an important role in this investigation, the R η values for these filters have been similarly extrapolated, resulting in R F336W = 5.148 and R F438W = 4.147 at T eff = 10, 000 K.) Once initial estimates of (m − M) V have been derived for the most metal-deficient group of clusters from overlays of their blue HBs onto appropriate ZAHB models, assuming dust-map reddenings, their upper-MS fiducial sequences are intercompared to determine whether they superimpose one another sufficiently well to be consistent with the theoretical results that are presented in Fig. 3.If this constraint is not satisfied, the initial estimates of (m − M) V and/or E(B − V) are adjusted in order to obtain the expected consistency.Fortunately, the second step in this two-step process usually yields just a minor refinement of the cluster properties (see the next section), which indicates that the ZAHB models do quite a good job of matching blue HBs without having to take into account the MS constraints.In fact, consistent results for the HB and MS populations of most of the GCs considered in this study could be obtained on the assumption of the reddenings from dust maps that are within the ranges implied by their 1 σ uncertainties, which are typically < ∼ 0.002 mag for nearly unreddened systems.
For those GCs with red HB populations, the derived values of (m − M) V that are obtained from the superpositions of the observed HB stars onto appropriate ZAHB models have very little dependence on the adopted reddenings (because ZAHB loci and observed HBs are nearly horizontal) or on the assumed metallicities.Indeed, comparisons of the cluster CMDs in the vicinity of the TO are not needed to constrain the fits of cluster HBs to ZAHB models when the observed HBs span broad colour ranges, though they do provide a useful check of the relative cluster metallicities given by CBG09.The most challenging clusters to analyze are those with the bluest HBs, such as M 13 and NGC 6752.However, it turns out that (M F336W − M F606W ) 0 colours provide valuable constraints on the relative reddenings and apparent distance moduli of such systems.This will be demonstrated in the next section, where many intercomparisons of UV-optical CMDs are presented and discussed.5 illustrates the results that are obtained for GCs in the two lowest metallicity bins when (i) their HBs are fitted to ZAHBs for [Fe/H] = −2.4 and −2.2 (in panels a-f) or those for −2.1 and −1.8 (in panels h-m), and (ii) their MS fiducials superimpose one another quite well in the magnitude range 5.0 M F606W 6.0; see panels (g) and (n).Given the rather small separations of ZAHBs arising from a 0.2 dex difference in the assumed [Fe/H] values, uncertainties in the cluster metallicities amounting to ±0.1 dex clearly have relatively minor consequences for the apparent distance moduli that are derived from ZAHB models.Note that adjacent tick marks in the bottom panels correspond to a colour difference of only 0.005 mag, which is comparable to or greater than the widths of the bands that encompass the upper-MS fiducial sequences of most of the GCs in each bin.To facilitate examinations of this and similar plots, the average reddenings from the SF11 and SFD98 dust maps and the cluster metallicities from CBG09 are specified in small boxes in each of the smaller panels.According to Carretta et al., for instance, the most metal-deficient clusters that have been considered (M 68, NGC 5053, . ..) all have the same [Fe/H] values to within 0.06 dex.
When considering GCs that have predominately blue HBs, the clusters in each metallicity bin that have the lowest reddenings and/or the reddest HB populations are of particular importance because they provide the most robust fits to the ZAHB models and the most secure definition of the CMD location of the MS fiducial that should be common (or nearly so) to all of the clusters.Fortunately, most of the GCs with [Fe/H] ≈ −2.3 have low reddenings, and it is possible to obtain satisfactory fits of the cluster HBs to ZAHB models, simultaneously with a near coincidence of their MS fiducials, if dust map reddenings, to within their 1 σ uncertainties are adopted.However, it proved to be more difficult to obtain such consistency in the case of NGC 5466.Even if its HB stars are fitted to a ZAHB for [Fe/H] ≈ −2.20 (see panel c), instead of one for the metallicity given by CBG09, its MS is offset from those of the other GCs that have similar metallicities by about 0.005 mag (see panel g).Although this discrepancy could be reduced if a higher reddening were adopted, it is also possible that NGC 5466 has a metallicity closer to [Fe/H] = −2.0than to −2.3, as found in the recent spectroscopic analysis by Lamb et al. (2015).Regardless, the adopted ZAHB-based apparent distance modulus, (m − M) V = 16.03, would not change by more than ≈ 0.02-0.03mag in view of the relatively weak sensitivity of the ZAHB models to [Fe/H].
M 92 also has a very low reddening, and if the best estimate of E(B − V) from dust maps is adopted, its MS stars are found to be just slightly redder than those in M 68, NGC 5053, and M 30 despite having a similar, or slightly lower, metal abundance.Indeed, since M 92 appears to contain stars with enhanced He abundances (see Ziliotto et al. 2023), it probably also has a higher mean value of Y than the other three GCs, which could explain why its HB spans a wider range in colour and extends to bluer colours.Higher Y should result in a bluer MS, though only by two or three thousandths of a magnitude if the mean enhancement is as large as ∆ Y ≈ 0.01 (recall Fig. 3).While this discrepancy may simply be an indication that the reddening of M 92 from dust maps is too low by > ∼ 0.003 mag, an intrinsically slightly redder-than-expected MS could arise if the core H-burning stars rotate sufficiently rapidly; see § 5 for some discussion of this possibility.
The requirement that the MS fiducial seqences of GCs superimpose, or are offset by a fixed amount from, one another places rather tight constraints on the ZAHB-based apparent distance moduli.Suppose, for instance, that a higher or lower reddening by 0.005 mag were adopted for M 92.In order to recover the same CMD location that its MS had on the assumption of E(B − V) = 0.022, it would be necessary to increase or decrease, respectively, the apparent distance modulus by approximately 0.05 mag.However, doing so would cause a significant discrepancy between the cluster HB stars and the appropriate ZAHB models, as illustrated in the two left-hand panels of Figure 6.Clearly, there are too many stars below the ZAHBs in panel (a), whereas the HB stars are somewhat too bright relative to the ZAHBs in panel (b).This occurs because  f) and (h-(i): the adopted fits of the HB populations in several GCs to ZAHB loci (solid curves) that have been generated for [Fe/H] values corresponding to the minimum and maximum metallicities that define each of two bins (identified at the top of the plot).These fits and the superpositions of the cluster MS fiducials in panels (g) and (n) are obtained when the indicated values of E(B − V) and (m − M) V are assumed.The small boxes in each of the smaller panels give, in turn, the cluster metallicities from the spectroscopic survey by CBG09 and the best estimates of the E(B − V) values from dust maps.
the change in the distance modulus that would be needed to compensate for a change in the reddening is in the opposite sense for the fitting of the HB stars to a ZAHB than for the recovery of the original MS location.To improve upon the results that are shown in panels (a) and (b), one would need to adopt intermediate values of E(B − V) and (m − M) V , as in Fig. 5d.Thus, the MS constraint limits the uncertainty of the derived distance modulus to ∼ ±0.03-0.04mag.(The right-hand panels similarly show the implications for the fitting of the HB stars in M 30 to ZAHB models when the cluster parameters are derived solely from the MS fits.)Of all of the GCs in this group, M 15 is the most likely one to have large star-to-star He abundance variations given that such variations would provide a natural explanation for its very extended blue HB.This is suggested by the simulations of the similarly extended M 13 HB carried out by DVKF17, who were able to reproduce the observed morphology if M 13 has a spread in Y amounting to ∆ Y ∼ 0.08 and an average He abundance close to Y = 0.285.If M 15 is similar to M 13 in this respect, its median MS fiducial should be significantly bluer at a given absolute magnitude than those of the other GCs with similar metallicities, and indeed this is found to be the case if E(B−V) = 0.101 (from dust maps) is adopted for M 15 (see Fig. 5g); the 1 σ uncertainty of this determination is If these reddenings are adopted, the superpositions of the cluster MS fiducials are comparable to those shown in Fig. 5 only if the specified values of (m − M) V are adopted.The resultant cluster parameters imply the fits of the cluster HBs to the ZAHB models that are illustrated.
Figure 7.As in Fig. 5e, except for small differences in the adopted cluster parameters (as indicated).0.003 mag.In fact, a moderately large variation in Y is suggested by so-called "chromosome maps" (Milone et al. 2018)  4 and by the analysis of the properties of the many RR Lyrae variables in this cluster by VandenBerg et al. (2016).
M 15 may have a lower reddening, which would reduce, possibly even eliminate, the separation of its MS relative to those of other GCs of similar [Fe/H], but as shown in Figure 7, this would alter the derived value of (m − M) V by only 0.01-0.02mag.The reason why the derived distance modulus is not particularly dependent on E(B − V) is the presence of HB stars with (M F606W − M F814W ) 0 > ∼ 0.05, where ZAHBs have relatively shallow slopes.Even in the small versions of the plots that are shown, it is clear that the ZAHB provide identical fits to the faintest stars with red colours as in Fig. 5e.Indeed, (m − M) V = 15.41 appears to be quite a robust determination of the distance modulus of M 15.
For the 6 GCs considered in this investigation that have −2.1 < ∼ [Fe/H] < ∼ −1.8 (see Fig. 5), the assumption of dust-map reddenings 4 There are some concerns about the reliability of the He abundance spreads that are inferred from such maps.Whereas chromosome maps imply ∆ Y ∼ 0.10 for the 1G stars in M 3 (see Milone et al. 2018) A satisfactory fit of the NGC 6101 HB to ZAHB models could not be obtained on the assumption of E(B − V) = 0.097 (from dust maps), which has a 1 σ uncertainty of 0.005 mag.Such a low reddening would result in the cluster MS being redder than the location of the orange curve in Fig. 5n by more than 0.02 mag.Note that a lower reddening must be accompanied by an increased distance modulus in order to fit the cluster HB stars to the ZAHBs, which exacerabates the discrepancy.)As there is no obvious way of explaining such a red MS, especially when the HB morphology of NGC 6101 is so similar to those of NGC 4147, M 53, and NGC 6397, it would appear that the reddening from dust maps is too low.Better overall consistency is obtained if NGC 6101 has E(B − V) = 0.115 and (m − M) V = 16.11, as shown in panels (j) and (n).Indeed, the UV-optical CMDs that are discussed in the next section support these values of the cluster parameters.
Both NGC 6541 and M 55 have extended blue HBs (especially the former; see Brown et al. 2018); consequently, it can be anticipated that they have higher mean He abundances than the other GCs in their respective metallicity groups.This is, in fact, consistent with the indications from chromosome maps (see Milone et al. 2018).Given the similarity of the blue HBs in NGC 6541 and M 15, in particular, it can be expected that both of their median MS fiducials will be bluer by ≈ 0.01 mag than they would be if all of the member stars had Y ≈ Y P .This is approximately the offset that is obtained between the MS of NGC 6541 and other GCs that have [Fe/H] ∼ −2.0 if NGC 6541 has E(B − V) = 0.134, which is less than the dust-map determination by only 1 σ.
A reddening as high as E(B − V) = 0.145 (from dust maps) seems unlikely because this would result in a near superposition of the MS fiducials of NGC 6541 and M 15.This is untenable because of the ∼ 0.5 dex difference in their metallicities and because the adopted properties of M 15 already imply that its MS is appreciably bluer than those of, e.g., NGC 5053 and M 30 as a consequence of helium abundance differences.On the other hand, if E(B − V) ≈ 0.125 (nearly a 2 σ reduction) and the same distance modulus (i.e., m − M) V = 14.75) were adopted, the fit of the cluster HB stars to the ZAHB models would be somewhat improved over the fit that is shown in Fig. 5l, at the cost of reducing the offset between the MS fiducial of NGC 6541 and those of NGC 4147, M 53, and NGC 6397 to ∼ 0.003 mag.No colour offset whatsoever would be obtained if NGC 6541 has E(B − V) = 0.125 and (m − M) V = 14.77.Thus, the assumed He abundance of NGC 6541 has larger implications for the cluster reddening than for the apparent distance modulus that is derived from the intercomparisons of the main sequences of GCs of similar [Fe/H] and the simultaneous fitting of the observed HB stars to ZAHB models.
M 55 has already been the subject of a thorough investigation of its CMD, its RR Lyrae variables, and its eclipsing binaries by VandenBerg & Denissenkov (2018) using the same or very similar stellar models, from which it was concluded that this cluster has E(B − V) ≈ 0.12 and (m − M) V = 13.95 ± 0.05.Simulations of the M 55 HB, which were also presented in this paper, indicated that the He abundance varies by ∆ Y ≈ 0.02, with a median enhancement of < ∼ 0.01.It can therefore be expected that the cluster MS should be about 0.005 mag bluer than those of, say, NGC 4147 and NGC 6397, and indeed, an offset of about this amount is obtained if the cluster parameters derived by VandenBerg & Denissenkov are adopted (see Fig. 5n).

Constraints from HST UV Legacy Photometry
During the course of this investigation, it was discovered that F336W, F438W observations from the HST UV Legacy Survey provide the means to validate or to refine the best estimates of the relative cluster parameters that are derived from F606W, F814W photometry.This is illustrated in Figure 8, which superimposes the HB populations of M 92 and M 15 on four of the CMDs that can be generated from the Wide Field Camera 3 (WFC3) and ACS observations provided by NLP18.In contrast with optical CMDs, in which the morphologies of observed HBs have steep blue tails at M F606W > ∼ 0.7 (indicated by the horizontal dashed line) and quite shallow slopes at redder colours, CMDs in which the colours involve F336W magnitudes have nearly the same slopes over the full colour ranges that have been considered. 5oreover, the first two panels show that observed HB populations of the two clusters define quite sharp, nearly linear lower boundaries which coincide almost exactly when the specified reddenings and apparent distance moduli are adopted.Indeed, the superpositions of the HBs of M 92 and M 15 in all four CMDs could hardly be improved upon.Achieving such consistency across several different colour planes provides strong support of the relative values of E(B − V) and (m − M) V that have been determined.(Because the CMDs in panels (a) and (d) are qualitatively very similar to those shown in panels (b) and (c), respectively, the former have been dropped from consideration in the additional intercomparisons of NLP18 photometry that are presented and discussed below.It is also a considerable space-saving measure to plot two panels for each cluster, instead of four.) Similar comparisons of the HB of M 92 with those of the other GCs with [Fe/H] ∼ −2.3 are presented in Figure 9.In contrast with the HB populations of M 92 and M 15 (see Fig. 8), the HB stars in NGC 4590 (M 68), NGC 5053, NGC 5466, and NGC 7099 (M 30) exhibit almost no scatter in M F606W at a given (M F336W − M F606W ) 0 colour.A possible, if not the most probable, explanation of the brighter HB stars in M 92 and M 15 is that they are helium-enhanced stars that have evolved from ZAHB locations at (M F336W − M F606W ) 0 < ∼ −0.5 (i.e., further down their blue HB tails).The faintest HB stars in all 6 GCs that superimpose each other along the red edges of the (M F336W − M F606W ) 0 , M F006W diagram when the specified values of E(B − V) and (m − M) V are  adopted probably have very similar He abundances, likely close to the primordial abundance.Clearly, there are no obvious problems with the adopted cluster parameters (in a relative sense) in any of the panels that comprise Fig. 9.
The UV-optical CMDs of GCs with −2.1 < ∼ [Fe/H] < ∼ −1.8, using NGC 5024 (M 53) as the reference cluster, are similarly intercompared in Figure 10.(Plots are not provided for NGC 4147 because this system does not appear to have been included in the HST UV Legacy Survey; see NLP18.)The HB populations are obviously well centered on one another in the (M F438W − M F606W ) 0 , M F006W diagrams, which serves to confirm the relative reddenings, in particular.In fact, the bluest HB stars would be noticeably offset from their counterparts in NGC 5024 if the adopted reddenings of NGC 6101, NGC 6397, NGC 6541, and NGC 6809 differed from the adopted values by more than 0.005 mag, assuming the same distance moduli.
Encouragingly, even though the HBs of NGC 6541 and NGC 6809 extend to considerably fainter absolute magnitudes than the HB stars in NGC 5024, it is quite evident that they superimpose one another remarkably well over the full ranges in their respective luminosities.It is interesting that most of the HB stars in NGC 6397 define a fairly tight sequence in Fig. 10c when the photometric scatter of the same stars is so large in panel (d).Although a few stars in NGC 6397 fall below the distribution of HB stars in NGC 5024 at (M F336W − M F606W ) 0 ≈ −0.15, the majority of them support the specified cluster parameters.This serves to confirm the adopted fit of the cluster HB to appropriate ZAHB models in Fig. 5k.
According to Baumgardt & Vasiliev (2021), the best estimate of the distance of NGC 6397 from Gaia EDR3 parallaxes is 2.458 ± 0.059 kpc, which corresponds to (m − M) 0 = 11.95 ± 0.05.Brown et al. (2018) had previously obtained a true distance modulus of 11.89 ± 0.07 from a trigonometric parallax based on HST WFC3 observations.If E(B − V) = 0.175 and A V = 3.13 E(B − V) (Casagrande & VandenBerg 2014), the apparent distance modulus derived here (12.49) implies a true distance modulus of 11.94, which is in excellent agreement with both of the direct geometric determinations.Provided that the stellar models accurately reproduce the properties of observed HB stars, the 1 σ uncertainties of ZAHB-based distance moduli for the majority of the lowest metallicity GCs should not be larger than ∼ ±0.03 mag, especially if they contain HB stars that populate the transitions of ZAHBs to redder colours; otherwise it is not possible to achieve consistent simultaneous fits of both their HB and MS observations.This conclusion is supported by the intercomparisons of UV-optical CMDs.
4.2 GCs that have −1.7 < ∼ [Fe/H] < ∼ −1.5 Figure 11 presents the fits to ZAHB models of the observed HBs of several GCs with metallicities within the range −1.7 < ∼ [Fe/H] < ∼ −1.5.Most of the clusters that are considered in the top half of the plot have moderately large numbers of both blue and red HB stars, spread out over wide colour ranges, that can be matched to ZAHBs quite easily. 6Even if their metal abundances (from CBG09) are in error by ±0.10 dex, which is possible, the distance moduli that are derived from such fits would not change by more than ±0.02 mag.Errors in the adopted E(B − V) values or in the predicted temperatures or colours of the stellar models are of little consequence.Whether or not the MS fiducials of clusters with the same or similar metallicities superimpose one another is largely irrelevant for the derived distance moduli as well, though any inconsistencies that are found would imply that our understanding of the problematic GCs is deficient in some respect.
According to Table 2, the difference in the (M F606W − M F814W ) 0 colour between isochrones for [Fe/H] = −1.7 and −1.5 is approximately 0.006 mag at M F606W = 5.5.This is comparable with the width of the band in panel (g) that encompases the cluster MS fiducial sequences when the specified values of E(B − V) in panels (a)-(f) are adopted.These estimates are within the 1 σ uncertainties of the reddenings from dust maps, except in the case of NGC 3201 and NGC 6584, for which the dust-map determinations are uncertain by ±0.016 and ±0.006 mag, respectively.To further investigate this discrepancy, isochrones were fitted to the cluster turnoffs assuming the adopted reddenings and distance moduli.The resultant plots revealed that NGC 3201, M 3, and NGC 6584 are coeval to within ±0.4 Gyr and, moreover, that they have the same intrinsic TO colours to within 0.003 mag -as expected given the similarity of their metallicities and ages.This test provides compelling support for the adopted reddenings of NGC 3201 and NGC 6584 over the E(B − V) values determined from dust maps.Worth pointing out is that, when the dust-map reddening is assumed, the MS of M 10 is bluer by about 0.006 mag than those of the other GCs at the same absolute magnitude (see panel g), which suggests that it may have a high median value of Y.Although the fitting of the HB stars in M 10 to the ZAHB models is especially uncertain, UV-optical CMDs (to be discussed shortly) support the adopted cluster parameters.
The 6 GCs considered in panels (h)-(n) of Fig. 11 all have very extended blue HBs that, in optical CMDs, reach fainter absolute magnitudes than their TO stars.They are among the clusters that were given the highest, "Category 5" ranking by Brown et al. (2016) in recognition of their extreme HB morphologies.As already mentioned in § 2, a considerable effort has been made by researchers over the years to determine why the HBs of some clusters 6 M 10 was included in this group mainly to fill the available space, but also because its HB does not extend as far to the blue as those of the clusters that are considered in panels (h) to (m).
with apparently very similar metallicities, such as M 3 and M 13, are so different.At the moment, helium seems to be the leading contender for the so-called "second parameter" (after metallicity, which is the first parameter) insofar as it is possible to simulate observed HBs rather well if star-to-star spreads in Y vary from cluster to cluster -though variations in mass loss also appear to play a critical role in the resolution of this problem; see DVKF17, who also provide quite an extensive review of this subject.
The HBs of NGC 5286 and NGC 5986 resemble those of M 15 and NGC 6541 -all of which are Category 5 clusters.If, within these systems, Y varies from ≈ 0.25 (close to the primordial value) to ∼ 0.30-0.33,such that the median He abundance is ∼ 0.28, their upper-MS stars can be expected to have significantly bluer (M M606W − M F814W ) 0 colours, at the same M F606W , than those of GCs with small He abundance variations.Relatively large offsets are obtained, in fact, if the reddenings of NGC 5286 and NGC 5986 are close to dust-map determinations.On the other hand, chromosome maps indicate that the average He abundances of the 2G and 1G populations in these systems differ by ∆ Y < ∼ 0.007, as compared with ∆ Y ∼ 0.021 in the case of M 15 (Milone et al. 2018).If these findings are trustworthy, the CMD locations of the MS fiducials of NGC 5286 and NGC 5986 should not be very different from those of NGC 5272 and NGC 6584, which can be accomplished if 1 σ reductions of the E(B − V) values from dust maps are adopted.As it turns out, such reductions result in quite satisfactory fits of the cluster HB stars to ZAHB models, as shown in Fig. 11h,i.Note that, whether or not the lower or higher reddenings are assumed, the derived apparent distance moduli do not vary by more than 0.02-0.03mag, especially in the case of NGC 5286 because its HB extends to fairly red colours.
If the reddenings from dust maps to within their 1 σ uncertainties are adopted for M 13, M 70, NGC 6752, and M 2, the median MS fiducial sequences of all four GCs would be nearly coincident or intrinsically very slightly redder than that of M 3, despite having lower metallicities, when their HB populations are fitted to ZAHB models.(If the assumed reddenings of NGC 5286 and NGC 5986 were reduced by a further 0.007 mag, their MS fiducials would also superimpose those of the other four GCs.)Because of the constraints provided by UV-optical CMDs (see below), the decision was made to adopt a higher reddening for M 2 by 0.01 mag, in which case the fit of the cluster HB population to ZAHB models yields (m − M) V = 15.46 (see Fig. 11m).A plausible justification for a blueward offset of its MS (see panel n) is the discovery by Milone et al. (2015a) of at least seven chemically distinct stellar populations in M 2 with He abundance variations as high as ∆ Y ∼ 0.07.However, according to chromosome maps (see Milone et al. 2018, their Table 4), there are no statistically significant differences in the average He abundances between the 2G and 1G stars of M 2, M 3, or M 13.
On the other hand, HB simulations are able to reproduce the observed HBs of M 3 and M 13 if the former has a mean He abundance Y avg = 0.255 and a spread ∆ Y ≈ 0.01 while the latter has Y avg = 0.284 together with ∆ Y ≈ 0.08 (see DVKF17).In this scenario, the MS stars in M 13 and other clusters that have similar HB morphologies should be intrinsically bluer by ∼ 0.01 mag at M V > ∼ 5.0 than the MS stars in M 3 at the same absolute magnitudes.However, to achieve this, it would be necessary to adopt E(B − V) values that are higher than dust-map estimates by approximately 0.01 mag even though the 1 σ uncertainties of these determinations are only 0.001 mag for M 13 and NGC 6752 (and M 2) as compared with 0.005 mag for M 70.This seems rather unlikely, especially as the adopted values of E(B − V) for nearly all of the clusters con- sidered thus far have been in good agreement with, or less than, the reddenings from dust maps.To reiterate this point: it would be quite odd if M 2, M 13, M 70, and NGC 6752, which all happen to have exceedingly blue HBs, all have reddenings that exceed dustmap determinations by such large amounts.This anomaly calls into question the possibility that the very extended blue HBs that are characteristic of second-parameter clusters are mainly due to large star-to-star He abundance variations.the same age, helium abundance, metallicity, and C+N+O abundance, but whereas the former assumes the a4CNN mixture with reduced C by 0.3 dex and increased N by 1.13 dex, the latter (i.e., the a4ONN mix) assumes lower abundances of both C and O by 0.8 dex and higher N by 1.48 dex.Note that the solid black curve is shifted to the location of the green curve if Y is increased from a value of 0.251 to 0.29, or to the location of the dotted blue curve if the metallicity is reduced from [Fe/H] = −1.50 to −1.70.
Empirical support for the metallicity dependence is provided in panel (a), which shows that, at a fixed absolute magnitude, the MS of NGC 1261 is appreciably redder than that of M 3 (NGC 5272), which, in turn, is significantly redder than the MS of M 55 (NGC 6809.This is consistent with a variation in their [Fe/H] values from ≈ −1.3 (NGC 1261) to ≈ −1.5 (M 3) to ≈ −1.9 (M 55).According to the stellar models (see panel d), an increased He abundance by ∆ Y ≈ 0.04 has the same effect on the colours of MS stars at M F606W > ∼ 5.8 as a reduction in the metallicity by ∆ [Fe/H] ≈ 0.2 dex.It is also apparent in this panel that some mixtures of C and N can affect (M F336W − M F606W ) 0 colours in a way that would tend to compensate for the effects on the colours due to enhanced helium or a reduced metallicity.
Consider M 13 (NGC 6205) and M 2 (NGC 7089), and suppose that the basic parameters of M 13 are such that its MS is about 0.01 mag bluer than the MS of M 3 at a given absolute magnitude, similar to what is shown for M 2. Since most spectroscopic studies have found that these two clusters are somewhat more metal deficient than M 3 (e.g., Kraft & Ivans 2003, CBG09), and if both M 2 and M 13 have higher mean He abundances than M 3 by ∆ Y ∼ 0.035 to explain their very blue HB morphologies, their main sequences should be bluer, at M F606W > ∼ 5.8, than the MS of M 3 (see Fig. 12d).However, the observations indicate otherwise.The MS fiducials of M 3 and M 13, in particular, are essentially coincident (see panel b), which suggests that there is little or no difference in either their metallicities or their helium contents -though this conclusion obviously depends on the relative distance moduli of the two GCs.If M 13 has (m − M) V ≈ 14.45 and E(B − V) = 0.017 (see Fig. 11j), its MS would lie slightly above the M 3 MS at redder colours.This might be explained, at least in part, by the observation that there are stars in M 13 with higher C+N abundances than in M 3 (see Cohen & Meléndez 2005), even though both clusters are known to have the same C+N+O abundance to within measurement errors (also see Smith et al. 1996).However, it would be suprising if the difference in the mean CN strengths of the MS stars in M 3 and M 13 is large enough to cause very much of an offset between the median fiducial sequences of these two clusters on the (M F336W − M F606W ) 0 , M F606W diagram.
While the separation between the M 3 and M 2 fiducial sequences at M F606W < ∼ 5.2 is consistent with about a 0.2 dex difference in [Fe/H] (see panel c), it does not continue to fainter magnitudes, in contrast with the M 3-M 55-NGC 1261 intercomparisons in panel (a).This may be suggesting that M 2 has a different abundance of C+N+O (or perhaps a range in the C+N+O abundance) and/or different variations in the ratios of C:N:O than M 3.In fact, Yong et al. (2014, also see Lardo et al. 2012) have found extreme variations in the observed CN and CH line strengths in spectra of M 2 stars.Unfortunately, due to the current lack of model atmospheres and synthetic spectra that assume wider ranges in both the C+N+O abundance and the ratios of C:N:O than those considered by VandenBerg et al. (2022a), it is not possible to generate suitable isochrones to further investigate this issue.Note that, although the differences between the M 2 and M 3 CMDs in panel (c) could be an indication that the distance modulus of M 2 is too high, any reduction in the value of (m− M) V would present a problem for the fitting of the reddest HB stars in M 2 to ZAHB models (see Fig. 11m).
The intercomparisons of blue HB populations on UV-optical CMDs provide valuable consistency checks of the relative cluster properties that are implied by fits of observed HB populations to ZAHB models.In particular, such plots provide important connections between GCs that have, and those which do not have, red HB components that are easily fitted to ZAHB models.As shown in panel (a) of Figure 13, the blue HB stars in M 13 and M 3 superimpose each other almost exactly if M 13 has E(B − V) = 0.017 and (m − M) V = 14.45 and M 3 has the indicated properties.However, this does not rule out the possibility that M 13 has E(B − V) = 0.024 and (m − M) V = 14.42 because an increased reddening by 0.007 mag compensates for a 0.03 mag reduction in the apparent distance modulus, thereby displacing the HB stars along the the same line as the M 3 HB.To choose between these two or alternative possi- bilities, it is necessary to know a priori the relative CMD locations of the MS fiducials of M 13 and M 3 at a fixed absolute magnitude.
The near coincidences of the the blue HB stars in NGC 6681 and NGC 6752 onto their counterparts in M 13 (panels i, k) are really quite remarkable.Even at M F606W > 1.5, they lie on top of one another and exhibit exactly the same morphologies.While there are no obvious difficulties in similar plots for NGC 6254 and NGC 6934, one has the visual impression that better matches of the NGC 5286 and NGC 5986 HBs to that of M 13 would be obtained if the former had higher reddenings and/or a larger distance moduli.However, as already mentioned, such changes to the basic cluster parameters would result in somewhat less satisfactory fits of the cluster HB stars to the ZAHB models than those shown in Fig. 11h,i.Some of the scatter of the NGC 5286 HB can probably be attributed to the star-to-star metal abundance variations that were recently detected by Marino et al. (2015) or, in the case of NGC 5986, to the possible presence of differential reddening.Regardless, the comparisons of the (M F336W − M F606W ) 0 colours of the cluster HB stars that are presented in Fig. 13 are generally very supportive of the adopted reddenings and distance moduli.
The same can be said of the (M F336W − M F438W ) 0 , M F606W diagrams, except for the one given in panel (p), which shows that there is a significant offset between the M 13 and M 2 HBs.This difficulty cannot be resolved by simply increasing the assumed reddening of M 2 because a higher value of E(B− V) would introduce a significant discrepancy between the cluster HBs in panel (o).Curiously, a similar comparison on the (M F606W − M F814W ) 0 , M F606W diagram (not shown) suggests that, if anything, M 2 has E(B − V) < 0.052; i.e., the discrepancy between the HB stars in M 13 and M 2 is in the opposite sense, though only marginally.Thus, whatever is responsible for the apparent separation of the HBs of M 13 and M 2 in panel (p) has something to do with the F438W photometry.The apparent mismatch of the (M F438W − M F438W ) 0 colours might be due to chemical abundance differences since M 2 is an especially peculiar GC in terms of the number of chemically distinct stellar populations that it contains and the wide range of its chemical properties (Milone et al. 2015a).It should also be kept in mind that small cluster-to-cluster differences in the photometric zero points of the NLP18 photometry may have some impact on the intercomparisons of UV-optical CMDs that are presented in this study.

GCs that have
The fitting of the HB populations in 5 GCs that have −1.4 < ∼ [Fe/H] < ∼ −1.2 to the relevant ZAHB sequences are presented in Figure 14.Aside from NGC 288 and M 12, they have significant numbers of red HB stars that are easily matched to ZAHB models.The inferred values of (m − M) V from such fits clearly have very little dependence on the assumed reddenings, though the lat- ter are needed for the determination of true distance moduli; consequently, it is of some importance to adopt the best possible estimates of the E(B − V) values.However, as for the more metaldeficient GCs that have been considered in this study, quite agreeable results are obtained when the reddenings from dust maps to within their 1 σ uncertainties are adopted for most of the clusters.As shown in panel (h), the MS fiducial sequences of NGC 362, NGC 1261, NGC 1851, and M 5 define quite a narrow band on the (M F606W − M F814W ) 0 , M F606W diagram, which is to be expected given they have nearly the same metallicities.
The MS of NGC 2808 is the only one that is significantly offset from the others, but this is to be expected as well.NGC 2808 is another Category 5 cluster with a very extended blue HB that has clearly had a very complex chemical evolution history (see Milone et al. 2015b).Moreover, it has spectroscopically confirmed He abundance variations by up to ∆ Y = 0.09, relative to the primordial helium abundance (Marino et al. 2014).The MS of NGC 2808 should, therefore, be bluer than those of other GCs with similar or even somewhat lower metallicities when the cluster HB is fitted to appropriate ZAHB models, and this is precisely what is obtained when the dust-map reddening (to within 1 σ) is adopted.However, the offset of the blue dashed curve relative to the others that have been plotted in Fig. 14n is larger than one would expect, particularly if NGC 2808 has a lower metallicity than most of the other clusters by ≈ 0.1 dex.Better consistency would be obtained if it has [Fe/H] = −1.29 (Kraft & Ivans 2003) and a smaller reddening, say E(B − V) = 0.200, which is still within the 2 σ error bar of the dust-map value.
The CMDs of NGC 288 and M 12 are much less straightforward to understand.Because these Category 4 clusters have very blue HBs, their apparent distance moduli as derived from fits of their HB stars to ZAHB models are very dependent on the cluster reddenings.If the E(B − V) values from dust maps and ZAHBbased distance moduli are adopted, the MS stars in NGC 288 and M 12 would be redder than their counterparts in M 5, which has the same [Fe/H] value to within 0.01 dex according to CBG09.This adds to the considerable evidence presented in the previous section that second-parameter clusters appear to have anomalously red main sequences for their metallicities.In order for the MS fiducials of NGC 288 amd M 12 to be in close promixity to those of the other GCs with [Fe/H] ∼ −2.3, except NGC 2808, which is known to have high Y, it is necessary to adopt E(B − V) ≈ 0.017 and (m − M) V ≈ 14.82 for NGC 288 and E(B − V) ≈ 0.203 and (m − M) V ≈ 14.12 for M 12 (see panels f-h).Larger reddenings by ∼ 0.01 mag would be needed to produce the sufficiently blue main sequences that would be expected if enhanced He abundances are primarily responsible for their extremely blue HBs.
Valuable support for the adopted parameter values is provided by comparisons of the cluster HBs on UV-optical CMDs.Fortunately, all of the GCs in this group have at least a few very blue HB stars; consequently, such intercomparisons may be used to check the relative values of E(B − V) and (m − M) V .In fact, the fits of the reddest HB stars to ZAHB models in those GCs that have red HB components should yield apparent distance moduli that are accurate in both an absolute and relative sense.As a result, the UV-optical CMDs primarily provide helpful constraints on the relative reddenings of such clusters.M 5 is the obvious choice for the reference cluster because its HB spans a very wide colour range and because it has a relatively low reddening.The various (M F336W − M F606W ) 0 , M F606W diagrams in Figure 15 show that the bluest HB stars in NGC 1261, NGC 1851, and NGC 2808 superimpose their counterparts in M 5 very well when the specified values of E(B− V) and (m− M) V are adopted and M 5 has E(B− V) = 0.034 and (m − M) V = 14.39.7 (As noted above, NGC 2808 may have a metallicity closer to [Fe/H] = −1.3than to −1.2, in which case, the fit of its HB to ZAHB models would yield (m − M) V ≈ 15.60, and it would be necessary to adopt E(B − V) ≈ 0.200 in order to obtain NGC 362 has only a few blue HB stars that apparently straddle the M 5 HB.An improved fit of the four stars which lie just below the band that constitutes the M 5 HB could be obtained if the apparent distance modulus of NGC 362 were increased by ∼ 0.02 mag, though Fig. 14a suggests that (m − M) V = 14.85 would be too high.This slight inconsistency might be due to small errors in the photometric zero points.It is interesting that most of the bluest HB stars in NGC 362 are brighter than the majority of the core He-burning stars in M 5, presumably due to enhanced He abundances.
Panels (i) and (k) of Fig. 15 show that the blue HB populations of NGC 288 and NGC 6218 (M 12) superimpose the HB of M 5 almost exactly when the indicated cluster parameters are adopted.
The fact that there is very little scatter about the tight, nearly linear distributions of their HB stars, in stark contrast with NGC 2808 (see panel g), suggests that they contain chemically relatively simple stellar populations.It is surprising that NGC 288, in particular, appears to require a higher reddening by ≈ 0.005 mag than the dustmap determination, which has a 1 σ uncertainty of 0.001 mag, in order to obtain an acceptable fit of its HB stars to ZAHB models simultaneously with a close match of its MS with those of M 5 and NGC 362 (Fig. 14f,h).Although an even higher reddening and/or an increased apparent distance modulus would tend to improve the fit of the HB stars in NGC 288 with M F606W ≈ 1.2 to ZAHB models, an increased value of (m − M) V by as little as 0.03 mag results in a significant offset between its HB and the HB of M 5 -as illustrated in Figure 16a.A higher reddening by ∼ 0.007 mag would have similar consequences, unless the apparent distance modulus is also decreased by about 0.03 mag.The parameter values that were assumed in Fig. 15i are clearly preferable.
For the most part, the comparisons of cluster HBs on (M F438W − M F606W ) 0 , M F606W diagrams support the adopted cluster parameters, though visual inspections of panels (d) and (j) suggest that the E(B − V) values of NGC 1261 and NGC 288 may be too high.However, as in the case of M 2, which showed a similar discrepancy but in the opposite sense relative to the reference cluster M 13 (see Fig. 13l), no such problems are found when the cluster HBs are compared on (M F606W − M F814W ) 0 , M F606W diagrams.An example of such a plot is presented in panel (b) of Fig. 16, which shows that the bluest HB stars in M 5 and NGC 288 superimpose each other exceedingly well, regardless of small differences in the assumed distance moduli.Evolutionary effects are almost certainly responsible for the enhanced brightnesses of the reddest HB stars in NGC 288 relative to the stars in M 5 with the same intrinsic colours.
The discrepant (M F438W − M F606W ) 0 colours are apparently connected with the F438W photometry.While this problem would be well worth investigating, it does not call into question the results of this investigation, which rely primarily on F336W, F606W, F814W observations.

GCs that have
The fits to ZAHB models of the HB populations of the most metalrich GCs that have been considered in this investigation are shown in Figure 17.Since most of the stars are concentrated in red clumps, it is a relatively straightforward to derive their apparent distance moduli on the assumption of well-supported estimates of their metallicities, though the resultant values of (m − M) V for the clusters with the reddest HBs will have larger uncertainties because their HB stars are matched to the portions of ZAHBs that bend upwards at their red ends.As a result, such fittings have an increased dependence on the uncertainties associated with the reddenings and with the colours predicted by the stellar models.
NGC 6362 and NGC 6723 have almost the same [Fe/H] values according to CBG09 and if the dust-map reddening is adopted for NGC 6362 and E(B − V) = 0.072 for NGC 6723, their MS fiducials are nearly coincident if the ZAHB-based distance moduli are also adopted, as shown in panel (c).VBLC13 found from a similar fitting of the NGC 6723 HB to their ZAHB models that the reddening of this GC according to dust maps has to be too large by about a factor of two; also see Gontcharov et al. (2023), who derived E(B − V) = 0.068 ± 0.01 ± 0.02 (statistical and systematic uncertainties) for NGC 6723 from an analysis of UV to mid-infrared observations.
With regard to more metal-rich systems, there are other con-cerns aside from those mentioned above.Because the effects of ±0.1 dex uncertainties in GC metallicities on the predicted locations of upper-MS stars increase with increasing [Fe/H], the superposition of the fiducial sequences for the cluster upper-MS stars provides a progressively weaker constraint on the reddenings and apparent distance moduli as the metallicity increases.Moreover, comparisons of the cluster HBs on (M F336W − M F606W ) 0 , M F606W diagrams cannot be used to constrain the basic properties of clusters with with [Fe/H] > ∼ −1.0 because such systems contain very few, if any, HB stars on the blue side of the instabiity strip.Fortunately, 47 Tucanae is sufficiently well understood that it can be used as a bridge between clusters of both lower and higher metallicities.
The analysis of the eclipsing binaries in 47 Tuc by Brogaard et al. (2017) indicated a preference for [Fe/H] ≈ −0.70 instead of the CBG09 determination of −0.76.In fact, the higher metallicity is quite possibly the best available estimate given that Kraft & Ivans (2003) obtained [Fe/H] = −0.70 from the equivalent widths of Fe II lines, which are much less affected, if at all, by departures from local thermodynamic equilibrium than Fe I lines.Brogaard et al. also concluded from their study of the binary V69 that 47 Tuc has (m − M) V = 13.30.This is only 0.03 mag smaller than the distance modulus that was subsequently derived from Gaia DR2 parallaxes by Chen et al. (2018) if E(B− V) = 0.030 (from dust maps) is adopted.On the other hand, a somewhat lower value, (m − M) V = 13.27, is favoured by simulations of the cluster HB that successfully reproduce the observed HB on the (M F606W − M F814W ) 0 , M F606W diagram if the He abundance is assumed to vary from Y = 0.257 to 0.287 (see DVKF17).As shown in Fig. 17d, (m − M) V = 13.30and E(B − V) = 0.030 results in a good fit of the faintest and reddest HB stars in 47 Tuc to ZAHB models for [Fe/H] ≈ −0.70.
The horizontal dotted line that has been drawn through these stars in panel (d) has been reproduced in panels (e)-(g).Since M 69 is only slightly more metal rich than 47 Tuc, those HB stars that are closest to their ZAHB locations should lie just below the dotted line, resulting in (m − M) V ≈ 15.20.It turns out that, if E(B − V) = 0.154 (from dust maps) is also assumed, the reddest of these stars have colours that are similar to the ZAHB models for [Fe/H] = −0.60,which is consistent with the metallicity of M 69 that was derived by CBG09.Similarly, the faintest HB stars in NGC 6652 and in M 71 should be matched to the dotted line or located slightly above it, respectively, given that these clusters appear to have the same or a somewhat lower [Fe/H] value than 47 Tuc.Although the assumption of the dust-map reddening to within its 1 σ uncertainty leads to a satisfactory match of the HB stars in NGC 6652 to ZAHB models (panel f), the reddening of M 71 has to be much less than the dust-map determination in order to obtain a similar result for this cluster (panel g).Indeed, a reddening near E(B − V) = 0.23 is required in order for its MS stars to have similar CMD locations as those of other GCs in this group, as shown in panel (h).A higher median He abundance due to chemical selfenrichment presumably explains most of the blueward extension of the MS of 47 Tuc relative to the others.

ON THE IMPORTANCE OF STELLAR ROTATION AND MASS LOSS IN GCS
Some further discussion is warranted concerning the finding that, when the best estimates of the reddenings from dust maps are adopted, the GCs with the bluest HBs in many of the secondparameter pairs, including M 3-M 13 and NGC 288-NGC 362, have somewhat redder main sequences than the clusters in each pair with HBs that are more typical of their shared metallicities.Cluster-tocluster differences in the distributions of stellar rotation can potentially explain this anomaly (see Mengel & Gross 1976), though more recent stellar models that incorporate a much more sophisticated treatment of rotation are not supportive of this possibility (see Deliyannis et al. 1989).While there is little doubt of the importance of rotation in modulating the mass loss that occurs during RGB evolution (Renzini 1977) and/or at the helium flash (Peterson 1982), thereby impacting HB morphologies, it is not expected to have any effect on the temperatures of cluster MS stars.On the other hand, the modeling of rotation involves free parameters that need to be calibrated, and various assumptions must be made concerning the transport of angular momentum in stellar interiors, the loss of angular momentum via stellar winds, etc. (see, e.g., Pinsonneault et al. 1991).Consequently, one cannot say with absolute certainty that the median fiducial sequences for the MS stars in all GCs, especially some of those with extended blue HBs, are unaffected by rotation.
To be sure, it is an important prediction of the models com-puted by Pinsonneault et al. (1991) that metal-poor stars are able to retain sufficient internal angular momentum to explain the rapid rotations that have been determined in several of the blue HB stars in M 13 (Peterson 1983) and in NGC 288 (Peterson 1985).M 13 is quite a remarkable cluster in this regard insofar as approximately one-third of the 32 HB stars from the Peterson (1983) and Peterson et al. (1995) surveys, with temperatures in the range 7000 < T eff < ∼ 11, 000 K, have 20 < ∼ v sin i < ∼ 40 km s −1 ; the mean value of v sin i for the entire sample is 18.1 km s −1 .By comparison, Peterson (1985) obtained values of v sin i ranging from 12 to 22 km s −1 ( v sin i = 16.3 km s −1 ) for six of the seven HB stars that she studied, all of which have temperatures between 7000 and 9000 K.However, all 16 stars with 9000 < T eff < 11, 000 K that were subsequently investigated by Peterson et al. (1995) have v sin i < ∼ 12 km s −1 , resulting in v sin i = 7.5 km s −1 .
It is not clear what to make of the apparent variation of v sin i with T eff in NGC 288, as there is no indication of a similar trend along the M 13 HB.Regardless, the rotational velocities in NGC 288 are much smaller than in M 13 and, in fact, they are not very different from the rotation rates that have been determined for the 22 blue HB stars in M 3 that were studied by Peterson et al. (1995); they have temperatures that are mostly between 7000 and 9000 K (like the cooler group of stars in NGC 288) and v sin i values that range from 2 to 21 km s −1 ( v sin i = 12.4 km s −1 ).This would seem to argue against rotation being the main driver of the very blue HB of NGC 288.However, the latter has a significantly higher metallicity than M 13 and M 3, and more metal-rich giants can be expected to lose more mass than those of lower [Fe/H], perhaps especially if they are rotating, because they evolve along cooler tracks and they reach higher luminosities prior to the He flash.According to, e.g., the empirically based Reimers (1975) formula, which is still widely used in stellar evolutionary computations (e.g., Pietrinferni et al. 2021), the rate at which low-mass giants lose mass is proportional to L 1.5 /T eff 2 .Indeed, if rotation is of sufficient importance, it could delay the He flash (Mengel & Gross 1976), which would allow more mass to be lost (Renzini 1977) and possibly result in somewhat brighter HBs as the result of increased helium core masses at the RGB tip (Sweigart & Catelan 1998).
As shown by Recio-Blanco et al. (2002) and particularly by Behr (2003), globular cluster HB stars with T eff > ∼ 11, 000-11,500 K are slow rotators with few, if any, exceptions.The fairly sharp transition at this temperature from relatively high values of v sin i in cooler HB stars to < ∼ 5 km s −1 in hotter stars -see, e.g., Behr's results for M 13 and NGC 288 in his Fig. 23 -occurs at the same T eff as the so-called "Grundalh jump", where radiative levitations greatly enhance the surface abundances of the metals, causing a jump in the Strömgren u magnitude (see Grundahl et al. 1999).This feature is common to all GCs with HB stars that have T eff > ∼ 11, 500 K, irrespective of metallicity.As suggested by Vink & Cassisi (2002), the increased metal abundances can be expected to result in enhancements of both the mass-loss rates and the concomitant losses of angular momentum via stellar winds.Thus, the hottest HB stars along blue tails that are now observed to be slow rotators probably had much higher rotational velocities at the beginning of their core He-burning lifetimes.
On the other hand, some GCs with extended blue HB populations do not seem to contain any fast rotators with T eff < 11, 500 K, including NGC 2808 and NGC 6093 (M 80); see Recio-Blanco et al. (2002).In the case of NGC 2808, the spectroscopic study by Marino et al. (2014) found that helium varies by up to ∆ Y ≈ 0.09, which can explain the extended HB of this Category 5 cluster.As there are no indications from chromosome maps that such large He abundance variations exist in M 80 (Milone et al. 2018), which is also a Category 5 GC, it is possible that both rotation and helium (to a lesser extent than in NGC 2808) play a role in producing its HB.Recio-Blanco et al. included only three HB stars with T eff < ∼ 10, 000 K in their survey, and since it is not uncommon to find a wide range in the measured v sin i values in some GCs (see Behr 2003), it would be worthwhile to measure the rotational velocities of many more HB stars in this cluster, especially those with lower temperatures.
It is also noteworthy that rotation has been found to be more important in M 92 than in M 68 or in M 15 (Behr 2003), which has the most extended HB of the three GCs.However, just as NGC 2808 has a much bluer MS than other clusters of similar metallicity with redder HB populations, when the dust-map reddening to within 1-2 σ is adopted, so is the MS of M 15 significantly bluer than those of M 92 and M 68 if dust-map determinations of E(B − V) are adopted for all three clusters (see Fig. 5g).This is consistent with M 15 having significantly higher Y than the other two GCs, as found from analyses of chromosome maps (Milone et al. 2018)which also suggest that M 92 has a somewhat higher He abundance, in the mean, than M 68.Although this could explain why the HB of M 92 has a greater blueward extension than that of M 68, the observed differences in their rotational properties and the associated mass loss could also be a factor.Indeed, if M 92 has a higher He abundance, why does it have a slightly redder MS than M 68 when dust-map reddenings are adopted for both GCs?Although the answer to this question could simply be that dust maps underestimate the reddening of M 92, it would be surprising that the dust-map E(B − V) value is problematic for the cluster in the most metaldeficient group that seems to have the highest fraction of rapidly rotating blue HB stars, but not for any other GC in that group (see Fig. 5).
The cluster that is closest to being a "smoking gun" with regard to the notion that rotation and mass loss may be primarily responsible for the very blue HBs in several clusters is NGC 288.Because this system has a higher metallicity than most of the GCs with extremely blue HBs, its RGB stars must undergo especially large amounts of mass loss in order to wind up on the blue HB after the helium flash (recall Fig. 2).At M F606W ≈ 1.0, which is near the top of the blue HB of NGC 288 (see Fig. 14f), a ZAHB star is predicted to have a mass of 0.597M ⊙ , according to the present models for [Fe/H] = −1.3,Y = 0.252, [O/Fe] = +0.6, and [m/Fe] = +0.4 for all other α elements.This may be compared with a mass of 0.876M ⊙ or 0.812M ⊙ , which are the predicted masses of RGB tip stars at an age of 10.0 Gyr or 13.0 Gyr, respectively, if no mass loss occurs.(The difference in mass due to an age difference as large as 3.0 Gyr is clearly largely irrelevant.)NGC 288 has few HB stars fainter than M F606W ≈ 2.7, where the predicted ZAHB mass is 0.542M ⊙ .Hence, the entire observed HB of NGC 288 can be explained if stars in this cluster lose between ≈ 0.25 and ≈ 0.31M ⊙ prior to reaching the HB, assuming a mass of 0.85M ⊙ for the RGB tip precursors.
The very tight sequence of the blue HB stars in NGC 288 on the UV-optical CMD (Fig. 15i) provides compelling evidence that this cluster contains chemically relatively simple stellar populations.Its CMD is in stark contrast with that of NGC 2808 (Fig. 15g), which has large star-to-star He abundance variations and other chemical peculiarities (Marino et al. 2014).Lardo et al. (2018) also found from their examination of chromosome maps that stars in NGC 288 appear to have very homogeneous helium and light element abundances.Since age can play no more than a minor role in producing blue HBs at higher metallicities (as noted above), mass loss, presumably facilitated by high rotational velocities, would appear to be the primary cause of the observed HB of NGC 288.(The same can be said of M 12, given the close similarity of its HB with that of NGC 288; see Fig. 15k.) DVKF17 have already shown that at least part of the explanation for the very different HB morphologies of M 3 and M 13 is a large difference in the mass that is lost from the giants in the two clusters.According to the present models for [Fe/H] ∼ −1.55, which is approximately the metallicity of M 13, its RGB stars would need to lose between ∼ 0.20 and ∼ 0.30M ⊙ in order to account for the observed HB.Although DVKF17 also showed that the full extension of the M 13 HB can be explained if the helium abundances of member stars vary by ∆ Y ∼ 0.08, as compared with ∆ Y < ∼ 0.01 in the case of M 3, this possibility is not supported by analyses of chromosome maps (Milone et al. 2018) or by the fact that the MS stars in M 13 are nearly coincident with, if not slightly redder than, the MS stars in M 3, at the same absolute magnitudes, if it has E(B − V) = 0.017 (from dust maps).The same difficulty applies to NGC 6681 and NGC 6752, which have M 13-like HBs (see Fig. 11).Thus, insofar as the CMDs of at least these three GCs, as well as NGC 288, are concerned, mass loss would appear to be the most important "second parameter".
However, there are some observations that appear to favour a much higher reddening for M 13 than the dust-map estimate.As reported by DVKF17, stellar models are unable to predict the periods of the mostly c-type RR Lyrae variables in M 13 satisfactorily, even if they have E(B − V) = 0.025 and (m − M) V = 14.42, which are close to the parameter values that are needed to reconcile the CMDs of M 3 and M 13 with the differences in Y that are implied by HB simulations.(No such difficulties were found in the analyses of the M 3 RR Lyrae stars in the previous study by VandenBerg et al. 2016.)Because pulsation periods vary inversely with T eff and directly with luminosity (Marconi et al. 2015), improved consistency would be obtained if a higher reddening and/or a smaller distance modulus were assumed for M 13.Clearly, the smaller value of E(B − V) = 0.017 from dust maps and the consequent ZAHB-based distance modulus that has been derived in this investigation, (m − M) V = 14.45, exacerbate the problem.
Fortunately, the permitted values of the basic cluster parameters are constrained by the intercomparisons of their respective HB populations on UV-optical CMDs.For instance, it would generally not be possible to obtain a near coincidence of the blue HB stars in M 3 and M 13, similar to that shown in Fig. 13a on the assumption of smaller values of (m − M) V for M 13, unless suitable increases in the adopted reddening are also assumed.In fact, a visually indistinguishable overlay of the M 3 and M 13 HBs would be obtained if M 13 has E(B− V) = 0.032 and (m− M) V = 14.40, which would also provide a reasonably satisfactory resolution of the RR Lyrae problem.Of course, these results assume that the properties of M 3 are well determined, though similar conclusions are obtained if other GCs are considered.In particular, the apparent distance modulus of M 2 cannot be much smaller than (m − M) V = 15.46 without causing an unsatisfactory fit of its reddest HB stars to the ZAHB models (see Fig. 11m).Reduced distance moduli for NGC 6681 and NGC 6752 are also unlikely because the adopted values of (m− M) V already imply that they are among the oldest of the Galactic GCs (see the next section).
Although the high reddening and short distance modulus seem to be preferred from the perspective of the M 13 RR Lyrae variables, they result in a less than satisfactory fit of the cluster HB stars to ZAHB models at M F606W > 1.3, as illustrated in Figure 18 and highlighted by a comparison of this plot with Fig. 11j.In addition, they cause a blueward offset of the MS of M 13 relative to that of M 3 by ∼ 0.02 mag, making it considerably bluer than the M 2 MS despite the latter having a lower [Fe/H] value and CMD locations of its MS stars that are consistent with enhanced He abundances (see Fig. 11n).Such a large offset would be comparable with the separation in colour between the MS portions of isochrones that have been computed for Y = 0.25 and 0.32, which is larger by about a factor of two than the difference in the mean He abundances of M 3 and M 13 according to the HB simulations carried out by DVKF17.This argues against the parameter values that were assumed in the contruction of Fig. 18.It is more likely that the actual reddening and distance modulus of M 13 lie within the ranges 0.016 E(B − V) 0.024 and 14.42 (m − M) V 14.45, as favoured by DVKF17 and the present study.
Given that stellar models are generally quite successful in reproducing the pulsational and evolutionary properties of globular cluster RR Lyrae stars (see, e.g., VandenBerg et al. 2016VandenBerg et al. , 2018)), why do they fail to do so in the case of M 13?Is this is another consequence of the high number of rapid rotators in this system?Indeed, it would be worthwhile to explore the implications of rotating HB models, not only for variable stars, but also for fits of GC HB populations to ZAHB models.The matching of the NGC 288 HB to the ZAHBs, in particular, is especially problematic because a relatively large number of the HB stars with M V ≈ 1.25 lie below the model computations (see Fig. 14f).It is quite possible that the adopted reddening is too low or that there is a problem with the photometry, but stellar rotation might also have some impact on the luminosities, temperatures, and colours of these stars.

SUMMARY AND DISCUSSION
New grids of ZAHB models for chemical abundances that are relevant to the Galactic GCs have been presented in this study.They have been applied to the HB populations of 37 clusters in order to derive their apparent distance moduli.The very weak dependence of the location of MS stars on the (M F606W − M F814W ) 0 , M F606W diagram, especially at the lowest metallicities, have been used to constrain the absolute values of (m − M) V .Comparisons of the bluest HB stars on UV-optical CMDs have been found to provide particularly valuable constraints on the relative apparent distance moduli of clusters of similar [Fe/H].Although it was not anticipated at the outset of this investigation, good fits of the observed HBs to ZAHB sequences could be obtained for ∼ 75% of the GCs on the assumption of dust-map reddenings to within their 1 σ uncertainties.This provides quite a strong indication that the E(B − V) values from dust maps are generally very accurate.
The main results of this study are listed in Table 3.This gives, for the GCs that are identified in the first two columns, the best estimates of the reddenings and their uncertainties from dust maps, the adopted values of E(B − V), and the derived ZAHB-based apparent distance moduli.True distance moduli can be readily cal- (Casagrande & VandenBerg 2014) using the reddenings given in the fourth column.They differ significantly from dust-map determinations only in the case of a few moderately reddened systems, such as NGC 3201 and NGC 6723, for which the adopted values of E(B − V) are expected to be more accurate.The resultant values of (m − M) 0 have not been tabulated because they are less reliable than the apparent distance moduli that have been determined in this study as a result of their greater dependence on E(B − V).For in- stance, the (m − M) V values that are found from the fitting to ZAHB models of the HB populations in GCs that have significant numbers of red HB stars depend only weakly, if at all, on the reddening.Baumgardt & Vasiliev (2021, hereafter BV21) have recently derived the distances to 162 GCs by combining the results that they obtained using several methods with ∼ 1300 published distance determinations.Only a small fraction of their findings are independent of the reddening; e.g., distances based on Gaia EDR3 parallaxes.In general, they adopted the E(B − V) values from the papers that were included in their survey or from the 2010 edition of the Harris (1996) catalogue of cluster parameters.However, they do not provide any reddening information or their best estimates of (m − M) V ; consequently, there is no other option but to compare (m − M) 0 values.The uppermost panel of Figure 19 shows that the true distance moduli that are implied by the adopted reddenings and the derived (m − M) V values in Table 3 for the 37 GCs that were considered in this investigation are smaller, in the mean, by 0.034 mag than those reported by BV21 for the same clusters.This would imply an age difference of only ≈ 0.34 Gyr if all other factors that affect age determinations are unchanged.As indicated in  Baumgardt & Vasiliev (2021) and the results of this study for the GCs that are listed in Table 3.The few clusters that are explicitly identified are discussed in the text.Panel (b): Differences between the apparent distance moduli derived by VBLC13 and the present findings.The most discrepant results were obtained for M 2, which is explictly identified.panel (a), the standard deviation of the mean is 0.032 mag, which is comparable with the estimated uncertainties of the ZAHB-based apparent distance moduli.
While this level of agreement is really quite good, the scatter in the results is relatively large, especially at [Fe/H] ∼ −1.5.This could be a reflection of the inhomogeneity of the BV21 findings.For instance, both M 5 and NGC 362 have red HB populations that are easily matched to ZAHB models.Whereas BV21 report a distance for NGC 362 that is in excellent agreement with the inferred value from appropriate ZAHB models, their distance for M 5 implies that its HB stars are ∼ 0.08 mag brighter than the same ZAHB.This inconsistency is presumably the consequence of taking an average of many distance determinations over the years that have employed a variety of methods and made different assumptions concerning the cluster reddenings and chemical abundances.Indeed, prior to the revision of GC metallicities by CBG09, the [Fe/H] values given by Carretta & Gratton (1997), which are higher by about 0.2 dex for many clusters, were frequently adopted.This would have the consequence of, e.g., increasing the distances that are derived via the classic MS-fitting technique.The present work has the important advantage that the derived cluster properties are very homogeneous as they are based on the same methods, the same ZAHB models, and the same [Fe/H] scale (CBG09).It is entirely possible that the ZAHB-based distance moduli are too large or too small by a few hundredths of a magnitude, but the difference in the distance moduli of two clusters that have almost the same metallicities and well-defined, nearly horizontal distributions of HB stars, such as M 5 and NGC 362, should be a robust result.
The ∆ (m − M) 0 values for NGC 6934, NGC 6752, and M 13, which have the same metallicities to within 0.03 dex according to CBG09, also span a wide range in Fig. 19a.Of these three clusters, only the HB of NGC 6934 extends to very red colours, with the consequence that the fitting of these stars to the relevant ZAHB, and the resultant value of (m− M) V , involve rather little uncertainty.Fortunately, NGC 6934 also has a sufficient number of very blue HB stars that they may be easily compared with similar stars in M 13 on the (M F336W − M F606W ) 0 , M F606W diagram.As shown in Fig. 13m, they superimpose one another very well if M 13 has (m − M) V = 14.45 and E(B − V) = 0.017 (from dust maps).It would be possible to obtain the same distance modulus that was derived by Figure 20.Fits of isochrones for Y = 0.26 and the indicated ages and cluster metallicities (from CBG09) to the median MS fiducial sequences of M 3, M 70, and NGC 6752 on the assumption of the specified cluster parameters.The M 3 CMD has been reproduced in the middle and right-hand panels to illustrate its location relative to those of the other two GCs.The δ X values specify the colour offsets that were applied to the isochrones in order to obtain the fits to the upper-MS and turnoff observations that are shown; these offsets are discussed in § 3.
BV21 and a comparable match of the blue HB stars in M 13 and NGC 6934 if M 13 has (m − M) V = 14.40 and E(B − V) = 0.030, but such parameter values would present problems for the relative locations of the M 13 and M 3 main sequences (see § 5).Note that the distance modulus of NGC 6752 as determined in this study and by BV21 differ by only 0.01 mag.M 92 and 47 Tuc have been explicitly identified in Fig. 19a because they stand apart from the other clusters in their respective metallicity groups.While an equally good fit of the HB stars in M 92 to the ZAHB models could be obtained on the assumption of a larger distance modulus by ∼ 0.03 mag, which would move the filled circle that represents M 92 close to the points that represent the other GCs with [Fe/H] ∼ −2.3, this would have the consequence of making its MS appreciably redder at a given absolute magnitude than the MS fiducials of, e.g., M 68, M 30, and NGC 5053.This would be contrary to expectations if, as appears to be the case (see Ziliotto et al. 2023), M 92 has a somewhat higher median He abundance than the other clusters.Assuming an increased reddening to compensate for this discrepancy would result in an unacceptable fit of the cluster HB stars to the ZAHB models; recall the discussion of Fig. 6 in § 4.1.Thus, provided that the ZAHB models are trustworthy, the apparent distance modulus of M 92 should be very close to (m − M) V = 14.72, which is obtained, in fact, if the distance of M 92 is 8.501 kpc (from BV21) and it has a reddening E(B − V) = 0.022 (from dust maps).
The previous section has already described the considerable evidence in support of (m − M) V = 13.30for 47 Tuc, but the main point that is relevant for this paper is that this determination is favoured by ZAHB models when current best estimates of the reddening and metallicity are assumed.It is also worth pointing out that, as reported by BV21, Gaia EDR3 parallaxes, corrected for systematic errors, yield a distance of 4.367 ± 0.18 kpc, which corresponds to (m − M) 0 = 13.20 ± 0.09.This is nearly identical with the true modulus that is derived from (m − M) V = 13.30if the dustmap reddening, E(B − V) = 0.030, is adopted.Unfortunately, there are only a few GCs with sufficiently accurate parallax-based distances that yield true distance moduli with uncertainties < ∼ ±0.10 mag.In all such cases, the ZAHB-based distance moduli derived in this study (and by BV21) agree with those determinations to within their uncertainties.
The bottom panel of Fig. 19 shows that the apparent distance moduli derived by VBLC13 differ from the present results by 0.014 mag, on average, for 27 GCs in common to the two investigations.As indicated, the standard deviation of the mean is 0.026 mag.Note that the plots provided by VBLC13 give the values of (m− M) F606W that were derived from fits of the cluster HBs to ZAHB models; they were converted to (m − M) V using the R λ values given by Casagrande & VandenBerg 2014.VBLC13 did not attempt to derive the apparent distance moduli of several clusters with extremely blue HBs, such as NGC 6397 and NGC 6752, but instead determined their ages relative to those of clusters with HBs that could be reliably fitted to ZAHB models using the so-called "horizontal method" of determining relative cluster ages (see VandenBerg et al. 1990).This makes use of the predictions from isochrones for the difference in colour between the TO and the lower RGB as a function of age.
It turns out that very close to the same ages are obtained when isochrones are fitted to the turnoff observations on the assumption of ZAHB-based distance moduli.As shown in Figure 20, M 3 is predicted to have an age near 11.8 Gyr, as compared with ages of ≈ 12.8 Gyr for M 70 and NGC 6752.For the same three clusters, VBLC13 obtained ages of 11.75, 12.75, and 12.50 Gyr, respectively, with 1 σ uncertainties of ±0.25-0.38Gyr.Although not shown, similar plots were produced for 5 other clusters that, like M 70 and NGC 6752, have HBs which are entirely to the blue of the instability strip; except for M 2, their ages also differ by 0.3 Gyr from the determinations made by VBLC13.Such differences are clearly too small to significantly alter the GC age-metallicity relation that was derived by VBLC13 and discussed at some length by Leaman et al. (2013).This is to be expected given the similarity of the derived distance moduli for most of the clusters in common to the two studies (see Fig. 19b).It should be kept in mind, however, that ages are quite sensitive to the assumed chemical abundances; e.g., DVKF17 obtained a higher age for M 3 (12.6Gyr) primarily because they adopted lower values of [Fe/H] and [O/Fe].
The discordant results for M 2 are mainly due to the difference in the adopted values of (m − M) V values (see Fig. 19b).VBLC13 found a larger distance modulus for M 2 by about 0.08 mag because they adopted a lower reddening (E(B − V) = 0.046 from the SDF98 dust maps).Had this reddening been assumed in the present study, the fitting of the cluster HB stars to the ZAHB models would have yielded a distance modulus very similar to the value that was derived by VBLC13.However, it is not possible to obtain a satisfactory superposition of the blue HB stars in M 2 and M 13 on the (M F336W − M F606W ) 0 , M F606W diagram if the cluster parameters given by VBLC13 are adopted.There is no such difficulty if M 2 has E(B − V) = 0.052 and (m − M) V = 15.46 (see Fig. 13o).This example illustrates the valuable constraints on our understanding of GCs that are provided by UV-optical CMDs.
VBLC13 also found somewhat larger distances moduli for the most-metal rich clusters due to a preference for (m − M) V = 14.35 for 47 Tuc from a consideration of its eclipsing binary V69 and an assumption that mass loss along the RGB did not exceed 0.20M ⊙ .Given the ambiguities of fitting the observed HBs to the red ends of ZAHBs, VBLC13 acknowledge that their estimates of (m − M) F606W for clusters with [Fe/H] > ∼ −0.8 could easily be in error by ±0.05 mag.The fact that the distance moduli derived in this paper for three of the four metal-rich clusters are in good agreement with the distances reported by BV21 (see Fig. 19a), which are based on a number of different methods, indicates a preference for the present results for GCs with [Fe/H] > −1.0 over those given by VBLC13.
What is potentially one of the most important results of this investigation is the discovery that the main sequences of many (most?)second-parameter clusters, including M 2, M 12, M 13, M 70, NGC 6752, and NGC 288, nearly coincide with, or are slightly redder than, the MSs of GCs that have almost the same metallicities but much redder HB populations, such as M 3, M 5, and NGC 362, if dust-map reddenings and distance moduli based on ZAHB models are adopted.The same thing would be found for the moderately reddened (E(B − V) ≈ 0.30) clusters NGC 5286 and NGC 5986, which also have extended blue HBs, if they have E(B − V) values that are less than dust-map determinations by only 1.5 σ.
Examples of these findings are presented in Fig. 20, which shows that the indicated cluster parameters lead to very slight redward offsets of the MSs of both M 70 and NGC 6752, which have extended blue HBs, relative to the M 3 MS, even though the latter is the most metal-rich of the three GCs by δ [Fe/H] ∼ 0.1 dex.Provided that dust-map reddenings are reliable, as they appear to be, all three clusters would appear to have similar He abundances, in the mean, as otherwise the MSs of M 70 and NGC 6752 would be significantly bluer than that of M 3. Thus, something other than helium abundance variations must be the primary cause of the very different HB morphologies of clusters that comprise such famous second-parameter pairs as M 3-M 13 and NGC 288-NGC 362though such variations appear to have produced the HBs of, e.g., M 15 and NGC 2808.
Higher ages would tend to promote bluer HBs, though the difference in mass associated with an age difference as high as 2-3 Gyr is much less than the amount of mass that must be lost for stars with [Fe/H] ∼ −1.5 to end up on the blue HB after the He flash (recall Fig. 2).In any case, the extensive surveys of GC ages by Marín-Franch et al. (2009) and VBLC13 both concluded that M 3 and M 13 are nearly coeval and that NGC 288 is older than NGC 362 by < 1 Gyr.Hence, age probably plays no more than a minor role in accounting for the HB morphologies of second-parameter GCs.The same can be said of CNO since spectroscopic work has established that M 3 and M 13 have very similar C+N+O abundances (e.g., Smith et al. 1996, Cohen & Meléndez 2005), as do NGC 288 and NGC 362 (Caldwell & Dickens 1988).
Because (i) mass loss is inconsequential for the CMD locations of MS stars, (ii) stellar rotation can be expected to increase the mass-loss rates along the upper RGB, and (iii) the frequency of rapid rotators has been observed to vary from cluster to cluster (Peterson et al. 1995, Behr 2003), it would not be a surprise if mass loss, facilitated by rotation, is mainly responsible for the observed variations in the HB morphologies of clusters that have essentially the same metallicities -especially given the lack of viable alternative explanations.This would mean that stellar rotation is the dominant second parameter.Whether or not the mean temperatures of the MS stars at a given absolute magnitude are affected by rotation in some of the GCs that have very blue HBs is extremely difficult to determine because such effects are likely to be smaller than those arising from reddening and chemical composition uncertainties.Nevertheless, the present work does provide some tantalizing indications in support of this possibility.

Figure 1 .
Figure 1.Panel (a): grid of ZAHB models for the indicated initial chemical abundances.The dashed vertical lines have been plotted simply to illustrate the approximate temperature range at which interpolations at constant log T eff are used to obtain ZAHB loci for any metallicity within the range −2.5 [Fe/H] −0.5 (see the text).Panel (b): Comparison of BaSTI ZAHB models (Pietrinferni et al. 2021) for the specified [Fe/H] values with interpolated ZAHBs from this study for the same metallicities.Both sets of models assume Y ≈ 0.250 and [m/Fe] = +0.4 for the α elements, including oxygen.

Figure 2 .
Figure 2. Panel (a): comparison of ZAHB loci for the indicated chemical abundances.The reddest points that are represented by the various symbols (i.e., filled circles, open circles, etc.) give the predicted ZAHB locations of 12.5 Gyr stars, assuming that no mass loss occurred during the preceding evolution.The bluer points along each ZAHB similarly illustrate, in turn, the consequences of mass loss amounting to 0.04, 0.08, 0.12, . . .M ⊙ .Panel (b): as in the top panel, except that the effects of reducing the O abundance from [O/Fe] = +0.60 (solid curves) to +0.40 (dotted curves) are shown.

Figure 3 .
Figure 3.Comparison of the upper-MS and TO portions of isochrones for the indicated ages and chemical abundances.The dot-dashed curve gives the location of an isochrone that is otherwise the same as the adjacent solid curve (for [Fe/H] = −2.5)except that it assumes Y = 0.270 instead of 0.250.An isochrone for [Fe/H] = −2.5, but with [O/Fe] = +0.4(instead of +0.6) appears as the dotted curve in purple.The horizontal dashed lines enclose the magnitude range over which the fiducial sequences of GCs are superimposed (in § 4) to constrain the cluster distances and reddenings.

Figure 4 .
Figure 4. Variations of R F606W (upper panel), R F814W (middle panel) and R F606W − R F814W (lower panel) as a function of T eff .The small filled circles are based on BCs for E(B − V) = 0.0 and 0.20 that have been derived by interpolations in the tables given by Casagrande & VandenBerg (2014) at the temperatures and gravities along a ZAHB for Y = 0.25 and [Fe/H] = −2.0.These results were fitted by quadratic equations, which were then evaluated at the temperatures that are indicated by the open circles.The extrapolations to temperatures greater than 8000 K are represented by the dashed curves.

Figure 5 .
Figure 5. Panels (a)-(f) and (h-(i): the adopted fits of the HB populations in several GCs to ZAHB loci (solid curves) that have been generated for [Fe/H] values corresponding to the minimum and maximum metallicities that define each of two bins (identified at the top of the plot).These fits and the superpositions of the cluster MS fiducials in panels (g) and (n) are obtained when the indicated values of E(B − V) and (m − M) V are assumed.The small boxes in each of the smaller panels give, in turn, the cluster metallicities from the spectroscopic survey by CBG09 and the best estimates of the E(B − V) values from dust maps.

Figure 6 .
Figure 6.Overlays of the same ZAHBs for [Fe/H] = −2.4 and −2.2 that appear in the previous figure onto the HB stars in M 92 and M 30 on the assumption of the indicated values of E(B − V) and (m − M) V .The CMDs plotted in orange and black assume reduced or increased reddenings, respectively, by 0.005 mag.If these reddenings are adopted, the superpositions of the cluster MS fiducials are comparable to those shown in Fig.5only if the specified values of (m − M) V are adopted.The resultant cluster parameters imply the fits of the cluster HBs to the ZAHB models that are illustrated.

Figure 8 .
Figure 8. Superposition on 4 different CMDs of the HB stars (black filled circles) in NGC 7078 (M 15) onto the HB population (orange filled circles) of NGC 6341 (M 92).The adopted values of E(B − V) and (m − M) V , which are specified in the left-hand panel, are identical to the values that were adopted in the construction of Figs.5d.The horizontal dashed line gives the approximate location at which a transition is made between the steep blue tails and the more horizontal parts of ZAHBs when plotted on optical CMDs (panels c and d).

Figure 9 .
Figure 9. Similar to panels (b) and (c) in the previous figure, except that the HB stars in NGC 4590 (M 68), NGC 5053, NGC 5466, and NGC 7099 (M 30) have been superimposed onto the HB population of NGC 6341 (M 92).The adopted cluster parameters are the same as in Fig. 5.

Figure 10 .
Figure 10.Similar to panels (b) and (c) in Fig. 8, except that the HB stars in NGC 6101, NGC 6397, NGC 6541, and NGC 6809 (M 55) have been superimposed onto the HB population of NGC 5024 (M 53).The adopted cluster parameters are the same as in Fig. 5.

Figure 11 .
Figure 11.As in Fig. 5, except that the HB populations of GCs with −1.7 < ∼ [Fe/H] < ∼ −1.5 have been fitted to ZAHBs for the minimum and maximum metallicities in this range.
WFC3 UV-optical CMDs for the MS stars should be able to shed some light on the possibility that these GCs have high Y.As shown by VandenBerg et al. (2022b, see their Fig.13), the MS portions of isochrones on (M F336W − M F438W ), M F606W diagrams at M F606W > ∼ 5.5 are fairly sensitive functions of [Fe/H] and Y.They also depend on the abundances of C and N, particularly when they are typical of the abundances found in CN-strong stars; see VandenBerg et al. (2022a, their Fig.10).Indeed, of the mixtures considered in the latter study, the a4CNN and the a4ONN mixes have the strongest effects on the intrinsic (M F336W − M F438W ) 0 colours along the MS.Examples of these findings are illustrated in the right-hand panel of Figure 12.The solid black curve represents a 12.5 Gyr isochrone for the indicated values of Y, [Fe/H], and the standard a4xO_p2 mix, which assumes scaled-solar abundances, but with [O/Fe] = +0.60 and [m/Fe] = +0.40 for all other α elements.The orange and dashed isochrones have been generated for

Figure 12 .
Figure 12.Comparisons of the median fiducial sequences for the MS and TO stars in NGC 5272 (M 3) with those of NGC 6809 (M 55) and NGC 1261 in panel (a), NGC 6205 (M 13) in panel (b), and NGC 7089 (M 2) in panel (c), on the assumption of the indicated values of E(B − V) and (m − M) V .Panel (d) plots the CMD locations of 12.5 Gyr isochrones for the a4xO_p2, a4CNN, and a4ONN mixtures of the metals, as described in the text, assuming Y = 0.251 and [Fe/H] = −1.50.Otherwise identical isochrones for the a4xO_p2 mix, but for Y = 0.29 instead of 0.251 and for [Fe/H] = −1.70instead of −1.50 are represented by the solid green and dotted blue curves, respectively.

Figure 13 .
Figure 13.As in Figs. 9 and 10, except that the HBs of most of the GCs with −1.7 < ∼ [Fe/H] < ∼ −1.5 (see Fig. 11) have been superimposed on the HB of NGC 6205 (M 13), assuming E(B − V) = 0.017 and (m − M) V = 14.45 for the latter and the specified cluster parameters for the other GCs.

Figure 14 .
Figure 14.As in Fig. 5 and 11, except that the HB populations of GCs with −1.4 < ∼ [Fe/H] < ∼ −1.2 have been fitted to ZAHBs for the minimum and maximum metallicities in these ranges.

Figure 15 .
Figure 15.As in Figs. 9, 10, and 13, except that the HBs of GCs with −1.4 < ∼ [Fe/H] < ∼ −1.2 (see Fig. 14) have been superimposed on the HB of NGC 5904 (M 5), assuming E(B − V) = 0.045 and (m − M) V = 14.39 for the latter and the indicated cluster parameters for the other GCs.

Figure 16 .
Figure 16.Panel (a): as in panel (i) of the previous figure except that the apparent distance modulus of NGC 288 has been increased by 0.03 mag.Panel (b): a comparison of the HBs of NGC 288 and M 5 on the (M F606W − M F814W ) 0 , M F606W diagram.The small filled and open circles assume E(B − V) = 0.017 and (m − M) V = 14.85 and 14.82, respectively, for NGC 288.

Figure 17 .
Figure 17.As in Figs. 5, 11, and 14, except that the HB populations of GCs with −1.1 < ∼ [Fe/H] < ∼ −0.9 (panels a and b) and −0.8 < ∼ [Fe/H] < ∼ −0.6 (panels d-g) have been fitted to ZAHBs for the minimum and maximum metallicities in these ranges.The faintest HB stars in 47 Tuc are represented by the horizontal dotted line (see panel d).This line is reproduced in panels (c)-(g) to facilitate comparisons with the faintest stars in M 69, NGC 6652, and M 71.

Figure 18 .
Figure18.Similar to Fig.11jexcept that a higher reddening and a smaller value of (m − M) V have been assumed, as indicated.

Figure 19 .
Figure19.Panel (a): Differences between the true distance moduli determined byBaumgardt & Vasiliev (2021) and the results of this study for the GCs that are listed in Table3.The few clusters that are explicitly identified are discussed in the text.Panel (b): Differences between the apparent distance moduli derived by VBLC13 and the present findings.The most discrepant results were obtained for M 2, which is explictly identified.

Table 1 .
Basic Stellar Physics Implemented in the Victoria Code

Table 2 .
Colour Offsets at M F606W = 5.5 between Isochrones for different values of [Fe/H] and Y a

Table 3 .
Summary of the Results