# A Comparative Study Between M30 and M92 : M92 Is A Merger Remnant With A
Large Helium Enhancement

Jae-Woo Lee Department of Physics and Astronomy, Sejong University, 209
Neungdong-ro, Gwangjin-Gu, Seoul 05006, Republic of Korea,
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We perform a comparative study of the ex–situ second–parameter pair globular
clusters (GCs) M30 and M92, having similar metallicities but different
horizontal branch morphologies. We obtain the similar mean primordial carbon
abundances for both clusters. However, M92 shows a large dispersion in carbon
due to a more extended C–N anticorrelation, while M30 exhibits a higher
primordial nitrogen abundance, suggesting that they have different chemical
enrichment histories. Our new results confirm our previous result that M92 is
a metal–complex GC showing a bimodal metallicity distribution. We also find
that the metal–rich group of stars in M92 shows a helium enhancement as large
as $\Delta Y$ $\sim$ 0.05 from the red giant branch bump (RGBB) $V$
magnitudes, which can also be supported by (i) a lack of bright RGB stars,
(ii) synthetic evolutionary HB population models and (iii) the more extended
spatial distribution due to different degree of the diffusion process from
their lower masses. We reinterpret the [Eu/Fe] measurements by other, finding
that the two metallicity groups of stars in M92 have significantly different
[Eu/Fe] abundances with small scatters. This strongly suggests that they
formed independently out of well mixed interstellar media in different
environments. We suggest that M92 is a more complex system than a normal GC,
most likely a merger remnant of two GCs or a even more complex system. In
Appendix, we address the problems with the recently developed
color–temperature relations and the usage of broadband photometry in the
populational taggings.

Stellar populations (1622); Population II stars (1284); Hertzsprung Russell
diagram (725); Globular star clusters (656); Chemical abundances (224);
Stellar evolution (1599); Red giant branch (1368); Red giant bump (1369);
Horizontal branch (2048)

††facilities: CTIO: 1.0 m (STA), WIYN: 0.9 m (HDI, S2KB), Gaia††software:
ATLAS12 (Kurucz, 2011), MOOG_SCAT (Sobeck & Sneden, 2023), Linemake (Placco et
al., 2021)

##  1 INTRODUCTION

Galactic globular clusters (GCs) had played critical roles in stellar and
Galactic astronomy for several decades. Due to the seminal discovery of
multiple stellar populations (MSPs) in GCs (D’Ercole et al., 2008; Carretta et
al., 2009; Lee et al., 2009; Milone et al., 2017; Gratton et al., 2019;
Cassisi & Salaris, 2020), the present situation may not be as lucid as before,
since the homogeneous chemical composition of a given GC is no longer valid.
Nonetheless, they are still the unique object to calibrate the theoretical
models for the structure and evolution of low–mass stars with various chemical
compositions (e.g., see Cassisi & Salaris, 2013; VandenBerg, Casagrande, &
Edvardsson, 2022).

Galactic GCs have also been extensively studied to figure out the formation
and evolution of our Galaxy. During the past several decades, not only the
in–situ but also the ex–situ GCs have been identified (e.g., Searle & Zinn,
1978; Lee & Carney, 1999), which can nicely be interpreted with a cold dark
matter cosmology that predicts a hierarchical structure formation in the
universe.

In this study, we perform a comparative study of two ex–situ metal–poor GCs,
M92 (NGC 6341) and M30 (NGC 7099), which have long been considered as standard
metal–poor GCs. Recent studies of M92 discovered a stellar stream related to
M92 (Sollima, 2020; Thomas et al., 2020). Furthermore, Thomas et al. (2020)
argued that M92 could be a GC that was brought into our Galaxy by a dwarf
galaxy or a remnant nucleus of the progenitor dwarf galaxy.111Note that M92 is
known to be associated to Gaia–Enceladus (Forbes, 2020; Callingham et al.,
2022). In our previous study, we discussed that M92 is a metal–complex GC with
an atypical primordial population (Lee, 2023a), in sharp contrast to a long-
held belief of a prototypical mono–metallic metal–poor GC. As Bekki &
Tsujimoto (2016) proposed, for example, GCs in dwarf galaxies with slightly
different formation epochs can merge into a metal–complex GC, such as M92.
Interestingly, Kirby, Ji, & Kovalev (2023) found the $r$–process elemental
abundance variations only in the first generation (FG)222In our work, the FG
denotes GC stars having the similar chemical compositions as the Galactic
field stars with the same metallicity, while the second generation (SG) of
stars showing the chemical compositions that experienced proton capture
processes at high temperature. of stars, apparently constraining temporal
separation between the FG and the SG of stars. On the other hand, M30 is known
to be a member of the Gaia–Enceladus system, that merged into our Galaxy about
10 Gyr ago leading to the dynamical heating of the precursor of the Galactic
thick disk (Helmi et al., 2018).

Figure 1: A comparison of M30 and M92 CMDs. The $V$ magnitude and $(b-y)$
color for M92 are adjusted to match with those of M30. Figure 2: (Top) CMDs of
M30 proper motion member stars. (Bottom) CMDs for M92 proper motion member
stars.

In Figure 1, we show a comparison of $(b-y)$ color–magnitude diagrams (CMDs)
of M30 and M92 by putting them together at the same distance.333 In the
figure, we applied $\Delta(b-y)$ = +0.0084 mag and $\Delta V$ = $-$0.01 mag
for M92 with respect to M30 to account the differences in the foreground
reddening values and the apparent visual distance moduli between the two
clusters (Harris, 1996). Note that $E(B-V)$ = 0.03 and 0.02 for M30 and M92,
respectively, and we adopted $E(b-y)$ = 0.84 $E(B-V)$ from our previous study
(Lee, 2023b). The apparent visual distance moduli are very similar, 14.64 and
14.65 mag for M30 and M92, respectively. The figure clearly show the
difference in the blue HB (BHB) morphology, while the lack of difference in
the red HB and asymptotic giant branch (AGB). M92 and M30 have similar ages
and metallicities (Sandquist et al., 1999; Dotter et al., 2010) but different
horizontal branch (HB) morphologies, making an exemplary second–parameter pair
(e.g., Arp & Johnson, 1955; Sandage & Wildey, 1967). Since mid 1950’s, a
strong correlation between the HB morphology and the metallicity of the
Galactic GCs has been recognized (e.g., Arp, 1955): The metallicity is the
first parameter that governs the HB morphology. However, it was soon noticed
that the HB morphology of some fraction of Galactic GCs cannot be
monotonically parameterized by the metallicity alone, requiring additional
parameters. Among several candidates, the age (e.g., Sandage, 1962; Searle &
Zinn, 1978; Lee, Demarque, & Zinn, 1994; Lee & Carney, 1999) and the helium
abundance (e.g., Arp, 1955; Faulkner, 1966; Sandage & Wildey, 1967) have been
frequently considered as strong candidates for the second–parameter, although
the helium abundance is extremely difficult to measure in the GC stars due to
the lack of measurable absorption lines in the cool red giant branch (RGB)
stars and metal levitation and helium settling in the hot HB stars (e.g.,
Grundahl et al., 1999). Furthermore, the HB morphology should be understood in
the context of the GC MSPs (e.g., see Jang & Lee, 2015; Lee, 2022).

We will argue that a large amount of the helium enhancement, $\Delta Y$ $\sim$
0.050, in the SG of the metal-rich groups of stars is responsible for the
extended BHB population in M92. We reinterpret the [Eu/Fe] measurements by
Kirby, Ji, & Kovalev (2023) and find that the metal–poor (MP) and metal–rich
(MR) groups of stars in M92 have significantly different [Eu/Fe] abundances
with small scatters. This strongly suggests that they are independently formed
out of well mixed interstellar media in different environments. In addition to
our previous study (Lee, 2023a), our current study may suggest that M92 is not
a typical globular cluster but a more complex system, perhaps a merger remnant
of two GCs or a remnant nucleus of the metal–poor progenitor dwarf galaxy.

In appendix, we will show comparisons of our photometric [Fe/H] and [C/Fe] of
the M92 RGB stars with those from high resolution spectroscopy by Kirby, Ji, &
Kovalev (2023), finding that our [Fe/H]hk and [C/Fe]ch are in excellent
correlations with spectroscopic measurements. We discuss a potential problem
with the color–temperature relations by Mucciarelli, Bellazzini, & Massari
(2021), which predict significantly low effective temperatures for M92. We
also address the faulty performance of the $(U-B)-(B-I)$ color index in
populational taggings of the metal–poor GCs.

##  2 OBSERVATIONS AND DATA REDUCTION

The journal of observations for M92 is given in Lee (2023a). Observations for
M30 were made in 33 nights in 13 runs from 2007 July to 2019 September using
the CTIO 1.0 m and KPNO 0.9 m telescopes. Until 2010, we used the filters that
were provided by the CTIO. As we noted in our previous work, some of the
CTIO’s filter were deteriorated due to aging effect (e.g., Lee, 2019b). We
used our own filters since 2011 both in the CTIO and KPNO, and our work
presented here will rely on photometric data using our own filters (Lee, 2017,
2019a, 2019b; Lee & Sneden, 2021). The total integration times for M30 using
our own filters were 6680, 13160, 27400, 12500, 9600, 7600 s for Strömgren
$y$, $b$, $Ca_{\rm JWL}$, JWL39, JWL43, and JWL34, respectively.

The raw data handling was described in detail in our previous works (Lee,
2015, 2017). The photometry of M30 and standard stars were analyzed using
DAOPHOTII, DAOGROW, ALLSTAR and ALLFRAME, and the relevant packages (Stetson,
1987, 1994). The astrometric solutions for individual stars were calculated
using the Gaia Data Release 3 (DR3; Gaia Collaboration, 2022) and the IRAF
IMCOORS package.

In order to select M30 member stars, we made use of the proper-motion study
from the Gaia DR3. We derived the mean proper-motion values using iterative
sigma-clipping calculations, finding that, in units of milliarcsecond per
year, ($\mu_{\rm RA}\times\cos\delta$, $\mu_{\rm decl.}$) = (0.721, $-$7.245)
with standard deviations along the major axis of the ellipse of 0.986 mas yr-1
and along the minor axis of 0.856 mas yr-1. We considered stars within
3$\sigma$ from the mean values to be M30 proper-motion member stars.

In Figure 2, we compare CMDs of M30 and M92 using our color indices (see Lee,
2019a; Lee & Sneden, 2021; Lee, 2022, for our color indices). In the figure,
we shows member stars only from the Gaia proper motions. Both clusters have
very small interstellar reddening values, and our color indices are
insensitive to the variations in the interstellar reddening values (Lee,
2023b), and the differential reddening effects across our science fields are
negligibly small. Therefore, the broad RGB widths in the $ch_{\rm JWL}$,
$cn_{\rm JWL}$ and $nh_{\rm JWL}$ are due to the internal variations of the
carbon and nitrogen abundances in each GC. Our results suggest that both M30
and M92 have almost identical CMDs in individual color indices except for the
extended BHB population of M92. As mentioned above, both clusters have similar
metallicities, [Fe/H] = $-$2.2 and $-$2.3 dex for M30 and M92, respectively
(Harris, 1996). The difference in the HB morphology between the two clusters
make them a good example of the second–parameter pair, i.e., GCs with
different HB morphologies with similar metallicities.

Table 1: Photometric data of RGB Stars in M30 and M92

Name | ID | $V$ | $(b-y)$ | $cn_{\rm JWL}$ | $ch_{\rm JWL}$ | $nh_{\rm JWL}$ | $hk_{\rm JWL}$ | [Fe/H]hk | [C/Fe]ch | [N/Fe]nh | R.A. | decl.
---|---|---|---|---|---|---|---|---|---|---|---|---
M30 | 26 | 12.407 | 0.839 |  $-$0.118 |  $-$0.596 | 0.282 | 0.844 |  $-$2.2042 |  $-$0.5351 | 1.4363 | 325.07913208 |  $-$23.18136024
M30 | 29 | 12.546 | 0.813 |  $-$0.103 |  $-$0.610 | 0.275 | 0.816 |  $-$2.1909 |  $-$0.6246 | 1.7438 | 325.07095337 |  $-$23.16836166
M30 | 30 | 12.764 | 0.766 |  $-$0.105 |  $-$0.638 | 0.074 | 0.746 |  $-$2.2058 |  $-$0.7122 | 1.3039 | 325.10092163 |  $-$23.18638992
M30 | 31 | 12.787 | 0.768 |  $-$0.105 |  $-$0.610 | 0.092 | 0.730 |  $-$2.2289 |  $-$0.5553 | 1.5341 | 325.11093140 |  $-$23.24319458
M30 | 35 | 12.947 | 0.738 |  $-$0.115 |  $-$0.605 |  $-$0.095 | 0.673 |  $-$2.2545 |  $-$0.4593 | 1.0460 | 325.04934692 |  $-$23.20969391
M30 | 41 | 13.006 | 0.726 |  $-$0.108 |  $-$0.579 |  $-$0.007 | 0.690 |  $-$2.2154 |  $-$0.3842 | 1.5129 | 325.00881958 |  $-$23.21672249
M30 | 46 | 13.055 | 0.717 |  $-$0.109 |  $-$0.626 |  $-$0.060 | 0.661 |  $-$2.2393 |  $-$0.5789 | 1.3987 | 325.05804443 |  $-$23.15091705
M30 | 50 | 13.160 | 0.706 |  $-$0.117 |  $-$0.580 |  $-$0.235 | 0.655 |  $-$2.2069 |  $-$0.3534 | 0.6801 | 325.08374023 |  $-$23.18958282
M30 | 53 | 13.178 | 0.703 |  $-$0.110 |  $-$0.597 |  $-$0.189 | 0.664 |  $-$2.1794 |  $-$0.4611 | 0.8898 | 325.09738159 |  $-$23.16900063
M30 | 52 | 13.259 | 0.688 |  $-$0.107 |  $-$0.582 |  $-$0.291 | 0.640 |  $-$2.1970 |  $-$0.3546 | 0.5741 | 325.08929443 |  $-$23.17413902
M30 | 59 | 13.306 | 0.681 |  $-$0.097 |  $-$0.650 |  $-$0.327 | 0.629 |  $-$2.1849 |  $-$0.7155 | 0.4486 | 325.09591675 |  $-$23.18000031
M30 | 60 | 13.390 | 0.675 |  $-$0.111 |  $-$0.595 |  $-$0.315 | 0.613 |  $-$2.1971 |  $-$0.3877 | 0.7174 | 325.10140991 |  $-$23.16088867

Note - This table is available in its entirely in a machine-readable form in
the online journal. A portion is shown here for guidance regarding its form
and contents.

Figure 3: (Top) Distributions of photometric elemental abundances against the
$V$ magnitude of M30 RGB stars. The green and purple colors denote the FG and
SG RGB stars in M30. (Bottom) Same as top panels, but for M92. The blue,
olive, and red colors denote the FG, IG, and SG of the M92 RGB population.

##  3 Metallicity, Carbon, and Nitrogen Abundances

We derive photometric [Fe/H], [C/Fe], and [N/Fe] for M30 using the upgraded
method that we applied for M92 (Lee, 2023a). First, we calculate photometric
metallicity, [Fe/H]hk, of individual RGB stars. We retrieved the model
isochrones for [Fe/H] = $-$2.5, $-$2.3, and $-$2.1 dex, $Y$ = 0.247, 0.275,
and 0.300 with [$\alpha$/Fe] = +0.4 dex, and the age of 12.5 Gyr from a Bag of
Stellar Tracks and Isochrones (BaSTI; Pietrinferni et al., 2021). We adopted
different CNO abundances, [C/Fe] = ($-$0.6, $\Delta$[C/Fe] = 0.2 ,0.6), [N/Fe]
= ($-$0.8, $\Delta$[N/Fe] = 0.4 , 1.6), and [O/Fe] = (0.1, 0.3, 0.5) for each
model grid. We constructed 97 model atmospheres and synthetic spectra for each
chemical composition from the lower main sequences to the tip of RGB sequences
using ATLAS12 (Kurucz, 2011), the latest versions of MOOG_SCAT (Sneden, 1974;
Sobeck & Sneden, 2023), and Linemake (Placco et al., 2021). As discussed in
our previous studies (Lee & Sneden, 2021; Lee, 2023a), Rayleigh scattering
from the neutral hydrogen (RSNH) atoms in the metal–poor GC RGB stars is the
dominant source of the continuum opacity in the blue and ultra-violet
wavelength regions. MOOG_SCAT takes proper care of RSNH with an non-local
thermodynamic equilibrium (non-LTE) treatment of the source function. Without
the proper care of the RSNH during the synthetic spectrum calculations, some
of our color indices, in particular $nh_{\rm JWL}$, cannot be reproduced
correctly.

In our current study, we upgraded our codes to derive the photometric
elemental abundances of individual RGB stars in a more sophisticated way. For
photometric metallicity, we assumed the following relation,

$\displaystyle{\rm[Fe/H]}_{hk}$ $\displaystyle\approx$ $\displaystyle
f_{1}({{hk}}_{\mathrm{JWL}},~{}Y,~{}\mathrm{[C,N,O/Fe]},~{}M_{V}).$ (1)

This relation makes use of more input parameters than that used in our
previous study (Lee & Sneden, 2021),

$\displaystyle{\rm[Fe/H]}_{hk}$ $\displaystyle\approx$ $\displaystyle
f_{1}^{\prime}({{hk}}_{\mathrm{JWL}},~{}M_{V}).$ (2)

At first, the CNO and helium abundances are assumed to be the same for all RGB
stars. Then the carbon and nitrogen abundances are updated using the [C/Fe]ch
and [N/Fe]nh from the relations given below and the helium abundances
estimated from their RGB bump (RGBB) $V$ magnitudes. Since we do not have
oxygen abundances of individual stars, we assigned [O/Fe] = 0.5 dex for the FG
and 0.0 dex for the SG. We note that the presumed oxygen abundance does not
significantly affect our results. We iterated these processes until the
derived metallicity converges within $\leq$ 0.01 dex. In most cases, no more
than 3 iterations is necessary. The difference in the [Fe/H]hk value from our
current study for M92 and that of our previous study (Lee, 2023a) is
$\Delta$[Fe/H]hk = $-$0.005 $\pm$ 0.002 dex ($\sigma$ = 0.057), in the sense
of the current minus the previous work.

We obtained [Fe/H]hk = $-$2.194$\pm$0.002 dex ($\sigma$ = 0.041) for M30. Our
metallicity for M30 is about 0.1 dex larger than that of M92 (see Table 2),
$-$2.316$\pm$0.003 dex ($\sigma$ = 0.062). Unlike M92, M30 appears to have an
unimodal metallicity distribution as shown in Figure 3. Note that the
metallicity dispersion of M30 is significantly smaller than that of M92. It is
similar to those of two metallicity groups (metal–poor:MP, metal–rich:MR) in
M92, $\sigma$ = 0.044 and 0.042 dex (see Table 1 of Lee, 2023a).

Table 2:  Comparisons of Elemental Abundances of RGB Stars in M30 and M92 | [Fe/H]hk | [C/Fe]chaaFor RGB stars fainter than their RGBB $V$ magnitudes. | [N/Fe]nhaaFor RGB stars fainter than their RGBB $V$ magnitudes.
---|---|---|---
M30 |  $-$2.194$\pm$0.002($\pm$0.041) | 0.053$\pm$0.009($\pm$0.134) | 0.657$\pm$0.021($\pm$0.325)
M92 |  $-$2.316$\pm$0.003($\pm$0.062) | 0.031$\pm$0.008($\pm$0.169) | 0.443$\pm$0.015($\pm$0.323)
M92-MP |  $-$2.414$\pm$0.004($\pm$0.044) | 0.072$\pm$0.019($\pm$0.173) | 0.404$\pm$0.038($\pm$0.343)
M92-MR |  $-$2.295$\pm$0.002($\pm$0.042) | 0.023$\pm$0.009($\pm$0.167) | 0.452$\pm$0.016($\pm$0.318)

The photometric carbon and nitrogen abundances for M30 RGB stars were
estimated using the following upgraded relations,

$\displaystyle{\rm[C/Fe]}_{ch}$ $\displaystyle\approx$ $\displaystyle
f_{2}({{ch}}_{\mathrm{JWL}},~{}{\mathrm{[Fe/H]}_{hk}},~{}Y,~{}\mathrm{[N,O/Fe]},~{}M_{V}),$
(3) $\displaystyle{\rm[N/Fe]}_{nh}$ $\displaystyle\approx$ $\displaystyle
f_{3}({{nh}}_{\mathrm{JWL}},~{}{\mathrm{[Fe/H]}_{hk}},~{}Y,~{}\mathrm{[C,O/Fe]},~{}M_{V}).$
(4)

The [C/Fe]ch and [N/Fe]nh returned from our current relations provide slightly
different results than those from our previous study. For M92, the differences
are $\Delta$[C/Fe] = 0.016 $\pm$ 0.007 dex ($\sigma$ = 0.181) and
$\Delta$[N/Fe] = 0.017 $\pm$ 0.009 dex ($\sigma$ = 0.226). In Table 1, we show
our photometric elemental abundances along with our photometry for RGB stars
with $-$2.5 mag $\leq$ $V-V_{\rm HB}$ $\leq$ 2.0 mag in M30 and M92. In Figure
3, the evolution of the carbon and nitrogen abundances against the $V$
magnitude can be clearly seen due to the deep mixing accompanied by the CN
hydrogen shell burning. In Table 2, we show the mean and dispersion of
[C/Fe]ch and [N/Fe]nh for the two clusters. The mean [C/Fe]ch values of the
RGB stars fainter than their RGBB $V$ magnitudes are similar for both
clusters, suggesting that their primordial mean carbon abundances are the
same. However, the [C/Fe] dispersion of M92 is slightly larger than that of
M30, which reflects the fact that M92 has a more extremely enhanced population
as shown below. On the other hand, the mean primordial nitrogen abundance of
the RGB stars in M30 is $\Delta$[N/Fe] $\sim$ 0.2 dex larger than those in
M92, indicating that M30 and M92 formed out of interstellar media experienced
different chemical enrichment histories.

In Appendix A, we show comparisons of our [Fe/H] and [C/Fe] of M92 RGB stars
with those from high resolution spectroscopy by Kirby, Ji, & Kovalev (2023).
We obtained that our [Fe/H]hk is nicely correlated with their [Fe/H], finding
$\langle$[Fe/H]hk $-$ [Fe/H]${}_{\rm K23}\rangle$ = 0.183$\pm$0.007 dex
($\sigma$ = 0.031 dex, Figure 12(g)–(i)), and a Pearson correlation
coefficient of 0.825. We note the small dispersion in the mean difference, a
strong indication of the reliability of our measurements. The rather large
offset between the two studies is partially due to inappropriate effective
temperatures used by Kirby, Ji, & Kovalev (2023) as discussed in Appendix B.

For our carbon abundance measurements, we achieved an excellent agreement with
those of Kirby, Ji, & Kovalev (2023), finding that $\langle$[C/Fe]ch $-$
[C/Fe]${}_{\rm K23}\rangle$ = $-0.103\pm 0.016$ dex ($\sigma$ = 0.073 dex,
Figure 12(j)–(l)), with a Pearson correlation coefficient of 0.981. Again, it
is believed that the inappropriate effective temperature adopted by (Kirby,
Ji, & Kovalev, 2023) is responsible for the offset between the two
measurements, at least in some part. Again, we stress that the scatter in the
$\langle$[C/Fe]ch $-$ [C/Fe]${}_{\rm K23}\rangle$ around the mean value is
very small. Therefore, it should be fair to say that our photometric carbon
abundances is very accurate in a relative sense, at least.

Figure 4: (a) A plot of $\parallel$ [C/Fe]ch vs. $\parallel$ [N/Fe]nh of the
M30 RGB stars with $-$2.5 mag $\leq$ $V-V_{\rm HB}$ $\leq$ 2.0 mag. The green
and purple colors denote the FG and SG RGB stars in M30. (b) A plot of
$\parallel$ [C/Fe]ch $-$ $\parallel$ [N/Fe]nh vs. $V$ of the M30 RGB stars.
(c) A histogram for the $\parallel$ [C/Fe]ch $-$ $\parallel$ [N/Fe]nh
distribution. The green and purple solid lines denote those returned from our
EM estimator, while the gray solid line is for the total RGB population.
(d)–(f) Same as (a)–(c), but for M92. The blue, olive, and red colors denote
the FG, IG, and SG of the M92 RGB population. (e) The bright yellow shaded box
indicates the region where no RGB stars present, due to high helium abundance
of the M92 SG. (g)–(i) Same as (a)–(c), but for the M92 MP population. The
yellow, mint, and orange colors denote the FG, IG, and SG of the M92 MP RGB
population. (j)–(l) Same as (a)–(c), but for the M92 MR population. The blue,
olive, and red colors denote the FG, IG, and SG of the M92 MR RGB population.

##  4 Populational Taggings

Due to internal mixing episodes accompanied by the CN cycle in the hydrogen
shell burning region, the surface carbon and nitrogen abundances of the RGB
stars brighter than their RGBBs are significantly changed from their initial
abundances, as Figure 3 represents well. Furthermore, the mixing efficiency
sensitively depends on metallicity since the hydrogen shell will be hotter and
wider with decreasing metallicity (e.g., see Lee, 2023b, and references
therein). Therefore, it is difficult to classify individual populations for
the bright RGB stars from their [C/Fe] and [N/Fe] as already shown in Figure
3. Similar to our previous approach, we make use of the normalized abundance
ratios $\parallel$ [C/Fe]ch and $\parallel$ [N/Fe]nh. We derived the outer
envelopes of our [C/Fe]ch and [N/Fe]nh against $V$ magnitude bins and they
were fitted with the second order polynomials for those of fainter and
brighter than the RGBB (see also Milone et al., 2017). Then the normalized
[C/Fe]ch and [N/Fe]nh are calculated as

$\displaystyle\parallel{\rm[C/Fe]}_{\rm ch}$ $\displaystyle\equiv$
$\displaystyle\frac{{\rm[C/Fe]}_{\rm ch}-{\rm[C/Fe]}_{\rm
ch,max}}{{\rm[C/Fe]}_{\rm ch,max}-{\rm[C/Fe]}_{\rm ch,min}},$ (5)
$\displaystyle\parallel{\rm[N/Fe]}_{\rm nh}$ $\displaystyle\equiv$
$\displaystyle\frac{{\rm[N/Fe]}_{\rm nh}-{\rm[N/Fe]}_{\rm
nh,max}}{{\rm[N/Fe]}_{\rm nh,max}-{\rm[N/Fe]}_{\rm nh,min}},$ (6)

where the subscripts denote the fiducials of the maximum and the minimum
sequences of the [C/Fe]ch and [N/Fe]nh.

In Figure 4, we show plots of the $\parallel$ [C/Fe]ch versus $\parallel$
[N/Fe]nh and the $\parallel$ [C/Fe]ch $-$ $\parallel$ [N/Fe]nh versus $V$
magnitude for both clusters. In Figure 4(e), we marked a yellow box where no
bright RGB stars are present. As we will show later, this yellow box region is
supposed to be the region where bright RGB stars with enhanced helium
abundance (the M92–SG) by $\Delta Y$ $\sim$ 0.05 would pass through. However,
the helium enhanced stars have a denser core with a higher central temperature
due to a larger mean molecular weight. As a consequence, the helium flash
occurs at a smaller core mass, resulting in fainter RGB tip luminosity as the
helium abundance increases (e.g., see Valcarce, Catelan & Sweigart, 2012). We
believe that the lack of bright RGB stars in the yellow box is a strong
evidence of a large helium enhancement in M92.

Finally, by generating histograms for the ($\parallel$ [C/Fe]ch $-$
$\parallel$ [N/Fe]nh), we applied an expectation–maximization (EM) algorithm
for two–component Gaussian mixture models for the M30, while we applied a
three–component Gaussian mixture model for the M92 (e.g., see Lee, 2023a).
Through these calculations, we obtained populational number ratios of
$n$(FG):$n$(SG) = 50.2:49.8 ($\pm$5.1) for M30. Our result is in good
agreement with that by Carretta et al. (2009), who obtained $n$(FG):$n$(SG) =
41($\pm$12):58($\pm$14) from the [Na/Fe] and [O/Fe] measurements, and
Leitinger et al. (2023), who obtained $n$(FG):$n$(SG) = 46:54($\pm$5) from a
combined photometric of Stetson et al. (2019) and Nardiello et al. (2018). In
Appendix C, we discuss the problem of applying the $C_{\rm UBI}$ index for the
metal-poor GCs. As shown in Figure 16, the populational number ratio by
Leitinger et al. (2023) could be a coincidence. The critical factor that we
have to pursue is the proper population assignment of the individual stars,
not the hazy overall populational number ratios. On the other, our
populational ratio is slightly different from that by Milone et al. (2017),
38.0:62.0 ($\pm$2.8), and is significantly different from that derived from
Washington photometry by Frelijj et al. (2017), $n$(FG):$n$(SG) $\sim$ 15:85.
It is not surprising because, Washington photometry is not suitable to study
GC MPs, in particular for those in metal–poor regimes same as $UBI$
photometry, as we discussed in Appendix C. We also note that Carretta et al.
(2009) classified 3$\pm$3 extreme component RGB stars out of 99 stars in M30,
but our $\parallel$ [C/Fe]ch $-$ $\parallel$ [N/Fe]nh histogram does not
clearly indicate such population.

For M92, we obtained a $n$(FG):$n$(IG):$n$(SG)444Our FG, IG, and SG are
approximately equivalent to the primordial, intermediate, and extreme
populations devised by Carretta et al. (2009). See also Lee (2021). =
38.1:35.5:26.4 ($\pm$2.7), where the IG denotes the intermediate generation of
stars. Note that our current estimation for M92 is slightly different from our
previous value from the RGB stars fainter than the RGBB, 32.2:31.6:36.2
($\pm$2.4) (Lee, 2023a), for two reasons. First, in our current study, we
upgraded our computer programs to calculate photometric abundances in a more
sophisticated way and our current photometric elemental abundances for M92 RGB
stars are slightly different from those in our previous study (Lee, 2023a).
Second, in our previous study, we relied on the ([C/Fe]ch $-$ [N/Fe]nh), not
the ($\parallel$ [C/Fe]ch $-$ $\parallel$ [N/Fe]nh), since we were interested
in the RGB stars fainter than the RGBB and the normalization processes such as
in Equations 5 and 6 were not applied in our previous study.

Our populational number ratio here is different from that of Milone et al.
(2017), $n$(FG):$n$(SG) = 30.4:69.6($\pm$1.5), who decomposed the FG and SG
only assuming that M92 is a mono-metallic GC, and that of Leitinger et al.
(2023), who obtained $n$(FG):$n$(SG) = 45:55($\pm$3) for M92.

We note that the populational number ratios of the MP and MR populations in
M92 are significantly different: $n$(FG):$n$(IG):$n$(SG) = 49:28:23 for the MP
and 36:37:27 for the MR. As we pointed out in our previous work (Lee, 2023a),
given the current mass of M92 (Baumgardt & Hilker, 2018), the total mass of
the M92–MP population is $\sim$ 6$\times$104$M_{\rm\odot}$, comparable to
those of less massive Galactic GCs with the FG population dominance (e.g., see
Milone et al., 2020).

In Figure 5, we show plots of [C/Fe] versus [N/Fe] of individual populations
in M30 and M92. It is clear that M92 exhibits a more extended C–N
anticorrelation than M30 does, which is related to the extended BHB population
in M92.

Figure 5:  Plots of [C/Fe] vs. [N/Fe] of M30 and M92. (a) A plot of [C/Fe]ch
vs. [N/Fe]nh for bright RGB stars (i.e., those brighter than the RGBB) in M30
(green:FG, purple:SG). (b) Same as (a) but for the faint RGB stars (i.e.,
those fainter than the RGBB) in M30. (c) Same as (a) but for M92 (yellow:MP-
FG, mint:MP-SG, orange:MP-SG, blue:MR-FG, olive:MR-IG, red:MR-SG). (d) Same as
(a) but for the faint RGB stars in the M92–MP. (e) Same as (a) but for the
faint RGB stars in the M92–MR.  Figure 6:  (Top) Plots of the $by5$ vs. $V$
CMDs and smoothed differential LFs for the M30–FG and SG. The gray dashed
lines indicate the RGBB $V$ magnitudes of individual populations. (Bottom)
Same as the top panels but for M92 (M92–1:MP–FG, 2:MP–IG, 3:MP–SG, 4:MR–FG,
5:MR–IG, and 6:MR–SG).  Figure 7: The CMD of the M30 HB region. We show HB
stars with helium abundances (Mucciarelli et al., 2014). Thin solid lines are
for the ZAHB loci for $Y$ = 0.247 and 0.275 (Pietrinferni et al., 2021). Note
that we exclude the star 30016296, which has a very low helium abundance, $Y$
= 0.088$\pm$0.022.

##  5 Red-Giant Branch Bump

During the course of low–mass star evolution, RGB stars experience a temporary
drop in luminosity when the very thin hydrogen burning shell crosses the
discontinuity in the chemical composition and lowered mean molecular weight
left by the deepest penetration of the convective envelope during the ascent
of the RGB. In such evolutionary phases, RGB stars have to cross three times
the same luminosity interval, leaving a distinctive feature, the so-called
RGBB (e.g., see Cassisi & Salaris, 2013). It is well recognized that, at a
given age, the RGBB luminosity increases with helium abundance and decreases
with metallicity (e.g., see Cassisi & Salaris, 1997).

The prominence of the RGBB also depends on metallicity, in the sense that the
RGBB bump feature becomes more distinguishable with increasing metallicity
(e.g., see Fusi Pecci et al., 1990; Valcarce, Catelan & Sweigart, 2012). At
the same time, the number of RGB stars in M30 is much smaller than those in
M92, for example, as shown in Figure 2.555The current mass of M30 is about a
half of that of M92 (Baumgardt & Hilker, 2018). The numbers of RGB stars used
by Milone et al. (2017) and Leithinger et al. (2023) for their studies also
reflect the mass difference between the two clusters: $n$(M92):$n$(M30) =
795:323 (Milone et al., 2017) and 740:295 for the HST and 238:110 for the
ground-based catalogues that Leitinger et al. (2023) used. Note that we used
$n$(M92):$n$(M30) = 592:271. We suspect that, due to these reasons, Sandquist
et al. (1999) did not detect clear signs of the RGBB in M30.

In Figure 6, we show the CMDs and smoothed differential luminosity functions
(LFs) with Gaussian kernels around the RGBB regimes of individual populations
in M30 and M92. In the figure, we used a modified color $by5$ (=
$5\times(b-y)-2.2$) for the clarity of the RGB distributions. In order to
interpret the observed RGBBs, we relied on Monte Carlo simulations by
constructing evolutionary population synthesis models (e.g., Lee, 2022). We
retrieved BaSTI isochrones for [Fe/H] = $-$2.5, $-$2.3, and $-$2.1 and $Y$ =
0.247, 0.275, and 0.300 with the age of 12.5 Gyr (e.g., Dotter et al., 2010).
To determine the $V_{\rm bump}$ of individual model isochrones, we populated
200,000 artificial stars and we generated generalized histograms to derive the
$V_{\rm bump}$ as we did for our observed data. We obtained $\Delta V_{\rm
bump}$/$\Delta$[Fe/H] $\propto$ 0.560 $\pm$ 0.002 and $\Delta V_{\rm
bump}$/$\Delta Y$ $\propto$ $-$1.786 $\pm$ 0.002. Using these relations, we
calculate the RGBB $V$ magnitude differences in terms of relative metallicity
and helium abundance. We summarize our results as follow:

  * •

The RGBB $V$ magnitude difference $\Delta V_{\rm bump}$ = 0.032$\pm$0.028
mag666The apparent distance modulus of M92 is 0.01 mag larger than that of M30
(Harris, 1996) between the M30 FG and M92 MR–FG can be interpreted by the
metallicity difference between the two populations, $\Delta$[Fe/H]bump =
0.057$\pm$0.051 dex, assuming they have the identical helium abundance, $Y$ =
0.247. The metallicity difference between M30 FG and M92 MR using the RGBB $V$
magnitude is in agreement with that from our photometric metallicity,
$\Delta$[Fe/H]hk = 0.101$\pm$0.004 dex.

  * •

$\Delta V_{\rm bump}$ = 0.044 mag between the M30 FG and SG can be explained
by the helium abundance difference of $\Delta Y$bump = 0.025$\pm$0.016, which
is in excellent agreement with that of Milone et al. (2018), $\delta Y_{\rm
max}$ = 0.022$\pm$0.010.

  * •

$\Delta V_{\rm bump}$ = 0.114 mag between the M92 MP and MR–FG can be
explained by the metallicity difference of $\Delta$[Fe/H]bump =
0.200$\pm$0.051 dex, assuming they have the identical helium abundance, $Y$ =
0.247. In our previous study (Lee, 2023a), we argued that M92 exhibits a
bimodal metallicity distribution with $\langle$$\Delta$[Fe/H]$\rangle$ =
0.119$\pm$0.004 dex ($\sigma$ = 0.061), which is slightly smaller than the
current estimate using the RGBB $V$ magnitude difference between the M92 MP
and MR–FG.

  * •

$\Delta V_{\rm bump}$ = 0.020 and 0.092 mag between the M92 MR–FG and IG, SG
can be translated by the helium abundance difference of $\Delta Y$bump = 0.011
and 0.052$\pm$0.016, which is marginally in agreement with those of Milone et
al. (2018), $\Delta Y$max = 0.039$\pm$0.016 (color spreads in the RGB),
VandenBerg, Casagrande, & Edvardsson (2022), $\Delta Y$ $\approx$ 0.04
(isochrone fittings), and Ziliotto et al. (2023), $\Delta Y$ $\approx$
0.01–0.04 (color spreads in the lower main–sequence).

Figure 8:  RGB tip masses against [Fe/H]. The short and long dashed, and solid lines denote for $Y$ = 0.247, 0.275, and 0.300, respectively. The colors are the same as Figure 5.  Table 3:  Input Parameters Used in Our Synthetic HB Model Calculations Cluster | Pop. | [Fe/H] |  $\sigma$[Fe/H] | $Y$ | $\sigma Y$ | $\langle M_{\odot}\rangle$ | $\sigma M_{\odot}$ | frac.(%)
---|---|---|---|---|---|---|---|---
M30 | FG |  $-$2.19 | 0.044 | 0.247 | 0.0025 | 0.687 | 0.020 | 50
M30 | SG |  $-$2.19 | 0.038 | 0.272 | 0.0025 | 0.654 | 0.020 | 50
M92 | MP |  $-$2.41 | 0.044 | 0.247 | 0.0025 | 0.691 | 0.020 | 19
M92 | MR–FG |  $-$2.29 | 0.042 | 0.247 | 0.0025 | 0.689 | 0.021 | 32
M92 | MR–IG |  $-$2.29 | 0.042 | 0.258 | 0.0025 | 0.674 | 0.020 | 31
M92 | MR–SG |  $-$2.29 | 0.042 | 0.299 | 0.0025 | 0.621 | 0.018 | 18
Figure 9:  (a) A comparison of observed (gray dots) and synthetic (the same
color scheme as Figure 5) CMDs of M30 HB stars. (b) The yellow shaded
histogram with error bars denotes the observed ($b-y$) distribution of M30 HB
stars, while the green, purple, and black histograms denote the mean
histograms of 100 trials for the synthetic FG, SG, and all HB stars. (c) Same
as (b) but for the $V$ magnitude. (d) Same as (a) but for M92. The yellow,
mint, blue, olive, and red plus signs denote the synthetic MP–FG, MP–SG,
MR–FG, MR–IG, and MR–SG HB stars. (e) Same as (b), but for M92. (f) Same as
(c), but for M92.

##  6 Horizontal Branch

HB stars can play a significantly role to probe the metallicity and helium
abundance in GCs, due to their rather sensitive morphological dependence on
chemical compositions. Figure 7 shows the CMD of the M30 HB region along with
BHB stars with helium abundance measurements from intermediate-resolution
($R\sim$ 18,000) spectroscopy by Mucciarelli et al. (2014). In the figure, we
also show the zero-age HB (ZAHB) loci for $Y$ = 0.247 and 0.275 (Pietrinferni
et al., 2021). The figure clearly shows that the M30 BHB stars are segregated
by their helium abundances, in the sense that the BHB stars with lower helium
abundances occupy the redder part of the BHB domain at a given $V$ magnitude,
consistent with the model prediction. This also suggests that our estimates of
the helium abundance dispersion from the RGBB $V$ magnitudes are fairly
reasonable.

Encouraged with Figure 7, we constructed synthetic HB models following the
similar methods that we developed in our previous studies (e.g., Lee & Carney,
1999; Lee, 2022), in order to interpret the observed HB morphologies of M30
and M92.

First, we obtained the RGB tip masses for individual populations using the
BaSTI isochrones with the Reimer’s index of $\eta$ = 0.3 and [$\alpha$/Fe] =
+0.4. For this purpose, we adopt relative helium abundances returned from our
RGBB $V$ magnitudes. In Figure 8, we show the RGB tip masses with different
helium abundances as functions of metallicity. In the figure, we also show the
mean masses of individual populations in M30 and M92 (the error bars indicate
$\pm\sigma$). Next, we populate the synthetic HB stars with our evolutionary
population synthesis models using the input parameters as shown in Table 3.

In Figure 9, we show our results for synthetic HB models along with the
observed HB stars in both clusters. In the figure, our synthetic histograms
are the mean values returned from 100 trials, while the synthetic CMDs are
those of one specific trial. We note that our simulations reproduce the
observed HB distributions of both clusters reasonably well. We also calculated
the synthetic HB models for M92 using a mildly enhanced helium abundance for
the MR–SG, $\Delta Y$ = 0.02–0.03, which fail to reproduce the observed
extended BHB distribution with $V$ $\gtrsim$ 16.1 mag in M92.777Note that, at
[Fe/H] $\sim$ $-$2.20 dex and $Y$ = 0.300, the HB stars less massive than
$\sim$ 0.510 $M_{\rm\odot}$ fail to evolve into the normal AGB phases and
become AGB-manqué stars. Their masses are significantly smaller than the HB
masses presented in Table 3, and the emergence of the AGB-manqué does not
affect our simulations.

The fraction of the underlying populations may also suggest that the extended
BHB is related to the helium enhanced RGB population. The fraction of the M92
HB stars fainter than $V$ = 16.1 mag is about 0.20($\pm$0.03), consistent with
the fraction of M92–MR–SG, 0.18($\pm$0.02).

We conclude that not only the RGBB $V$ magnitude of the M92 MR–SG but also the
presence of the extended BHB population with $V$ $\gtrsim$ 16.1 mag strongly
suggest that M92 contains an extreme enhanced population with $\Delta Y$
$\approx$ 0.05.

Figure 10:  CRDs of individual populations in M30 and M92 RGB stars, and M92
HB stars. Colors for RGB stars are the same as Figure 5.

##  7 Cumulative Radial Distributions

The observed cumulative radial distributions (CRDs) of individual GC MSPs may
depend on many physical properties at the time of GC formation and provide
critical information on the dynamical evolution of GCs. Previous studies on
the formation of the MSPs in GCs suggested that the SG populations may form in
the innermost part of the cluster in a more extended FG system (e.g., see
D’Ercole et al., 2008; Bekki, 2019), where the degree of the helium
enhancement of the SG can be dependent on the external gas density (Calura et
al., 2019). The initial structural difference between the FG and SG
populations can be gradually erased with time, due to the result of the
preferential loss of the FG stars during the cluster’s dynamical evolution
(e.g., see Vesperini et al., 2021). It is also likely that the current
location can be affected by the different evolutionary stellar masses of
individual populations due to different degree of diffusion processes. Over
the Hubble time, for example, the radial distributions of the later
generations of stars with enhanced helium abundances can expand outward due to
the smaller stellar masses resulted from their fast stellar evolution (e.g.,
see Fare, Webb, & Sills, 2018).

In this regards, we derive the CRDs of individual MSPs, and we show our
results in Figure 10 and Table 4. The figure shows that the M30 FG is
distributed more spatially concentrated than the M30 SG is. However, it does
not appear to be statistically significant. We performed the
Kolmogorov–Smirnov (K–S) tests and we obtained the $p$ value of $p$ = 0.127,
indicating that they may be drawn from the same parent distribution.

The CRDs for the M92 MSPs are more interesting. The M92–MP is more centrally
concentrated than the M92–MR is, with the $p$ value of $p$ = 1.98$\times
10^{-4}$, strongly suggesting that they are statistically independent groups
of stars. This is apparently consistent with the idea that M92 may be a merger
remnant of the two former dwarf galaxy GCs with slightly different formation
epoch and structural parameters (Lee, 2023a). Among the M92–MR populations,
the FG is more centrally concentrated than the SG, with a $p$ = 0.003, which
appears to be consistent with the recent idea proposed by Fare, Webb, & Sills
(2018) that the secular evolution of the CRDs in GC MSPs can be governed by
their stellar masses with different helium contents. The small current stellar
masses of the M92–MR–SG due to a significant enhanced helium abundance
($\Delta Y$ $\sim$ 0.05) is thought to be responsible for its more centrally
extended distribution than other populations (also see Table 3).

We note that Leitinger et al. (2023) claimed that the M92 SG is more centrally
concentrated than the FG is, in sharp contrast to our result. Again, we
believe that this vividly shows the reason for not using broadband photometry
for the metal–poor GC MSPs as discussed in Appendix C.

We also compared the CRDs of the M92 HB stars. As before, we divide the HB
stars with $V$ = 16.1 mag; the bright HB and faint HB. The faint HB is more
centrally extended than the bright HB is, with a $p$ value of 6.79$\times
10^{-4}$ returned from the K–S test, strongly suggest that they are drawn from
different parent distributions. Our K–S test indicates that CRDs of M92–MR–SG
and the faint HB are very similar with the $p$ value of 0.980. Our
investigation of CRDs clearly tells that the faint HB population is progeny of
the helium enhanced population M92–MR–SG.

Table 4:  $p$ Values Returned from the K-S Tests for the CRDs of Individual Populations Cluster | Pop. 1 | Pop. 2 |  $p$ Value
---|---|---|---
M30 | FG | SG | 0.127
M92 | MP | MR | 1.98$\times 10^{-4}$
M92 | MP–FG | MP–SG | 0.064
M92 | MR–FG | MR–IG | 0.447
M92 | MR–FG | MR–SG | 0.003
M92 | MR–IG | MR–SG | 3.62$\times 10^{-3}$
M92 | Bright HB | Faint HB | 6.79$\times 10^{-4}$
M92 | MR–SG | Bright HB | 1.33$\times 10^{-4}$
M92 | MR–SG | Faint HB | 0.980
Figure 11:  Plots of the [Fe/H]hk vs. [Eu/Fe] by Kirby, Ji, & Kovalev (2023)
for the FG and SG RGB stars. The vertical gray lines denote the boundaries
between the M92–MP and MR populations in our study (Lee, 2023a). (a) For the
FG both from the M92–MP and MR populations. (b) For the SG RGB stars both from
the MP and MR. Note that the MP–SG shown with the light green diamonds are
classified as the AGB stars by Kirby, Ji, & Kovalev (2023). (c) Combined plot
using (a) and (b). The horizontal cyan solid and dashed lines denote the mean
[Eu/Fe] and the standard deviations.

##  8 The [Eu/Fe] abundances in M92 as an evidence of a merger remnant

As we mentioned above, Kirby, Ji, & Kovalev (2023) discovered the variations
in the $r$-process neutron capture elemental abundances only in the M92 FG,
while no such variation in the SG, which led them to suggest a minimum
temporal separation of 0.8 Myr ($\sim$ several crossing times of M92) between
the FG and SG. Here, we reinterpret their results in the context of our
framework of M92 as a merger remnant of the two GCs.

In Figure 11, we show plots of our [Fe/H]hk versus [Eu/Fe] by Kirby, Ji, &
Kovalev (2023) for the M92 RGB stars, where we classify individual RGB stars
by their [Fe/H]hk (for the MP and MR), [C/Fe]ch and [N/Fe]nh (for the FG and
SG). We note that our FG and SG populations are about equivalent to those
based on the [Na/Fe] and [Mg/Fe] abundances by Kirby, Ji, & Kovalev (2023). We
also note that all four MP–SG stars shown with light green diamonds in the
panel (b) were classified as AGB stars by Kirby, Ji, & Kovalev (2023).
However, our multi–color photometry suggests that they are indeed MP–SG RGB
stars. Our populational number ratio for the M92–MP population,
$n$(FG):$n$(IG+SG) $\sim$ 49:41($\pm$4), is in agreement with the number ratio
of the spectroscopic targets, $n$(MP–FG):$n$(MP–SG) = 3:4. Therefore, it
should be reasonable to consider these four stars as MP–SG RGB stars.

Figure 11(a) shows a large hiatus between the [Eu/F] of the MP–FG and MR–FG,
which led Kirby, Ji, & Kovalev (2023) concluded that the M92 FG stars show
variation in the $r$-process neutron capture elemental abundances. In Table 5,
we show the mean [Eu/Fe] values and dispersions for individual populations.
The [Eu/Fe] dispersion of the FG is about three times larger than that of the
SG. However, we stress again that the MP and MR are independent populations
with different abundances and structural parameters.

When we rearrange individual stars with their metallicity groups, the scatters
in [Eu/Fe] become smaller as shown in Figure 11(c). The mean [Eu/Fe] values do
not agree each other, with $\Delta$[Eu/Fe] = 0.198$\pm$0.038. Note that the
difference is more than 5$\sigma$ levels. The figure also suggests that there
is no [Eu/Fe] evolution within the MP and MR populations.

We believe that the different [Eu/Fe] abundances between the MP and MR
populations provides a strong observational constraint on the true nature of
M92 as the merger remnant. Our current result may suggest that they were
independently formed out of chemically well mixed interstellar media.

Table 5: Mean [Eu/Fe] values |  $\langle$[Eu/Fe]$\rangle$ | n
---|---|---
FG | 0.452$\pm$0.073($\pm$0.178) | 6
SG | 0.363$\pm$0.016($\pm$0.055) | 12
MP | 0.546$\pm$0.034($\pm$0.089) | 7
MR | 0.348$\pm$0.017($\pm$0.066) | 15

##  9 SUMMARY

We performed a comparative study of M30 and M92 using our own photometric
system. For M30, we obtained the mean photometric metallicity of [Fe/H]hk =
$-$2.194$\pm$0.002 dex ($\sigma$ = 0.041), which is about 0.1 dex more
metal–rich than M92 is. Our photometric metallicities are in good agreement
with those in literatures (e.g., 2010 version of Harris, 1996)

We derived the photometric carbon and nitrogen abundances for both clusters.
The mean primordial [C/Fe]ch abundances are in good agreement in both
clusters, although the [C/Fe]ch dispersion of M92 slightly larger than that of
M30, consistent with the fact that M92 has a more extended C-N anticorrelation
with a more helium enhanced population. On the other hand, M30 appears to have
a more enhanced primordial nitrogen abundance by $\Delta$[N/Fe] = 0.2 dex than
M92 does. It is believed that M30 and M92 formed out of interstellar media
experienced different chemical enrichment histories.

We investigated the metallicity and helium abundance differences between
individual populations using our RGBB $V$ magnitudes, which sensitively depend
on elemental abundances. We obtained the helium abundance difference of
$\Delta Y$ = 0.025$\pm$0.016 between M30 FG and SG, which is in excellent
agreement with that of Milone et al. (2018), $\Delta Y$max = 0.022$\pm$0.010.

By assuming the same helium abundance, we obtained the metallicity difference
of $\Delta$[Fe/H] = 0.200$\pm$0.051 dex between M92–MP and MR from the RGBB
$V$ magnitudes, which is marginally in agreement with that of photometric
metallicities, $\langle$$\Delta$[Fe/H]$\rangle$ = 0.119$\pm$0.004($\pm$0.061).

The most intriguing result is a large difference in the helium abundance
between the M92 MR–FG and SG, $\Delta Y$ = 0.052$\pm$0.016. The lack of bright
RGB stars in M92 MR–SG population also support the idea that it is indeed
significantly enhanced in helium. It is believed that our result is marginally
in agreement with those by others (e.g., Milone et al., 2018; VandenBerg,
Casagrande, & Edvardsson, 2022; Ziliotto et al., 2023).

We constructed evolutionary HB population synthesis models, finding that our
metallicities and relative helium abundances satisfactorily reproduce the
observed HB distributions for both clusters. A mild helium enhancement of
$\Delta Y$ = 0.020–0.030 in the M92 MR–SG population failed to reproduce the
extended BHB population with $V$ $\geq$ 16.1 mag in M92.

The CRDs may also suggested that the M92–MR–SG is significantly more helium
enhanced than other populations and it is the progenitor of the faint HB. The
most spatially extended CRD of the M92–MR–SG is due to their small stellar
masses resulted from their fast stellar evolution with an significantly
enhanced helium abundance by $\Delta Y$ $\sim$ 0.05. The CRDs of the M92–MR–SG
and the faint HB are very similar with a $p$ value of 0.980 from a K–S test.
The fractions of the M92–MR–SG and the faint HB with respect to the total RGB
and HB populations are also very similar, 0.18($\pm$0.02) and 0.20($\pm$0.03),
respectively, strongly indicating that the faint HB stars with $V$ $\geq$ 16.1
mag are progeny of the M92–MR–SG RGB stars.

In our previous study (Lee, 2023a), we argued that M92 is a metal–complex GC,
with a bimodal metallicity distribution and an atypical FG population,
requiring significant magnesium and oxygen depletion. Here, we discussed a
strong observational line of evidence of a large helium enhancement in MR–SG
population. We also showed that M92 MP and MR populations have significantly
different populational number ratios and spatial distributions. Furthermore,
our reanalysis of [Eu/Fe] measurements by Kirby, Ji, & Kovalev (2023)
suggested that the MP and MR populations are chemically separate populations,
suggesting that both the MP and MR populations in M92 were most likely formed
out of chemically well mixed interstellar media. Our results may suggest that
M92 is not a typical globular cluster but a more complex system, perhaps a
merger remnant of two GCs or a remnant nucleus of the metal–poor progenitor
dwarf galaxy (e.g., see Thomas et al., 2020; Lee, 2023a, and references
therein).

J.-W. Lee thanks Christopher Sneden for his comments on the early version of
the manuscript and the anonymous referee for the useful suggestions. We
acknowledge financial support from the Basic Science Research Program (grant
No. 2019R1A2C2086290) through the National Research Foundation of Korea (NRF)
and from the faculty research fund of Sejong University in 2022.

Figure 12:  (a–c) Comparisons of our [Fe/H]hk versus [Fe/H] by Roederer &
Sneden (2011). In (a) the thick grey line denote the linear fit and the number
in the parenthesis is the slope of the relation. In (b–c), the thick solid
lines are the mean and the thick dashed grey lines denote the standard
deviations ($\pm\sigma$). In (c), the red vertical lines denote the $V$
magnitude span of the RGB stars common in our study and that of Roederer &
Sneden (2011). (d–f) Comparisons of our [Fe/H]hk versus the luminosity effect
uncorrected [Fe/H] by Kirby, Ji, & Kovalev (2023) for RGB stars of our
interest. (g–i) Same as (d–f) but for the the luminosity effect uncorrected
[Fe/H]. (j–l) Comparisons of our [C/Fe]ch versus the luminosity effect
uncorrected [C/Fe] by Kirby, Ji, & Kovalev (2023).  Figure 13:  (a)
Differences in effective temperatures between our evolutionary population
synthesis models and those of Kirby, Ji, & Kovalev (2023). (b) Run of the
uncorrected $A$(FeI) against $M_{G,0}$ of Kirby, Ji, & Kovalev (2023). The
blue solid line denotes the polynomial fit given Figure 4 of Kirby, Ji, &
Kovalev (2023). (c) The black crosses denote the abundance corrections due to
the difference in the effective temperature in (a), while the red crosses
those due to difference in the surface gravity. The blue solid line is for the
polynomial fit in (b) minus 5.02. (d) The differential temperature corrected
[Fe/H]. (e) The luminosity-based gradient corrected [Fe/H] by Kirby, Ji, &
Kovalev (2023).  Figure 14:  (a) Differences in effective temperatures between
our evolutionary population synthesis models and those of Roederer & Sneden
(2011). (b) Run of the uncorrected [Fe/H] against $V$ magnitude. (c) The
abundance corrections based on the panel (a). The red crosses denote the
corrections due to different adopted surface gravity, which does not greatly
affect the results. (d) The differential temperature corrected [Fe/H].

##  Appendix A Comparisons of our [Fe/H]hk and [C/Fe]ch of M92 RGB stars to
the [Fe/H] and [C/Fe] by Kirby, Ji, & Kovalev

In our previous study for M92 (Lee, 2023a), we showed comparisons of our
[Fe/H]hk measurements of the M92 RGB stars to those of the differential
spectroscopic analysis by Roederer & Sneden (2011), finding a large mean
metallicity offset, $\langle$[Fe/H]hk $-$ [Fe/H]${}_{\rm RS11}\rangle$ =
0.314$\pm$0.039 dex ($\sigma$ = 0.128 dex), and a strong luminosity or
temperature gradient.888In Figure 3 of Lee (2023a), $\Delta$[Fe/H] denotes the
residuals around the fitted lines, not the differences between our [Fe/H]hk
and [Fe/H] by Roederer & Sneden (2011), [Fe/H]hk $-$ [Fe/H]RS11. We showed the
residuals around the fitted line to demonstrate the luminosity effect in the
[Fe/H] measurements by Roederer & Sneden (2011).

In Figure 12(a–c), we show comparisons of our [Fe/H]hk versus [Fe/H] by
Roederer & Sneden (2011). We note a strong gradient in the metallicity
difference between our measurements and those of Roederer & Sneden (2011)
against the $V$ magnitude in the panel (c), in the sense that the [Fe/H]
values of Roederer & Sneden (2011) become significantly smaller with
increasing luminosity. As shown in Figure 3, our [Fe/H]hk does not show any
gradient against the $V$ magnitude and, therefore, the gradient in (c) should
be arisen from the [Fe/H] measurements by Roederer & Sneden (2011).

Recently, Kirby, Ji, & Kovalev (2023) performed precision high resolution
spectroscopy of M92 stars, spanning from the RGB-tip to the subgiant, with
much better spectra qualities than those used by Roederer & Sneden (2011). In
Figure 12(d–f), we show comparison of our [Fe/H]hk versus uncorrected [Fe/H]
measurements by Kirby, Ji, & Kovalev (2023, see their Table 5). The
correlation between the two datasets does not look good with the Pearson
correlation coefficient of 0.106. The strong abundance gradient against
luminosity persists in the panel (f), although the mean difference is somewhat
smaller than that from Roederer & Sneden (2011), $\langle$[Fe/H]hk $-$
[Fe/H]${}_{\rm K23,uncor}\rangle$ = 0.191$\pm$0.020 dex ($\sigma$ = 0.095
dex).

Kirby, Ji, & Kovalev (2023) noticed considerable elemental abundance trends
with the stellar luminosity even after applying non-LTE corrections for some
elements. In order to remove the unexplained trend, they interpreted these
trends as artifacts of their measurements. They derived polynomial fits of
elemental abundances against the $M_{G,0}$ magnitude and they showed their
results in their Figure 4. In Figure 12(g–i), we show comparison of our
[Fe/H]hk versus the luminosity-based gradient corrected [Fe/H] measurements by
Kirby, Ji, & Kovalev (2023). As shown in the panel (g), the correlation
between the two datasets is excellent with a correlation coefficient of 0.824.
The panel (i) clearly showed that the elemental abundance gradient against
luminosity disappears. We find that $\langle$[Fe/H]hk $-$ [Fe/H]${}_{\rm
K23,cor}\rangle$ = 0.183$\pm$0.007 dex ($\sigma$ = 0.031 dex). Note that the
dispersion is also significantly improved, which is what we demonstrated in
our previous study in comparison with the results by Roederer & Sneden (2011).
Below, we will argue that the inappropriate temperature scales are responsible
for this discrepancy, at least a part.

Kirby, Ji, & Kovalev (2023) also measured carbon abundances by employing
spectral synthesis of the CH G-band features. As kindly noted by Dr. Kirby
that the [C/Fe] values in their Table 6 are based on the uncorrected [C/H] and
the luminosity-based gradient corrected [FeI/H]. When we compare our [C/Fe]ch
with the [C/Fe] values in their Table 6, a strong gradient in the difference
in carbon abundances ([C/Fe]ch $-$ [C/Fe]) against the luminosity, similar to
our Figure 12(f), can be seen. As recommended by Dr. Kirby, we derived [C/Fe]
based on their uncorrected [C/H] and [Fe/H] measurements, [C/Fe]K23,uncor, and
we show our comparisons in Figure 12(j–l). Our [C/Fe]ch is nicely correlated
with the spectroscopic measurements by Kirby, Ji, & Kovalev (2023) without any
perceptible trend against luminosity, with the Pearson correlation coefficient
of 0.981. The mean difference between the two measurements is
$\langle$[C/Fe]ch $-$ [C/Fe]${}_{\rm K23,uncor}\rangle$ = $-0.103\pm 0.016$
dex ($\sigma$ = 0.073 dex).

As we will discuss below, the absence of any gradient against luminosity in
[C/Fe]ch $-$ [C/Fe] shown in Figure 12(l) can be explained with variations in
effective temperature of the input atmosphere models. Qualitatively, for
example, in metal–poor stars, as effective temperature increases the number
densities of CH and Fe I both decrease while at the same time the H- continuum
opacity increases due to increasing electron density from ionizing metals. As
a consequence, the correlation between uncorrected [C/H] and [Fe/H] with the
effective temperature should exist. Therefore any artificial trends from
inappropriate input effective temperature should be canceled out in the
uncorrected [C/Fe]. This is the reason why we do not see any gradient in our
Figure 12(l). We simulate this with MOOG_SCAT for the star V–45, which appears
to have the best spectrum quality among other stars. Using the equivalent
width measurements by Kirby, Ji, & Kovalev (2023), we obtained
$\delta$[Fe/H]/$\delta$$T_{\rm eff}$(+100K) = 0.12 dex K-1. Since Kirby, Ji, &
Kovalev (2023) relied on the synthetic spectra for their carbon abundance
measurements, we generated synthetic spectra around the CH G band region, and
we obtained $\delta$[C/H]/$\delta$$T_{\rm eff}$(+100K) = 0.15 dex K-1,
confirming our qualitative reasoning that the temperature dependency of the CH
G band absorption strength is about the similar to that of the neutral iron
atoms.

We conclude that our [C/Fe]ch measurements are in excellent agreement with
those from the high resolution spectral syntheses of the CH G band. It should
not be surprising that both results are based on the same spectral features
and analyzing tools, but different approaches; photometric versus
spectroscopic.

##  Appendix B Effective Temperatures: Origin of the elemental gradient
against luminosity in high resolution spectroscopic study

As mentioned above, Kirby, Ji, & Kovalev (2023) noticed considerable elemental
abundance gradients against the stellar luminosity. They applied non-LTE
corrections for some elements, but these gradients persist. This is at odd. We
used the similar LTE approach with the same tools as those adopted by Kirby,
Ji, & Kovalev (2023), but our [Fe/H]hk does not show any perceptible gradient
with the stellar luminosity, while the results of Kirby, Ji, & Kovalev (2023)
show considerable quadratic trends in elemental abundances against luminosity.
The major difference between the two studies is the adopted input stellar
parameters, in particular the effective temperature. We relied on the
evolutionary effective temperatures, while Kirby, Ji, & Kovalev (2023) used
those estimated from the color–temperature relations by Mucciarelli,
Bellazzini, & Massari (2021). We argue that their additional luminosity-based
gradient is arisen from their inappropriate temperature scales.

In Figure 13(a), we compare our evolutionary $T_{\rm eff}$ and photometric
$T_{\rm eff}$ adopted by Kirby, Ji, & Kovalev (2023). As we discussed above,
our evolutionary population synthesis models are based on the latest BaSTI
(Pietrinferni et al., 2021). We emphasize again that our [Fe/H]hk measurements
do not show any gradient against the $V$ magnitude as in Figure 3, strongly
suggesting that our input parameters, such as the effective temperature and
surface gravity, are self-consistent. Below and above $M_{\rm G,o}$ $\sim$ 0.0
mag, the effective temperatures adopted by Kirby, Ji, & Kovalev (2023) are
significantly smaller than those from our evolutionary models. At given
observed equivalent widths of neutral iron atoms, lowering input effective
temperatures in the LTE analysis will result in lowering output [Fe/H] values
in RGB stars.

In Figure 13(b), we show the run of uncorrected $A$(FeI) by Kirby, Ji, &
Kovalev (2023) against the $M_{G,0}$ magnitude, showing a considerable
curvature. Also shown is their polynomial fit to the iron abundances to remove
the trend.

In Figure 13(c), we show the amount of the iron abundances that were under- or
over-estimated due to the temperature differences in the panel (a). To
calculate the correction value, we adopted $\delta$[Fe/H]/$\delta$$T_{\rm
eff}$(+100K) = 0.12 dex K-1 from our result above. We also calculated those
from the different surface gravity, but their contributions are insignificant
as shown in the panel (c). In the figure, we show the modified polynomial fit,
which is the polynomial fit by Kirby, Ji, & Kovalev (2023) minus 5.02. In the
panel (c), the modified polynomial fit nicely traces the [Fe/H] correction
values due to temperature differences of individual stars. We applied these
correction values to individual stars as shown in the panel (d), and we
obtained the mean temperature and surface gravity difference corrected [Fe/H]
value of $\langle$[Fe/H]${}_{\rm Teff,logg,cor}\rangle$ = $-2.429\pm 0.013$
dex ($\sigma$ = 0.061 dex). Note that our mean [Fe/H]hk value from these
common stars is $\langle$[Fe/H]hk$\rangle$ = $-2.309\pm 0.010$ dex ($\sigma$ =
0.047 dex) and the discrepancy in the metallicity difference between the two
studies is slightly mitigated. Our new [Fe/H] value is about 0.06 dex larger
than that from the luminosity-based polynomial fit by Kirby, Ji, & Kovalev
(2023), $\langle$[Fe/H]${}_{\rm Teff,cor}\rangle$ = $-2.491\pm 0.011$ dex
($\sigma$ = 0.054 dex), as shown in (e).

We conclude that the luminosity-based elemental abundance trends seen by
Kirby, Ji, & Kovalev (2023) are due to the inappropriate temperature scale
adopted by them, which strongly indicating that the color–temperature
relations by Mucciarelli, Bellazzini, & Massari (2021) fail to work, at least
for M92. We note that the distance modulus and interstellar reddening value
are well known for M92 and the quadratic trend around $M_{G,0}$ cannot be
explained by them.

We applied the same procedures for the [Fe/H] measurements by Roederer &
Sneden (2011). Again, the photometric effective temperatures adopted by
Roederer & Sneden (2011) are cooler than our evolutionary effective
temperatures, $\langle$$T_{\rm eff}$$\rangle$ = $118\pm 18$ K ($\sigma$ = 60
K, 11 stars). The temperature correction will increase their mean metallicity
by $\langle$[Fe/H]${}_{\rm Teff,cor}-$[Fe/H]${}_{\rm uncor}\rangle$ =
$0.141\pm 0.022$ dex ($\sigma$ = 0.072 dex) but it does not completely remove
the trend against luminosity as shown in Figure 14(d), which may require
additional explanation.999The referee also confirmed a strong trend of
metallicity from spectroscopy by Roederer & Sneden (2011) as a function of
magnitude and temperature along the RGB in M92, with warmer and fainter stars
showing higher metallicities. The referee pointed out that this trend reflects
a similar trend as a function of S/N and may then be driven by (or, at least,
due in part to) the quality of spectra. This could be maybe the additional
explanation invoked by us, beside the lowering of [Fe/H] values due to the
lowering input effective temperatures.

Figure 15:  Plots of [C/Fe] vs. [N/Fe], $C_{\rm UBI}$(= $(U-B)-(B-I)$) vs.
$V$, and $cn_{\rm JWL}$ vs. $V$ of RGB stars fainter than their RGBBs in M5,
M30, and M92. We use the $UBI$ photometry by Stetson et al. (2019). The blue
and red dots are for the FG and SG populations classified based on the
photometric elemental abundances in our previous study (Lee, 2021, for M5) and
our current study (for M30 and M92). Our results show that the $C_{\rm UBI}$
is not capable of distinguishing MPs in metal–poor GCs. Figure 16:  (a) A plot
$\parallel\,C_{\rm UBI}$ vs. $V$ of RGB stars in M30. The blue dots are for
the FG and the red dots are for the SG based on the $\parallel\,C_{\rm UBI}$
distribution. (b) A histogram for the $\parallel\,C_{\rm UBI}$ distribution.
The blue and red solid lines denote the FG and SG returned from our EM
analysis of the $\parallel\,C_{\rm UBI}$ distribution, while the gray solid
line is for the total RGB population. (c) A plot $\parallel\,C_{\rm UBI}$ vs.
$V$ of the FG (green) and SG (purple) RGB stars in M30 from our current study,
showing a gradual transition from one population to the other. (d)–(f) Same as
(a)–(c) but for M92

##  Appendix C On the use of the $C_{\mathrm{U}BI}$ index in the metal-poor
regimes

It seems very attractive to make use of the broadband $UBI$ photometry and the
relevant pseudo color index, $C_{\rm UBI}$ = $(U-B)-(B-I)$, in the GC MSP
study, since a huge amount of $UBI$ photometric data has been already
accumulated and publicly available over the years, such as nicely presented by
Stetson et al. (2019). However, it is a great tragedy that many astronomers
claimed their own results for the GC MSP characteristics, such as populational
number ratios, CRDs, and etc., by simply employing the $C_{\rm UBI}$ without
any careful validation.

In our previous study (Lee, 2019a), we demonstrated that the $C_{\rm UBI}$
index in the GC MSP study is not reliable due to the inherited trait of the
broadband photometry. Using three intermediate metallicity GCs, M5, M3, and
NGC 6752, we compared performances of populational taggings from our $cn_{\rm
JWL}$ and the $C_{\rm UBI}$ indices, finding that the results from the $C_{\rm
UBI}$ are easily in error. We proposed further that the $C_{\rm UBI}$ should
not be used in the metal-poor domain (e.g., see Lee, 2023a). It is because the
$C_{\rm UBI}$ makes great use of the CN absorption band features as shown in
Figure 1 of Lee (2019a). However, due to a double–metal molecule nature of CN,
its absorption strength rapidly decline with decreasing metallicity and CN is
almost absent in the metal-poor RGB stars, such as in M30 and M92, especially
the CN red system resided in the $I$ passband.

In Figure 15, we show plots of [C/Fe] vs. [N/Fe], $V$ vs. $C_{\rm UBI}$, and
$V$ vs. $cn_{\rm JWL}$ of RGB stars fainter than their RGBBs in M5 (Lee,
2021), M30, and M92 (Lee, 2017, 2019a, 2023a). In the figure, we classified
individual populations following our tagging scheme described in §4. For M5
and M92, the SG includes both the IG and SG for M92 and the IG and the Extreme
population for M5 (Lee, 2021). For the $C_{\rm UBI}$ index, we used the $UBI$
photometry by Stetson et al. (2019). In the figure, we distinguish the FG and
SG only for clarity. In the intermediate metallicity GC M5, our $cn_{\rm JWL}$
index provides two discrete populations. On the other hand, the boundaries of
the two populations are overlapped in the $C_{\rm UBI}$ index and caution
should be used to separate the correct population in the mixed zone.

For metal–poor GCs M30 and M92, both the $cn_{\rm JWL}$ and $C_{\rm UBI}$ are
not capable of distinguishing individual populations. This is especially true
for our $cn_{\rm JWL}$, since it mainly relies on the CN band absorption
feature at $\lambda$3883Å. Variations in the nitrogen and carbon abundances
among MSPs can produce some gradual change of $C_{\rm UBI}$, but the discrete
populational separation cannot be seen. Figure 16 shows the situation more
clearly. We normalized the $C_{\rm UBI}$ indices for M30 and M92,
$\parallel\,C_{\rm UBI}$, and we performed an EM analysis on their
$\parallel\,C_{\rm UBI}$ distributions, finding the populational number ratio
of $n$(FG):$n$(SG) = 50:50($\pm$5) for M30 and $n$(FG):$n$(SG) = 63:37($\pm$4)
for M92. This populational number ratio of M30 returned from the
$\parallel\,C_{\rm UBI}$ is apparently exactly the same value as that from the
($\parallel$ [C/Fe]ch $-$ $\parallel$ [N/Fe]nh) distribution, 50:50($\pm$5),
of our current study. However, the populational tagging of the individual
stars tells that it is a coincidence. As Figure 16(c) and (f) show, the FG
stars classified from our photometric abundance overlap with the SG stars,
i.e., a gradual transition from one population to the other in metal-poor GCs.
Therefore, the populational taggings cannot be done correctly based on the
$C_{\rm UBI}$ alone. For this reason, we performed populational tagging based
on our $\parallel$ [C/Fe]ch and $\parallel$ [N/Fe]nh domain, which seems to
work excellently.

Finally, we show plots of $V$ versus $C_{\rm UBI}$ of six representative GCs
in Figure 17. We show the FG and SG RGB stars classified with the [Na/Fe]
measurements by Carretta et al. (2009); [Na/Fe] $\leq$ 0.2 dex for the FG and
[Na/Fe] ¿ 0.2 dex for the SG. For NGC 104 and NGC 6254, we adopted slightly
different [Na/Fe] values to secure proper numbers in the FG. We note that our
adopted [Na/Fe] does not affect our main issue addressing here. For the
metal–rich GC NGC 104 and the intermediate metallicity GC NGC 5904, the
$C_{\rm UBI}$ index may be a barometer of the populational tagging of the RGB
stars. However, when it comes to metal–poor GCs, the $C_{\rm UBI}$ becomes a
hopelessly terrible index, due to the reasons that we mentioned above. Figure
17 clearly shows severe confusions in populational taggings with the $C_{\rm
UBI}$ emerge in the whole RGB luminosity level in metal–poor regimes, a strong
evidence of an intrinsic drawback of the $C_{\rm UBI}$. We stress again that
the populational taggings cannot be done with the $C_{\rm UBI}$ alone in the
metal–poor domain, and more dimensions are definitely needed to do the job
correctly.

Figure 17: Plots $C_{\rm UBI}$(= $(U-B)-(B-I)$) vs. $V$ of RGB stars in six
GCs. The blue dots are for the FG and the red dots are for the SG based on
[Na/Fe] of Carretta et al. (2009). Yellow dots are for all stars by Stetson et
al. (2019). The horizontal dark green lines denote the mean $V$ magnitude of
the HB stars in each GC. On the top of each box, we show the name, [Fe/H], and
[Na/Fe] for the FS and SG separation.

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