# Multiple Stellar Populations of Globular Clusters From Homogeneous
Ca–CN–CH–NH Photometry. VII. Metal-Poor Populations in 47 Tucanae (NGC 104)
111Based on observations made with the Cerro Tololo Inter-American Observatory
(CTIO) 1 m telescope, which is operated by the SMARTS consortium. 222This work
presents results from the European Space Agency (ESA) space mission Gaia. Gaia
data are being processed by the Gaia Data Processing and Analysis Consortium
(DPAC). Funding for the DPAC is provided by national institutions, in
particular the institutions participating in the Gaia MultiLateral Agreement
(MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia. The Gaia
archive website is https://archives.esac.esa.int/gaia.

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 present new large field-of-view ($\sim$1°$\times$1°) Ca-CN photometry of
the prototypical metal-rich globular cluster 47 Tucanae (NGC 104). Our results
are the following. (1) The populational number ratios of the red giant branch
(RGB) and red horizontal branch (RHB) are in excellent agreement:
$n$(CN-w):$n$(CN-s) = 30:70 ($\pm$1–2), where the CN-w and CN-s stand for the
CN-weak and CN-strong populations, respectively. Both the CN-s RGB and RHB
populations are more centrally concentrated than those of CN-w populations
are. (2) Our photometric metallicities of individual RGB stars in each
population can be well described by bimodal distributions with two metallicity
peaks, [Fe/H] $\sim$ $-$0.72 and $-$0.92 dex, where the metal-poor components
occupy $\sim$ 13% of the total RGB stars. The metal-poor populations are more
significantly centrally concentrated than the metal-rich populations, showing
a similar result that we found in M3. (3) The RGB bump $V$ magnitudes of
individual populations indicate that there is no difference in the helium
abundance between the two metal-poor populations, while the helium enhancement
of $\Delta Y$ $\sim$ 0.02–0.03 is required between the the two metal-rich
populations. (4) The RHB morphology of 47 Tuc appears to support our idea of
the bimodal metallicity distribution of the cluster. We suggest that 47 Tuc
could be another example of merger remnants of two globular clusters, similar
to M3 and M22.

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 stars (746)

##  1 INTRODUCTION

The ubiquitous nature of the multiple populations (MPs) in globular clusters
(GCs) is an enigmatic phenomenon that cannot be easily understood in modern
astrophysics (e.g., see Bastian & Lardo, 2018; Gratton et al., 2019; Cassisi &
Salaris, 2021). Once thought to be simple groups of old stars, not only the
Galactic but also the extra-Galactic GCs show MPs containing at least two
populations: One is called first generation (FG) with chemical compositions
similar to abundance patterns observed in Galactic field stars with the same
metallicity, while the other is called a second generation (SG) showing the
chemical compositions that experienced proton capture processes at high
temperature. Most notably, the so-called Na–O and C–N anticorrelations are
thought to be a strong evidence of the MPs in GCs (Carretta et al., 2009; Lee,
2010).

Recent development of numerical simulations may seem promising to delineate
the formation of GCs with MPs (e.g., D’ercole et al., 2008; Bekki, 2019;
Calura et al., 2019; McKenzie & Bekki, 2021; Lacchin et al., 2021), however,
some fundamental processes that induce the chemical evolution between MPs are
yet to be understood (e.g., see Bastian et al., 2015; Renzini et al., 2015;
Bastian & Lardo, 2018; Gratton et al., 2019)

47 Tuc (NGC 104) is a massive nearby GC, that has long been considered as a
prototypical metal-rich GC (e.g., Hesser et al., 1987). Numerous photometric
and spectroscopic studies have been conducted for this cluster. Elemental
abundance variations in the red giant branch (RGB), red horizontal branch
(RHB), and main-sequence (MS) stars in 47 Tuc had been known for decades long
before the establishment of the MPs in GCs during the last decade.

Hesser (1978) employed low-resolution spectroscopy, and he found that there
exist CN abundance variations from RGB to MS stars in 47 Tuc (see also Briley
et al., 1991, 2004; Cannon et al., 1998; Harbeck, 2003, for the abundance
variation in MS stars in 47 Tuc). Norris & Freeman (1979) confirmed this
result, and they concluded that the primordial elemental abundance difference
is responsible for such CN anomalies (see also Cottrell & Da Costa, 1981, who
drew the same conclusion from the Na–CN positive correlation). The primordial
origin of MPs in 47 Tuc was securely confirmed by Carretta et al. (2004), who
revealed the Na–O anticorrelation in the turn-off and the base of RGB stars in
47 Tuc. Later, Milone et al. (2012) nicely demonstrated that the MS split in
47 Tuc can be ascribed to the different helium and light elemental abundances,
such as CNO, that can be seen in GC MPs. They also found a radial gradient in
the CN abundance, in the sense that the CN-s RGB stars are more centrally
concentrated than the CN-w RGB stars are, where the CN weak and strong stars
are defined to be stars with weak and strong CN absorption bands at
$\lambda$3883 at a given $V$ magnitude (see also Chun & Freeman, 1979; Smith,
1979; Briley, 1997; Nataf et al., 2011).

Not only the RGB stars but also the RHB stars in 47 Tuc also display the
prototypical characteristics of MPs. Norris & Freeman (1982) found that there
exist a CN–CH anticorrelation in the RHB stars in 47 Tuc, and $V$ magnitude of
the CN-s RHB stars is $\Delta V\sim$ 0.04 mag brighter than the CN-w
counterparts, which led them to conclude that differences in helium abundance
or helium core mass are responsible for this luminosity difference between the
two groups. Briley (1997) confirmed this result, and they also noted that the
CN-s RHB stars tend to be bluer than the CN-w RHB stars (see also Nataf et
al., 2011). In a similarly context, Gratton et al. (2013) found that sodium
abundances increase toward bluer color along the RHB in 47 Tuc, which can be
understood from the existence of the Na–CN positive correlation as we
mentioned above.

The presence of the subgiant branch (SGB) split in 47 Tuc also indicates that
it is not composed of a simple stellar population. After the monumental
discovery of the double SGB sequence in NGC 1851 (Milone et al., 2008; Cassisi
et al., 2008), Anderson et al. (2009) reanalyzed a large number of Hubble
Space Telescope (HST) archival data, and they found the existence of the
double SGB in 47 Tuc, where the second SGB sequence is $\sim$ 0.05 mag fainter
than the main SGB body and occupies $\sim$ 10% of the stars. They also noted
that there exists intrinsic color dispersion in the MS stars, which can be
ascribed to the metallicity spread (with standard helium abundance) or a
helium abundance dispersion of $\Delta Y$ = 0.026 (see also Milone et al.,
2012). Di Criscienzo et al. (2010) interpreted that, in addition to the helium
abundance variations in the 47 Tuc, the difference in the C+N+O abundance
could be responsible for the double SGB in 47 Tuc, in the sense that the
fainter SGB stars contain higher overall C+N+O abundance than the brighter
ones. Later, Marino et al. (2016) performed a spectroscopic study for SGB
stars, finding no concrete evidence of a C+N+O enhancement in the faint SGB
stars.

Finally, the RGB bump (RGBB) of 47 Tuc also indicates that it is a very
intriguing cluster. Nataf et al. (2011) found that RGBB $V$ magnitude becomes
fainter with the radial distance from the cluster’s center.

In our previous study, we showed that M3, previously considered as a
monometallic prototypical GC, has two groups of stars with different
metallicity, $\Delta$[Fe/H] $\sim$ 0.15 dex. More recently, independent
studies showed some evidence of a metallicitiy spread in the FG in previously
considered monometallic GCs and different metallicity distributions between
the FG and SG (e.g., Legnardi et al., 2022; Lardo et al., 2022), making the
understanding of the formation of GCs a more formidable task.

In this paper, we investigate 47 Tuc using our own photometric system that is
optimized to study the GC MPs. So far, almost all the previous studies of the
cluster assumed that 47 Tuc shows an unimodal metallicity distribution (see
also Gratton et al., 2019). Interestingly, our results will show that 47 Tuc
shows a bimodal metallicity distribution with a metallicity difference of
$\Delta$[Fe/H] $\sim$ 0.2 dex, which can be also supported by the RGBB
magnitude and the RHB morphology. Our finding will set a crucial constraint to
understand the formation of 47 Tuc.

Figure 1:  (a) Comparison of [Fe/H] of Carretta et al. (2009) against that of
Codero et al. (2014) for 28 RGB stars in 47 Tuc in common between the two
studies. The filled red circles indicate the RGB stars using the same
instrument between the two studies. (b) Histogram of the [Fe/H] distribution
by Codero et al. (2014). (c) Histogram of the [Fe/H] distribution by Carretta
et al. (2009). (d) Histograms of the [Fe/H] measurements from literature. The
red, blue, and green solid lines are for the RGB stars by Carretta et al.
(2009), Codero et al. (2014), and Wang et al. (2017). The thick red dashed
line is for the RHB stars by Gratton et al. (2013), while the thick blue
dashed line is for the SGB stars by Marino et al. (2016). Except for the RGB
[Fe/H] distribution by Carretta et al. (2009), other four measurements show
nonunimodal [Fe/H] distributions with large metallicity dispersions.

##  2 Spectroscopic Metallicity Distributions of 47 Tuc

Thanks to the advance in high throughput (multiobject) spectrographs combined
with large aperture telescopes, utility of high-resolution spectroscopy
becomes more important. However, there appears to exist some limitations that
significantly reduce the capability of spectroscopic study of GCs. For
example, RGB stars in metal-rich GC 47 Tuc show very strong absorption lines
combined with severe degree of line blendings that make accurate elemental
abundance measurements somewhat difficult (see also Lee, 2010, 2016). Not
surprisingly, unequivocal results among different research groups can often be
witnessed.

In Figure 1(a), we show a comparison of the [Fe/H] measurements for 28 RGB
stars in 47 Tuc in common with Carretta et al. (2009) and Codero et al.
(2014). Using these 28 stars, we obtained the similar mean [Fe/H] values
between the two studies: $\langle$[Fe/H]$\rangle$ =
$-$0.739$\pm$0.028$\pm$0.005 dex from Carretta et al. (2009) and
$-$0.769$\pm$0.075$\pm$0.014 dex from Codero et al. (2014). However, the
[Fe/H] distributions of individual stars may tell a completely different story
as shown in the figure. The [Fe/H] distribution of Carretta et al. (2009)
appears to be an unimodal, while that of Codero et al. (2014) is a nonunimodal
with a significantly large dispersion. We performed statistical tests to see
if these two [Fe/H] distributions are unimodal. The $p$-value returned from
our dip test is 1.85$\times$10-3 for Codero et al. (2014), indicating that it
is nonunimodal. On the other hand, the $p$-value of that of Carretta et al.
(2009) is 0.922, and it is most likely a unimodal, as our histogram suggested.
In the figure, the red-filled circles show the RGB stars measured by using the
same instrument (FLAMES-GIRAFFE on the Very Large Telescope Kueyen telescope)
between the two studies, exhibiting large differences. Therefore, the
discrepancy between the two studies is not instrument dependent, but it may
have something to do with data analysis procedures in each research group.

In Figure 1(d), we show histograms for [Fe/H] distributions of individual
stars in 47 Tuc from various studies (Carretta et al., 2009; Codero et al.,
2014; Gratton et al., 2013; Marino et al., 2016; Wang et al., 2017). The
figure clearly shows that the [Fe/H] distribution of 47 Tuc is not likely
unimodal with large metallicity dispersions, except for that of Carretta et
al. (2009).

We are left with the impression that $\Delta$[Fe/H] $\sim$ 0.1 – 0.2 dex from
high-resolution spectroscopy can be introduced, which may hamper revealing
small-scale structures in the metallicity distribution of given stellar
populations (see also Lee, 2010, 2016; Lee & Sneden, 2021). In this regard,
our narrow- and intermediate-band photometry can be a valuable asset to
investigate metallicity distributions, for example, of the MPs in GCs.

Figure 2:  Gaia EDR3 proper-motions of our FOV. The red ellipse indicates the
boundary (3$\sigma$) of the 47 Tuc member stars shown with blue dots. The red
dots are stars located within 300″ from 47 Tuc, and they are classified as
nonmember stars based on the proper-motion study.  Figure 3:  (a) Positions of
bright stars with $V\leq$ 17 mag that match with the Gaia EDR3 and classified
as 47 Tuc’s members. (b) Positions of bright stars with $V\leq$ 17 mag that
match with the Gaia EDR3 but classified as 47 Tuc’s nonmembers. The thick red
circle indicates the radial distance of 300″ from the cluster’s center. (c)
Positions of bright stars with $V\leq$ 17 mag that have no counterpart in the
Gaia EDR3. (d) $(b-y)$ versus $V$ CMD for (a). (e) $(b-y)$ versus $V$ CMD for
(b). Note the presence of the SMC stars in the faint regime. The red dots show
the CMD from stars located within 300″ from the center that are considered as
proper-motion nonmembers of the cluster. (f) $(b-y)$ versus $V$ CMD for stars
with no Gaia EDR3 counterpart.

##  3 OBSERVATIONS AND DATA REDUCTION

Observations for 47 Tuc were carried out in 54 nights, 27 of which were
photometric, in 9 runs from September 2007 to August 2013 using the CTIO 1.0m
telescope. The CTIO 1.0m telescope was equipped with a STA 4k $\times$ 4k CCD
camera, providing a plate scale of 0$\farcs$289 pixel-1 and a field-of-view
(FOV) of about 20′ $\times$ 20′. The detailed discussion of our new filter
system can be found in Lee (2017). During the whole seasons, the average
seeing of our 47 Tuc science frames was 1$\farcs$37 $\pm$ 0$\farcs$20\. The
effective FOV of our 47 Tuc science field was about 1° $\times$ 1° except for
the JWL39 filter. We employed our JWL39 filter only in 2013, and it mostly
covered an FOV of 20′ $\times$ 20′ of the central part of the cluster.

The detailed procedures for the raw data handling were described in our
previous works (Lee et al., 2014; Lee, 2015, 2017; Lee & Pogge, 2016). The
photometry of the cluster and photometric standard frames were analyzed using
DAOPHOTII, DAOGROW, ALLSTAR and ALLFRAME, and associate packages (Stetson,
1987, 1994; Lee & Carney, 1999). The total number of stars measured from our
ALLFRAME run was about 111,000.

Astrometric solutions for individual stars in our field were derived using the
data extracted from the Gaia EDR3 (Gaia Collaboration, 2021) and the IRAF
IMCOORS package. Then the astrometric solution was applied to calculate the
equatorial coordinates for all stars measured in our science frames.

Figure 4: CMDs for proper-motion membership stars in 47 Tuc. We also show the
mean measurement uncertainties at a given $V$ magnitude bin with blue error
bars. Note the broad RGB sequences in $m1$, $cy$ and the discrete double RGB
sequences in $cn_{\rm JWL}$ and $cn^{\prime}_{\rm JWL}$ CMDs.  Figure 5:
Parallelized CMDs with histograms for various color indices used in our study.
Note that our $\parallel$$cn_{\rm JWL}$ and $\parallel$$cn^{\prime}_{\rm JWL}$
indices show discrete double RGB sequences in 47 Tuc. We also note that both
the $\parallel$$(b-y)$ and $\parallel$$hk_{\rm JWL}$, both of which are good
metallicity indicators, show asymmetric distributions toward smaller color
indices.

##  4 Membership Selection

The Galactic latitude of 47 Tuc is large, $b$ = $-$45°, and the contamination
from the foreground off-cluster Galactic field stars is expected to be small.
On the other hand, due to the proximity to the Small Magellanic Cloud (SMC),
the contamination from the SMC can be severe, in particular for the faint
regime. In Figure 2, we show the proper-motions for our 47 Tuc’s FOV from the
Gaia EDR3 (Gaia Collaboration, 2021), where two groups of stars, one for 47
Tuc and the other for the SMC, can be clearly seen.

In order to select proper-motion member stars, we derived the mean values of
proper-motions of 47 Tuc using iterative $\sigma$-clipping calculations,
finding that, in units of milliarcsecond per year, ($\mu_{\rm
RA}\times\cos\delta$, $\mu_{\rm decl.}$) = (5.253, $-$2.543) with standard
deviations along the major axis of the ellipse of 0.666 mas yr-1 and along the
minor axis of 0.659 mas yr-1. We considered that stars within 3$\sigma$ from
the mean values are 47 Tuc proper-motion member stars as shown with blue dots
in Figure 2.

In Figure 3, we show color-magnitude diagrams (CMDs) for proper-motion members
and nonmembers of 47 Tuc. Also shown are the stars without the Gaia EDR3
counterparts. We believe that, due to large measurement uncertainties rising
from crowdedness, a significant fraction of 47 Tuc member stars in the central
part deviates large in the proper-motion measurements of the Gaia EDR3 as
shown in Figure 2. Furthermore, there exists a large number of stars located
in the central part of cluster, owing to crowdedness, without proper-motion
measurements. In our current paper, we consider RGB stars with $-$1.5 $\leq$
$V-V_{\rm HB}$ $\leq$ 2.5 mag in two cases separately: (1) using proper-motion
member stars only (case (1)), where our sample is not complete and tends to be
slightly biased toward the outer part of the cluster; (2) using stars selected
from the multicolor CMDs (case (2)), which include the stars in case (1) and
missing member stars by Gaia EDR3 as shown in Figure 2.

##  5 Photometric Indices and Color-Magnitude Diagrams

We used the filters provided by the CTIO from 2007 to 2010, and we used our
own filters since 2012. As we already noted in our previous study (Lee, 2015,
2019c), the original $Ca_{\rm CTIO}$ was designed to have a filter bandwidth
and pivot wavelength very similar to that of the $Ca_{\rm AT}$ defined by
Anthony-Twarog et al. (1991). Due to aging, the $Ca_{\rm CTIO}$ had been
degraded, and its original transmission curve had been significantly altered
to the shorter wavelength, resulting in having the passband similar to our
JWL39 filter (e.g., see Figure 1 of Lee, 2019c). Therefore, we do not use the
$Ca_{\rm CTIO}$ to measure metallicity. Instead, we used the $Ca_{\rm CTIO}$
as a good proxy of the JWL39 filter in our current study.

Throughout this work, we will use our own photometric indices (see also Lee,
2019c; Lee & Sneden, 2021), defined as

$\displaystyle hk_{\rm JWL}$ $\displaystyle=$ $\displaystyle(Ca_{\rm
JWL}-b)-(b-y).$ (1) $\displaystyle cn_{\rm JWL}$ $\displaystyle=$
$\displaystyle JWL39-Ca_{\rm JWL},$ (2) $\displaystyle cn^{\prime}_{\rm JWL}$
$\displaystyle=$ $\displaystyle Ca_{\rm CTIO}-Ca_{\rm JWL}.$ (3)

The $hk_{\rm JWL}$ index is a good photometric measure of metallicity (e.g.,
Anthony-Twarog et al., 1991; Lee et al., 2009; Lee, 2015), assuming a constant
[Ca/Fe] among GC membership stars (Carney, 1996; Marino et al., 2019). As we
already discussed (Lee, 2019c), the $cn^{\prime}_{\rm JWL}$ can be used as an
excellent proxy of the $cn_{\rm JWL}$ index. Hence, both the $cn_{\rm JWL}$
and $cn^{\prime}_{\rm JWL}$ indices are photometric measures of the CN band at
$\lambda$3883.

In Figure 4, we show our CMDs. The discrete RGB sequences can be clearly seen
both in our $cn_{\rm JWL}$ and $cn^{\prime}_{\rm JWL}$ CMDs. On the other
hand, both the $m1$ and $cy$ CMDs show splits in the upper RGB sequences but
not in the lower RGB sequences.

Figure 6:  (a) A $\parallel$$cn^{\prime}_{\rm JWL}$ vs. $V$ CMD for the
proper-motion member RGB stars (case (1)). The blue and red dots denote the
CN-w and CN-s RGB stars in 47 Tuc returned from our EM estimator. (b) The
black line denotes the $cn^{\prime}_{\rm JWL}$ histogram of 47 Tuc RGB stars.
The blue and red solid lines are the histograms returned from our EM
estimator, and the gray solid line is for that of the total RGB population.
(c) Same as (a) but for all RGB stars (case (2)). (d) Same as (b) but for all
RGB stars.  Figure 7:  Metallicity distributions of individual populations.
The pink, navy, orange, and dark-green colors denote the CN-wMP, CN-wMR, CN-
sMP, and CN-sMR, respectively. Gray color denotes the whole population. The
error bars are for $\pm 1\sigma$. In upper left corner of the panel (a), we
show the measurement uncertainties ($\pm 1\sigma$) using the bootstrap method,
$\pm$0.019 dex.

##  6 Red Giant Branch

###  6.1 Populational Tagging

We followed the similar method that we developed in our previous studies of
other GCs with MPs (e.g., see Lee & Sneden, 2021; Lee, 2021). In order to
remove the luminosity effect in our analysis, the RGB sequences in the
individual color indices were parallelized using the following relation (also
see Lee, 2019a, c);

$\parallel{\rm CI}(x)\equiv\frac{{\rm CI}(x)-{\rm CI}_{\rm red}}{{\rm CI}_{\rm
red}-{\rm CI}_{\rm blue}},$ (4)

where CI$(x)$ is the color index of the individual stars, and CIred, CIblue
are color indices for the fiducials of the red and the blue sequences of
individual color indices, respectively (see also Milone et al., 2017). We
derived fourth-order polynomial fits for individual color indices, and we show
our parallelized CMDs and histograms in Figure 5. Note that our
$\parallel$$cn_{\rm JWL}$ and $\parallel$$cn^{\prime}_{\rm JWL}$ show discrete
bimodal RGB distributions, a strong evidence of MPs in 47 Tuc. As we showed in
Figure 4, our measurement uncertainties are significantly smaller than the
widths of RGB sequences in individual color indices. Therefore, double RGB
sequences in $\parallel$$cn_{\rm JWL}$ and $\parallel$$cn^{\prime}_{\rm JWL}$
are thought to be real, reflecting a bimodal CN distribution of RGB stars in
47 Tuc (e.g., Hesser, 1978; Norris & Freeman, 1979; Briley et al., 1991; Lee,
2019b). The $\parallel$$m1$ and $\parallel$$cy$ CMDs also suggest MPs in 47
Tuc, but the populational separations in the lower part of the RGB sequence
are somewhat ambiguous as we already pointed out. We also note asymmetric
distributions of RGB stars in the $\parallel$$(b-y)$ and $\parallel$$hk_{\rm
JWL}$, extending toward smaller color index values, most likely due to the
difference in metallicity in RGB stars in 47 Tuc as we will discuss below.

In our previous studies (Lee, 2017, 2018, 2019a, 2019c, 2020; Lee & Sneden,
2021), we showed that both the $cn_{\rm JWL}$ and $cn^{\prime}_{\rm JWL}$
indices are very powerful tools to classify MPs in GCs. As we already
mentioned, our $cn_{\rm JWL}$ observations were limited to the central part of
the cluster, and we decided to use the $cn^{\prime}_{\rm JWL}$ to classify MPs
in 47 Tuc in our current study. Assuming a bimodal
$\parallel$$cn^{\prime}_{\rm JWL}$ distribution as already shown in Figure 5,
we applied the expectation maximization (EM) algorithm for a two-component
Gaussian mixture model on our $\parallel$$cn^{\prime}_{\rm JWL}$ index in
order to perform a populational tagging of RGB stars in 47 Tuc. The stars with
$P$($\parallel$$cn^{\prime}_{\rm JWL}$$|x_{i})\geq$ 0.5 from the EM estimator
correspond to the CN-w population, where $x_{i}$ denotes the individual RGB
stars, while those with $P$($\parallel$$cn^{\prime}_{\rm JWL}$$|x_{i})$ $<$
0.5 correspond to the CN-s population. In an iterative manner, we calculated
the probability of individual stars for being the CN-w and the CN-s
populations. Through this process, we obtained the number ratio between the
two populations $n$(CN-w):$n$(CN-s) = 32:68 ($\pm$1) for the proper-motion
member RGB stars (case (1)) and 30:70 ($\pm$1) for all RGB stars (case (2)).
We show our result in Figure 6. As we will show later, the CN-s population is
more centrally concentrated than the CN-w is, and the incomplete detection is
more severe for the CN-s stars in the central part of cluster, resulting in a
slightly smaller CN-s fraction when the proper-motion-selected RGB stars were
used. Note that our result is significantly different from that of Milone et
al. (2017). They relied on the HST photometry with a small FOV, $\sim$
3′$\times$3′, compared to our observation, $\sim$ 1°$\times$1°, and they
obtained $N_{1}/N_{\rm tot}$ = 0.175$\pm$0.009, where the $N_{1}$ denotes the
number of the FG of stars.

We emphasize that the FG of stars defined by Milone et al. (2017) is
corresponding to our CN-w population (see Lee, 2017). The discrepancy between
our result and that of Milone et al. (2017) is a natural consequence of a
small FOV of the HST observations of Milone et al. (2017) with a strong
populational radial gradient in 47 Tuc. We note that Milone et al. (2012)
obtained $\sim$ 30% of the FG of stars by combining the ground-based
observations with a large FOV, consistent with our results.

Table 1: Abundances for input model atmospheres. Element | Abundance | Note
---|---|---
[Fe/H] |  $-$0.5, $-$0.7, $-$0.9 |
$Y$ | 0.25, 0.28 |
([C, N, O/Fe]) | (0.00, 0.00, 0.30), ($-$0.30, 1.00, 0.00) | CNO1
| (0.15, 0.15, 0.50), ($-$0.45, 1.35, 0.10) | CNO2
Table 2: Mean [Fe/H] values. | Dartmouth | $Y^{2}$ | PGPUC | Freq.
---|---|---|---|---
|  $\langle$[Fe/H]hk$\rangle$ |  $\langle$[Fe/H]hk$\rangle$ |  $\langle$[Fe/H]hk$\rangle$ | (%)
Case (1): Proper-motion member (Gaia EDR3)
Mean |  $-$0.753$\pm$0.075$\pm$0.001 |  $-$0.734$\pm$0.074$\pm$0.001 |  $-$0.730$\pm$0.074$\pm$0.001 |
CN-w |  $-$0.728$\pm$0.072$\pm$0.002 |  $-$0.731$\pm$0.073$\pm$0.002 |  $-$0.719$\pm$0.073$\pm$0.002 | 32.3
CN-wMP |  $-$0.903$\pm$0.049$\pm$0.005 |  $-$0.900$\pm$0.046$\pm$0.005 |  $-$0.901$\pm$0.049$\pm$0.006 | 2.3
CN-wMR |  $-$0.714$\pm$0.053$\pm$0.002 |  $-$0.717$\pm$0.055$\pm$0.002 |  $-$0.705$\pm$0.053$\pm$0.002 | 30.0
CN-s |  $-$0.766$\pm$0.074$\pm$0.002 |  $-$0.736$\pm$0.074$\pm$0.002 |  $-$0.735$\pm$0.075$\pm$0.002 | 67.7
CN-sMP |  $-$0.936$\pm$0.044$\pm$0.003 |  $-$0.911$\pm$0.037$\pm$0.003 |  $-$0.914$\pm$0.040$\pm$0.003 | 6.6
CN-sMR |  $-$0.748$\pm$0.052$\pm$0.001 |  $-$0.723$\pm$0.058$\pm$0.001 |  $-$0.722$\pm$0.057$\pm$0.001 | 61.1
Case (2): All RGBs
Mean |  $-$0.766$\pm$0.088$\pm$0.001 |  $-$0.746$\pm$0.085$\pm$0.001 |  $-$0.742$\pm$0.087$\pm$0.001 |
CN-w |  $-$0.735$\pm$0.081$\pm$0.002 |  $-$0.738$\pm$0.082$\pm$0.002 |  $-$0.726$\pm$0.083$\pm$0.002 | 29.7
CN-wMP |  $-$0.916$\pm$0.060$\pm$0.006 |  $-$0.911$\pm$0.055$\pm$0.005 |  $-$0.915$\pm$0.060$\pm$0.006 | 2.8
CN-wMR |  $-$0.715$\pm$0.055$\pm$0.002 |  $-$0.717$\pm$0.057$\pm$0.002 |  $-$0.706$\pm$0.056$\pm$0.002 | 26.9
CN-s |  $-$0.779$\pm$0.087$\pm$0.002 |  $-$0.749$\pm$0.086$\pm$0.002 |  $-$0.749$\pm$0.088$\pm$0.002 | 70.3
CN-sMP |  $-$0.929$\pm$0.055$\pm$0.003 |  $-$0.913$\pm$0.048$\pm$0.003 |  $-$0.916$\pm$0.052$\pm$0.003 | 9.8
CN-sMR |  $-$0.747$\pm$0.052$\pm$0.001 |  $-$0.724$\pm$0.058$\pm$0.001 |  $-$0.722$\pm$0.057$\pm$0.001 | 60.5
Table 3: Differences in Mean [Fe/H] Values Using Different Isochrones. |  $\Delta\langle$[Fe/H]$\rangle$
---|---
$\Delta$[Fe/H]hk(Dartmouth $-$ $Y^{2}$) | 0.003$\pm$0.008$\pm$0.000
$\Delta$[Fe/H]hk(Dartmouth $-$ PGPUC) |  $-$0.020$\pm$0.012$\pm$0.000
$\Delta$[Fe/H]hk($Y^{2}$ $-$ PGPUC) |  $-$0.006$\pm$0.008$\pm$0.000
Table 4: Differences in Mean [Fe/H] Values Using Different CNO Abundances. |  $\langle$[Fe/H]${}_{\rm CNO1}-$[Fe/H]${}_{\rm CNO2}\rangle$
---|---
| All | CN-w | CN-s
Dartmouth |  $-$0.003$\pm$0.007$\pm$0.000 | 0.006$\pm$0.002$\pm$0.000 |  $-$0.007$\pm$0.008$\pm$0.000
PGPUC |  $-$0.003$\pm$0.006$\pm$0.000 | 0.006$\pm$0.002$\pm$0.000 |  $-$0.007$\pm$0.008$\pm$0.000
YAPSI |  $-$0.003$\pm$0.007$\pm$0.000 | 0.007$\pm$0.001$\pm$0.000 |  $-$0.008$\pm$0.009$\pm$0.000
Figure 8:  The $(B-V)$, $(B-I)$, and $(U-V)$ CMDs using the multi-color photometry of Stetson et al. (2019) with three different radial zones, 60″ $\leq$ $r$ $<$ 200″ (zone (1)), 200″ $\leq$ $r$ $<$ 400″ (zone (2)), and 400″ $\leq$ $r$ $<$ 600″ (zone (3)), in 47 Tuc. Note that we used stars with low photometric measurement uncertainties, $\sigma(B-V)\leq 0.01$ mag. The pink, navy, orange, and dark-green colors denote model isochrones with ([Fe/H], $Y$, CNO) = ($-$0.90 dex, 0.25, CN-w), ($-$0.70 dex, 0.25, CN-w), ($-$0.90 dex, 0.25, CN-s), and ($-$0.70 dex, 0.28, CN-s), respectively, for 12 Gyr.  Table 5: RGBB $V$ magnitudes. Pop. | This Study | Stetson et al. (2019)
---|---|---
CN-wMP | 14.399 ($\pm$0.040) | 14.408 ($\pm$0.040)
CN-wMR | 14.559 ($\pm$0.025) | 14.530 ($\pm$0.025)
CN-sMP | 14.385 ($\pm$0.035) | 14.358 ($\pm$0.030)
CN-sMR | 14.492 ($\pm$0.020) | 14.525 ($\pm$0.020)

###  6.2 Metallicity Distributions

As we mentioned earlier, our $hk_{\rm JWL}$ index is an excellent measure of
the Ca II H & K lines (see Anthony-Twarog et al., 1991; Lee et al., 2009; Lee,
2015, and references therein). Providing a constant [Ca/Fe] value in a given
GC (e.g., Carney, 1996; Marino et al., 2019), the $hk_{\rm JWL}$ index can be
a good measure of metallicity. In our previous studies, we obtained the [Fe/H]
values of individual RGB stars in M3 and M5 using our $hk_{\rm JWL}$ (Lee &
Sneden, 2021; Lee, 2021). Here, we applied same procedure to derive
metallicities of individual RGB stars in 47 Tuc.

To derive photometric metallicity of individual RGB stars, we used three
different model isochrones, the Dartmouth (Dotter et al., 2008), PGPUC
(Valcarce et al., 2012), and $Y^{2}$ (Yi et al., 2008). We obtained model
grids for [Fe/H] = $-$0.9, $-$0.7, and $-$0.5 dex with [$\alpha$/Fe] = +0.3
dex, $Y$ = 0.25 and 0.28, and the age of 12 Gyr (e.g., see Brogaard et al.,
2017). We constructed series of model atmospheres and synthetic spectra using
ATLAS12 and SYNTHE (Kurucz, 2011). During our calculations, we adopted two
sets of CNO abundances between the CN-w and CN-s populations to explore how
the CNO abundances may affect our results. For one set, we adopted ([C/Fe],
[N/Fe], [O/Fe]) of (0.00, 0.00, 0.30) for the CN-w and ($-$0.30, 1.00, 0.00)
for the CN-s (CNO1, see Table 1). For the other, we adopted (0.15, 0.15, 0.50)
for the CN-w and ($-$0.45, 1.35, 0.10) for the CN-s (CNO2). Individual
synthetic spectra were convolved with our filter transmission functions to be
converted to our photometric system. Finally we calculated bolometric
corrections for individual colors using the same method described by Girardi
et al. (2002).

The photometric metallicity of individual RGB stars can be calculated using
the following relation (e.g., see Lee & Sneden, 2021);

$\displaystyle{\rm[Fe/H]}_{hk}$ $\displaystyle\approx$ $\displaystyle
f(hk_{\rm JWL},~{}M_{V}),$ (5)

and we show our results in Table 2. We obtained the mean photometric
metallicity of $\langle$[Fe/H]hk$\rangle$ $\approx$ $-$0.75 dex for 47 Tuc,
and our result is consistent with that of previous results by others (e.g.,
see Carretta et al., 2009; Codero et al., 2014; Marino et al., 2016) to within
measurement uncertainties. Not surprisingly, the metallicity from the proper-
motion member RGB stars (case (1)) and that from all RGB stars (case (2)) are
in excellent agreement. As shown in Tables 2 and 3, the differences in the
photometric metallicity returned from different model isochrones are too small
to affect our results, $\Delta$[Fe/H] $\leq$ 0.02 dex. We would like to
emphasize that we are interested in the relative [Fe/H] values among RGB
stars, and the differences in the relative [Fe/H] distributions from different
model isochrones are negligibly small. Also importantly, the differences in
metallicity from different CNO abundances are also negligibly small. In Table
4, we show $\Delta$[Fe/H] between those returned from the CNO1 and CNO2
compositions. We obtained the metallicity difference of $\Delta$[Fe/H] $\leq$
0.01 dex, which may imply that the metallicity spread in 47 Tuc RGB stars
described below may not be related to our adopted CNO abundances of individual
populations. In the following analysis, we will take the [Fe/H]hk returned
from the Dartmouth isochrones as our reference photometric metallicity in
order to maintain consistency with our previous works on M3 and M5 (Lee &
Sneden, 2021; Lee, 2021). We emphasize again that the choice of our reference
values does not affect our results.

In Figure 7, we show the metallicity distributions for the CN-w and CN-s
populations. As already shown, there is no metallicity difference between the
CN-w and CN-s populations, same as in our previous results for M3 and M5 (Lee
& Sneden, 2021; Lee, 2021). On the other hand, both the CN-w and CN-s
populations exhibit asymmetric metallicity distributions with long metal-poor
tails, suggestive of multimodal metallicity distributions. We estimated
measurement uncertainties using the bootstrap method, and we obtained
$\pm$0.019 dex for both populations. As shown in Figure 7, our estimated
measurement error is too small to fully explain the metallicity spread seen in
each population, and we believe that the intrinsic metallicity differences
among constituent stars are responsible for the asymmetric metallicity
distributions. In order to examine multimodal metallicity distributions of 47
Tuc, we applied an EM analysis with a two-component Gaussian mixture model for
the metallicity distributions of both the CN-w and CN-s populations. We show
our results in Table 2 and Figure 7, suggesting that each population can be
well described by bimodal metallicity distributions, reminiscent of M3 (Lee &
Sneden, 2021). The metal poor components (the CN-wMP and CN-sMP), which
constitute about 10 – 13% of the total RGB population, are about 0.2 dex more
metal-poor than the metal-rich components (the CN-wMR and CN-sMR).

The $UBVI$ photometry of Stetson et al. (2019) appears to support our results
that 47 Tuc contains metal-poor components. In Figure 8, we show the $(B-V)$,
$(B-I)$, and $(U-V)$ CMDs with three different radial zones, 60″ $\leq$ $r$
$<$ 200″ (zone (1)), 200″ $\leq$ $r$ $<$ 400″ (zone (2)), and 400″ $\leq$ $r$
$<$ 600″ (zone (3)), in 47 Tuc. Note that only the $V$ passband is not
affected by the carbon and nitrogen abundance variations. Other photometric
passbands, $U$, $B$, and $I$ contain CN and CH molecular bands, and therefore,
their magnitudes can be altered depending on their CNO abundances (i.e., the
CN-w and CN-s). Interestingly, $B$ passband contains both the CN and CH bands
(see Figure 1 of Lee, 2019c). In GC stars, where the carbon and nitrogen
abundances are anticorrelated, the contribution of the CN molecules tends to
be canceled out by that of the CH molecules (see also Sbordone et al., 2011) .
Therefore, $(B-V)$ color is almost unaffected by the carbon and nitrogen
abundance variations. In Figure 8, one can find that the $(B-V)$ colors of the
model isochrone for the CN-wMP are almost identical to those of the CN-sMP.
The same is true for the CN-wMR and CN-sMR. The figure suggests that 47 Tuc
contains metal-poor population in the central part of the cluster with a
radial gradient, supporting our results discussed above.

In Figure 9, we show comparisons of individual parallelized color indices of
the proper-motion membership RGB stars in 47 Tuc. The figure clearly suggests
that the four groups of stars that we obtained are truly separate populations
occupying their own characteristic domains.

Figure 9:  Comparisons of parallelized color indices of the proper-motion membership 47 Tuc RGB stars with $-$1.5 $\leq$ $V-V_{\rm HB}$ $\leq$ 2.5 mag, showing that our four groups of stars are different stellar populations. Colors are the same as Figure 7.  Figure 10:  Plots of $by_{5}$ vs. $V$ CMDs and differential LFs of individual populations. We show the mean RGBB $V$ magnitudes of individual populations with gray dotted lines and those of individual populations with dotted lines with respective colors and yellow-shaded boxes. Note that the $by_{5}$ values are defined to be $by_{5}=5\times(b-y)-2.5$ for the clarity of the figure.  Figure 11:  (a) Plots of $(B-V)$ vs. $V$ CMD for the CN-w population using the photometric data by Stetson et al. (2019). The blue and red dots denote the CN-wMP and CN-wMR stars based on the $\parallel(B-V)$ distribution. The horizontal long-dashed lines and yellow boxes denote the $V_{\rm bump}$ and measurement uncertainties for the CN-wMP (pink) and CN-wMR (navy) populations based on our own photometry. (b) Differential LFs of the CN-wMP (blue) and CN-wMR (red). The error bars denote the measurement uncertainties using the data by Stetson et al. (2019). (c) Same as (a) but for the CN-s population. (d) Same as (b) but for the CN-s population.  Figure 12:  (a) A plot of $V_{\rm bump}$ versus [Fe/H] using the Y2 isochrones for $Y$ = 0.250, 0.275, and 0.300 along with those of four populations in 47 Tuc. The $\Delta V$ indicates the $V$ magnitude offset in model isochrones to match our $V_{\rm bump}$ magnitude the CN-wMR population. The colors are the same as Figure 7. (b) Same as (a) but using the PGPUC isochrones. (c) A plot of $V_{\rm bump}$ versus [Fe/H] using the BaSTI isochrones for $Y$ = 0.255, 0.275, and 0.300. (d) A plot of ($V_{\rm bump}-V_{\rm TO}$) versus [Fe/H] using the $Y^{2}$ isochrones. (e) Same as (d) but using the PGPUC isochrones. (f) Same as (d) but using the BaSTI isochrones.  Table 6: Slopes in $V_{\rm bump}$ magnitudes versus [Fe/H] and $Y$. Relation | Slope $a$
---|---
($\Delta$ Abundance) | $Y^{2}$ | PGPUC | BaSTI
(1) $V_{\rm bump}$ $\propto$ $a\times Y$ |  $-$2.279$\pm$0.028 |  $-$2.425$\pm$0.033 |  $-$1.571$\pm$0.134
$\Delta Y$11The difference in helium abundance between the CN-wMR and CN-sMR. | 0.029$\pm$0.014 | 0.028$\pm$0.013 | 0.043$\pm$0.021
(2) ($V_{\rm bump}$ $-V_{\rm TO}$) $\propto$ $a\times Y$ |  $-$4.267$\pm$0.163 |  $-$3.947$\pm$0.192 |  $-$3.790$\pm$0.504
$\Delta Y$11The difference in helium abundance between the CN-wMR and CN-sMR. | 0.016$\pm$0.008 | 0.017$\pm$0.008 | 0.018$\pm$0.009
(3) $V_{\rm bump}$ $\propto$ $a\times$[Fe/H] | 1.124$\pm$0.015 | 1.221$\pm$0.021 | 1.036$\pm$0.006
$\Delta$[Fe/H]22The difference in [Fe/H] between the CN-wMR and CN-wMP. | 0.142$\pm$0.042 | 0.131$\pm$0.039 | 0.154$\pm$0.045
(4) ($V_{\rm bump}$ $-V_{\rm TO}$) $\propto$ $a\times$[Fe/H] | 0.814$\pm$0.019 | 0.972$\pm$0.017 | 0.835$\pm$0.036
$\Delta$[Fe/H]22The difference in [Fe/H] between the CN-wMR and CN-wMP. | 0.197$\pm$0.058 | 0.165$\pm$0.048 | 0.192$\pm$0.057

###  6.3 Red Giant Branch Bumps

During the course of low-mass star evolution, RGB stars experience a temporary
drop in luminosity when the very thin H-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. RGB stars in such evolutionary phases have to cross three times
the same luminosity interval, leaving a distinctive feature, the so-called
RGBB (e.g., see Renzini & Fusi Pecci, 1988; 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). Accurate differential photometry can be possible in the GC MP study,
and therefore, one can precisely estimate relative helium contents among MPs
in a given GC with prior metallicity information (e.g., see Bragaglia et al.,
2010; Lee, 2015, 2017, 2018; Milone et al., 2018b; Lagioia et al., 2018; Lee &
Sneden, 2021).

In order to derive the RGBB $V$ magnitudes, we calculated the generalized
differential luminosity functions (LFs) using a Gaussian kernel density
estimation for individual populations in 47 Tuc. We determined the peak values
as our RGBB $V$ magnitudes for individual populations. In Figure 10 and Table
5, we show our results, obtaining the RGBB $V$ magnitudes of 14.399 ($\pm$
0.040), 14.559 ($\pm$ 0.025), 14.385 ($\pm$ 0.035), 14.492 ($\pm$0.020) mag
for the CN-wMP, CN-wMR, CN-sMP, and CN-sMR, respectively. We caution that the
number of RGBB stars, in particular, for the CN-wMP population is very small
and it is worrisome that our RGBB $V$ magnitude may not be accurate.

We applied the same procedures for the $UBVRI$ photometry by Stetson et al.
(2019), to see if their data show the same trend in the RGBB $V$ magnitudes.
First, we select the proper-motion member stars using the Gaia EDR3. Then we
separated the CN-w and CN-s populations using the $\Delta C_{UBI}$ index,

$\Delta C_{UBI}=\frac{C_{UBI}-C_{UBI,r}}{C_{UBI,r}-C_{UBI,b}},$ (6)

where $C_{UBI}=(U-B)-(B-I)$, while $C_{UBI,b}$ and $C_{UBI,r}$ denote the
fiducials for the $C_{UBI}$-blue and $C_{UBI}$-red sequences, respectively, at
a given magnitude (Milone et al., 2012; Lee, 2019a). To perform a populational
tagging, we applied the EM algorithms for a two-component Gaussian mixture
model. Through this process, we obtained populational number ratio of
$n$(CN-w):$n$(CN-s) = 32:68 ($\pm$1), consistent with our result based on the
proper-motion member stars. As already shown in Figure 8, the $(B-V)$ color
depends more sensitively on metallicities than the helium abundances or the
variations in the CNO abundances (also see, Sbordone et al., 2011). Therefore,
the $(B-V)$ color can be used as a metallicity indicator. We calculated the
$(B-V)$ color difference between individual RGB stars and the mean $(B-V)$
fiducial sequence, $\Delta(B-V)$, which shows asymmetric distributions toward
the small $(B-V)$ values for both populations. We performed the populational
tagging for the CN-w and CN-s populations by applying the EM algorithm for a
two-component Gaussian mixture model on the $\Delta(B-V)$ distributions. We
obtained $\Delta(B-V)_{\rm blue}:\Delta(B-V)_{\rm red}$ = 6:94 ($\pm$2) for
the CN-w population and 9:91 ($\pm$2) for the CN-w population, where
$\Delta(B-V)_{\rm blue}$ and $\Delta(B-V)_{\rm red}$ denote RGB stars bluer or
redder than the mean fiducial sequence at a given $V$ magnitude. We note that
these values are consistent with those presented in Table 2 using our own
photometry. Given the similar populational number ratios and the metallicity
sensitivity of the $(B-V)$ color, we argue that $\Delta(B-V)_{\rm blue}$ and
$\Delta(B-V)_{\rm red}$ are corresponding to the metal-poor and metal-rich
subpopulations in the CN-w and CN-s populations, respectively. We calculated
generalized differential LFs using a Gaussian kernel density estimation for
the photometry by Stetson et al. (2019) and we obtained the RGBB $V$
magnitudes of 14.408 ($\pm$ 0.040), 14.530 ($\pm$ 0.025), 14.358 ($\pm$
0.030), 14.525 ($\pm$0.020) mag for the CN-wMP, CN-wMR, CN-sMP, and CN-sMR. We
show our results in Figure 11 and Table 5. We emphasize that the RGBB $V$
magnitudes of individual populations using the photometric data by Stetson et
al. (2019) are in good agreement with those from our results. The number of
RGBB stars of the CN-wMP population of Stetson et al. (2019) is still small,
but it is strongly believed that the RGBB at $V$ $\sim$ 14.40 mag for the CN-
wMP population is a real feature.

In order to derive the relative helium abundances and metallicities, we
compare our RGBB $V$ magnitudes with those from model isochrones. In Figure
12, we show plots of the $V_{\rm bump}$ and ($V_{\rm bump}-V_{\rm TO}$)
against [Fe/H] for model isochrones with $Y$ = 0.250 (0.257), 0.275, and
0.300. We used three different isochrones currently available: Y2 (Yi et al.,
2008), the PGPUC (Valcarce et al., 2012), and the Bag of Stellar Tracks and
Isochrones (BaSTI; Pietrinferni et al., 2021). To determine the $V_{\rm bump}$
of individual model isochrones, we performed Monte Carlo simulations by
constructing evolutionary population synthesis models, similar to those of our
previous studies (e.g., see Lee & Sneden, 2021, and references therein), with
various metallicity and helium contents. We populated 200,000 artificial stars
for individual model isochrones with different chemical abundances, and we
generated generalized histograms to derive the $V_{\rm bump}$ as we did for
our observed data. We note that the $V_{\rm bump}$ of each model isochrone is
brighter than our observations. Therefore, we subtracted 0.17, 0.22 and 0.28
mag from the $V_{\rm bump}$ of the $Y^{2}$, PGPUC and BaSTI model isochrones,
respectively, to match observed [Fe/H], $\sim$ $-$0.72 dex, and presumed
helium abundance, $Y$ $\sim$ 0.25, for the CN-wMR(RGB) population (see also,
Lee, 2015; Lee & Sneden, 2021, for the discrepancies in the RGBB magnitudes in
model isochrones). In Table 6, we show the results for the $V_{\rm bump}$ and
$V_{\rm bump}-V_{\rm TO}$ dependencies on [Fe/H] and $Y$. The difference in
the RGBB $V$ magnitude of 0.067($\pm$0.032) mag between the CN-wMR and CN-wMP
can be translated into the helium abundance difference of $\Delta Y$ =
0.029($\pm$0.014), 0.028($\pm$0.013) and 0.043($\pm$0.021) from relation (1)
in Table 6 using the $Y^{2}$, PGPUC and BaSTI isochrones, respectively. Note
that the result from the BaSTI model isochrone is larger than the other two
isochrones.

Due to the well-known discrepancy between theory and observations concerning
the RGBB brightness, it was kindly recommended by the referee that comparisons
of the difference in brightness between the RGBB magnitude associated with the
individual populations and the corresponding theoretical quantities would be
more reasonable. In this respect, we obtained the turn-off $V$ magnitude of
17.588($\pm$0.020) mag for 47 Tuc, and we calculated the ($V_{\rm bump}-V_{\rm
TO}$) of individual populations assuming they have the same turn-off
magnitude. Note that this assumption may not be appropriate unless they have
slightly different ages, since, at a given age, the turn-off $V$ magnitude
becomes brighter with decreasing metallicity and increasing helium abundance.
Using relation (2) in Table 6, we obtained the helium abundance difference of
$\Delta Y$ = 0.016($\pm$0.008), 0.017($\pm$0.008) and 0.018($\pm$0.009) from
the $Y^{2}$, PGPUC, and BaSTI isochrones, respectively. Note that all of three
estimates provide consistent degrees of helium enhancement, but they are much
smaller than those from relation (1) above. Our estimates of helium abundance
difference between the CN-wMR and CN-sMR are $\Delta Y$ = 0.016 – 0.043, which
are in good agreement with those of Anderson et al. (2009, for MS), Di
Criscienzo et al. (2010, for SGB), and Nataf et al. (2011, for RGBB), $\Delta
Y$ $\sim$ 0.03. On the other hand, the relative helium abundance estimates
based on the HST observations of the RGBB by Milone et al. (2018b, $\Delta Y$
= 0.011$\pm$0.005) and Lagioia et al. (2018, $\Delta Y$ = 0.010$\pm$0.004)
appear to be too small compared to our results, although their estimates are
marginally agree with our results from relation (2) within the uncertainties.

We also attempt to calculate the metallicity difference between the CN-wMR and
CN-wMP based on their RGBB $V$ magnitudes. Using relation (3) in Table 6, we
obtained 0.142($\pm$0.042), 0.131($\pm$0.039) and 0.154($\pm$0.045) from the
$Y^{2}$, PGPUC and BaSTI isochrones, respectively, all of which are slightly
smaller than the metallicity difference from our $hk_{\rm JWL}$ photometry.
When we used relation (4), we obtained consistent results with that from the
$hk_{\rm JWL}$ photometry, finding 0.197($\pm$0.058), 0.165($\pm$0.048) and
0.192($\pm$0.057) from the $Y^{2}$, PGPUC and BaSTI isochrones, respectively

As shown in Figures 10 and 11, the number of the CN-sMP RGB stars around the
RGBB regime is greater than that of the CN-wMP, in particular for the case (2)
(using the photometrically selected RGB stars). Therefore, it can be thought
that using the CN-sMP RGBB $V$ magnitude would be more reliable to estimate
the metallicity difference between the metal-rich and metal-poor components.
The RGBB $V$ magnitude of the CN-sMP, $V$ = 14.385($\pm$0.035), can be
interpreted in the following three ways:

  * 1.

The CN-sMP population has the same helium abundance as the CN-wMR but lower
[Fe/H].

  * 2.

The CN-sMP population has the same helium abundance as the CN-sMR but lower
[Fe/H].

  * 3.

The CN-sMP population has the same [Fe/H] as the CN-wMR and CN-sMR but
enhanced helium abundance.

In case (1) above, we obtained that the CN-sMP population should be
0.155($\pm$0.010, relation (3)) and 0.200($\pm$0.015, relation (4)) dex lower
than the CN-wMR. Not surprisingly, these are the similar metallicities that we
derived for the CN-wMP. In case (2), the CN-sMP should be $\Delta$[Fe/H] =
0.095($\pm$0.006, relation (3)) and 0.123($\pm$0.009, relation (4)) dex more
metal-poor than the CN-sMR. Finally, in case (3), the CN-sMP should be $\Delta
Y$ = 0.086($\pm$0.017, relation (1)) and 0.044($\pm$0.002, relation (2)) dex
more helium enhanced than the CN-wMR.

As we will show below, the RHB morphology of 47 Tuc can be explained best by
case (1), i.e., the CN-sMP RHB populations has the same helium abundance as
the CN-wMR population and a lower metallicity by $\Delta$[Fe/H] $\sim$ $-$0.20
dex. Therefore, our results may suggest that both the CN-wMP and CN-sMP have
the same metallicity and helium abundance.

Figure 13:  CRDs of individual RGB populations. Colors for individual populations are the same as Figures 7 and 9, while the thick blue and red solid lines denote the whole CN-w and CN-s populations. The error bars show the fraction of the CN-w population with a binning radial distance of the half-light radius. The vertical gray dashed lines denote the core and the half-light radii of 47 Tuc. The green-shaded box indicates the fraction of the first generation by Milone et al. (2017), 0.175$\pm$0.009, measured in the central part of the cluster, $\sim$ 3′ $\times$ 3′.  Table 7: $p$ Values Returned from the K-S Tests for the CRDs of individual RGB populations | CN-s | CN-wMP | CN-wMR | CN-sMP | CN-sMR
---|---|---|---|---|---
CN-w |  $<$ 1.00$\times 10^{-15}$ |  $<$ 1.00$\times 10^{-15}$ | 0.234 |  $<$ 1.00$\times 10^{-15}$ |  $<$ 1.00$\times 10^{-15}$
CN-s |  | 0.180 |  $<$ 1.00$\times 10^{-15}$ |  $<$ 1.00$\times 10^{-15}$ | 1.88$\times 10^{-2}$
CN-wMP |  |  |  $<$ 1.00$\times 10^{-15}$ | 2.80$\times 10^{-6}$ | 3.21$\times 10^{-2}$
CN-wMR |  |  |  |  $<$ 1.00$\times 10^{-15}$ |  $<$ 1.00$\times 10^{-15}$
CN-sMP |  |  |  |  |  $<$ 1.00$\times 10^{-15}$

###  6.4 Cumulative Radial Distributions

The cumulative radial distributions (CRDs) of individual MPs may provide
crucial information on the formation and dynamical evolution of MPs. Many
previous studies on the formation of GCs with MPs suggested that the CN-s
populations form in the innermost part of the cluster in a more extended CN-w
system (e.g., see D’ercole et al., 2008; Bekki, 2019; Cassisi & Salaris,
2021). Moreover, the degree of helium enhancement or the multiphase formations
of the later generation of stars can be dependent on the external gas density
(Calura et al., 2019). The initial structural difference between the CN-w and
CN-s populations can be gradually erased with time, due to the result of the
preferential loss of the CN-w stars during the cluster’s dynamical evolution
(e.g., see Vesperini et al., 2021). It is also plausible that the current
location can be affected by the stellar masses due to different degree of
diffusion processes. For example, the later generations of stars with enhanced
helium abundances have smaller stellar masses due to their fast evolution.
Over the Hubble time, the radial distributions of the later generations of
stars can expand outward (e.g., see Fare et al., 2018; Calura et al., 2019).

In this context, we derive the CRDs of individual MPs in 47 Tuc, and we show
our results in Figure 13. It is very interesting to note that the metal-poor
components, the CN-wMP and CN-sMP, have significantly more centrally
concentrated CRDs than the metal-rich components, the CN-wMR and CN-sMR,
similar to those in M3 (Lee & Sneden, 2021). Furthermore, the CN-sMP
population is more centrally concentrated than the CN-wMP suggesting that the
metal-poor populations mimic the general trend of normal GCs, i.e., the more
centrally concentrated nature of the CN-s population. We performed the
Kolmogorov–Smirnov (K-S) tests, and our results show that the CRDs of
individual populations are statistically independent each other as shown in
Table 7.

In the figure, we also show the fraction of the FG of stars (equivalent to our
CN-w) from the HST photometry by Milone et al. (2017), 0.175$\pm$0.009. Their
FOV of HST observation was $\sim$ 3′ $\times$ 3′, significantly smaller than
that of our study, $\sim$ 1° $\times$ 1°. Due to a strong radial gradient of
the populational ratio, it is most likely that Milone et al. (2017)
underestimated the fraction of the FG of stars. We derived the fraction of the
CN-w population within the radial distance of one half-light radius ($\sim$
3′.2), 0.159$\pm$0.013, and our result is consistent with that of Milone et
al. (2017), 0.175$\pm$0.009. Our exercise shows the importance of securing a
large FOV when deriving populational number ratios of GCs with MPs.

###  6.5 Radial Abundance Gradients

Since the late 70s, it has been known that 47 Tuc shows the radial abundance
and color gradients. For example, from their pioneering works, Chun & Freeman
(1979), and Norris & Freeman (1979) found that the CN-s RGB stars are more
centrally concentrated than the CN-w stars are. Using our $cn^{\prime}_{\rm
JWL}$ index, which is a measure of the CN band strengths at $\lambda$3883, we
examined radial variation of CN strengths. We calculated the moving average of
the radially adjacent 50 RGB stars, and we show our results in Figure 14. The
figure shows that the average $cn^{\prime}_{\rm JWL}$ value is significantly
larger, i.e., more CN-rich, in the central part of the cluster, and it
decreases with radial distance up to about 10′. This is consistent with the
fact that the CN-s stars are more centrally concentrated as we already
discussed earlier. When we use all RGB stars, which are photometrically
selected and contain more complete samples in the central part of the cluster,
the slope of the abundance gradient becomes steeper due to inclusion of more
CN-s RGB stars in the central part of the cluster. Beyond the radial distance
of 10′, the $cn^{\prime}_{\rm JWL}$ value maintains a constant level due to
the domination of the CN-w population in the outer regime of the cluster.

We also investigated the radial [Fe/H]hk variation of the cluster. As shown in
Figure 14, the mean [Fe/H]hk value is smaller in the central part, and then it
increases with the radial distance up to about 10′ and maintains a constant
level, consistent with the fact that the metal-poor stars are significantly
more centrally concentrated than the metal-rich stars, reminiscent of M3 (Lee
& Sneden, 2021). This is also supported by the $(B-V)$ color gradient of 47
Tuc RGB stars as we showed in Figure 8.

Figure 14:  (a) The moving average of the adjacent 50 points for the
$\parallel$$cn^{\prime}_{\rm JWL}$ index of the proper-motion member RGB stars
plotted as functions of radial distance. The vertical thin gray error bars
denote the standard error of the mean, and the vertical red dashed lines
denote the core and the half-light radii of 47 Tuc. (b) Same as the top panel,
but for [Fe/H]. (c) Same as (a) but for using all RGB stars. (d) Same as (b)
but for using all RGB stars.

##  7 Red Horizontal Branch

Since the GC RHB stars are not warm enough to completely suppress the
formation of diatomic molecules, such as NH, CN, and CH, in their surface, our
color indices can be useful to perform populational tagging for RHB stars (Lee
& Sneden, 2021, and references therein). In their pioneering work, Norris &
Freeman (1982) studied RHB in 47 Tuc, finding the CN-s RHB stars are enhanced
in nitrogen by $\Delta$[N/Fe] $\approx$ 0.9 dex and depleted in carbon by
$\Delta$[C/Fe] $\approx$ 0.3 dex with respect to the CN-w RHB stars. They also
argued the $V$ magnitude difference of 0.04 mag between the two RHB groups,
and they attributed it to difference in helium abundance.

Here, we investigate the RHB population in 47 Tuc using our color indices,
finding that differences not only in helium abundance but also in metallicity
are necessary to explain the observed RHB CMDs.

Figure 15:  (a) A plot of [O/Fe] versus [Na/Fe] of the RHB stars in 47 Tuc (Gratton et al., 2013). The blue and red dots denote RHB stars with [Na/O] $\leq$ 0.2 (i.e., equivalent to the CN-w), and $>$ 0.2 dex (i.e., the CN-s), respectively. (b) A plot of $(b-y)$ versus $cn^{\prime}_{\rm JWL}$ of the RHB stars in 47 Tuc along with RHB stars with [O/Fe] and [Na/Fe] measurements.  Figure 16:  (a) A plot of $hk_{\rm JWL}$ versus $cn^{\prime}_{\rm JWL}$ of the RHB stars in 47 Tuc, showing clustered distributions of the RHB stars. The gray solid line denotes the boundary between the CN-w(RHB) and CN-s(RHB) populations. The navy, dark-green, and orange plus signs are for the CN-w(RHB), CN-sMR(RHB), and CN-sMP(RHB), respectively. (b) The RHB distribution on the rotated plane, where the boundary between the two main bodies of the RHB stars in panel (a) is parallel to the vertical axis. (c) The histogram of the RHB stars along the $x$-axis, showing a distinctive bimodal distribution. We also show the results returned from our EM estimator. (d) The histogram of the CN-s(RHB) along the $y$-axis, showing an asymmetric distribution. We show the results from our EM estimator  Figure 17:  (a) A $(b-y)$ versus $V$ CMD of the CN-sMP RHB stars in 47 Tuc. The vertical and horizontal thin lines are for the mean $(b-y)$ and $V$ of each population: orange (CN-sMP), dark-green (CN-sMR), and navy (CN-w). The thick magenta solid line represents the BaSTI HB isochrone for ([Fe/H], $Y$, age) = ($-$0.9, 0.257, 12.0 Gyr), while the green and yellow solid lines are for ($-$0.8, 0.275, 12.0 Gyr) and ($-$0.7, 0.300, 12.0 Gyr). We used the distance modulus of $(m-M)_{0}$ = 13.21 mag and the interstellar reddening of $E(B-V)$ = 0.03 mag (Brogaard et al., 2017). (b) Same as (a) but for the CN-sMR RHB stars. The thick cyan solid line represents the BaSTI HB isochrone for ($-$0.7, 0.275, 12.0 Gyr). (c) Same as (c) but for the CN-w RHB stars. The thick gray solid line represents the BaSTI HB isochrone for ($-$0.7, 0.257, 12.0 Gyr). (d)–(f) The $(b-y)$ distributions of each population: orange (CN-sMP), dark-green (CN-sMR), and navy (CN-w). The dashed vertical lines denote the peak $(b-y)$ values. (g) The $V$ distributions of each population. The dashed horizontal lines denote the peak $V$ magnitudes. (h) Synthetic HB models for the CN-sMP RHB stars using the BaSTI model isochrone for ([Fe/H], $Y$, age) = ($-$0.9, 0.257, 12.0 Gyr). (i) Same as (h) but for CN-sMR RHB stars using the BaSTI model isochrone for ($-$0.7, 0.275, 12.0 Gyr). (j) Same as (h) but for CN-w RHB stars using the BaSTI model isochrone for ($-$0.7, 0.257, 12.0 Gyr). (k)–(m) Same as (d)–(f) but for the mean distributions of 1000 synthetic model simulation runs. (n) Same as (g) but for the mean distributions of 1000 synthetic model simulation runs.  Table 8: Mean and peak RHB $V$ and $(b-y)$ from Our Observations and HB Model Simulations. | CN-w | CN-s | CN-sMP | CN-sMR
---|---|---|---|---
Mean $V$11From our observations. | 14.052$\pm$0.052$\pm$0.003 | 14.017$\pm$0.050$\pm$0.002 | 14.019$\pm$0.049$\pm$0.002 | 14.002$\pm$0.042$\pm$0.005
Peak $V$11From our observations. | 14.085$\pm$0.010 | 14.039$\pm$0.010 | 14.042$\pm$0.010 | 14.021$\pm$0.010
Peak $V$22From synthetic HB model simulations shown in Figure 17. | 14.077$\pm$0.010 | 14.026$\pm$0.010 | 14.027$\pm$0.010 | 14.015$\pm$0.010
Peak $V$33From synthetic HB model simulation using the model isochrone for ([Fe/H], $Y$, age) = ($-$0.8, 0.275, 12.0 Gyr): case (2) in § 6.3. |  |  | 13.991$\pm$0.010 |
Peak $V$44From synthetic HB model simulation using the model isochrone for ([Fe/H], $Y$, age) = ($-$0.7, 0.300, 12.0 Gyr): case (3) in § 6.3. |  |  | 13.965$\pm$0.010 |
Mean $(b-y)$11From our observations. | 0.538$\pm$0.015$\pm$0.001 | 0.523$\pm$0.018$\pm$0.001 | 0.526$\pm$0.015$\pm$0.001 | 0.493$\pm$0.014$\pm$0.001
Peak $(b-y)$11From our observations. | 0.538$\pm$0.010 | 0.529$\pm$0.010 | 0.529$\pm$0.010 | 0.500$\pm$0.010
Peak $(b-y)$22From synthetic HB model simulations shown in Figure 17. | 0.535$\pm$0.010 | 0.520$\pm$0.010 | 0.520$\pm$0.010 | 0.490$\pm$0.010
Peak $(b-y)$33From synthetic HB model simulation using the model isochrone for ([Fe/H], $Y$, age) = ($-$0.8, 0.275, 12.0 Gyr): case (2) in § 6.3. |  |  | 0.494$\pm$0.010 |
Peak $(b-y)$44From synthetic HB model simulation using the model isochrone for ([Fe/H], $Y$, age) = ($-$0.7, 0.300, 12.0 Gyr): case (3) in § 6.3. |  |  | 0.500$\pm$0.010 |

###  7.1 Populational Tagging

In Figure 15(a), we show a plot of the [O/Fe] versus [Na/Fe] of the RHB stars
in 47 Tuc (Gratton et al., 2013). In the figure, the blue- and red-filled
circles denote the RHB stars with [Na/O] $\leq$ 0.2 dex (the Na-normal and
equivalent to the CN-w) and $>$ 0.2 dex (the Na-enhanced and equivalent to the
CN-s), respectively. In panel (b), we show a plot of $hk_{\rm JWL}$ versus
$cn^{\prime}_{\rm JWL}$ for the RHB stars in 47 Tuc. At the given $hk_{\rm
JWL}$ values, our $cn^{\prime}_{\rm JWL}$ index is well correlated with the
sodium abundance, due to the existence of a correlation between the CN and the
sodium abundances, confirming our previous result that our $cn_{\rm JWL}$ and
$cn^{\prime}_{\rm JWL}$ indices can be a powerful tool to investigate the MP
of the RHB stars. In the figure, the $cn^{\prime}_{\rm JWL}$ gradient against
the $hk_{\rm JWL}$ is partially due to the temperature effect along the RHB,
in a sense that as the effective temperature of RHB stars increases the
suppression of the CN band at $\lambda$3883 becomes greater resulting in small
$cn^{\prime}_{\rm JWL}$ values. When the effective temperature increases even
higher, the influence of the Hζ and Hη at 3889.05 and 3835.38Å becomes
important, resulting in larger $cn^{\prime}_{\rm JWL}$ values with
temperatures as can be seen in the M3 RHB stars (e.g., see Lee & Sneden,
2021).

We performed the populational tagging of the RHB stars in 47 Tuc using the
similar method that we applied for the RGB stars. In Figure 16(a), we show a
plot of $hk_{\rm JWL}$ versus $cn^{\prime}_{\rm JWL}$ of the RHB stars in 47
Tuc. We set a boundary between the two groups of the RHB stars and rotate the
$hk_{\rm JWL}$ versus $cn^{\prime}_{\rm JWL}$ plane so that the boundary line
becomes parallel to the vertical axis as shown in Figure 16(b). Then we
applied the EM analysis with a Gaussian mixture models along the $x$-axis,
which is perpendicular to the boundary line. Through this process, we obtained
the populational number ratio of $n$(CN-w):$n$(CN-s) = 30:70 ($\pm$2),
consistent with that of RGB. Note that our result does not agree with those of
Milone et al. (2012) or Dondoglio et al. (2021), who employed the HST
photometry and obtained $n$(CN-w):$n$(CN-s) $\sim$ 20:80. Again, their results
are based on the central part of the cluster owing to the small FOV of the HST
observations. Therefore, it is natural to have a larger fraction ($\sim$80%)
of the CN-s population due to a strong radial populational gradient of 47 Tuc.

We also examined the $y$ distributions of each RHB population. As shown in
Figure 16(d), the $y$ range of the CN-s(RHB) appears to be very large compared
to that of the CN-w(RHB). Moreover, the $y$ distribution of the CN-s(RHB)
exhibits an asymmetric distribution, suggesting that the CN-s(RHB) may contain
multiple subpopulations as can be seen in the RGB stars. Assuming a bimodal
distribution of the CN-s(RHB) population, we applied the EM analysis with a
two-component Gaussian mixture model along the $y$-axis for the CN-s(RHB)
stars. We obtained the populational number ratio of $n$[CN-sMP(RHB)]:$n$[CN-
sMR(RHB)] = 10:90 ($\pm$2), where the CN-sMP(RHB) and CN-sMR(RHB) refer the
CN-s RHB stars with high and low $y$ values, respectively. We note that the
$y$-axis is roughly the inverse of the $hk_{\rm JWL}$ index. Therefore, the
CN-sMP(RHB) stars are those with smaller $hk_{\rm JWL}$, the photometric
measure of the metallicity at a given effective temperature in the RHB
temperature range. Also it will be shown below that the CN-sMP(RHB) population
can be explained best with the lower metallicity by $\Delta$[Fe/H] $\sim$
$-$0.2 dex as can be seen in the RGB populations.

We show $(b-y)$ CMDs of the 47 Tuc RHB stars in Figure 17 along with the BaSTI
models (Pietrinferni et al., 2021), by using the distance modulus of
$(m-M)_{0}$ = 13.21 mag and the interstellar reddening of $E(B-V)$ = 0.03 mag
(Brogaard et al., 2017). The HB isochrones with constant mass loss for
([Fe/H], Y, age) = ($-$0.7, 0.257, 12.0 Gyr), ($-$0.7, 0.275, 12.0 Gyr), and
($-$0.9, 0.257, 12.0 Gyr) can nicely explain the color and magnitude
distributions of the CN-w(RHB), CN-sMR(RHB) and CN-sMP(RHB), respectively. We
note that the CN-sMP(RHB) occupies about 7 ($\pm$2)% of the total RHB stars,
which is roughly consistent with that of the CN-sMP RGB stars, $\sim$ 10
($\pm$2)%, within uncertainties. On the other hand, the fraction of the CN-wMP
RGB population takes about 3% of the total RGB stars, whose RHB counterpart we
do not identify as a separate population in our analysis. The distribution of
individual RHB populations in our ($b-y$) CMD is consistent with the result of
Gratton et al. (2013) that the sodium abundance increases as the $(B-V)$ color
decreases, due to the correlation between the CN and sodium abundances in the
cluster (e.g., Cottrell & Da Costa, 1981).

In Figure 17(a), we also show isochrones for ([Fe/H], Y, age) = ($-$0.8,
0.275, 12.0 Gyr) and ($-$0.7, 0.300, 12.0 Gyr) to test our ideas for the cases
(2) and (3) discussed in §6.3. As shown, these two RHB isochrones fail to
explain the CN-sMP(RHB) distribution with satisfaction, and one can rule out
these two assumptions. In Figures 17(b)–(c), we show isochrones for ([Fe/H],
Y, age) = ($-$0.7, 0.275, 12.0 Gyr) and ($-$0.7, 0.257, 12.0 Gyr), and they
appear to explain the CN-sMR and CN-w RHB populations well.

We calculate the mean and peak $V$ magnitudes of individual RHB populations,
and we show our results in Table 8. Note that the peak magnitudes are
determined from the greatest values in their smoothed distribution function
with a Gaussian density estimation. The mean color and magnitude of individual
populations can be affected by the highly evolved RHB stars. It is believed
that the peak color and magnitude determined by their distributions could be
more practical to be compared with the model predictions as we will show
below. The CN-s(RHB) stars are about 0.3 – 0.4 mag brighter than the CN-w(RHB)
stars as we already showed in Figure 17. Our results are in excellent
agreement with those by Norris & Freeman (1982), and Briley (1997), $\sim$
0.04 – 0.05 mag.

The mean $(b-y)$ color of the CN-s(RHB) stars is $\sim$ 0.01 – 0.02 mag bluer
than that of the CN-w(RHB) stars, consistent with the idea that the bulk of
the CN-s RHB stars are more helium-enhanced than the CN-w RHB stars are. Since
the helium-enhanced RHB population will have smaller total mass and smaller
ratio of the envelope mass to the total mass, they will be bluer than those
with normal helium abundances (e.g., see Cassisi & Salaris, 2021). We also
note that the range of the $(b-y)$ distribution of the CN-s(RHB) stars is
larger than that of the CN-w(RHB) stars, indicating that the CN-s(RHB) stars
experience a larger mass-loss dispersion during their RGB evolution, which
seems to be odd. It could be also possible that the CN-s(RHB) population
contains multiple subpopulations, as we discussed for the RGB stars, the CN-
sMP and CN-sMR.

To explore our idea of the existence of the metal-poor RHB component, we
constructed the synthetic evolutionary population models using the ([Fe/H], Y,
age) = ($-$0.7, 0.257, 12.0 Gyr), ($-$0.7, 0.275, 12.0 Gyr), and ($-$0.9,
0.257, 12.0 Gyr) model isochrones for the CN-w, CN-sMR and CN-sMP populations.
We performed 1000 trials to calculate the mean $(b-y)$ color and $V$
distributions for individual populations along with the peak $(b-y)$ colors
and $V$ magnitudes. We show our results in Table 8. Although crude, our
synthetic models can reproduce the observed peak $(b-y)$ colors and $V$
magnitudes. In the table, we also show the results from the ($-$0.7, 0.300,
12.0 Gyr) and ($-$0.8, 0.275, 12.0 Gyr) models for alternative explanations of
the CN-sMP population. Our simulations with such models predict brighter peak
$V$ magnitudes and slightly bluer $(b-y)$ colors than our observations. In
Figure 17(h–j), we show the synthetic $(b-y)$ versus $V$ CMDs of individual
population using one specific trial. We note that the distributions of
synthetic models appear to be good for the CN-w and CN-sMP populations.
However, the wedge shape of the observed CN-sMR population in the faint
magnitude regime cannot be reproduced with a simple stellar population that we
used in our study. In the future, more sophisticated HB model simulation would
be very desirable (e.g., see Salaris et al., 2016).

Figure 18:  (a) CRDs of individual RHB populations (solid lines) along with
the whole RGB populations (dashed lines). The blue, red and pink solid lines
are for the CN-w(RHB), CN-sMR(RHB) and CN-sMP(RHB) while the blue and red
dashed lines are for the CN-w(RGB) and CN-s(RGB). The gray error bars show the
fractions of the CN-w RHB population with a binning radial distance of the
half-light radius, while the pink error bars show those of the CN-w RGB
population. The vertical gray dashed lines denote the core and half-light
radii of 47 Tuc. The green-shaded box indicates the fraction of the first
generation by Milone et al. (2017) measured in the central part of the
cluster. (b) Same as (a) but for the bright RGB stars with $V\leq$ 15.0 mag.

###  7.2 Cumulative Radial Distributions

In Figure 18, we show the CRDs of individual RHB populations. As can be seen
in the RGB, the CN-s RHB population is more centrally concentrated than the
CN-w RHB, which is not a surprise since the CN-w and CN-s RHB stars are
progenies of the CN-w and CN-s RGB stars, respectively. We note that the more
centrally concentrated nature of the CN-s RHB stars was also found by Milone
et al. (2012).

To quantitatively examine similarity of individual CRDs, we performed the K-S
tests, and we show our results is Table 9. When the RGB stars with the whole
magnitude range of our interest, $-$1.5 $\leq$ $V-V_{\rm HB}$ $\leq$ 2.5 mag,
were used, the $p$-values related to the CN-w(RHB) population are
significantly small, and it can be interpreted that none of the CRDs of the
RGB populations were drawn from the same distribution of the CN-w(RHB)
population. On the other hand, the CN-s, CN-sMR and CN-sMP RHB populations
show large $p$-values with the CN-wMP.

It is thought that the biased sampling in our work may be responsible for this
discrepancy. RGB stars were selected from 2.5 mag fainter than the RHB stars
are. At the central part of the cluster, where the CN-s population dominates,
the degree of incomplete detection of the faint stars would be great and must
have affected our individual CRDs. When we restricted the bright RGB stars
with $V\geq$ 15.0 mag, the agreement in individual CRDs between the RHB and
RGB populations becomes improved, suggesting that each RHB population is the
progeny of the corresponding RGB population.

Table 9: $p$ Values Returned from the K-S Tests for the CRDs of individual RHB and RGB populations | CN-w | CN-wMP | CN-wMR | CN-s | CN-sMP | CN-sMR
---|---|---|---|---|---|---
All
CN-w(RHB) | 0.026 | 3.07$\times 10^{-9}$ | 0.001 |  $<$ 1.00$\times 10^{-15}$ |  $<$ 1.00$\times 10^{-15}$ | 9.03$\times 10^{-9}$
CN-s(RHB) |  $<$ 1.00$\times 10^{-15}$ | 0.425 |  $<$ 1.00$\times 10^{-15}$ | 2.34$\times 10^{-4}$ |  $<$ 1.00$\times 10^{-15}$ |  $<$ 1.46$\times 10^{-9}$
CN-sMP(RHB) | 1.94$\times 10^{-11}$ | 0.457 | 3.95$\times 10^{-13}$ | 0.007 | 4.76$\times 10^{-4}$ | 5.91$\times 10^{-4}$
CN-sMR(RHB) |  $<$ 1.00$\times 10^{-15}$ | 0.383 |  $<$ 1.00$\times 10^{-15}$ | 0.003 |  $<$ 1.00$\times 10^{-15}$ | 1.04$\times 10^{-7}$
Bright RGB ($V\leq$ 15.0 mag)
CN-w(RHB) | 0.455 | 1.73$\times 10^{-3}$ | 0.253 |  $<$ 1.00$\times 10^{-15}$ |  $<$ 1.00$\times 10^{-15}$ | 1.77$\times 10^{-12}$
CN-s(RHB) |  $<$ 1.00$\times 10^{-15}$ | 0.142 |  $<$ 1.00$\times 10^{-15}$ | 0.074 |  $<$ 1.00$\times 10^{-15}$ | 0.219
CN-sMP(RHB) | 9.52$\times 10^{-8}$ | 0.719 | 2.31$\times 10^{-8}$ | 0.668 | 1.47$\times 10^{-7}$ | 0.129
CN-sMR(RHB) |  $<$ 1.00$\times 10^{-15}$ | 0.118 |  $<$ 1.00$\times 10^{-15}$ | 0.041 |  $<$ 1.00$\times 10^{-15}$ | 0.319

##  8 SUMMARY

In this paper, we presented new large FOV ($\sim$1°$\times$1°) Ca-CN
photometry of 47 Tuc (NGC 104). We found that the CN-s population is the major
component of the cluster and it is more centrally concentrated than the CN-w
population, similar to other Galactic GCs. We obtained the populational number
ratios of the RGB and RHB: $n$(CN-w):$n$(CN-s) = 30:70 ($\pm$1-2). Our results
are in good agreement with those of previous results by others (e.g. Nataf et
al., 2011; Milone et al., 2012).

Following the similar method that we developed in our previous studies (Lee,
2021; Lee & Sneden, 2021), we derived the photometric metallicity of
individual RGB stars from our $hk_{\rm JWL}$ index, finding that both the CN-w
and CN-s populations show asymmetric metallicity distributions toward metal-
poor regime, and they are well described by bimodal metallicity distributions.
In both the CN-w and CN-s populations, the metal-poor components are
$\Delta$[Fe/H] $\sim$ 0.20 dex more metal-poor than the metal-rich components,
where the metal-rich populations are main components of the clusters. The
metal-poor components constitute $\sim$3% (CN-wMP) and $\sim$10% (CN-sMP) of
the total RGB population. The metal-poor components (i.e., CN-wMP and CN-sMP)
are significantly more centrally concentrated than the metal-rich components
(i.e., CN-wMR and CN-sMR). In addition, the CN-sMP is more centrally
concentrated than the CN-wMP. Therefore, the metal-poor populations themselves
mimic the general trend of normal GCs that the CN-s population is the more
centrally concentrated than the CN-w.

We investigated the RGBB $V$ magnitudes, finding that $V$ = 14.399 ($\pm$
0.040), 14.559 ($\pm$ 0.025), 14.385 ($\pm$ 0.035), 14.492 ($\pm$0.020) mag
for the CN-wMP, CN-wMR, CN-sMP, and CN-sMR, respectively, which are in
excellent agreement with those analyzed using the photometric data of Stetson
et al. (2019). We performed Monte Carlo simulations by means of the
evolutionary population synthesis models, showing that the observed RGBB $V$
magnitudes are consistent with the metallicity difference of $\Delta$[Fe/H]
$\sim$ 0.15 – 0.20 dex between the metal-poor and metal-rich components and
the helium abundance difference of $\Delta Y$ $\sim$ 0.02 – 0.03 between the
CN-wMR and CN-sMR.

Using our photometric indices, we decomposed three RHB populations in 47 Tuc.
They are thought to be progenies of the CN-wMR, CN-sMP, and CN-sMR based on
the CN strengths and comparisons with the RHB model isochrones. Good
agreements in the CRDs of RHB stars with those of the bright RGB stars may
support our conclusion. It is believed that, due to its very small fraction,
the RHB counterpart of the CN-wMP cannot be clearly seen.

Recently, McKenzie & Bekki (2021) conducted numerical simulations for the
formation of a 47 Tuc-like GC in a dwarf galaxy environment. Their result
indicated that newly formed GCs should be able to retain some low-mass field
stars from their parent galaxy. The metal-poor components (i.e., CN-wMP and
CN-sMP) that we found in this work could be field stars surrounding the CN-wMR
and CN-sMR. However, the carbon and nitrogen abundance difference between the
CN-wMP and CN-sMP cannot be easily explained in the context of the chemical
evolution of the disk stars in dwarf galaxies, where the C–N anticorrelation
does not appear to exist. Instead, the metal-poor components could be
understood by an independent GC system that merged with the more massive
metal-rich component as we suggested for M22 and M3 (e.g., Lee, 2015, 2020;
Lee & Sneden, 2021). The fraction of the metal-poor stars in our study is
about 13%, which can be translated into the total mass of $\sim$ 1.0$\times
10^{4}$ $M_{\rm\odot}$ (Baumgardt & Hilker, 2018). In such a small mass
system, it would be natural to expect no helium enhancement between the CN-wMP
and CN-sMP populations (e.g., see Milone et al., 2018b), which is exactly what
we observed in 47 Tuc.

J.-W.L. thanks Professor Y.-C. Kim at Yonsei University for the discussion on
the $Y^{2}$ isochrones and Drs. A.A.R. Valcarce and M. Catelan at Pontificia
Universidad Católica de Chile for reviving the PGPUC online service after the
tragic fire accident that heavily affected the Institute for Astrophysics. He
also thanks an anonymous referee for a careful review of the manuscript and
many priceless suggestions. J.-W.L. acknowledges 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 2019.

## References

  * Anthony-Twarog et al. (1991) Anthony-Twarog, B. J., Laird, J. N., Payne, D., & Twarog, B. A. 1991, AJ, 101, 1902
  * Anderson et al. (2009) Anderson, J., Piotto, G., King, I. R., Bedin, L. R., Guhathakurta, P. 2009, ApJ, 697, L58
  * Bastian et al. (2015) Bastian, N., Cabrera-Ziri, I., & Salaris, M. 2015, MNRAS, 449, 3333
  * Bastian & Lardo (2018) Bastian, N., & Lardo, C. 2018, ARA&A, 56, 3
  * Bekki (2019) Bekki, K. 2019, A&A, 622, 53
  * Bragaglia et al. (2010) Bragaglia, A., Carretta, E., Gratton, R., et al. 2010 A&A, 519, 60
  * Baumgardt & Hilker (2018) Baumgardt, H., & Hilker, M. 2018, MNRAS, 478, 1520
  * Briley (1997) Briley, M. M. 1997, AJ, 114, 1051
  * Briley et al. (2004) Briley, M. M., Harbeck, D., Smith, G. H., Grebel, E. K. 2004, AJ, 127, 1588
  * Briley et al. (1991) Briley, M. M., Hesser, J. E., & Bell, R. A. 1991, ApJ, 373, 482
  * Brogaard et al. (2017) Brogaard, K., VandenBerg, D. A., Bedin, L. R., et al. 2017, MNRAS, 468, 645
  * Calura et al. (2019) Calura, F., D’Ercole, A., Vesperini, E., Vanzella, E., & Sollima, A. 2019, MNRAS, 489, 3269
  * Cannon et al. (1998) Cannon, R. D., Croke, B. F. W., Bell, R. A. et al. 1998, MNRAS, 298, 601
  * Carney (1996) Carney, B. W. 1996, PASP, 108, 900
  * Carretta et al. (2009) Carretta E., Bragaglia, A., Gratton, R.G., et al. 2009, A&A, 505, 117
  * Carretta et al. (2004) Carretta E., Gratton, R.G., Bragaglia, A., et al. 2004, A&A, 416, 925
  * Cassisi & Salaris (1997) Cassisi, S., & Salaris, M. 1997, MNRAS, 285, 593
  * Cassisi & Salaris (2013) Cassisi, S., & Salaris, M. 2013, Old Stellar Populations: how to study the fossil record of galaxy formation (Berlin:Wiley-VCH)
  * Cassisi & Salaris (2021) Cassisi, S., & Salaris, M. 2020, A&AS, 28, 5
  * Cassisi et al. (2008) Cassisi, S., & Salaris, M., Pietriferni, A. et al. 2008, ApJ, 672, L115
  * Chun & Freeman (1979) Chun, M. S., & Freeman, K. C. 1979, ApJ, 227, 93
  * Cottrell & Da Costa (1981) Cottrell, P. L., & Da Costa, G. S. 1981, ApJ, 245, L79
  * Codero et al. (2014) Codero, M. J., Pilachowski, C. A., Johnson, C. I., et al. 2014, ApJ, 780, 94
  * D’ercole et al. (2008) D’Ercole, A., Vesperini, E., D’Antona, F., McMillan, S. L. W., & Recchi, S. 2008, MNRAS, 391, 825
  * Di Criscienzo et al. (2010) Di Criscienzo, M., Veutura, P., D’Antona, F. et al. 2010, MNRAS, 408, 999
  * Dondoglio et al. (2021) Dondoglio, E., Milone, A. P., Lagioia, E. P. et al. 2121, ApJ, 906, 21
  * Dotter et al. (2008) Dotter, A., Chaboyer, B., Jevremović, D., Kostov, V., Baron, E., Ferguson, J. W. 2008, ApJS, 178, 89
  * Fare et al. (2018) Fare, A., Webb, J. J., & Sills, A. 2018, MNRAS, 481, 3027
  * Gaia Collaboration (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1
  * Girardi et al. (2002) Girardi, L., Bertelli, G., Bressan, A. et al. 2002, A&A, 391, 195
  * Gratton et al. (2019) Gratton, G. R., Bragaglia, A., Carretta, E. et al. 2019, A&AS, 27, 8
  * Gratton et al. (2013) Gratton, G. R., Lucatello, S., Sollima A., et al. 2013, A&A, 549, A41
  * Harbeck (2003) Harbeck, D., Smith, G. H, Grebel, E. K. 2003, AJ, 125, 197
  * Hesser et al. (1987) Hesser, J. E., Harris, W. E., Vandenberg, D. A., et al. 1987, PASP, 99, 739
  * Hesser (1978) Hesser, J. E. 1978, ApJ, 223, L117
  * Kurucz (2011) Kurucz, R. L. 2011, Can. J. Phys., 89, 417
  * Lacchin et al. (2021) Lacchin, E., Calura, F., & Vesperini, E. 2021, MNRAS, 506, 5951
  * Lagioia et al. (2018) Lagioia, E. P., Milone, A. P., Marino, A. F., et al. 2018, MNRAS, 475, 4088
  * Lardo et al. (2022) Lardo, C., Salaris, M., Cassisi, S., & Bastian, N. 2022, A&A, 662, 117
  * Lee (2010) Lee, J.-W. 2010, MNRAS, 405, L36
  * Lee (2015) Lee, J.-W. 2015, ApJS, 219, 7
  * Lee (2016) Lee, J.-W. 2016, ApJS, 226, 16
  * Lee (2017) Lee, J.-W. 2017, ApJ, 844, 77
  * Lee (2018) Lee, J.-W. 2018, ApJS, 238, 24
  * Lee (2019a) Lee, J.-W. 2019a, ApJ, 872, 41
  * Lee (2019b) Lee, J.-W. 2019b, ApJ, 875, L27
  * Lee (2019c) Lee, J.-W. 2019c, ApJ, 883, 166
  * Lee (2020) Lee, J.-W. 2020, ApJ, 888, L6
  * Lee (2021) Lee, J.-W. 2021, ApJ, 918, L24
  * Lee & Carney (1999) Lee, J.-W., & Carney, B. W. 1999, AJ, 117, 2868
  * Lee et al. (2009) Lee, J.-W., Kang, Y.-W., Lee, J., & Lee, Y.-W. 2009, Nature, 462, 480
  * Lee et al. (2014) Lee, J.-W., López-Morales, M., Hong, K. et al. 2014, ApJS, 210, 6
  * Lee & Pogge (2016) Lee, J.-W., & Pogge, R. 2016, JKAS, 49, 289
  * Lee & Sneden (2021) Lee, J.-W., & Sneden, C. 2021, ApJ, 909, 167
  * Legnardi et al. (2022) Legnardi, M. V., Milone, A. P., Armillotta, L., et al. 2022, MNRAS, 513, 735
  * Marino et al. (2016) Marino, A. F., Milone, A. P., Casagrande, L., et al. 2016, MNRAS, 459, 610
  * Marino et al. (2019) Marino, A. F., Milone, A. P., Renzini, A., et al. 2019, MNRAS, 487, 3815
  * McKenzie & Bekki (2021) McKenzie, M., & Bekki, K. 2021, MNRAS, 500, 4578
  * Milone et al. (2008) Milone A. P., Bedin, L.R., Piotto, G. et al. 2008, ApJ, 673, 241
  * Milone et al. (2018a) Milone A. P., Marino, A. F., Mastrobuno-Battisti, A., & Laagioia, E. P. 2018a, MNRAS, 479, 5005
  * Milone et al. (2018b) Milone A. P., Marino, A. F., Renzini, A. et al. 2018b, MNRAS, 481, 5098
  * Milone et al. (2012) Milone A. P., Piotto, G., Bedin, L.R. et al. 2012, ApJ, 744, 58
  * Milone et al. (2017) Milone A. P., Piotto, G., Renzini, A. et al. 2017, MNRAS, 464, 3636
  * Nataf et al. (2011) Nataf, D. M., Gould, A., Pinsonneault, M. H., & Stetson, P. B. 2011, ApJ, 736, 94
  * Norris & Freeman (1979) Norris, J., & Freeman, K. C. 1979, ApJ, 230, L179
  * Norris & Freeman (1982) Norris, J., & Freeman, K. C. 1982, ApJ, 254, 143
  * Pietrinferni et al. (2021) Pietrinferni, A., Hidalgo, S., Cassisi, S., Salaris, M., et al. 2021, ApJ, 908, 102
  * Renzini et al. (2015) Renzini, A., D’Antona, F., Cassisi, S. et al. 2015, MNRAS, 454, 4197
  * Renzini & Fusi Pecci (1988) Renzini, A., & Fusi Pecci, F. 1988, ARA&A, 26, 199
  * Salaris et al. (2016) Sarlaris, M., Cassisi, S., & Pietrinferni, A. 2016, A&A, 590, 64
  * Sbordone et al. (2011) Sbordone, L., Salaris, M., Weiss, A., & Cassisi, S. 2011, A&A, 534, A9
  * Smith (1979) Smith, H. A. 1979, AJ, 84, 176
  * Stetson (1987) Stetson P. B. 1987, PASP, 99, 191
  * Stetson (1994) Stetson P. B. 1994, PASP, 106, 250
  * Stetson et al. (2019) Stetson P. B., Pancino, E., Zocchi, A. et al. 2019, MNRAS, 485, 3042
  * Vesperini et al. (2021) Vesperini, E., Hong, J., Giersz, M., & Hypki, A. 2021, MNRAS, 502, 4290
  * Valcarce et al. (2012)
  * Wang et al. (2017) Wang, Y., Primas, F., Charbonnel, C. et al. 2017, A&A, 607, 135
  * Yi et al. (2008) Yi, S., Kim, Y.-C., Demarque, P., et al. 2008, in Proc. IAU Symp. 252, The Art of Modeling Stars in the 21st Century, ed. L. Deng & K. L. Chan (Cambridge: Cambridge Univ. Press), 413