# Determination of Sodium Abundance Ratio from Low-Resolution Stellar Spectra
and Its Applications

Jae-Rim Koo11affiliationmark:  22affiliationmark:  , Young Sun
Lee11affiliationmark:  , Hye-Jin Park33affiliationmark:  , Young Kwang
Kim11affiliationmark:  , and Timothy C. Beers44affiliationmark:
1Department of Astronomy and Space Science, Chungnam National University,
Daejeon 34134, Republic of Korea<EMAIL_ADDRESS>
2Department of Astronomy and Atmospheric Sciences, Kyungpook National
University, Daegu 41566, Republic of Korea
3Department of Physics and Astronomy, Sejong University, Seoul 05006, Republic
of Korea
4Department of Physics and JINA Center for the Evolution of the Elements,
University of Notre Dame, IN 46556, USA 1 2 3 4

###### Abstract

We present a method to determine sodium-abundance ratios ([Na/Fe]) using the
Na I D doublet lines in low-resolution ($R\sim$ 2000) stellar spectra. As
stellar Na I D lines are blended with those produced by the interstellar
medium (ISM), we developed a technique for removing the interstellar Na I D
lines using the relationship between extinction, which is proportional to
$E(B-V)$, and the equivalent width (EW) of the interstellar Na I D absorption
lines. When measuring [Na/Fe], we also considered corrections for non-local
thermodynamic equilibrium (NLTE) effects. Comparisons with data from high-
resolution spectroscopic surveys suggest that the expected precision of
[Na/Fe] from low-resolution spectra is better than 0.3 dex for stars with
[Fe/H] $>$ –3.0. We also present a simple application employing the estimated
[Na/Fe] values for a large number of stellar spectra from the Sloan Digital
Sky Survey (SDSS). After classifying the SDSS stars into Na-normal, Na-high,
and Na-extreme, we explore their relation to stars in Galactic globular
clusters (GCs). We find that, while the Na-high SDSS stars exhibit a similar
metallicity distribution function (MDF) to that of the GCs, indicating that
the majority of such stars may have originated from GC debris, the MDF of the
Na-normal SDSS stars follows that of typical disk and halo stars. As there is
a high fraction of carbon-enhanced metal-poor stars among the Na-extreme
stars, they may have a non-GC origin, perhaps due to mass-transfer events from
evolved binary companions.

###### Subject headings:

methods: data analysis — technique: spectroscopy — galaxy: halo — stars:
abundances — stars: kinematics and dynamics

##  1\. Introduction

A non-zero fraction of field stars in the Milky Way (MW) are known to exhibit
large abundances of light elements such as N, Na, and Al (Carretta et al.,
2010; Fernández-Trincado et al., 2016; Martell et al., 2016; Schiavon et al.,
2017a; Pereira et al., 2019). Globular clusters (GCs) in the MW also display
significant abundance variations of the light elements, as well as strong
anti-correlations between the abundances of Na and O, N and C, and Al and Mg
(Carretta et al., 2009), which numerous authors have taken to be associated
with the existence of multiple populations of (first- and second-generation)
stars in GCs.

Because the second-generation stars in MW GCs exhibit relatively high
abundances of N, Na, and Al, the field stars enhanced with such elements are
often regarded as objects that have been accreted from tidally disrupted GCs
(Carretta et al., 2009; Carretta, 2016; Martell et al., 2016; Schiavon et al.,
2017a; Horta et al., 2021; Tang et al., 2021). From this perspective, it is
expected that the debris from partially or fully disrupted GCs contributed in
some degree to the build-up of the Galactic halo (Thomas et al., 2020; Wan et
al., 2020) and bulge (Lee et al., 2019; Lim et al., 2021), in addition to
stars accreted from major and minor merger events such as Gaia-Sausage (GS;
Belokurov et al., 2018), Gaia-Enceladus (GE; Helmi et al., 2018), Sequioa
(Myeong et al., 2019), Thamnos 1 & 2 (Koppelman et al., 2019), and the
currently disrupting Sagittarius dwarf galaxy (Ibata et al., 1994; Majewski et
al., 2003).

Several previous studies have been carried out to evaluate the significance of
the contribution of the GC-origin stars to the Galactic halo. For example,
Martell & Grebel (2010) searched for halo giants that originated from GCs by
identifying N-rich stars on the basis of CN-band strengths in low-resolution
stellar spectra from the Sloan Digital Sky Survey (SDSS; York et al. 2000), in
particular the stellar specific sub-survey Sloan Extension for Galactic
Understanding and Exploration (SEGUE; Yanny et al. 2009), and reported a
fraction of 2.5% of N-rich stars in the halo populations. Horta et al. (2021)
also attempted to estimate the contribution of the GCs to the halo using the
nitrogen abundances of giant stars from the Apache Point Observatory Galactic
Evolution Experiment (APOGEE; Majewski et al., 2017), and derived a similar
fraction, 2.7%.

In addition to N, Na has become a key element for recognizing the population
of stars that have escaped from MW GCs and to estimate the contribution of the
GC-origin stars in the local halo, as it is relatively straightforward to
measure its abundance in optical and near-IR spectra for dwarfs and giants.
The Galactic field halo stars usually have sodium-abundance ratios ([Na/Fe])
in the range $-0.5\lesssim{\rm[Na/Fe]}\lesssim+0.5$ (Suda et al., 2008), while
higher abundance ratios are observed among members of the Galactic GCs
(Carretta et al., 2010). Based on these results, several studies (Carretta et
al., 2010; Martell et al., 2011; Ramírez et al., 2012) attempted to estimate
the fraction of the Na-rich stars in the MW halo, reporting fractions from
2.8% to 17%. The clear interpretation is that there is a non-negligible
contribution of disrupted GCs to the assembly of the Galactic halo. However,
since most of these studies are based on small numbers of relatively bright
stars (with [Na/Fe] obtained from high-resolution spectroscopy), they are
limited to relatively local samples. In order to obtain a better understanding
of the connection of the GC debris to the assembly of the Galactic halo, it is
necessary to study a much large number of homogeneously analyzed stars
reaching farther into the Galactic halo.

Fortunately, spectroscopic surveys such as legacy SDSS, SEGUE, and the Large
Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST; Cui et al., 2012)
provide unprecedented numbers of stellar spectra, with resolution sufficiently
high to identify the Na I D resonance lines. However, in order to estimate
[Na/Fe] from a low-resolution stellar spectrum, one challenge is the presence
of sodium from the integrated interstellar medium (ISM) between the Earth and
the stars. The Na I D lines produced in the stellar atmosphere are blended
with those from Na in the ISM, and they are not easily distinguishable. As a
result, the Na I D lines in a low-resolution spectrum appear stronger than the
intrinsic lines, and [Na/Fe] is generally over-estimated; the problem becomes
even more severe at low Galactic latitudes. Note that throughout this paper,
we refer to the Na I D lines produced by Na in the ISM as interstellar sodium
(ISS) lines.

However, if one can remove the effects of ISS lines in a low-resolution
spectrum, it is possible to obtain reasonably well-determined [Na/Fe] for very
large numbers of stellar spectra from SDSS and LAMOST. In this paper, we
collectively refer to all of the stellar spectra from SDSS, SEGUE, Baryon
Oscillation Spectroscopic Survey (BOSS; Dawson et al., 2013), and extended
Baryon Oscillation Spectroscopic Survey (eBOSS; Blanton et al., 2017) as SDSS
stars. Based on these large samples, one can better understand the
distribution and properties of the Na-rich stars and their relationship to MW
GCs, and derive statistically well-estimated fractions of these stars in the
Galactic halo. This paper presents a method for removing the effect of ISS
from low-resolution SDSS stellar spectra, and determine estimates of their
[Na/Fe] abundance ratios.

This paper is organized as follows. Section 2 describes the method for the
removal of the ISS lines from the SDSS spectra and the determination of
[Na/Fe]. Validation of the estimated [Na/Fe] ratios, along with systematic and
random errors, and the impact of the signal-to-noise ratio (SNR) of a stellar
spectrum on the measured [Na/Fe], are presented in Section 3. In Section 4, we
present one application of sodium-abundance ratios to understand their linkage
with GC debris. A summary is given in Section 5.

##  2\. Methodology for Estimation of [Na/Fe]

###  2.1. Impacts of Interstellar Sodium Absorption

As stellar light travels through interstellar space, its intensity is
attenuated by absorption or scattering by the ISM. As one result, in
particular for certain chemical elements (e.g., Na and Ca), we expect ISM
lines to be present in a stellar spectrum. The strength of these lines is
correlated with the amount of dust or gas along the line-of-sight. The
absorption lines produced by the ISM are normally narrow and sharp due to the
small internal velocity dispersion of gas in the ISM. Thus, we can identify
and remove them from the spectrum of an astronomical object with relatively
small differences in the radial velocities between the object and the ISM gas
clouds, especially for high-resolution spectra.

However, at low resolution, such as for SDSS stellar spectra, the stellar
absorption lines are blended with the ISM absorption lines. In this case, they
are not easily distinguishable from the intrinsic stellar lines, even for a
relatively high-velocity star. When measuring the abundance of a given
chemical element, the overall effect of the unresolved ISM lines is to produce
an over-estimate of the stellar elemental abundance. Sodium is a classic
example of this difficulty in optical spectra.

The impact of the ISS lines on a stellar spectrum is illustrated in Figure 1.
The top-left panel of Figure 1 displays a high-resolution spectrum in the Na I
D doublet region of the extremely metal-poor star BD+44∘ 493, which was
observed with Gemini/GRACES (Chene et al., 2014). From inspection, one can
easily identify the strong ISS lines shifted to the red by about 3 Å from the
weaker stellar lines. The red spectrum in the bottom-left panel shows a
degraded version (to the SDSS resolving power of about $R\sim 2000$) of the
high-resolution spectrum; the stellar and the ISS lines are indistinguishable.
The top-right panel of the figure exhibits the high-resolution spectrum after
excising the ISS lines; the bottom-right panel shows the degraded spectrum
(red line). We clearly see the much weaker strength of the intrinsic lines
compared to the ISS lines. The blue spectrum in the panels is a synthetic one
generated with the stellar parameters ($T_{\rm{eff}}$ = 5430 K, log $g$ = 3.4,
and [Fe/H] = –3.8) and sodium-abundance ratio ([Na/Fe] = +0.3), reported by
Ito et al. (2013). Comparison of the bottom-left and bottom-right panels makes
it obvious that the sodium-abundance ratio determined from low-resolution
stellar spectra would result in a gross over-estimation without accounting for
the presence of the ISS lines.

Figure 1.— Example spectra of the Na I D doublet region before (left column)
and after (right column) removal of the ISS lines for the star BD$+$44∘ 493\.
The upper panels show a high-resolution ($R$ = 45,000) spectrum obtained with
Gemini/GRACES, whereas the red spectra in the bottom panels show its degraded
spectrum to $R$ = 2000. The blue line is the synthetic spectrum generated with
$T_{\rm{eff}}$ = 5430 K, log $g$ = 3.4, [Fe/H] = –3.8, and [Na/Fe] = 0.3,
derived from the high-resolution spectroscopic analysis of this star by Ito et
al. (2013). Figure 2.— An example of our ISS line-removal method applied to
the spectrum of an SDSS star (J185226.5$+$201240). The left panel shows the
spectrum before the removal of the ISS lines, while the right panel
illustrates the spectrum after their removal. The black line indicates the
observed spectrum; the red lines are the best-matching synthetic spectrum. The
stellar parameters from the SSPP and [Na/Fe] (derived from the best-matching
synthetic spectrum) are listed at the top of each panel. The much higher
sodium-abundance ratio (by 1.0 dex) for the spectrum without removal of the
the ISS lines is evident.

###  2.2. Removal of Interstellar Sodium Absorption

As demonstrated above, the removal of the Na I D lines produced by ISM is
straightforward for high-resolution spectra, since we can easily identify the
lines and excise them. In the case of low-resolution spectra, the situation is
more complicated because the ISS lines are unresolved. Therefore, we require a
method for eliminating the ISS lines before the estimation of [Na/Fe], and
have developed the following approach.

We first estimated the strength of the ISS lines in terms of the expected
equivalent width (EW) along the line-of-sight to a given star, and subtracted
it from the measured Na I D lines present in a stellar spectrum. In order to
calculate the EW of the ISS lines at a given Galactic coordinate, we adopted
from the literature the relations between the reddening, $E(B-V)$, and the
strength of the Na I D absorption lines by the ISM. Poznanski et al. (2012)
derived a correlation between the reddening estimated by Schlegel et al.
(1998) and the EW of the ISS lines, using high-resolution spectra of quasars
(QSOs) and low-resolution spectra of QSOs and galaxies obtained by SDSS. Murga
et al. (2015) also presented a relation between the reddening and the strength
of the interstellar absorption of the Na I D and Ca II H & K lines, which were
derived using extra-galactic sources from SDSS. They provide a best-fit
relation for the range of $0\leq E(B-V)<0.08$. We employed the results from
both studies. For stars with $0\leq E(B-V)<0.08$ we adopted the relation of
Murga et al. (2015):

$\displaystyle\mathrm{EW}(E(B-V))_{\rm{D2}}=\left(\frac{E(B-V)}{0.39}\right)^{0.63}\,\mathrm{\AA}$
$\displaystyle\mathrm{EW}(E(B-V))_{\rm{D1}}=\left(\frac{E(B-V)}{0.26}\right)^{1.06}\,\mathrm{\AA}$

For stars with $E(B-V)\geq 0.08$ we adopt the relation of Poznanski et al.
(2012):

$\displaystyle\mathrm{EW}(E(B-V))_{\rm{D2}}=\frac{\mathrm{log_{10}}(E(B-V))+1.91}{2.16}\,\mathrm{\AA}$
$\displaystyle\mathrm{EW}(E(B-V))_{\rm{D1}}=\frac{\mathrm{log_{10}}(E(B-V))+1.76}{2.47}\,\mathrm{\AA}.$

The subscript D1 in the equations indicates the Na I D line at 5896 Å, while
D2 applies for the line at 5890 Å.

Figure 3.— Comparison of [Na/Fe] determined from the high-resolution (HR)
analysis to that from our analysis of the degraded low-resolution (LR)
spectrum before (upper panel) and after (lower panel) the ISS line removal. To
remove the ISS lines, we first measured their EWs in the original high-
resolution spectrum, and derived the Gaussian line profiles by following the
method described in Section 2.2. The derived line profiles were degraded and
subtracted from the degraded high-resolution spectrum to obtain a LR spectrum
with the ISS lines removed. The color bar at the top represents the SNR of the
high-resolution spectrum. It is clear that the difference becomes much smaller
after the ISS lines are removed. There are no apparent trends with SNR.

To remove the unresolved ISS lines in the low-resolution SDSS stellar spectra,
from the above equations we first estimated the EW of the Na I D doublet lines
produced by the ISM after computing the $E(B-V)$ value at a star’s Galactic
coordinates. Because the $E(B-V)$ value derived from Schlegel et al. (1998) is
the total reddening in a given direction, but the stars have different
distances, we used instead the 3D reddening map from Green et al. (2019) to
determine the $E(B-V)$ value for a star using the python package DUSTMAPS
(Green, 2018). The distance of a given star primarily comes from the parallax
provided by $Gaia$ Early Data Release 3 (EDR3; Gaia Collaboration et al.,
2021).

After obtaining the EW of the Na I D lines caused by the ISM along the line-
of-sight to a star, we generated very narrow artificial Na I D doublet lines
by forcing two Gaussian functions to have the same area as the calculated EW.
We then degraded the Gaussian profiles to the SDSS resolution of $R$ = 2000,
and subtracted it from the observed Na I D lines in a normalized SDSS spectrum
to create an approximately ISS-free spectrum. While fitting the two Gaussians,
we set the full width at half maximum (FWHM) to 0.1 Å. It turned out that the
choice of the FWHM parameter is not critical, as we degrade the artificially
created lines to a much lower resolution. In addition, we simply ignored the
velocity of ISM, even though it can cause the ISS lines to be shifted, not
only because it is infeasible to derive the radial velocities of the gas
clouds in the ISM that contribute to the ISS lines, but it is known that the
mean shift of the ISS lines in the Northern Hemisphere is about –3 km s-1
(Murga et al., 2015), which is a very small offset in low-resolution spectra.

Figure 2 shows an example of the removed ISS lines from a relatively metal-
rich SDSS stellar spectrum (SDSS J185226.49$+$201240.2), following the method
described above. The left panel of the figure shows the observed spectrum with
the ISS lines included, while the right panel is the same spectrum after their
removal. The effect of the removal is clear, with a substantially weaker Na
doublet visible in the right panel. The red line in each panel is the best-fit
model spectrum. The stellar atmospheric parameters ($T_{\rm{eff}}$, log $g$,
and [Fe/H]) and the sodium-abundance ratio used to generate the synthetic
spectrum are listed at the top of each panel. The stellar atmospheric
parameters for the SDSS spectrum were determined using the latest version of
the SEGUE Stellar Parameter Pipeline (SSPP; Allende Prieto et al., 2008; Lee
et al., 2008a, b, 2011; Smolinski et al., 2011). The sodium-abundance ratio
was estimated by following the method described in Section 2.3. It is obvious
that a much higher estimate of [Na/Fe] is obtained for the spectrum including
the ISS lines, as expected.

Figure 4.— Same as in Figure 3, but for comparison with APOGEE. The color bar
at the top left is added and represents the 3D reddening value derived from
Green et al. (2019) after adopting $Gaia$ parallaxes. There are 299 stars
included.

###  2.3. Method for Determination of [Na/Fe]

In order to estimate [Na/Fe] from SDSS stellar spectra with the ISS lines
removed, we applied a similar spectral-matching technique as that currently
used in the SSPP (Lee et al., 2011, 2013). Briefly, we first generated a grid
of synthetic spectra with wide ranges of stellar parameters and [Na/Fe]
values. We employed the Kurucz NEWODF model atmospheres (Castelli & Kurucz,
2003), which assumes plane-parallel line-blanketed model structures in one-
dimensional local thermodynamical equilibrium (LTE). Then, we utilized the
TURBOSPECTRUM synthesis code (Alvarez & Plez, 1998), which adopts the solar
abundances of Asplund et al. (2005), to create a synthetic spectrum in the
wavelength range between 5500 Å and 6500 Å at a resolving power of $R$ =
500,000. The generated grid covers 4000 K $\leq T_{\rm{eff}}\leq$ 7000 K in
steps of 250 K, $0.0\leq$ log $g$ $\leq 5.0$ in steps of 0.5 dex, –5.0 $\leq$
[Fe/H] $\leq$ +1.0 in steps of 0.5 dex, and –1.0 $\leq$ [Na/Fe] $\leq$ +3.0 in
steps of 0.5 dex. A total of 16,731 synthetic spectra were generated. Then,
these spectra were degraded to the SDSS resolution ($R$ = 2000) and normalized
with a pseudo continuum, which was derived from iterative polynomial fitting
to a spectrum.

After the radial-velocity correction is made for an observed spectrum, we
applied the same continuum routine used for the synthetic grid to the observed
SDSS spectrum to obtain a normalized spectrum. From the normalized spectrum,
we removed the ISS lines following the methodology described in Section 2.2.
The best-matching model spectrum was searched for by minimizing the residuals
in the wavelength range of 5880 Å – 5910 Å between the normalized observed
spectrum and the synthetic template, using the IDL/MPFIT routine (Markwardt,
2009). The spectral range considered is sufficiently wide to define an
accurate continuum and include the Na I D lines. During the minimization, we
held $T_{\rm{eff}}$, log $g$, and [Fe/H] constant, and only varied [Na/Fe] to
search for the best-fit model. The stellar parameters come from the latest
version of the SSPP.

Table 1Systematic and Random Errors of [Na/Fe] due to Errors in Stellar Parameters |  $T_{\rm{eff}}$ |  $\Delta$[Na/Fe] |  log $g$ |  $\Delta$[Na/Fe] |  [Fe/H] |  $\Delta$[Na/Fe]
---|---|---|---|---|---|---
|  Error | $\mu$ | $\sigma$ |  Error | $\mu$ | $\sigma$ |  Error | $\mu$ | $\sigma$
Type |  (K) | (dex) | (dex) |  (dex) | (dex) | (dex) |  (dex) | (dex) | (dex)
Giant |  +100 | +0.161 | 0.035 |  +0.2 |  $-$0.023 | 0.047 |  +0.2 |  $-$0.210 | 0.037
Cool dwarf |  +100 | +0.155 | 0.035 |  +0.2 |  $-$0.055 | 0.093 |  +0.2 |  $-$0.182 | 0.041
Hot dwarf |  +100 | +0.106 | 0.019 |  +0.2 |  $-$0.083 | 0.119 |  +0.2 |  $-$0.190 | 0.019
Giant |  $-$100 |  $-$0.164 | 0.039 |  $-$0.2 | +0.017 | 0.023 |  $-$0.2 | +0.201 | 0.041
Cool dwarf |  $-$100 |  $-$0.156 | 0.040 |  $-$0.2 | +0.093 | 0.047 |  $-$0.2 | +0.183 | 0.041
Hot dwarf |  $-$100 |  $-$0.107 | 0.016 |  $-$0.2 | +0.097 | 0.023 |  $-$0.2 | +0.188 | 0.024

###  2.4. Evaluation of the Interstellar Sodium Line-removal Scheme with High-
resolution Spectra

####  2.4.1 Comparison with [Na/Fe] Derived from Gemini/GRACE Stellar Spectra

As part of a study of the early chemical evolution of the MW, we have obtained
high-resolution Gemini/GRACES spectra for several dozen extremely metal-poor
(EMP; [Fe/H] $<$ –3.0) candidates selected from SDSS and LAMOST, and analyzed
these spectra with MOOG (Sneden, 1973) to derive stellar parameters and
abundances for various chemical elements, including Na (M. Jeong et al. in
preparation). Eleven of the EMP candidates have [Na/Fe] $>$ –0.5, and we make
use of them to evaluate the performance of our ISS line-removal scheme by
comparing our [Na/Fe] estimates from the low-resolution spectra before and
after removing the ISS lines with those of the high-resolution analysis.

To remove the ISS lines from the Gemini/GRACES spectra in a similar fashion as
we do for observed SDSS stars, we first measured their EWs in the original
high-resolution spectrum, and derived the two Gaussian line profiles following
the method described in Section 2.2. These derived line profiles were
convolved to the SDSS resolution and subtracted from the degraded high-
resolution spectrum. Following this step, we determined [Na/Fe] via our
spectral-fitting technique. While determining [Na/Fe], we fixed the stellar
parameters adopted from the high-resolution analysis. We also determined
[Na/Fe] from the degraded high-resolution spectrum including the ISS lines.

Figure 3 shows the results of this exercise. The color bar shown above the top
panel represents the SNRs of the high-resolution spectra. The upper panel
shows the difference in the derived [Na/Fe] between our low-resolution
estimate and that of the high-resolution analysis before the ISS line removal.
It is apparent that the ISS lines overwhelm the intrinsic stellar Na I D
lines, resulting in an estimated [Na/Fe] that is systematically higher
compared to the high-resolution spectra, by over 1 dex. In the lower panel,
which applies to the situation after the removal of the ISS lines, there is a
much smaller offset (less than 0.5 dex for most of the stars). Although we did
not find any trend with SNR, our estimate of [Na/Fe] is slightly larger, on
average, by about 0.2 dex.

We also found that the EWs computed from the ISS lines in the high-resolution
spectra are actually quite close to those predicted from the relations between
the reddening and EWs of the ISS lines described above. The sum of EWs of the
doublets calculated from the 3D extinction map is on average about 6% larger
than that measured for the high-resolution spectrum. This result shows that
our method for ISS line removal works quite well.

####  2.4.2 Comparison with [Na/Fe] Derived from APOGEE

We also carried out a similar set of evaluations using data from APOGEE. The
APOGEE survey obtains H-band spectra with a resolving power of $R\sim$ 22,500.
Even though the APOGEE Data Release 16 (DR16; Ahumada et al., 2020) lists
[Na/Fe] abundances determined from different Na lines (16378 Å and 16393 Å),
this test still provides a valuable comparison. Note that the Na abundance we
adopted from APOGEE is based on the LTE assumption.

Figure 4 shows the difference between the APOGEE results and our measured
[Na/Fe] from the Na I D lines with (top panels) and without (bottom panels)
the ISS lines, similar to Figure 3. From inspection, it is obvious that our
estimated [Na/Fe] is consistently higher in the top panels, in particular for
the stars with the higher reddening, as expected. We also note from inspection
of the bottom-left panel that our ISS removal scheme has a tendency to
slightly over-estimate the EW of the ISS for high-reddening stars, as they
exhibit lower [Na/Fe], on average. This behavior is opposite to that seen in
Figure 3, where we removed the predicted ISS lines from the high-resolution
spectrum. For our present investigation (comparison of field stars with MW
GCs, as described below), we are more interested in stars with [Fe/H] $>$
–2.5; thus we rely on the comparison with the APOGEE for further calibration
(see Section 3.4). We note that no significant trends with [Fe/H] or SNR are
apparent in the right-column panels.

Table 2Impacts of SNR for Low-Resolution Spectra on Determination of [Na/Fe] |  $\Delta$[Na/Fe]
---|---
| Giant | Cool dwarf | Hot dwarf
SNR | $\mu$ | $\sigma$ | $\mu$ | $\sigma$ | $\mu$ | $\sigma$
10 |  0.022 | 0.036 |  0.039 | 0.049 |  0.076 | 0.075
15 |  0.006 | 0.029 |  0.013 | 0.036 |  0.034 | 0.058
20 |  $-$0.002 | 0.025 |  0.002 | 0.030 |  0.017 | 0.054
30 |  $-$0.009 | 0.020 |  $-$0.003 | 0.026 |  $-$0.000 | 0.038
40 |  $-$0.011 | 0.017 |  $-$0.006 | 0.020 |  $-$0.006 | 0.031
50 |  $-$0.012 | 0.017 |  $-$0.005 | 0.019 |  $-$0.010 | 0.027

##  3\. Validation of [Na/Fe] Determinations from Low-Resolution Spectra

###  3.1. Random and Systematic Errors on [Na/Fe]

The strength of the Na I D lines increases at higher [Fe/H] and [Na/Fe], as
expected, and are weaker for giants compared to dwarfs. Thus, a degeneracy
could exist in the strength of the Na I D lines depending on stellar
parameters, which could result in a systematic error in the estimated [Na/Fe]
by our spectral-fitting method. In addition, as we fixed $T_{\rm eff}$,
$\log~{}g$, and [Fe/H] while estimating [Na/Fe], the existence of any
systematic errors in $T_{\rm eff}$, $\log~{}g$, and [Fe/H] could produce a
systematic error in the determined [Na/Fe]. Therefore, we need to check how
the systematic errors in the adopted stellar parameters and the possible
degeneracy of the Na I D line strengths affects the estimated [Na/Fe].

To carry out this test, we first processed the synthetic spectra through our
methods to obtain an estimate of the (known) [Na/Fe] under various conditions.
While estimating [Na/Fe], we shifted the model parameters by $\pm$100 K for
$T_{\rm{eff}}$ and $\pm$0.2 dex for log $g$ and [Fe/H], respectively, in order
to estimate how much the derived [Na/Fe] can be affected by errors of the
adopted stellar parameters.

After completing the estimates of [Na/Fe] to check the collective behavior of
the determined [Na/Fe], we divided the measured [Na/Fe] values into three
groups: giants for $T_{\rm eff}$ $<$ 5500 K and $\log~{}g$ $<$ 3.5, cool
dwarfs for $T_{\rm eff}$ $<$ 5500 K and $\log~{}g$ $\geq$ 3.5, and hot dwarfs
for $T_{\rm eff}$ $\geq$ 5500 K and $\log~{}g$ $\geq$ 3.5. We did not include
the spectra for models with [Fe/H] $<$ –3.0, because not only are such objects
rare, but we are here primarily interested in the escaped stars from the
Galactic GCs or the debris from the already-disrupted GCs, which presumably
have [Fe/H] $>$ –3.0.

Table 1 summarizes the results of the above exercise. It generally indicates
that the systematic offsets of the measured [Na/Fe] do no vary much among the
different groups, and that the random scatters are also small for all groups.
Specifically, the incorrect assignment of surface gravity does not much affect
the measured [Na/Fe], as the mean offset is less than 0.1 dex. The errors
caused by the $T_{\rm eff}$ and [Fe/H] offsets become a little larger, up to
$\pm$0.16 dex and $\pm$0.2 dex, respectively, but are still not significant.
The mean $\mu$ and standard deviation $\sigma$ in the table are determined
from a Gaussian fit to the differences between our estimates and the model
values. Considering that the typical uncertainty of the elemental abundance
from the low-resolution spectroscopy is about 0.3 dex, we conclude that the
systematic errors in the stellar parameters do not significantly affect the
estimated [Na/Fe].

Figure 5.— Comparison of our derived sodium-abundance ratios with those from
the APOGEE (left panels) and GALAH (right panels) surveys, which are based on
high-resolution near-IR and optical spectroscopy, respectively. In the upper
panels, the mean offset ($\mu$) and the standard deviation ($\sigma$) were
obtained by fitting a Gaussian to both samples. The comparison with the APOGEE
data exhibit no significant trend with $\log~{}g$, so we decided to adjust the
offset (–0.16 dex) to the [Na/Fe] value before applying the NLTE corrections
described in the text (see Section 3.3. Since [Na/Fe] from the GALAH survey is
based on NLTE, we compared our values of [Na/Fe] after the NLTE correction to
those from the GALAH stars. The top-right panel indicates a small offset (less
than 0.1 dex). However, we note a negative trend with surface gravity in the
bottom-right panel. We fitted a linear function to the residuals to obtain a
slope of –0.21 dex dex-1 and intercept of 0.88 dex. We applied this correction
function to our NLTE corrected [Na/Fe], and use this value for further
analysis. Figure 6.— Comparison of the stellar parameters from the SSPP with
those from the APOGEE (upper panels) and GALAH (lower panels) data based on
high-resolution spectroscopic analysis. The mean offset and the scatter for
each parameter are derived from a Gaussian fit. Overall, we observe relatively
small offsets and scatters from both samples for all three parameters. This
figure only includes the stars with [Na/Fe] available in the APOGEE and GALAH
data sets, as we calibrate our estimates of [Na/Fe] with those from these two
surveys. Note that we used the NLTE [Fe/H] values from the GALAH survey; see
the text for details on the validity of its use.

###  3.2. Impact of SNR on Estimated [Na/Fe]

Stellar spectra from large sky surveys such as SDSS and LAMOST cover a wide
range of SNRs. Because the noise in a spectrum is able to induce systematic
and random errors, we also check on how the SNR affects the estimated [Na/Fe].

To evaluate the impact of the SNR on the abundance estimate for sodium,
following Lee et al. (2008a), we injected different levels of random noise
into the grid of synthetic spectra used for the spectral fitting. In this
process, we generated a set of 10 different noise-added synthetic spectra for
SNR = 10, 15, 20, 30, 40, and 50 per given model spectrum. Then, we applied
our method to the noise-added synthetic spectra to recover [Na/Fe]. In our
analysis, we took an average value of 10 [Na/Fe] values from 10 simulated
noise-injected spectra at a given SNR. To analyze the results of this test,
once again we divided the test sample into three groups as in Section 3.1:
giants, cool dwarfs, and hot dwarfs.

Table 2 summarizes the results of the above exercise. It indicates that, even
though there is a tendency that the mean offset and scatter increase with
decreasing SNR, as expected, their size is relatively small. The mean $\mu$
and standard deviation $\sigma$ in the table are determined from a Gaussian
fit to the residuals between our estimates and the model parameters for
[Fe/H]$\geq$ –3.0. Furthermore, we observe that, as the metallicity decrease
and temperature increases, the mean difference and their scatter become
larger, presumably due to the weaker Na I D lines. Nonetheless, the mean
offset and scatter are significantly less than 0.1 dex for all three groups
and SNRs. Consequently, this exercise confirms that our method is very robust
to the SNR of the spectrum.

###  3.3. Non-LTE Corrections

It is known that the [Na/Fe] abundance ratio can vary, by as much as 0.5 dex,
under the non-LTE (NLTE) assumption, especially when the EW of the Na I D
lines is around 150 to 200 mÅ (Gratton et al., 1999; Lind et al., 2011). The
correction is generally larger for giants than dwarfs (Amarsi et al., 2020).
Therefore, we need to apply NLTE corrections to our measured [Na/Fe]. Gratton
et al. (1999) reported the NLTE corrections for lines of Fe I, O I, Na I, and
Mg I, and provided the correction values for stars with $T_{\rm eff}$ from
4000 K to 7000 K with 1000 K steps, over a broad range of gravities
($\log~{}g$ = 1.5, 3.0, 4.5) and metallicities ([Fe/H] = 0, –1, –2, –3). For
the Na I D lines the correction grid is valid for the EW range of 100 mÅ –
1350 mÅ. We adopted this grid to derive the NLTE correction value by linear
interpolation over the stellar parameters and the EW of the Na I D lines. When
obtaining the correction value from this interpolation, we used the EW of the
Na I D lines calculated from the synthetic spectrum, which is generated with
the estimated $T_{\rm eff}$, $\log~{}g$, [Fe/H], and [Na/Fe] for each star,
because it provides much clearer lines to calculate the EW. We simply took the
value from the grid that is the nearest to $T_{\rm eff}$, $\log~{}g$, [Na/Fe],
and EW for the star whose parameters are outside the grid.

###  3.4. Calibration of Estimated [Na/Fe] with High-resolution Surveys

Comparison of our derived [Na/Fe] with that estimated based on high-resolution
spectroscopy provides a direct measure of the reliability of our estimated
[Na/Fe]. We employed stars observed in two surveys, APOGEE and the Galactic
Archaeology with HERMES 111https://galah-survey.org/ (GALAH; Zucker et al.,
2012) survey. The GALAH survey obtained optical spectra with a resolving power
of $R\sim$ 28,000. Although the APOGEE DR16 and the GALAH Data Release 3
(Buder et al., 2021) contain [Na/Fe] abundances determined from different Na
lines: 16378 Å and 16393 Å for APOGEE and 5688 Å, 6154 Å, and 6169 Å for
GALAH, we expect the derived Na abundance ratio should be the same independent
of the lines employed, unless some unexpected problem exists. Below we
describe a comparison exercise between the two surveys.

Cross matches (within 1$\arcsec$) were used to identify 318 and 116 stars in
common between SDSS and APOGEE, and SDSS and GALAH, respectively. We selected
only the stars with $T_{\rm eff}$ = 4000 – 7000 K and the uncertainties of the
measured [Na/Fe] less than 0.1 dex in the comparison-survey data. These stars
also met the requirement of 10 km s-1 in radial-velocity differences. For the
SDSS spectra, we removed the ISS lines following the scheme described in
Section 2.2. We adopted for each star the inverse parallax from $Gaia$ EDR3 as
the distance used to calculate the strength of the ISS lines. The sodium-
abundance ratio was then determined by fixing the atmospheric parameters
($T_{\rm eff}$, $\log~{}g$, and [Fe/H]) obtained from the SSPP.

Note that the [Na/Fe] abundance ratio from APOGEE is based on an LTE
assumption, while that from the GALAH is based on the NLTE assumption. To
carry out the calibration properly, we first compared our measured value
before the NLTE correction with that of APOGEE to check if any systematic
error exists in the LTE value. The left column of panels in Figure 5 show
histograms of the [Na/Fe] differences (top panel) between our low-resolution
(LR) and the APOGEE high-resolution (HR) value, and their trend with
$\log~{}g$ (bottom panel). We found a systematic offset of –0.16 dex and a
standard deviation of 0.27 dex, which were calculated with a Gaussian fit to
the residuals. The bottom panel suggests no significant trend of the mean
offset with $\log~{}g$. From these findings, we decided to adjust our
estimated LR LTE [Na/Fe] by 0.16 dex to place it on the same abundance scale
as for the HR LTE APOGEE sample.

To compare with the GALAH data, after the 0.16 dex offset described above is
applied to our LR LTE estimates, we corrected them to the NLTE scale following
the recipe described in Section 3.3. The Na abundance differences with respect
to GALAH and their trend with $\log~{}g$ are displayed in the right column of
panels shown in Figure 5. Our LR NLTE Na abundances exhibit a very small
systematic offset of 0.04 dex, as shown in the top panel. The lower panel,
however, indicates a negative trend with $\log~{}g$. In order to eliminate
this trend, we used a linear fit, with slope and intercept of –0.21 dex dex-1
and 0.88 dex, respectively, and applied it to our LR NLTE values. We noticed
no significant trend with $T_{\rm eff}$ or [Fe/H] after this correction was
applied.

To summarize our approach for determination of the final adopted [Na/Fe]
estimates, we first add 0.16 dex to our measured LR [Na/Fe], obtain the NLTE-
corrected value, and apply the correction function found in the bottom-right
panel of Figure 5 to determine the final LR NLTE [Na/Fe] estimate. We use this
final value for the following analysis. The comparisons with the high-
resolution data and the other two tests described in Sections 3.1 and 3.2
suggest that the expected precision of the estimated [Na/Fe] is better than
0.3 dex for [Fe/H] $>$ –2.0 and SNR $>$ 10.

###  3.5. Accuracy of the Stellar Parameters Determined by the SSPP

We also compared the stellar parameters delivered by the SSPP to those from
the APOGEE and GALAH surveys to check their accuracy and uncertainties,
because any systematic error in the SSPP stellar parameters can result in
systematic deviations in the determined [Na/Fe], as demonstrated in Section
3.1. Note that we employed the SDSS stars in common with stars from the APOGEE
and GALAH surveys with [Na/Fe] available, as we used them to calibrate our
estimates of [Na/Fe]. Figure 6 shows the distributions of the differences in
$T_{\rm eff}$ (left), $\log~{}g$ (middle), and [Fe/H] (right) for the APOGEE
(upper panels) and GALAH (lower panels) data. The mean difference ($\mu$) and
the standard deviation ($\sigma$) are derived from a Gaussian fit to each
distribution. We note the very small mean offsets and scatters for both
surveys. Specifically, the temperature comparison indicates that the zero-
point offset is less than 30 K, with a scatter less than 90 K. The gravity
comparison shows $\mu<$ 0.1 dex with $\sigma<$ 0.25 dex, and $\mu<$ 0.05 dex
with $\sigma<$ 0.15 dex for the metallicity. Note that the metallicity from
the GALAH survey is the NLTE value. As the NLTE correction for [Fe/H] is very
small (mostly less than 0.1 dex) for the stars with [Fe/H] $>$ –1.0 (Lind et
al., 2012) and most of the GALAH stars have [Fe/H] $>$ –1.0, we simply used
the NLTE value of [Fe/H] for this comparison.

Taking into account that the SDSS stellar spectrum is low resolution, these
mean offsets and scatters are quite small, suggesting that the SSPP recovers
estimates of the stellar parameters very well.

Figure 7.— Density map of our sample stars in $Z_{\rm{max}}$ versus
circularity space. Positive and negative circularity values correspond to
prograde and retrograde motions, respectively. The stars inside the blue box
are regarded to be of $in\,situ$ origin, while the objects outside the box are
taken to be of accreted origin. Note that, although a number of stars with
$r_{\rm{max}}<3.5$ kpc are located outside the blue box, they are regarded as
belonging to the $in~{}situ$ origin (see text). Figure 8.— Metallicity
distribution functions (MDFs) of the $in\,situ$ (black) and the accreted (red)
components, which are grouped by the dividing scheme of Massari et al. (2019).
The histogram is normalized by the total number of stars in each group. The
top-left panel shows the MDFs of the GCs. The $in~{}situ$ GCs display two
peaks around [Fe/H] = –1.3 and –0.6, respectively, whereas the accreted
component exhibits one peak at [Fe/H] = –1.6. The two vertical dotted
reference lines are provided at [Fe/H] = –1.3 and –0.6, respectively. The
other panels present the MDFs of our sample of stars in different ranges of
[Na/Fe]. Notice the rise of the second peak in the $in~{}situ$ sample from
[Na/Fe] = +0.6; as the Na-abundance ratio increases, the portion of the metal-
poor stars increases as well. In comparison, the accreted population displays
two peaks at low [Na/Fe], and they disappear above [Na/Fe] $\sim$ +0.5; its
MDF is gradually shifted to the low-metallicity regime with increasing
[Na/Fe].

##  4\. An Example Application

Here we describe an example application of our sodium-abundance ratio
estimates for a large sample of stars with available low-resolution spectra
from SDSS. Other applications are clearly of interest, which we will pursue in
future papers. When the currently observed GCs in the MW are divided into the
$in~{}situ$ and accreted component by their dynamical properties (e.g.,
Massari et al., 2019), their MDFs are dissimilar, with distinct
characteristics. As the Na-rich stars are believed to have originated from
stripped or totally disrupted GCs, we might expect that their dynamical
properties and chemical characteristics may be similar to those observed in
the surviving GCs, but different for the Na-normal stars. We explore this
hypothesis below.

###  4.1. The Field Star Sample

We applied our method for determining [Na/Fe] to 539,920 stellar spectra with
SNR $\geq$ 10, which were assembled from the SDSS stars. We adopted parallaxes
from $Gaia$ EDR3 to calculate the distances, which we require to estimate the
strength of the ISS lines from the 3D reddening map. When calculating the
distance, we corrected the global parallax offset of $+$0.017 mas (Lindegren
et al., 2021). After the corrections for NLTE effects and the systematic
offset described in Sections 3.3 and 3.4, respectively, we are left with a
total number of 427,799 stars with available LR NLTE [Na/Fe] values.

###  4.2. Calculation of Space Velocities and Orbital Parameters

For the dynamical study of our program stars we require information on their
velocity components and orbital parameters. We employed the proper motions and
parallaxes from $Gaia$ EDR3 and the radial velocities from the SDSS spectra,
and computed the $U,V,W$ velocity components and dynamical parameters,
including the maximum ($r_{\rm max}$) and minimum ($r_{\rm min}$) distances
from the Galactic center, maximum distance ($Z_{\rm max}$) from the Galactic
plane, orbital eccentricity ($e$), angular momentum, and energy, etc., using
the galpy Python package222http://github.com/jobovy/galpy (Bovy, 2015),
adopting the Galactic gravitational potential model of McMillan (2017). During
these calculations, we integrated the orbit of each star over 13 Gyr with 1
Myr steps, and assumed $V_{\rm{LSR}}$ = 233.1 km s-1 for the Local Standard of
Rest, a solar peculiar motion of ($U,V,W$)⊙ = (11.1, 12.24, 7.25) km s-1
(Schönrich et al., 2010), and the distance of the Sun from the Galactic center
of $R_{\odot}$ = 8.21 kpc. The orbital parameters of Galactic GCs were also
computed by adopting their positions, proper motions, distances, and radial
velocities from Vasiliev (2019).

Figure 9.— Metallicity distribution functions (MDFs) of our sample according
to their level of Na enhancement: all (upper left), [Na/Fe] $<$ +0.6 (upper
right), +0.6 $\leq$ [Na/Fe] $<$ +1.0 (lower left), and [Na/Fe] $\geq$ +1.0
(lower right). It is clear to see in the upper-left panel that all three
subgroups have different peaks and shapes of their MDFs. About 85 % of our
sample stars belong to the Na-normal subgroup, and only 15 % of stars have
[Na/Fe] $\geq+0.6$. The other panels show the MDFs of the $in\,situ$ and
accreted stars for the three subgroups. The fraction of the accreted stars
clearly increases with increasing [Na/Fe].

###  4.3. Results and Discussion

Among the SDSS stars with estimated [Na/Fe] and available orbital parameters,
we selected the stars with 4400 K $\leq$ $T_{\rm{eff}}$ $\leq$ 7000 K, –3
$\leq$ [Fe/H] $\leq 0$, and SNR $\geq$ 25.0 to ensure the most reliable
measurement of [Na/Fe]. We also imposed on our sample stars the requirement of
the renormalized unit weight error less than 1.4 from $Gaia$ EDR3 to select
stars with good astrometric solutions. For stars with derived $r_{\rm max}$
$\leq$ 200 kpc, we ended up with a final sample of 188,216 stars.

Provided that the Na-rich stars originated from GCs or disrupted GCs, we may
expect that their MDF and dynamical properties would be similar to those of
the GCs. In addition, they would exhibit distinct features in their MDF,
compared to those from the Na-normal stars. To identify their connection to
the GCs, we followed a similar approach as Massari et al. (2019). They divided
the GCs into $in\,situ$ (bulge and disk clusters) and accreted clusters, using
their dynamical properties. They defined that the bulge clusters have small
orbits with $r_{\rm max}$ $<3.5$ kpc, and disk clusters have orbital
circularity larger than 0.5 and $Z_{\rm{max}}<5$ kpc. Note that the orbital
circularity is defined as $L_{Z}/L_{Z,\rm{circ}}$, where $L_{Z,\rm{circ}}$ is
the angular momentum of a circular orbit with a cluster’s energy. Figure 7
displays the density map of our sample stars on $Z_{\rm{max}}$ versus
circularity space. In the figure, positive and negative circularity values
mean prograde and retrograde orbits, respectively. The stars inside the blue
box are classified as of the $in\,situ$ origin, while outside the box they are
considered of accretion origin.

After dividing into the two groups, we first inspected the MDFs of each group
in different bins of [Na/Fe] values, as shown in Figure 8. The top-left panel
of the figure shows the MDFs of the GCs in the MW. The black histogram is for
the $in\,situ$ GCs, while the red histogram is for the accreted GCs, divided
by following the work by Massari et al. (2019). The [Fe/H] values of the GCs
were adopted from the Harris (1996, 2010) catalog. The notable features in
this panel are that the MDF of the accreted GCs has only a single peak at
[Fe/H] $\sim$ –1.6, with more objects populating the more metal-poor region,
while the MDF of the $in\,situ$ GCs exhibits two peaks at [Fe/H] = –1.3 and
–0.6, respectively, indicated by the two vertical dotted lines. The rest of
the panels in the figure show the MDFs of our program stars.

Inspection of the MDFs in each panel reveals that the MDF for the $in~{}situ$
population of stars peaks at between [Fe/H] = –0.6 and –0.4, with a long tail
toward the low-metallicity region, up to [Na/Fe] = +0.6. These Na-normal,
metal-rich stars are the dominant sources of the canonical Thick Disk (TD) and
Metal-Weak Thick Disk (MWTD; Carollo et al., 2019). The second peak at [Fe/H]
= –1.3 starts to rise from [Na/Fe] = +0.6, and the first peak remains the
same, at [Fe/H] = –0.6, for [Na/Fe] $>$ +0.6. These features qualitatively
agree with those of the $in~{}situ$ GCs, as can be seen in the top-left panel.
From [Na/Fe] = +1.0 to [Na/Fe] = +1.5, the second peak becomes more obvious,
with a much broader, but more metal-poor, distribution, while the fist peak
still exists. The much larger scatter and more metal-poor feature in the
second peak may imply that more than one population contributes to the second
peak. Even though they are defined as the $in~{}situ$ component, it is
possible that some of the metal-poor component came from the accreted GCs, as
claimed by Woody & Schlaufman (2021), who argue that the metal-poor GCs are
mostly accreted from disrupted dwarf galaxies.

The accreted population of our sample fairly well follows the shape of the MDF
of the accreted GCs above [Na/Fe] = +0.5. For [Na/Fe] $<$ +0.5, we can also
observe a second peak at [Fe/H] = –0.6. Considering their metallicity and
distinct dynamical status, this group of stars may consist of the GE (Helmi et
al., 2018) and Splashed Disk (SD; Belokurov et al., 2020), and the rest of the
more metal-poor stars may contribute to the Sequioa (Myeong et al., 2019) and
Thamnos 1 & 2 (Koppelman et al., 2019) structures. We note in Figure 8 that
the low-metallicty tails of the MDFs of the $in~{}situ$ and the accreted
populations in the range of [Na/Fe] $>$ +0.9 exhibit similar peaks and shapes.
This analogy may suggest that the low-metallicity, Na-rich stars may share
chemically similar GC progenitors.

For the ensemble view of the $in~{}situ$ and the accreted populations of our
sample, based on the MDF characteristics found in Figure 8, we divided our
sample into three subgroups: Na-normal stars with [Na/Fe]$<$ +0.6, Na-high
stars with +0.6 $\leq$ [Na/Fe] $<$ +1.0, and Na-extreme stars with [Na/Fe]
$\geq$ +1.0, and explored differences in their MDFs, as shown in Figure 9.

The upper-left panel of Figure 9 displays the MDFs of the three subgroups,
normalized by the total number of stars in each subgroup. Different peaks and
morphologies for the MDFs are evident. The general behavior is that the higher
the sodium-abundance ratio, the lower the metallicity peak. About 85% of our
sample stars belong to the Na-normal subgroup, and only 15% of stars have
[Na/Fe] $\geq$ +0.6. The other three panels exhibit the MDFs of the $in\,situ$
(black) and accreted (red) stars for each subgroup. The $in\,situ$ stars of
the Na-normal subgroup (upper right) exhibits a single peak at [Fe/H] $\sim$
–0.5 with a long metal-poor wing. This group of stars comprises mainly the
canonical TD and MWTD. On the other hand, the accreted subgroup displays a
single peak at [Fe/H] $\sim$ –1.3, without much skewness. This subgroup
comprises the GS, GE, and SD stars. The two MDFs are reminiscent of the local
disk and halo populations in the MW.

About 9.2% of stars (lower-left panel of Figure 9) among our sample belong to
the Na-high subgroup. The shapes of the MDFs of the $in\,situ$ and accreted
stars in this subgroup are very similar to those of the GCs (see the upper-
left panel of Figure 8) — a single peak around [Fe/H] = –1.6 for the accreted
population and the double peaks around [Fe/H] = –1.3 and –0.6 for the
$in\,situ$ population. As Galactic field stars mostly have [Na/Fe] $<$ +0.5
(Suda et al., 2008), and the Na-high stars have the same characteristics in
their MDF as the Galactic GCs, we can confirm that the stars in the Na-high
subgroup are likely the escaped stars from present GCs.

The MDFs of the Na-extreme subgroup (lower-right panel of Figure 9) present
some distinct features compared to the Na-normal and Na-high subgroups.
Although the $in~{}situ$ population still has two peaks, the metal-poor stars
are more prominent. The accreted component is also more metal-poor and has a
broader distribution. One notable feature is that the metal-poor component
([Fe/H] $<$ –1.2) of the MDF for the $in~{}situ$ population generally follows
that of the accreted component, a characteristic that does not appear for the
surviving GCs. Consequently, it is plausible that the metal-poor stars in the
Na-extreme subgroup may have originated from already disrupted metal-poor GCs,
and experienced similar chemical evolution, but different dynamical evolution.
Another possible interpretation is that the $in~{}situ$ metal-poor stars may
not be the stars formed in $in~{}situ$ GCs. According to the analysis of
Piatti (2019), among the prograde GCs, which are assumed to be of the
$in~{}situ$ origin, the number of accreted GCs is similar to that of the
$in~{}situ$ ones. Following this reasoning, some of the stars in our metal-
poor, Na-extreme $in~{}situ$ population may originate from accreted GCs. This
interpretation is in line with the claim by Woody & Schlaufman (2021).

Regarding the metal-rich ([Fe/H] $>$ –1.0) $in~{}situ$ population in the Na-
high and Na-extreme subgroups, we note that these may be related to the metal-
rich ([Fe/H] $>$ –0.7) population recently discovered from APOGEE data by
Fernández-Trincado et al. (2021). These authors claimed that this population
is likely to be stellar debris from partially or fully disrupted metal-rich
GCs.

Another point to consider is that some fraction of Carbon-Enhanced Metal-Poor
(CEMP; Beers & Christlieb, 2005) stars exhibit enhancements of sodium although
many of the CEMP stars exhibit [Na/Fe] $<$ 0.5 (Spite et al., 2014; Hansen et
al., 2015; Yoon et al., 2016; Purandardas & Goswami, 2021). Thus, our Na-
extreme stars may also be related to CEMP stars. Indeed, when we considered as
CEMP stars the objects with [Fe/H] $<$ –1.0 and [C/Fe] $>$ +0.7, we found from
the Na-extreme subgroup that the fraction of CEMP stars for [Fe/H] $<$ –1.0 is
12.0%; it increases to 18.7% for [Fe/H] $<$ –2.0. This proportion is much
higher than those of previous studies (e.g., Lee et al., 2013). We also found
that the CEMP fraction (19.1%) of the accreted component in the Na-extreme
subgroup is higher than that (16.3%) of the $in~{}situ$ component for [Fe/H]
$<$ –2.0. In the metallicity range of [Fe/H] $>$ –2.5, the CEMP-$s$ (those
enhanced with slow neutron-capture elements) stars are dominant among the
different CEMP subclasses (Aoki et al. 2007; see also Yoon et al. 2016, 2018);
hence their likely progenitors are asymptotic giant branch (AGB) stars.
Furthermore, the second-generation stars observed in GCs exhibit low carbon
and high nitrogen abundances. These results suggest that many of the Na-
extreme stars probably did not originate from GCs.

Taking into account only the Na-high stars, we obtained the fraction of 9.2%
(17257/188216), which is within the fractions reported by previous studies
(Carretta et al., 2010; Martell et al., 2011; Ramírez et al., 2012). This
fraction implies a non-negligible contribution from the GCs to the build-up of
the Galactic halo.

To sum up, since the $in~{}situ$ and accreted stars of our Na-high stars
exhibit similar MDFs to those of present GCs, they may have originated from
these, but this is not the case for the Na-extreme stars. As the fraction of
the Na-high stars is 9.2%, the contribution from GC-origin stars to the build-
up of the Galactic halo appears non-negligible. Furthermore, because the
metal-poor stars in the $in~{}situ$ and accreted populations exhibit similar
MDFs, we cannot rule out that they may share the progenitors with similar
chemical-evolution histories. Indeed, a K-S two-sample test between MDFs for
the $in~{}situ$ and accreted stars with [Fe/H] $<$ –1.2 in the Na-extreme
subgroup yielded a $p$-value of 0.0722, which is small, but only permits a
marginal rejection of their selection from the same parent population.

##  5\. Summary

We have presented a method for the estimation of [Na/Fe] from low-resolution
stellar spectra. To determine [Na/Fe] more accurately, we developed a
technique of removing the ISS lines in a stellar spectrum, using the
correlation between the Galactic reddening, $E(B-V)$, and the strength of Na I
D doublet lines produced by the ISM. In addition, we applied an NLTE
correction to determine [Na/Fe] more accurately and precisely. We also
investigated the MDFs of the Na-rich stars, and compare with those of Galactic
GCs, in order to better understand the connection of the Na-rich stars with
the buildup of the Galactic halo. In a subsequent paper, we will carry out a
more thorough analysis of the Na-rich stars gathered from SDSS and LAMOST,
using their chemical and dynamical properties to unravel their relation to the
GCs and the Galactic halo assembly.

We also note that our measured [Na/Fe] will be extremely useful to calibrate
photometrically estimated [Na/Fe] for stars from narrow-band photometric
surveys such as Javalambre Photometric Local Universe Survey (J-PLUS; Cenarro
et al., 2019) and the Southern Photometric Local Universe Survey (S-PLUS;
Mendes de Oliveira et al., 2019). Eventually, these efforts should result in
tens to hundreds of million stars with measured [Na/Fe] to explore various
stellar populations in the MW.

Y.S.L. acknowledges support from the National Research Foundation (NRF) of
Korea grant funded by the Ministry of Science and ICT (NRF-2021R1A2C1008679).
J.-R.K. acknowledges support from Basic Science Research Program through the
National Research Foundation of Korea(NRF) funded by the Ministry of Education
(NRF-2019R1I1A3A02062242). T.C.B. acknowledges partial support for this work
from grant PHY 14-30152; Physics Frontier Center/JINA Center for the Evolution
of the Elements (JINA-CEE), awarded by the U.S. National Science Foundation.
This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. Funding for the Sloan
Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the
U.S. Department of Energy Office of Science, and the Participating
Institutions. SDSS-IV acknowledges support and resources from the Center for
High Performance Computing at the University of Utah. The SDSS website is
www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for
the Participating Institutions of the SDSS Collaboration including the
Brazilian Participation Group, the Carnegie Institution for Science, Carnegie
Mellon University, Center for Astrophysics — Harvard & Smithsonian, the
Chilean Participation Group, the French Participation Group, Instituto de
Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the
Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the
Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz
Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie
(MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-
Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatories of China, New Mexico State University, New York University,
University of Notre Dame, Observatário Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University.

## References

  * Ahumada et al. (2020) Ahumada, R., Prieto, C. A., Almeida, A., et al. 2020, ApJS, 249, 3
  * Allende Prieto et al. (2008) Allende Prieto, C., Sivarani, T., Beers, T. C., et al. 2008, AJ, 136, 2070
  * Alvarez & Plez (1998) Alvarez, R. & Plez, B. 1998, A&A, 330, 1109
  * Amarsi et al. (2020) Amarsi, A. M., Lind, K., Osorio, Y., et al. 2020, A&A, 642, 62
  * Aoki et al. (2007) Aoki, W., Beers, T. C., Christilieb, N., et al. 2007, ApJ, 655, 492
  * Asplund et al. (2005) Asplund, M., Grevesse, N., & Sauval, A. J. 2005, Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis, 336, 25
  * Beers & Christlieb (2005) Beers, T. C., & Christlieb, N. 2005, ARA&A, 43, 531
  * Belokurov et al. (2018) Belokurov, V., Erkal, D., Evans, N. W., et al. 2018, MNRAS, 478, 611
  * Belokurov et al. (2020) Belokurov, V., Sanders, J. L., Fattahi, A., et al. 2020, MNRAS, 494, 3880
  * Blanton et al. (2017) Blanton, M. R., Bershady, M. A., Abolfathi, B., et al. 2017, AJ, 154, 28
  * Bovy (2015) Bovy, J. 2015, ApJS, 216, 29
  * Buder et al. (2021) Buder, S., Sharma, S., Kos, J., et al. 2021, MNRAS
  * Carollo et al. (2019) Carollo, D., Chiba, M., Ishigaki, M., et al. 2019, ApJ, 887, 22
  * Carretta (2016) Carretta, E. 2016, arXiv:1611.04728
  * Carretta et al. (2009) Carretta, E., Bragaglia, A., Gratton, R. G., et al. 2009, A&A, 505, 117
  * Carretta et al. (2010) Carretta, E., Bragaglia, A., Gratton, R. G., et al. 2010, A&A, 516, A55
  * Castelli & Kurucz (2003) Castelli, F. & Kurucz, R. L. 2003, Modelling of Stellar Atmospheres, 210, A20
  * Cenarro et al. (2019) Cenarro, A. J., Moles, M., Cristóbal-Hornillos, D., et al. 2019, A&A, 622, A176
  * Chene et al. (2014) Chene, A. -N, et al. 2014, GRACES: Gemini remote access to CFHT ESPaDOnS spectrograph through the longest astronomical fiber ever made: experimental phase completed. p. 915147, doi:10.1117/12.2057417
  * Cui et al. (2012) Cui, X.-Q., Zhao, Y.-H., Chu, Y.-Q., et al. 2012, Research in Astronomy and Astrophysics, 12, 1197
  * D’Antona et al. (2016) D’Antona, F., Vesperini, E., D’Ercole, A., et al. 2016, MNRAS, 458, 2122
  * D’Ercole et al. (2010) D’Ercole, A., D’Antona, F., Ventura, P., et al. 2010, MNRAS, 407, 854
  * Dawson et al. (2013) Dawson, K. S., Schlegel, D. J., Ahn, C. P., et al. 2013, AJ, 145, 10
  * Fernández-Trincado et al. (2016) Fernández-Trincado, J. G., Robin, A. C., Moreno, E., et al. 2016, ApJ, 833, 132
  * Fernández-Trincado et al. (2017) Fernández-Trincado, J. G., Zamora, O., García-Hernández, D. A., et al. 2017, ApJ, 846, L2
  * Fernández-Trincado et al. (2021) Fernández-Trincado, J. G., Beers, T. C., Queiroz, A. B. A., et al. 2021, ApJ, accepted
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
  * Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1
  * Gieles et al. (2018) Gieles, M., Charbonnel, C., Krause, M. G. H., et al. 2018, MNRAS, 478, 2461
  * Gratton et al. (1999) Gratton, R. G., Carretta, E., Eriksson, K., et al. 1999, A&A, 350, 955
  * Green (2018) Green, G. M. 2018, The Journal of Open Source Software, 3, 695
  * Green et al. (2019) Green, G. M., Schlafly, E., Zucker, C., et al. 2019, ApJ, 887, 93
  * Hansen et al. (2015) Hansen, T., Hansen, C. J., Christlieb, N., et al. 2015, ApJ, 807, 173
  * Harris (1996) Harris, W. E. 1996, AJ, 112, 1487
  * Harris (2010) Harris, W. E. 2010, arXiv:1012.3224
  * Helmi et al. (2018) Helmi, A., Babusiaux, C., Koppelman, H. H., et al. 2018, Nature, 563, 85
  * Horta et al. (2021) Horta, D., Mackereth, J. T., Schiavon, R. P., et al. 2021, MNRAS, 500, 5462
  * Ibata et al. (1994) Ibata, R. A., Gilmore, G., & Irwin, M. J. 1994, Nature, 370, 194
  * Ito et al. (2013) Ito, H., Aoki, W., Beers, T. C., et al. 2013, ApJ, 773, 33
  * Jeong et al. (2021) Jeong, Miji, Lee, Y. S., Kim, Y., Beers, T. C. 2021, in prep
  * Koppelman et al. (2019) Koppelman, H. H., Helmi, A., Massari, D., et al. 2019, A&A, 631, L9
  * Lee et al. (2011) Lee, Y. S., Beers, T. C., Allende Prieto, C., et al. 2011, AJ, 141, 90
  * Lee et al. (2013) Lee, Y. S., Beers, T. C., Masseron, T., et al. 2013, AJ, 146, 132
  * Lee et al. (2008a) Lee, Y. S., Beers, T. C., Sivarani, T., et al. 2008a, AJ, 136, 2022
  * Lee et al. (2008b) Lee, Y. S., Beers, T. C., Sivarani, T., et al. 2008b, AJ, 136, 2050
  * Lee et al. (2019) Lee, Y.-W., Kim, J. J., Johnson, C. I., et al. 2019, ApJ, 878, L2
  * Lim et al. (2021) Lim, D., Lee, Y.-W., Koch, A., et al. 2021, ApJ, 907, 47
  * Lind et al. (2011) Lind, K., Asplund, M., Barklem, P. S., et al. 2011, A&A, 528, A103
  * Lind et al. (2012) Lind, K., Bergemann, M., & Asplund, M. 2012, MNRAS, 427, 50
  * Lindegren et al. (2021) Lindegren, L., Klioner, S. A., Hernández, J., et al. 2021, A&A, 649, A2
  * Majewski et al. (2017) Majewski, S. R., Schiavon, R. P., Frinchaboy, P. M., et al. 2017, AJ, 154, 94
  * Majewski et al. (2003) Majewski, S. R., Skrutskie, M. F., Weinberg, M. D., et al. 2003, ApJ, 599, 1082
  * Markwardt (2009) Markwardt, C. B. 2009, Astronomical Data Analysis Software and Systems XVIII, 251
  * Martell & Grebel (2010) Martell, S. L. & Grebel, E. K. 2010, A&A, 519, A14
  * Martell et al. (2016) Martell, S. L., Shetrone, M. D., Lucatello, S., et al. 2016, ApJ, 825, 146
  * Martell et al. (2011) Martell, S. L., Smolinski, J. P., Beers, T. C., et al. 2011, A&A, 534, A136
  * McMillan (2017) McMillan, P. J. 2017, MNRAS, 465, 76
  * Massari et al. (2019) Massari, D., Koppelman, H. H., & Helmi, A. 2019, A&A, 630, L4
  * Mendes de Oliveira et al. (2019) Mendes de Oliveira, C., Ribeiro, T., Schoenell, W., et al. 2019, MNRAS, 1111, 241
  * Murga et al. (2015) Mu rga, M., Zhu, G., Ménard, B., et al. 2015, MNRAS, 452, 511
  * Myeong et al. (2019) Myeong, G. C., Vasiliev, E., Iorio, G., et al. 2019, MNRAS, 488, 1235
  * Pereira et al. (2019) Pereira, C. B., Holanda, N., Drake, N. A., et al. 2019, AJ, 157, 70
  * Piatti (2019) Piatti, A. E. 2019, ApJ, 882, 98
  * Piatti (2020) Piatti, A. E. 2020, A&A, 643, A77
  * Poznanski et al. (2012) Poznanski, D., Prochaska, J. X., & Bloom, J. S. 2012, MNRAS, 426, 1465
  * Purandardas & Goswami (2021) Purandardas, M, & Goswami, A. 2021, ApJ, in press, arXiv:2108.06075
  * Ramírez et al. (2012) Ramírez, I., Meléndez, J., & Chanamé, J. 2012, ApJ, 757, 164
  * Schiavon et al. (2017a) Schiavon, R. P., Johnson, J. A., Frinchaboy, P. M., et al. 2017a, MNRAS, 466, 1010
  * Schiavon et al. (2017b) Schiavon, R. P., Zamora, O., Carrera, R., et al. 2017b, MNRAS, 465, 501
  * Schlegel et al. (1998) Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525
  * Schönrich et al. (2010) Schönrich, R., Binney, J., & Dehnen, W. 2010, MNRAS, 403, 1829
  * Smolinski et al. (2011) Smolinski, J. P., Lee, Y. S., Beers, T. C., et al. 2011, AJ, 141, 89
  * Sneden (1973) Sneden, C. A. 1973, Ph.D. Thesis, The Univ. of Texas at Austin
  * Spite et al. (2014) Spite, M., Spite, F., Bonifacio, P., et al. 2014, A&A, 571, A40
  * Suda et al. (2008) Suda, T., Katsuta, Y., Yamada, S., et al. 2008, PASJ, 60, 1159
  * Tang et al. (2017) Tang, B., Cohen, R. E., Geisler, D., et al. 2017, MNRAS, 465, 19
  * Tang et al. (2021) Tang, B., Wang, Y., Huang, R., et al. 2021, ApJ, 908, 220
  * Thomas et al. (2020) Thomas, G. F., Jensen, J., McConnachie, A., et al. 2020, ApJ, 902, 89
  * Vasiliev (2019) Vasiliev, E. 2019, MNRAS, 484, 2832
  * Wan et al. (2020) Wan, Z., Lewis, G. F., Li, T. S., et al. 2020, Nature, 583, 768
  * Woody & Schlaufman (2021) Woody, T. & Schlaufman, K. C. 2021, ApJ, 162, 42
  * Woosley & Weaver (1995) Woosley, S. E. & Weaver, T. A. 1995, ApJS, 101, 181
  * Yanny et al. (2009) Yanny, B., Rockosi, C., Newberg, H. J., et al. 2009, AJ, 137, 4377
  * Yoon et al. (2016) Yoon, J., Beers, T. C., Placco, V. M., et al. 2016, ApJ, 833, 20
  * Yoon et al. (2018) Yoon, J., Beers, T. C., Dietz, S., et al. 2018, ApJ, 861, 146
  * York et al. (2000) York, D. G., Adelman, J., Anderson, J. E., et al. 2000, AJ, 120, 1579
  * Zucker et al. (2012) Zucker, D. B., de Silva, G., Freeman, K., et al. 2012, Galactic Archaeology: Near-field Cosmology and the Formation of the Milky Way, 421