NLTE Analysis of High-Resolution H-Band Spectra, V: Neutral Sodium

Abstract

In order to derive sodium abundances and investigate the effects of non-local thermodynamic equilibrium (NLTE) on the formation of H-band Na I lines, we update the sodium atomic model by incorporating collision rates with hydrogen from new quantum-mechanical calculations. The differential Na abundances for 13 sample stars are obtained by analyzing high-resolution H-band spectra from the Apache Point Observatory Galactic Evolution Experiment (APOGEE) and optical spectra under both local thermodynamic equilibrium (LTE) and NLTE conditions. Consistent abundances from both bands suggest that our updated atomic model is valid for studying the formation of H-band Na I lines. Our calculations show that, in our stellar parameter space, NLTE effects are negative and can result in corrections larger than −0.4 dex on optical lines. The corrections on H-band Na I lines are typically small, within about 0.05 dex, but not negligible if accurate sodium abundance is desired. We note that the [Na/Fe] ratios favor the theoretical galactic chemical model.

1. Introduction

Detailed chemical abundances are essential for interpreting nucleosynthesis processes in stellar interiors and the chemical evolution of the galaxy [1]. Sodium is of particular interest because it is a primary element, while its production shows significant secondary behavior [2]. Nearly all the sodium is synthesized through hydrostatic carbon fusion in massive stars, but its final survival is sensitive to neutron excess [3]. It can also be synthesized through NeNa cycle (proton-capture reactions) in stars with $M>$ 1.5 $M_{\odot }$ [4,5], $\alpha$-reactions in asymptotic-giant-branch (AGB) stars, and s-process [3,6].

Observational studies have shown that Na I lines are present in stars of wide spectral types and metallicities, making sodium adequate for spectroscopic analyses of the galactic histories (e.g., [7]). Some LTE studies have shown that the galactic [Na/Fe] ratio remains generally solar across metallicities (see, e.g., literature reviewed by [8,9]), while [10,11] reported enhancements of [Na/Fe] ratios at both [Fe/H] ends of their sample stars. The increasing trend with increasing metallicity at the metal-rich end was confirmed by [12,13], while [14] found a mild enhancement of sodium in their thick disk stars with [Fe/H] from −1.2 to −0.3 dex, whereas both [7,15] found deficiencies among metal-poor stars. The behavior of [Na/Fe] remains a subject of controversy and unresolved at present. However, it is worth noting that the majority of these analyses have been conducted under the assumption of local thermodynamic equilibrium (LTE). By performing a non-local thermodynamic equilibrium (NLTE) analysis of sodium in cool metal-poor stars, ref. [16] reported a steady trend of decreasing [Na/Fe] with decreasing metallicity and considerable NLTE corrections for the Na I D lines. Recently, several NLTE studies of the formation of sodium lines have been conducted [13,17,18,19,20,21,22,23,24], which demonstrate that it is imperative to account for NLTE effects for the accurate study of stellar sodium abundances.

Previous abundance studies of sodium are focused on analysis of its optical lines, and are primarily restricted to solar neighbourhood because of dust extinction. However, with the advent of the Apache Point Observatory Galactic Evolution Experiment (APOGEE) survey, the situation has been changed. For example, a few studies (e.g., [25,26]) in recent years have analyzed sodium based on the results or high-resolution H-band spectra from APOGEE. APOGEE is a large-scale spectroscopic survey that aims to obtain high-resolution ($R\sim$ 22,500), high signal-to-noise ratio (S/N ≥ 100), near-infrared H-band spectra data of stars across the Milky Way with fiber-fed multiobject spectrographs located at Apache Point and du Pont Observatories [27]. Details of its instrumental design and performance were provided by [28,29]. Taking advantage of the reduced extinction effect in the H-band, APOGEE is capable of penetrating dust-obscured regions, including particularly low galactic latitudes [30], and can map the chemo-dynamics of all subcomponents of Milky Way galaxy. The seventeenth data release (DR17) from Sloan Digital Sky Surveys (SDSS) contains APOGEE infrared spectra of over 650,000 stars [31], providing a substantial contribution to our knowledge of the universe.

This work is part of a series of NLTE analyses for several key elements related to H-band, together with the in-depth analyses and discussions by [32,33,34,35]. We update the sodium atomic model from quantum-mechanical data on collision rates with neutral hydrogen and charge transfer, which are essential when carrying out NLTE analyses [36]. To study the evolution of the Galaxy with the APOGEE data in later studies, the present investigation aims to validate the comprehensive Na atomic model and to investigate how sodium abundances derived from H-band spectra deviate from LTE.

The organization of this analysis is as follows. In Section 2, we briefly summarize the observations and atmospheric parameters of our sample stars. The method to derive sodium abundances is described in Section 3. The Na abundances derived from the optical and H-band spectra under both LTE and NLTE conditions are presented and discussed in Section 4. Finally, the conclusions are summarized in Section 5.

2. Observations and Parameters

Our sample stars investigated here have been described comprehensively in [32,33]. For better spectral quality, ref. [34] updated the optical high-resolution spectra of nine stars with the Xinglong 2.16 m and APO 3.5 m telescopes. The raw data were reduced using the Interactive Data Language (IDL)-based programs [37] and the Image Reduction and Analysis Facility (IRAF) software [38,39] according to standard spectral reduction procedures. The resolving powers of the spectrographs are between 50,000 and 60,000, and the S/N are above 150. To check the influence of instruments on deriving consistent sodium abundances, HD121370 was observed with both the 1.8 m telescope at Lijiang Observatory and the 2.16 m telescope at Xinglong Observatory. For the H band, we adopt the combined high-resolution spectra from SDSS-IV/APOGEE-2 DR17 with the exception of HD177249, for which we utilize the spectrum and LTE sodium abundance from SDSS DR15 as it is not included in the DR17 catalogue. The spectral SNRs are between 100 and 550.

The fundamental parameters (the effective temperature $T_{\mathrm{eff}}$, the surface gravity log g, the metallicity [Fe/H], and the micro-turbulent velocity ${\xi }_{\mathrm{t}}$) of the program stars are determined through spectroscopic approach and NLTE effects on the iron lines are taken into account. Details about the determination have been described in [32] and references therein. The general uncertainties of $T_{\mathrm{eff}}$, log g, [Fe/H] and ${\xi }_{\mathrm{t}}$ are ±80 K, 0.1 dex, 0.1 dex and 0.2 km s${}^{-1}$, respectively. The stellar parameters are tabulated with sodium abundances, which will be discussed later.

3. Method

We perform spectral synthesis techniques to determine LTE and NLTE sodium abundances. This approach interactively identifies the optimal matches between the observed and synthetic spectra. The synthetic line profiles are computed with an IDL/Fortran-based program of Spectrum Investigation Utility (SIU, [40]). During the fitting procedures, free parameters are sodium abundance and a Gaussian broadening profile, which accounts for broadenings caused by instrument, rotation, and macroturbulence. Besides the stellar spectra and their parameters, three other crucial aspects are necessarily designated to accurately measure the Na abundances:

• Stellar Model atmospheres.
• Sodium atomic model.
• Sodium line data.

3.1. Stellar Model Atmospheres

All the stellar model atmospheres in this study are interpolated based on grids downloaded from the 1D MARCS-OS atmosphere database1. The commonly used MARCS-OS model atmospheres are line-blanketed LTE model grids and computed either in plane-parallel or spherically symmetrical geometries [41]. To interpolate models for our sample stars, the FORTRAN-based routine2 written by Thomas Masseron is used. During the interpolation process, the geometry and $\alpha$-enhancement are adopted according to [32].

3.2. Sodium Model Atom

The sodium atomic model employed here is updated from the one that has been presented in detail by [13], including how to treat fine structures and data on level energies, radiative transitions, and collisions with electrons. It consists of 58 Na I energy levels and the Na II ground state. The accuracy of NLTE analyses heavily rely on the completeness of the model atoms. Especially in late-type stellar atmospheres, interactions with hydrogen influence statistical equilibrium (e.g., [36,42]) and are important for strong lines (e.g., [13,20,43]). However, the classical Drawinian formula overestimates the rates [44,45]. We update the quantum-mechanical-based rate coefficients for inelastic collisions with neutral hydrogen and charge exchange processes from [46].

All calculations including the statistical equilibrium, the radiative transfer, and the departure coefficients of energy levels are conducted with a revised version of the DETAIL program [47] using accelerated $\mathrm{\Lambda }$-iteration (see [48] and references therein for details).

Figure 1 shows the distribution of departure coefficients3 with log ${\tau }_{5000}$4 for HD58367. It is found that the NLTE corrections of Na I result from overpopulation of lower levels and the departure coefficients for the H-band upper levels are near thermal. Taking the line pair 6154/6160 as an example, we denote $b_i$ and $b_j$ as the departure coefficients of the lower and upper sub-levels, respectively. Notably, the lower sub-level has $b_i>$ 1, and in $S_{ji}\approx \frac{b_j}{b_i}{B}_{\nu }\left(T\right)$, it is evident that $b_i$ is larger than $b_j$. Both factors contribute to increasing line intensities, leading us to expect that NLTE will enhance the line profiles and NLTE corrections would be negative.

3.3. Line Data

In our research, we utilize the optical sodium line list from [13] and calculate the van der Waals damping constant log $C_6$ based on the dataset featured in [43]. The H-band line table is sourced from [49], and alterations are made to log $gf$ according to the optical mean abundances. All the adopted sodium line data are listed in

Table 1. Sodium line data.

$\mathit{\lambda }$/Å	Transition	$\mathit{\chi }$/eV	log gf	log C6
Optical Na I
5682.633	$3{\mathrm{p}}^2{\mathrm{P}}_{1/2}^{\circ }-4{\mathrm{d}}^2{\mathrm{D}}_{3/2}$	2.102	−0.720	−29.426
5688.205	$3{\mathrm{p}}^2{\mathrm{P}}_{3/2}^{\circ }-4{\mathrm{d}}^2{\mathrm{D}}_{5/2}$	2.104	−0.470	−29.426
5889.951	$3{\mathrm{s}}^2{\mathrm{S}}_{1/2}-3{\mathrm{p}}^2{\mathrm{P}}_{1/2}^{\circ }$	0.000	0.110	−31.120
5895.924	$3{\mathrm{s}}^2{\mathrm{S}}_{1/2}-3{\mathrm{p}}^2{\mathrm{P}}_{3/2}^{\circ }$	0.000	−0.184	−31.120
6154.226	$3{\mathrm{p}}^2{\mathrm{P}}_{1/2}^{\circ }-5{\mathrm{s}}^2{\mathrm{S}}_{1/2}$	2.102	−1.570	−29.426
6160.747	$3{\mathrm{p}}^2{\mathrm{P}}_{3/2}^{\circ }-5{\mathrm{s}}^2{\mathrm{S}}_{1/2}$	2.104	−1.280	−29.426
8183.255	$3{\mathrm{p}}^2{\mathrm{P}}_{1/2}^{\circ }-3{\mathrm{d}}^2{\mathrm{D}}_{3/2}$	2.102	0.280	−30.381
8194.824	$3{\mathrm{p}}^2{\mathrm{P}}_{3/2}^{\circ }-3{\mathrm{d}}^2{\mathrm{D}}_{5/2}$	2.104	0.490	−30.381
H-band Na I
16,373.853	$4{\mathrm{p}}^2{\mathrm{P}}_{1/2}^{\circ }-6{\mathrm{s}}^2{\mathrm{S}}_{1/2}$	3.753	−1.400	−29.208
16,388.857	$4{\mathrm{p}}^2{\mathrm{P}}_{3/2}^{\circ }-6{\mathrm{s}}^2{\mathrm{S}}_{1/2}$	3.753	−1.040	−29.208

. The transition parameters of nearby lines are taken from the National Institute of Standards and Technology (NIST) Atomic Spectra Database5 and the Vienna Atomic Line Database6 (VALD).

The accuracy of measuring elemental abundances in infrared spectral region may be affected due to potential interference by molecule lines. APOGEE employs molecular features including predominantly CN, CO, and OH to derive individual abundances. Detailed information on the construction of the APOGEE line list can be found in [50], and the recent updates to the line list have been described in [51]. We adopt the H-band molecular line list7 near our target line features used by the APOGEE Stellar Parameter and Chemical Abundances Pipeline (ASPCAP) for DR17.

4. Results and Disscussion

4.1. Sodium Abundances

The optical and H-band sodium abundances are obtained under both LTE and NLTE conditions.

Table 2. Stellar parameters and obtained sodium abundances.

Name	Teff	log g	[Fe/H]	${\mathit{\xi }}_{\mathbf{t}}$	spec.	Opt.-Band		H-Band
	(K)			(km s${}^{-1}$)	Type	LTE	NLTE	LTE	NLTE	ASPCAP
Sun	5777	4.44	0.00	0.90	G2V	6.30 ± 0.05	6.21 ± 0.04	6.22 ± 0.01	6.21 ± 0.00	…
Arcturus${}^H$	4275	1.67	−0.58	1.60	K1.5III	0.11 ± 0.11	0.01 ± 0.07	0.05 ± 0.03	0.05 ± 0.02	…
Arcturus${}^A$	4275	1.67	−0.58	1.60	K1.5III	0.11 ± 0.11	0.01 ± 0.07	0.06 ± 0.04	0.05 ± 0.03	0.05
HD87	5053	2.71	−0.10	1.35	G5III	0.15 ± 0.07	0.07 ± 0.05	0.12 ± 0.01	0.11 ± 0.03	−0.05
HD6582	5390	4.42	−0.81	0.90	G5Vb	0.08 ± 0.03	0.10 ± 0.02	…	…	−1.63
HD6920	5845	3.45	−0.06	1.40	F8V	0.05 ± 0.12	−0.04 ± 0.02	−0.01	−0.01	−1.20
HD22675	4901	2.76	−0.05	1.30	G8.5IIIb	0.08 ± 0.08	0.01 ± 0.05	0.02	0.02	−0.73
HD31501	5320	4.45	−0.40	1.00	G8V	0.09 ± 0.06	0.10 ± 0.05	0.14	0.15	0.03
HD58367	4932	1.79	−0.18	2.00	G6IIb	0.39 ± 0.10	0.27 ± 0.04	0.29 ± 0.05	0.26 ± 0.05	0.19
HD67447	4933	2.17	−0.05	2.12	G7IIb	0.32 ± 0.03	0.16 ± 0.03	0.17 ± 0.01	0.16 ± 0.01	0.16
HD102870	6070	4.08	0.20	1.20	F9V	−0.03 ± 0.08	−0.06 ± 0.04	0.00	0.00	0.07
HD103095	5085	4.65	−1.35	0.80	K1V	−0.25 ± 0.05	−0.22 ± 0.04	…	…	0.48
HD121370	6020	3.80	0.28	1.40	G0IV	0.32 ± 0.06	0.25 ± 0.04	0.20 ± 0.02	0.19 ± 0.01	0.43
HD148816	5830	4.10	−0.73	1.40	F7V	0.09 ± 0.06	0.05 ± 0.05	…	…	0.42
HD177249	5273	2.66	0.03	1.65	G6IIb	0.25 ± 0.08	0.12 ± 0.05	0.12 ± 0.00	0.10 ± 0.01	0.14

Arcturus${}^H$: H-band abundances of Arcturus are derived based on the spectrum from [52]. Arcturus${}^A$: H-band abundances of Arcturus are derived based on the APOGEE spectrum. spec. type: The spectral types are from SIMBAD website. ASPCAP: the NLTE sodium abundances from ASPCAP. All the data are from SDSS DR17 except HD177249, for which the LTE sodium abundance is from DR15.

provides all the determined mean abundances along with the standard deviations for our sample stars. The individual line abundances are listed in

Table A1. Individual sodium line abundances.

Star	5682		5688		5889		5895		6154		6160		8183		8194		16,373		16,388
	LTE	NLTE	LTE	NLTE	LTE	NLTE	LTE	NLTE	LTE	NLTE	LTE	NLTE	LTE	NLTE	LTE	NLTE	LTE	NLTE	LTE	NLTE
Sun	6.29	6.19	6.31	6.21	6.23	6.16	6.23	6.17	6.31	6.27	6.31	6.26	6.4	6.23	6.33	6.19	6.21	6.21	6.22	6.21
Arcturus${}^H$	0.14	−0.0	0.14	0.0	−0.11	−0.08	…	…	0.11	0.03	0.14	0.03	0.12	−0.03	0.26	0.14	0.07	0.06	0.03	0.03
Arcturus${}^A$	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	…	0.08	0.07	0.03	0.03
HD87	0.12	−0.01	0.19	0.03	0.05	0.08	0.07	0.05	0.19	0.12	0.24	0.13	0.2	0.06	…	…	0.11	0.09	0.13	0.13
HD6582	0.06	0.11	0.03	0.06	0.11	0.09	…	…	0.1	0.11	0.1	0.11	…	…	…	…	…	…	…	…
HD6920	−0.03	−0.07	0.06	−0.04	−0.08	−0.04	0.02	−0.05	−0.03	−0.05	−0.01	−0.05	0.21	−0.02	0.24	0.01	…	…	−0.01	−0.01
HD22675	0.06	−0.06	0.18	0.06	−0.04	−0.01	−0.01	−0.02	0.15	0.07	0.14	0.03	0.11	0.02	…	…	…	…	0.02	0.02
HD31501	0.1	0.12	0.06	0.08	0.03	0.08	0.1	0.09	0.22	0.21	0.13	0.13	0.02	0.04	0.09	0.08	…	…	0.14	0.15
HD58367	0.39	0.2	0.46	0.25	0.29	0.31	0.29	0.3	0.36	0.27	0.39	0.25	0.57	0.3	0.71	0.52	0.32	0.3	0.25	0.23
HD67447	0.3	0.14	0.32	0.13	…	…	…	…	0.29	0.19	0.33	0.19	0.37	0.15	0.62	0.47	0.18	0.17	0.17	0.16
HD102870	−0.05	−0.08	0.06	−0.02	−0.15	−0.1	−0.09	−0.11	−0.06	−0.07	−0.06	−0.08	0.07	0.01	0.04	−0.04	…	…	0.0	0.0
HD103095	−0.29	−0.2	−0.29	−0.22	−0.24	−0.24	−0.21	−0.27	−0.18	−0.16	−0.27	−0.24	…	…	…	…	…	…	…	…
HD121370	0.4	0.31	0.37	0.28	0.24	0.21	0.26	0.22	0.32	0.25	0.28	0.2	0.37	0.26	0.36	0.28	0.21	0.2	0.18	0.18
HD148816	0.06	0.09	0.1	0.1	−0.03	0.01	0.06	−0.02	0.1	0.09	0.07	0.08	0.17	0.05	0.16	0.01	…	…	…	…
HD177249	0.26	0.09	0.26	0.07	0.13	0.16	0.16	0.18	0.26	0.17	0.28	0.15	0.36	0.06	0.32	0.12	0.12	0.1	0.12	0.11

Arcturus${}^H$ and Arcturus${}^A$ refer to the same as those in Table 2.

.

Representative of the general profile fittings are shown in Figure 2 for Arcturus in optical and H bands. In Panel (a), the line intensity is so strong that the LTE assumption shows sign of saturation and does not represent the observation when modelling the spectrum, while NLTE fitting provides a better representation. In this case, the LTE abundance is determined by equating its equivalent width (EW) with that obtained from NLTE calculation. Panels (b) and (c) show the profile fittings to the Fourier Transform Spectrometer (FTS) and APOGEE spectra of Arcturus for the same line in the H band. It is obvious that the line intensity is relatively weak and both syntheses can represent the observations well.

In astrophysics, the reliability of elemental abundance determinations depends on various factors, such as the quality of the observed spectra and the accuracy of the theoretical models. Discussions concerning the optical sodium abundances have been extensively drawn (e.g., [13,19,22]), this work focuses on the H-band lines. Based on the parameter spaces of our sample stars and the line fittings, the H-band sodium abundances from stars with metallicities lower than −0.73 dex can not be fitted. Because the lines are so weak that abundances may not be reliable and may come from fitting noises rather than from fitting the actual spectral lines.

4.2. Abundance Uncertainties

Uncertainties can affect the accuracy of the methodology for the determination of stellar chemical abundances. Therefore, it is important to discuss the potential uncertainties associated with the sodium abundance measurement. In this study, several potential sources of uncertainties can involve: (1) the instruments, (2) uncertainties in the derived stellar atmosphere parameters, (3) the fitting procedures.

Different instruments have different responses and spectral resolutions, which may affect the measurement of line strengths. This study uses spectra observed with several telescopes. To check the possible effects in determining sodium abundances, we analyze the optical spectra of HD121370 from both the Lijiang 1.8 m and the Xinglong 2.16 m telescopes, and compare their NLTE abundances. The NLTE abundance differences derived from individual lines are given in

Table 3. Uncertainties from instruments.

$\mathit{\lambda }$/Å	$\mathbf{\Delta }\mathbf{Na}$
5889.951	0.03
5895.924	0.00
6154.226	0.00
6160.747	0.00

$\Delta \mathrm{Na}$: Sodium abundance differences between NLTE abundances determined with spectra of HD121370 obtained by 2.16 m and 1.8 m telescopes. We note that the reason for comparing only four lines here is due to the narrower spectral wavelength coverage (from 5740 to 7940 Å) of the 1.8 m telescope.

. The consistent abundances indicate negligible effects from instruments on the sodium abundances determination in this study.

Uncertainties in the stellar atmosphere parameters can propagate into the determination of stellar abundances. We carefully evaluate the uncertainties associated with the adopted stellar parameters by conducting a shift to the four parameters separately. How the uncertainties affect the sodium abundances of the Sun, Arcturus and HD58367 are shown in

Table 4. Uncertainties from the stellar parameters.

Star	Opt.-Band					H-Band
	$\Delta$Teff	$\Delta$log g	$\Delta$[Fe/H]	$\Delta$${\mathbf{\xi }}_{\mathrm{t}}$	Total	$\Delta$Teff	$\Delta$log  g	$\Delta$[Fe/H]	$\Delta$${\mathbf{\xi }}_{\mathrm{t}}$	Total
	80 K	0.1 dex	0.1 dex	0.2 km s${}^{-\mathbf{1}}$		80 K	0.1 dex	0.1 dex	0.2 km s${}^{-\mathbf{1}}$
Sun	0.04	0.00	−0.09	−0.01	0.10	0.03	0.01	−0.10	0.00	0.10
Arcturus	0.07	0.00	−0.10	−0.02	0.12	0.04	0.00	−0.10	0.00	0.11
HD58367	0.05	0.00	−0.10	−0.01	0.11	0.04	0.00	−0.09	0.00	0.10

Uncertainties in optical and H bands are determined from lines at 6154.226 and 16,388.857 Å, respectively. The total uncertainties are calculated on the assumption that uncertainties in atmospheric parameters are independent.

. The total uncertainty is approximately 0.11 dex. As has been discussed by [19,22], the accuracy of sodium abundance measurements is less dependent on $T_{\mathrm{eff}}$ and log g.

Finally, it is important to carefully test and consider the sensitivity of sodium abundances to the potential uncertainties from the spectral fitting processes. One potential uncertainty stems from determining the best fit for the observed spectra. In checking this case, sodium abundance alterations of ±0.05 dex are made to see how the best fit changes. The left panel of Figure 3 illustrates the best fit changes after the alterations and suggests that this fitting uncertainty is reliable within 0.05 dex for APOGEE spectra to some extent. Another potential uncertainty arises from placement of spectral continuum. In the line-profile fitting procedures, continuum placement is an important step for chemical composition determinations, as it involves identifying the unit where the spectral lines to be measured are located. Accurate placement is crucial especially for weak lines, as the unit level can significantly affect the strengths in percentage. To examine the potential effects, shifts of ±0.01 are applied to the normalized spectrum for the H-band line at 16,388 Å, and the resulting abundances are compared. The fitting results are illustrated in the middle and right panels of Figure 3. As we can see, this shift is large enough to judge whether the local continuum is placed well. Offsets of −0.18 and 0.14 dex are required to compensate the ±1% shifts in the continuum.

4.3. Abundance Comparisons

Abundance comparisons are made for our determined sodium abundances in mainly three ways: optical solar abundance comparison with other work, abundance consistency between the optical and H bands, and abundance comparison with the results of ASPCAP.

First, we aim to compare our solar optical results with the values reported in the literature. Based on the line-profile abundance determination and the updated quantum collision rates with hydrogen, our eight optical lines yield a NLTE solar sodium abundance of 6.21 ± 0.04. Our analysis confirms the consistency of our solar value with the sodium abundances reported by ([53,54] 6.24 ± 0.04 and 6.22 ± 0.03, respectively). However, when compared with other studies we do observe some differences. For example, our solar abundance is slightly lower than that given in the CI chondritic meteorite [55]. Additionally, the sodium abundance we obtained is higher than the value reported by [56], which was 6.11 ± 0.11. The authors of [57] conducted simultaneous NLTE calculations of mutiple elements for APOGEE, including Na, Mg, K, and Ca, and reported a solar abundance of 6.15. APOGEE uses solar Na abundance of 6.17 from [58], which is the same given by [59].

When studying the abundance of sodium in stars, it is important to ensure consistency between different spectral regions. To validate the adopted sodium atomic model, Figure 4 displays the results of our analysis, comparing the sodium abundances derived from the optical and H-band spectra. Overall, the NLTE abundances provide a better match between the optical and H bands than the LTE results. In fact, our analysis shows that the NLTE abundance differences are within ±0.1 dex. It should be noted that the H-band spectral line data initially used in this work are obtained from [49]. The values of log $gf$ are adjusted from being compared to the optical NLTE abundances. Our derived log $gf$ for 16,373 Å is slightly lower in comparison to the data used by [25,57], while our log $gf$ for 16,388 Å is almost identical to theirs.

It is worthwhile to compare our manually derived abundances with those obtained by ASPCAP since both are NLTE abundances derived from the same observed spectra. ASPCAP is an automated pipeline that determines the stellar parameters and up to 24 chemical abundances by identifying the optimal match between the observed and synthetic spectral grids. To obtain a general description of ASPCAP and its modifications across SDSS DRs, we recommend readers to some of the literature, including but not limited to [60,61,62,63]. The ASPCAP sodium abundances are given in the last column of Table 2. The differences between the mean H-band [Na/Fe] ratios derived by our work and those from ASPCAP are shown against the metallicity in Figure 5. It is seen that the differences are within about ±0.2 dex except for two stars, HD22675 and HD6920, which deviate 0.75 and 1.21 dex, respectively. Their stellar parameters used by ASPCAP are examined and found to be 4974/2.79/−0.01/1.15 and 5772/3.54/−0.08/1.66. Although our temperatures differ by nearly 70 K, this alone can not account for the significant differences, which can also be indicated by Table 4. The H-band line fittings are also checked and plotted in Figure 6. Abundances reported in the literature (e.g., [64,65]) are not as low as those obtained by ASPCAP, but are comparable to ours. We suggest that one possible cause of the observed discrepancy may be related to differences in continuum placement. As depicted in Figure 2, Figure 3 and Figure 6, the sodium lines in the H-band spectra are rather weak, thus requiring careful attention to continuum determination to avoid significant impacts on the final Na abundances.

4.4. NLTE Effects

The NLTE effects on spectral lines are not uniform and vary depending on the line properties. Refs. [13,18] conducted extensive sodium NLTE analyses, including the impact of NLTE effects on the optical lines. Generally, the NLTE effects of the optical line pairs 5682/5688 and 6154/6160 are relatively small, while the line pairs 5889/5895 and 8183/8194 are important for determining the sodium abundances of metal-poor stars and require consideration of NLTE corrections.

To explore the dependence of H-band NLTE corrections on stellar parameters, the relationships for the Na I line at 16,388 Å are presented in Figure 7. Regarding this line, the NLTE corrections are small and do not exceed −0.05 dex. The negative NLTE corrections observed for the H-band lines suggest that the sodium abundance may be overestimated when the LTE hypothesis is considered. It appears that the impact of NLTE effects is more linked to log g than $T_{\mathrm{eff}}$ and [Fe/H]. The left panel reveals that there is no distinct trend between NLTE corrections and $T_{\mathrm{eff}}$. Stars with $T_{\mathrm{eff}}$ near 4900 K have roughly similar metallicities, and their NLTE corrections are mainly caused by log g. The middle panel illustrates a subtle general increase in the NLTE corrections for the sodium abundance of stars with similar metallicities as log g decreases, particularly with a log g below approximate 2.8. While for stars with a log g higher than 3.0, the NLTE corrections tend to remain at the same level. The right panel does not display a clear trend for the NLTE corrections with respect to [Fe/H]. However, it shows a decrease as [Fe/H] decreases for stars with similar log g. This may occur due to the the relatively decreased line intensity with decreasing metallicity.

4.5. Evolutionary Trend of [Na/Fe]

Here, we specifically investigate how NLTE sodium abundance ratios vary with respect to the stellar metallicity, and compare our results with Galactic Chemical Evolution (GCE) models. Our abundances and two GCE models from [1,66] are presented in Figure 8. The results are consistent with the data provided in Figure 10 of [23]. Our results show a consistency in the [Na/Fe] ratios as a function of [Fe/H] for stars with both optical and H-band measurements. It is insufficient to reproduce the increasing trend for metal-rich part (see e.g., [10]) due to the lack of stars with [Fe/H] exceeding 0.1 dex. Despite the limited data points in the metal-poor range, a decreasing trend in optical NLTE [Na/Fe] ratios with decreasing metallicity seems emerging. It indicates the so-called secondary effect showing dependence of Na production on the metallicity of progenitors.

Numerous studies (see e.g., [8]) have been conducted to model the GCE of elements including sodium, using a variety of observational data and theoretical calculations to explore chemical distribution and formation history. Due to the lack of sufficient data, we are unable to make comparisons for stars with metallicity below −1.0 dex. It appears that our [Na/Fe] ratios are generally consistent with the model from [1] when [Fe/H] exceeds −1.0 dex. However, additional data and investigations are needed to fully understand the behavior of sodium. For a more comprehensive understanding of the [Na/Fe] ratio distributions, we suggest referring to the more detailed NLTE analyses conducted by [13,23] for metal-rich stars, and [23,24,67,68,69] for (extremely) metal-poor stars.

5. Conclusions

The sodium atomic model is updated with inelastic collisions of Na I + H I and Na${}^{+}$ + H${}^{-}$. Our aim is to inspect: (1) the validity of the adopted model atom and extending our NLTE methodology to APOGEE H-band spectra; (2) the NLTE corrections and their variations with respect to stellar parameters in H band. For these purposes, we conducted a systematic investigation respecting the determinations of photospheric sodium abundances for 14 giants and dwarfs (plus the Sun) through V- and H-band spectra under LTE and NLTE assumptions. Our results lead us to the following conclusions:

• The consistent sodium abundances between H-band and optical lines for our sample stars suggest that our Na atomic model is valid for studying the formation of the H-band Na I lines.
• The NLTE effects vary across different lines. The corrections for our selected optical lines could be as large as around −0.4 dex, while the NLTE effects for H-band lines, although relatively small, cannot be negligible. It is observed that the impact of NLTE effects is more associated with log g than $T_{\mathrm{eff}}$ and [Fe/H] in H band.
• The [Na/Fe] ratios are consistent with the theoretical GCE model and may indicate a secondary effect.
• For stars with low metallicities, caution should be exercised in obtaining sodium abundances with the APOGEE spectra.