He-rich Hot Subdwarf Stars Observed in Gaia DR3 and LAMOST DR7: Carbon and Nitrogen Abundances and Kinematics

The sample studied in this investigation was derived from the catalogs of spectroscopically identified hot subdwarf stars listed in Gaia DR2, combined with spectra from LAMOST survey releases DR5, DR6, and DR7 (Lei et al. 2018, 2019,2020; Luo et al. 2019, 2021). Initially, 350 He-rich hot subdwarf stars were collected from these catalogs. Subsequently, the sample was expanded to encompass entries from the catalog of known hot subdwarf stars (Geier 2020), which provides crucial spectroscopic parameters such as effective temperatures, surface gravities, and helium abundances for 2187 hot subdwarf stars. Cross-referencing this catalog with LAMOST DR7 led to the inclusion of an additional 41 He-rich hot subdwarf stars in the analyzed sample.

After crossmatching the selected sample from Gaia DR3 and LAMOST DR7, a total of 391 He-rich hot subdwarf stars were identified in both Gaia DR3 and LAMOST DR7. To investigate C and N abundances, the final sample was refined to include 210 He-rich hot subdwarf stars with LAMOST spectra having a signal-to-noise ratio (S/N) greater than 25 in the g band. Figure 1 shows the positions of this sample in the Gaia H-R diagram.

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In our previous work (Luo et al. 2021), we demonstrated that hot subdwarf stars observed in the LAMOST DR7 and Gaia DR2 samples, located between 500 and 1500 pc away, exhibit a very similar distribution function in their absolute Gaia magnitudes. We proposed that the LAMOST DR7 hot subdwarf star sample is a representative and unbiased subsample of the volume-complete Gaia sample. Figure 2 illustrates the nearly identical distribution functions for all He-rich hot subdwarfs in LAMOST DR7 and our final sample concerning absolute Gaia magnitudes (M_G) and distances.

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We utilized the spectra from LAMOST DR7 to determine the atmospheric parameters of the selected objects. These parameters include effective temperature T_eff, surface gravity $\mathrm{log}g$, He abundance $\mathrm{log}(n\mathrm{He}/n{\rm{H}})$, C abundance $\mathrm{log}(n{\rm{C}}/n{\rm{H}})$, and N abundance $\mathrm{log}(n{\rm{N}}/n{\rm{H}})$ and radial velocity. The LAMOST spectra, akin to Sloan Digital Sky Survey data, have a spectral resolution of R ∼ 1800 and cover a wavelength range from 3750 to 9100 Å. More detailed information about the data can be found at http://www.lamost.org/dr7/v2.0/.

Gaia DR3 (Gaia Collaboration et al. 2023) provided precise positions (α and δ), proper motions (${\mu }_{\alpha }\cos \delta$ and μ_δ ), and parallaxes ($\bar{\omega }$), as well as three broadband magnitudes (G, G_BP, and G_RP) for all 210 He-rich hot subdwarf stars. Distances (D) were computed by simply inverting the parallax. For two stars with unreliable parallaxes and 48 stars with a parallax_over_error < 5, their distances were substituted with the estimated values from the Gaia DR3 distance catalog (Bailer-Jones et al. 2021).

Atmospheric parameters were derived by fitting non–local thermodynamic equilibrium Tlusty (v207) and Synspec (v53) (Hubeny & Lanz 1995, 2017) models to LAMOST spectra. We employed the iterative data-driven spectral analysis procedure XTgrid (Németh et al. 2012) to optimize all free parameters simultaneously and determine the most suitable parameter values that provide the best fit. The models included H, He, C, and N chemical elements self-consistently in both the atmospheric model calculations and spectrum synthesis. Details regarding the ionization stages for which detailed model atoms are used in the atmospheric model calculations can be found in Table 1. To ascertain the effective temperature and surface gravity and concurrently look for the abundances of He, C, and N, we employed a global minimization without individually tracing the surface parameters or the ionization balance of separate ions. The global minimization assumes that a single model is able to reproduce the observation. This approach ensures that all parameter correlations and interconnections are considered. However, at the same time, it is sensitive to inconsistencies in or a lack of atomic data, which becomes apparent only in the hottest stars of our sample. Figure 3 shows a global fit to a single spectrum, and Figure 4 displays fits to the LAMOST LRS spectra of eight hot subdwarf stars with T_eff ranging from 30,000 to 51,000 K. The LAMOST LRS observations encompass most of the reliable diagnostic lines of the He, C, and N ions.

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All models were calculated in real time, adjusting the model parameters through comparison of the synthetic spectra with the observed spectra within the 3780–6850 Å wavelength range. The synthetic spectra were convolved with an instrumental profile to match the resolution of the LAMOST data and normalized to the flux of the LAMOST spectra within 80 Å segments. Additionally, the radial velocity was determined by measuring the Doppler shift between the observed data and the model.

As also described by Schindewolf et al. (2018), many C and/or N lines are blended with He lines in He-rich hot subdwarfs. Metal lines are often occurring in the wings of H/He lines (see Figure 3), and these overlaps significantly impact the determination of the atmospheric parameters from low-resolution spectroscopy such as the LAMOST LRS. Table 2 lists the spectral lines that contributed the most in determining the atmospheric parameters. In instances when multiple ionization stages are present, such as the occurrence of both C ii and C iii lines, the fitting of these lines yields crucial insights into the C abundance and the atmospheric conditions, particularly of T_eff stratification.

XTgrid iterates the global minimization until all model parameter variations and the chi-square variation remain below 0.5% over three consecutive iterations. Next, one-dimensional parameter errors are calculated by mapping the chi-square surface near the best solution until the 1σ confidence limit is reached. Atmospheric models and synthetic spectra are calculated self-consistently in the error analysis.

In Table 3, we have compiled the effective temperature T_eff, surface gravity $\mathrm{log}g$, He abundance $\mathrm{log}(n\mathrm{He}/n{\rm{H}})$, C abundance $\mathrm{log}(n{\rm{C}}/n{\rm{H}})$, N abundance $\mathrm{log}(n{\rm{N}}/n{\rm{H}})$, and radial velocity RV. Their errors are 1σ statistical errors. The typical values of the errors are 499 K, 0.09, 0.06, 0.19, and 0.23 for T_eff, $\mathrm{log}g$, $\mathrm{log}(n\mathrm{He}/n{\rm{H}})$, $\mathrm{log}(n{\rm{C}}/n{\rm{H}})$, and $\mathrm{log}(n{\rm{N}}/n{\rm{H}})$. However, the error budget is dominated by systematic uncertainties. The average systematic uncertainties were estimated by comparing our results to those obtained from independent analyses of high-resolution spectra. Five of our target stars (J092440.10+305013.1 and J160131.27+044027.0, Naslim et al. 2020; J091251.66+272031.4, Wild & Jeffery 2018; J113726.37+141014.1, Latour et al. 2019, Dorsch et al. 2020; and J132335.26+360759.5, Dorsch et al. 2019) were observed with high-resolution spectroscopy. Figure 5 presents a comparison between the atmospheric parameters of these stars from our study and the results of the analysis at high spectral resolution. The average systematic shifts are −112 ± 1950 K, −0.10 ± 0.12 cm s⁻², −0.03 ± 0.14, −0.14 ± 0.41, and 0.14 ± 0.43 for T_eff, $\mathrm{log}g$, $\mathrm{log}(n\mathrm{He}/n{\rm{H}})$, $\mathrm{log}(n{\rm{C}}/n{\rm{H}})$, and $\mathrm{log}(n{\rm{N}}/n{\rm{H}})$, respectively. These comparisons indicate that our results are consistent with published parameters. However, for one star, J160131.27+044027.0 (Naslim et al. 2020), we found a larger discrepancy in the value of $\mathrm{log}g$, which is most likely due to the missing blue part of the spectrum. The processed observation covers only the 4400–6800 Å range.

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Table 3. Atmospheric Parameters, Galactic Velocities, and Orbital Parameters for 210 He-rich Hot Subdwarf Stars Observed in Gaia DR3 and LAMOST DR7

No.	Label	Explanations
1	LAMOST	LAMOST target
2	RAdeg	Barycentric R.A. (J2000) a
3	DEdeg	Barycentric decl. (J2000) a
4	Teff	Stellar effective temperature
5	e_Teff	Standard error in Teff
6	$\mathrm{log}g$	Stellar surface gravity
7	$e\_\mathrm{log}g$	Standard error of stellar surface gravity
8	$\mathrm{log}(n\mathrm{He}/n{\rm{H}})$	Stellar surface He abundance
9	$e\_\mathrm{log}(n\mathrm{He}/n{\rm{H}})$	Standard error in $\mathrm{log}(n\mathrm{He}/n{\rm{H}})$
10	$\mathrm{log}(n{\rm{C}}/n{\rm{H}})$	Stellar surface C abundance
11	$e\_\mathrm{log}(n{\rm{C}}/n{\rm{H}})$	Standard error in $\mathrm{log}(n{\rm{C}}/n{\rm{H}})$
12	$\mathrm{log}(n{\rm{N}}/n{\rm{H}})$	Stellar surface N abundance
13	$e\_\mathrm{log}(n{\rm{N}}/n{\rm{H}})$	Standard error in $\mathrm{log}(n{\rm{N}}/n{\rm{H}})$
14	$\mathrm{log}{\beta }_{\mathrm{He}}$	Logarithmic mass fraction of surface He abundance b
15	$e\_\mathrm{log}{\beta }_{\mathrm{He}}$	Standard error in $\mathrm{log}{\beta }_{\mathrm{He}}$
16	$\mathrm{log}{\beta }_{{\rm{C}}}$	Logarithmic mass fraction of surface C abundance b
17	$e\_\mathrm{log}{\beta }_{{\rm{C}}}$	Standard error in $\mathrm{log}{\beta }_{{\rm{C}}}$
18	$\mathrm{log}{\beta }_{{\rm{N}}}$	Logarithmic mass fraction of surface N abundance b
19	$e\_\mathrm{log}{\beta }_{{\rm{N}}}$	Standard error in $\mathrm{log}{\beta }_{{\rm{N}}}$
20	vsini	Projected rotational velocity, $v\sin i$
21	e_vsini	Standard error in $v\sin i$
22	pmRA	Gaia EDR3 proper motion in R.A.
23	e_pmRA	Standard error in pmRA
24	pmDE	Gaia EDR3 proper motion in decl.
25	e_pmDE	Standard error in pmDE
26	D	Gaia EDR3 stellar distance
27	e_D	Standard error in stellar distance
28	RVel	Radial velocity from LAMOST spectra
29	e_RVel	Standard error in radial velocity
30	U	Galactic radial velocity positive toward Galactic center
31	e_U	Standard error in U
32	V	Galactic rotational velocity along Galactic rotation
33	e_V	Standard error in V
34	W	Galactic velocity toward north Galactic pole
35	e_W	Standard error in W
36	Rap	Apocenter radius c
37	e_Rap	Standard error in Rap
38	Rperi	Pericenter radius c
39	e_Rperi	Standard error in Rperi
40	${z}_{\max }$	Maximum vertical height c
41	$e\_{z}_{\max }$	Standard error in ${z}_{\max }$
42	e	Eccentricity c
43	e_e	Standard error in e
44	Jz	Z-component of angular momentum c
45	e_Jz	Standard error in Jz
46	Pops	Population classification d
47	PTH	Probability in thin disk
48	PTK	Probability in thick disk
49	PH	Probability in halo

Notes.

^a At epoch 2000.0 (ICRS). ^b $\mathrm{log}{\beta }_{{\rm{i}}}=\tfrac{{m}_{{\rm{i}}}}{{m}_{{\rm{H}}}+{m}_{\mathrm{He}}+{m}_{{\rm{C}}}+{m}_{{\rm{N}}}}.$ ^c Form the numerical orbit integration. ^d H = halo; TK = thick disk; TH = thin disk.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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We performed a comparative analysis of the atmospheric parameters for 15 stars within our sample, comparing them with those reported in a low-resolution spectroscopic survey of hot subluminous stars in the Galaxy Evolution Explorer survey (Németh et al. 2012). The comparisons of atmospheric parameters are presented in Figure 5, where objects indicating upper limits are denoted by blue arrows pointing downward. The average systematic shits are 700 ± 3554 K, −0.39 ±0.34 K, −0.00 ± 0.44 K, −0.46 ± 0.55 K, and −0.40 ±0.71 K for T_eff, $\mathrm{log}g$, $\mathrm{log}(n\mathrm{He}/n{\rm{H}})$, $\mathrm{log}(n{\rm{C}}/n{\rm{H}})$, and $\mathrm{log}(n{\rm{N}}/n{\rm{H}})$, respectively. The comparisons indicate that T_eff and $\mathrm{log}(n\mathrm{He}/n{\rm{H}})$ align consistently with the current parameters. However, the systematic shifts in $\mathrm{log}g$, $\mathrm{log}(n{\rm{C}}/n{\rm{H}})$, and $\mathrm{log}(n{\rm{N}}/n{\rm{H}})$ are attributed to the different coverage and spectral resolution of the observations and the inclusion of rotational broadening of line profiles during the LAMOST spectral analysis. Moreover, among the six stars with available upper limits for $\mathrm{log}(n{\rm{C}}/n{\rm{H}})$ and $\mathrm{log}(n{\rm{N}}/n{\rm{H}})$, a larger dispersion was noted compared to other objects. Nonetheless, both $\mathrm{log}(n{\rm{C}}/n{\rm{H}})$ and $\mathrm{log}(n{\rm{N}}/n{\rm{H}})$ for the entire sample agree with our results.

Projected rotational velocities ($v\sin i$) were also determined from the LAMOST spectral analysis, and the corresponding $v\sin i$ values are detailed in Table 3. Among our sample, more than half (134 out of 210 spectra) exhibit consistency with no measurable $v\sin i$. The subset of most extreme "rotators" (16 out of 210 spectra) comprises highly unusual cases such as misclassified WDs, cataclysmic variables, potential close binaries, and similar outliers.

In LAMOST LRS spectroscopy, individual observations are coadded, introducing orbital smearing as additional line broadening in the final spectra for binaries. Therefore, our $v\sin i$ values serve as indicators of such extra broadening in spectral lines. This may encompass various factors like orbital smearing, inconsistencies in instrumental broadening, turbulent broadening, Stark line broadening, etc., extending beyond the actual projected rotation.

We used Gaia DR3 for proper motions and distances and utilized LAMOST spectra for radial velocities to obtain the space velocity components and Galactic orbit parameters. To compute the three space velocity components U, V, and W, corresponding to the directions of the Galactic center, Galactic rotation, and north Galactic pole, respectively, we employed the software package Astropy. The parameters within Astropy were configured to set the distance from the Sun to the Galactic center at 8.4 kpc, the Cartesian velocity of the Sun in the Galactocentric frame as (11.1, 12.24, 7.25) km s⁻¹, and the velocity of the local standard of rest (LSR) at 242 km s⁻¹ (Schönrich et al. 2010; Irrgang et al. 2013).

We utilized the Galpy Python package to compute the Galactic orbits. Our calculations were based on the Milky Way potential model "MWpotential2014," encompassing a power-law bulge with an exponential cutoff, an exponential disk, and power-law halo components (Bovy 2015). Similar to the calculation of Galactic UVW velocities mentioned earlier, we employed the Galactocentric distance of the Sun as 8.4 kpc and the LSR velocity of 242 km s⁻¹ (Irrgang et al. 2013). The orbit integration spanned from 0 to 3.5 Gyr in steps of 1 Myr. Extracting the Galactic orbital parameters of He-rich hot subdwarfs involved determining their apocenter (R_ap), pericenter (R_peri), eccentricity (e), maximum vertical amplitude (${z}_{\max }$), and z-component of the angular momentum (J_z ) from the integration of their orbital paths, where R_ap and R_peri respectively indicate the maximum and minimum distances of an orbit from the Galactic center. The eccentricity was defined as

$$\begin{eqnarray}&&e=\displaystyle \frac{{R}_{\mathrm{ap}}-{R}_{\mathrm{peri}}}{{R}_{\mathrm{ap}}+{R}_{\mathrm{peri}}}.\end{eqnarray} \tag{ 1 }$$

The space velocity components and the orbital parameters can be found in Table 3. The uncertainties associated with these parameters were estimated using the Monte Carlo method. We generated 1000 random input values per star, following a Gaussian distribution. Subsequently, the output parameters and their errors were computed. Additional details on the calculation of parameter errors can be found in Luo et al. (2019).

To determine the Galactic population memberships of He-rich hot subdwarf stars, we followed the classification scheme outlined in Martin et al. (2017). This scheme builds upon the one proposed by Pauli et al. (2003, 2006) with a minor adjustment. Martin et al. (2017) introduced a slight modification by introducing the vertical scale height, setting it at Z ∼ 1.5 kpc for thick-disk stars as the cutoff height of the thin disk (Ma et al. 2017).

Our primary method for distinguishing the populations of He-rich hot subdwarf stars involved utilizing the U − V diagram, the J_z − e diagram, and the maximum vertical amplitude ${z}_{\max }$. To ensure accurate population assignments, all orbits were visually inspected to complement the automated classifications. The U − V and J_z − e diagrams can be observed in Figure 6. Following the approach in Martin et al. (2017), we established the 3σ limits for thin-disk and thick-disk WDs (Pauli et al. 2006) in the U − V diagram using two dotted ellipses. Furthermore, we separated the positions of the thin-disk, thick-disk, and halo populations in the J_z − e diagram using three distinct regions, referred to as regions A, B, and C, respectively.

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As detailed in Luo et al. (2020, 2021), thin-disk stars are identified within the 3σ thin-disk contour in the U − V diagram and fall within region A in the J_z − e diagram. Their orbits exhibit minor excursions in Galactocentric distance (R) and from the Galactic plane in the Z-direction, typically confined in the range of $| {z}_{\max }| \lt 1.5\,\mathrm{kpc}$. Thick-disk stars are situated within the 3σ thick-disk contour and in region B. These stars display greater orbit extension in both R and Z compared to thin-disk stars but less than that of halo stars. Halo stars reside outside regions A and B, as well as beyond the 3σ thick-disk contour. Their orbits show substantial differences in R and Z, with some halos exhibiting an extension in Galactocentric distance (R) larger than 18 kpc or a vertical distance from the Galactic plane (Z) larger than 6 kpc.

Figure 7 displays the distributions of 210 program stars in the ${T}_{\mathrm{eff}}-\mathrm{log}(n\mathrm{He}/n{\rm{H}})$ panel, representing the thin-disk, thick-disk, and halo populations. Based on our helium classification scheme (see Luo et al. 2021), 191 stars are identified as He-rich hot subdwarf stars. Among these He-rich hot subdwarf stars, 127 were further categorized as eHe, while 64 were classified as iHe with $\mathrm{log}(n\mathrm{He}/n{\rm{H}})=0$.

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Figure 7 reveals a trend showing a gradual decrease in helium abundance with increasing temperature, notably observed among the eHe stars. This pattern appears to persist across the thin-disk, thick-disk, and halo populations. Interestingly, we did not identify the two distinct eHe subclasses previously reported by Luo et al. (2021). However, notable distinctions in the effective temperature distribution for the eHe stars in the halo population were observed when compared to those in the thin and thick disks. Particularly, eHe stars in the halo exhibit a distinct gap near T_eff = 46,000 K in the ${T}_{\mathrm{eff}}-\mathrm{log}(n\mathrm{He}/n{\rm{H}})$ panel. The number ratio of eHe stars with T_eff > 46,000 K to those with T_eff < 46,000 K is 16:6 for the halo population, 26:26 for the thick-disk population, and 21:31 for the thin-disk population. This observation does not explicitly support the existence of two distinct subclasses of eHe stars, especially in the thin-disk and thick-disk populations. However, the number ratio of eHe stars with T_eff > 46,000 K to those with T_eff < 46,000 K appears to gradually increase from the thin to the thick disk and further to the halo populations, using T_eff = 46,000 K as the classification criterion, as previously outlined by Luo et al. (2021).

Figure 7 shows the presence of two separate iHe groups, as previously identified by Luo et al. (2021). These groups are represented by gray ellipses in the figure. According to the findings of Luo et al. (2021), group iHe-1 exhibits a notably distinct Galactic population fraction in comparison to group iHe-2. Specifically, the former presents a thin-disk population fraction of 63% ± 9%, while the latter demonstrates a thin-disk population fraction of 80% ± 13%.

Figure 8 displays the distributions of He-rich hot subdwarf stars in both the $\mathrm{log}(n\mathrm{He}/n{\rm{H}})-\mathrm{log}(n{\rm{C}}/n{\rm{H}})$ and $\mathrm{log}(n\mathrm{He}/n{\rm{H}})\,-\mathrm{log}(n{\rm{N}}/n{\rm{H}})$ panels. These trends are represented by dashed lines in the top panel for C abundances and in the bottom panel for N abundances, as shown in Figure 7. Our sample exhibits similar patterns of C and N as reported in Németh et al. (2012), demonstrating a distinct trend with higher average C and N concentrations at higher He abundances. In Figure 8, two best-fitting trends are plotted for C abundances using dashed lines in the top panel,

$$\begin{eqnarray}&&\mathrm{log}(n{\rm{C}}/n{\rm{H}})=1.72(\pm 0.23)\mathrm{log}(n\mathrm{He}/n{\rm{H}})-3.55(\pm 0.17),\end{eqnarray} \tag{ 2 }$$

and for N abundances in the bottom panel,

$$\begin{eqnarray}&&\mathrm{log}(n{\rm{N}}/n{\rm{H}})=1.00(\pm 0.13)\mathrm{log}(n\mathrm{He}/n{\rm{H}})-3.16(\pm 0.18),\end{eqnarray} \tag{ 3 }$$

respectively. These two trends are taken from Németh et al. (2012).

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In the top panel of Figure 8, two probable C–He sequences are observed for eHe hot subdwarf stars. Approximately two-thirds of these stars align within the first C–He sequence, consistent with the current best-fit trend. The second C–He sequence consists of eHe hot subdwarfs characterized by low carbon abundances. This suggests that eHe hot subdwarf stars belonging to these two C–He sequences likely stem from entirely distinct formation channels. Likewise, among iHe hot subdwarf stars, we also observe two different C groups. Those within the high C group do not align with the first C–He sequence of eHe hot subdwarf stars, indicating that the origin of high carbon stars in both the eHe and iHe subtypes might lack an evolutionary correlation.

The high carbon group of iHe hot subdwarf stars does not align with the first C–He sequence observed in eHe hot subdwarf stars, suggesting a lack of correlation in the evolutionary origins of high C stars between the eHe and iHe subtypes. However, iHe hot subdwarf stars in the low C group show a good agreement with the second C–He sequence of eHe hot subdwarf stars, implying a potential inherent correlation between low C stars in both the eHe and iHe groups regarding their origins. Nevertheless, it was observed that iHe hot subdwarf stars in the low C group align well with the second C–He sequence of eHe hot subdwarf stars, suggesting that stars exhibiting low C abundance in both the eHe and iHe subtypes may share an inherent correlation in their origins.

The bottom panel of Figure 8 shows that both eHe and iHe hot subdwarf stars do not exhibit clear differences in the N–He distribution. Most of these stars appear to follow a N–He sequence that aligns well with the best-fit trend identified by Németh et al. (2012).

For a statistical comparison with the SPY sample of Hirsch (2009), the determined abundances are defined by mass fraction: $\mathrm{log}{\beta }_{{\rm{i}}}=\mathrm{log}\tfrac{{m}_{{\rm{i}}}}{{m}_{{\rm{H}}}+{m}_{\mathrm{He}}+{m}_{{\rm{C}}}+{m}_{{\rm{N}}}}$, where i represents He, C, and N. Figure 9 shows the distributions of He-rich hot subdwarf stars in the ${T}_{\mathrm{eff}}-\mathrm{log}{\beta }_{\mathrm{He}}$, ${T}_{\mathrm{eff}}-\mathrm{log}{\beta }_{{\rm{C}}}$, and ${T}_{\mathrm{eff}}-\mathrm{log}{\beta }_{{\rm{N}}}$ panels. In Figure 9, the SPY sample from Hirsch (2009) is denoted by magenta dots. Moreover, Schindewolf et al. (2018) conducted an analysis of high-resolution visual and ultraviolet spectra of four He-rich sdO stars. These four stars are marked in Figure 9 with red dots. Our findings show that both the SPY sample and the four He-rich sdO stars studied by Schindewolf et al. (2018) fall within the range of our sample in both the ${T}_{\mathrm{eff}}-\mathrm{log}{\beta }_{{\rm{C}}}$ and ${T}_{\mathrm{eff}}-\mathrm{log}{\beta }_{{\rm{C}}}$ diagrams.

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In the ${T}_{\mathrm{eff}}-\mathrm{log}{\beta }_{{\rm{C}}}$ panel, our sample exhibits a distinctly bimodal distribution of C abundances, aligning with the observations of Hirsch (2009). This distribution can be categorized into two primary groups: the C-enriched and C-deficient groups, distinguished at $\mathrm{log}{\beta }_{{\rm{C}}}=-3.5$. The C-enriched subgroup further divides into C-enriched eHe and C-enriched stars at T_eff = 43,000 K. Meanwhile, the carbon-deficient group comprises a mixture of eHe and iHe hot subdwarf stars. Consequently, three distinct groups exist in the ${T}_{\mathrm{eff}}-\mathrm{log}{\beta }_{{\rm{C}}}$ panel. The first group comprises C-enriched eHe hot subdwarf stars, showing a trend where C abundances decrease from supersolar values to below the solar value as the effective temperature increases. Specifically, for T_eff > 47,000 K, the first group demonstrates C abundances below the solar value. In contrast, for T_eff < 47,000 K, this group presents C abundances above the solar value. Notably, this region encompasses nearly all the stars from both the SPY sample by Hirsch (2009) and the sample by Schindewolf et al. (2018). The second group predominantly includes C-enriched iHe hot subdwarf stars spanning a wide temperature range from 16,000 to 40,000 K. Our sample indicates that the third group aligns with the findings of Hirsch (2009). This group comprises a mix of eHe and iHe stars with approximately half of the iHe stars falling into this category. The distributions of these stars suggest that eHe and iHe stars within this third group likely share similar formation pathways.

In the ${T}_{\mathrm{eff}}-\mathrm{log}{\beta }_{{\rm{N}}}$ diagram, there is no clear division observed in the N abundances. More than 82% of the He-rich stars exhibit N abundances surpassing the solar value, which is proportionally higher than the two-thirds value found in the SPY sample by Hirsch (2009). Notably, all four He-rich sdO stars studied by Schindewolf et al. (2018) also show N abundances above the solar value. Moreover, a discernible pattern is observed where N appears to be more abundant in objects with a lower C content compared to those with higher C abundances. It was also observed that the average N abundance decreased with increasing temperatures in eHe stars, in line with the findings of Hirsch (2009).

Figure 10 presents the distributions of C and N abundances within the Galactic thin-disk, thick-disk, and halo populations, aimed at unveiling insights into He-rich hot subdwarfs with distinct C subclasses based on kinematics. A notable observation is the bimodal distribution of C abundance evident in both eHe and iHe hot subdwarf stars within the thin- and thick-disk populations, while C-deficient stars are notably scarce in the halo population. The proportion of C-enriched eHe and iHe stars progressively increases from the thin-disk to the halo population, in contrast to carbon-deficient stars. The stark absence of C-deficient stars within the halo population suggests that C-enriched and C-deficient stars have different origins. Regarding the N abundance distributions, no significant differences are observed between eHe and iHe stars across the thin-disk, thick-disk, and halo populations.

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In our prior investigation (Luo et al. 2021), we categorized eHe hot subdwarf stars into two groups, labeled eHe-1 and eHe-2, within the ${T}_{\mathrm{eff}}-\mathrm{log}(n\mathrm{He}/n{\rm{H}})$ panel. The classification criterion set at T_eff = 46,000 K allowed for this distinction. Our galactic population analysis revealed a higher prevalence of the eHe-2 group within the thin-disk population compared to the eHe-1 group. However, in our present study, we were unable to discern the two distinct eHe subclasses reported by Luo et al. (2021). Instead, our sample exhibits a continuous trend displaying a decreasing He abundance with increasing temperature. While this consistent trend of decreasing helium abundance with temperature persisted across the thin-disk, thick-disk, and halo populations, an observable gap emerged among eHe stars in the halo population near T_eff = 46,000 K within the ${T}_{\mathrm{eff}}-\mathrm{log}(n\mathrm{He}/n{\rm{H}})$ panel. Although our data did not support the existence of the two distinct subclasses of eHe stars described by Luo et al. (2021), particularly within the thin- and thick-disk populations, we did note a gradual increase in the ratio of eHe stars with T_eff > 46,000 K to those with T_eff < 46,000 K. This trend was consistent from the thin disk to the thick disk and then to the halo populations, aligning with the findings of Luo et al. (2021). Moreover, our observations indicated that C-deficient stars were primarily located at T_eff < 46,000 K, and their prevalence decreased gradually from the thin disk to the thick disk and halo populations. These results suggest that the presence of C-deficient stars could impact the distribution of eHe stars within the various Galactic populations.

The merger of double HeWDs is considered a key scenario in explaining the formation of isolated He-rich hot subdwarf stars (Han et al. 2002). Zhang & Jeffery (2012) proposed three merger models of double HeWDs: slow accretion from debris, fast accretion from a corona, and a combination of both processes. Their findings suggested that fast hot mergers result in C-rich, N-poor surfaces; slow cold mergers lead to N-rich surfaces; and composite models yield C- and N-rich surfaces. Yu et al. (2021) demonstrated that the merger of double HeWDs can account for the bimodal distribution of C abundances, where the C-enriched eHe stars are formed by high-mass HeWDs and the C-deficient ones by low-mass HeWDs. This finding aligns with the observations of eHe hot subdwarf stars in the Galactic field (Luo et al. 2019), supporting the predictions stemming from the double HeWD merger scenario. The fraction of C-enriched eHe hot subdwarf stars exhibits a consistent, monotonic increase from the thin disk to the halo population, aligning with the predictions from the merger models of double HeWDs (Yu et al. 2021). However, the distribution of C-deficient eHe hot subdwarf stars demonstrates an opposite tendency, inconsistent with the expected outcome from double HeWD merger models. Moreover, Hirsch (2009) observed different behaviors in the rotational velocities of C-enriched and C-deficient eHe hot subdwarf stars. The former displayed significantly higher projected rotational velocity compared to the latter, providing further support for the double HeWD merger channel in the formation of C-enriched eHe hot subdwarf stars.

The late hot flasher scenario (Miller Bertolami et al. 2008) provides a potential explanation for He-rich hot subdwarf stars. According to Luo et al. (2021), the evolutionary tracks within this scenario can align with eHe stars having T_eff < 46,000 K and iHe stars in the ${T}_{\mathrm{eff}}-\mathrm{log}g$ diagram. However, upon detailed comparisons with the findings of Hirsch (2009), it was concluded that the late hot flasher scenario does not account for the presence of C-deficient He-rich hot subdwarf stars. Additionally, Battich et al. (2018) conducted calculations of surface abundances for various late hot flasher scenarios, incorporating updated electron-conduction opacities introduced by Cassisi et al. (2007). Their results indicated a qualitative agreement of surface abundances with those proposed by Miller Bertolami et al. (2008). Despite this, further investigation is required to elucidate the formation channel of C-deficient He-rich hot subdwarf stars.