Mass Distribution for Single-lined Hot Subdwarf Stars in LAMOST

Once the synthetic SED of a star is known, we can obtain the model flux density at the stellar surface, F(λ). On the other hand, the observed flux density at the Earth, f(λ), can be obtained by converting photometric magnitudes into flux density (see Heber et al. 2018; Baran et al. 2021; Schaffenroth et al. 2022 for more information). With known F(λ), f(λ), and distance from the Earth, d, one can easily obtain the stellar angular diameter, Θ, and stellar radius, R, using the following equation:

$$\begin{eqnarray}&&\displaystyle \frac{{\text{}}f(\lambda )}{{\text{}}F(\lambda )}=\displaystyle \frac{{{\rm{\Theta }}}^{2}}{4}=\displaystyle \frac{{R}^{2}}{{d}^{2}}.\end{eqnarray} \tag{ 1 }$$

Then, the stellar mass of a hot subdwarf star can be obtained by

$$\begin{eqnarray}&&M=\displaystyle \frac{{{gR}}^{2}}{G},\end{eqnarray} \tag{ 2 }$$

where g is the surface gravity and G is the gravitational constant.

With the useful information from Gaia DR2 (Gaia Collaboration et al. 2018), e.g., magnitudes, parallaxes, colors, etc., Lei et al. (2018, 2019b, 2020) identified 864 single-lined hot subdwarf stars in LAMOST. Since all of these hot subdwarf stars have good quality LAMOST spectra (e.g., signal-to-noise ratio (S/N) greater than 10.0 in the u band), they obtained reliable atmospheric parameters (e.g., T_eff, $\mathrm{log}g$ and $\mathrm{log}(n\mathrm{He}/n{\rm{H}})$) by fitting H and He profiles with synthetic spectra, which were calculated with Synspec (version 49; Lanz & Hubeny 2007) from non–local thermodynamic equilibrium (NLTE) Tlusty model atmospheres (version 204; Hubeny & Lanz 2017). Based on the obtained atmospheric parameters, synthetic SEDs were also calculated for each star in the same way as the synthetic spectra.

Gaia EDR3 (Gaia Collaboration et al. 2021) provided astrometry and photometry for 1.8 billion objects collected in the first 34 months of the satellite mission, among which 1.5 billion sources have parallaxes, proper motions, and G, G_BP, and G_RP magnitudes. With precise parallaxes, we could calculate distances for a huge number of objects that are bright and relatively nearby to the Sun. However, distances cannot be obtained simply by directly reversing parallaxes for faint and distant objects; since the parallax uncertainties are large, even negative parallaxes can occur. To provide more reliable distances for as many stars in Gaia EDR3 as possible, Bailer-Jones et al. (2021) used a probabilistic approach, which employed a prior constructed from a three-dimensional model of our Galaxy, and estimated stellar distances based on Gaia parallaxes and photometry.

In this study, 864 single-lined hot subdwarf stars identified in Lei et al. (2018, 2019b, 2020) were crossmatched with Gaia EDR3 data (Gaia Collaboration et al. 2021) to obtain their parallaxes. By inspecting quasar data in Gaia EDR3, Lindegren et al. (2021b) found that parallaxes can be biased. Lindegren et al. (2021a) investigated the variation of the parallax bias for Gaia EDR3 data with magnitude, color, and ecliptic latitude. They provided a Python implementation ⁶ to correct the bias for Gaia EDR3 parallaxes, which was also adopted for our selected sample in this study. Since the uncertainty of parallax can affect the precision of distance directly, thus affecting the determination of stellar radius and mass (Irrgang et al. 2021), only objects with positive parallax and a relative parallax uncertainty of less than 20% (e.g., ϖ > 0 and σ_ϖ /ϖ ≤ 0.2; see Raddi et al. 2022) were reserved for the following analysis. After this step, 727 hot subdwarf stars with Gaia EDR3 parallax remained in our selection, and their distances were obtained directly by using the inverse of parallaxes after applying a zero-point correction.

We used the VO Sed Analyzer ⁷ (VOSA; Bayo et al. 2008) of the Spanish Virtual Observatory (SVO) to search for photometric data for our sample and convert observed magnitudes to fluxes. VOSA is a public web tool designed to help users search for observed photometric data from VO services and build SEDs. It can compare observed photometry with synthetic photometry from theoretical models to obtain important physical parameters for stars, such as T_eff, $\mathrm{log}g$, metallicity, radius, luminosity, etc. We obtained interstellar extinctions in the visual band, A_V, for the selected sample through Galactic dust reddening maps ⁸ (Schlegel et al. 1998; Schlafly & Finkbeiner 2011) with an extinction parameter R_V = 3.1. All this information together with coordinates (e.g., R.A. and decl.) of the sample was uploaded into VOSA, and then the conversion of photometric fluxes from different filters could be obtained for each sample star.

To compare synthetic SEDs with observed photometric fluxes, one needs to convolve synthetic SEDs with the transmission curve of the specific filter. Thanks to the Filter Profile Service ⁹ in SVO (Rodrigo et al. 2012; Rodrigo & Solano 2020), which provided information for more than 2300 astronomical filters, including transmission curves and zero-points in the Vega, AB, and ST photometric systems, we obtained all the transmission curves and zero-point information of the astronomical filters listed in the Filter Profile Service. Then, we convolved the synthetic SEDs with the filter transmission curves and compared them with the observed photometric fluxes obtained from VOSA. By doing this, we obtained the dilution factor M_d = R²/d² and stellar angular diameter Θ (see Equation (1)) for our selected sample.

Figure 1 gives an example of a comparison between synthetic SED and observed photometric fluxes for one of the selected stars, LAMOST_obsid = 580907209. The observed photometric data were collected from about 100 filters within different passbands equipped on various photometric systems, such as the Galaxy Evolution Explorer (GALEX; Martin et al. 2005), Gaia mission (Gaia Collaboration et al. 2016), Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006), Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010), etc. Figure 1 shows that most of the observed photometric fluxes (e.g., magenta crosses) are consistent with the synthetic SED (e.g., black solid curve).

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Once the stellar radii and effective temperatures are known, one can calculate the luminosity of a star by comparing the two parameters with the solar standard model using the following equation:

$$\begin{eqnarray}&&\displaystyle \frac{L}{{L}_{\odot }}={\left(\displaystyle \frac{R}{{R}_{\odot }}\right)}^{2}{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\odot }}\right)}^{4},\end{eqnarray} \tag{ 3 }$$

where R_⊙ and T_⊙ are the radius and surface effective temperature of the Sun, which were set to 6.959 × 10⁸ m and 5777 K, respectively.

With a known radius and $\mathrm{log}g$, we can calculate the stellar masses for the selected sample through Equation (2). To estimate mass uncertainties, a Monte Carlo (MC) approach was adopted in the calculations. Based on the values and uncertainties for the parameters involved in mass determination (i.e., parallaxes, gravity, etc.), a set of values was generated for each parameter by using a Gaussian distribution. With these values, the same number of masses was calculated for each sample star. In the calculation, with the symmetric uncertainty, we assumed a Gaussian distribution of parallax for each star. However, the transformation from parallax to distance is nonlinear, thus the distances obtained for each star are not a Gaussian-like distribution and their uncertainties are asymmetric (for a detailed discussion see Bailer-Jones 2015; Astraatmadja & Bailer-Jones 2016; Bailer-Jones et al. 2021). This is also valid for the obtained mass values. Therefore, for each sample star, we calculated the median (e.g., 50th percentile) mass value as the final mass. Moreover, the 16th and 84th percentile mass values form a 68% confidence interval around the median mass, and the differences between them and the median mass are considered as the asymmetric uncertainties of the mass (see column (14) in Table 1).

Table 1. Main Parameters for the 664 Hot Subdwarfs Selected in This Study

R.A.	Decl.	obs_id	source_id	G	Parallax	Teff	$\mathrm{log}\ g$	spclass	E(B − V)	Angular Diameter	Radius	Luminosity	Mass
LAMOST	LAMOST	LAMOST	Gaia EDR3	Gaia EDR3 (mag)	Gaia EDR3 (mas)	(K)	(cm s−2)			log(rad)	R/R⊙	L/L⊙	M/M⊙
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)
0.0767	22.30079	385816105	2847977322031768832	14.219558 ± 0.002796	1.2039 ± 0.0359	28810 ± 630	5.57 ± 0.06	sdB	0.058 ± 0.004	−11.013${}_{-0.006}^{+0.004}$	${0.176}_{-0.004}^{+0.006}$	${19.2}_{-1.8}^{+2.3}$	${0.42}_{-0.06}^{+0.08}$
0.278039	11.010084	66605009	2765454164004531328	13.589941 ± 0.002805	1.6454 ± 0.0352	27030 ± 602	5.48 ± 0.05	sdB	0.071 ± 0.002	−10.856${}_{-0.003}^{+0.003}$	${0.191}_{-0.004}^{+0.004}$	${17.4}_{-1.6}^{+1.9}$	${0.40}_{-0.05}^{+0.06}$
0.535283	19.987016	385805170	2846319155418023424	15.581263 ± 0.002853	0.4198 ± 0.0496	31560 ± 722	5.58 ± 0.08	sdB	0.035 ± 0.002	−11.348${}_{-0.008}^{+0.008}$	${0.233}_{-0.024}^{+0.035}$	${48.5}_{-10.4}^{+16.3}$	${0.77}_{-0.20}^{+0.29}$
0.981681	27.810404	492112073	2854091981071865344	13.308936 ± 0.002796	1.7692 ± 0.0329	28000 ± 572	5.56 ± 0.06	sdB	0.048 ± 0.002	−10.826${}_{-0.010}^{+0.041}$	${0.188}_{-0.004}^{+0.003}$	${19.4}_{-1.7}^{+1.7}$	${0.47}_{-0.06}^{+0.08}$
1.8907183	13.5993244	619614193	2767874292175410560	13.047603 ± 0.002814	2.0899 ± 0.0285	29560 ± 577	5.41 ± 0.05	sdB	0.11 ± 0.005	−10.746${}_{-0.029}^{+0.015}$	${0.188}_{-0.003}^{+0.003}$	${24.0}_{-1.7}^{+2.4}$	${0.33}_{-0.04}^{+0.05}$
2.1021972	49.083822	593009050	393589879591384576	15.888843 ± 0.002816	0.6389 ± 0.0432	26640 ± 899	5.53 ± 0.09	sdB	0.108 ± 0.002	−11.293${}_{-0.006}^{+0.010}$	${0.178}_{-0.013}^{+0.014}$	${14.3}_{-2.4}^{+3.5}$	${0.40}_{-0.09}^{+0.14}$
...	...			...			...			...			...
359.439198	24.43959	492005208	2851511427281425536	16.330795 ± 0.002858	0.4081 ± 0.0662	53970 ± 6514	5.61 ± 0.08	sdO	0.117 ± 0.009	−11.578${}_{-0.011}^{+0.011}$	${0.144}_{-0.020}^{+0.031}$	${160.3}_{-71.5}^{+135.6}$	${0.31}_{-0.09}^{+0.18}$

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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Atmospheric parameters for hot subdwarf stars in Lei et al. (2018, 2019b, 2020) were obtained with NLTE model atmospheres and only statistical uncertainties were presented, while no systematic errors were considered in their studies. A realistic assessment of systematics must be based on independent studies; therefore, we estimated systematic errors for T_eff and $\mathrm{log}g$ by comparing atmospheric parameters in Lei et al. (2018, 2019b, 2020) with the results from Schaffenroth et al. (2022). Finally, we found systematic differences of about 560 K and 0.013 dex for the two parameters, respectively. Moreover, different atmospheric models could also lead to systematic errors in parameter determination. Heber et al. (2000) determined atmospheric parameters for three pulsating sdB stars by using blanketed NLTE and LTE model atmospheres, respectively. They found that there is no obvious difference in T_eff determination between NLTE and LTE models, but NLTE $\mathrm{log}g$ is slightly lower than LTE ones by at least 0.05 dex (see Section 4.1 in their study for a detailed discussion). In this study, the systematic errors for T_eff and $\mathrm{log}g$ discussed above were added to the statistical uncertainties in quadrature for the selected hot subdwarf stars to estimate their mass uncertainties.

Based on the method described above, we obtained masses for 727 hot subdwarf stars from Lei et al. (2018, 2019b, 2020). For some of these hot subdwarf stars, we got excessively large mass values (e.g., larger than 1.0 M_⊙), which are not likely real hot subdwarfs, or their mass values are unreliable due to large uncertainties of distances and/or gravity. On the other hand, a few stars turned out to have very low mass values (e.g., less than 0.1 M_⊙). These stars may be extremely low-mass white dwarfs or the mass values are unreliable due to large uncertainties from related parameters as well. Such extreme cases are not reported here. In total, 664 hot subdwarf stars with mass values between 0.1 and 1.0 M_⊙ are reported and analyzed in the following sections.

The four panels in Figure 2 show the relationships we found between masses and atmospheric parameters (e.g., T_eff, $\mathrm{log}g$, $\mathrm{log}(n\mathrm{He}/n{\rm{H}})$, and luminosity) for the selected hot subdwarfs. As one can see, no obvious characteristics are contained in the mass–T_eff plane (upper left panel), mass–$\mathrm{log}g$ plane (upper right panel), and mass–$\mathrm{log}(n\mathrm{He}/n{\rm{H}})$ plane (bottom left panel). Note that the detailed discussion of the relationships among atmospheric parameters for the selected hot subdwarfs can be found in Section 3.2 of Lei et al. (2018) and Section 4.1 of Lei et al. (2019b).

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On the other hand, as shown in the mass–luminosity plane (bottom right panel), though a wide mass distribution (e.g., roughly from 0.1 to 1.0 M_⊙; see Section 3.3 for detailed discussion) is presented, most of sdB (black circles) and sdOB (blue triangles) stars settle in a narrow range at a relatively low luminosity (e.g., roughly from $\mathrm{log}(L/{L}_{\odot })=0.5$ to 1.5), which demonstrates that these stars are still at the core He burning stage.

However, most sdO (green squares) and He-rich stars, i.e., He-sdO (aqua right triangles), He-sdOB (red diamonds), and He-sdB (magenta stars), show a much higher and wider range of luminosities than sdB and sdOB stars. Some of them even present luminosity near 3000 L_⊙ (e.g., $\mathrm{log}(L/{L}_{\odot })=3.5$). These results demonstrate that they could either have evolved off the core He-burning stage and now are on the way to the WD cooling curve, or could be pre-EHB stars that just cross the hot subdwarf region. This result is also clearly shown in Figure 3, in which the relationships between T_eff and luminosity for the selected hot subdwarfs are presented. As shown in the figure, most sdB and sdOB stars show a much narrower T_eff and luminosity range than that of sdO and He-rich hot subdwarfs (also see Figure 10 in Schaffenroth et al. 2022) and are located well in the region defined by the zero-age horizontal branch (ZAHB) and the terminal-age horizontal branch (TAHB) lines.

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Figure 4 shows the relationship between mass and radius for our selected sample. $\mathrm{log}g$ values for each object were denoted by the color gradient shown on the right. Most of the hot subdwarf stars in our sample have radii between 0.1 and 0.4 R_⊙, while most of the objects have $\mathrm{log}g$ values between 5.0 and 6.3 dex. The black dashed curves reveal the relationship between mass and radius that the square of the radius has a positive correlation with stellar mass when $\mathrm{log}g$ is a keen constant (see Equation (2)). Since all spectral types of hot subdwarfs present relative wide distributions of $\mathrm{log}g$ (see the upper right panel in Figure 2), it is not easy to distinguish them in this figure. Therefore, stars with different spectral classifications were not denoted by different labels as in Figures 2 and 3.

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To check the reliability of the mass obtained in this study, we crossmatched the selected stars with the sample analyzed by Schaffenroth et al. (2022) and Fontaine et al. (2012) (see Section 1), and found 18 and 6 common objects, respectively. In Figure 5, we compared masses obtained in this study with the values of common objects in Schaffenroth et al. (2022; left panel) and Fontaine et al. (2012; right panel). Although considerable dispersion is present in both panels, the masses obtained in this study are consistent with the two former studies when the uncertainties are also considered. Note that the uncertainties for our mass values are mainly from the large uncertainties of $\mathrm{log}g$ and distances. Considering that we have a large number of samples (e.g., 664 hot subdwarf stars), it will not influence much the analysis and interpretation of the mass distribution in this study.

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In Figure 6, we also compared the radius and luminosity obtained in this study with the values by Schaffenroth et al. (2022). As shown in the two panels, since $\mathrm{log}g$ values were not used when calculating radius and luminosity, both the radius and luminosity values show much better consistency than the direct mass comparison between the two studies.

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To study the mass distribution at different parallax precisions, we selected three groups of hot subdwarf stars from our sample for a mass distribution analysis, i.e., objects with σ_ϖ /ϖ ≤ 0.2 were selected as Group 1, objects with σ_ϖ /ϖ ≤ 0.1 were selected as Group 2, and objects with σ_ϖ /ϖ ≤ 0.05 were selected as Group 3. The statistical information for subsets with different spectral types in the three groups is listed in Table 2. From left to right, it gives the group name, subset name, number count, peak mass of the distribution, mean mass, number of stars in the massive sample, and the relative fraction of the massive sample to the total, respectively.

With an increasing parallax precision, the number of stars within each group decreases. Group 1 contains all of the 664 selected hot subdwarfs, Group 2 has 503 hot subdwarfs, and Group 3 consists of 289 hot subdwarfs. The number fractions of each subset with different spectral types are consistent among the three groups, e.g., sdB/sdOB stars are the largest population in each group, which represents about 73% of the total, while He-rich stars (He-rich sdB/sdOB/sdO) and sdO stars in each group present about 16% and 10%, respectively. Furthermore, the peak of the mass distribution for each subset in the three groups is nearly the same, e.g., sdB/sdOB stars present a peak mass of about 0.46 M_⊙ (also see Figure 8), He-rich stars present a peak mass at 0.4 M_⊙, while sdO stars present an inconspicuous peak mass of about 0.18 M_⊙.

The left panel in Figure 7 shows the mass distributions for the whole sample in each group. The mass distributions are very similar among the three groups, most of the hot subdwarf stars have masses between 0.37 and 0.5 M_⊙, and a sharp peak at 0.46 M_⊙ is present. However, Group 3 (i.e., red dotted histogram) seems to have much fewer stars (with a mass between 0.2 and 0.37 M_⊙) than Group 1 (i.e., black solid histogram) and Group 2 (i.e., blue dashed histogram). This feature is shown more clearly in the right panel of Figure 7, in which the number counts in each bin were normalized. One can see that with the parallax (distance) precision increasing, the relative number of less massive objects (e.g., mass between 0.2 and 0.37 M_⊙) decreases, which leads to a sharper peak at the low mass side. It means that with larger uncertainties in parallax (distance), the relative contribution of low-mass objects shows an increase. On the other hand, massive stars (e.g., M >0.6 M_⊙) present nearly the same mass distribution among the three groups, independent of the parallax precision.

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The left panel in Figure 8 shows the mass distributions of hot subdwarfs with different spectral types in Group 1. One can see that the whole sample presents a very wide mass distribution (black solid histogram) from 0.1 to 1.0 M_⊙, with a sharp peak (mean mass) at 0.46 M_⊙. Since sdB and sdOB stars have the largest fraction in the group (e.g., 483 objects in a total of 664, or about 73% in percentage), they show very similar mass distributions (magenta dashed–dotted histogram) to the whole sample. They also present a sharp mass peak around 0.46 M_⊙ (see Table 2).

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On the other hand, both He-rich stars (e.g., He-sdB, He-sdO, and He-sdOB) and sdO stars present much flatter mass distributions (denoted by blue dashed and green dotted histograms, respectively) when compared with sdB/sdOB stars. Moreover, He-rich stars show a small mass peak near 0.40 M_⊙ and have a mean mass of about 0.46 M_⊙. Meanwhile, sdO stars present an inconspicuous peak mass at around 0.18 M_⊙ and a mean mass of about 0.34 M_⊙, which is obviously smaller than the values of sdB/sdO and He-rich stars. If we consider stars with a mass greater than 0.6 M_⊙ as massive hot subdwarfs, the fraction of massive stars in the sdO sample is about 10% (e.g., 7 out of 71; see Table 2), while this fraction among other types of hot subdwarfs (e.g., sdB/sdOB and He-rich sample) are both around at 20%. Note that, according to the results of Han et al. (2002), the minimum core mass for He ignition in a star is about 0.3 M_⊙ (see Table 1 in their study). This means that the low-mass sdO stars found in our sample probably did not undergo a core He burning phase (see detailed discussion in Section 4.4).

The middle and right panels of Figure 8 show the mass distribution of hot subdwarfs with different spectral types in Groups 2 and 3, respectively. Mass distributions for the various subsets in the two groups are very similar to the distributions of the corresponding subsets in Group 1, except that less massive stars become fewer with an increasing parallax precision (distance) from Group 1 to Group 3, as shown in Figure 7. Considering that higher parallax precision (distance) means more reliable mass determination, we used hot subdwarf stars in Group 3 to compare the mass distribution with other studies in the following sections.

Combining high-quality light curves from the Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015) and the K2 space mission (Howell et al. 2014) with the fit of SED, distance from Gaia, and atmospheric parameters from the literature, Schaffenroth et al. (2022) not only derived the properties of primary and secondary stars in about 200 sdB binaries but also obtained absolute masses, radii, and luminosities for 68 sdB stars with different types of companions. They found a broad mass distribution for sdB stars with cool, low-mass stellar or substellar companions (i.e., sdB+dM/BD systems), which present a peak mass at 0.46 M_⊙. Though the mass distribution of sdB stars with WD companions (i.e., sdB+WD systems) also has a similar width as sdB+dM/BD systems, they have a much smaller peak mass of around 0.38 M_⊙ (see Section 3.2 and Figure 12 in their study). Based on this result, they concluded that sdB+dM/BD and sdB+WD systems originate from different populations.

In the left panel of Figure 9, we compared the mass distributions for sdB+dM/BD and sdB+WD systems obtained in Schaffenroth et al. (2022) separately with the mass distribution for sdB/sdOB stars of Group 3 in this study. All the sample numbers in each bin are normalized to the maximum bin number, which has been set to 1. One can see that the mass distribution for sdB/sdOB stars in Group 3 (magenta dashed–dotted curve, e.g., roughly from 0.1 to 1.0 M_⊙) is generally wider than the mass distribution for both sdB+dM/BD (black solid curve, e.g., roughly from 0.3 to 0.74 M_⊙) and sdB+WD systems (black dashed curve, e.g., from 0.28 to 0.68 M_⊙). Note that the atmospheric parameters for our sample are obtained by fitting H and He profiles in Lei et al. (2018, 2019b, 2020) with no metal opacities included. The inclusion of metals can potentially change the T_eff and $\mathrm{log}g$ values, thus resulting in slightly different masses. Moreover, the number of sdB stars analyzed in this study is much bigger than that in Schaffenroth et al. (2022), e.g., 220 versus 68. Though these factors would act toward a wider mass distribution, they will not influence the observed mass distribution tendencies as significantly. The peak of the mass distribution for our sdB/sdOB stars is nearly the same as the one for sdB+dM/BD systems (at 0.46 M_⊙). Furthermore, there is also a second mass peak around 0.38 M_⊙ for our sdB/sdOB stars, which corresponds to the peak mass of sdB+WD systems in Schaffenroth et al. (2022).

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Note that we cannot separate sdB+dM/BD and sdB+WD systems in our sample without additional information, e.g., light curves. Therefore, we compared the total mass distribution in the right panel of Figure 9 for all sdB binaries in Schaffenroth et al. (2022; black solid curve) with the mass distribution for our sdB/sdOB stars in Group 3 (magenta dashed–dotted curve). We find the two mass distributions to be consistent with each other, especially on the low-mass side. However, the mass distribution in Schaffenroth et al. (2022) seems to present a little wider peak and have more stars with masses around 0.5 M_⊙ than ours. It could be due to differences in the atmospheric parameters and their uncertainties adopted in the two studies, especially for $\mathrm{log}g$.

Han et al. (2003) carried out a detailed binary population synthesis to study the formation of sdB stars. Three main formation channels were investigated in their study, i.e., RLOF, CE ejection, and merger of two He-WDs. sdB stars produced from the best-fitting model of these channels could satisfy most observational characteristics, such as P − M_comp plane, ${T}_{\mathrm{eff}}-\mathrm{log}g$ plane, orbital period distribution, mass function distribution, birth rates, etc. To study the effects of input parameters (e.g., metallicity, initial mass–ratio distribution, the critical mass ratio for stable RLOF q_crit, CE ejection efficiency α_CE, and thermal contribution to the binding energy of the envelope α_th, etc.) on the formation of sdB stars, they conducted 12 sets of MC simulations by varying the parameters in a reasonable range (see Tables 1 and 2 in their study). Han et al. (2003) give mass distributions for the 12 sets of simulations and chose set 2 as their best-fitting models after comparing their results with observations (see Figure 11 in their study). These results provide a great convenience to study the mass distribution of sdB stars, and thus their formation channels.

In Figure 10, we compared the mass distribution of our sdB/sdOB stars from Group 3 with mass distributions for the 12 sets of simulations in Han et al. (2003). In the upper left panel, the mass distribution for our sdB/sdOB stars from Group 3 (e.g., magenta dashed–dotted curve) is generally consistent with the mass distribution predicted by set 1, 2, and 3 simulations in Han et al. (2003), which are denoted by black solid, dashed, and dotted curves, respectively. Furthermore, the mass distribution peaks predicted by Han et al. (2003), i.e., at 0.46 M_⊙, agree quite well with the one presented in this study. However, the sdB mass distribution in Han et al. (2003) displays a much steeper fall for masses larger than 0.48 M_⊙. It seems to demonstrate that the main channels in Han et al. (2003) predicted fewer sdB stars above 0.48 M_⊙ than the results from this study. Note that, to compare with the observational results of Maxted et al. (2001), which are biased against sdB binaries with bright F/G type companions, the GK selection in Han et al. (2003) removed sdB binary systems with companions with T_eff higher than 4000 K or brighter than the sdB primaries. Due to the strict selection criterion, we did not compare the results with corrections for the GK selection effects in Han et al. (2003) with the results obtained in this study directly. Since the minimum core mass for igniting He in a nondegenerate core is about 0.3 M_⊙, while the minimum core mass to ignite He in degenerate stars could be larger (see Tables 1 and 2 in Han et al. 2002), sdB stars have a low mass cutoff around 0.3 M_⊙ in Han et al. (2003). There are still some stars less massive than 0.3 M_⊙ in our sdB/sdOB sample, which can be low-mass WDs or pre-ELM WDs.

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The three other (e.g., bottom left, upper right, and bottom right) panels in Figure 10 present the mass distribution comparison between our sdB/sdOB stars from Group 3 and the simulations of sets 4–12 in Han et al. (2003), respectively. Though the peaks of mass distributions are roughly consistent between the two studies, models from sets 4 to 12 in Han et al. (2003) predicted obviously fewer stars at the low-mass side (e.g., 0.3–0.42 M_⊙) than in this study. Therefore, the mass distribution for sdB/sdOB stars from Group 3 in this study prefers the model predictions of sets 1–3 in Han et al. (2003), especially for set 2 (e.g., the black dashed curve in the upper left panel, with Z = .02, a flat initial mass ratio distribution, q_crit= 1.5, and α_CE = α_th = 0.75) even by visual inspections.

To study how parameter variations in binary evolution impact the production of sdB stars, Clausen et al. (2012) also conducted a grid of binary population synthesis models with different parameter assumptions, such as minimum core mass for He ignition, envelope binding energy, CE ejection efficiency, the amount of mass and angular momentum loss during stable mass transfer, and the criteria for a stable mass transfer on the RGB and in the Hertzsprung gap, etc. (see Table 1 in their study). They found that the variations of these parameters separately or together can significantly change the production of sdB binaries. They present mass distributions of sdB stars for 14 sets of simulations in Figure 14 of their study, most of which predicted similar mass distributions as in Han et al. (2003), especially for their Run 6, which used very similar parameters as the best-fitting model (set 2) in Han et al. (2003). However, most of the models in Clausen et al. (2012) predicted a most probable mass for sdB stars around 0.48 M_⊙, which is higher by about 0.02 M_⊙ than the value obtained in this study and by Han et al. (2003). They also predicted a population of post-RGB stars in their models, which has a lower mean stellar mass (e.g., less than 0.22 M_⊙; see Figure 14 in their study) than sdB stars and failed to ignite He burning in the cores due to an excessive mass transfer on the RGB, but still crossed the sdB region in T_eff–$\mathrm{log}g$ plane. This population also appears in our results as we mentioned above, e.g., stars less massive than 0.3 M_⊙ in Group 3 (see Figure 8). Furthermore, the models by Clausen et al. (2012) predicted an obvious mass gap between the sdB and post-RGB population between 0.22 and 0.26 M_⊙ (see Figure 14 in their study), but this feature is not present in our sdB/sdOB stars (see Figure 8). It could be averaged out due to a different bin size adopted in our sample, and the overlap of different types of hot subdwarf stars would also mask this feature in our mass distributions. We will continue to discuss the nature of this population in Section 4.4.

Geier et al. (2022) performed an RV variability study for 646 single-lined hot subdwarfs with multiepoch observed spectra in SDSS and LAMOST. The distribution of the maximum RV variations, (i.e., ${\rm{\Delta }}{{RV}}_{\max }$, defined as the difference between the maximum and minimum RV values of a star, see Section 2.3 in their study), was used as diagnostics to study the RV variability, and thus the binarity of hot subdwarfs. If a false-detection probability calculated in their study, p is smaller than 0.01% ($\mathrm{log}p\lt -4.0$), the detection of RV variability was considered to be significant, and thus a close hot subdwarf binary should be expected. Employing this method, they confirmed 164 hot subdwarf stars in the sample showing significant RV variations. They also found a distinctive difference between He-poor and He-rich hot subdwarfs, where the former presents a high fraction of close binaries, while nearly no significant RV variations were detected for the latter. This result made them conclude that there is no evolutionary connection between He-poor and He-rich hot subdwarfs. Moreover, their results also indicated that He-rich hot subdwarfs should be formed from the binary merger channels when considering the results from Pelisoli et al. (2020) and that binary interaction is always required to form hot subdwarf stars.

To investigate the binarity of hot subdwarf stars in our study, we crossmatched the selected stars in the three groups with the hot subdwarfs studied by Geier et al. (2022), respectively. The statistical properties of the common objects are shown in Table 3. From left to right, it gives the group name, subset name, number counts, close binary number counts, and close binary fractions, respectively. There are, in total, 216 hot subdwarfs in Group 1 with RV variation values detected by Geier et al. (2022), among which 86 stars present significant RV variations (e.g., $\mathrm{log}p\lt -4.0$). This means that about 40% of hot subdwarfs are expected to be in close binaries in this group. Specifically, sdB/sdOB stars present a binary fraction of 42% and sdO stars present a binary fraction of 48%. He-rich (e.g., He-sdB, He-sdOB, and He-sdO) stars in our sample present a much lower close binary fraction of 6%, and only one star has a significant RV variation among the 16 He-rich stars. With an increasing parallax precision from Group 1 to 3, the fractions of close binaries in each subset for the three groups are roughly consistent. Furthermore, though both He-poor and He-rich stars in this study present a little higher close binary fraction than the counterparts in Geier et al. (2022), e.g., 30% ± 2% for He-poor stars and about 3% for He-rich stars (see Section 3.1 in their study for details), these results are still consistent with each other when considering the randomness of crossmatching between the two sets of samples.

Figure 11 shows the relationship between mass distribution and the maximum RV variations for the 96 common objects between Group 3 in this study and Geier et al. (2022). As clearly shown in the figure, there are a considerable number of objects that present large maximum RV variations and have very small false-detection probabilities (denoted by the color bar on the right of the figure, e.g., $\mathrm{log}p\lt -4.0$ means significant RV variability) for all samples (upper left panel), sdB/sdOB stars (upper right panel), and sdO stars (bottom right panel), which indicates a moderate close binary fraction among these types of hot subdwarfs (e.g., higher than 40%; see Table 3). While in the bottom left panel, He-rich stars present nearly no significant RV variations, which is very different from sdB/sdOB and sdO stars. Based on this feature, Geier et al. (2022) came to the conclusion that there is no evolutionary connection between He-rich and He-poor hot subdwarf stars, and He-rich stars are formed by the binary merger channels. If this is the case, He-rich hot subdwarfs should present a higher fraction of massive stars than He-poor hot subdwarfs, since hot subdwarf stars formed from the merger channel shall have larger masses on average according to the model prediction of Han et al. (2002, 2003)—also see the results from Zhang & Jeffery (2012) and Zhang et al. (2017). However, based on our results presented in Table 2, He-rich stars have nearly the same fraction of massive stars (e.g., about 20%) as He-poor sdB/sdOB stars, which does not support binary mergers as the only formation channel for He-rich hot subdwarfs.

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To understand this issue more clearly, in Figure 12, we compared the mass distribution of He-rich stars from Group 3 in this study with the results predicted by a two He-WD merger channel from the set 2 simulation in Han et al. (2003). As seen in the figure, the two He-WD merger channel (black dotted curve) predicts a wide mass distribution of hot subdwarfs, e.g., roughly from 0.42 to 0.76 M_⊙, with a fairly wide peak between 0.5 and 0.6 M_⊙. On the other hand, He-rich stars in this study (blue dashed–dotted curve) present a wider mass distribution, e.g., from 0.3 to nearly 1.0 M_⊙, with a sharp peak around 0.42 M_⊙, ¹¹ which is much lower than the wide peak predicted by the binary merger channel. Furthermore, our He-rich sample presents a much lower relative contribution than the model predicts in the broad mass peak area (e.g., 0.5–0.6 M_⊙). Zhang et al. (2017) also studied the merger of a He-WD with an MS star, which can produce iHe-rich (e.g., $-1.0\lt \mathrm{log}(n\mathrm{He}/n{\rm{H}})\lt 1.0$) hot subdwarfs. However, most of the iHe-rich hot subdwarfs formed through the He-WD+MS merger channel have masses in the range of 0.48–0.50 M_⊙, and a few of them have masses up to 0.52 M_⊙, which is also more massive than the mass peak in our He-rich hot subdwarf sample.

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As discussed above, these comparisons of results indicated that the two He-WD or the He-WD+MS merger channels cannot be the only formation channels for He-rich hot subdwarfs; some other mechanisms must exist and make contributions to form these kinds of stars. Meng & Luo (2021) proposed that the MS companions of massive WDs can survive a Type Ia supernova (SN Ia) explosions and evolve into hot subdwarfs. The hot subdwarfs formed in this channel are also single and He enriched. Moreover, the results based on binary population synthesis can explain some observed features of iHe-rich hot subdwarfs, especially when spin-up/spin-down models were considered. Surprisingly, the spin-up/spin-down models predict the production of hot subdwarfs in the mass range from 0.35 to 1.0 M_⊙, and a distinct mass peak around 0.4 M_⊙ (see Figure 9 in their study), which is consistent with the mass distribution of He-rich stars obtained in this study. However, the same model predicted a low Galactic birth rate for He-rich hot subdwarf stars (e.g., 2.3–6 × 10⁻⁴yr⁻¹; see Section 4.1 in their study), which demonstrates that some other channels still contribute to the formation of He-rich hot subdwarfs.

On the other hand, Werner et al. (2022) discovered two He-sdO stars with unusually strong carbon and oxygen lines (named CO-sdO stars in their study). This new type of hot subdwarf star can neither be explained by a late hot He-core flash (Battich et al. 2018) nor by the merger of two He-WDs, which are considered as the formation channels for canonical He-sdO stars. Miller Bertolami et al. (2022) proposed a new formation channel for the two CO-sdO stars discovered by Werner et al. (2022). In this scenario, the merger of a more massive He-WD and a less massive CO-WD could ignite He core or shell burning and become a CO-sdO star. The material of the CO-WD could be accreted on the top of the He-WD and lead to the CO-enriched envelope. This scenario explains the surface parameters and composition of CO-sdO stars. However, Miller Bertolami et al. (2022) did not give the mass distribution for this formation channel, and thus we cannot compare our results with it directly. As discussed in Miller Bertolami et al. (2022), the scenario proposed in their study can be a small subchannel, since the parameter space of the progenitor system needs to be well defined (see Section 2 in their study for a detailed discussion) and CO-sdO stars are just a minority compared to the hot subdwarf population. Therefore, this scenario plays a limited role in the formation of He-sdO stars.

Based on the results mentioned above, one can conclude that in addition to the two He-WD merger channels, there should be some other channels that still make contributions to the formation of He-rich hot subdwarf stars, e.g., the SNe Ia explosion channel could make some contributions on the formation of less massive He-rich hot subdwarfs (e.g., less than 0.44 M_⊙; see Figure 12), while the binary merger channel could dominate the formation of massive He-rich hot subdwarfs (e.g., larger than 0.44 M_⊙).

The minimum core mass for He ignition predicted by models is about 0.3 M_⊙ (Han et al. 2002; Clausen et al. 2012), which could be a little different depending on the metallicity and initial mass at zero-age main sequence. Therefore, hot subdwarf stars less massive than 0.3 M_⊙ found in this study are potential post-RGB stars (e.g., low-mass WDs or pre-ELM WDs) that failed to ignite He burning in their cores.

It is believed that ELM WDs cannot be formed by the evolution of single stars since the evolutionary timescale of such low-mass stars would exceed the Hubble time. Istrate et al. (2016) studied the effects of element diffusion and rotational mixing on the evolution of pre-ELM WDs with various metallicities. They found that element diffusion plays a significant role in the evolution of pre-ELM WDs that experience hydrogen shell flashes. Moreover, rotational mixing plays a key role in determining their surface chemical abundances, but it does not influence significantly the number of hydrogen shell flashes and the hydrogen envelope mass at the beginning of the cooling track. They calculated a large number of evolutionary tracks of pre-ELM WDs within a mass range of 0.16–0.45 M_⊙, which provides great convenience to analyze the nature of low-mass hot subdwarf stars in our sample.

Figure 13 shows the T_eff–luminosity (left panel) and T_eff–$\mathrm{log}g$ planes (right panel) for the low-mass hot subdwarfs (e.g., M ≤0.3M_⊙) in Group 3, together with three evolutionary tracks of pre-ELM WDs at Z = 0.001 from Istrate et al. (2016). As can be seen in the figure, the positions of all low-mass sdB/sdOB stars (blue pluses), most of the low-mass sdO stars (red circles), and the He-rich stars (green squares) can be covered by the evolutionary tracks of pre-ELM WDs with higher stellar masses, e.g., 0.25 M_⊙ (black solid curve) and 0.303 M_⊙ (black dotted curve) both in the T_eff–luminosity plane and the T_eff–$\mathrm{log}g$ plane. These models experience several hydrogen shell flashes before entering the final cooling curve. On the other hand, our low-mass stars cannot be covered by the less massive evolutionary track of M = 0.183 M_⊙, which did not experience any hydrogen shell flashes due to the thick shell (see Section 2.1 in Istrate et al. 2016 for a detailed discussion). Furthermore, we checked the mass distribution of low-mass hot subdwarfs in Group 3 and found that most of the low-mass sdB/sdOB stars and He-rich stars have masses in the range of 0.25–0.30 M_⊙, while most of the low-mass sdO stars have masses around 0.18 M_⊙, as described in Table 1.

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The results described above indicate that low-mass sdB/sdOB stars and He-rich stars should be post-RGB stars, which would lose too much envelope mass at the RGB stage to ignite He burning in their cores, and now are on the way toward the ELM WD cooling curves. Heber et al. (2003) discovered a pre-ELM WD in a close binary system, named HD 188112, with a mass of 0.24 M_⊙. The primary in that system presents T_eff = 21,500 ± 500 K and $\mathrm{log}g$=5.66 ± 0.05, which is located at a similar region as our low-mass sdB/sdOB stars in the T_eff–$\mathrm{log}g$ plane. However, pre-ELM models seem unable to explain our low-mass sdO stars because the evolutionary tracks with the same mass (e.g., about 0.18 M_⊙; black dashed curve in Figure 13) predict much lower effective temperatures than our low-mass sdO stars (red circles in Figure 13; also see Brown et al. 2022). Thus, this result demonstrates that some other unknown mechanisms could play roles in the formation and evolution of these low-mass sdO stars, which need to be studied in the future.