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\begin{document} 


    \title{From synthetic SEDs to stellar origins: A deep learning model for physical parameter retrieval in hot subdwarf stars}
    \titlerunning{Deep learning SED modeling of hot subdwarfs}

   \author{Mengqi Feng\inst{1}
          \and
          Zhenxin Lei\inst{1}\fnmsep\thanks{e-mail: \href{mailtoleizhenxin2060@163.com}
          {leizhenxin2060@163.com}}
          \and
         Bokai Kou \inst{1}
         \and
          Yangyang Dong\inst{1}
          \and
          Ke Hu\inst{1}
          \and
          Huaping Xiao\inst{1}
          \and
          Jingkun Zhao\inst{2}
          }

   \institute{Key Laboratory of Stars and Interstellar Medium, Xiangtan University, Xiangtan 411105, People's Republic of China. 
         \and
   Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China.
    }

    \authorrunning{Feng et al.}
  \abstract
  % context heading (optional)
   {}
  % aims heading (mandatory)
{The formation mechanisms of spectrally diverse hot subdwarfs remain unclear. While existing mass distribution analyses suggest additional channels beyond helium white dwarf (He-WD) mergers contribute to He-rich subdwarf formation, these conclusions are constrained by the limited sample sizes of mass-measured He-rich objects.}
  % methods heading (mandatory)
{We developed a deep leaning model that combines a convolution neural network (CNN) with a squeeze-and-excitation (SE) block to calculate synthetic spectral energy distributions (SEDs) for 1,012 spectroscopically confirmed hot subdwarfs. By directly comparing synthetic SEDs and the observed flux density, we derived stellar parameters (mass, radius, and luminosity) for an unprecedented number of hot subdwarf stars, enabling more conclusive channel discrimination than prior studies.}
  % results heading (mandatory)
{The mass distribution of sdB/sdOB stars confirmed the results from model predictions of binary population synthesis (BPS). A primary and secondary peak (i.e., around 0.56 and 0.4 ${\rm M}_{\odot}$) is obviously presented in the mass distribution of He-rich  hot subdwarf stars. By comparing this with the results from the predictions of the recent BPS model, we propose that the merger of two He-WDs could produce most of the observed He-rich hot subdwarf stars, but the mass transfer during the stable Roche lobe overflow phase in binary evolution should be partially conserved.}
  % conclusions heading (optional), leave it empty if necessary 
   {}
   {}

   \keywords{(Stars:) subdwarfs;
                binaries (including multiple): close;
                Stars: evolution;
                Stars: fundamental parameters
               }

   \maketitle
%
%________________________________________________________________

\section{Introduction}
Hot subdwarf stars are positioned between main-sequence (MS) and white dwarf (WD) regions in the Hertzsprung-Russell diagram (HRD) \citep{1986A&Aheber}. Hot subdwarf stars have very low stellar masses (e.g., 0.5 ${\rm M}_{\odot}$) and thin envelopes ($M_{\text{env}} \le 0.02 M_{\odot}$), but exhibit very high effective temperatures (e.g., $20,000~\mathrm{K} \leq T_{\text{eff}} \leq 70,000~\mathrm{K}$) and large surface gravities (e.g., $5.0 \leq \log g \leq 6.5$). These blue stars have a diversity of atmospheric compositions that could present atmospheres from pure hydrogen (H) to pure helium (He). According to their spectral line features, hot subdwarfs can be classified into sdB, sdO, sdOB, He-sdB, He-sdO, and He-sdOB types \citep{1990_A&AS_moehler,2017_A&A_geier, 2018ApJ...868...70L}. In addition, \citet{2013A&A...551A..31D} designed an MK-like classification scheme by which hot subdwarf stars could be classified in a similar way as normal MS stars (also see \citealt{2021MNRAS.501..623J, 2024PASJ...76.1084Z}).  SdB stars are also referred to as extreme horizontal branch (EHB) stars due to their location at the bluest end of the horizontal branch (HB
). More characteristics of hot subdwarf stars were described in recent excellent reviews of \citet{2009ARA&A..47..211H, 2016PASPHeber}.

Generally, single stars with such low stellar masses as hot subdwarfs cannot evolve into core He burning or even a late stage within Hubble time. Although formation channels connected with single stars were proposed by several authors (\citealt{1993ApJ...407..649C, 2008A&A...491..253M}), \citet{2020A&A...642A.180P} found that binary interactions are always required in the formation of hot subdwarf stars, including those in wide binary systems.
Furthermore, most of the sdB-type hot subdwarfs were found in close binaries \citep{2001MNRASmaxted, 2004Ap&SSnapiwotzki,2011MNRAScopperwheat, 2025A&A...693A.121H}. These results indicate that binary evolution could provide natural explanations for the formation of hot subdwarfs. 

\citet{2002MNRASHan, 2003MNRASHan} investigated the characteristics of hot subdwarfs formed through binary evolution using the binary population synthesis (BPS) method. They found that  stable Roche lobe overflow (RLOF), common envelope (CE) ejection, and the merger of two helium white dwarfs (He-WDs) could produce sdB stars in long-period binaries, short-period binaries, and single sdB stars, respectively. Most of the observed characteristics for sdB stars could be satisfied by their model predictions. \citet{2012MNRASZhang} and \citet{2017ApJ...835..242Z} studied in detail the mergers of two He-WDs and a WD + MS star, respectively. They found that both channels could produce He-rich sdB stars such that their observed features could be satisfied. 
In addition, \citet{2022MNRASmiller} proposed that the merger of carbon-oxygen WDs with He WDs could form He-rich hot subdwarfs with atmospheres strongly enriched by carbon and oxygen. Recently, \citet{2021MNRAS.507.4603M} and \citet{2024RAA....24e5003J} proposed that the MS companion of massive WD could survive the type Ia supernova (SN) explosions and become intermediate He-rich hot subdwarf stars if their envelopes are ejected during the explosion. Their model predicted that the peak of mass distribution for survived stars is around 0.4 $\rm M_{\odot}$, which is consistent with the results in \citet{2023ApJLei}. However, the birth rate of this channel would be too low compared to the number of He-rich hot subdwarf stars discovered up to now \citep{2021MNRAS.507.4603M}. \citet{2024ApJ...964...22L} proposed that massive sdO/B stars could be produced from CE ejection process with an asymptotic giant branch (AGB) star. This model could explain the recently discovered sdO binary star SMSS-J1920 \citep{2022MNRAS.515.3370L}, for which strong Ca H and K lines are presented in its spectrum. 

With a large number of new hot subdwarf stars having been discovered using the new data release of spectroscopic surveys \citep{2017OAst...26..164G, 2019MNRAS.486.2169K, 2018ApJ...868...70L, 2019ApJ...881..135L, 2020ApJ...889..117L, 2023ApJ...942..109L, 2019ApJ...881....7L, 2021ApJS..256...28L,2019A&A...630A..80S, 2021MNRAS.501..623J}, also with the help of the Gaia mission Early Data Release 3 (Gaia EDR3, \citealt{2021A&A...649A...1G}),  \citet{2022A&ACulpan} compiled two catalogs for known hot subdwarf stars and their candidates, respectively (also see the serial studies, \citealt{2017_A&A_geier, 2019A&A...621A..38G, 2020A&A...635A.193G}). The catalog of known hot subdwarfs contains more than 6000 identified hot subdwarf stars, in which more than 3000 stars have atmospheric parameters and 2,700 stars have radial velocities. The candidate catalog contains more than 60,000 objects, which can be used as reliable input catalogs for follow-up analysis with new observed data from spectroscopy. 

Driven by the availability of large observational datasets, machine learning has been extensively applied in astronomical research, including studies of hot subdwarfs.  \citet{2019ApJbu} combined a convolution neural network (CNN) with a support vector machine (SVM) to classify hot subdwarf stars in the dataset of Large Sky Area Multi-Fiber Spectroscopic Telescope (LAMOST, \citealt{2012RAA....12.1197C, 2012RAA....12..723Z}). \citet{2022ApJStan} also used CNN to search for hot subdwarf candidates in a larger LAMOST spectra dataset. More recently, \citet{2024PASJliu} applied the machine learning method to identify hot subdwarfs using photometric data, while  \citet{2024ApJSchen}  constructed a Se-ResNet+SVM model to search for hot subdwarf stars and predicted their atmospheric parameters. 

Despite considerable progress having been made both in simulation and observation of hot subdwarf stars, questions remain regarding the formation channels for those stars. Some notable challenges stem from the determination of physical parameters, such as masses, radii, and luminosities. Accurate mass estimation methods include asteroseismology and Eclipse binary. However, their applications are limited by the strict requirements of observed data that only dozens of hot subdwarfs have accurate masses determined by these methods \citep{2012A&AFontaine}. To analyze the mass distribution of hot subdwarfs with different companion types, \citet{2022A&ASchaffenroth} obtained masses, radii, and luminosities for 68 hot subdwarf stars by comparing observed fluxes with synthetic fluxes calculated from spectral energy distributions (SEDs). With the help of these parameters, and with the help of space-based light curves from the Transiting Exoplanet Survey Satellite (TESS,  \citealt{2015JATIS...1a4003R}) and the K2 space mission \citep{2014PASP..126..398H}, \citet{2023A&A...673A..90S} further obtained the masses of 27 companions for these hot subdwarf binaries. 

Using a similar method, \citet{2023ApJLei} obtained masses, radii, and luminosities for 664 single-lined hot subdwarfs with the spectra of LAMOST. Based on this large size of sample, they obtained the mass distributions for hot subdwarf stars with different spectral types, and compared them with the results from model predictions. Although the mass distribution of sdB/sdOB stars is consistent with the prevailing model of \citet{2002MNRASHan, 2003MNRASHan}, they found that He-rich hot subdwarf stars have an obviously lower peak of mass distribution (e.g., around 0.42 ${\rm M}_{\odot}$) than model predictions (e.g., 0.5 - 0.6 ${\rm M}_{\odot}$, see \Fig  12 in \citealt{2023ApJLei}). Based on these comparisons, they proposed that besides the two He-WDs merging channel, there should be other important formation channels for He-rich hot subdwarfs. However, only 39 He-rich stars were used in their statistical analysis, and such a small size sample would influence the final conclusions. 

As a further study of \citet{2023ApJLei}, we constructed a deep learning architecture that integrates a CNN with a squeeze-and-excitation (SE) block to calculate the synthetic SEDs for more than 2000 known hot subdwarfs selected in \citet{2022A&ACulpan} with the spectra of LAMOST and Sloan Digital Sky Survey (SDSS, \citealt{2020ApJS..249....3A}). 
Using these SEDs, we calculated the synthetic fluxes at stellar surface and compared them with fluxes observed on the Earth to obtain masses, radii, and luminosities for the selected hot subdwarf stars. These parameters were used in following statistical analysis to help us understand the formation of hot subdwarfs more deeply. The structure of this paper is as follows. Sample selection and methods are described in Section 2. Section 3 presents our main results. A detailed discussion for mass distribution with a large size of sample is given in Section 4. A brief summary in Section 5 closes this paper. 

\section{Methodology and sample selection}
\subsection{Deep learning model SENN }

\begin{figure*}
    \centering
    \includegraphics[width=0.8\linewidth]{picture/SENN.pdf}
    \caption{Structure of deep learning model SENN. See section 2.1 for detailed description.  }
    \label{Fig 1}
\end{figure*}

We designed a deep learning model, which integrates a CNN neural network with a SE block (hereafter SENN), to calculate synthetic SEDs for selected hot subdwarf stars. As is shown in \Fig \ref{Fig 1}, SENN accepts normalized stellar spectra with a shape of 2,984 $\times$ 1 as inputs, and processes them through a hierarchical architecture to calculate SEDs. The model comprises three modules. The initial feature embedding module uses a 1D convolution layer (64 filters, 3 kernels, and ReLU activation) to capture local spectral features, and reshape the output into a 64 $\times$ 2982 tensor. The SE feature refinement module is then employed, in which a global average pooling (GAP) layer compresses global spatial information into a channel-wise descriptor. This descriptor is connected by two dense layers to fully capture channel-wise dependencies and compute channel-specific weights. These weights are then applied to the original feature map, producing the final weighted feature map. Finally, the SED regression module is used to flatten the refined features, pass them through a 64-neuron dense layer, and generate SEDs through an output layer. 

 The SENN model takes observed spectra as input features and their corresponding synthetic SEDs as training labels. Through training processes, SENN 
established a mapping relationship between observed spectra and synthetic SEDs, enabling it to calculated the synthetic SED associated with a given observed spectrum. With these synthetic SED and the observed magnitudes retrieved from the virtual observatory (VO), we could calculate the masses, radii and luminosities of hot subdwarf stars reported in \citet{2022A&ACulpan}, if their atmospheric parameters are known (e.g., $T_{\text{eff}}$ and $\log g$. see Section 2.3).

\subsection{Training dataset}

In previous studies (\citealt{2018ApJ...868...70L,2019ApJ...881..135L, 2020ApJ...889..117L, 2023ApJ...942..109L}),  950 single-lined hot subdwarf stars were identified with LAMOST spectra. Their atmospheric parameters, i.e., effective temperature ($T_{\text{eff}}$), surface gravity (${\rm log} g$), and logarithmic number ratio of helium to hydrogen ($\log(n_{\text{He}}/n_{\text{H}})$), were determined by fitting H and He profiles with synthetic spectra. The synthetic spectra and SEDs of these stars were generated using \texttt{Synspec} (version~49; \citealt{2007ApJSLanz}) with nonlocal thermodynamic equilibrium (NLTE) Tlusty atmospheres (version 204; \citealt{2017arXivHubeny}). See \citet{2018ApJ...868...70L} and \citet{2014ASPC..481...95N} for more details about the synthetic spectra and SED calculation. 

In this study, the synthetic SEDs of the 950 stars mentioned above were used as a training sample for the SENN model, while the observed spectra of these stars were adopted as SENN inputs. 80\% of the selected stars were used as training dataset, while 20\% of them were used as test dataset. 

\subsection{Known hot subdwarfs and their spectra}
\citet{2022A&ACulpan} compiled a catalog of the known hot subdwarf stars that contains 6,616 unique sources collected from the literature. This catalog provides atmospheric parameters for 3,087 stars and radial velocities for 2,791 stars. To obtain the observed spectra for these stars, we cross-matched the catalog with the dataset of LAMOST DR11 and SDSS DR18, and then totally obtained 4,289 spectra. After removing duplicated sources 
and low-quality spectra (e.g., that had a signal-to-noise ratio in the u band of less than 5.0), 3,411 unique spectra of known hot subdwarfs were left. Since $T_{\text{eff}}$ and $\log g$ are necessary parameters to determine stellar masses and luminosities in this study, only stars with the two parameters reported in the catalog of \cite{2022A&ACulpan} were finally selected for the following analysis, and it contains 2047 hot subdwarf stars.

 Within the catalog of \citet{2022A&ACulpan}, the atmospheric parameters of hot subdwarfs were collected from different studies; these were predominantly derived through spectral fitting. However, as we lack the specific atmospheric models and parameters used in these works to calculate synthetic spectra for hot subdwarf stars, we did not employ the conventional method of deriving the SEDs for this selected sample with known atmospheric parameters. Instead, we utilized the highly efficient SENN model to compute the SEDs for these hot subdwarfs with corresponding observed spectra.

Using the SENN model constructed in Section 2.1 and the training dataset selected in Section 2.2, synthetic SEDs for 2047 selected hot subdwarf stars were calculated; these were then used to obtain stellar masses, radii, and luminosities, as is described in the next section. Note that the SENN model is much more efficient at calculating synthetic SEDs than traditional methods. For example, it just needs tens of seconds to obtain synthetic SEDs for 2000 stars, and would be more powerful when confronting with huge numbers of stars. 

 In Fig \ref{Fig 2}, we compare the ‌synthetic SED of a hot subdwarf star calculated using the ‌SENN model (dashed blue curve) with that computed directly from the ‌Tlusty model (solid red curve). The near-identical agreement between the two SEDs ‌fully demonstrates that utilizing the deep learning method  to calculate theoretical SEDs for hot subdwarfs is both ‌feasible and reliable.


\subsection{Determination of masses, radii, and luminosities for selected hot subdwarf stars}
Using the synthetic SEDs calculated by SENN model, we obtained the flux density at the stellar surface within a specific pass-band. The synthetic flux density can be compared with the flux density observed on the Earth to determine the stellar angular diameters. The synthetic flux density at stellar surface, observed flux density on the Earth, stellar angular diameter, radius, and distance of the star have the following relationships \citep{2018OAst...27...35H, 2023ApJLei}: 
\begin{equation}
    \frac{f\left ( \lambda  \right ) }{F\left ( \lambda  \right ) } =\frac{\theta ^{2} }{4} =\frac{R^{2} }{d^{2}}
    \label{eq 1}
,\end{equation}

where $f(\lambda)$ is the flux density observed on the Earth,
$F(\lambda)$ is the synthetic flux density at stellar surface, $\theta$ is the angular diameter of the star, $R$ is the radius of the star, and $d$ is the stellar distance from the Earth. 

If the distance of a star is known, the stellar radius can be determined according to Equation (1). Then, with the help of  atmospheric parameters (e.g., $\log g$ and $T_{\text{eff}}$), we can obtain the mass and luminosity of the star using the following equations: 

\begin{align}
M & = \frac{gR^{2}}{G} \label{eq 2} \\
\frac{L}{L_{\odot} } & = \left ( \frac{R}{R_{\odot}} \right ) ^{2}\left ( \frac{T_{\rm eff}}{T_{\odot}} \right ) ^{4} \label{eq 3}
,\end{align}
 where $L_{\odot}$, $R_{\odot}$, and $T_{\odot}$ are the luminosity, radius, and effective temperature of the Sun, respectively. 

Distances for the stars selected in this study were calculated using Gaia DR3 parallaxes \citep{2023A&Agaiadr3} after zero-point correction \citep{2021A&A...649A...4L}. The flux density observed on the Earth were obtained by searching for photometric data using VO SED Analyzer (VOSA, \citealt{2008_A&A_bayo}) of the Spanish Virtual Observatory (SVO) and converting observed magnitudes into fluxes. 

When converting the observed magnitude into the corresponding observed flux on the Earth, an extinction correction must be applied. To determine the visual extinction, $A_{\rm V}$, we fit the synthetic SEDs with the observed fluxes by treating $A_{\rm V}$ as a free parameter. For each trial value of $A_{\rm V}$, the wavelength-dependent extinction, $A_\lambda$, was computed as
\begin{equation}
A_\lambda = A_{\rm V} \times (k_\lambda / k_V),
\end{equation}
where $k_\lambda / k_{\rm V}$ represents the normalized extinction law. The functional form of $k_\lambda / k_V$ was adopted from \citet{1999PASP..111...63F} with $R_{\rm V} = 3.1$ and $k_{\rm V} = 211.4$. The observed fluxes, $f_{\text{obs}}(\lambda)$, were then dereddened using
\begin{equation}
f(\lambda) = f_{\text{obs}}(\lambda) \times 10^{\,A_\lambda / 2.5},
\end{equation}
which yielding the intrinsic flux density, $f(\lambda)$. The best-fit $A_{\rm V}$ were determined simultaneously through $\chi^2$ minimization between the dereddened fluxes and the synthetic SEDs.  

\begin{figure}
    \centering
    \includegraphics[width=1.0\linewidth]{picture/spec-57919-HD173133N504043V01_sp07-209.fits.pdf}
    \caption{Comparison between the synthetic SED calculated with Tlusty model (solid red curve) and the SED calculated by SENN model (dashed blue curve). The bottom panel displays the residual differences between the two SEDs.}
    \label{Fig 2}
\end{figure}

 Using the method described above, we obtained masses, radii, and luminosities for the selected 2,047 hot  subdwarf stars. The stars with mass values lower than 0.1 $M_\odot$ or higher than 1.0 $M_\odot$ are not reported here, since these values could be significantly influenced by large uncertainties of input parameters, and are not reasonable for normal hot subdwarf stars. Furthermore, only stars with reliable parallaxes (e.g., $\sigma_{\varpi}/\varpi\leq$ 0.2) are included in the following analysis. Thus, we finally have 1,012 hot subdwarf stars for which the main parameters are reported in this study (see Table \ref{tab1}), consisting of 606 sdB, 64 sdO, 206 sdOB, 9 He-sdB, 66 He-sdO, and 61 He-sdOB stars, respectively. The obtained parameter values are presented in Table \ref{tab1}. 

 Using Monte Carlo methods, we obtained the statistical  errors ($\sigma_{sta}$) for mass, radius, and luminosity of the selected stars. Specifically, we calculated the median values for the three parameters, along with their values at the 16th and 84th percentiles. This defines a 68\% confidence interval around the median values and provides the corresponding asymmetric uncertainties.
 
 On the other hand, we cross-matched the sample from this work with the sample from \citet{2022A&ASchaffenroth}, and identified 24 common sources. By comparing the masses, radii, and luminosities for these 24 sources, we determined the  systematic errors ($\sigma_{sys}$), which were calculated using the following formula:
\begin{equation*}
\sigma_{sys}=\frac{\sum(x_{i}-y_{i}) }{N}   
,\end{equation*}
 where $x_{i}$ is the parameter value calculated in this work, $y_{i}$ is the parameter value calculated in \citet{2022A&ASchaffenroth}, and $N$ is the number of common sources. The final errors ($\sigma$) was then calculated as
 \begin{equation}
    \sigma=\sqrt{\sigma_{sta}^2+\sigma_{sys}^2}    
 .\end{equation}
 Using the method described above, we obtained systematic errors of ‌0.02 $\rm M_{\odot}$ , ‌0.002 $\rm R_{\odot}$, and ‌1.4 $\rm L_{\odot}$ for mass, radius and luminosity, respectively. In Table \ref{tab1}, we presented the ‌final errors for these three parameters, incorporating both statistical errors and systematic errors.

\section{Results}
\subsection{Main parameters for selected hot subdwarf stars}

\begin{figure*}
    \centering
    \includegraphics[width=0.85\linewidth]{picture/mass_vs_parameters.pdf}
    \caption{Relationship between mass and atmospheric parameters for the selected hot subdwarfs. Labels with different colors indicate the spectral classification from \citet{2022A&ACulpan}.}
    \label{Fig 3}
\end{figure*}

The masses, radii, luminosities, and other important parameters for the 1,012 selected hot subdwarf stars are presented in Table \ref{tab1}. From left to right, it gives object names, Gaia source\_id, spectral classification, $T_{\text{eff}}$, ${\rm log} g$, extinction coefficient ($A_{\rm V}$), Gaia parallax after zero-point correction, and four parameters obtained in this study, i.e., angular diameter, radius, luminosity, and mass. Of these stars,  601 have had their masses reported in previous studies. This means that the masses of 411 hot subdwarfs are reported for the first time in this study. 

The four panels in \Fig \ref{Fig 3} present the relationships between obtained masses and atmospheric parameters; for example, from top left to bottom right, it gives planes of mass versus $T_{\text{eff}}$, ${\rm log} g$, $\log (n_{\text{He}}/n_{\text{H}})$, and $\log L$, respectively. It can be observed that most of the selected hot subdwarfs have radii of between 0.1 and 0.3 $\rm R_{\odot}$, luminosities between 0.5 and 2.5 $\rm L_{\odot}$, and masses between 0.2 and 0.8  ${\rm M}_{\odot}$, respectively. These results are consistent with those of \citet{2023ApJLei}; see also \Fig  3 in their study. 

\subsection{Comparison with previous study}
In order to evaluate the reliability of the masses, radii, and luminosities obtained in this study, we cross-matched the selected hot subdwarfs with the stars analyzed in \citet{2022A&ASchaffenroth} and \citet{2023ApJLei}, respectively. Comparisons of masses, radii, and luminosities for the common stars are shown in the panels of \Fig  \ref{Fig 4}. As can be see in the figure, there is a notable agreement between the results of the present study and previous research, particularly with respect to radius and luminosity. However, the comparison of mass exhibits a slightly larger dispersion than the other two parameters. This is mainly due to the fact that the large uncertainties of surface gravity would directly influence stellar mass determination (see Equation \ref{eq 2}), while it is not used in the determination of radius and luminosity (see Equation \ref{eq 1} and \ref{eq 3}). 
\begin{figure*}
    \centering
    \includegraphics[width=0.3\linewidth]{picture/contrast_Schaffenroth_radius.pdf}
    \includegraphics[width=0.3\linewidth]{picture/contrast_Schaffenroth_mass.pdf}
    \includegraphics[width=0.3\linewidth]{picture/contrast_Schaffenroth_lumin.pdf} 
    
    \includegraphics[width=0.3\linewidth]{picture/contrast_Lei_radius.pdf}
    \includegraphics[width=0.3\linewidth]{picture/contrast_Lei_mass.pdf}
    \includegraphics[width=0.3\linewidth]{picture/contrast_Lei_lumin.pdf}
    \caption{Top panels: Comparison of radii, masses, and luminosities between this study and \citet{2022A&ASchaffenroth}. Bottom panels: Comparison of radii, masses, and luminosities between this study and \citet{2023ApJLei}. }
    \label{Fig 4}
\end{figure*}

\subsection{Mass distribution of hot subdwarf stars}

\begin{figure*}
    \centering
    \includegraphics[width=0.45\linewidth]{picture/distribute_mass_plx.pdf}
    \includegraphics[width=0.45\linewidth]{picture/distribute_mass_logg.pdf}
    \caption{Mass distribution for selected hot subdwarfs with different precisions of parallax (left panel) and surface gravity (right panel). See context for details.}
    \label{Fig 5}
\end{figure*}

Since the precision of masses obtained in this study depends mainly on the precision of the parallax (i.e., distance) and surface gravity, we show the mass distribution with different uncertainties of parallax and surface gravity in \Fig  \ref{Fig 5}. In the left panel, the entire sample has been split into three groups according to the relative parallax uncertainty (e.g., $\sigma_{\varpi}/\varpi\leq$ 0.2, 0.1 and 0.05, respectively). As expected, with the parallax precision increasing from 0.2 to 0.05, the number of hot subdwarf stars in each mass bin decreases. However, the three groups of hot subdwarfs show very similar mass distributions. As can be seen, all groups have a wide mass range of 0.1 - 1.0 ${\rm M}_{\odot}$ and present two distinct peaks: a primary peak at 0.47 ${\rm M}_{\odot}$ and a secondary peak at  0.37 ${\rm M}_{\odot}$ (see detailed discussion on the mass distribution in Section 4). On the other hand, the entire sample has been separated into three other groups in the right panel according to the relative uncertainty of surface gravity (e.g., uncertainty of ${\rm log} g$ over ${\rm log} g$ less than 0.09, 0.02, and 0.007, respectively). The three groups in this panel show very similar mass distributions to those in the left panel, which also present two distinct peaks at the same mass values. Considering similar mass distributions for different precisions of parameters, all the selected hot subdwarf stars were adopted in the following analysis. 

\begin{figure}
    \centering
    \includegraphics[width=1\linewidth]{picture/distribute_mass_spclass.pdf}
    \caption{Mass distributions for selected hot subdwarf stars with different spectral classifications.}
    \label{Fig 6}
\end{figure}
In \Fig  \ref{Fig 6}, we present the mass distributions for hot subdwarf stars according to their spectral classifications; for example, He-poor sdB/sdOB (dashed red histogram), He-rich stars (blue histogram, including He-sdB, He-sdO, and He-sdOB), and He-poor sdO (green histogram). The mass distribution of He-poor sdB/sdOB stars, which make up the majority of the hot subdwarf population, closely mirrors that of the entire sample, with two peaks at  0.47 and  0.37  ${\rm M}_{\odot}$. For He-rich hot subdwarfs, two peaks are also present, but the values differ from those seen in sdB/sdOB, occurring at 0.4 and 0.58  ${\rm M}_{\odot}$ (see the discussion in Section 4.3), respectively. However, He-poor sdO stars have a much flatter mass distribution and consist primarily of low-mass stars. 

\section{Discussion}
\subsection{Comparison of mass distribution with previous study}

We compared the mass distribution of the entire sample with that of \citet{2023ApJLei} in \Fig  \ref{Fig 7}. Both mass distributions exhibit a wide mass range of 0.1 - 1.0 ${\rm M}_{\odot}$. This is likely due to the large uncertainties of the input parameters (e.g., parallax and surface gravity) that were used in both studies to estimate the mass of hot subdwarf stars. In general, the two mass distributions show a basic consistency in that a primary peak appears around 0.47 ${\rm M}_{\odot}$, and the distribution trend for the low-mass and high-mass sides is nearly the same. As is described in Section 3.3, in addition to the primary one, a second mass peak (around 0.37 ${\rm M}_{\odot}$) is also clearly presented in the mass distribution of this study (red histogram). Nevertheless, this distinct feature is much more obscure in the mass distribution of \citet{2023ApJLei} (dashed blue histogram). Note that the input surface gravities used to calculate mass for the two studies are from different catalogs. The parameters used in this study are from the catalog of \citet{2022A&ACulpan}, which were collected from other literature \citep{2020A&A...635A.193G, 2021ApJS..256...28L, 2021MNRAS.501..623J}. While the parameters used in \citet{2023ApJLei} are from \citet{2018ApJ...868...70L, 2019ApJ...881..135L, 2020ApJ...889..117L, 2023ApJ...942..109L}. The method of spectral analysis and the observed spectra used in these studies could be different, which would lead to the differences in obtained parameters. For example, the systematic errors of surface gravity between \citet{2021ApJS..256...28L} and Lei et al. are around 0.07 dex. Thus, these differences would cause an inconsistency in the mass distribution, especially for some local features.  

\begin{figure}
    \centering
    \includegraphics[width=1\linewidth]{picture/distribute_mass_with_previos_work.pdf} 
    \caption{Comparison of mass distribution between this study (red histogram) and \citet{2023ApJLei} (dashed blue histogram).}
    \label{Fig 7} 
\end{figure}

\subsection{Mass distribution of He-poor sdB/sdOB stars} 

\citet{2003MNRASHan} (hereafter Han03) have carried out a detailed BPS study on the formation of sdB-type hot subdwarfs. In their study, 12 sets of models were constructed with various input parameters to investigate their impacts on the formation of sdB stars. Among these models, the model of set 2 (with $Z=0.02$, a flat initial mass ratio distribution,  $q_{\rm crit}$ = 1.5, $\alpha_{\rm CE}$ = $\alpha_{\rm th}$ = 0.75) was chosen as the best-fitting model in their study, capable of satisfying most of the observed properties for sdB stars. These results from BPS models are highly convenient for making direct comparisons with observations. 

The left panel of \Fig  \ref{Fig 8} exhibits comparisons between the mass distribution of set 2 in Han03 (solid black curve) and that of this study (gray histogram). As is discussed in Section 4.1, the mass distribution of sdB/sdOB stars in this study shows a wide range from 0.1 to 1.0  ${\rm M}_{\odot}$ due to the large uncertainties of the surface gravity and parallax, while the mass distribution of Han03 ranges from 0.3 to 0.8 ${\rm M}_{\odot}$, which is narrower than that of this study. Since the minimum core mass for He ignition is about 0.3  ${\rm M}_{\odot}$ (see Table 1 and 2 in \citealt{2002MNRASHan}), there is a cutoff of mass around 0.3  ${\rm M}_{\odot}$ for the low-mass side in the distribution of Han03. Therefore, the stars with masses of less than 0.3 ${\rm M}_{\odot}$ in this study could be low-mass WDs or extremely low-mass (ELM) WDs (also see the discussion in Section 4.4 of \citealt{2023ApJLei}). 

Furthermore, three distinct peaks are presented in the mass distribution of Han03, at around 0.46, 0.4, and 0.36 ${\rm M}_{\odot}$. These were generated through the combination of the first CE, the second CE, and the first stable RLOF channels (see \Fig  12 and Section 7.3 in their study). As can clearly be seen in the panel, the primary mass peak (around  0.47 ${\rm M}_{\odot}$) in our study is consistent very well with the model prediction of Han03. The secondary mass peak in our study (e.g., around 0.36  ${\rm M}_{\odot}$) corresponds to the left mass peak in Han03, which demonstrates that these stars are mainly produced by the stable RLOF channel with their progenitors at the Hertzsprung gap evolution stage. On the other hand, the middle mass peak (e.g., around 0.4 ${\rm M}_{\odot}$) in Han03 is obscure in our mass distribution, for which the stars are mainly produced from the first and the second CE channel with the mass of progenitors around 1.9 ${\rm M}_{\odot}$ (see Table 1 in \citealt{2002MNRASHan}). 


\begin{figure*}
    \centering
    \includegraphics[width=0.4\linewidth]{picture/distribute_mass_sdB_sdOB_with_hanset2.pdf}
    \includegraphics[width=0.4\linewidth]{picture/distribute_mass_sdB_sdOB_with_segovia2024_all.pdf}
    \caption{Left panel: Comparison of mass distribution for sdB/sdOB stars between this study (gray histogram) and Han03 (solid black curve). Right panel: Comparison of mass distribution for sdB/sdOB stars between this study (gray histogram) and \citet{2025PASA...42...12R} (solid black curve).}
    \label{Fig 8}
\end{figure*}

\citet{2025PASA...42...12R} also studied the impacts of various parameters on the formation of sdB stars through BPS methods. Unlike previous BPS models (e.g., Han03, \citealt{2012ApJ...746..186C}), they considered H-rich shells using an analytic prescription in the study that gets more closer to the real evolution of sdB stars. In the right panel of \Fig  \ref{Fig 8}, we compare the mass distribution predicted by \citet{2025PASA...42...12R} with that of this study. The mass range of sdB stars predicted by \citet{2025PASA...42...12R} (shown by the solid black curve in the right panel) is roughly from 0.3 to 0.62 ${\rm M}_{\odot}$ and is a little narrower than that of Han03 and this study. Note that the results of two He-WDs merging are not included in the BPS code COMPAS \citep{2022ApJSRiley} that was used in their study\footnote{Although the results of two He-WDs merging are not taken into account in COMPAS, \citet{2025PASA...42...12R} still give a simplified and qualitative analysis on this channel; see Section 3.2.4 in their study and Section 4.3 in this study for details.}; therefore, some massive hot subdwarf stars are absent in the mass distribution of \citet{2025PASA...42...12R}. As is shown in the panel, a sharp primary peak around 0.47 ${\rm M}_{\odot}$ is clearly presented in both two mass distributions, which indicates a good agreement between the predictions of BPS model and observations. Furthermore, a second peak of around 0.37 ${\rm M}_{\odot}$ is also presented in the mass distribution of \citet{2025PASA...42...12R} (see also Fig 7 in \citealt{2025MNRAS.539.3273R}). Although the relative number of stars around this peak in their study is less than that of present study, it shows a one-to-one correspondence between the two studies. 

\subsection{Mass distribution for He-rich Hot subdwarfs}
Two He-WDs merger is considered as the dominant formation channel for He-rich hot subdwarf stars  (e.g., $\log(n{\rm He}/n{\rm H})$ $>$ -1.0). The mass range of He-rich hot subdwarfs produced by this channel in Han03 is roughly from 0.42 to 0.76 ${\rm M}_{\odot}$, with a relatively flat peak between 0.5 and 0.6 ${\rm M}_{\odot}$. \citet{2012MNRASZhang} studied three different merging models for two He-WDs, and found that the results could explain most of observed gravities, effective temperatures, and surface chemical abundance for He-rich stars. Furthermore, \citet{2017ApJ...835..242Z} found that the mergers of He-WDs with low-mass MS stars could contribute to the formation of intermediate He-rich (iHe-rich) hot subdwarfs. On the other hand, \citet{2021MNRAS.507.4603M} proposed that MS companions could survive Ia SN explosions and evolve as He-rich hot subdwarfs (see also \citealt{2024RAA....24e5003J}). They obtained a wide mass range of 0.35 to 1.0 ${\rm M}_{\odot}$ for this formation channel, with a mass peak at around 0.4 ${\rm M}_{\odot}$. 

As was mentioned above, \citet{2023ApJLei} obtained a wide mass range and a sharp mass peak around 0.42 ${\rm M}_{\odot}$ for He-rich hot subdwarf stars observed by LAMOST. The mass peak in their study is  much lower than the prediction of Han03 and \citet{2012MNRASZhang}. On the other hand, the mass peak in \citet{2023ApJLei} is basically consistent with the value predicted by \citet{2021MNRAS.507.4603M}, but the birth rate for this channel could be too low to explain the discovered He-rich hot subdwarfs (see Section 4.1 of \citealt{2021MNRAS.507.4603M}). According to these results, \citet{2023ApJLei} proposed that, besides the two He-WDs merger, some other channels could contribute to the formation of He-rich hot subdwarfs. \citet{2024ApJS..271...21L} came to a similar conclusion after analyzing the surface chemical abundances of He-rich hot subdwarf stars. 

\begin{figure*}
    \centering
    \includegraphics[width=1\linewidth]{picture/distribute_mass_He_rich.pdf}
    \caption{Left panel: Comparison of mass distribution for He-rich stars between this study (gray histogram) and \citet{2023ApJLei} (dotted red curve). Middle panel: Comparison of mass distribution for He-rich stars between this study and Han03 (solid black curve). Right panel: Comparison of mass distribution for He-rich stars between this study and \citet{2025PASA...42...12R} (dash-dotted blue curve). }
    \label{Fig 9}
\end{figure*}

However, due to the limited size of the sample analyzed in \citet{2023ApJLei}, for example only 39 He-rich stars in group 3 of their sample (see Table 2 in their study), it would influence the statistical results for mass distribution without completeness of sample. Thanks to the deep learning model SENN designed in the present study (see Section 2.1), more than 2000 synthetic SEDs of the selected hot subdwarfs were calculated. Using these information, the masses of 131 He-rich stars were determined in this study, with the sample size being much larger than that analyzed in \citet{2023ApJLei}. Therefore, a comparison of the mass distribution between this study and previous studies could help us better understand the formation of He-rich hot subdwarf stars. 

In the left panel of \Fig \ref{Fig 9}, we compare the mass distribution of He-rich stars obtained in the present study with that of in \citet{2023ApJLei}. As can be seen in the panel, it exhibits a relatively wide mass distribution that is roughly from 0.3 to 0.9 ${\rm M}_{\odot}$ for both two studies (Note that hot subdwarf stars with masses of less than 0.3 ${\rm M}_{\odot}$ cannot be produced in model predictions; thus, we excluded these stars from our results for the comparison). Two distinct mass peaks are presented in the mass distribution of this study (gray histogram): a primary peak around 0.56 ${\rm M}_{\odot}$ and a secondary peak around 0.4 ${\rm M}_{\odot}$. As was mentioned above, a sharp mass peak around 0.42 ${\rm M}_{\odot}$ is clearly shown in the mass distribution of \citet{2023ApJLei} (dotted red curve), which corresponds to the secondary mass peak in this study. However, it could be due to the small sample size used in the study of \citet{2023ApJLei} that no distinct mass peak between 0.5 and 0.6 ${\rm M}_{\odot}$ can be distinguished in their mass distribution. 

In the middle panel of \Fig  \ref{Fig 9}, we compare the mass distribution of He-rich stars in this study with that of Han03. As can be seen clearly in the panel, our mass distribution is much wider than that in Han03 (e.g., roughly from 0.42 to 0.76 ${\rm M}_{\odot}$, solid black curve). As was discussed in Section 4.1, this could be due to the large uncertainties of the parallax and gravity, which  significantly affect the precision of mass determination in this study. He-rich stars produced through two He-WDs merger in Han03 present a flat mass peak between 0.5 and 0.6 ${\rm M}_{\odot}$, which could corresponding roughly to the primary mass peak around 0.56 ${\rm M}_{\odot}$ in this study. However, the secondary mass peak around 0.4 ${\rm M}_{\odot}$ is not predicted by BPS models of Han03, which would demonstrate that two He-WDs merging channel cannot explain the formation of all He-rich hot subdwarf stars.       

Nevertheless, it leads to a little different conclusion when comparing our results with recent BPS model predictions from \citet{2025PASA...42...12R} (see the right panel in \Fig  \ref{Fig 9}). The mass distribution of He-rich stars from the merging channel of \citet{2025PASA...42...12R} (dash-dotted blue curve) presents two distinct mass peaks around 0.55 ${\rm M}_{\odot}$ and 0.41 ${\rm M}_{\odot}$ (see \Fig  14 in their study), which correspond perfectly to the two peaks shown in our study (gray histogram). However, the relative number of He-rich stars around the secondary mass peak (e.g., 0.41 ${\rm M}_{\odot}$) predicted by \citet{2025PASA...42...12R} is less than in this study. As is discussed in Section 3.2.4 of their study, the secondary mass peak heavily depends on mass-transfer accretion efficiency at stable RLOF phase. It would disappear when the accretion efficiency equals 0,   and it would merge with the primary peak and form a symmetrical distribution with a peak around 0.47 ${\rm M}_{\odot}$ if the accretion efficiency equals 1. 

 Note that the formation of two He-WD binaries\footnote{In \citet{2025PASA...42...12R}, the treatment of the two He-WDs' merging channel was very sample. They did not evolve the two He-WDs binary into merging phase, but instead considered the binary system as hot subdwarfs if they could merge within the age of our Universe (e.g., 14 Gyr).} requires two RLOF phases. The whole envelope of the primary star will be removed through stable mass transfer during the first RLOF phase, while the two components will be surrounded by a CE formed due to unstable mass transfer during the second RLOF phase, and the CE should be ejected before the two He-WDs binary is formed \citep{1984ApJ...277..355W}. The mass-transfer accretion efficiency during the first RLOF phase has significant effects on the loss of angular momentum due to mass loss from the system, which in turn determines the subsequent evolution of the orbital separation. Therefore, a lower accretion efficiency during the first RLOF phase would result in either systems not triggering CE events (too far from each other) or merging (too close), both resulting in a final configuration different from the two He-WD binaries. 

 Based on the discussion mentioned above, if the second mass peak (around 0.4 ${\rm M}_{\odot}$) is a real feature, the two He-WDs merging channel could produce most of the observed He-rich hot subdwarf  stars on the condition that mass transfer during the first RLOF phase to produce the two He-WDs' binary should be partially conserved, and the accretion efficiency is not too low.  

\section{Summary}

In this study, a deep learning model SENN was designed to calculate synthetic SEDs for a large number of known hot subdwarfs with the help of spectra from LAMOST and SDSS. Through the VOSA service of the Spanish Virtual Observatory, the physical parameters (e.g., masses, radii, and luminosities) of 1,012 hot subdwarfs were obtained by comparing the synthetic SEDs with the observed flux density. With the obtained parameters for a large sample size, comparisons between observations and model predictions would help us understand the formation of hot subdwarf stars more deeply and comprehensively. 

The two mass peaks (e.g., 0.47 and 0.37) in the mass distribution of sdB/sdOB stars are consistent with the prediction from BPS models, which corresponds to the production of CE ejection and stable RLOF respectively. There is also a primary peak (around 0.56 ${\rm M}_{\odot}$) and a secondary peak (around 0.4 ${\rm M}_{\odot}$) clearly visible in the mass distribution of He-rich stars. Although the primary mass peak is well predicted by BPS models, the occurrence of the secondary peak in the models depends heavily on the mass-transfer accretion efficiency at stable RLOF stage in binary evolution. These results indicate that the two He-WDs' merging channel could produce most of the observed He-rich hot subdwarf stars, but mass transfer should be partially conserved during the stable RLOF phase. Considering the large uncertainties of the masses obtained in this study, other alternative formation channels for He-rich hot subdwarfs would not be completely excluded. More accurate determinations of the physical parameters of hot subdwarf stars are urgently needed.      

It is important to note that although we could efficiently obtain synthetic SEDs of hot subdwarfs using the SENN model, deriving their masses, radii, and luminosities still requires atmospheric parameters ($T_{\text{eff}}$ and ${\rm log} g$). Accurate atmospheric parameters, however, must still be acquired through the conventional method of synthetic spectral fitting.‌

Fortunately, we are actively exploring the use of machine learning methods to predict the atmospheric parameters of hot subdwarfs \citep{2026ApJS..283...16L}. This approach, in the near future, should enable us to directly obtain both the atmospheric parameters and masses of hot subdwarfs using artificial intelligence techniques, without having to compute  stellar atmosphere models.

\section{Data availability}

Table \ref{tab1} is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.

\begin{acknowledgements}

 We greatly thank the referee for his/her valuable comments and useful suggestions, which help improve the manuscript significantly. We also thank Nicolás Rodríguez-Segovia for his detailed suggestions on the discussion of Section 4.3. 
This work acknowledges support from the National Natural Science Foundation of China (Nos. 12073020, 12588202 and  12273055), Scientific Research Fund of Hunan Provincial Education Department 
 grant No. 20K124 and 23A0132, the Strategic Priority Research Program of Chinese Academy of Sciences, grant No. XDB1160301. K.H. acknowledges support from the Scientific Research Funds of Hunan Provincial Education Department (Nos. 22A0099 and 24A0101). 
 This research has made use of the Spanish Virtual Observatory (\url{https://svo.cab.inta-csic.es}) project funded by MCIN/AEI/10.13039/501100011025/ through grant PID2020-112949GB-I00. Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.
\end{acknowledgements}

\bibliography{bib}
\bibliographystyle{aa}
\begin{appendix}
\onecolumn
\section{Main parameters for the selected hot subdwarf stars}
\begin{table}[!ht]
    \centering
    \caption{Main parameters for part of 1,012 selected hot subdwarf stars}
    {\fontsize{6}{10}\selectfont
    \setlength{\tabcolsep}{2pt}
    \begin{tabular}{lllllllllll}
    \hline
    \centering
    Name & \multicolumn{1}{c}{Source\_id} & \multicolumn{1}{c}{sp\_class} & \multicolumn{1}{c}{$T_{\rm eff}$} & \multicolumn{1}{c}{${\rm log} g$} & \multicolumn{1}{c}{$A_{\rm V}$} & \multicolumn{1}{c}{plx\_zp } & \multicolumn{1}{c}{Angular diameter} & \multicolumn{1}{c}{Radius} & \multicolumn{1}{c}{Luminosity} & \multicolumn{1}{c}{Mass} \\
    \multicolumn{1}{c}{} & \multicolumn{1}{c}{Gaia DR3} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{K} & \multicolumn{1}{c}{(cm\ s$^{-2}$)} & \multicolumn{1}{c}{mag} & \multicolumn{1}{c}{mas} & \multicolumn{1}{c}{log(rad)} & \multicolumn{1}{c}{R/R$_{\odot}$} & \multicolumn{1}{c}{L/L$_{\odot}$} & \multicolumn{1}{c}{M/M$_{\odot}$} \\ 
    \hline
    BD+423250 & 2105469320138052864 & sdB & $28700\pm1000$ & $5.08\pm0.02$ & 0.422 & $4.1508\pm0.0319$ & $-10.268^{+0.036}_{-0.020}$ & $0.287^{+0.003}_{-0.003}$ & $50.2^{+7.5}_{-6.8}$ & $0.36^{+0.03}_{-0.03}$ \\
BD+481777 & 825628714431805056 & sdO & $47565\pm251$ & $5.84\pm0.04$ & 0.010 & $4.2560\pm0.0617$ & $-10.503^{+0.019}_{-0.014}$ & $0.163^{+0.003}_{-0.003}$ & $122.7^{+4.7}_{-4.5}$ & $0.67^{+0.07}_{-0.07}$ \\
BD-32179 & 3068744791442939904 & sdO & $59121\pm686$ & $4.55\pm0.03$ & 0.010 & $1.0124\pm0.0666$ & $-10.458^{+0.020}_{-0.029}$ & $0.767^{+0.054}_{-0.047}$ & $6445.7^{+992.0}_{-811.9}$ & $0.76^{+0.13}_{-0.10}$ \\
BPSCS22890-74 & 4420741294390728448 & sdB & $24560\pm567$ & $5.14\pm0.06$ & 1.135 & $1.0170\pm0.0312$ & $-10.690^{+0.064}_{-0.016}$ & $0.430^{+0.014}_{-0.013}$ & $60.4^{+7.1}_{-6.4}$ & $0.93^{+0.15}_{-0.13}$ \\
Bal90100005 & 2845710850610464640 & sdB & $32789\pm123$ & $5.66\pm0.02$ & 0.302 & $1.2124\pm0.0358$ & $-11.098^{+0.011}_{-0.052}$ & $0.146^{+0.005}_{-0.005}$ & $22.2^{+2.0}_{-1.9}$ & $0.36^{+0.04}_{-0.03}$ \\
Bal90900004 & 1874534323107953664 & sdB & $42860\pm1590$ & $5.51\pm0.12$ & 0.168 & $1.3991\pm0.0553$ & $-10.883^{+0.018}_{-0.016}$ & $0.207^{+0.009}_{-0.008}$ & $129.9^{+23.4}_{-20.3}$ & $0.51^{+0.17}_{-0.13}$ \\
CBS250 & 1373881186687373696 & sdB & $32280\pm140$ & $5.76\pm0.03$ & 0.052 & $0.8909\pm0.0328$ & $-11.194^{+0.010}_{-0.007}$ & $0.159^{+0.006}_{-0.006}$ & $24.6^{+2.4}_{-2.2}$ & $0.53^{+0.06}_{-0.05}$ \\
CBS274 & 1605239049014274560 & sdOB & $36833\pm2000$ & $5.83\pm0.20$ & 0.030 & $0.3374\pm0.0601$ & $-11.708^{+0.014}_{-0.005}$ & $0.128^{+0.028}_{-0.020}$ & $27.3^{+15.3}_{-9.1}$ & $0.41^{+0.34}_{-0.18}$ \\
CBS282 & 1591755325606144640 & sdOB & $36728\pm932$ & $6.02\pm0.14$ & 0.457 & $0.3160\pm0.0622$ & $-11.696^{+0.024}_{-0.010}$ & $0.139^{+0.034}_{-0.023}$ & $31.5^{+17.6}_{-9.9}$ & $0.75^{+0.53}_{-0.29}$ \\
CBS374 & 1471988375850922368 & sdOB & $35110\pm73$ & $6.08\pm0.02$ & 0.093 & $1.0166\pm0.0376$ & $-11.195^{+0.025}_{-0.006}$ & $0.139^{+0.006}_{-0.005}$ & $26.3^{+2.5}_{-2.3}$ & $0.85^{+0.08}_{-0.07}$ \\
CBS98 & 810691780249660800 & sdOB & $33588\pm147$ & $5.39\pm0.03$ & 0.029 & $0.6313\pm0.0665$ & $-11.408^{+0.015}_{-0.008}$ & $0.136^{+0.016}_{-0.013}$ & $21.0^{+5.4}_{-4.1}$ & $0.16^{+0.05}_{-0.04}$ \\
EGGR113 & 4422600328035803264 & sdO & $61075\pm1182$ & $5.77\pm0.05$ & 0.146 & $0.7145\pm0.0416$ & $-11.398^{+0.005}_{-0.034}$ & $0.124^{+0.008}_{-0.007}$ & $192.9^{+29.2}_{-24.8}$ & $0.33^{+0.06}_{-0.05}$ \\
FBS0021+418 & 382086995098288896 & sdB & $29032\pm477$ & $5.12\pm0.06$ & 0.224 & $0.3996\pm0.0475$ & $-11.341^{+0.013}_{-0.004}$ & $0.252^{+0.034}_{-0.027}$ & $40.7^{+12.1}_{-8.6}$ & $0.31^{+0.10}_{-0.07}$ \\
FBS0033+376 & 368263110278860544 & sdB & $38970\pm1330$ & $5.89\pm0.10$ & 0.076 & $1.4164\pm0.0409$ & $-11.144^{+0.008}_{-0.006}$ & $0.112^{+0.004}_{-0.004}$ & $26.2^{+4.3}_{-3.9}$ & $0.36^{+0.10}_{-0.08}$ \\
FBS0051+377 & 364742852004132736 & sdOB & $35103\pm116$ & $5.91\pm0.02$ & 0.080 & $1.0341\pm0.0418$ & $-11.181^{+0.018}_{-0.006}$ & $0.142^{+0.006}_{-0.006}$ & $27.3^{+2.8}_{-2.5}$ & $0.59^{+0.06}_{-0.06}$ \\
FBS0102+362 & 369576820516013824 & sdB & $32628\pm98$ & $5.68\pm0.02$ & 0.011 & $2.6839\pm0.0523$ & $-10.729^{+0.007}_{-0.007}$ & $0.154^{+0.004}_{-0.004}$ & $24.1^{+1.7}_{-1.7}$ & $0.41^{+0.03}_{-0.03}$ \\
FBS0104+367 & 369621277722573568 & sdB & $31086\pm120$ & $5.94\pm0.04$ & 0.157 & $0.7730\pm0.0421$ & $-11.236^{+0.044}_{-0.008}$ & $0.166^{+0.010}_{-0.009}$ & $23.0^{+3.1}_{-2.7}$ & $0.87^{+0.14}_{-0.12}$ \\
FBS0106+374 & 369573831218501376 & sdB & $30340\pm218$ & $5.75\pm0.05$ & 0.141 & $1.1993\pm0.0391$ & $-10.953^{+0.005}_{-0.006}$ & $0.207^{+0.007}_{-0.007}$ & $32.6^{+2.8}_{-2.6}$ & $0.88^{+0.13}_{-0.11}$ \\
FBS0117+396 & 371711041304831616 & sdB & $28500\pm100$ & $5.42\pm0.01$ & 0.060 & $0.5853\pm0.0462$ & $-11.279^{+0.035}_{-0.007}$ & $0.198^{+0.017}_{-0.014}$ & $23.3^{+4.4}_{-3.5}$ & $0.38^{+0.07}_{-0.06}$ \\
FBS0156+439 & 346721341030064640 & He-sdOB & $43970\pm1190$ & $5.72\pm0.16$ & 0.166 & $0.5015\pm0.0491$ & $-11.401^{+0.031}_{-0.009}$ & $0.176^{+0.019}_{-0.016}$ & $103.6^{+26.9}_{-20.1}$ & $0.59^{+0.31}_{-0.20}$ \\
FBS0203+374 & 330722897089067392 & sdB & $28744\pm303$ & $5.47\pm0.04$ & 0.150 & $1.0701\pm0.0367$ & $-10.984^{+0.044}_{-0.004}$ & $0.214^{+0.008}_{-0.007}$ & $28.0^{+2.7}_{-2.6}$ & $0.49^{+0.06}_{-0.06}$ \\
FBS0206+428 & 351633546665907200 & sdB & $25735\pm251$ & $5.14\pm0.03$ & 0.135 & $0.9280\pm0.0366$ & $-11.019^{+0.006}_{-0.007}$ & $0.229^{+0.009}_{-0.009}$ & $20.6^{+2.4}_{-2.2}$ & $0.26^{+0.04}_{-0.03}$ \\
FBS0208+390 & 332606704106366848 & sdB & $38743\pm269$ & $6.15\pm0.04$ & 0.106 & $0.7313\pm0.0419$ & $-11.439^{+0.008}_{-0.011}$ & $0.111^{+0.007}_{-0.006}$ & $24.8^{+3.5}_{-3.0}$ & $0.63^{+0.10}_{-0.09}$ \\
FBS0212+334 & 326009496898928896 & sdB & $29697\pm197$ & $5.63\pm0.04$ & 0.254 & $1.2116\pm0.0350$ & $-10.875^{+0.012}_{-0.006}$ & $0.243^{+0.007}_{-0.007}$ & $41.3^{+3.0}_{-2.9}$ & $0.92^{+0.11}_{-0.10}$ \\
FBS0314+372 & 235323736345986176 & sdB & $29283\pm192$ & $5.42\pm0.03$ & 0.902 & $0.8740\pm0.0403$ & $-11.085^{+0.007}_{-0.029}$ & $0.209^{+0.010}_{-0.009}$ & $28.8^{+3.3}_{-2.9}$ & $0.42^{+0.06}_{-0.05}$ \\
FBS0324+379 & 234861632224343424 & sdB & $24809\pm298$ & $5.27\pm0.03$ & 0.473 & $1.3008\pm0.0340$ & $-10.908^{+0.009}_{-0.007}$ & $0.210^{+0.006}_{-0.006}$ & $15.0^{+1.8}_{-1.7}$ & $0.30^{+0.03}_{-0.03}$ \\
FBS0638+428 & 963881581386403072 & sdB & $28090\pm161$ & $5.29\pm0.02$ & 0.229 & $1.1992\pm0.0344$ & $-10.985^{+0.042}_{-0.010}$ & $0.191^{+0.006}_{-0.006}$ & $20.4^{+1.9}_{-1.9}$ & $0.26^{+0.03}_{-0.03}$ \\
FBS0639+391 & 944892878136403712 & sdB & $29658\pm509$ & $5.53\pm0.08$ & 0.372 & $0.8262\pm0.0358$ & $-11.070^{+0.008}_{-0.007}$ & $0.229^{+0.010}_{-0.010}$ & $36.3^{+4.5}_{-4.0}$ & $0.65^{+0.15}_{-0.12}$ \\
FBS0646+386 & 944390774983674496 & sdB & $33238\pm190$ & $5.63\pm0.03$ & 0.234 & $0.7018\pm0.0473$ & $-11.323^{+0.020}_{-0.012}$ & $0.150^{+0.011}_{-0.010}$ & $24.6^{+3.9}_{-3.4}$ & $0.35^{+0.06}_{-0.05}$ \\
FBS1400+389 & 1495942160078351744 & sdOB & $18800\pm1000$ & $4.71\pm0.10$ & 0.029 & $1.4515\pm0.0345$ & $-10.534^{+0.020}_{-0.045}$ & $0.449^{+0.011}_{-0.011}$ & $22.6^{+5.5}_{-4.7}$ & $0.38^{+0.10}_{-0.08}$ \\
FBS1531+381 & 1375814952762454272 & sdB & $29230\pm125$ & $5.58\pm0.03$ & 0.029 & $1.9294\pm0.0345$ & $-10.805^{+0.013}_{-0.164}$ & $0.181^{+0.004}_{-0.004}$ & $21.5^{+1.7}_{-1.6}$ & $0.46^{+0.04}_{-0.04}$ \\
FBS1715+424 & 1360139937040515072 & sdO & $58438\pm514$ & $6.11\pm0.03$ & 0.010 & $2.3950\pm0.0387$ & $-10.911^{+0.011}_{-0.009}$ & $0.114^{+0.003}_{-0.003}$ & $135.4^{+6.8}_{-6.4}$ & $0.61^{+0.05}_{-0.05}$ \\
FBS1745+437 & 1349088676950060672 & sdB & $33601\pm109$ & $5.56\pm0.02$ & 0.073 & $1.0659\pm0.0262$ & $-11.132^{+0.008}_{-0.006}$ & $0.153^{+0.004}_{-0.004}$ & $26.9^{+2.0}_{-1.9}$ & $0.31^{+0.03}_{-0.03}$ \\
FBS2255+404 & 1930945626165804032 & He-sdOB & $48127\pm277$ & $5.92\pm0.04$ & 0.355 & $0.4056\pm0.0444$ & $-11.490^{+0.049}_{-0.010}$ & $0.176^{+0.022}_{-0.017}$ & $149.9^{+39.4}_{-28.1}$ & $0.95^{+0.27}_{-0.19}$ \\
FBS2307+338 & 1911572540520290048 & He-sdO & $50664\pm390$ & $5.85\pm0.04$ & 0.219 & $0.4957\pm0.0502$ & $-11.435^{+0.006}_{-0.011}$ & $0.165^{+0.018}_{-0.015}$ & $160.2^{+38.3}_{-28.2}$ & $0.70^{+0.18}_{-0.14}$ \\
FBS2347+385 & 2881551562420806784 & sdB & $23770\pm350$ & $5.38\pm0.06$ & 0.269 & $3.9390\pm0.0341$ & $-10.434^{+0.026}_{-0.011}$ & $0.207^{+0.003}_{-0.003}$ & $12.3^{+1.6}_{-1.6}$ & $0.37^{+0.06}_{-0.05}$ \\
Feige114 & 1858674589447492608 & sdOB & $37689\pm189$ & $5.56\pm0.02$ & 0.148 & $1.2354\pm0.0317$ & $-10.983^{+0.023}_{-0.006}$ & $0.186^{+0.005}_{-0.005}$ & $62.5^{+3.8}_{-3.6}$ & $0.46^{+0.04}_{-0.04}$ \\
Feige26 & 2503706418759709184 & He-sdO & $65608\pm912$ & $5.46\pm0.04$ & 0.010 & $0.5763\pm0.0471$ & $-11.247^{+0.007}_{-0.010}$ & $0.218^{+0.019}_{-0.017}$ & $793.7^{+154.1}_{-121.7}$ & $0.50^{+0.11}_{-0.09}$ \\
Feige34 & 781164326766404736 & sdO & $62400\pm330$ & $5.95\pm0.02$ & 0.145 & $4.4105\pm0.1039$ & $-10.645^{+0.069}_{-0.018}$ & $0.113^{+0.003}_{-0.003}$ & $174.4^{+9.3}_{-8.8}$ & $0.42^{+0.04}_{-0.03}$ \\
Feige67 & 3935488605023787392 & sdO & $61114\pm318$ & $5.92\pm0.01$ & 0.105 & $2.9833\pm0.0735$ & $-10.739^{+0.010}_{-0.039}$ & $0.136^{+0.004}_{-0.004}$ & $232.9^{+13.0}_{-12.0}$ & $0.56^{+0.04}_{-0.04}$ \\
GALEXJ00468+1956 & 2801277176253968640 & sdB & $32891\pm111$ & $5.84\pm0.02$ & 0.200 & $0.9281\pm0.0413$ & $-11.213^{+0.041}_{-0.008}$ & $0.146^{+0.007}_{-0.006}$ & $22.2^{+2.5}_{-2.3}$ & $0.53^{+0.06}_{-0.05}$ \\
GALEXJ00476+3123 & 360511037906831872 & sdB & $31069\pm166$ & $5.65\pm0.03$ & 0.207 & $1.0593\pm0.0383$ & $-11.149^{+0.042}_{-0.011}$ & $0.148^{+0.006}_{-0.006}$ & $18.4^{+2.0}_{-1.9}$ & $0.36^{+0.04}_{-0.04}$ \\
GALEXJ00498+3522 & 363795760175504768 & sdOB & $36551\pm243$ & $5.74\pm0.04$ & 0.063 & $0.8767\pm0.0412$ & $-11.251^{+0.010}_{-0.016}$ & $0.142^{+0.007}_{-0.007}$ & $32.5^{+3.7}_{-3.3}$ & $0.41^{+0.06}_{-0.05}$ \\
GALEXJ01012+3125 & 312628749626419328 & sdB & $28202\pm155$ & $5.45\pm0.02$ & 0.063 & $1.8070\pm0.0362$ & $-10.788^{+0.010}_{-0.017}$ & $0.201^{+0.005}_{-0.004}$ & $22.9^{+1.8}_{-1.7}$ & $0.41^{+0.03}_{-0.03}$ \\
GALEXJ01194+4901 & 400211958948711552 & He-sdOB & $43720\pm510$ & $5.86\pm0.21$ & 0.313 & $1.2614\pm0.0326$ & $-10.965^{+0.017}_{-0.012}$ & $0.189^{+0.005}_{-0.005}$ & $117.7^{+8.6}_{-8.0}$ & $0.95^{+0.59}_{-0.37}$ \\
GALEXJ01509+3107 & 304311150320629376 & sdB & $30183\pm189$ & $5.84\pm0.04$ & 0.130 & $1.3955\pm0.0367$ & $-11.041^{+0.004}_{-0.008}$ & $0.145^{+0.004}_{-0.004}$ & $15.6^{+1.7}_{-1.7}$ & $0.53^{+0.06}_{-0.06}$ \\
GALEXJ01543+4905 & 357580873780641920 & sdB & $32100\pm410$ & $5.50\pm0.32$ & 0.452 & $1.7860\pm0.0250$ & $-10.871^{+0.008}_{-0.008}$ & $0.166^{+0.003}_{-0.003}$ & $26.4^{+2.1}_{-2.1}$ & $0.32^{+0.35}_{-0.17}$ \\
GALEXJ02069+1438 & 77515058657678592 & sdB & $30310\pm660$ & $5.77\pm0.06$ & 0.132 & $2.2104\pm0.0447$ & $-10.881^{+0.010}_{-0.028}$ & $0.133^{+0.003}_{-0.003}$ & $13.3^{+1.9}_{-1.9}$ & $0.38^{+0.06}_{-0.05}$ \\
GALEXJ02163+2759 & 107585292928397568 & sdB & $25480\pm450$ & $5.35\pm0.05$ & 0.155 & $1.5540\pm0.0313$ & $-10.767^{+0.028}_{-0.013}$ & $0.244^{+0.005}_{-0.005}$ & $22.5^{+2.4}_{-2.3}$ & $0.49^{+0.07}_{-0.06}$ \\
GALEXJ02328+4411 & 341195058150383104 & sdB & $33260\pm420$ & $5.73\pm0.10$ & 0.237 & $1.2611\pm0.0577$ & $-11.082^{+0.011}_{-0.007}$ & $0.145^{+0.007}_{-0.007}$ & $23.1^{+2.9}_{-2.7}$ & $0.41^{+0.12}_{-0.09}$ \\
GALEXJ04158+0154 & 3259060049366022400 & sdBV & $34773\pm123$ & $5.79\pm0.02$ & 0.359 & $1.5335\pm0.0402$ & $-11.003^{+0.014}_{-0.006}$ & $0.144^{+0.004}_{-0.004}$ & $27.1^{+2.1}_{-2.0}$ & $0.46^{+0.04}_{-0.04}$ \\
GALEXJ04205+0120 & 3255780171819962496 & He-sdO & $46050\pm910$ & $5.97\pm0.16$ & 0.081 & $2.4658\pm0.0620$ & $-10.816^{+0.012}_{-0.006}$ & $0.137^{+0.004}_{-0.004}$ & $76.2^{+7.5}_{-6.9}$ & $0.64^{+0.29}_{-0.20}$ \\
GALEXJ04593-0228 & 3225359193340812288 & sdB & $30613\pm608$ & $5.70\pm0.12$ & 0.062 & $1.1390\pm0.0412$ & $-11.197^{+0.033}_{-0.008}$ & $0.124^{+0.005}_{-0.005}$ & $12.0^{+2.0}_{-1.9}$ & $0.28^{+0.09}_{-0.07}$ \\
GALEXJ05369+3955 & 190970433715259904 & He-sdOB & $38898\pm169$ & $5.79\pm0.04$ & 0.507 & $1.3436\pm0.0332$ & $-11.007^{+0.006}_{-0.008}$ & $0.164^{+0.005}_{-0.004}$ & $55.1^{+3.3}_{-3.1}$ & $0.60^{+0.07}_{-0.06}$ \\
GALEXJ05557+6408 & 287018271958898560 & sdO & $45600\pm3380$ & $5.73\pm0.39$ & 0.462 & $1.9389\pm0.0248$ & $-10.932^{+0.007}_{-0.023}$ & $0.134^{+0.003}_{-0.003}$ & $70.1^{+23.2}_{-18.7}$ & $0.35^{+0.51}_{-0.21}$ \\
GALEXJ06134+3420 & 3452237404778065152 & sdOB & $34250\pm390$ & $5.75\pm0.10$ & 0.432 & $1.7373\pm0.0329$ & $-10.935^{+0.010}_{-0.010}$ & $0.148^{+0.003}_{-0.003}$ & $27.1^{+2.2}_{-2.1}$ & $0.45^{+0.12}_{-0.10}$ \\
GALEXJ063952.00+515658.00 & 992534888766785024 & sdB & $30400\pm340$ & $5.58\pm0.13$ & 0.065 & $3.0845\pm0.0455$ & $-10.552^{+0.031}_{-0.006}$ & $0.202^{+0.004}_{-0.004}$ & $31.2^{+2.2}_{-2.1}$ & $0.56^{+0.20}_{-0.15}$ \\
GALEXJ06437+3201 & 937248901501446272 & sdB & $26956\pm303$ & $5.12\pm0.04$ & 0.331 & $1.0644\pm0.0448$ & $-10.873^{+0.061}_{-0.008}$ & $0.276^{+0.012}_{-0.011}$ & $36.0^{+3.9}_{-3.5}$ & $0.37^{+0.05}_{-0.05}$ \\
GALEXJ06463+2920 & 3386622158606040960 & sdOB & $41489\pm229$ & $6.16\pm0.03$ & 0.073 & $2.1178\pm0.0437$ & $-10.998^{+0.009}_{-0.016}$ & $0.105^{+0.003}_{-0.003}$ & $29.3^{+2.0}_{-1.9}$ & $0.58^{+0.05}_{-0.05}$ \\
GALEXJ07013+5329 & 981727582818800128 & He-sdO & $67230\pm3880$ & $5.60\pm0.22$ & 0.117 & $0.3925\pm0.0431$ & $-11.375^{+0.012}_{-0.005}$ & $0.238^{+0.029}_{-0.023}$ & $1038.8^{+393.0}_{-282.0}$ & $0.83^{+0.61}_{-0.35}$ \\
GALEXJ07104+2333 & 3368383116127317120 & sdB & $55420\pm24300$ & $5.21\pm0.43$ & 0.724 & $0.5599\pm0.0242$ & $-11.119^{+0.071}_{-0.016}$ & $0.299^{+0.013}_{-0.013}$ & $756.7^{+2464.1}_{-679.2}$ & $0.53^{+0.90}_{-0.33}$ \\
GALEXJ07108+5624 & 988436459174352512 & sdB & $39970\pm980$ & $5.91\pm0.14$ & 0.094 & $0.9733\pm0.0521$ & $-11.251^{+0.009}_{-0.005}$ & $0.128^{+0.007}_{-0.007}$ & $37.5^{+6.1}_{-5.2}$ & $0.49^{+0.20}_{-0.14}$ \\
GALEXJ07248+1923 & 3362730178953977472 & sdB & $25777\pm334$ & $5.25\pm0.03$ & 0.172 & $0.9219\pm0.0378$ & $-11.069^{+0.010}_{-0.006}$ & $0.205^{+0.009}_{-0.008}$ & $16.7^{+2.2}_{-2.1}$ & $0.27^{+0.04}_{-0.04}$ \\
    \hline
    \end{tabular}}
    \label{tab1}
\end{table}

\end{appendix}
\end{document}
