The measurement of masses of OB-type stars from LAMOST DR5

The Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST), which is also called the Guo Shou Jing telescope, innovatively presents the world’s unique and largest aperture large field telescope. The vast majority of LAMOST spectra come from objects in the Milky Way. Since 2011, the LAMOST survey has released more than 20 million astronomical spectra covering the wavelength range of 369-910  nm with a resolution of 1800, building the world’s largest database of the astronomical spectra (Cui et al., 2012; Zhao et al., 2012; Deng et al., 2012).

We use the OB catalog of Liu et al. (2019), which contains 16,032 OB-type stars selected from LAMOST DR5 based on the stellar spectral line indices, as our initial sample data. Guo et al. (2021) used the data-driven technique Stellar LAbel Machine (SLAM) to derive their stellar atmospheric parameters ($T_{\rm eff}$, log $g$, [M/H] and $v$ sin $i$), based on the non-LTE TLUSTY synthetic spectra.

Considering the results of extrapolation and the uncertainties of stellar parameters from Guo et al. (2021), the effect from the spectral signal-to-noise of g-band (S/N)_g, and the limitation of the TLUSTY model grids, we used the following criteria to select sample further:

Applying the above criteria (1), we obtain 2,946 normal OB-type stars including 78 O-type stars and 2,868 B-type stars ¹¹1We define the massive OB-type stars as the “normal” OB-type stars, and our sample does not contain 57 OB-type stars, which are not within PARSEC model, and special OB-type stars, such as: sdOB, Oe/Be, WD.. Figure 1 shows the distribution of the stellar parameters of 2,946 OB-type stars, we can see that most of the stars have surface gravity values (log $g$ =4.0 dex) of main-sequence stars and lower metallicity than the Sun. In other words, the majority of our selected sample are the main-sequence OB-type stars.

PAdova and TRieste Stellar Evolution Code (PARSEC), which is widely used by the astronomical community (Bressan et al., 1993; Gafeira, Patacas & Fernandes., 2012), is the first applied by Bressan et al. (2012), and Chen et al. (2014, 2015); Tang et al. (2014) has extended this model. The PARSEC model covers the metallicity of -2.2 $\leq$ [M/H] $\leq$ +0.5, the mass range 0.1 M_☉ $\leq$ M $\leq$ 150 M_☉ and a broad age distribution from pre-main sequence to 30 Gyr very low-mass stars. The feature of the PARSEC model can make us determine the masses and luminosities of the OB-type stars.

We use the stellar parameters ($T_{\rm eff}$, log $g$ and [M/H]) of 2,946 OB-type stars, which is from Guo et al. (2021), to interpolate the PARSEC model, and determine their masses. Considering the stellar parameters of OB-type stars derived using their LAMOST low-resolution spectra (R $\sim$1800), we use the uncertainties of atmosphere parameters ($\Delta$$T_{\rm eff}$=1600 K, $\Delta$log $g$=0.25 dex) and metallicity of Guo et al. (2021). The uncertainties of stellar masses are also obtained by applying Monte Carlo sampling by assuming a Gaussian distribution with the uncertainties of stellar parameters ($T_{\rm eff}$, log $g$ and [M/H]). The final results are listed in Table 1. Rolleston et al. (2000) used about 80 B-type main-sequence stars in Galactic open clusters$/$associations to obtain the metallicity gradient of -0.07 dex/kpc. Ideally, the metallicity gradient distribution is -0.3$<$[M/H]$<$+0.2 for our OB-type stars with galactocentric distances of approx 5.5-12.2 kpc. However, most of our OB-type stars are at located in galactocentric distances of 9-12 kpc. Moreover, there may be a systematic difference between the metallicity of Rolleston et al. (2000) and the statistical metallicity of Guo et al. (2021). Taking into account the above conditions, we also obtain the masses of OB-type stars at solar metallicity and expressed as $M_{2}$ in Table 1.

To verify the effect of the degeneracy of stellar parameters on stellar masses, we apply Monte Carlo simulation (MCMC, Foreman-Mackey et al. (2013)) to examine the potential distribution of stellar masses. Figure 10 in the section appendix shows the distribution of mass of a B-type star by assuming a Gaussian distribution of its stellar parameters ($T_{\rm eff}$, log $g$ and [M/H]), and we can see that it can give a relatively reliable mass by using MCMC due to the relatively concentrated distribution of stellar mass.

To inspect the reliability of the method obtaining stellar masses, we randomly select 600 grid points with different $T_{\rm eff}$, log $g$, [M/H], masses from the evolutionary model. The masses of 600 grid points are estimated by using the above method again, based on the typical uncertainties of their stellar parameters (1600 K for $T_{\rm eff}$, 0.25 dex for log $g$ and 0.1 dex for [M/H]).

As shown in Figure 2, for the OB-type stars with masses less than 40 M_☉, there is a good consistency between the masses calculated by the theoretical model and the masses obtained by using our method. For some OB-type supergiants with masses larger than 40 M_☉, our results give larger masses than those from the theoretical model because they are located at the edge of the PARSEC grid. Considering our sample only contains 6 OB-type stars with masses larger than 40 M_☉, our method does not have an effect on our current results, and that also indicates that our method is self-consistent.

Figure 3 shows the distribution of 2,893 OB-type stars with masses ²²2All the masses involved inside our hereafter discussion are M. less than 20 M_☉, including 2,825 B-type stars and 68 O-type stars, in mass versus relative error of mass ($\Delta$ M/M) panel, and O-type stars have larger masses than B-type stars. From the top histogram, it is seen that the masses of 84 $\%$ OB-type stars are less than 8.0 M_☉, the mean value of the masses is about 5.4 M_☉, the masses of 16 $\%$ OB-type stars are less than 4.2 M_☉, and the stellar masses range are mainly concentrated in 4-10 M_☉ with the peak distribution in 5-6 M_☉. This is due to our sample selection effect that the stars are mainly distributed in the range of $T_{\rm eff}$ $<$ 20 kK and 3.5 $<$ log $g$ $<$ 4.5 dex (see Figure 1). From the right histogram, it can see that $\Delta$ M/M of 84 $\%$ OB-type stars is also less than 25 $\%$. Moreover, some OB-type stars with stellar winds, such as B-type supergiants and O-type stars, are not suitable to use the plane-parallel Tlusty model to derive their stellar parameters, which require spherical model atmospheres (e.g., FASTWIND, CMFGEN, PoWR) (Hawcroft et al., 2021; Crowther et al., 2006). Thus, the relative errors in the parameters of such kind of stars are large and less accurate, and we also marked these stars in Table 1. The subgraph of Figure 3, which contains all 2,946 OB-type stars, shows that the stars with masses larger than 20 M_☉ are relatively few. This is also consistent with the distribution of stellar effective temperature.

The majority of massive stars are members of binary systems so the majority of LAMOST targets will probably be binaries, which maybe impact on $T_{\rm eff}$, log $g$  and luminosity inferred for each star (Sana, 2012; Moe & Di Stefano et al, 2017). However, the results of Guo et al. (2021) indicate that the predicted atmospheric parameters of early-type stars are in good agreement with the results obtained by using their high-resolution spectra within the errors, which imply that the binary have no obvious effect on the stellar parameters of OB-type stars of Guo et al. (2021).

Figure 4 shows the spatial distribution of 2,946 OB-type stars. The color code represents the masses of the stars. The OB-type stars with larger masses are mainly located in the low Galactic latitude region, which is consistent with the distribution of the Galactic neutral atomic hydrogen (H I) column density (HI4PI Collaboration et al., 2016).

In general, the solution of the orbital motion of binary stars is a reliable method for determining stellar masses (Torres et al., 2010). To validate the reliability of mass, we cross-correlated the binary mass sample of Xiong et al. (2023), the dynamical mass sample of Eker et al. (2014), Bondarenko & Perevozkina (1996) and Surkova & Svechnikov (2004), respectively. Figure 5 shows the comparison of our results and their results for 7 common stars. We can see that most of the stars give a good agreement, except two stars HD 136175 and BD+41 3353, and we find that the different masses of these two stars are due to the difference between our stellar parameters and those of the literature (atmospheric parameters from photometric data ). This also supports the reliability of our estimated masses for the OB-type stars.

In addition, we also note that Hohle et al. (2010) also provides a mass sample of OB-type stars by matching the stellar theoretical models, which is similar to the method adopted in this work. By cross-matching the coordinates, out of these 2946 objects, we have 9 objects in common with Hohle et al. (2010). The isochronous models and stellar atmosphere parameters utilized in Hohle et al. (2010) differ from those used in this work, which will result in a significant discrepancy in the masses for the same OB-type star. We do not use their results as the test sample as a result.

To verify the reliability of masses of OB-type stars obtained by using their stellar parameters, we cross-matching our data with the $Gaia$ DR3 data and randomly select 100 OB-type stars with parallax$\_$error$/$parallax$\leq$ 20 $\%$, RUWE$<$ 1.4, and
distances$<$ 4 kpc. We use the $Gaia$ DR3 luminosities of 100 OB-type stars to calculate their masses (L = 4$\pi$  R² $\sigma$T⁴ and $g$ = GM/R²). The left panel of Figure 6 shows a comparison between the luminosities predicted by using the stellar parameters for 100 OB-type stars and the luminosities of $Gaia$ DR3, and it is seen that the luminosities of $Gaia$ DR3 are generally larger than our predicted luminosities, which suggest that the stellar luminosities of $Gaia$ DR3 may be affected by binary systems. The right panel of Figure 6 shows a comparison between the masses obtained by using the PARSEC model for 100 OB-type stars and the masses obtained using their $Gaia$ DR3 data. It can be seen that when the stellar mass is larger than 5 M_☉, the mass obtained from the stellar luminosity is biased towards the mass obtained from the stellar atmospheric parameters, which is because more massive stars have a greater binary ratio (Sana, 2012). Therefore, our method can obtain the masses of more OB-type stars with less influence from binary stars.

Based on the stellar parameters from Guo et al. (2021), the luminosities and radii of 2,946 OB-type stars are estimated by using the PARSEC isochrones and the relationship between luminosity and radius (L = 4$\pi$ R² $\sigma$T⁴), respectively.

Figure 7 shows the distribution of OB-type stars in the luminosity ( log (L/$\rm{L_{\sun}}$)) vs. luminosity relative error ($\Delta$ L/L), the  log (L/$\rm{L_{\sun}}$) of OB-type stars with masses less than 20 M_☉ ranges from 2 to 4. From the top histogram, it can be seen that the 84, 50, 16 percentiles of the distribution are less than 3.9, 3.2, 2.7, respectively. However, we also note that there is a larger luminosity error due to the relatively larger error of stellar parameters (the uncertainty of the surface gravity). Figure 8 shows the distribution of radii (R) and relative errors ($\Delta$ R/R) of OB-type stars, and OB-type stars with masses less than 20 M_☉ are mainly located in the region of R $<$10 R_☉ and $\Delta$ R/R $<$ 0.4. From the top histogram, we can see that the radii the distribution of 84 $\%$ OB-type stars are less than 5.9 R_☉, the mean value of the radii are 3.6 R_☉, and there are only 16$\%$ stars with R $>$ 5.9 R_☉.

In addition, we also derive the stellar photometry luminosity using photometric and astrometric data of $Gaia$ DR3 (Gaia Collaboration et al., 2020), based on bellow bolometric magnitude.

where M_b,☉ is the absolute bolometric magnitude of the Sun. The M_b represents stellar absolute bolometric magnitude. The $G$ is the $G$-band magnitude, and the $d$ is the distance in pc from Early Gaia Data Release (EDR3) Distance (Bailer-Jones et al., 2021). The extinction A_G is given by the blue-edge method (Wang & Jiang, 2014; Xue et al., 2016; Jian et al., 2017; Zhao et al., 2018, 2020). The bolometric corrections of stars are obtained from Jordi et al. (2010) and Bessell et al. (1998), respectively. We have also listed the luminosities in Table 1.

Figure 9 shows the distribution of our 2,825 B-type stars (M $\leq$ 20 M_☉) in the mass-luminosity ($M-L$) diagram. In Figure 9, the different metallicity ([M/H]) zero-age main sequence (ZAMS) lines from the PARSEC model are marked using the different color solid lines, and the dots represent the 2,825 B-type stars. The color code is the metallicity of the stars. The almost independent between [M/H] and the $M-L$ relation is also consistent with the stellar model shown by the ZAMS lines.

According to the polynomial model of the $M-L$ -[M/H] relation from Xiong et al. (2023), we also find the best-fit coefficients for this model using our OB-type stars. The polynomial model is written as:

where the values of a1, a2, a3, and a4 are -0.06, 4.399, 0.203, and -0.293, respectively. We use the polynomial of the model to fit the coefficients of the third- and fourth-order terms of $logM$, and show in Figure 9.