The Eclipsing Binaries from the LAMOST Medium-resolution Survey. III. A High-precision Empirical Stellar Mass Library

Stars are the main ingredient of galaxies and play a critical role in their structure and evolution. An accurate understanding of the mass, radius, luminosity, chemical composition, and age of stars is therefore essential. In general, stellar mass and radius can mainly be derived from models of stellar structure and evolution, e.g., PARSEC (Bressan et al. 2012), Y² (Demarque et al. 2004, 2008), Dartmouth (Dotter et al. 2008), MIST (Dotter 2016), etc., empirical relations, e.g., mass–luminosity relation (MLR), mass–radius relation (MRR), and mass–temperature relation (MTR; Eker et al. 2014, 2015, 2018), or asteroseismic techniques (Chaplin et al. 2014; Marcy et al. 2014). Among these methods, the dynamical mass of detached eclipsing binaries is not only the method that relies least on stellar models, but it also reaches a precision of around 1% (Torres et al. 2010). Therefore, it can be used as a calibrator for the other approaches to mass estimation. Indeed, most of the stellar models and empirical relations rely on the dynamical masses and radii for calibration (Hoxie 1973; Cox 2000; Torres et al. 2010; Eker et al. 2015, 2018; Southworth 2015).

However, to date, only ∼700 binaries have been provided with accurate masses and radii. Among them, only <200 binary systems have comprehensive atmospheric parameters (T_eff, logg, [M/H]). Metallicity is an especially important parameter to accompany stellar mass and radius, since it can help to break the mass–metallicity degeneracy at around the turnoff point (Serenelli et al. 2021). With metallicity, one can derive the luminosity–mass and other relations for different metallicity so that the empirical mass can be well compared with stellar models.

It is clear that enlarging the data set of stars with accurate stellar mass and metallicity can significantly improve the calibration of stellar models. Recently, significant progress has been made in spectroscopic surveys, e.g., RAVE (Steinmetz et al. 2006, 2020), SDSS/SEGUE (Yanny et al. 2009), LAMOST (Cui et al. 2012; Deng et al. 2012; Zhao et al. 2012; Luo et al. 2015), APOGEE (Majewski et al. 2017), and GALAH (De Silva et al. 2015), and time-domain photometric surveys, e.g., Gaia (Katz et al. 2004; Gilmore et al. 2012; Cropper et al. 2018), NASA/Kepler (Batalha et al. 2010; Borucki et al. 2010; Abdul-Masih et al. 2016), TESS (Ricker et al. 2015), ZTF (Bellm et al. 2019), and ASAS-SN (Kochanek et al. 2017; Jayasinghe et al. 2019).

In particular, LAMOST launched a medium-resolution survey (MRS, R ∼ 7500), which includes both time-domain and usual spectroscopic observations (Liu et al. 2020), in 2018 October. Moreover, relatively high-quality parameters have been obtained from the spectra. For LAMOST low-resolution spectra (LRS with R ∼ 1800), the accuracies of radial velocity, T_eff, logg, and [Fe/H] are around 5 km s⁻¹, 150 K, 0.25 dex, and 0.15 dex, respectively (Xiang et al. 2015). For LAMOST MRS, the accuracy of radial velocity can reach around 1 km s⁻¹ (Wang et al. 2019; Zhang et al. 2021; Xiong et al. 2021). The accuracies of T_eff, logg, and [Fe/H] are around 119 K, 0.17 dex, and 0.06–0.12 dex, respectively (Wang et al. 2020). Furthermore, many eclipsing binaries with light curves (Slawson et al. 2011; Chen et al. 2020; Jayasinghe et al. 2021) are identified from the LAMOST survey (Li et al. 2021a; Zhang et al. 2022). These published data offer a unique opportunity to measure the dynamic mass and radius of more eclipsing binaries.

This work aims to present a high-precision empirical stellar mass library that includes an accurate dynamical mass and independently measured atmospheric parameters (T_eff, logg, [M/H]) from observed spectra for main-sequence stars. The compilation of literature data and the LAMOST data acquisition and light curve are described in Section 2. The method of measuring accurate mass for LAMOST binaries is described in Section 3. The results are presented in Section 4. The comparison of dynamical mass and radius derived from different measurement approaches is discussed in Section 5. Finally, we summarize in Section 6.

2.1. Compilation of Literature Data

2.2. LAMOST Data with Light Curves

3.1. Brief Description

For each star, the light curve with eclipses and the radial velocity curves of two companions covering the whole period are used to derive the orbital solution. Light curves with eclipses are able to constrain the inclination and relative radii of both companions with respect to the semimajor axis (R/a). However, the stellar mass cannot be solely determined without double-line radial velocity curves. In this subsection we describe the general method to derive the stellar masses and radii of two companions in a binary system via the orbital solution.

The radial velocity of either of the companions in a spectroscopic binary system can be written as

$$\begin{eqnarray}&&{{RV}}_{i}=\displaystyle \frac{2\pi {a}_{i}\sin i}{P{\left(1-{e}^{2}\right)}^{1/2}}[\cos (\theta +\omega )+e\cos \omega ]+\gamma ,\end{eqnarray} \tag{ 1 }$$

where P is the orbital period, θ is the angular position, ω is the longitude of the periastron, e is the orbital eccentricity, a_i is the semimajor distance from the ith companion to the barycenter, i is the inclination of the orbit, and γ is the systemic velocity. The coefficient of the right-hand side is actually the semiamplitude of the radial velocity curve and is usually denoted as

$$\begin{eqnarray}\begin{array}{rcl}{K}_{1} & = & \displaystyle \frac{2\pi {a}_{1}\sin i}{P{\left(1-{e}^{2}\right)}^{1/2}},\\ {K}_{2} & = & \displaystyle \frac{2\pi {a}_{2}\sin i}{P{\left(1-{e}^{2}\right)}^{1/2}}.\end{array}\end{eqnarray} \tag{ 2 }$$

K_i and P are associated with the stellar masses via the mass function, which can be written as

$$\begin{eqnarray}\begin{array}{rcl}f({M}_{1}) & = & \displaystyle \frac{{M}_{1}^{3}{\sin }^{3}i}{{\left({M}_{1}+{M}_{2}\right)}^{2}}\\ & = & (1.0361\times {10}^{-7}){\left(1-{e}^{2}\right)}^{3/2}{K}_{2}^{3}P\,{M}_{\odot },\\ f({M}_{2}) & = & \displaystyle \frac{{M}_{2}^{3}{\sin }^{3}i}{{\left({M}_{1}+{M}_{2}\right)}^{2}}\\ & = & (1.0361\times {10}^{-7}){\left(1-{e}^{2}\right)}^{3/2}{K}_{1}^{3}P\,{M}_{\odot }.\end{array}\end{eqnarray} \tag{ 3 }$$

Although the radial velocity curves can give K₁ and K₂, they cannot directly solve M₁ and M₂ without the help of light curves.

We then seek the orbit solution with the combination of the radial velocities and light curves using Physics of Eclipsing Binaries (PHOEBE 2.2 ⁶ ; Prša & Zwitter 2005; Prša et al. 2016; Jones et al. 2020), which is based on the WD code (Wilson & Devinney 1971).

The atmospheric parameters of the F-, G-, and K-type stars among the 56 stars are from the LAMOST pipeline (Wu et al. 2014), while those of the M-dwarf and OB-type stars are from Li et al. (2021b) and Guo et al. (2021), respectively. For the secondary star, T_eff and logg are found from the best orbit fitting from PHOEBE.

In this work, the effective temperature and logg of the primary stars are estimated from spectra observed near the secondary minimum eclipse, at which a secondary star makes the minimum contribution to the flux. For instance, the secondary only contributes about 11% of the primary flux near the secondary minimum eclipse when q = 0.7 (El-Badry et al. 2018). These parameters are then dominated by primary stars and used as inputs in PHOEBE for orbital solution.

El-Badry et al. (2018) performed experiments with synthetic spectra to investigate the systematic biases of atmospheric parameters for unresolved main-sequence binaries on spectral fitting with single-star models; they modeled spectra similar to those collected by the APOGEE, GALAH, and LAMOST surveys. They found that, when an unresolved binary was considered as a single star, the typical errors in estimates of T_eff, logg, and [Fe/H] from the combined spectra are <200 K , <0.1 dex, and <0.1 dex for LAMOST low-resolution spectra. These systematic errors are analogous to the measurement error and hence would not significantly affect the final results of the mass and radius estimation.

A Markov Chain Monte Carlo (MCMC; ⁷ Foreman-Mackey et al. 2013) sampling is then applied on PHOEBE to solve all orbital parameters. The likelihood used in the MCMC sampling is a joint chi-square value from the residuals of the fits with the observational light curve and radial velocity curves. The following priors are also added to restrict the MCMC sampling:

• 1.  
  mass ratio (q): 0 < q < 2
• 2.  
  orbital inclination (i): 0° < i < 90°
• 3.  
  eccentricity (e): 0 < e < 1
• 4.  
  argument of periastron (ω): 0° < ω < 360°
• 5.  
  semimajor axis of orbit (a): 0 < a/R_⊙ < 100
• 6.  
  systemic velocity (V_γ ): −200 < V_γ /(km s⁻¹) < 200
• 7.  
  effective temperature of secondary (T₂): $3000\lt {T}_{2}/{\rm{K}}\lt {T}_{2}+3{\sigma }_{{T}_{1}}$
• 8.  
  equivalent radius of primary (R₁): 0 < R₁/R_⊙ < 10
• 9.  
  equivalent radius of secondary (R₂): 0 < R₂/R_⊙ < 10

We initialized the MCMC chains using random values drawn from uniform distributions with the restricting ranges defined above. The surface gravity and reflection coefficient are fixed in PHOEBE unless T_eff of the primary star is larger than 8000 K. For each star, we run the model with 80 walkers, each of which takes 2000 steps. Finally, we calculate the peak value of the probability distribution and the standard deviations of randomly drawn points as the best-fit parameters and their uncertainties.

3.2. Method Verification

As a verification of the method, we solve the orbit for KIC 5359678 and compare with the results of Wang et al. (2021). Figure 1 shows our MCMC result for KIC 5359678. In the corner plot, the histograms show the probability distributions of q, i, $\sqrt{e}\cos \omega$, $\sqrt{e}\sin \omega$, a, V_γ , T₂, R₁, and R₂.

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Figure 2(a) shows the observed and the best-fit radial velocity curves. The residuals of the velocity indicate that the uncertainty of the best-fit radial velocity curve is around 4.24 km s⁻¹. The top of panel (b) shows the best-fit light curve model by PHOEBE (red line) and the corresponding observations (black dots), and the bottom of the panel shows the residuals of the light curve between the model fit and observations. This illustrates that the best-fit model of the radial velocity curves and the light curve match well with the observed data.

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The masses and radii of the primary and secondary of KIC 5359678 that we obtain from PHOEBE with the MCMC method are M₁ = 1.361 ± 0.008 M_⊙, R₁ = 1.49 ± 0.08 R_⊙, M₂ = 1.121 ± 0.008 M_⊙, and R₂ = 1.075 ± 0.130 R_⊙. All best-fit parameters of the MCMC method and those obtained by Wang et al. (2021) are listed and compared in Table 1. This illustrates that the orbital parameters obtained with PHOEBE+MCMC are consistent with those of Wang et al. (2021) within the uncertainties.

Table 1. Comparison of the Orbital Parameters of KIC 5359678 between This Work and Wang et al. (2021)

Parameters	Wang et al. (2021)	This Work
e	0.00032 ± 0.00006	0.0024 ± 0.0057
ω a (deg)	−89.55 ± 1.05	88.00 ± 6.03
i (deg)	85.56 ± 0.10	85.56 ± 0.37
q	0.851 ± 0.10	0.825 ± 0.011
Vγ (km s−1)	−29.26 ± 0.19	−28.29 ± 0.34
a (R⊙)	19.19	19.28 ± 0.13
T1 (K)	6500 ± 50	6501.04 ± 47.94
T2 (K)	5980 ± 22	6040.84 ± 40.75
M1 (M⊙)	1.320 ± 0.060	1.361 ± 0.008
M2 (M⊙)	1.12 ±	1.121 ± 0.008
R1 (R⊙)	1.52 ± 0.04	1.490 ± 0.080
R2 (R⊙)	1.05 ± 0.05	1.075 ± 0.130
logg	⋯	4.200 ± 0.079
[M/H]	⋯	−0.104 ± 0.046

Note.

^a Wang et al. (2021) derived ω from $e\cos \omega =(\pi /2)[({\phi }_{2}-{\phi }_{1})-0.5]$ and $e\sin \omega =({w}_{2}-{w}_{1})/({w}_{2}+{w}_{1})$. ϕ₂ is the phase of the secondary eclipses, ϕ₁ = 0, and w₁ and w₂ are widths of primary and secondary eclipses in phase, respectively (Kjurkchieva & Vasileva 2015).

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We compile 128 detached binary systems with atmospheric parameters and accurate masses and radii. Ten of the 128 detached eclipsing binaries are presented in Table 2, and the whole catalog can be found in China-VO: doi:10.12149/101147.

Table 2. List of 10 of the Samples Compiled from Previous Studies with Accurate Masses, Radii, and Atmospheric Parameters (T_eff, logg, [M/H])

Name	Teff (K)	logg (dex)	[M/H] (dex)	${\rm{log}}(L/{L}_{\odot })$	M (M⊙)	R (R⊙)	Reference
47 Tuc E32	6025.60 ± 18.08	4.227 ± 0.002	−0.71 ± 0.10	0.221 ± 0.012	0.862 ± 0.002	1.183 ± 0.001	Thompson et al. (2020)
	5956.62 ± 17.87	4.352 ± 0.003	−0.71 ± 0.10	0.059 ± 0.012	0.827 ± 0.002	1.004 ± 0.002
47 Tuc V69	5956.62 ± 17.86	4.143 ± 0.003	−0.71 ± 0.1	0.293 ± 0.012	0.876 ± 0.002	1.315 ± 0.002	Thompson et al. (2020)
	5984.12 ± 17.95	4.242 ± 0.003	−0.71 ± 0.10	0.193 ± 0.013	0.859 ± 0.003	1.162 ± 0.003	Brogaard et al. (2017)
AD Boo	6575.0 ± 120.0	4.173 ± 0.008	0.10 ± 0.15	0.642 ± 0.075	1.414 ± 0.008	1.613 ± 0.014	Clausen et al. (2008)
	6145.0 ± 120.0	4.350 ± 0.007	0.10 ± 0.15	0.279 ± 0.080	1.209 ± 0.006	1.216 ± 0.010
AH Cep	30,690.22 ± 245.52	4.019 ± 0.012	0.0 ± 0.0	4.530 ± 0.034	16.140 ± 0.113	6.510 ± 0.044	Popper & Hill (1991)
	28,773.98 ± 230.19	4.073 ± 0.018	0.0 ± 0.0	4.294 ± 0.036	13.690 ± 0.092	5.640 ± 0.048	Pavlovski et al. (2018)
AI Phe	5010.0 ± 120.0	3.595 ± 0.014	−0.14 ± 0.10	0.689 ± 0.101	1.234 ± 0.005	2.932 ± 0.048	Andersen et al. (1988)
	6310.0 ± 150.0	3.996 ± 0.011	−0.14 ± 0.10	0.674 ± 0.098	1.193 ± 0.004	1.818 ± 0.024
AL Ari	6367.95 ± 25.47	4.229 ± 0.005	−0.42 ± 0.08	0.446 ± 0.017	1.164 ± 0.001	1.372 ± 0.004	Graczyk et al. (2021)
	5559.04 ± 27.80	4.458 ± 0.008	−0.42 ± 0.08	−0.152 ± 0.021	0.911 ± 0.000	0.905 ± 0.003
AL Dor	6053.41 ± 30.27	4.404 ± 0.002	−0.10 ± 0.04	0.159 ± 0.020	1.102 ± 0.000	1.092 ± 0.001	Gallenne et al. (2019)
	6053.41 ± 30.27	4.399 ± 0.002	−0.10 ± 0.04	0.164 ± 0.020	1.103 ± 0.000	1.098 ± 0.001	Graczyk et al. (2021)
ASAS J045021+2300.4	5662.39 ± 28.31	4.360 ± 0.018	−0.26 ± 0.26	0.016 ± 0.027	0.934 ± 0.007	1.058 ± 0.010	Graczyk et al. (2021)
	3589.22 ± 43.07	4.829 ± 0.019	−0.26 ± 0.26	−1.604 ± 0.052	0.409 ± 0.002	0.408 ± 0.004
ASAS J051753-5406.0	5984.12 ± 89.76	3.982 ± 0.006	−0.1 ± 0.13	0.636 ± 0.061	1.311 ± 0.015	1.935 ± 0.009	Miller et al. (2022)
	5847.90 ± 81.87	4.332 ± 0.008	−0.1 ± 0.13	0.167 ± 0.057	1.093 ± 0.013	1.181 ± 0.006
ASAS J052821+0338.5	5105.05 ± 45.95	4.05 ± 0.01	−0.15 ± 0.14	0.312 ± 0.036	1.375 ± 0.005	1.830 ± 0.004	Stempels et al. (2008)
	4709.77 ± 42.39	4.08 ± 0.01	−0.15 ± 0.14	0.123 ± 0.036	1.329 ± 0.003	1.730 ± 0.004

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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In Table 3, the masses, radii, and atmospheric parameters of 58 double-line eclipsing binaries selected from LAMOST MRS are displayed. It contains 56 new orbital solutions of binaries. In Table 3, we also provide the parameters of secondary stars obtained from the orbital solution with PHOEBE. Although the internal errors of the parameters of the secondary stars are similar to those of the primary, T_eff and logg may have larger uncertainties in the measurement.

Table 3. Summary of 10 of the Binary Samples Obtained from LAMOST MRS with the Masses, Radii, and Atmospheric Parameters (T_eff, $\mathrm{log}g$, [M/H])

No.	gaia_source_id	Teff (K)	$\mathrm{log}g$ (dex)	${\rm{log}}(L/{L}_{\odot })$	M (M⊙)	R (R⊙)	[M/H] (dex)	$v\sin i$ (km s−1)
1 a	2101510803402761344	6500.00 ± 50.00	4.200 ± 0.079	0.570 ± 0.061	1.320 ± 0.060	1.520 ± 0.040	−0.104 ± 0.046	79.310
		5980.00 ± 22.00	4.544 ± 0.070	0.104 ± 0.096	1.120 ± 0.000	1.050 ± 0.050	−0.104 ± 0.046	⋯
2 b	2126356983051726720	6144.00 ± 100.00	4.220 ± 0.010	0.431 ± 0.066	1.290 ± 0.020	1.450 ± 0.010	−0.019 ± 0.035	288.310
		5966.00 ± 97.00	4.330 ± 0.020	0.216 ± 0.067	1.110 ± 0.050	1.200 ± 0.010	−0.019 ± 0.035	⋯
3	2101192082470870912	6012.74 ± 23.74	4.105 ± 0.022	0.413 ± 0.144	1.208 ± 0.003	1.482 ± 0.106	0.142 ± 0.014	89.190
		5767.75 ± 21.55	3.985 ± 0.051	0.559 ± 0.088	1.281 ± 0.003	1.905 ± 0.083	0.142 ± 0.014	⋯
4	3811041442290209024	5940.85 ± 32.41	4.157 ± 0.053	0.237 ± 0.164	1.065 ± 0.031	1.240 ± 0.101	−0.303 ± 0.031	54.630
		4482.10 ± 131.76	3.547 ± 0.080	0.463 ± 0.222	1.027 ± 0.031	2.824 ± 0.266	−0.303 ± 0.031	⋯
5	3423803170795356800	16,074.66 ± 100.00	3.656 ± 0.300	2.838 ± 0.059	3.452 ± 0.044	3.384 ± 0.091	−0.350 ± 0.100	⋯
		16,186.85 ± 200.69	4.019 ± 0.100	2.724 ± 0.143	3.272 ± 0.044	2.927 ± 0.197	−0.350 ± 0.100	⋯
6	3378228658638299136	9824.11 ± 1000.00	3.659 ± 0.100	1.676 ± 0.423	2.137 ± 0.060	2.375 ± 0.136	−0.482 ± 0.100	⋯
		9423.69 ± 249.40	4.024 ± 0.121	1.544 ± 0.218	1.901 ± 0.060	2.219 ± 0.211	−0.482 ± 0.100	⋯
7	604716006409635456	5529.74 ± 45.16	4.210 ± 0.100	0.238 ± 0.320	1.594 ± 0.013	1.432 ± 0.228	0.030 ± 0.130	66.400
		6598.67 ± 519.58	4.000 ± 0.238	0.826 ± 0.506	1.432 ± 0.013	1.980 ± 0.392	0.030 ± 0.130	⋯
8	598940203109496064	4697.68 ± 43.23	4.130 ± 0.200	−0.717 ± 0.254	0.746 ± 0.011	0.661 ± 0.083	−0.224 ± 0.170	⋯
		4658.78 ± 123.60	4.434 ± 0.260	−0.490 ± 0.410	0.756 ± 0.011	0.873 ± 0.173	−0.224 ± 0.170	⋯
9	609180023619173632	5971.97 ± 10.79	3.800 ± 0.040	0.696 ± 0.101	1.148 ± 0.009	2.081 ± 0.105	−0.234 ± 0.010	126.360
		5665.94 ± 84.29	3.836 ± 0.078	0.633 ± 0.161	1.157 ± 0.009	2.150 ± 0.161	−0.234 ± 0.010	⋯
10	658309364243893504	5840.19 ± 128.78	3.972 ± 0.107	0.688 ± 0.311	0.825 ± 0.030	2.156 ± 0.322	0.034 ± 0.067	83.920
		5343.71 ± 240.28	3.770 ± 0.153	0.468 ± 0.378	0.859 ± 0.030	1.998 ± 0.332	0.034 ± 0.067	⋯

Notes.

^a Wang et al. (2021). ^b Pan et al. (2020).

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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In this work, a total of 184 binaries with dynamic masses, radii, T_eff, logg, and [M/H] are compiled, including 128 binaries composed of two main-sequence stellar companions from literature data and 56 newly orbited solutions of binaries from LAMOST DR8 MRS. Figure 3 shows the [M/H] distribution of these stars. The range of [M/H] is between −1.86 and 0.61 dex. The filled histogram is the [M/H] distribution of samples from the literature, while the open red histogram indicates the distribution of the samples from LAMOST DR8 MRS. It is seen that, while most of the LAMOST samples are concentrated around the solar metallicity, similar to the previous samples, a few of them located at lower metallicity help to densifying the distribution of metallicity.

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Figure 4 shows the T_eff–logg relation of the stars in different [M/H] bins. Most of the samples are in the main-sequence stage. At a temperature around 6000 K, some older stars just about to turn off from the main sequence. The red rectangles show that the new LAMOST samples well fill up the metal-poor regime (see the top left panel). Meanwhile, because the hot stars are more sensitive to the age, the new LAMOST samples also provide more sampling points at different ages in the regime of high T_eff.

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4.1. Comparison with PARSEC

4.1.1.  T_eff–M Relations in Different Ranges of [M/H]

Figure 5 shows the T_eff–M relation of the samples. The samples cover a temperature range from 2600 to 38,000 K. In panel (a), the solid lines indicate the zero-age main sequence (ZAMS) with color-coded [M/H] from the PARSEC model. Although the model isochrones show a metallicity difference in the T_eff–M plane, the stellar samples overlapped with the isochrones do not show a clear gradient in metallicity with T_eff > 6000 K. This is likely caused by the relatively smaller difference in mass and T_eff rather than the measurement uncertainties of the two parameters. However, for the stars with T_eff < 6000 K, it seems that the metallicity gradient can be seen in the observed samples. The orange and red dots, which represent high metallicity, are located above the bluish symbols, which stand for low-metallicity stars.

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We separately show the T_eff–M relationships in different metallicity bins in panels (b)–(e) to enable detailed comparison between the PARSEC model and the observed stars.

It is seen that most of the stars are well consistent with the stellar models except for the cold stars with T_eff < 5000 K. The cold, low-mass stars show a steeper slope than the stellar models (dashed lines in the panels) in the variation of the stellar mass when T_eff declines. This implies that the mass directly estimated from the effective temperature of these cold stars may be overestimated when M ∼ 0.1 M_⊙ and underestimated when M ∼ 0.5 M_⊙. Although such systematic error may only around a few hundredths of M_⊙, it may potentially flatten the slope of the initial mass function.

Nevertheless, Figure 5 implies that the effective temperature combined with a quite coarse estimation of metallicity can approximate the stellar mass in a quite robust way. Therefore, we provide an empirical polynomial model of the stellar mass depending on T_eff and [M/H]. It is not trivial to fit the mass as a function of T_eff and [M/H] with totally free constraints in the coefficients of a polynomial. Hence, we first fit a quartic polynomial to the stellar model and then fix the coefficients of the third- and fourth-order terms of logT_eff obtained from the first step and fit the remaining coefficients to the data. After a few tests, we adopt the following form for the quartic polynomial:

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{log}(M)={a}_{1}+{a}_{2}\mathrm{log}(T)+{a}_{3}\mathrm{log}{\left(T\right)}^{2}+18.398\mathrm{log}{\left(T\right)}^{3}\\ \,-\,0.998\mathrm{log}{\left(T\right)}^{4}+{a}_{4}Z,\end{array}\end{eqnarray} \tag{ 4 }$$

where T and Z represent T_eff and [M/H]. The best-fit a₁, a₂, a₃, and a₄ are 414.761, 372.928, −124.902, and 12.088, respectively. It is noted that the relationship derived among these various parameters only applies to unevolved stars. The gray solid lines in panels (b)–(e) indicate the best-fit polynomial model. The relative residuals shown at the bottom of panels (b)–(e) are around 11%–18%.

In panel (b), benefiting from the observation of lower-metallicity stars in the outer disk of the Milky Way by LAMOST, we can see that there is an extension for high T_eff.

In panels (c) and (d), which correspond to −0.3 < [M/H] ≤ −0.1 and −0.1 < [M/H] ≤ +0.1, the numbers of samples from LAMOST fill some gaps in T_eff so that the coverage of T_eff becomes more continuous. In panel (e), for the stars with [M/H] > +0.1, the LAMOST samples also extend the range of T_eff to higher value.

4.1.2.  T_eff–R Relations in Different Ranges of [M/H]

Figure 6 shows the T_eff–R relation of the samples. Similar to Figure 5, we draw the relation for all the stars in panel (a) and for different metallicity ranges in the other four panels. Unlike the T_eff–M relation, the T_eff–R relation depends on the age of stars. Therefore, stars in panel (a) are mostly located above the ZAMS lines. The gradient of metallicity in the T_eff–R plane is not seen in the stellar samples since that R is affected either by age or by uncertainties in the R and T_eff estimates.

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Panels (b)–(e) display the T_eff–R relations with different ranges of [M/H]. The solid lines indicate the ZAMS lines at different [M/H], while dotted–dashed, dotted, and dashed lines with color-coded ages are the isochrones with ages of 1, 5, and 10 Gyr, respectively. The influence of age, which moves older stars upward out of the ZAMS, especially near T_eff ∼ 6000 K, is clearly seen in all the panels. Comparing to the theoretical isochrones, this implies that, with accurate measurement of radius and effective temperature, one could also determine the age of stars, not only for turnoff stars (which are one of the most sensitive kinds), but also for stars over a large range of T_eff.

In panel (d), for stars with T_eff < 4000 K, a clear deviation between the observed stars and the theoretical ZAMS is found, as seen in Figure 4(c). This may be caused by the inaccurate atmospheric model for cool stars.

4.1.3.  M–R Relations in Different Ranges of [M/H]

Figure 7 shows the M–R relations of our samples including both compiled and LAMOST MRS data. In principle, the trend is quite consistent with that presented by Eker et al. (2018). In panel (a), the ZAMS lines show that the difference in M–R relations for different [M/H] is small, at least in the range −0.58 < [M/H]< +0.07. In particular, when M < 0.5, the ZAMS lines for different [M/H] almost overlap the observed data.

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Panels (b)–(e) separately display the M–R relations in different ranges of [M/H]. Compared to the T_eff–R relations, they show that the M–R relation of the observed samples is well consistent with the stellar models, especially at M < 1 M_⊙, in different ranges of [M/H].

Almost all the data samples are located on or above the theoretical ZAMS lines. Although the masses of stars do not change significantly from the ZAMS, their radii enlarge to the extent that we are able to measure. In particular, for stars with mass around 2 M_⊙, the radii can extend by a factor of a few from the initial value at the ZAMS with the same mass. The main explanation for this phenomenon is the evolution of stars (Torres et al. 2010), ages, rotation (Kraus et al. 2011; Irwin et al. 2011), and/or magnetic fields (Spruit & Weiss 1986; Feiden & Chaboyer 2012; MacDonald & Mullan 2013).

4.1.4.  M–L Diagrams in Different Ranges of [M/H]

The luminosity, which is defined as $L=4\pi {R}^{2}\times \sigma {T}_{{\rm{eff}}}^{4}$, is derived from T_eff and R. In logarithmic form, $\mathrm{log}L\,=4\mathrm{log}{T}_{\mathrm{eff}}+2\mathrm{log}R-15.045$, in which L and R are in solar units.

Figure 8 shows the M–L diagram. In panel (a), the color-coded solid lines are the ZAMS lines with different [M/H], the dots are stars from the literature, and the rectangles are from LAMOST MRS. The colors of these symbols represent the metallicity with same code as the ZAMS lines. It is seen that [M/H] is almost independent of the M–L relation, which is consistent with the stellar model shown by ZAMS lines. Furthermore, at around 1 M_⊙, a larger dispersion in observed data can be clearly seen in the panel. This is likely due to the influence of age.

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The M–L diagrams with different [M/H] are shown in panels (b)–(e). They show that, for each metallicity bin, the observed data are all well consistent with the stellar models. It is noted that at the low-mass end with M < 1 M_⊙, the data also well agree with the models, unlike the T_eff–M and T_eff–R relations. This means that, although R is not well predicted by the stellar model for low-mass stars, the luminosity predicted by the model seems quite good.

We also constructed a polynomial model of the M–L–[M/H] relation and find the best-fit coefficients of the model using the stellar samples. The following relationship also only applies for unevolved stars. The polynomial model is written as

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}(L) & = & {a}_{1}+{a}_{2}\mathrm{log}(M)+{a}_{3}\mathrm{log}{\left(M\right)}^{2}\\ & & -0.755\mathrm{log}{\left(M\right)}^{3}+0.189\mathrm{log}{\left(M\right)}^{4}+{a}_{4}Z,\end{array}\end{eqnarray} \tag{ 5 }$$

in which a₁, a₂, a₃, and a₄ are 0.066, 4.141, 0.314, and −0.245, respectively. The coefficients of the third- and fourth-order terms of $\mathrm{log}M$ are determined from the polynomial fitting to the model. The best-fit M–L polynomial model is shown by thick solid lines for different [M/H] in panels (b)–(e). The relative difference between observed data and model (L_fit), D_relative = (L − L_fit)/L_fit, is shown at the bottom of each panel.

The high-precision, model-independent mass and radius derived from binaries allow us to calibrate different methods of estimation of mass and radius for single stars from observational atmospheric parameters (e.g., T_eff, $\mathrm{log}g$, and [M/H]). In this section, we compare M and R derived from orbital solutions with those from other measurements.

There are lots of approaches to obtaining stellar mass and radius. Figure 9 summarizes the usual paths to the two parameters.

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In principle, there are three kinds of approaches, which are drawn as three rectangles in the second layer, to estimate M and R. Each approach requires observed parameters drawn in the third layer in Figure 9. The solid blocks indicate the parameters obtained directly from observation and the dotted blocks are the parameters derived from some known relationships, which are shown in the green circles with dotted lines.

Other than orbital solution, the stellar mass can also be determined using isochrone fitting with T_eff, logg, and metallicity as inputs. Luminosity can also be used in the isochrone fitting for the stellar mass. In total, there are three usual methods for mass estimation: orbital solution, isochrone fitting with T_eff, logg, and Z, and isochrone fitting with additional $\mathrm{log}L$ derived from parallax. These three methods are listed in Table 4.

Table 4. Summary of the Usual Methods for Measuring Mass and Radius

Parameters	Definition	Method	Input Parameters	Mean Relative Difference a	Random Error
Mass	M	Orbited solution	RVs (RV1 and RV2), LC, Teff (primary star)	⋯	⋯
	Mg	Isochrone fitting	Teff, logg, [M/H]	0.010	0.143
	MLG	Isochrone fitting	Teff, logg, [M/H]; log L	−0.289	0.188
Radius	R	Orbited solution	RVs (RV1 and RV2), LC, Teff (primary star)	⋯	⋯
	Rg	Isochrone fitting	Teff, logg, [M/H]	0.011	0.164
	RLG	Isochrone fitting	Teff, logg, [M/H], log L	−0.322	0.191
	RSED	SED fitting	Teff (T1 and T2), q, distance and photometry from Gaia	−0.029	0.205
	RMg	$\sqrt{{MG}/g}$	mass, logg	0.027	0.292

Note.

^a Relative difference: (M – M_x )/M_x , x are various methods.

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To estimate the stellar radius, one usually has four methods other than the orbital solution. The first two use isochrone fitting with or without luminosity. The luminosity required in radius estimation should be determined from parallax. The other two methods are based on spectral energy distribution (SED) fitting either with multiband photometry, T_eff, and distance or with additional mass ratio, q, from other methods.

Table 4 summarizes all these methods with different input parameters and the results of the comparison with the binary orbital solution. We compare the results for mass and radius from various methods and assess the performance of them based on the results from the orbital solutions.

In general, all isochrone fitting approaches use a similar process, that is, to find the best-fit physical parameters in synthetic stellar models with the most similar observed atmospheric parameters and/or luminosity.

In this study, nearest neighbor search (NNS) is applied to find the best-match stellar atmospheric parameters and/or luminosity in the PARSEC (Bressan et al. 2012) theoretical stellar model grid. The atmospheric parameters and/or luminosity to be matched are defined as a vector θ = (T_eff, logg, [M/H]) or (T_eff, logg, [M/H], L), in which L is derived from parallax. The PARSEC isochrone grid with [M/H] between −2.0 dex and +0.5 dex and ages between 10 Myr and 10 Gyr sets up the search space S = {s₁, s₂, s₃, ..., s_n } with various combination of T_eff, logg, and [M/H]. First, we generate a searching point (θ_th) following a uniform distribution within the range of ±3 times measurement errors for θ. Second, the distance between the PARSEC isochrone grid and θ_th is calculated such that

$$\begin{eqnarray}&&d({\theta }_{\mathrm{th},i},{s}_{n,i})=\sqrt{\sum _{i=1}^{m}{\left(({s}_{n,i}-{{\boldsymbol{\theta }}}_{\mathrm{th},i})/{C}_{i}\right)}^{2}},\end{eqnarray} \tag{ 6 }$$

where i represents T_eff, logg, [M/H] or with additional L and m = 3 or 4 depending whether luminosity is among the input parameters. C is a custom coefficient to unify the scale of the parameters. We empirically adopt C as 1000 for T_eff, 0.25 for logg and L, and 0.1 for [M/H]. Third, we search for the closest point to θ in S and adopt the mass and radius of the nearest neighbor point as the best estimates. Finally, we obtain the mass and radius estimation with measurement error by repeating the above steps 5000 times. Each time, the input θ are randomly drawn from a Gaussian distribution adopting the uncertainties of θ as scales. During this process, T_eff, logg, and [M/H] are obtained from the observed spectra. L is measured using the bolometric magnitude, which is

$$\begin{eqnarray}&&\mathrm{log}L=0.45({M}_{b,\odot }-{M}_{b}),\end{eqnarray} \tag{ 7 }$$

where M_b,⊙ is the absolute bolometric magnitude of the Sun. The absolute bolometric magnitude M_b of each star is derived from

$$\begin{eqnarray}&&{M}_{b}=M{{\prime} }_{G}-{A}_{G}+{{BC}}_{G},\end{eqnarray} \tag{ 8 }$$

where $M{{\prime} }_{G}$ is derived from the following equation (Gaia Collaboration et al. 2018):

$$\begin{eqnarray}&&M{{\prime} }_{G}=G+5+5{\mathrm{log}}_{10}(\varpi /1000)\end{eqnarray} \tag{ 9 }$$

based on photometric and astrometric data of Gaia EDR3 (Gaia Collaboration et al. 2020). In Equation (9), G is G-band magnitude and ϖ is the parallax in milliarcseconds. The extinction A_V is given by the 3D dust map provided by dustmaps bayestar (Green et al. 2019). A_G is further derived from A_V using the extinction coefficient from Wang & Chen (2019). Finally, the bolometric corrections of stars are obtained from Chen et al. (2019).

Figure 10(a) shows the histogram of the relative residuals between dynamical mass (M) and other mass estimates. Figure 10(b) displays the histogram of the relative residuals between dynamical radius and other radius estimates.

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It is seen from the red line that, with T_eff, logg, and [M/H], the stellar mass estimated from isochrone fitting can reach a random error of about 14.3%. The systematic bias in M_g is only 1.0%, which is quite small. This implies that the stellar model applied in the isochrone is quite accurate compared to the dynamical mass.

The blue dashed line shows the performance of M_LG estimates, which is derived from isochrone fitting of the LAMOST MRS samples with renormalized unit weight error (Lindegren et al. 2021) < 1.5. The luminosities are estimated based on Gaia parallax. The relative systematic bias of M_LG is −28.9%, which is slightly larger than the random error, which is 18.8%.

It seems that the isochrone fitting without luminosity can well reproduce stellar mass, while with luminosity it tends to overestimate stellar mass. The systematic bias and larger random error of M_LG are probably due to the large uncertainties of bolometric correction in the G band.

Figure 10(b) shows the difference between dynamical radius (R) and that derived from four other methods. The red solid line shows that the isochrone-derived radius R_g using T_eff, logg, and [M/H] as inputs closely follows the dynamical radius. The systematic difference between R_g and R is only 1.1% with a random error of 16.4%.

The blue dashed line shows the comparison between dynamical R and R_LG, which is obtained from isochrone fitting using T_eff, logg, [M/H], and L. Similar to the mass estimate M_LG, a significant overestimation of 32.2% occurs in R_LG. This is again likely caused by the larger uncertainty of bolometric correction.

The orange dotted line shows that the SED-derived radius R_SED is quite consistent with the dynamical values. The mean difference is only −2.9% with a random error of 20.5%. The surface gravity-derived radius R_Mg shows a similar performance to that of the black dotted–dashed line. The bias is 2.7% and the random error is 29.2%, slightly larger than that for R_SED. Note that R_SED adopts an accurate distance derived from parallax given by Gaia and R_Mg uses dynamical mass in the calculation. These accurate parameters help to constrain the accuracy of R_SED and R_Mg.

The accuracy of atmospheric stellar parameters T_eff, logg, [M/H], and L is critical in the precision of mass and radius estimation from the stellar model. If these parameters cannot be measured to high precision, the mass and radius cannot be accurately constrained.

In this work, we publish 56 new detached binaries selected from the LAMOST MRS survey combined with 128 detached eclipsing binaries with independent atmospheric parameters (T_eff, logg, [M/H]) and accurate masses and radii compiled from previous studies. For the 56 detached binaries observed by LAMOST, we perform the MCMC method with PHOEBE to obtain the orbital solutions. With the additional LAMOST stars, we are able to increase the samples in each [M/H] bin. In particular, we extend the samples to lower [M/H] for stars with high T_eff. Hence, the distribution of [M/H] and T_eff becomes more continuous and densified. In total, we provide a catalog containing 184 binaries as the benchmark for stellar mass and radius covering a wide range of stellar parameters. The measurement uncertainties of masses and radii are within 5%.

We compared the samples with the PARSEC model for different [M/H] and found that the observed data, including the 56 new LAMOST stars and the 128 stars from the literature, essentially match well with the PARSEC isochrones. In addition, we also find that the enlarged radii at around the turnoff point depend more on stellar ages than on rotational velocity. Therefore, the radius estimates of the turnoff stars can also be potentially used as an indicator of age.

The comparisons of dynamical masses and radii with those derived from the stellar models show that the precision of model-estimated mass is >10% and that of model-estimated radius is >15% based on atmospheric parameters.

This work is supported by the National Key R&D Program of China No. 2019YFA0405500. C.L. thanks the National Natural Science Foundation of China (NSFC) with grant Nos. 11835057 (C.L.), 12173047(X.D.C.), and 12073047 (Z.H.C). The 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.