CALIBRATION OF LAMOST STELLAR SURFACE GRAVITIES USING THE KEPLER ASTEROSEISMIC DATA

LAMOST, also known as the "Guoshoujing Telescope," is a reflecting Schmidt telescope with an effective aperture of ∼4 m and a field of view (FOV) of 20 deg². Four thousand fiber units in its focal plane and 16 multi-object spectrographs make it highly efficient for spectroscopic surveys. During the first three years of operation, from 2011 October to 2014 June, LAMOST collected over 4.1 million spectra with resolving power ($R=\lambda /{\rm{\Delta }}\lambda$) of 1800, and public access to these spectra has been granted in the second data release (DR2).⁵ We cross-matched the DR2 and DR3 Quarter 1 (DR3Q1) catalogs with the KIC, and found 87,834 spectra of 70,703 common objects within 36 exposures in the LAMOST-Kepler project (De Cat et al. 2015). Atmospheric parameters for 48,486 stars out of these objects have been determined by LASP (Wu et al. 2011, 2014). The median uncertainties of T_eff, log g, and [Fe/H] were 128 K, 0.47 dex, and 0.15 dex for spectra with signal-to-noise ratios (S/N) of ∼50 at 477 nm, and 101 K, 0.44 dex, and 0.12 dex for spectra with S/N of ∼100.

The asteroseismic scaling relations (see Section 3.1) require T_eff as an input parameter. Thus, it is necessary to compare the LAMOST results with those of high-resolution spectroscopy (HRS). However, such a comparison for a large sample of stars is not always feasible because most targets of the LAMOST observing plan are not sufficiently bright, and thus are lacking of HR studies. Fortunately, the wealth of planet candidate hosts and other stars with noticeable values from the Kepler mission has generated significant interest in ground-based follow-up observations, and many of these are performed using HR spectrographs on large telescopes, such as the 10 m-aperture Keck I telescope and the Subaru telescope. As a result, accurate stellar parameters for hundreds of FGK stars in the Kepler field have been determined using various techniques, providing a good opportunity to test the LAMOST low-resolution spectra parameters.

Bruntt et al. (2012, hereafter, Br2012) and Thygesen et al. (2012, hereafter, Th2012) observed 93 solar-like and 82 red giant stars using high-resolution spectrographs. They determined the log g values for these stars from global oscillation parameters, while the other atmospheric parameters T_eff, [Fe/H], and ξ (micro-turbulent velocity) were determined using the spectroscopic method. Molenda-Żakowicz et al. (2013, hereafter, MZ2013) also analyzed 169 Kepler targets using spectral synthesis based on high-resolution spectra collected by different ground-based telescopes. Moreover, Buchhave et al. (2012, hereafter, Bu2012) studied the HR spectra of 152 planet-host stars using Stellar Parameter Classification (SPC), which is also a realization of spectral synthesis with a grid of template spectra. To validate and characterize the planetary properties, Marcy et al. (2014 hereafter, M2014) published stellar parameters of 22 Kepler Objects of Interests (KOIs) using the reconnaissance spectra obtained using the HIRES spectrometer (Vogt et al. 1994). Hirano et al. (2012, 2014, hereafter, Hi2014) also derived stellar parameters for 40 KOIs using the excitation/ionization equilibrium of Fe i and Fe ii lines. We divided the above samples into two groups—depending on how the values of log g were derived—using either the asteroseismology method or purely by using spectroscopic techniques.

Using the LAMOST AFGK-type star parameters catalog, we found 26, 41, 49, 39, 13, and 21 common stars with Br2012, Th2012, MZ2013, Bu2012, M2014, and Hi2014, respectively. In Figure 1, we compare the stellar parameters extracted from literature and the LAMOST catalog. The mean differences between the LAMOST and high-resolution spectroscopy parameters were $\langle {\rm{\Delta }}{T}_{{\rm{eff}}}\rangle =-1\pm 71$ K, $\langle {\rm{\Delta }}\mathrm{log}g\rangle =0.06\pm 0.17$ dex, and $\langle {\rm{\Delta }}{\rm{[Fe/H]}}\rangle =-0.03\pm 0.12$ dex for 73 stars in the asteroseismic group (red points in Figures 1), and $\langle {\rm{\Delta }}{T}_{{\rm{eff}}}\rangle =19\;\pm 100$ K, $\langle {\rm{\Delta }}\mathrm{log}g\rangle =0.02\pm 0.19$ dex, and $\langle {\rm{\Delta }}{\rm{[Fe/H]}}\rangle =0.00\;\pm 0.09$ dex for 116 stars in the spectroscopic group (blue points in Figure 1). For all the common stars, the mean differences were $\langle {\rm{\Delta }}{T}_{{\rm{eff}}}\rangle =11\pm 90$ K, $\langle {\rm{\Delta }}\mathrm{log}g\rangle =0.01\pm 0.18$ dex, and $\langle {\rm{\Delta }}{\rm{[Fe/H]}}\rangle =-0.01\pm 0.10$ dex. The temperature values obtained by using the LAMOST catalog were in good agreement with those obtained from high-resolution spectra (with $\langle {\rm{\Delta }}{T}_{{\rm{eff}}}\rangle \;=-14\pm 86$ K) for stars with T_eff < 5500 K; however, the difference was slightly higher (31 ± 89 K) for hotter stars. Figure 1 also shows that ${\rm{\Delta }}\mathrm{log}g\;(\mathrm{LAMOST}-\mathrm{HRS})$ tend to increase with decreasing log g for $\mathrm{log}g\lesssim 2.5$, and with decreasing T_eff for T_eff < 5,000 K, where the HR samples were mostly from giant stars studied by Th2012, for which log g values were derived using the asteroseismology method. Th2012 presented the stellar parameters based on pure spectroscopic methods as well. In Figure 2 we plot the differences between the LAMOST log g and the asteroseismic and spectroscopic log g in Th2012, as functions of T_eff. It is obvious that, for both cases, the trends of ${\rm{\Delta }}\mathrm{log}g$ are quite similar. Moreover, previous studies (e.g., Th2012, Takeda & Tajitsu 2015) have shown that log g obtained by the two methods are satisfactorily similar for giants. These facts suggest that LASP overestimated log g by up to 0.5 dex for cool giants. For metallicity, the scatter tends to increase with decreasing T_eff and log g.

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Solar-like oscillations are excited by the near-surface turbulent convection in a star, which is characterized by the global oscillation parameters Δν, corresponding to the average frequency separation between oscillation modes with consecutive radial orders n and the same spherical degree l, and ν_max, the frequency at which the oscillation power is maximum. The parameter Δν is proportional to the square root of the mean stellar density (ρ) and is therefore given by (Ulrich 1986)

$$\begin{eqnarray}&&{\rm{\Delta }}\nu =\sqrt{\displaystyle \frac{M/{M}_{\odot }}{{(R/{R}_{\odot })}^{3}}}{\rm{\Delta }}{\nu }_{\odot }\end{eqnarray} \tag{ 1 }$$

with respect to the Sun. The parameter ν_max is assumed to be scaled with the acoustic cutoff frequency (Brown et al. 1991), and Kjeldsen & Bedding (1995) used this assumption to relate ν_max to the fundamental stellar parameters as follows.

$$\begin{eqnarray}&&{\nu }_{{\rm{max}}}=\displaystyle \frac{M/{M}_{\odot }}{{(R/{R}_{\odot })}^{2}\sqrt{{T}_{{\rm{eff}}}/{T}_{\mathrm{eff},\odot }}}{\nu }_{\mathrm{max},\odot }.\end{eqnarray} \tag{ 2 }$$

By solving Equations (1) and (2), one can obtain the relations linking the stellar mass M, radius R, mean density ρ, and surface gravity log g with the global oscillation parameters Δν and ν_max. It is noted that log g only depends on ν_max for a given T_eff.

The values of Δν and ν_max for different types of stars in the Kepler field have been used to estimate M, R, ρ, and logg in various studies. For instance, Kallinger et al. (2010a) determined the parameters for >1000 red giants based on the first 138 days of the Kepler photometric data. Hekker et al. (2011) used the data of the first 33 days to characterize more than 10,000 giants for which solar-like oscillations have been detected. This work was later refined by Stello et al. (2013) using the Kepler data with a longer time baseline of 681 days. The Kepler mission also detected solar-like oscillations for 500 out of 2000 pre-selected main-sequence and sub-giant stars during the first 10 months of its scientific operation (Chaplin et al. 2011). The fundamental parameters of these stars were published in Chaplin et al. (2014) and led to better characterization of planets (Huber et al. 2013) and their host stars (Mathur et al. 2011; Johnson et al. 2014). Huber et al. (2014) presented the revised catalog of parameters for more than 190,000 stars for the Kepler Quarter 1–16 data.

Although stellar parameters can be directly derived Equations (1) and (2) as

$$\begin{eqnarray}&&\left(\displaystyle \frac{M}{{M}_{\odot }}\right)={\left(\displaystyle \frac{{\nu }_{\mathrm{max}}}{{\nu }_{\mathrm{max},\odot }}\right)}^{3}{\left(\displaystyle \frac{{\rm{\Delta }}\nu }{{\rm{\Delta }}{\nu }_{\odot }}\right)}^{-4}{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\mathrm{eff},\odot }}\right)}^{3/2}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{R}{{R}_{\odot }}\right)=\left(\displaystyle \frac{{\nu }_{\mathrm{max}}}{{\nu }_{\mathrm{max},\odot }}\right){\left(\displaystyle \frac{{\rm{\Delta }}\nu }{{\rm{\Delta }}{\nu }_{\odot }}\right)}^{-2}{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\mathrm{eff},\odot }}\right)}^{1/2}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{\rho }{{\rho }_{\odot }}\right)={\left(\displaystyle \frac{{\rm{\Delta }}\nu }{{\rm{\Delta }}{\nu }_{\odot }}\right)}^{2}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\mathrm{log}g=\mathrm{log}{g}_{\odot }+\mathrm{log}\left(\displaystyle \frac{{\nu }_{\mathrm{max}}}{{\nu }_{\mathrm{max},\odot }}\right)+\displaystyle \frac{1}{2}\mathrm{log}\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{T}_{\mathrm{eff},\odot }}\right)\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\left(\displaystyle \frac{L}{{L}_{\odot }}\right)={\left(\displaystyle \frac{{\nu }_{{\rm{max}}}}{{\nu }_{\mathrm{max},\odot }}\right)}^{2}{\left(\displaystyle \frac{{\rm{\Delta }}\nu }{{\rm{\Delta }}{\nu }_{\odot }}\right)}^{-4}{\left(\displaystyle \frac{{T}_{{\rm{eff}}}}{{T}_{\mathrm{eff},\odot }}\right)}^{5},\end{eqnarray} \tag{ 7 }$$

some sets of (M, R, T_eff) for a given metallicity are not permitted according to the stellar evolution theories. Grid-based methods containing a significantly large number of parameters (M, R, T_eff, [Fe/H]) returned by stellar evolution programs have been widely used to find the best match to the observed parameters (see Chaplin et al. 2013, and references therein). We adopted the Geneva stellar evolutionary tracks (Lejeune & Schaerer 2001), which cover a wide range of mass and metallicity (Z) values. The values of high-temperature opacities were taken from the OPAL data (Iglesias & Rogers 1996), and those of low-temperature opacities were taken from Kurucz (1991) or Alexander & Ferguson (1994). For stars with M ≤ 1.5 M_⊙, a core overshooting parameter of $d/{H}_{{\rm{P}}}=0.2$ was adopted. Mass loss of Reimers (1975) and de Jager et al. (1988) were taken into account. In previous grid-based analyses (e.g., Basu et al. 2010; Kallinger et al. 2010a; Huber et al. 2014), some widely used stellar models, such as the YREC (Yale Stellar Evolution Code; Demarque et al. 2008), DSEP (Dartmouth Stellar Evolution Program; Dotter et al. 2008), and BaSTI (Bag of Stellar Tracks and Isochrones; Piersanti et al. 2004) models, did not account for the evolutionary stages after the helium flash. On the contrary, the evolution phases of Geneva database were calculated to the end of the early asymptotic giant branch (EAGB) phase for intermediate-mass stars ($2\leqslant M/{M}_{\odot }\leqslant 5$), and to the end of the carbon burning phase for larger mass stars ($M/{M}_{\odot }\geqslant 7$). Therefore, the evolutionary stages following the helium flash were included for stars with M > 2 M_⊙. As a substantial number of our samples met the above condition, we considered that our method naturally eliminates the systematic bias toward larger masses for giants in Huber et al. (2014), where post-helium flash data were not included for calculations.

To ensure that at least 10² models are available for the final probability density function (PDF) of each star, we generated a dense grid by interpolating the evolutionary tracks in steps of 0.02 dex for [Fe/H], ranging from −2.0 to +1.5, and steps of 0.02 M_⊙ for the initial mass (M₀), ranging from 0.8 to 5.0 M_⊙. For each track, the Geneva database contained at most 51 groups of data points with T_eff, L, age, and M. Here, M is the stellar mass, varying with time due to the mass loss. We interpolated 500 points along the entire time span, and calculated R and log g using the basic physical relations, along with Δν and ν_max that were calculated according to the scaling relations in Equations (1) and (2), for each interpolated point. We adopted the solar seismic parameters ${\rm{\Delta }}{\nu }_{\odot }=135.1\pm 0.1\;\mu \mathrm{Hz}$ and ${\nu }_{{\rm{max}},\odot }=3090\pm 30\;\mu \mathrm{Hz}$ that were based on the data collected by VIRGO aboard SOHO spacecraft during ∼11,000 days (Huber et al. 2011). Our complete grid had a total of ∼1.8 × 10⁷ points, each containing nine parameters, T_eff, Z, M, R, L, log g, age, Δν, and ν_max.

Stellar fundamental parameters can be subsequently derived from the observed oscillation parameters (Δν and ν_max) using the Bayesian approach, if T_eff and [Fe/H] are known. The Bayes' theorem can be stated as

$$\begin{eqnarray}&&p({\boldsymbol{\theta }}| {\boldsymbol{d}},M)=\displaystyle \frac{p({\boldsymbol{\theta }}| M)p({\boldsymbol{d}}| {\boldsymbol{\theta }},M)}{p({\boldsymbol{d}}| M)},\end{eqnarray} \tag{ 8 }$$

where $p({\boldsymbol{\theta }}| {\boldsymbol{d}},M)$ is the posterior probability distribution of parameters ${\boldsymbol{\theta }}$ for a certain model M, based on the observational data ${\boldsymbol{d}}$. The model M stands for an individual datum corresponding to an evolutionary status in our grid. The distribution $p({\boldsymbol{\theta }}| M)$ is the prior probability distribution of ${\boldsymbol{\theta }}$, and the likelihood function $p({\boldsymbol{d}}| {\boldsymbol{\theta }},M)$ is the probability of obtaining ${\boldsymbol{d}}$, given the parameters ${\boldsymbol{\theta }}$ for model M. The quantity $1/p({\boldsymbol{d}}| M)$ is the normalization term. In our case, the observational data set is ${\boldsymbol{d}}=({T}_{{\rm{eff}}},{\rm{[Fe/H]}},{\rm{\Delta }}\nu ,{\nu }_{{\rm{max}}})$, and

$$\begin{eqnarray}&&p({\boldsymbol{d}}| {\boldsymbol{\theta }},M)={{ \mathcal L }}_{{T}_{{\rm{eff}}}}{{ \mathcal L }}_{{\rm{[Fe/H]}}}{{ \mathcal L }}_{{\rm{\Delta }}\nu }{{ \mathcal L }}_{{\nu }_{{\rm{max}}}}.\end{eqnarray} \tag{ 9 }$$

The likelihood functions of each parameter are calculated to match the observational ones by assuming independent Gaussian-distributed errors. Therefore, we have

$$\begin{eqnarray}&&{{ \mathcal L }}_{d}=\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{d}}\mathrm{exp}\left[-\displaystyle \frac{{({d}_{{\rm{obs}}}-{d}_{{\rm{model}}})}^{2}}{2{\sigma }_{d}^{2}}\right].\end{eqnarray} \tag{ 10 }$$

Some previous studies adopted uniform priors $p({\boldsymbol{\theta }}| M)$ for all models in the grid (e.g., Kallinger et al. 2010a). However, it should be noted that for a star with given (${M}_{0},Z$), the probability of its physical quantities being (${T}_{{\rm{eff}}},R,L,\mathrm{log}g$) when the star is being observed is inversely proportional to the star's evolutionary speed in its current stage. Otherwise, the resulting stellar parameters would be biased toward the rapid evolution phases (see the description of the GOE pipeline in Chaplin et al. 2014). In our approach, the differential age of a track with a given (${M}_{0},Z$) can well represent the reciprocals of the evolutionary speeds; thus,

$$\begin{eqnarray}&&p({\boldsymbol{\theta }}| {M}_{i,j})=C\displaystyle \frac{{a}_{i+1,j}-{a}_{i,j}}{{a}_{n,j}-{a}_{1,j}}\quad i=1,2,\cdots n-1,\end{eqnarray} \tag{ 11 }$$

where ${a}_{i,j}$ is the age of the ith interpolated point in the jth track, C denotes the normalization factor, and n = 500 is the number of interpolated points along each track. In the above equation, the time span of two adjacent points (${a}_{i+1,j}-{a}_{i,j}$) is normalized by the total time (${a}_{n,j}-{a}_{1,j}$) of the jth track; otherwise, the posterior probability distributions would be biased toward low-mass stars. Although larger-mass stars have shorter lifetimes than less massive stars, and hence, have lower probabilities of being observed as their higher luminosities make them visible over longer distances to a magnitude-limited survey, which, to some extent, cancels out the above age selection effect. Therefore, aim of Equation (11) only corrects the bias caused by different evolutionary speeds at different stages, rather than lifetimes, as a function of the stellar masses. In our study, uniform probabilities for stars with different (M₀, Z) were assumed, because our observed data ${\boldsymbol{d}}$ were accurate, and the prior probabilities of (M₀, Z) were not expected to vary significantly over such a relatively narrow parametric range.

All the sample stars in this work have been monitored by the Kepler space telescope with extremely high photometric precision during its scientific operation. Several research groups have devoted attention to extracting the values of Δν and ν_max from the Kepler light curves using various techniques (e.g., Huber et al. 2009; Mosser & Appourchaux 2009; Hekker et al. 2010; Kallinger et al. 2010b). We employed the parameters from different literature sources, as listed in Table 1. For nonseismic parameters T_eff and [Fe/H], we used the values returned by the LASP in the LAMOST AFGK-type star parameters catalog.

In our work, the derived log g obtained using the above approach could differ from spectroscopically obtained values by as much as 0.5 dex (see Section 4.3), which could in turn yield a significant bias in T_eff and [Fe/H]. Therefore, we determined our spectroscopic parameters (T_eff, [Fe/H], and log g) iteratively. First, asteroseismic log g values (hereafter, $\mathrm{log}{g}_{{\rm{iter0}}}$) were obtained by using the above-mentioned grid method, with T_eff and [Fe/H] listed in the LAMOST catalog (hereafter, ${T}_{{\rm{eff,}}{\rm{LASP}}}$, and [Fe/H]${}_{{\rm{LASP}}}$), and oscillation parameters Δν and ν_max. Then, the LAMOST spectra for all the sample stars were reanalyzed by LASP with fixed $\mathrm{log}{g}_{{\rm{iter0}}}$, to acquire new ${T}_{{\rm{eff,}}{\rm{iter1}}}$ and [Fe/H]_iter1 values, which were then used for calculating asteroseismic $\mathrm{log}{g}_{{\rm{iter1}}}$. We found that, in our sample, a change of +0.1 dex in log g resulted in ΔT_eff ∼ +27 K and Δ[Fe/H] ∼ +0.02 dex for giants, and in ΔT_eff ∼ +36 K and Δ[Fe/H] ∼ +0.01 dex for dwarfs. The differences between $\mathrm{log}{g}_{{\rm{iter1}}}$ and $\mathrm{log}{g}_{{\rm{iter0}}}$ were within ±0.03 dex for 99% of our giants, and ±0.01 dex for all of our dwarfs, except for only one star. These small changes in log g after the first iteration had negligible effects on T_eff and [Fe/H] compared with the observational uncertainties because, according to Equation (6), asteroseismic log g only depends weakly on T_eff. Consequently, our results regarding atmospheric parameters converged after one iteration.

We applied the grid-based method described in Section 3.1 to derive the PDFs of M, R, log g, L, and age, for 3060 stars with S/N >30 spectra in the LAMOST-DR2 and DR3 Quarter 1 catalog. For each PDF, we report its mean as the result, and use standard deviation as a measure of uncertainty. Figure 3 shows examples of the PDFs of M, R, log g, age, and L for one typical main-sequence star and two evolved stars. For comparison, we plot the time-weighted and non-weighted PDFs by using red and black solid curves, respectively. These results show that by taking into account the evolution speed effect as discussed in Section 3.1, the values of M, R, and L shift toward higher values, whereas the resulting age becomes smaller. This is expected because the weights of the phases are reduced after evolving off the main sequence.

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Stellar properties of planet candidate hosts are of particular interest because they are directly related to the planetary radii and masses in transit and Doppler detections. Serious uncertainties in metallicities, surface gravities, and radii, mostly based on broadband photometry, have been found in the KIC (e.g., Verner et al. 2011; Dong et al. 2014), while high-resolution spectra are expensive for most of the Kepler planet hosts with K_p < 13 (e.g., Marcy et al. 2014). Alternatively, asteroseismology with spectroscopic inputs has been used for characterizing these planetary systems (e.g., Huber et al. 2013; Chaplin et al. 2013). There were 60 KOIs in our catalog, including 15 confirmed planet-host stars, 23 "false positives," and 22 host candidates awaiting validation. In Table 2, we list the results for the confirmed and candidate hosts. The entire sample is available via an online catalog, and the first five rows are shown in Table 3 to illustrate the format.

Table 2.  Stellar Parameters of Kepler Planet Candidate Hosts

KOI	KIC	Kepler name	Teff (K)	log g	[Fe/H]	${M}_{\star }/{M}_{\odot }$	${R}_{\star }/{\text{}}{R}_{\odot }$	${L}_{\star }/{L}_{\odot }$	Age (Gyr)
5	8554498	⋯	5863 ± 91	4.015 ± 0.013	+0.130 ± 0.110	1.23 ± 0.09	1.80 ± 0.05	3.55 ± 0.30	6.10 ± 2.44
7	11853905	Kepler-4	5820 ± 93	4.110 ± 0.010	+0.170 ± 0.110	1.15 ± 0.07	1.56 ± 0.04	2.62 ± 0.22	7.11 ± 2.08
41	6521045	Kepler-100	5888 ± 91	4.130 ± 0.008	+0.090 ± 0.110	1.14 ± 0.06	1.52 ± 0.04	2.59 ± 0.19	7.23 ± 1.88
75	7199397	⋯	5913 ± 88	3.758 ± 0.008	−0.110 ± 0.120	1.36 ± 0.05	2.55 ± 0.04	7.09 ± 0.54	3.51 ± 0.26
85	5866724	Kepler-65	6264 ± 101	4.232 ± 0.006	+0.130 ± 0.110	1.25 ± 0.05	1.42 ± 0.02	2.81 ± 0.24	3.20 ± 0.97
108	4914423	Kepler-103	5901 ± 98	4.167 ± 0.010	+0.100 ± 0.120	1.14 ± 0.06	1.46 ± 0.03	2.38 ± 0.19	6.80 ± 1.78
122	8349582	Kepler-95	5652 ± 113	4.168 ± 0.012	+0.220 ± 0.150	1.07 ± 0.07	1.41 ± 0.03	1.96 ± 0.23	8.42 ± 2.15
123	5094751	Kepler-109	5992 ± 107	4.216 ± 0.014	−0.030 ± 0.130	1.09 ± 0.06	1.35 ± 0.03	2.15 ± 0.20	7.42 ± 1.52
262	11807274	Kepler-50	6271 ± 101	4.132 ± 0.008	−0.090 ± 0.120	1.23 ± 0.07	1.58 ± 0.03	3.41 ± 0.34	4.30 ± 1.49
269	7670943	⋯	6481 ± 133	4.233 ± 0.010	−0.060 ± 0.140	1.29 ± 0.06	1.43 ± 0.03	3.28 ± 0.36	2.46 ± 0.87
271	9451706	Kepler-127	6142 ± 100	4.261 ± 0.010	+0.250 ± 0.110	1.22 ± 0.05	1.35 ± 0.02	2.36 ± 0.22	3.55 ± 1.15
275	10586004	Kepler-129	5802 ± 93	4.086 ± 0.009	+0.230 ± 0.110	1.23 ± 0.06	1.66 ± 0.04	2.93 ± 0.21	5.33 ± 1.46
276	11133306	⋯	6013 ± 99	4.322 ± 0.010	−0.030 ± 0.120	1.08 ± 0.05	1.19 ± 0.02	1.70 ± 0.16	5.93 ± 1.41
280	4141376	⋯	6171 ± 132	4.420 ± 0.009	−0.320 ± 0.160	1.07 ± 0.06	1.05 ± 0.02	1.48 ± 0.15	2.84 ± 1.81
281	4143755	⋯	5699 ± 118	4.108 ± 0.010	−0.480 ± 0.180	0.97 ± 0.04	1.43 ± 0.03	2.09 ± 0.21	10.18 ± 0.91
285	6196457	Kepler-92	5908 ± 89	4.059 ± 0.010	+0.150 ± 0.110	1.23 ± 0.07	1.71 ± 0.04	3.26 ± 0.24	5.56 ± 1.85
319	8684730	⋯	5879 ± 88	3.930 ± 0.015	+0.080 ± 0.110	1.29 ± 0.12	2.04 ± 0.09	4.66 ± 0.58	5.51 ± 2.57
370	8494142	Kepler-145	6102 ± 102	4.016 ± 0.012	+0.030 ± 0.130	1.25 ± 0.09	1.82 ± 0.05	4.08 ± 0.38	5.38 ± 2.07
623	12068975	Kepler-197	6129 ± 104	4.315 ± 0.014	−0.420 ± 0.140	1.00 ± 0.06	1.15 ± 0.03	1.73 ± 0.17	6.67 ± 1.91
674	7277317	⋯	4750 ± 107	3.636 ± 0.011	+0.100 ± 0.140	1.10 ± 0.10	2.64 ± 0.10	3.67 ± 0.44	9.28 ± 2.57
1221	3640905	Kepler-278	4891 ± 84	3.612 ± 0.006	+0.270 ± 0.110	1.27 ± 0.06	2.91 ± 0.05	4.63 ± 0.28	5.37 ± 1.12
1621	5561278	⋯	6115 ± 107	3.969 ± 0.012	−0.010 ± 0.130	1.32 ± 0.10	1.97 ± 0.07	4.83 ± 0.55	4.40 ± 1.90
1925	9955598	Kepler-409	5421 ± 93	4.496 ± 0.005	+0.130 ± 0.130	0.91 ± 0.03	0.89 ± 0.01	0.64 ± 0.05	7.39 ± 1.83
2133	8219268	Kepler-91	4464 ± 80	2.933 ± 0.010	+0.270 ± 0.090	1.32 ± 0.09	6.47 ± 0.19	17.56 ± 1.39	4.75 ± 1.30
2706	9697131	⋯	6313 ± 141	4.022 ± 0.013	+0.120 ± 0.140	1.42 ± 0.08	1.92 ± 0.05	5.26 ± 0.66	2.79 ± 1.13
2792	11127479	⋯	5982 ± 97	4.229 ± 0.012	+0.210 ± 0.110	1.15 ± 0.06	1.36 ± 0.03	2.13 ± 0.21	5.69 ± 1.68
5110	4953262	⋯	4899 ± 104	3.186 ± 0.036	−0.130 ± 0.140	1.38 ± 0.23	4.94 ± 0.44	12.58 ± 2.68	4.32 ± 2.81
5174	5524229	⋯	4971 ± 87	3.102 ± 0.035	−0.400 ± 0.130	1.41 ± 0.23	5.50 ± 0.49	16.33 ± 3.45	3.39 ± 1.88
5223	6023571	⋯	4610 ± 105	2.984 ± 0.020	+0.180 ± 0.120	1.26 ± 0.17	5.96 ± 0.32	15.59 ± 2.01	6.12 ± 3.08
5281	6425377	⋯	4941 ± 148	3.107 ± 0.013	−0.010 ± 0.200	1.27 ± 0.15	5.20 ± 0.26	13.11 ± 1.59	5.23 ± 2.57
5287	6509282	⋯	4706 ± 133	2.528 ± 0.035	−0.110 ± 0.160	2.50 ± 0.42	14.20 ± 1.34	96.87 ± 22.13	0.81 ± 0.74
5322	6756369	⋯	4808 ± 114	3.228 ± 0.012	−0.190 ± 0.150	1.26 ± 0.13	4.51 ± 0.19	10.31 ± 1.13	5.04 ± 2.15
5543	8560475	⋯	4798 ± 108	2.448 ± 0.023	−0.160 ± 0.130	2.23 ± 0.43	14.67 ± 1.08	98.49 ± 23.31	1.02 ± 0.52
5578	8868481	⋯	5582 ± 90	3.774 ± 0.014	−0.040 ± 0.130	1.26 ± 0.07	2.41 ± 0.08	5.05 ± 0.51	5.25 ± 1.74
5782	10272858	⋯	4992 ± 76	3.448 ± 0.009	+0.020 ± 0.100	1.34 ± 0.14	3.61 ± 0.18	7.06 ± 0.89	4.46 ± 2.09
6877	7431665	⋯	4661 ± 82	2.645 ± 0.025	−0.090 ± 0.100	1.75 ± 0.29	10.35 ± 0.64	47.06 ± 8.90	2.09 ± 1.33
7296	10214328	⋯	6624 ± 141	4.017 ± 0.014	−0.350 ± 0.140	1.36 ± 0.08	1.89 ± 0.05	6.16 ± 0.74	2.77 ± 0.61

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Figure 4 compares the stellar parameters of the KOIs obtained in our work with those obtained in the previous studies that employed high-resolution spectroscopy. There are five stars in common with M2014, and all of them show good agreements in terms of T_eff, log g, [Fe/H], M, and R. Our derived age values were systematically higher than those in M2014, which is likely owing to the different theoretical evolution tracks used in these two studies (Y² in M14, and Geneva model in our study). We also analyzed four common KOIs with Hi2012 and Hi2014, for which, the values of T_eff and [Fe/H] obtained were systematically lower while the values of M and R were higher than the previously reported results. The stellar ages of two KOIs (KOI-269 and KOI-262) agreed within the corresponding error ranges, while the age of KOI-280 determined by us was lower. Because the main-sequence stars evolve slowly on the HR diagram compared with the post-main-sequence phases, age estimation by fitting the isochrones or evolution tracks is difficult and model-dependent (see Soderblom 2010, and references therein). Our estimation based on global oscillation parameters remains meaningful because the stellar ages are further constrained by Δν and ν_max in addition to T_eff, log g, and [Fe/H]. A more detailed approach involves spectral analysis of excited oscillation modes (e.g., Silva Aguirre et al. 2015). Moreover, the field of gyrochronology, which has been developing with the help of the Kepler data, has made remarkable progress in refining the empirical relation between the stellar age and rotational period (e.g., García et al. 2014; Angus et al. 2015).

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Huber et al. (2014, hereafter, H2014) presented the stellar parameters for a large sample of Kepler stars observed in Quarter 1–16. Their catalog is composed of several sub-categories designated by C.1–C.14 (see their Table 1), depending on the sources of input parameters (T_eff, log g, and [Fe/H]). In this sub-section we focus on common stars within the H2014 sub-categories C.1, C.4, and C.5. All three of these data sets adopted asteroseismic log g, while T_eff and [Fe/H] were obtained using various techniques (spectroscopy, photometry, and KIC). The category C.1 contains most of the "gold-standard" samples of H2014, for which high-resolution spectroscopy was used for the best possible characterization. On the other hand, stars in C.4 and C.5 had no spectroscopic temperatures or metallicities. In such cases, the authors of H2014 used a revised temperature scale by Pinsonneault et al. (2012), and their [Fe/H] values were either fixed to −0.2 or obtained from KIC.

Figures 5–7 show the differences between the values of M, R, and L (panels (a)–(c)), as well as the atmospheric parameters T_eff, log g, and [Fe/H] (panels (d)–(f)) that were used as inputs. All of these reveal an excellent agreement between the values of log g in the two works, despite the fact that the values of T_eff and [Fe/H] were taken from different sources. In each of the above figures, we also plot the differences between positions of the stars in the Kiel diagram, where the colored circles correspond to our results, and the other sides of solid lines correspond to the values from H2014. The colors are coded with ΔM (this work—H2014). Geneva evolution tracks with M = 1.0, 2.0, and 3.0 M_⊙, and [Fe/H] = 0.0 (solid gray lines) and −0.5 (dashed gray lines) are also shown.

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Figure 5 shows the comparison of 76 common stars including dwarfs and giants, for which both H2014 and this work used spectroscopic T_eff and [Fe/H] as inputs for asteroseismic log g and other physical parameters. Our results are in a good agreement with the previous work, with mean differences of only 0.00 ± 0.02, −0.02 ± 0.06, and −0.04 ± 0.14, for log g, log R, and log L, respectively. All outlying points in panel (a) correspond to giants with $\mathrm{log}g\lt 3.5$, for which evolution tracks are highly degenerated in the HR diagram. We noted that our results on stellar masses for these giants are systematically lower than those from H2014. This can be explained by the bias toward higher mass in the previous studies, as discussed in Section 3.1.

Figure 6 compares the physical and spectroscopic parameters for 199 dwarfs and sub-giants in common with sub-category C.4 in H2014. Although for all of the stars in this sub-category the values of [Fe/H] were fixed at −0.2, a good agreement between the two studies was found in terms of the values of R, L, and log g. On average, our T_eff values were 91 ± 120 K lower than the previously reported values. The mean difference between the stellar mass values was 0.01 ± 0.10 M_⊙, as shown in panel (a).

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Figure 7 shows the same comparison, but for sub-category C.5 in H2014. In contrast to Figure 6, the stars in C.5 are giants, with $\mathrm{log}g\lt 3.5$. Our T_eff values were 226 ± 130 K lower than those reported in H2014, which subsequently significantly affected the stellar physical parameters. In terms of log g, our results were in good agreement with H2014, with the mean difference of only 0.00 ± 0.08 dex. The mean differences were −0.03 ± 0.23 dex for [Fe/H], −0.07 ± 0.10 for $\mathrm{log}R/{R}_{\odot }$, and −0.23 ± 0.23 for $\mathrm{log}L/{L}_{\odot }$. Moreover, the derived stellar masses were generally lower than those in H2014, and the comparison of stellar masses in panel (a) reveals a chaotic distribution. We note that most of the masses in H2014 are in the 0.8–3.7 M_⊙ range; however, the range was 0.9–3.0 M_⊙ in the present study, with only a few exceptions corresponding to M > 3.1 M_⊙. These deviations can be interpreted by the facts that (1) our spectroscopic T_eff values obtained from low-resolution spectra were systematically lower than those reported in H2014, which were photometric T_eff from Sloan Digital Sky Survey griz filters (Pinsonneault et al. 2012); and (2) our approach correct the bias caused by different evolution speeds. Therefore, the overall distribution shifted rightward toward the tracks of less massive evolved stars on the Kiel diagram. Despite this, as $M\propto {T}_{{\rm{eff}}}^{3/2}$ whereas $\mathrm{log}g\propto 0.5\mathrm{log}\;{T}_{{\rm{eff}}}$ according to Equations (3) and (6), and $\mathrm{log}g\propto \mathrm{log}M/{R}^{2}$ is explicitly related to ν_max in Equation (2), independent of metallicity. It is not surprising that a good agreement was found between the values of log g in the two studies.

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Figures 5–7 show that although asteroseismology yields satisfactory log g insensitive to T_eff, the determination of stellar masses, and especially for giants, remains a challenge without reliable T_eff and [Fe/H]. Because the masses and radii of extra-solar planets are usually measured in terms of ratios relative to their host stars, the influence of inaccurate stellar T_eff and [Fe/H] is inevitable. For giant stars, we estimate that an error of +100 K in T_eff results in a mass error of about +0.20 M_⊙ and a radius error of about +0.61 R_⊙ when using the asteroseismic grid-based method. In addition, an error of +0.1 dex in [Fe/H] results in a mass error of +0.23 M_⊙ and a radius error of about +0.74 R_⊙. This in turn emphasizes the importance of spectroscopic analysis of planet-hosting giant stars, for characterizing the planetary properties. However, this effect is not significant for dwarfs.

By comparing the LAMOST log g values with the asteroseismic log g values adopted in this work, we found that their difference exhibited a clear trend in the T_eff–log g plane, implying the possibility of establishing calibration relations for log g values of LAMOST samples. To obtain reliable relationships, we excluded all spectra with S/N < 50, and adopted the atmospheric parameters based on the spectra with highest S/N, if there were multiple observations for the same star in the LAMOST DR2 and DR3 Quarter 1 catalog. This left us 2289 samples, including 2094 giants and 195 dwarfs.

In Figure 8 we show the differences between $\mathrm{log}g{}_{{\rm{(LAMOST)}}}$ and $\mathrm{log}g{}_{{\rm{(Adopted)}}}$ as a function of T_eff, with color coded by LAMOST log g, for 2094 giants with T_eff < 5400 K and $\mathrm{log}g\lt 3.5$. We used a first order 2D polynomial function $f(x,y)={p}_{0}+{p}_{1}x+{p}_{2}y+{p}_{3}{xy}$ to model ${\rm{\Delta }}\mathrm{log}g$, where x is T_eff and y is log g. The coefficients ${p}_{0}\sim {p}_{3}$ were determined by least squares fitting. After the coefficients were determined, the residuals of the fitting for all the data points were calculated. In the next step, the points with residuals falling outside ±3σ were removed, and the least squares fitting was performed again. The procedure converged after two iterations, when all residuals were within ±3σ. There are 2,044 stars left out of 2094 giants. This means that ∼2% of the giants in Figure 8 are outliers that were not included in the fitting procedure. The final relation was

$$\begin{eqnarray}\mathrm{log}g{}_{{\rm{(Adopted)}}} & = & \mathrm{log}g{}_{{\rm{(LAMOST)}}}\\ & & -5.716\quad +1.283\times {T}_{3}\\ & & +1.188\times \mathrm{log}g{}_{{\rm{(LAMOST)}}}\\ & & -0.2882\times {T}_{3}\times \mathrm{log}g{}_{{\rm{(LAMOST)}}},\end{eqnarray} \tag{ 12 }$$

where ${T}_{3}={T}_{{\rm{eff}}}/{10}^{3}$ K is the normalized temperature from the LAMOST. The range of temperatures in which this relation is applicable is 3800 K ≤ T_eff ≤ 4500 K for stars with $+1.3\;\leqslant \mathrm{log}g\leqslant 2.2$, or 3800 K ≤ T_eff ≤ 5200 K for stars with $+2.2\;\leqslant \mathrm{log}g\leqslant 3.5$. In Figure 10, we show the residuals of fitting, namely, ${\rm{\Delta }}\mathrm{log}g\;{\rm{(Adopted)}}$ for giants against T_eff LAMOST within a range of 0.3 dex in each panel, with log g in the 1.7–3.5 range. The rms values of log g in each panel were comparable, varying from 0.07 to 0.12 dex. We also calculated the rms for these stars with T_eff in steps of 200 K, and obtained a 0.07–0.10 dex variation range. The overall rms for all giants, excluding the outliers, was 0.082 dex.

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Figure 9 shows the same relation for dwarfs with T_eff > 5400 K and $\mathrm{log}g\gt 3.5$. It is seen that ${\rm{\Delta }}\mathrm{log}g({\rm{LAMOST}}-{\rm{Adopted}})$ has a weak dependence on T_eff, but no dependence on log g. Therefore, we only performed a linear least squares fit, which yielded

$$\begin{eqnarray}&&\mathrm{log}g{}_{{\rm{(Adopted)}}}=\mathrm{log}g{}_{{\rm{(LAMOST)}}}+0.525-0.0902\times {T}_{3},\end{eqnarray} \tag{ 13 }$$

where T₃ and $\mathrm{log}g{}_{{\rm{(LAMOST)}}}$ are the same as those in Equation (12). The applicable range is 5400 K ≤ T_eff ≤ 7000 K, and $+3.5\leqslant \mathrm{log}g\leqslant +4.5$. The rms value was only 0.075 dex, and the residuals are plotted in Figure 10 (Right).

In Figure 11 we show the dependence of the relation on the stellar metallicity ([Fe/H]) as given in the LAMOST AFGK-type star parameters catalog. We separated the entire sample into four groups, corresponding to different metallicity ranges: [Fe/H] > +0.1; −0.1 < [Fe/H] < +0.1; −0.4 < [Fe/H] < −0.1; and [Fe/H] < −0.4, as shown, respectively, in Panels (b)–(e) of Figure 11. Both the giants and dwarfs for different metallicities ranging from −2.0 to +0.4 exhibited very similar trends. Although our entire sample covered a wide range of metallicity values, from −2.26 to +0.50, the number of stars at the metal-poor end was very small (only 10 stars with [Fe/H] < − 1.0). Therefore, the calibration relations are applicable for stars with −1.0 < [Fe/H] < +0.5, and care must be taken for stars with metallicities outside this range.

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As shown in Section 4.3, the correction of LAMOST log g reached 0.5 dex for cool giants among our sample stars. To quantify the impact of this change on T_eff and [Fe/H], we iteratively performed asteroseismic and spectroscopic analyses (see Section 3.2). Figures 12 and 13 show the variations in T_eff and [Fe/H] versus ${\rm{\Delta }}\mathrm{log}g({\rm{Adopted}}-{\rm{LAMOST}})$ for giants and dwarfs, respectively. It is clear that the giants in Figure can be divided into two groups. Giants with T_eff > 4500 K exhibit clear correlations of both ΔT_eff and Δ[Fe/H] with ${\rm{\Delta }}\mathrm{log}g$. Linear fits yielded

$$\begin{eqnarray}&&{\rm{\Delta }}{T}_{{\rm{eff}}}=284.9\times {\rm{\Delta }}\mathrm{log}g{}_{{\rm{(Adopted-LAMOST)}}}+3.8\end{eqnarray} \tag{ 14 }$$

and

$$\begin{eqnarray}&&{\rm{\Delta }}{\rm{[Fe/H]}}=0.216\times {\rm{\Delta }}\mathrm{log}g{}_{{\rm{(Adopted-LAMOST)}}}+0.008,\end{eqnarray} \tag{ 15 }$$

where the finally adopted equations were ${T}_{{\rm{eff}}}={T}_{\mathrm{eff}(\mathrm{LAMOST})}\;+{\rm{\Delta }}{T}_{{\rm{eff}}}$, and [Fe/H] = [Fe/H]_(LAMOST) + Δ[Fe/H]. The rms values for the above two equations were 18.4 K and 0.026 dex, respectively. On the other hand, the ΔT_eff for cooler giants (T_eff < 4500 K) exhibited no clear trends with ${\rm{\Delta }}\mathrm{log}g({\rm{Adopted}}\;-{\rm{LAMOST}})$.

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Figure 13 shows that ΔT_eff and Δ[Fe/H] are significantly correlated with ${\rm{\Delta }}\mathrm{log}g({\rm{Adopted}}-{\rm{LAMOST}})$ for dwarfs. Linear fits yielded

$$\begin{eqnarray}&&{\rm{\Delta }}{T}_{{\rm{eff}}}=292.4\times {\rm{\Delta }}\mathrm{log}g{}_{{\rm{(Adopted-LAMOST)}}}+14.3\end{eqnarray} \tag{ 16 }$$

and

$$\begin{eqnarray}&&{\rm{\Delta }}{\rm{[Fe/H]}}=0.103\times {\rm{\Delta }}\mathrm{log}g{}_{{\rm{(Adopted-LAMOST)}}}+0.009,\end{eqnarray} \tag{ 17 }$$

with rms values of 34.6 K and 0.024 dex, respectively.

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In this section, we propose empirical calibration relations for LAMOST log g in Equations (12) and (13) for giants and dwarfs, respectively. Generally speaking, the absolute values of log g corrections are much larger for giants than dwarfs, which reflects the difficulty associated with obtaining precise log g for evolved stars using low-resolution spectra. For the coolest giants in our sample, with temperatures around 4000 K, the magnitude of corrections reached 0.5 dex. Because no systematic bias between asteroseismic log g and spectroscopic log g values has been found previously, the deviations of the LAMOST log g values from the asteroseismic ones are likely attributed to the adopted pipeline. Because such major modifications of log g would inevitably affect the determination of both T_eff and [Fe/H], we provided the T_eff and [Fe/H] corrections in Equations (14)–(17). Although the relations were derived for the Kepler targets, they are applicable to any LAMOST stars with spectroscopic parameters in the ranges given in Sections 4.3 and 4.4. The corrections of log g, T_eff, and [Fe/H] should always be applied together.

Due to their high luminosity, giant stars are visible from longer distances than dwarfs and play important roles in probing the Galactic structure, kinetics, and chemical evolution. Because g ∼ MR⁻², an overestimation of 0.5 dex of log g for a K4 giant with T_eff of 4000 K implies that the radius is underestimated by $0.5\mathrm{ln}(10){\rm{\Delta }}\mathrm{log}g\simeq$, or 58% by assuming a fixed M. This in turn causes ∼115% underestimation of stellar luminosity as $L\sim {R}^{2}{T}_{{\rm{eff}}}^{4}$. Here T_eff is fixed because, according to Figure 12, it nearly does not change with log g. Furthermore, considering that $L=4\pi {D}^{2}F$, where D denotes the distance and F is the observed flux density, the luminosity distance is also underestimated by ∼58% if the interstellar extinction is ignored. For a typical K1 giant with T_eff of 4,600 K and log g of 2.8, LAMOST overestimates its actual log g by ∼0.22 dex; consequently, the values of R, L, and D will be underestimated by 25%, 50%, and 25%, respectively. However, given that T_eff is also reduced by ∼56 K (according to Equation (12)), the impact of increasing R on L will be offset by ∼5%. Therefore, the resulting luminosity and distance need to be increased by 45% and 22%, respectively. Another example is a red-clump giant with T_eff = 4900 K and log g = 2.6, for which the correction of log g is close to zero. Thus, previous works based on LAMOST red-clump giants (e.g., Wan et al. 2015) are nearly not affected by the systematic deviations of log g.