STELLAR PARAMETRIZATION OF LAMOST M DWARF STARS

Support-vector machine (SVM, also support-vector network; Cortes & Vapnik 1995) is one of the most important supervised machine learning algorithms to be used for classification and regression. The regression algorithms of SVM, named support-vector regression (SVR), has been used in many astronomical studies, particularly in spectral data analysis (Liu et al., 2012, 2014, 2015a, 2015b).

SLAM has three hyper-parameters, two of which are penalty level ($C$) and tube radius ($\epsilon$) coming from the genetic SVR algorithm. The third one ($\gamma$) indicates the width of the radial basis function (RBF), which is the kernel adopted by SLAM.

The architecture of SLAM consists of 3 steps:

• 1.

  Pre-processing. We normalize the spectra of the training data; in the mean time, we also standardize both stellar labels and spectral fluxes so that their mean is 0 and variance is 1;

• 2.

  Training. We train the SVR model with stellar parameters as independent variables and flux at given wavelength as dependent variable at each wavelength pixel using the training dataset;

• 3.

  Prediction. We apply the optimized SVR model to predict the stellar labels for observed spectra.

To choose the best-fit hyper-parameters at each wavelength, which is defined as training procedure, SLAM minimizes the $k$-fold cross-validated mean squared error (CV-MSE), which is defined as

where $\vec{\theta_{i}}=(T_{eff,i},\log{g}_{i},[M/H]_{i})$ denotes the stellar label vector of the $i$th star in the training data, $f_{j}(\vec{\theta_{i}})$ is the $j$th pixel of the training spectra as a function of $\vec{\theta_{i}}$. In prediction procedure, the posterior probability density function of $\vec{\theta}$ for an observed spectrum can be written as

where $p(f_{j,obs}|\vec{\theta})$ is the likelihood of the spectral flux $f_{j,obs}$ varing with $\vec{\theta}$ based on the trained SVR model and $p(\vec{\theta})$ is the prior of $\vec{\theta}$. The best estimate of stellar labels can be found at the maximum of the posterior probability $p(\vec{\theta}|f_{obs})$. In practice, the logarithmic form of Eq. (2) as in below is used by adopting a Gaussian likelihood:

where $f_{j,obs}$ is the $j$th pixel of the observed spectrum, $f_{j,model}(\vec{\theta})$ is the SVR model-predicted spectral flux corresponding to the stellar label vector $\vec{\theta}$. $\sigma_{j,obs}$ is the uncertainty of the $j$th pixel of the observed spectrum, and $\sigma_{j,model}(\vec{\theta})$ is the uncertainty of the $j$th pixel of the model-predicted spectrum given the stellar labels $\vec{\theta}$. In practice, $\sigma_{j,model}(\vec{\theta})$ is replaced with CV-MSE_j, which is independent of $\vec{\theta}$. SLAM adopts Maximum Likelihood Estimation (MLE) with Levenberg-Marquardt (LM, Moré 1978) least square optimizer as the optimization method to derive the most likely $\vec{\theta}$ for an observed spectrum.

LAMOST (Guo Shou Jing Telescope) is one of the most efficient spectroscopic survey telescopes providing 9,919,106 low-resolution (R$\sim$1800) optical spectra, among which 8,966,416 are stellar spectra, in its sixth data release (DR6) (Cui et al., 2012; Deng et al., 2012; Zhao et al., 2012). 607,142 spectra are published as the M dwarf catalog in LAMOST DR6.

We firstly select stars from LAMOST DR6¹¹1http://dr6.lamost.org M dwarf catalog (Yi et al., 2014; Guo et al., 2019) cross-matched with Gaia DR2 (Gaia Collaboration et al., 2018). Then samples are selected using the following criteria to obtain both reliable Gaia photometry ($G_{BP}-G_{RP}$ color and $G$-band magnitude), astrometry (parallax) and LAMOST spectra.

1. 1.

   parallax / parallax error > 5;

2. 2.

   phot_bp_mean_flux/phot_bp_mean_flux_error $>$ 20,
   phot_rp_mean$\_$flux/phot$\_$rp$\_$mean$\_$flux$\_$error $>$ 20,
   and phot$\_$g$\_$mean$\_$flux/phot$\_$g$\_$mean$\_$flux$\_$error $>$ 20;

3. 3.

   ruwe < 1.4

4. 4.

   signal-to-noise ratio (SNR) at $i$-band of LAMOST spectra is larger than 5.

The criteria 1-3 aim to select stars with both accurate photometry and astrometry. ruwe is the re-normalised unit-weight error which measures astrometric goodness-of-fit, crieria 3 is to select stars with small re-normalised unit-weight error. Criteria 4 aims to select spectra with clear stellar spectral signature. Similar to B20, criteria 5 aims to select the main-sequence M dwarfs as shown in Fig. 1.

We display the selected M dwarf samples in H-R diagrams (Fig. 1). The $G$-band absolute magnitude is estimated from the Bayesian distance from Bailer-Jones et al. (2018). To investigate the contamination of M giant stars, we draw 35,382 M giants on Fig. 1 (Zhong et al., 2019). We select $M_{G}+A_{G}\leq 5$ to remove the contaminations. Some of the M dwarf stars are not located at the main-sequence, but below and above the main-sequence. There are $\sim 7,000$ stars on the top side of the quadrangle which are likely pre-main sequence stars or binaries. And about 1,000 stars on the bottom side might be white dwarfs - MS binaries. We remove these stars and only select 379,258 M dwarf samples fell into the quadrangle for the estimation of the stellar parameters. Most of M dwarf stars are located within a few hundreds pc. Therefore, the interstellar extinction of M dwarfs are mostly very low. There may be a few stars with large extinction and thus location beyond the selection area. These stars may be removed mistakenly.

Since SLAM is a data-driven model, which assumes stellar labels as ground truth, reliable stellar parameters of training datasets are needed. To date, various methods and training datasets have been developed and introduced to measure labels (stellar parameters) for M dwarfs. In this work, we use APOGEE stellar parameters and BT-Settl synthetic spectra, respectively, as the training dataset separately described in the following two subsections 3.2.1 and 3.2.2.

Before estimating the metallicities of LAMOST M dwarfs, we first compared the [M/H] of APOGEE DR16 and [Fe/H] of B20, with a residual difference of only 0.1 dex, while B20 is overall more metal-rich than APOGEE by 0.2 dex. We find metallicity of APOGEE and B20 have better consistency where the metallicity abundance is higher. For stars with [M/H] larger than -0.25, the scatter between B20 and APOGEE is 0.07 dex with an offset of 0.15 dex. While for stars with [M/H] $<-0.25$, the scatter and bias are 0.15 and 0.25 dex, respectively.

M_H_AP is determined by [M/H] of APOGEE as the training dataset. The prediction of M_H_AP is estimated at the same time as the TEFF_AP using SLAM. We find M_H_AP agrees well with APOGEE labels (see the top-panel of Fig. 5). Their good agreement is exactly what we expected, since the APOGEE parameters are used as the training labels. Furthermore, M_H_AP is in agreement with B20 metallicity with a scatter of 0.16 dex as shown in the mid-panel of Fig. 5. We also find a similar scenario to the comparison between APOGEE and B20. For stars with [M/H] $>-0.25$, the scatter is $\sim$0.1 dex with a bias of 0.14 dex, but for stars with metallicities lower than -0.25 dex , the scatter is 0.2 dex with a 0.3 dex offset.

Metallicities could be also estimated by the BT-Settl models together with $T_{eff}$. However, the derived BT-Settl [M/H] (hereafter, M_H_BT) shows no clear correlation with observed data published in previous studies (see the bottom-panel of Fig 5). In the metal-poor regime ([M/H] $<-0.25$ dex), though with larger scatter, the correlation between M_H_BT and APOGEE metallicities is obvious. But for [M/H] $>-0.25$ dex, M_H_BT is not able to distinguish well the metallicities of stars. This is probably due to the lack of model grids with [M/H] $>0$. Therefore, the BT-Settl-trained [M/H] is not adopted in our catalog Table 4.

Fig. 6 displays the Hertzsprung-Russell Diagram (HRD) of the LAMOST M dwarfs in the Gaia $M_{G}$ versus logarithmic $T_{eff}$ (TEFF_AP) plane. Note that the $M_{G}$ is estimated from the Bayesian distance from Bailer-Jones et al. (2018) with extinction removed. 3D dust-reddening maps from Bayestar (Green et al., 2019) is used to estimate the visual extinction $A_{V}$ for each star. $A_{G}$ is further estimated from $A_{V}$ using the extinction factor from Wang & Chen (2019). Fig. 6 is the de-reddened HRD for a sub-sample of $\sim 9,000$ M dwarf stars in our catalog. As illustrated in the top-left panel, the metallicity (M_H_AP) can be clearly distinguished in HRD. Moreover, the top-right panel shows the color-magnitude diagram in $G_{BP}-G_{RP}$ versus $M_{G}$ plane, the gradient of metallicity is still very clear and shows the similar trend as in the top-left panel.

We further overlap the PARSEC (the Padova and Trieste Stellar Evolutionary Code) (Bressan et al., 2012) theoretical tracks with the age of 1 Gyr and find that they are well fit with each other as shown in the bottom panel of Fig. 6. The PARSEC version 1.2S⁵⁵5http://stev.oapd.inaf.it/cgi-bin/cmd (Chen et al., 2014) provides revisions on very-low-mass stars (VLMs) from the BT-Settl model with a wide range of metallicities from -2.19 to +0.70 dex. As displayed in Fig. 6, we found: a) The HRD given by the PARSEC stellar model shows good agreement with our observed HRD for TEFF_AP and TEFF_BT; b) Stars above the isochrone of [M/H] = 0.6 are likely to be in binary systems.

Considering active M dwarf is the only class of stars for which magnetic field affects overall stellar parameters systematically Kochukhov (2021). H$\alpha$ emission which can be the indicator of chromospheric activity might alter the reliability of the stellar parametrization. Guo et al. (2015) found that the fraction of active stars increases as spectral subtype becomes later. In our work, there is $\sim 8$ percent of M dwarf stars that have active magnetic fields. So during the procedure of data pre-processing, the pixels at the wavelength of H$\alpha$ emission are masked to improve the precision of stellar parameter estimation.

Table 2 shows the field definition of the stellar parameter catalog of our results. The important information of both the LAMOST and Gaia observations are presented in our catalog. TEFF_AP, TEFF_AP_ERR, M_H_AP and M_H_ERR are the ASPCAP-trained SLAM effective temperature and metallicity with the corresponding uncertainty. TEFF_BT and TEFF_BT_ERR are the BT-Settl-trained SLAM $T_{eff}$ with the estimated uncertainty. Note that the type column is adopted from Guo et al. (2015), which indicates the magnetic activity determined by measuring H$\alpha$ activity. The stellar parameters including $T_{eff}$ and [M/H] with the corresponding uncertainties of LAMOST M dwarfs estimated in this work are presented in Table 4.

To assess the accuracy of the metallicities estimated in our work, we select 23 Hyades member stars from our catalog. Hyades is the nearest open cluster, as far as we know (van Leeuwen, 2009), with a metallicity of about +0.13 dex (Schuler et al., 2006). We select all stars in a circle with a radius of 5 degrees, centered on right ascension of 66.725^∘ and declination of 15.867^∘ from Gaia DR2. Then, we adopt that stars located within $4<$pmra / parallax$<6$, $-2.5<$pmdec / parallax$<0$ and $0.04<$parallax$<0.05$, where pmra and pmdec are Gaia proper motions, are the member stars of Hyades. We cross-match our catalog with this sample and obtain 23 M dwarf member stars. We find that the distribution of M_H_AP of the members has a mean of 0.0 dex and a standard deviation of 0.1 dex, as shown in Fig. 7. This result tentatively verifies that the accuracy of M_H_AP is around 0.1 dex.

Furthermore, we cross-match our catalog with the Open Cluster Chemical Analysis and Mapping (OCCAM) survey (Donor et al., 2020) and obtain 138 member stars belonging to 15 open clusters. Fig. 8 compares [Fe/H] given by Donor et al. (2020) with M_H_AP. The left panel shows that M_H_AP matches very well with the metallicity of the corresponding clusters in the ranges from -0.7 to 0.4 dex. A few stars with higher metallicity in literature are estimated as lower metallicity using our method. This may implies some limit of the estimation in metal-poor regime. The right panel of Fig. 8 displays the distribution of residuals between M_H_AP and [Fe/H] of OCCAM. The mean value is 0.01 dex and the standard deviation is 0.1 dex. This illustrates that the uncertainty of the metallicity derived in this work is around 0.1 dex. Detailed information on the comparison of metallicities of cluster members is shown in Table. 3.

We further assess the precision of the metallicity using M dwarf - M dwarf wide binaries (Qiu et al., in prep). We start with the catalog of initial wide binaries candidates released by Tian et al. (2020), which contains 807,611 candidates, selected form Gaia DR2 within a distance of 4.0 kpc and a maximum projected separation $s=1.0$ pc. This catalog contains many types of wide binaries (e.g., main sequence - main sequence, main sequence - white dwarf, white dwarf - white dwarf etc.), and these wide binary stars may be polluted by visual binaries (chance alignments) at large separations, as described in Section 3.5 in Tian et al. (2020). We finally choose 92 pairs of binary stars with the separations less than 3,000 au to assess the precision of [M/H].

Fig. 9 compares the differences of the metallicities (M_H_AP) of the companions of these binaries. M_H_AP is expected to be consistent with each other if the companions are physically associated. The left panel shows the pairs have similar [M/H], only 2 of the 92 systems fall outside of 3-sigma range. These outliers are most likely not physical binary stars. The mean difference of metallicity of the primary and secondary companions is 0.01 dex with scatter of 0.23 dex. The right panel is the uncertainty-normalized metallicity difference; i.e., $\Delta{\rm[M/H]}/\sigma_{\Delta{\rm[M/H]}}=({\rm[M/H]_{1}}-{\rm[M/H]_{2}})/\sqrt{\sigma_{\rm[M/H]_{1}^{2}}+\sigma_{\rm[M/H]_{2}^{2}}}$. If the derived $\sigma_{\rm[M/H]}$ values are accurate, the uncertainty-normalized metallicity difference should be distributed as a Gaussian distribution with $\sigma=1$. As shown in the right panel, the distribution matches better with $\sigma\sim 0.8$, which suggests that M_H_AP uncertainties may be overestimated by $\sim 20\%$.

In this work, we have derived a spectroscopic catalog of stellar parameters for $\sim 300,000$ M dwarf stars. Precise effective temperatures and metallicities of M dwarf stars from LAMOST DR6 and Gaia DR2 with are given in this catalog. Stars located within the range of $2800<T_{eff}<4500$K (both of TEFF_AP and TEFF_BT) are finally adopted in our catalog. Two versions of effective temperatures are obtained with precisions of 40K for TEFF_BT and 50K for TEFF_AP at SNR $>50$. Particularly, TEFF_AP agrees with B20 with a 60K scatter and a 185K offset. The systematic errors come from different stellar atmospheric models, in this case, BT-Settl model and MARCS, respectively.

This study provides a method using BT-Settl model to obtain the parameters of LAMOST M dwarf stars. We also publish the code and the stellar parametrization pipeline on the website⁶⁶6https://github.com/jiadonglee/MDwarfMachine. Note that the data-driven method to derive stellar labels strongly relies on training datasets, for the estimation in this work is on BT-Settl model and stellar parameters of APOGEE. We address that SLAM can be used as the industrial framework against which to decode the stellar parameters of the cool atmosphere of M dwarf stars in the upcoming surveys such as SDSS-V (Kollmeier et al., 2017).