# Stellar Atmospheric Parameters of M-type Stars from LAMOST DR8

Ming-Yi Ding CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China  School of
Astronomy and Space Science, University of Chinese Academy of Sciences,
Beijing 100049, China  Jian-Rong Shi CAS Key Laboratory of Optical Astronomy,
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, China  School of Astronomy and Space Science, University of Chinese
Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS>Yue Wu  CAS Key
Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100101, China  Hugh R.A. Jones Centre for
Astrophysics Research, University of Hertfordshire, College Lane, AL10 9AB,
Hatfield, UK  Hong-liang Yan CAS Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
School of Astronomy and Space Science, University of Chinese Academy of
Sciences, Beijing 100049, China<EMAIL_ADDRESS>Chun-Qian Li CAS Key
Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100101, China  School of Astronomy and Space
Science, University of Chinese Academy of Sciences, Beijing 100049, China  Qi
Gao CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China  School of
Astronomy and Space Science, University of Chinese Academy of Sciences,
Beijing 100049, China  Tian-Yi Chen CAS Key Laboratory of Optical Astronomy,
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, China  School of Astronomy and Space Science, University of Chinese
Academy of Sciences, Beijing 100049, China  Jing-Hua Zhang CAS Key Laboratory
of Optical Astronomy, National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100101, China  Shuai Liu CAS Key Laboratory of Optical
Astronomy, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100101, China  School of Astronomy and Space Science, University of
Chinese Academy of Sciences, Beijing 100049, China  Tai-Sheng Yan CAS Key
Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100101, China  School of Astronomy and Space
Science, University of Chinese Academy of Sciences, Beijing 100049, China
Xiao-Jin Xie  CAS Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China  Tibet
University, Lhasa 850000, China

(Accepted 2022.04.14)

###### Abstract

The Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) Low
Resolution Spectroscopic Survey (LRS) provides massive spectroscopic data of
M-type stars, and the derived stellar parameters could bring vital help to
various studies. We adopt the ULySS package to perform $\chi^{2}$ minimization
with model spectra generated from the MILES interpolator, and determine the
stellar atmospheric parameters for the M-type stars from LAMOST LRS Data
Release (DR) 8. Comparison with the stellar parameters from APOGEE Stellar
Parameter and Chemical Abundance Pipeline (ASPCAP) suggests that most of our
results have good consistency. For M dwarfs, we achieve dispersions better
than 74 K, 0.19 dex and 0.16 dex for $T_{\rm eff}$, $\log{g}$ and [Fe/H],
while for M giants, the internal uncertainties are 58 K, 0.32 dex and 0.26
dex, respectively. Compared to ASPCAP we also find a systematic
underestimation of $\Delta{T_{\rm eff}}=$ $-$176 K for M dwarfs, and a
systematic overestimation of $\Delta{\log{g}}=$ 0.30 dex for M giants.
However, such differences are less significant when we make comparison with
common stars from other literature, which indicates that systematic biases
exist in the difference of ASPCAP and other measurements. A catalog of 763,136
spectra corresponding to 616,314 M-type stars with derived stellar parameters
is presented. We determine the stellar parameters for stars with $T_{\rm eff}$
higher than 2,900 K, with $\log{g}$ from -0.24 dex to 5.9 dex. The typical
precisions are 45 K, 0.25 dex and 0.22 dex, for $T_{\rm eff}$, $\log{g}$ and
[Fe/H], respectively, which are estimated from the duplicate observations of
the same stars.

stars: fundamental parameters — stars: late-type — methods: data analysis —
techniques: spectroscopic — surveys

††journal: ApJS

##  1 Introduction

Of the adopted astronomical spectral types, the M spectral type shows the most
wide-ranging properties. On the Hertzsprung-Russell (H-R) diagram, the M
dwarfs (dMs) and the M giants (gMs) showed the most extreme differences in
luminosity and radius (e.g., Gray & Corbally, 2009). The dMs are the faintest
of the core hydrogen burners (e.g., Kirkpatrick, 1992), while the gMs might
have the largest brightness variations. Studies of low-mass main-sequence dMs
show that dMs are the most common stars in the Galaxy, which comprise 70% of
all stars (e.g., Laughlin et al., 1997). The dMs are very important to
determine the initial mass function which constrains theoretical star
formation studies, and they are helpful to trace the chemical and dynamical
history of the Galaxy.

The main-sequence types—O, B, A, F, G and K stars are considered to be
completely hydrogen-burning stars, while the later types (giant – S and C
types, dwarf – L, T and Y type) stars appear more likely to be carbon stars or
brown dwarfs (Kirkpatrick et al., 1999). The turning point for these later
spectral types is the M-type stars, and they are a milestone in the study of
the chemical and dynamical evolution of the Galaxy.

M-type stars are iconic for their crowded molecular absorption bands (Gray &
Corbally, 2009), making their continuum difficult to define in the optical
region. Over the years, methods based on the measurement of the atomic and
molecular features in the optical and near-infrared region (6,300–9,000 Å),
such as the use of TiO, CaH and CaII triplet, are highlighted in stellar
atmospheric parameter determination for M-type stars (Bessell, 1991;
Kirkpatrick, 1992; Reid et al., 1995; Cenarro et al., 2001, etc.). In a work
of M-subdwarf (sdM) classification, Gizis (1997) showed the metallicity
dependency on the TiO and CaH features in the region of 6,200 to 7,400 Å.

Likewise, an empirical spectroscopic index has been suggested (Lepine et al.,
2007) for sdM classification, which is defined as the relative strength of
TiO5 and CaH molecular absorption bands. Mann et al. (2012) used six different
gravity-sensitive molecular and atomic indices to determine the luminosity
class for late-type Kepler exoplanet candidates. The spectral index of TiO5
and CaH2+CaH3 was used to separate dMs from gMs and sdMs in Large Sky Area
Multi-Object Fiber Spectroscopic Telescope (LAMOST) Data Release (DR) 1 (Luo
et al., 2015), and assemble these stars into a set of M-type spectral template
library (Zhong et al., 2015a, b). By employing this template, Zhang et al.
(2019) calibrated the spectroscopic index and applied a new separator to
identify 2,791 new sdMs from LAMOST DR4.

Currently, thanks to several ongoing large-scale spectroscopic surveys, large
numbers of spectra along with stellar parameters are available now. For
example, the Sloan Digital Sky Survey (SDSS) uses the Sloan Foundation 2.5 m
telescope at Apache Point Observatory (Gunn et al., 2006) to determine
spectroscopic abundances on a large scale. The Apache Point Observatory
Galactic Evolution Experiment (APOGEE, Majewski et al., 2017) is one of
subprojects of SDSS-III (Eisenstein et al., 2011), which provides high
resolution (R $\sim$ 22,500), high signal-to-noise ratio (SNR $>$ 100), near-
infrared spectra. During the second generation of the APOGEE project
(APOGEE-2) with the SDSS-IV, the latest DR17 presents spectra for about
650,000 stars. Meanwhile, a set of stellar parameters and chemical abundances
of up to 26 elements are also provided though the APOGEE Stellar Parameter and
Chemical Abundances Pipeline (ASPCAP) (García Pérez et al., 2016), which fits
the observed spectra via comparison to the synthetic spectra generated from
the MARCS stellar atmospheric model. Jönsson et al. (2020) updated ASPCAP with
the precalculated grids (Gustafsson et al., 2008), which improved the model
performance under the effective temperature of 3,500 K, and made the stellar
parameters for cool stars more accurate compared to the previous version.
Other spectroscopic surveys such as the Large Sky Area Multi-Object Fiber
Spectroscopic Telescope (LAMOST, Cui et al., 2012), the Radial Velocity
Experiment (RAVE, Steinmetz et al., 2006), the GALactic Archaeology with
HERMES (GALAH, De Silva et al., 2015), Gaia-ESO (Brown et al., 2018) and the
Calar Alto high-Resolution search for M-dwarfs with Exo-earths with Near-
infrared and optical Echelle Spectrographs (CARMENES, Reiners et al., 2018;
Quirrenbach et al., 2020), also provide relevant large-scale spectroscopic
data of M-type stars.

In stellar astrophysics research, stellar atmospheric parameters (effective
temperature $T_{\rm eff}$, surface gravity $\log{g}$, metallicity [Fe/H])
derived from high resolution and high signal-to-noise ratio (SNR) spectra are
important indicators. During the past few years, different methods have been
developed for determining the stellar parameters from high quality spectra.
Generally, these approaches could be simply divided into two categories (Wu et
al., 2011), the first one is the minimum distance method (MDM), including the
measurement of equivalent width (EWs) or the synthetic spectra based on
absorption lines (Jofré et al., 2019). For example, the $\chi^{2}$
minimization is a widely used MDM method, which searches the minimum
$\chi^{2}$ between the observed spectra and the templates or spectra generated
from stellar atmospheric models. The other one is so-called machine-learning
approaches such as the Artificial Neural Network (ANN) and non-linear
regression methods.

Mann et al. (2015) determined the precise stellar parameters for 183 nearby
dMs via MDM method and presented empirical relations between $T_{\rm eff}$,
absolute magnitude, radius, mass and bolometric flux. In a search for
exoplanets around dMs, Passegger et al. (2018) determined the stellar
atmospheric parameters for over 300 dMs by fitting the spectra generated from
the PHOENIX-ACES (Husser et al., 2013) models. Rajpurohit et al. (2018)
determined the stellar parameters for 292 early to late M-type stars by
comparing high resolution spectra with the synthetic spectra obtained from BT-
Settl (Allard et al., 2011, 2012, 2013). This model is also used to derive the
stellar atmospheric parameters and kinematics for sdMs from the LAMOST survey
by Zhang et al. (2021).

Similarly, Kesseli et al. (2019) obtained $T_{\rm eff}$ for 88 sdMs using the
BT-Settl grid and [Fe/H] values estimated from color empirical relation
(Newton et al., 2014). Hejazi et al. (2020) presented a catalog of 1,544
nearby dMs with stellar parameters determined from the latest version of BT-
Settl model. Sarmento et al. (2021) derived the stellar parameters for 313 dMs
through $\chi^{2}$ minimization using APOGEE H-band spectra. The LAMOST
Stellar Parameter Pipeline at Peking University (LSP3) adopts a cross-
correlation algorithm to determine stellar radial velocities (RV) and uses a
template matching method for stellar atmospheric parameter determination
(Xiang et al., 2015).

The machine-learning methods make stellar parameter predictions based on data-
driven models learned from large data set. The advantage of these data-driven
models is their flexibility of learning patterns from spectra and transforming
them into various stellar parameters. Maldonado et al. (2020) have presented a
catalog of $T_{\rm eff}$ and [Fe/H] via Principal Component Analysis (PCA) and
sparse Bayesian methods for 204 dMs from HARPS GTO M-dwarf survey (Bonfils et
al., 2013). Based on The Cannon (Ness et al., 2015; Casey et al., 2016; Ho et
al., 2017), Birky et al. (2020) trained a data-driven model with high
resolution spectra from APOGEE and derived stellar parameters ($T_{\rm eff}$
and [Fe/H]) for 5,875 M-type stars. Passegger et al. (2020) trained a
Convolutional Neural Network (CNN) based on synthetic spectra generated from
PHOENIX-ACES model to estimate the stellar parameters for CARMENES dMs.

The Stellar Parameters and Chemical Abundances Network (SPCANet, Wang et al.,
2020) have constructed a CNN with three convolutional layers trained on
spectra from the LAMOST Medium Resolution Spectroscopic survey (MRS, Liu et
al., 2020) with the APOGEE Payne (Ting et al., 2019) stellar labels. Based on
the Support Vector Regression (SVR, Smola & Schölkopf, 2004) method, Zhang et
al. (2020) developed a data-driven approach called the Stellar LAbel Machine
(SLAM), which performs a nonlinear regression on stellar labels and the
spectrum itself, and estimated the stellar labels for stars from LAMOST DR5
over a wide range of spectral types. Li et al. (2021) have adapted the SLAM
program by introducing the APOGEE stellar labels and synthetic spectra from
BT-Settl model as training sets, and obtained the stellar parameters for dMs
from LAMOST DR6.

The empirical spectral library is also commonly used for determining the
stellar parameters in large scale spectroscopic surveys, which directly or
indirectly compare the observed spectra with templates listed in the spectral
library. Yee et al. (2017) presented a high resolution, high SNR empirical
spectral library of 76 dMs, along with a parameterizing tool “Empirical
SpecMatch”, which estimates the stellar parameters for FGKM stars by comparing
them against this spectral library. Other empirical spectral libraries,
including STELIB (Le Borgne et al., 2003), ELODIE (Prugniel & Soubiran, 2001;
Prugniel et al., 2007), CFLIB (Indo-US library) (Valdes et al., 2004) and
MILES (Sanchez-Blazquez et al., 2006; Falcón-Barroso et al., 2011), are
composed of high quality spectra with good coverage of stellar atmospheric
parameter space. With the help of empirical libraries of homogeneous M-type
star templates, we are able to derive precise stellar atmospheric parameters
from matching our spectra to these templates.

In this study, we employ the ULySS (Koleva et al., 2009) package to perform
$\chi^{2}$ minimization between observed and model spectrum, and use the MILES
interpolator (Sharma et al., 2016) to estimate the stellar atmospheric
parameters for M-type stars from the LAMOST Low Resolution Spectroscopic
survey (LRS) DR8.

The structure of this paper is as follows. Section 2 introduces the
preparation of our spectra from the LAMOST survey, and Section 3 gives a brief
description of the interpolation model and the ULySS program. In Section 4, we
present the summary of our results and compare the stellar parameters derived
from our method with those in the literature. The deviation and precision are
discussed in Section 5. Finally, in Section 6 we show our conclusions.

##  2 Observational Data

The LAMOST spectral survey provides us with a massive number of medium (R
$\sim$ 7,500) and low (R $\sim$ 1,800) resolution spectra, which are collected
by the innovative active reflecting Schmidt telescope located in Xinglong
Observatory, China (Cui et al., 2012; Zhao et al., 2006; Deng et al., 2012).
The Schmidt optics system has a large aperture (effective aperture of 3.6 m
$\sim$ 4.9 m) with a wide field of view (FOV $\sim 5^{\circ}$). The
spectroscopic system contains 16 spectrographs with 32 integrated CCD cameras
(4K $\times$ 4K). A total of 4,000 spectra can be obtained simultaneously via
4,000 fibers plugged into the spectrographs.

Yi et al. (2014) utilized the spectra from LAMOST pilot survey to determine
the spectral subtype for 67,082 dMs by matching the relative strength of
atomic and molecular features in the spectral region from 6,000 Å to 9,000 Å.
Zhong et al. (2015a) classified 8,639 gMs and 101,690 dMs/sdMs in LAMOST DR1
using spectral templates. Guo et al. (2015) presented a catalog of 110,321
spectra for 93,619 dMs from LAMOST DR1, and separated gMs from dMs by spectral
features and 2MASS near-infrared photometry.

After the eighth year of the regular low-resolution survey, a total of
11,214,076 low resolution spectra is available in LAMOST
DR8111http://DR8.lamost.org/. This catalog contains 10,388,423 stellar spectra
with a resolution of R $\sim$ 1,800 at the 5,500 Å, among them 773,721 spectra
are from M-type stars, 520,934 of them have r-band SNR higher than 10. These
spectra cover a wavelength range from 3,690–9,100 Å, including the blue
channel which is optimized for 3,690–5,900 Å, and the red channel for
5,700–9,100 Å. In this study, we adopt the identified M-type stars from LAMOST
DR8 to derive the stellar parameters by applying the $\chi^{2}$ minimization
performed by the ULySS program.

##  3 Methodology

###  3.1 MILES Interpolator

Spectra libraries collect a set of widely-used templates and corresponding
stellar parameters to classify stars and synthesize stellar populations. For
example, the ELODIE library (Prugniel & Soubiran, 2001; Prugniel et al., 2007)
contains 1,962 spectra of 1,070 stars obtained from the ELODIE spectrograph
with a resolution of R $\sim$ 42,000. Soubiran et al. (2008) have determined
the stellar parameters of ELODIE stars using the TGMET code (Katz et al.,
1998). The MILES library (Sanchez-Blazquez et al., 2006; Falcón-Barroso et
al., 2011) consists of 985 stars at optical region with a resolution of R
$\sim$ 2,200, which were obtained from the 2.5 m Issac Newton Telescope (INT).
For stars in the MILES library, Cenarro et al. (2007) compiled and calibrated
stellar atmospheric parameters from the literature.

Thus, the MILES library is considered to be an ideal empirical library for
stellar atmospheric parameter determination. Prugniel et al. (2011)
redetermined a set of homogeneous stellar parameters for MILES stars, and
built an interpolator function (version 1, hereafter V1) based on MILES
spectra which can generate model spectra based on a function of stellar
parameters. The reliability of V1 has been proofed as it shows better
performance for both hot evolved and cool stars.

Based on V1, Sharma et al. (2016) extended the parameter space with several
cool stars and refined V1 interpolator with effective temperature downwards to
2,900 K. This new
interpolator222https://cdsarc.cds.unistra.fr/ftp/J/A+A/585/A64/miles_tgm2.fits.gz
(version 2, hereafter V2) recalculated the stellar parameters of V1 in the
well-populated regions of the parameter space, while in the sparse border
regions, it included external spectra for improvement. The V2 interpolator
also extended the validity of M-type stars, which used a fine-tuned 26 terms
polynomial to improve the modelling and decrease the biases. The valid
spectral range of V2 interpolator is 3,536 Å–7,410 Å, which is limited by the
wavelength coverage of the MILES library.

###  3.2 ULySS: Université de Lyon Spectroscopic analysis Software

ULySS333http://ulyss.univ-lyon1.fr/ (Koleva et al., 2009) is a full-spectrum
fitting package based on IDL/GDL language, which has been used in various
types of tasks. In this work, we employ ULySS to derive the stellar
atmospheric parameters by fitting a spectrum with a linear function of non-
linear models:

$\textit{F}_{\text{obs}}=\text{P}_{n}(\lambda)\times{\text{TGM}}(T_{\rm
eff},\log{g},\text{[Fe/H]},\lambda)\otimes{\text{G}}(v_{\text{sys}},\sigma)$
(1)

Where $\text{P}_{n}$ is an n-th order polynomial, TGM represents an
interpolator function modeling the spectrum with the variables denoting
effective temperature, surface gravity, metallicity and wavelength
respectively. $\text{G}(v_{\text{sys}},\sigma)$ describes the Gaussian
broadening caused by the systemic velocity $v_{\text{sys}}$, and the velocity
dispersion $\sigma$. The interpolator TGM is a function of the stellar
atmospheric parameters ($T_{\rm eff}$, $\log{g}$ and [Fe/H]) and wavelength
$\lambda$, providing a deduction of stellar flux distribution and $\chi^{2}$
minimization.

For M-type stars, the spectra are typically occupied by dense molecular
absorption bands, which make the continuum hard to determine. Because of this,
we do not calculate the pseudo-continuum for normalization. Instead, we adopt
a multiplicative polynomial $\text{P}_{n}$ as the scaling factor making the
theoretical spectrum comparable to the observed ones. This procedure achieves
the same effect as the ordinary continuum normalization. Due to the existence
of molecular absorption lines, a polynomial of lower orders may not fit the
features correctly, while a higher order of the multiplicative polynomial can
easily over-fit.

To find the best degree of polynomial $\text{P}_{n}$, we use a series of
$\text{P}_{n}$ with different orders in the fit procedure, calculate the loss
value, then evaluate the multiple sets of results obtained. The loss value is
defined as:

$\textit{loss}=\sqrt{\sum_{i=0}^{n}(x_{i}-x_{\text{ref}})^{2}}$ (2)

The $x_{i}$ represents each derived stellar parameter, and the
$x_{\text{ref}}$ is the corresponding parameter provided by ASPCAP.

To estimate the overall performance of these stellar parameters, we calculate
the total loss value by performing a weighted average for the loss values of
effective temperature ($\textit{loss}_{T}$), surface gravity
($\textit{loss}_{G}$) and metallicity ($\textit{loss}_{M}$):

$\textit{loss}_{\text{tot}}=\frac{w_{T}\times\textit{loss}_{T}+w_{G}\times\textit{loss}_{G}+w_{M}\times\textit{loss}_{M}}{w_{T}+w_{G}+w_{M}}$
(3)

In general, the Fe absorption lines significantly influence the determination
of [Fe/H], therefore we decide to empirically give higher weights to [Fe/H],
specifically, $w_{T}=1$, $w_{G}=1$ and $w_{M}=1.5$.

Figure 1 shows the loss function of each stellar parameter, as well as the
total loss value. The loss values of effective temperature and surface gravity
show positive correlations with the degree of polynomial $\text{P}_{n}$, while
the loss value of metallicity declines first, and then rises as the order N
increases.

Figure 1: The variations of loss value as a function of the degree of
$\text{P}_{n}$. Top panel : the variation trends of loss value for stellar
parameters : $T_{\rm eff}$ (red), $\log{g}$ (orange) and [Fe/H] (blue) by
changing the degree of the polynomial $\text{P}_{n}$. Bottom panel : the total
loss value (weighted average of three loss values above). To make the loss
values of different stellar parameters comparable, each loss value is
normalized to [0, 1].

Eventually, we decide to adopt a maximum degree $N=15$ for the polynomial
$\text{P}_{n}$, which ensures the spectra fitted properly, and does not hit
the marginal effect wall. Also, we apply “/CLEAN” option in ULySS program to
eliminate the spikes (emission lines, bad pixels) during the fitting
procedure.

After scaling the spectrum, we adopt a $\chi^{2}$ minimization approach to
deduce the stellar parameters. The $\chi^{2}$ is defined as:

$\chi^{2}=\sum_{i=1}^{n_{pxl}}(\frac{O_{i}-S_{i}}{\sigma_{i}})^{2}$ (4)

Here, $n_{pxl}$ stands for the number of pixels of the spectrum, while $O_{i}$
and $S_{i}$ represent the flux at the i-th pixel of the observed spectrum and
the synthetic spectrum. The $\sigma_{i}$ is the standard deviation of flux
measurement at the i-th pixel of the observed spectrum.

More specifically, we iteratively change the input parameters until the
reduced $\chi^{2}$ reaches a local minimum. The reduced $\chi^{2}$ is defined
as :

$\chi^{2}_{\nu}=\frac{\chi^{2}}{\nu}$ (5)

Here, $\nu$ is the degree of freedom (DOF), $\nu=n_{pxl}-n_{para}$, which
equals the difference between the number of pixels ($n_{pxl}$) and the number
of free parameters ($n_{para}$).

Table 1 lists the initial guess grid of stellar parameters adopted in this
work. It should be noted that, the initial guess values provide our program
with a grid of start points, and the final results are determined by the
$\chi^{2}$ minimization algorithm which may exceed the grid region (see Figure
2).

Table 1: The initial guess grid of stellar parameters. Variable | Initial guesses
---|---
$T_{\rm eff}$ (K) | 3100 | 3300 | 3500 | 3700 | 3900 |
$\log{g}$ (dex) | 0.50 | 1.00 | 4.00 | 4.50 | 5.00 | 5.50
[Fe/H] (dex) | $-2.00$ | $-1.00$ | $-0.50$ | 0.00 | 0.50 |
Figure 2: The initial guess values of $T_{\rm eff}$, $\log{g}$ and [Fe/H].
Dots are color coded in different effective temperatures. Detailed information
is listed in Table 1.

In order to apply the V2 interpolator to LAMOST spectra, we have separately
estimated the stellar parameters for a group of spectra by fitting the blue
(3,690–5,700 Å) and the red (5,900–9,100 Å) segments of the spectra, as well
as the assembled (3,690–9,100 Å) spectra. After a series of experiments, we
found that rather than using the blue regions or the assembled spectra,
fitting the red regions usually shows more robust dispersion and lower
systematic error. The red segment generally has better SNR and the $\log{g}$
sensitive molecular bands such as TiO and CaH bands locate in this spectral
region (Gizis, 1997). Although some spectral signatures may be lost, we have
to manually set the upper limit of our spectra as 7,400 Å to avoid unreliable
extrapolation. Therefore, we decide to use the overlap parts (5,700 $\sim$
7,400 Å) of V2 spectral range and the red region of LAMOST spectrum during the
fitting.

Figure 3: Samples of our fitting results with different spectral subtypes
(from top to bottom: dM0, gM2, dM4 and gM6). Black and red lines are observed
and model spectra, respectively. The designations and corresponding stellar
parameters are listed, and the residuals between model and observed spectra
are shown at the bottom.

Finally, we construct a template matching routine for loading the
observational spectrum and finding the best solution, by using ULySS 1.3.1
combined with the V2 interpolator. We display four samples of our fitting
results with different spectral subtypes (dM0, gM2, dM4 and gM6) in Figure 3,
and the residuals exhibit good consistencies in our fitting region of
5,700–7,400 Å. However, the fitting is not as good as the others in the
spectral region near the red end, which indicates the instability of model
spectrum generation restricted by the interpolator.

##  4 Validation of Our Method

Before applying our method to the entire sample set, it is necessary to
validate this method using a spectrum subset equipped with reliable external
stellar parameters. Therefore, we cross-match with APOGEE DR17 (Ahumada et
al., 2020; Jönsson et al., 2020; Abdurro’uf et al., 2022) and 16 suitable
papers from the literature, then compare the stellar parameters for the common
stars.

###  4.1 Comparison with APOGEE DR17

We cross-match our sample set with APOGEE DR17 allStar-
file444https://dev.sdss.org/dr17/irspec/spectro_data/, following the criteria
below:

  1. 1.

$2900<T_{\text{eff}}<4200$ K;

  2. 2.

The spectra tpye property in the LAMOST DR8 catalog is classified as M-type by
LASP;

  3. 3.

The signal-to-noise ratio of r-band (SNR_r) for the observed spectra should
larger than 20;

  4. 4.

The SNR of u, g, r, i and z-band for the observed spectra should not equal to
$-9999$.

SNR_r is the SNR value at SDSS r-band filter (5550–6900 Å, Gunn et al., 1998)
and is defined as the median value of each pixel in this band. Note that
unless specified, all SNR presented hereafter refer to the SNR_r value. The
aim of this constraint is to ensure that our sample has good observational
quality in the r-band region where covers the distinctive spectral features.
The criterion of SNR $>$ 20 can exclude most of the misfits caused by low SNR,
which will be discussed in Section 5. To avoid the influence of poor data
quality, we discard the spectra whose SNR value have been artificially set as
$-9999$ in corresponding spectral region. In this way, we have selected 19,592
spectra of 14,532 unique stars after cross-matching. For targets with repeated
observations, we only keep the spectrum with highest SNR in order to avoid the
impact of low spectral quality. Thus, we obtain a total of 14,532 common
stars.

Figure 4:  Figure 5: The distribution of $T_{\rm eff}$ versus $\log{g}$ for
common stars from APOGEE DR17. Stars are color coded in different metallicity
groups.

Furthermore, we exclude the binaries by cross-matching with the binary
catalogs including the binaries of APOGEE (El-Badry et al., 2018), the
Washington Double Star Catalog (WDS, Mason et al., 2001), the third revision
of Kepler Eclipsing Binary Catalog (KEBIII, Kirk et al., 2016) and the Gaia-
ESO binary candidates (Merle et al., 2017). Similarly, we exclude the
variables by cross-matching with the General catalog of Variable Stars (GCVS,
Samus’ et al., 2017), the International Variable Star Index (VSX, Watson et
al., 2006) and the ASAS-SN catalog of variable stars (Jayasinghe et al.,
2021). After these cuts, there are 13,846 common M-type stars with APOGEE
DR17. In Figure 5, we present the parameter distribution ($T_{\rm eff}$ versus
$\log{g}$) of these stars.

Figure 6: The comparison of the effective temperatures derived from our method
with the APOGEE DR17 calibrated values. The top-left panel shows the $T_{\rm
eff}$ comparison for dMs, while the top-right panel for gMs. The bottom-left
and bottom-right panels show the residuals’ distributions for dMs and gMs,
respectively. The color scale as well as the contour lines indicate the number
density of each region.  Figure 7: Same as Figure 6, but for surface gravity.
Figure 8: Same as Figure 6, but for metallicity.

As shown in Figures 6–8, the comparisons of the three stellar parameters show
good agreements, although there are some systematic offsets. The bottom panels
are residuals between our results and those from APOGEE, where the mean value
of bias and the standard dispersion are also marked. The $T_{\rm eff}$
comparison with those of APOGEE DR17 is plotted in Figure 6, and a consistent
result is found despite systematic underestimations of 176 K and 140 K for dMs
and gMs, respectively. It needs to be pointed out that the APOGEE $T_{\rm
eff}$ used in Figure 6 is calibrated by
ASPCAP555https://dev.sdss.org/dr17/irspec/parameters/. To illustrate the
difference, we also compare with those of APOGEE spectroscopic $T_{\rm eff}$
in Figure 28, as a result, the systematic difference is significantly reduced
to 73 K and 22 K for dMs and gMs, respectively, which suggests that there are
systematic differences of $T_{\rm eff}$ between APOGEE and other calibrations
(also reported by Birky et al. 2020).

The comparison of $\log{g}$ is plotted in Figure 7 for dMs and gMs. The dMs
have a larger sample size, and the comparison show good agreement for the vast
majority of dMs with a small offset of 0.16 dex, while the surface gravities
of gMs seem slightly overestimated with a systematic offset of 0.30 dex. The
residuals generally exhibit a weak negative correlation comparing to those
from APOGEE—stars with higher surface gravities are underestimated, while
those with lower gravities are overestimated. Similarly, the comparison with
those of APOGEE spectroscopic $\log{g}$ is shown in Figure 29. The dispersions
of both dMs and gMs are worse than the calibrated $\log{g}$, and the
systematic difference becomes larger for dMs (0.52 dex) and slightly smaller
for gMs (0.25 dex).

When comparing our [Fe/H] and those from APOGEE DR17, we find a good
consistency: the systematic offsets as well as dispersions are $0.14\pm 0.16$
dex and $0.18\pm 0.26$ dex for dMs and gMs, respectively, albeit with a
negative correlation in residuals.

###  4.2 Comparison with Other Literature

While the comparison with APOGEE DR17 has displayed the good performance of
our method, we do find some systematic deviations, so it is necessary to make
comparisons with other literature. Therefore, we collect stellar parameters
from 16 other papers. They both worked on M-type stars stellar parameter
estimation using different methodologies. We can take advantage of these
results and make comparisons with well-determined stellar parameters.

Figure 9 displays the quantities of cross-matched stars for each literature’s
contribution in our reference set, detailed information can be found in Table
2.

Figure 9: The number of valid stellar parameters provided by 16 papers from
the literature. See Table 2 for detail.

We assemble this reference set with 114,144 spectra cross-matched from those
works, and exclude the binary and variable candidates, similar to Section 4.1.
Finally, we have 49,698 common stars in this reference set.

Figure 11 shows the parameter distribution ($T_{\rm eff}$ versus $\log{g}$) of
common stars in this reference set, which displays a similar but broader
parameter space comparing to Figure 5.

Table 2: Literature sources for comparison. Ref names | Common stars | Parameters
---|---|---
Sarmento et al. (2021) | 91 | $T_{\rm eff}$ | $\log{g}$ | [Fe/H]
Zhang et al. (2021) | 1579 | $T_{\rm eff}$ | $\log{g}$ | [Fe/H]
Birky et al. (2020) | 3047 | $T_{\rm eff}$ | — | [Fe/H]
Galgano et al. (2020) | 26502 | $T_{\rm eff}$ | — | —
Wang et al. (2020) | 2284 | $T_{\rm eff}$ | $\log{g}$ | [Fe/H]
Maldonado et al. (2020) | 17 | $T_{\rm eff}$ | $\log{g}$ | [Fe/H]
Hejazi et al. (2020) | 14 | $T_{\rm eff}$ | $\log{g}$ | [Fe/H]
Schweitzer et al. (2019) | 28 | $T_{\rm eff}$ | $\log{g}$ | [Fe/H]
Passegger et al. (2018) | 20 | $T_{\rm eff}$ | $\log{g}$ | [Fe/H]
Muirhead et al. (2018) | 79658 | $T_{\rm eff}$ | — | —
Terrien et al. (2015) | 237 | $T_{\rm eff}$ | — | [Fe/H]
Mann et al. (2015) | 14 | $T_{\rm eff}$ | — | [Fe/H]
Gaidos et al. (2014) | 355 | $T_{\rm eff}$ | — | [Fe/H]
Newton et al. (2014) | 1192 | $T_{\rm eff}$ | — | [Fe/H]
Khata et al. (2021) | 43 | $T_{\rm eff}$ | — | —
Souto et al. (2021) | 7 | $T_{\rm eff}$ | $\log{g}$ | [Fe/H]
Figure 10:  Figure 11: The distribution of $T_{\rm eff}$ versus $\log{g}$ for
common stars from the reference set. Stars are color coded in different
metallicity groups. Figure 12: The comparison of the effective temperatures
derived from our method with the literature values. The top-left panel shows
the $T_{\rm eff}$ comparison for dMs, while the top-right panel for gMs. The
bottom-left and bottom-right panels show the residuals’ distributions for dMs
and gMs, respectively. The color scale as well as the contour lines indicate
the number density of each region.  Figure 13: Same as Figure 12, but for
surface gravity.  Figure 14: Same as Figure 12, but for metallicity.

The comparisons of these stellar parameters are presented in Figures 12–14
with detailed offsets and standard deviations listed in the residual
distributions. In Figure 12, we have compared our $T_{\rm eff}$ with those in
the reference set, which benefits from a large effective temperature coverage
from 2,900 K to 4,200 K. The $T_{\rm eff}$ offset for dMs is $-$88 K, which is
less than the systematic offset appearing in the APOGEE comparison. For gMs,
the systematic offset is $-$152 K with a larger dispersion of 139 K. The
differences of residual offsets also indicate a systematic difference of
$\sim$ 100 K exists between $T_{\rm eff}$ from APOGEE and those from other
measurements.

In our reference set, the available $\log{g}$ and [Fe/H] measurements are not
as large as $T_{\rm eff}$, the comparisons are displayed in Figures 13 and 14.
The $\log{g}$ for dMs derived from our method shows a modest systematic offset
(0.05 dex) and a good consistency comparing with those from different
literature. For the gMs, we find a systematic offset of $-$0.14 dex, which is
less than the offset found in APOGEE comparison.

For [Fe/H], the residuals between our results and other measurements show
small systematic offset and dispersion, along with a weak overall decrease
similar to the APOGEE comparison. The systematic offsets are $-$0.26 dex and
0.16 dex for dMs and gMs, respectively, which suggests that our derived [Fe/H]
is generally underestimated for dMs, while the derived [Fe/H] seems
overestimated for gMs.

##  5 Error Estimation

The comparisons in Section 4 showed that the fitting errors of our results
could be influenced by stellar parameters to some extent. Besides, the quality
of the observed spectra can also affect the precision of stellar atmospheric
parameter measurement. With the help of the overlapping observations of LAMOST
survey, over 26% stars have multiple visits in LAMOST DR8, which allows us to
examine the error in stellar parameters, due to the SNR of spectra.

In this section, we estimate the errors, and investigate the precision of our
measurements by comparing the stellar parameters derived from these stars with
repeated observations.

###  5.1 Random Errors due to the Quality of Spectra

To evaluate the effect of spectral quality on the stellar parameters, we first
select spectra of repeated observations for the same stars, with a difference
in SNR should be lower than 20%. Figure 15 shows the residuals of stellar
parameters for these duplicate stars as a function of SNR. We use the standard
dispersion to represent the errors of stellar parameter determination, which
present declining tendencies as the SNR of spectra increase. We split our
sample into dMs (top panel) and gMs (bottom panel), as well as group stars
into different temperature bins.

The effective temperatures are less sensitive to SNR. Generally the cooler
stars tend to have better precision, except for the coolest stars ($T_{\rm
eff}$ $<$ 3,000 K), and there are not so many stars as other groups in our
sample set. For the spectra of SNR $\sim$ 5, the precision of $T_{\rm eff}$ is
better than 120 K, and it immediately decays to 45 K when SNR is higher than
20.

However, the surface gravities are very sensitive to the SNR of spectra, which
may be due to the molecular bands blurred by noise. For both dMs and gMs,
$\log{g}$ can be determined with a precision better than 0.2 dex for SNR $>$
20\. For the low SNR spectra, the measurement of $\log{g}$ is visually poor,
e.g. the dispersion of hot dwarfs ($T_{\rm eff}$ $>$ 3,900 K) can be as large
as 0.85 dex, which becomes a main reason to exclude them from our comparisons.

A similar situation can be found for metallicities, which shows a significant
dependency on SNR. In general, dMs can have more precise [Fe/H] than gMs. For
dMs in each temperature group, the precision of [Fe/H] is about 0.24 dex when
SNR is about 5, and rapidly decreases to 0.08 dex when SNR reaches 15. The
determination of [Fe/H] for gMs is relatively poor as the precision reaches
0.36 dex of SNR $\sim$ 5 and 0.2 dex of SNR $>$ 25, respectively. In each
temperature group, the [Fe/H] dispersion for gMs differs from the other
groups, indicating a dependency of [Fe/H] determination on $T_{\rm eff}$.

Figure 15: The random errors of stellar parameters as functions of SNR. The
dMs (top) and gMs (bottom) are grouped in different temperature bins. The dots
represent the standard dispersion in each SNR bin.

###  5.2 Systematic Errors due to SNR

Figure 16: The systematic errors of stellar parameters as functions of SNR.
Red error bars represent the standard deviations in each SNR bin, and the red
dots in the center are the means. The yellow dashed lines are linear
regressions of the means.

Furthermore, in order to investigate the possible systematic errors introduced
by low spectral quality, we select spectra of repeated observations for the
same targets with different SNR, of which the spectra with higher SNR should
be 40 higher than the spectra of low SNR. The residuals are represented by
$\Delta P=P_{\text{high}}-P_{\text{low}}$, where $P$ refers to the three
stellar parameters, $P_{\text{high}}$ stands for the one of higher SNR, while
$P_{\text{low}}$ is the one of lower SNR.

Figure 16 exhibits the difference between stellar parameters deduced from
spectra of repeated observations as a function of lower SNR. The asymmetry of
residual distributions indicates that systematic deviations exist between high
and low SNR results.

We use error bars to present the standard dispersion of each SNR bin and
perform linear regressions of mean values in each bin (see the yellow dashed
lines in Figure 16) to generalize the gradients of systematic deviations for
$T_{\rm eff}$, $\log{g}$ and [Fe/H]. Similarly, $T_{\rm eff}$ shows good
precision and is less sensitive to SNR of lower spectral quality, and can be
determined with a systematic difference better than 50 K at SNR $\sim$ 10\.
The systematic errors of $\log{g}$ are quite obvious in the low SNR region,
which indicates that surface gravities deduced from low SNR spectra are
systematically underestimated compared to those from high SNR. It should be
noted that, for some stars of SNR lower than 5, the differences between
surface gravities derived from high and low SNR spectra can be as bad as
$\sim$ 2.5 dex, which suggests that our program tend to misclassify some dMs
as gMs in the low SNR region. For [Fe/H], an obvious underestimation of $\sim$
0.7 dex appears at SNR $\sim$ 5, however, for sample stars with SNR higher
than 20, the systematic errors are less than 0.2 dex, which is less
significant.

###  5.3 Internal Uncertainty Analysis

Figure 17:  Figure 18:  Figure 19:

Figure 20: The internal uncertainties in different stellar parameters and SNR
values. The values in each box stand for the standard deviation of residuals
within each interval.

As we know, during the $\chi^{2}$ minimization, errors can be introduced due
to the fitting procedure. We evaluate the internal uncertainties of our
stellar parameter measurements in three steps. In the first step, we sample a
set of stellar parameter (2900 K–4500 K for $T_{\rm eff}$, 0.0–6.0 dex for
$\log{g}$ and $-2.5$ to $+1.0$ dex for [Fe/H]) based on the distribution of
stellar parameter space of our results. Second, we utilize these stellar
parameters to obtain the corresponding spectra generated from V2 interpolator,
and add Gaussian noises to simulate the distortion during observations. In the
third step, we redetermine the stellar parameters for all these generated
spectra from a single Markov Chain Monte Carlo (MCMC) simulation. We calculate
the residuals between the input stellar parameters in step 1 and the outputs
in step 3, and present the deviations of these residuals in Figure 20.

We notice that hot metal-poor giants generally have worse performance than the
others, especially in the low SNR region. For spectra of SNR $>$ 20, stellar
parameters can be determined with internal uncertainties better than 22 K,
0.06 dex and 0.06 dex for $T_{\rm eff}$, $\log{g}$ and [Fe/H], respectively.
It should be noted that the error obtained in this simulation is lower than
those in the external comparisons, because the theoretical spectrum with
randomly added noise is more ideal than the observed spectrum.

###  5.4 The Catalog of Stellar Atmospheric Parameters

We present the stellar atmospheric parameter catalog of our results, which
contains 763,136 spectra of 616,314 M-type stars from LAMOST LRS DR8. In Table
B, we show a sample set of our stellar parameter results and several important
attributes, including the celestial coordinates, spectral quality information:
SNR in u, g, r and i bands as well as the spectral identification information:
the LAMOST designation (desig), LAMOST unique spectra ID (obsid) and source
identifier (source_id) from Gaia Early Data Release 3 (Collaboration et al.,
2021).

In Figure 22, we present the distribution of SNR for the observational
spectra. Spectra with SNR below 20 account for more than half of our samples,
one should be careful using their stellar parameters. As discussed in Section
5, we recommend using the stellar parameters derived from spectra with SNR
higher than 20 for reliability and good consistency. Additionally, in Figure
25, we plot the results derived from our method along with theoretical
isochrones from the PAdova and TRieste Stellar Evolution Code (PARSEC, Bressan
et al., 2012) for stars with SNR above 20 and below 20. As illustrated in the
top panel, the stars of metallicity above the solar value ([Fe/H] $>$ 0.0 dex)
can be clearly distinguished via the isochrone. Moreover, the comparison
between the top and bottom panels show the improvement of SNR cut discussed
above. Conspicuous outliers are rare for stars of SNR $>$ 20.

Figure 21:  Figure 22: The percentage for the stars with SNR below 10, from 10
to 20, 20 to 30, 30 to 40 and above 40.

We note that the lower limit of derived $T_{\rm eff}$ is as low as 2,900 K.
For gMs, the lower limit of $\log{g}$ can reach -0.24 dex, and for dMs the
upper limit is 5.9 dex. Figure 27 displays the distributions of [Fe/H] values
for dMs and gMs, separately. The dMs show a more symmetrical distribution with
a peak [Fe/H] around $-$0.30 dex. For gMs, however, the peak is more skewed
towards the solar abundance.

###  5.5 Potential Future Uses

This work builds on a number of other parameter determination studies for cool
stars. Our study provides a significantly larger sample than the current
literature sources that we have presented detailed comparisons with in Table
2. This allows our catalog to provide well-characterized properties which
might be useful for a wide range of purposes. Objects that are found to be
time-variable in some way greatly benefit from having known properties. These
properties of cool stars may help with the immediate interpretation or prove
valuable for further follow-up studies. These might be individual systems that
are for example found to host a planet or perhaps stars where an asteroseismic
analysis enables the identification of interesting oscillation modes. These
two use cases are also appropriate for the case of larger samples. It might be
interesting to look at planetary formation as a function of metallicity (e.g.,
Lu et al., 2020) or metallicity might be related to asteroseismic evolutionary
information for large numbers of stars (e.g., Bellinger et al., 2019).

Our catalog can also be valuable as a means to recognize important outliers
among the two key populations of cool stars: the most numerous stars (dMs) and
most luminous stars (gMs). Such studies can provide valuable insights for
understanding and modelling the evolution of both stellar properties as well
as interferences about our galaxy and beyond. The properties of cool stars
with age can provide a range of different information useful in a range of
contexts. The slow evolution of dMs provides the most pristine view of
elements unchanged by their lack of evolution through only a small fraction of
their main sequence lifetime (e.g., Laughlin et al., 1997). On the other hand
gMs are useful probes of the mixing of evolved stars as well as dust. It is
not clear that these are fully consistent with the predictions of stellar
evolutionary models (e.g., Lançon et al., 2018). Such efforts together
underpin systematic empirical grids of stellar types which are in turn a vital
component of galaxy evolution models. It has long been known that the detailed
properties of gMs on the horizontal branch as a function of stellar properties
are vital (e.g., Worthey, 1994; Cabrera-Ziri & Conroy, 2022) though dMs can
also dominate the energy output of galaxies (e.g., Conroy & van Dokkum, 2012;
van Dokkum & Conroy, 2021). In the era of Gaia parameters, large scale
spectroscopic properties from surveys such as LAMOST can be key in a detailed
modern understanding of our galaxy. Our future work includes repeat
observations, examination and determinations of our sample to further enhance
our robust characterisation for the properties of M stars.

Figure 23:  Figure 24:

Figure 25: Left panel: the Kiel diagram colored by different metallicity
groups of the samples with SNR $>$ 20\. Lines correspond to 10 Gyr PARSEC
isochrones with [Fe/H] $=$ -0.6, -0.3 and 0.0 dex, respectively. Right panel:
Same Kiel diagram but for stars of SNR $<$ 20. Figure 26:  Figure 27: The
histograms of [Fe/H] for dMs (top panel) and gMs (bottom panel). The
histograms only display [Fe/H] range from -1.6 dex to 1.0 dex, as there are
only a few metal-poor stars ([Fe/H] $<$ -1.6 dex).

##  6 Conclusion

Utilizing the MILES V2 interpolator and ULySS, we derive the stellar
atmospheric parameters for 763,136 spectra of 616,314 M-type stars from LAMOST
DR8. We compare our stellar parameters with external references including
APOGEE DR17 and other literature, and good consistencies can be found. To
evaluate the precision of our results, we analyze the stars with repeated
observations. The precision of derived stellar parameters is mainly dependent
on the SNR and effective temperature. For the SNR higher than 20, the typical
precisions are better than 45 K, 0.25 dex and 0.22 dex for $T_{\rm eff}$,
$\log{g}$ and [Fe/H], respectively. A Monte-Carlo simulation is applied to
check the performance of our method, and the internal uncertainty of our
method is $\sigma_{T_{\rm eff}}<$ 22 K, $\sigma_{\log{g}}<$ 0.06 dex and
$\sigma_{\rm[Fe/H]}<$ 0.06 dex, for the spectra of SNR better than 20.

Our results supply LAMOST LRS DR8 with large numbers of well-determined M-type
star stellar atmospheric parameters, and our method could be applied to the
works of stellar parameter determination for M-type star in future low
resolution surveys, for example, LAMOST LRS DR9. Moreover, it is important to
further investigate the metallicity distributions and kinematic properties of
different Galactic populations.

We thank the referee for the helpful comments which have helped us to improve
the manuscript. Our research is supported by National Key R&D Program of China
No.2019YFA0405502, the National Natural Science Foundation of China under
grant Nos. 12090040, 12090044, 11833006, 12022304, 11973052, 11973042 and
U1931102. We acknowledge the science research grants from the China Manned
Space Project with NO.CMS-CSST-2021-B05. This work is supported by Chinese
Academy of Sciences President’s International Fellowship Initiative. Grant No.
2020VMA0033. H.-L.Y. acknowledges the supports from Youth Innovation Promotion
Association, Chinese Academy of Sciences (id. 2019060). 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.

##  Appendix A Appendix Figure

Figure 28: The comparison of the effective temperatures derived from our
method with the APOGEE DR17 spectroscopic values. The top-left panel shows the
$T_{\rm eff}$ comparison for dMs, while the top-right panel for gMs. The
bottom-left and bottom-right panels show the residuals’ distributions for dMs
and gMs, respectively. The color scale as well as the contour lines indicate
the number density of each region.  Figure 29: Same as Figure 28, but for
surface gravity.

##  Appendix B Appendix Table

Table 3: Stellar parameters of randomly selected objects.

desig | obsid | source_id | RA | DEC | SNR_r | RV | T_eff | log g | [Fe/H]
---|---|---|---|---|---|---|---|---|---
|  |  | hms (J2000) | dms (J2000) |  | (km$\cdot s^{-1}$) | (K) | (dex) | (dex)
J$003233.56+023235.4$ | $182907177$ | $2547554358360573184$ | $00:32:33.57$ | $+02:32:35.4$ | $21$ | $-0.6$ | $3865$ | $4.66$ | $-0.23$
J$005018.74+383849.1$ | $407239$ | $367934386366947456$ | $00:50:18.75$ | $+38:38:49.1$ | $12$ | $-38.7$ | $3699$ | $4.73$ | $-0.11$
J$005137.86+382236.2$ | $612710127$ | $367725002416518272$ | $00:51:37.87$ | $+38:22:36.3$ | $35$ | $-20.0$ | $3739$ | $4.73$ | $-0.40$
J$062058.46+210949.0$ | $437312190$ | $3375942048114941696$ | $04:11:37.63$ | $+33:59:16.6$ | $144$ | $-26.4$ | $3409$ | $0.52$ | $-0.11$
J$041137.62+335916.6$ | $200512017$ | $170751686193676288$ | $06:14:31.81$ | $+08:40:37.2$ | $46$ | $-63.6$ | $3755$ | $2.39$ | $-0.64$
J$061431.80+084037.2$ | $503710146$ | $3328420434007985280$ | $06:20:58.46$ | $+21:09:49.0$ | $22$ | $1.9$ | $3506$ | $1.58$ | $-0.52$
J$063953.53+534915.5$ | $546315051$ | $994279332683432320$ | $06:39:53.54$ | $+53:49:15.5$ | $14$ | $-23.7$ | $3782$ | $1.46$ | $-0.47$
J$065018.79+264542.8$ | $46204100$ | $3385128472059867904$ | $06:50:18.80$ | $+26:45:42.8$ | $20$ | $-33.3$ | $3907$ | $4.64$ | $-0.21$
J$071141.54+195445.8$ | $175006180$ | $3363557565751137152$ | $07:11:41.54$ | $+19:54:45.9$ | $42$ | $-32.5$ | $3813$ | $4.63$ | $-0.16$
J$082845.70+104545.1$ | $337405103$ | $600889912104424192$ | $08:28:45.71$ | $+10:45:45.1$ | $49$ | $15.5$ | $3915$ | $4.65$ | $-0.15$
J$085752.81+155051.6$ | $629901215$ | $610580732712396032$ | $08:57:52.82$ | $+15:50:51.6$ | $67$ | $27.7$ | $3657$ | $4.89$ | $-0.45$
J$103456.90-054946.3$ | $422001160$ | $3777400028612544640$ | $10:34:56.91$ | $-05:49:46.3$ | $17$ | $-52.6$ | $3947$ | $4.67$ | $-0.38$
J$104029.59+040235.2$ | $546101111$ | $3857828655644765440$ | $10:40:29.60$ | $+04:02:35.2$ | $15$ | $53.0$ | $3970$ | $4.50$ | $-0.11$
J$143657.78+285648.1$ | $566705048$ | $1281154251515481344$ | $14:36:57.78$ | $+28:56:48.2$ | $29$ | $-8.8$ | $3792$ | $4.75$ | $-0.37$
J$143700.28+313008.0$ | $319815214$ | $1286110746854188288$ | $14:37:00.29$ | $+31:30:08.0$ | $32$ | $14.3$ | $3765$ | $4.73$ | $-0.29$
J$144136.81+162250.9$ | $450707153$ | $1234646382833797632$ | $14:41:36.81$ | $+16:22:50.9$ | $17$ | $-52.4$ | $3574$ | $4.85$ | $-0.40$
J$144852.72-001936.8$ | $649007156$ | $3650703754716377344$ | $14:48:52.46$ | $-00:19:36.9$ | $209$ | $50.1$ | $3045$ | $0.68$ | $-0.09$
J$235515.89+042523.8$ | $505908050$ | $2743370468663782528$ | $23:55:15.90$ | $+04:25:23.9$ | $20$ | $23.5$ | $3791$ | $4.58$ | $-0.25$
… | … | … | … | … | … | … | … | … | …

Note. — The attribute column is not fully presented here. The complete catalog
is accessible in its entirety in a machine-readable form.

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REDISCUSSION OF ECLIPSING BINARIES. PAPER IX.

THE SOLAR-TYPE SYSTEM KIC 5359678

By John Southworth

Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK

KIC 5359678 is a 6.231-d period F-type eclipsing binary system whose component
stars both show starspot activity. It was observed by the Kepler satellite in
long cadence for the full four-year duration of the mission. Wang et al. 1
obtained radial velocity measurements of the two stars and analysed these plus
the Kepler data to study their spot activity and measure their physical
properties, but left several questions unanswered. We have performed an
independent analysis and determined the masses ($1.252\pm 0.018$ and $1.065\pm
0.013$$~{}{\rm M}_{\odot}$) and radii ($1.449\pm 0.012$ and $1.048\pm
0.017$$~{}{\rm R}_{\odot}$) of the stars to high precision. The distance we
find to the system is slightly shorter than that from Gaia EDR3 for unknown
reason(s). We also investigated the precision of the numerical integration
applied to the model light curve to match the 1765-s sampling cadence of the
Kepler observations. We found that ignoring this temporal smearing leads to
biased radius measurements for the stars: that for the primary is too small by
4$\sigma$ and that for the secondary is too large by 10$\sigma$. Doubling the
sampling rate of the model light curve is sufficient to remove most of this
bias, but for precise results a minimum of five samples per observed datapoint
is required.

## Introduction

Detached eclipsing binaries (dEBs) are crucial to stellar physics because the
properties of their component stars can be measured directly from observed
light and radial velocity (RV) curves 2; 3. These direct measurements can then
be used to constrain and to calibrate theoretical models of stellar evolution
4; 5; 6. Many thousands of dEBs are now known 7; 8; 9; 10. Particularly
important contributions to the numbers of dEBs for which good light curves are
available have recently been made by space missions such as CoRoT 11, Kepler
12 and TESS 13; 14; 15. An extensive review of the impact of space photometry
on binary star science can be found in Southworth 16.

KIC 5359678 was found to be a dEB using data from the Kepler satellite 12; 17;
18; 19 with a morphology value of 0.27 which indicates that it is well-
detached 20. Armstrong et al. 21 determined effective temperature ($T_{\rm
eff}$) values for the two components of $6713\pm 405$ K and $6237\pm 623$ K.
Qian et al. 22 determined the $T_{\rm eff}$ of the system to be $6510\pm 70$ K
and its spectral type to be F5 from a medium-resolution ($R=1800$) spectrum
obtained using the LAMOST spectroscopic telescope 23 survey of the Kepler
field 24.

Wang et al. 1 (hereafter W21) presented a detailed study of KIC 5359678 which
concentrated on the characteristics of the starspots on the stellar surfaces.
To determine the physical properties of the stars, W21 modelled the light
curve from the Kepler satellite together with RVs from a set of 58 medium-
resolution ($R=7500$) LAMOST spectra 25 using the phoebe2 code 26. They gave
two sets of physical properties for the system, for eccentric and circular
orbits, which are formally identical but with errorbars differing by as much
as a factor of three. They also quoted no errorbar for the mass of the
secondary star, and did not mention whether they accounted for the cadence of
the observations obtained by the Kepler satellite. For these reasons, and to
see if the properties of the system can be established to a precision of 2% or
better 2; 27, we present below a reanalysis of KIC 5359678. Basic information
on the system is summarised in Table I.

Table I: Basic information on KIC 5359678.  Property | Value | Reference
---|---|---
Kepler Input Catalog designation | KIC 5359678 | 28
Kepler Object of Interest designation | KOI 6569 |  29 and updates
Gaia EDR3 designation | 2101510803402761344 | 30
Gaia EDR3 parallax |  $0.6151\pm 0.0135$ mas | 30
$B$ magnitude | $14.905\pm 0.023$ | 31
$V$ magnitude | $14.209\pm 0.056$ | 31
$H$ magnitude | $12.927\pm 0.029$ | 32
$K_{s}$ magnitude | $12.862\pm 0.030$ | 32
Spectral type | F5 | 1

## Observational material

The Kepler satellite is a 0.95 m Schmidt reflecting telescope with a focal
plane contining 42 CCDs, launched by NASA in March 2009 and placed into an
Earth-trailing heliocentric orbit 33; 34. Its mission was to obtain
photometric observations of a single patch of sky (the Kepler Field) for four
years in order to find transits of extrasolar planets. Due to a limit on the
amount of data that could be returned to Earth, Kepler could only be used to
observe 170 000 objects simultaneously. Consecutive observations were summed
into single “long cadence” observations with an effective duration of 1765.5
s, and the count rates of pixels in the region of selected targets were
transmitted to Earth. A subset of 512 of these targets could be observed in
“short cadence”, where individual observations were summed into data with an
effective duration of 58.8 s. Kepler observations were divided into quarters
(three months) due to the rotation of the spacecraft around its optical axis
to keep its solar panels illuminated by the Sun. These data were reduced by
the Kepler mission Science Operations Center (SOC) 35.

Figure 1:  Kepler quarter 10 light curve of KIC 5359678. The spot activity can
be seen as continual variations in the data outside eclipse, and two jumps in
the data are also visible after short interruptions to the observations.

KIC 5359678 was observed for the entirety of the Kepler mission, from quarters
Q0 to Q17, in long cadence. The first datapoint was taken on 2009/05/13 and
the last on 2013/05/11. Small gaps occured within this time interval when
Kepler paused observations for technical reasons or to transmit data to Earth.
The data were downloaded from the MAST archive***Mikulski Archive for Space
Telescopes,
https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html and converted to
relative magnitude. Rejection of unreliable data (typically represented by
“Inf” or “Nan”) left 64 073 measurements. We chose to work with the standard
aperture photometry (SAP) rather than the pre-search data conditioning (PDC)
data, after verifying that the differences were negligible for our purposes.
An example light curve, chosen at random to be from Q10, is shown in Fig. 1.

## Analysis of the Kepler light curve

Figure 2:  The Kepler light curve of KIC 5359678 as analysed (top panel) with
the residuals of the best fit for a circular orbit (lower panel).

The majority of the photometric observations occur outside eclipse, so hold
negligible information on the physical properties of the stars. The starspot
variation also affects the brightness of the system during eclipse but occurs
on much longer timescales. We tackled both problems by extracting the data
around each eclipse from the overall light curve, fitting a polynomial to the
out-of-eclipse data, and subtracting the polynomial (in magnitude units) to
remove the slow variations due to starspots and instrumental effects. All
eclipses were manually inspected and those with insufficient data either
before or after, or with more than two datapoints missing, were rejected. This
left a total of 8106 observations with the main effects of spot activity
removed (Fig. 2).

We then proceeded to analyse the data with version 41 of the
jktebop†††http://www.astro.keele.ac.uk/jkt/codes/jktebop.html code 36; 37.
Fitted parameters included the sum and ratio of the fractional radii ($r_{\rm
A}+r_{\rm B}$ and $k=\frac{r_{\rm B}}{r_{\rm A}}$ where $r_{\rm
A}=\frac{R_{\rm A}}{a}$, $r_{\rm B}=\frac{R_{\rm B}}{a}$, $R_{\rm A}$ and
$R_{\rm B}$ are the radii of the stars, and $a$ is the semimajor axis of the
relative orbit), the orbital inclination ($i$) and period ($P$), a reference
time of primary minimum ($T_{0}$; when star A is eclipsed by star B), and the
central surface brightness ratio of the two stars ($J$).

For limb darkening (LD) we adopted the quadratic law, fitted the linear
coefficient for each star ($u_{\rm A}$ and $u_{\rm B}$) and fixed the
quadratic coefficients ($v_{\rm A}$ and $v_{\rm B}$) to theoeretical values
from Sing 38. We found third light to be very small so fixed it at zero. One
important consideration is the long sampling cadence – one observation every
1765 s – and we accounted for that by numerically integrating the model light
curve to match the observed one 39.

Figure 3:  The Kepler light curve of KIC 5359678 (filled circles) around the
primary (left) and secondary (right) eclipses. The best fit is not plotted as
it is indistinguishable from the data. The lower panels show the residuals of
the fit with the line of zero residual overplotted in white.

We obtained solutions for a circular and an eccentric orbit. In the latter
case we fitted for $e\cos\omega$ and $e\sin\omega$ where $e$ is the orbital
eccentricity and $\omega$ is the argument of periastron. In both cases we
included the RV measurements of the two stars published by W21 in order to
measure the masses and radii of the stars. The errorbars on the three datasets
(light curve, RVs of star A, RVs of star B) were each scaled to give a reduced
$\chi^{\,2}$ of $\chi_{\nu}^{\,2}=1$ for that dataset.

The best fit to the light curve for the circular orbit can been seen in Fig.
2. The residuals in this figure show systematic trends with time which occur
because our approach to dealing with the spot activity does not account for
the partial obscuration of spots during eclipses. Our best fit therefore
systematically over- or under-predicts the eclipse depth in a way which
changes on a characteristic timescale of approximately 200 days. This is much
longer than the rotation period or the timescale over which individual
starspots appear and disappear (see Fig. 1), but is much shorter than the time
interval covered by the data.

Another visualisation of the situation is shown in Fig. 3, where the data have
been converted into orbital phase and shown in close-up around the eclipses.
The best fit is clearly a good representation of the data, but the residuals
significantly increase during both eclipses. As changes in eclipse depth are
driven primarily by spots on the surface of the eclipsed star, this tells us
that both stars show spot activity and that their activity levels are
comparable.

Table II:  Parameters of the jktebop best fits to the Kepler light curve and published RVs of KIC 5359678. The uncertainties are 1$\sigma$ and are the larger of the Monte Carlo and residual-permutation options for each parameter. Parameter | Circular orbit | Eccentric orbit
---|---|---
Fitted parameters: |  |
Primary eclipse (BJD/TDB) | 2455402.26675$\pm$0.00002 | 2455402.26670$\pm$0.00002
Orbital period (d) | 6.23060994$\pm$0.00000024 | 6.23060991$\pm$0.00000028
Orbital inclination (∘) | $85.537\pm 0.042$ | $85.527\pm 0.049$
Sum of the fractional radii | $0.13246\pm 0.00044$ | $0.13255\pm 0.00049$
Ratio of the radii | $0.724\pm 0.022$ | $0.732\pm 0.027$
Central surface brightness ratio | $0.7324\pm 0.0015$ | $0.7253\pm 0.0129$
Third light | 0.0 (fixed) | 0.0 (fixed)
Linear LD coefficient star A | $0.293\pm 0.008$ | $0.297\pm 0.011$
Linear LD coefficient star B | $0.318\pm 0.009$ | $0.307\pm 0.023$
Quadratic LD coefficient star A | 0.31 (fixed) | 0.31 (fixed)
Quadratic LD coefficient star B | 0.29 (fixed) | 0.29 (fixed)
$e\cos\omega$ | 0.0 (fixed) | $0.000035\pm 0.000048$
$e\sin\omega$ | 0.0 (fixed) | $-0.0010\pm 0.0017$
Velocity amplitude star A ( km s-1) | $70.14\pm 0.34$ | $70.15\pm 0.37$
Velocity amplitude star B ( km s-1) | $82.49\pm 0.56$ | $82.50\pm 0.58$
Systemic velocity star A ( km s-1) | $-29.19\pm 0.14$ | $-29.19\pm 0.28$
Systemic velocity star B ( km s-1) | $-29.30\pm 0.21$ | $-29.30\pm 0.28$
Derived parameters: |  |
Fractional radius of star A | $0.07685\pm 0.00053$ | $0.07655\pm 0.00091$
Fractional radius of star B | $0.05561\pm 0.00085$ | $0.0560\pm 0.0014$
Orbital eccentricity | 0.0 (fixed) | $0.0009\pm 0.0012$
Argument of periastron (∘) | n/a | $272\pm 92$
Light ratio | $0.381\pm 0.028$ | $0.388\pm 0.028$

The uncertainties in the fitted and derived parameters were estimated using
Monte Carlo and residual-permutation simulations 36; 40. The Monte Carlo
algorithm requires the data errors to be of a correct size, which was achieved
by the scaling to force $\chi_{\nu}^{\,2}=1$. The residual-permutation
algorithm successively shifts the residuals of the best fit along the data
strings before refitting, so is by design sensitive to red noise and eclipse
depth variations. The larger of the two errorbars was adopted for each
parameter. Table II gives the best-fitting parameters and their uncertainties
for modelling runs assuming a circular or an eccentric orbit. Fig. 4 shows the
RVs and the fitted circular orbits. The parameter values for the circular- and
eccentric-orbit cases are almost identical, and their errorbars are similar,
so we adopt the circular orbit as our final result. Inspection of the Monte
Carlo and residual-permutation results shows that there is a very strong
correlation between the ratio of the radii and the light ratio of the system,
as is normally seen in cases where eclipses are shallow and partial (e.g. V455
Aur 41). The radius measurements could be improved by obtaining a
spectroscopic light ratio, although this will need a large telescope given the
relative faintness of KIC 5359678.

Figure 4:  The RVs of KIC 5359678 from W21 for star A (filled circles) and
star B (open circles) plotted versus orbital phase. The solid curves give the
best-fitting circular orbits, which are visually indistinguishable from the
best-fitting eccentric orbits. The lower panels show the residuals of the fits
for the two stars individually.

## How much numerical integration is needed?

The Kepler long-cadence light curve of KIC 5359678 is sampled at a cadence of
$t_{\rm samp}=1765.5$ s (Borucki 34) so each eclipse of this EB is covered by
only nine or ten datapoints. It is therefore important to account for this
when fitting the data, by numerically integrating the model light curve. The
method implemented in jktebop is to divide each datapoint up into $n_{\rm
samp}$ time intervals of equal duration $t_{\rm dur}=t_{\rm samp}/n_{\rm
samp}$, calculate the model light curve at the midpoint of each time interval,
and then take the mean of the $n_{\rm samp}$ fluxes as the predicted value for
that datapoint 39. An obvious question is: what is a suitable value of $n_{\rm
samp}$? Smaller values risk undersampling the light curve and biasing the
best-fitting parameters, whereas larger values take proportionally more
computing time. A value of $n_{\rm samp}=9$ was used in the jktebop analysis
above.

Figure 5:  Best-fitting radii of the stars (star A in the top panel and star B
in the bottom panel) with varying amounts of numerical integration. Errorbars
from Monte Carlo simulations are shown. The values and uncertainties found in
the current work are indicated with dotted lines.

We therefore ran a set of solutions of the Kepler light curve (eclipses only)
with $n_{\rm samp}$ running from one (equivalent to no numerical integration)
to nine (the value used for the main analysis). The results are shown in Fig.
5 for the radii of the stars, in the expectation that these are the properties
whose measurement is most affected by the sampling rate of the observations.
It can be seen that ignoring numerical integration leads to results that are
very biased: $R_{\rm A}$ is too small by 4.1$\sigma$ and $R_{\rm B}$ is too
large by 10.6$\sigma$. This arises because the smearing of the eclipse shape
is best matched using a larger $k$ and a slightly smaller $i$. Using $n_{\rm
samp}=2$ immediately leads to a big improvement in the results, and $n_{\rm
samp}=3$ is sufficient to bring both radius measurements within the errorbars
of the final result. Larger values of $n_{\rm samp}$, perhaps 5 or 8 depending
on one’s science goals, are needed to measure reliable parameter values.
However, the errorbars are reliably measured for much coarser sampling:
$n_{\rm samp}\geqslant 2$. It is therefore acceptable to measure parameter
values with a larger $n_{\rm samp}$, but then run the error analysis
algorithms with a lower $n_{\rm samp}$ to save computing time.

These conclusions are specific to KIC 5359678 but are likely to hold for most
EBs with eclipses of similar duration. They are comparable to what Southworth
39 found for the synthetic light curve of a transiting extrasolar planetary
system resembling Kepler-6. Similar conclusions were also reached by Kipping
42. The most important point is that if the data have been temporally averaged
then the model one fits to it must be treated in the same way.

## Physical properties of KIC 5359678

To determine the physical properties of the KIC 5359678 system we used the
values and uncertainties of $r_{\rm A}$, $r_{\rm B}$, $P$, $i$, $K_{\rm A}$
and $K_{\rm B}$ from Table II. The calculations were performed with the
jktabsdim code 43, which calculates the physical properties using standard
formulae and propagates uncertainties via perturbation. We adopted the $T_{\rm
eff}$ values of the stars from W21, but increased the errorbar for star B
because the quoted value is measured relative to the $T_{\rm eff}$ of star A
so therefore should include its uncertainty. The ratio of the $T_{\rm eff}$s
is consistent with the surface brightness ratio measured from the light curve
(Table II). The results are given in Table III.

To determine the distance to the system we used its apparent magnitudes in $B$
and $V$ from APASS DR9 and in $H$ and $K$ from 2MASS (see Table I); the 2MASS
$J$ magnitude is unreliable 32. The interstellar extinction was estimated as
$E(B-V)=0.07\pm 0.03$ mag obtained using the
stilism‡‡‡https://stilism.obspm.fr online tool (Lallement et al. 44; 45). Its
distance of $1534\pm 27$ pc was calculated using the surface brightness method
from Southworth et al. 43 and the $K$-band surface brightness calibrations
from Kervella et al. 46. This is significantly smaller than the distance of
$1626\pm 36$ pc from the parallax of the system in Gaia EDR3 30.

We are unable to deduce the reason why our distance is discrepant with that
from Gaia. The 2MASS apparent magnitudes are single-epoch so may have been
obtained at a time of particularly strong starspot activity – we have no data
available to provide an independent cross-check but the required change of
approximately $+0.1$ mag in $K$ is much larger than the brightness modulation
due to starspot activity seen in Fig. 1. The 2MASS observations were taken at
phase $0.9801\pm 0.0001$ based on our ephemeris in Table II, which is close to
but confidently outside primary eclipse (see Fig. 3). The APASS $BV$
magnitudes come from six epochs so are very unlikely to be made significantly
fainter by eclipses. A larger set of $T_{\rm eff}$ values allied with stronger
reddening also requires implausible changes – increases of 1300 K in both
$T_{\rm eff}$s and 0.23 mag in $E(B-V)$ – to fix the discrepancy. Stellar
radii larger by 5% would also be sufficient, but far beyond our 1$\sigma$
errorbars in Table III and significantly greater than those found by W21. We
leave this mystery for future study.

Table III: Physical properties of KIC 5359678 defined using the nominal solar units given by IAU 2015 Resolution B3 (Ref. 47). The $T_{\rm eff}$s are from W21.  Parameter | Star A | Star B
---|---|---
Mass ratio | $0.8503\pm 0.0071$
Semimajor axis of relative orbit ($\mathcal{R}^{\rm N}_{\odot}$) | $18.854\pm 0.081$
Mass ($\mathcal{M}^{\rm N}_{\odot}$) | 1.252 $\pm$ | 0.018 | 1.065 $\pm$ | 0.013
Radius ($\mathcal{R}^{\rm N}_{\odot}$) | 1.449 $\pm$ | 0.012 | 1.048 $\pm$ | 0.017
Surface gravity ($\log$[cgs]) | 4.214 $\pm$ | 0.007 | 4.424 $\pm$ | 0.013
Density ($~{}\rho_{\odot}$) | 0.412 $\pm$ | 0.009 | 0.924 $\pm$ | 0.043
Synchronous rotational velocity ( km s-1) | 11.76 $\pm$ | 0.10 | 8.51 $\pm$ | 0.14
Effective temperature (K) | 6500 $\pm$ | 50 | 5980 $\pm$ | 70
Luminosity $\log(L/\hbox{$\mathcal{L}^{\rm N}_{\odot}$})$ | 0.555 $\pm$ | 0.014 | 0.103 $\pm$ | 0.025
$M_{\rm bol}$ (mag) | 3.353 $\pm$ | 0.037 | 4.483 $\pm$ | 0.061
Distance (pc) | $1534\pm 27$

## Summary and conclusions

KIC 5359678 is a dEB containing two F-type stars that both show brightness
variations due to starspots. We have extracted the data around eclipse from
the light curve of this system obtained using the Kepler satellite and fitted
them with jktebop to determine the photometric properties of the system.
Numerical integration was used to account for the low sampling rate of the
Kepler long-cadence data. We included published RVs for the two stars to
determine their masses and radii. With the inclusion of spectroscopic
measurements of the $T_{\rm eff}$s of the two stars, we determined their
luminosities and the distance to the system. The distance measurement is
slightly shorter than that from Gaia EDR3 – at the level of 2.0$\sigma$ – and
we have not found a good explanation for this minor discrepancy.

We investigated the amount of numerical integration needed to fit the Kepler
data for KIC 5359678. We found that ignoring the sampling rate leads to radius
measurements that are wrong by $-4\sigma$ (for $R_{\rm A}$) and $+10\sigma$
(for $R_{\rm B}$). These biases are fixed by the application of only a small
amount of numerical integration to the model light curve during the fitting
process: merely doubling the model sampling rate is sufficient for approximate
solutions but more precision is needed for final results.

We can now turn to the original prompt of this work: the analysis of W21
presented very different errorbars for circular- and eccentric-orbit
solutions; no uncertainty was given for the mass of star B; and there was no
mention of the sampling cadence. For the first point, we find very similar
results for the two options and adopted the circular-orbit results for our own
calculations. For the second, the uncertainty in the mass of star B is 1.2%.
For the third, the results of W21 are much closer to our own when we used
numerical integration so we conclude that they did account for this effect in
their analysis.

A wider comparison between the results of W21 (adopting the circular-orbit
values) and our own shows good agreement for the radius of star B: $1.048\pm
0.017$$~{}{\rm R}_{\odot}$ (this work) versus $1.05\pm 0.03$$~{}{\rm
R}_{\odot}$ (W21); but not star A: $1.449\pm 0.012$$~{}{\rm R}_{\odot}$ (this
work) versus $1.52\pm 0.03$$~{}{\rm R}_{\odot}$ (W21). The reason is not clear
but could be related to the different ways in which the spot activity was
accounted for. Our mass measurements, though, are in unexpectedly poor accord:
$1.252\pm 0.018$$~{}{\rm M}_{\odot}$ versus $1.32\pm 0.02$$~{}{\rm M}_{\odot}$
for star A, and $1.065\pm 0.013$$~{}{\rm M}_{\odot}$ versus 1.12$~{}{\rm
M}_{\odot}$ (W21, no errorbar quoted) for star B. As the mass measurements are
primarily dependent on the RVs, and both studies used the same ones, it is not
clear how this discrepancy could have occurred. Our results have come from
extensively-tested codes and analysis methods 37; 48; 49 so should be
reliable.

The final motivation for the current study was to see if the available data
were sufficient to establish the masses and radii of the component stars of
KIC 5359678 to 2% or better. Table III shows that it was indeed possible. This
dEB has now been added to the Detached Eclipsing Binary Catalogue
(DEBCat§§§https://www.astro.keele.ac.uk/jkt/debcat/, Ref. 50) and can in
future be used to help calibrate theoretical models of stars with masses close
to that of our Sun.

## Acknowledgements

This paper includes data collected by the Kepler mission and obtained from the
MAST data archive at the Space Telescope Science Institute (STScI). Funding
for the Kepler mission is provided by the NASA Science Mission Directorate.
STScI is operated by the Association of Universities for Research in
Astronomy, Inc., under NASA contract NAS 5–26555. The following resources were
used in the course of this work: the NASA Astrophysics Data System; the SIMBAD
database operated at CDS, Strasbourg, France; and the ar$\chi$iv scientific
paper preprint service operated by Cornell University.

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