Vol.0 (20xx) No.0, 000–000

11institutetext: Department of Astronomy, Beijing Normal University, No. 19,
XinJieKouWai Street, Beijing 100875, China<EMAIL_ADDRESS>
22institutetext: Xinjiang Astronomical Observatory, Chinese Academy of
Sciences, No. 150, Science 1-Street, Urumqi 830011, Xinjiang, China
33institutetext: Key Laboratory of Optical Astronomy, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, China
44institutetext: School of Astronomy and Space Science, University of Chinese
Academy of Sciences, Beijing 10049, China

# Spot and Facula Activity Variations of the Eccentric Detached Eclipsing
Binary KIC 8098300 Based on the Time-series Orbital Solutions

Hubiao Niu 1122 Jianning Fu 11 Jiaxin Wang 11 Chunqian Li 3344

(Received 2021 04 30; accepted 2021 05 15)

###### Abstract

The LAMOST spectra and $\it{Kepler}$ light curves are combined to investigate
the detached eclipsing binary KIC 8098300, which shows the O’Connell effect
caused by spot/facula modulation. The radial velocity (RV) measurements are
derived by using the tomographic spectra disentangling technology. The mass
ratio $q=K_{1}/K_{2}=0.812\pm 0.007$, and the orbital semi-major axis
$a\mathrm{\sin}i=14.984\pm 0.048\ R_{\odot}$ are obtained by fitting the RV
curves. We optimize the binary model concerning the spot/facula activity with
the code PHOEBE and obtain precise parameters of the orbit including the
eccentricity $e=0.0217\pm 0.0008$, the inclination $i=87.71\pm 0.04^{\circ}$,
and the angle of periastron $\omega=284.1\pm 0.5^{\circ}$. The masses and
radii of the primary and secondary star are determined as $M_{1}=1.3467\pm
0.0001\ M_{\odot}$, $R_{1}=1.569\pm 0.003\ R_{\odot}$, and $M_{2}=1.0940\pm
0.0001\ M_{\odot}$, $R_{2}=1.078\pm 0.002\ R_{\odot}$, respectively. The ratio
of temperatures of the two component stars is $r_{teff}=0.924\pm 0.001$. We
also obtain the periastron precession speed of $0.000024\pm 0.000001\
\mathrm{d}\ cycle^{-1}$. The residuals of out-of-eclipse are analyzed using
the Auto-Correlation Function (ACF) and the Discrete Fourier Transform (DFT).
The spot/facula activity is relatively weaker, but the lifetime is longer than
that of most single main sequence stars in the same temperature range. The
average rotation period of the spots $P_{rot}=4.32\ d$ is slightly longer than
the orbital period, which may be caused by either the migration of
spots/faculae along the longitude or the latitudinal differential rotation.
The activity may be spot-dominated for the secondary star and facula-dominated
for the primary star. The method of this work can be used to analyze more
eclipsing binaries with the O’Connell effect in the $Kepler$ field to get the
precise parameters and investigate the difference of spot behavior between
binaries and single stars. It’s helpful for a deeper understanding of the
stellar magnetic activity and dynamo theory.

###### keywords:

instruments: Kepler — techniques: photometric — stars: variables: eclipsing
binary — stars: individual: KIC 8098300

##  1 Introduction

It has been well known that the Sun has an 11 year magnetic activity cycle
(Hathaway 2010). During the activity cycle, the flux increases gradually and
then falls back. The number and sizes of sunspots also increase and then
decrease. In contrast to expectations, the intensity brighten for about 0.1%
at sunspot maximum activity, which is due to the contribution of faculae
(Foukal & Vernazza 1979; Foukal et al. 2006). The distribution of sunspots
migrates gradually from mid-latitudes toward the equator, which produces the
well-known butterfly diagram (Maunder 1904). The Doppler imaging technique can
be used to derive the approximate spot sizes (Collier Cameron 1995; Barnes et
al. 2002). Through Doppler imaging, many very active, fast-rotating stars are
found to have large, dark polar spots (Vogt & Penrod 1983; Strassmeier 2009),
which can produce quasi-sinusoidal modulation. This modulation can be used to
study the rotations (McQuillan et al. 2014; Balona & Abedigamba 2016; Lurie et
al. 2017), the lifetime of activity region (Giles et al. 2017), and magnetic
activity cycles (Reinhold et al. 2017; Nielsen et al. 2019). Montet et al.
(2017) found a transition from spot-dominated to facula-dominated flux
variation between rotation periods of 15 and 25 days, which implied that the
transition was complete for stars at the age of the Sun.

Most of current studies of spot/facula activity are concentrated in single
stars. Thanks to the large scale photometry surveys like $\it{Kepler}$
(Borucki et al. 2010), TESS (Ricker et al. 2015) and $\it{Gaia}$ (Perryman et
al. 2001), more and more eclipsing binaries are discovered (Mowlavi et al.
2014). Lurie et al. (2017) reported rotation periods for 816 EBs with spot
modulation. Gu et al. (2003) analyzed high-resolution spectra of II Peg with
Doppler imaging. They found that the spots were in high-latitude and the sizes
of active regions changed significantly. The active longitude switches from
one hemisphere to the other. Xiang et al. (2020) presented the first Doppler
images of the active K-type component of the prototypical binary star RS Canum
Venaticorum. The maps showed that the spots located below a latitude of about
$70^{\circ}$, but distributed widely in longitude. They also estimated the
surface differential rotation rate of the K star. Özavcı et al. (2018)
reconstructed maps of the K1-type subgiant component of the eclipsing binary
KIC 11560447 with light-curve inversions (LCI) using $\it{Kepler}$ light
curves. Spots on the K1 star showed two preferred longitudes, the differential
rotation, and a 0.5-1.3-yr cycle. Bahar et al. (2020) found longitudinal spot
migration on the eclipsing binary KIC 9821078 with LCI. Spot activity behavior
in binaries can also be investigated by model fitting (Pi et al. 2019; Czesla
et al. 2019). Czesla et al. (2019) modeled light curves of the eclipsing
binary CoRoT 105895502, and found one short-lived ($\sim$ 40 days) but quasi-
stationary spot, and one spot drifting at a rate of $2.3^{\circ}\ d^{-1}$. Pi
et al. (2019) modeled light curves of the RS CVn–type eclipsing binary DV Psc.
They found two active regions at longitude belts of about $90^{\circ}$ and
$270^{\circ}$, with the activity cycles of 3.60 $\pm$ 0.03 yr and 3.42 $\pm$
0.02 yr, respectively. They also found a correlation between spots and flares.

The LAMOST-$\it{Kepler}$ project (De Cat et al. 2015; Zong et al. 2018; Fu et
al. 2020; Zong et al. 2020) performs spectroscopic follow-up observations for
targets in the field of the $\it{Kepler}$ mission with low resolution and
time-domain medium resolution spectra, respectively. The stellar atmospheric
parameters and radial velocities (RVs) are obtained. Due to the unprecedented
precision and almost continues of four years of the $\it{Kepler}$ light
curves, and the sufficient phase coverage of LAMOST spectra, there is a good
opportunity to precisely model the binaries, and further study the spot/facula
activity on eclipsing binaries. Wang et al. (2021) and Pan et al. (2020) have
investigated the spot activity of the short-period eclipsing binaries KIC
8301013 and KIC 5359678 by combining LAMOST spectra and $\it{Kepler}$ light
curves, respectively. KIC 8098300 is a detached eclipsing binary located in
the $\it{Kepler}$ field, and has been observed for 4 years by $\it{Kepler}$.
The magnitude is $K_{p}$=12.817 mag, and the orbital period is $P=4.3059177$
d. It has also been observed by the LAMOST-$\it{Kepler}$ project from 2015
September to 2020 September. The physical parameters of this binary can be
obtained from model solution by combining the light curves and spectra data.
The out-of-eclipse residuals and spot/facula models can also be analyzed to
understand the characteristic behavior of the spot/facula activity.

Photometric and spectroscopic observations of KIC 8098300 are introduced in
Section 2. O-C fitting and spectra disentangling are carried out in Section 3.
The binary model solution is optimized for each cycle of light curves in
Section 4. The out-of-eclipse residuals are analyzed to study the characters
of active regions in Section 5. The discussion and summary are given in
Section 6 and Section 7 respectively.

##  2 Observations

$\it{Kepler}$ had carried out almost continuous photometric observations for 4
years (from 2009 to 2013) and obtained high-precision light curves of KIC
8098300. There are a total of 18 quarters (Q0 – Q17) of long cadence data with
the sampling of 29.4 min, and 4 segments of short cadence data with the
sampling of 58.8 s (Prša et al. 2011; Slawson et al. 2011; Kirk et al. 2016).
Those data can be accessed via the Mikulksi Archive for Space Telescopes
(MAST) archive111https://archive.stsci.edu/kepler/kepler_fov/search.php. The
MAST website provides the Simple Aperture Photometry (SAP) light curves
obtained from the $\it{Kepler}$ data processing pipeline and the Presearch
Data Conditioning SAP (PDCSAP) light curves after detrending (Stumpe et al.
2012). Based on the SAP data, the Villinova website provides the Kepler
Eclipsing Binary Catalog (KEBC) and the light curves of detrending and
normalization222http://keplerebs.villanova.edu (Prša et al. 2011; Slawson et
al. 2011; Kirk et al. 2016). We use all of the 18 long cadence and 4 short
cadence light curves accessed via the Villinova website (Fig. 1). Outliers are
removed by sigma clipping after data smoothing and calculating the rolling
mean and the standard error.

Figure 1: Light curves of KIC 8098300 from $\it{Kepler}$ photometry. The upper
panel represents the long cadence data. The lower panel shows the short
cadence data. The vertical red dashed lines in each panel represent the bounds
of different quarters except for the first and last segments, each of which
consists of two quarters data sets, i.e. Q0-Q1 and Q16-Q17, respectively.

The spectra of KIC 8098300 were obtained by the LAMOST from 2015 September to
2020 September at Xinglong Station of the National Astronomical Observatories,
Chinese Academy of Sciences (NAOC). In the released LAMOST DR8 and DR9 V0-Q1
catalogs, a total of 2 low resolution spectra and 98 medium resolution spectra
(49 each at the red and blue arm of the spectrometers) were obtained,
respectively. The wavelength coverage of the low resolution (R $\sim$ 1800)
spectrum is 370-900 nm (Zhao et al. 2012; Cui et al. 2012; De Cat et al.
2015). The wavelength coverage of the medium resolution (R $\sim$ 7500)
spectrum is 495-535 nm at the blue arm, and 630-680 nm at the red arm,
respectively (Hou et al. 2018; Liu et al. 2020). Since there are more
absorption lines at the blue arm, we only use the 49 spectra from the blue
arm. The stellar atmospheric parameters are derived by LAMOST Stellar
Parameter pipeline (LASP) using the ULySS algorithm and the ELODIE stellar
library (Luo et al. 2004, 2012; Wu et al. 2014). We adopt the average
atmosphere parameters derived from the two low resolution spectra,
$T_{eff1}=6425\pm 24$ K, [Fe/H]$=-0.146\pm 0.025$. The spectra type is F5. The
observations of the low and medium resolution spectra are listed in Tables 1
and 2, respectively. As reported by Zong et al. (2020), the RV precision
increases with the increase of the signal to noise ratio (SNR). It can be
better than $1\ km\ s^{-1}$ when the SNR is higher than 10, which is below the
SNR of our spectra.

Table 1: LAMOST Low Resolution Spectra.

Date-Obs | BJD-2400000 | Phase | ExpTime | SNR | $T_{eff}$ | log $g$ | [Fe/H]
---|---|---|---|---|---|---|---
| (d) |  | (s) |  | (K) |  |
2015-09-21 | 57287.00035 | 0.17680 | 1800 | 103.8 | 6442 $\pm$ 24 | 4.216 $\pm$ 0.04 | -0.155 $\pm$ 0.023
2020-09-18 | 59111.01846 | 0.78409 | 3600 | 253.9 | 6407 $\pm$ 12 | 4.142 $\pm$ 0.017 | -0.136 $\pm$ 0.010

Table 2: LAMOST medium resolution spectra. The expose time is $1200$ s for all
the observations.

Date | Time | BJD-2400000 | Phase | SNR |
---|---|---|---|---|---
(UT) | (UT) | (d) |  |  |
2018-05-31 | 18:12:00 | 58270.26043 | 0.52768 | 25.9 |
2018-05-31 | 18:35:00 | 58270.27640 | 0.53139 | 29.1 |
2018-05-31 | 18:59:00 | 58270.29307 | 0.53526 | 17.9 |
2018-05-31 | 19:22:00 | 58270.30904 | 0.53897 | 27.1 |
2019-05-21 | 18:38:00 | 58625.27809 | 0.97646 | 12.9 |
2019-05-21 | 19:01:00 | 58625.29406 | 0.98017 | 16.3 |
2019-05-21 | 19:25:00 | 58625.31073 | 0.98404 | 03.5 |
2019-06-09 | 17:31:00 | 58644.23226 | 0.37834 | 22.1 |
2019-06-09 | 17:55:00 | 58644.24892 | 0.38222 | 23.4 |
2019-06-09 | 18:18:00 | 58644.26490 | 0.38593 | 21.6 |
2019-06-09 | 18:42:00 | 58644.28156 | 0.38980 | 23.2 |
2019-06-11 | 17:59:00 | 58646.25177 | 0.84735 | 11.1 |
2019-06-11 | 18:23:00 | 58646.26843 | 0.85122 | 13.1 |
2019-06-11 | 18:55:00 | 58646.29066 | 0.85638 | 11.6 |
2019-06-14 | 17:40:00 | 58649.23866 | 0.54103 | 38.5 |
2019-06-14 | 18:26:00 | 58649.27061 | 0.54844 | 38.5 |
2019-06-14 | 18:49:00 | 58649.28658 | 0.55215 | 41.7 |
2019-06-14 | 19:12:00 | 58649.30255 | 0.55586 | 39.3 |
2020-05-31 | 17:32:00 | 59001.23266 | 0.28759 | 17.7 |
2020-05-31 | 18:12:00 | 59001.26044 | 0.29405 | 42.7 |
2020-05-31 | 18:38:00 | 59001.27850 | 0.29824 | 38.0 |
2020-05-31 | 19:01:00 | 59001.29447 | 0.30195 | 34.9 |
2020-05-31 | 19:25:00 | 59001.31114 | 0.30582 | 17.0 |
2020-06-02 | 17:23:00 | 59003.22648 | 0.75064 | 26.8 |
2020-06-02 | 17:46:00 | 59003.24246 | 0.75435 | 32.5 |
2020-06-02 | 18:09:00 | 59003.25843 | 0.75806 | 32.0 |
2020-06-02 | 18:33:00 | 59003.27510 | 0.76193 | 34.0 |
2020-06-02 | 18:56:00 | 59003.29107 | 0.76564 | 36.9 |
2020-06-02 | 19:19:00 | 59003.30704 | 0.76935 | 33.5 |
2020-06-03 | 17:49:00 | 59004.24457 | 0.98708 | 30.4 |
2020-06-03 | 18:12:00 | 59004.26055 | 0.99079 | 25.4 |
2020-06-03 | 18:35:00 | 59004.27652 | 0.99450 | 24.5 |
2020-06-03 | 18:59:00 | 59004.29319 | 0.99837 | 23.4 |
2020-06-03 | 19:22:00 | 59004.30916 | 0.00208 | 21.2 |
2020-06-05 | 19:06:00 | 59006.29812 | 0.46399 | 40.1 |
2020-06-10 | 18:14:00 | 59011.26217 | 0.61683 | 23.2 |
2020-06-10 | 18:38:00 | 59011.27884 | 0.62070 | 19.7 |
2020-06-10 | 19:01:00 | 59011.29481 | 0.62441 | 13.1 |
2020-06-10 | 19:25:00 | 59011.31148 | 0.62828 | 11.7 |
2020-06-14 | 16:53:00 | 59015.20604 | 0.53275 | 42.4 |
2020-06-14 | 17:16:00 | 59015.22202 | 0.53646 | 45.2 |
2020-06-14 | 17:40:00 | 59015.23868 | 0.54033 | 46.4 |
2020-06-14 | 18:13:00 | 59015.26160 | 0.54565 | 46.9 |
2020-06-14 | 18:37:00 | 59015.27827 | 0.54953 | 45.8 |
2020-06-14 | 19:00:00 | 59015.29424 | 0.55323 | 38.2 |
2020-06-15 | 17:56:00 | 59016.24982 | 0.77516 | 36.2 |
2020-06-15 | 18:19:00 | 59016.26580 | 0.77887 | 27.6 |
2020-06-15 | 18:42:00 | 59016.28177 | 0.78258 | 28.1 |
2020-06-15 | 19:06:00 | 59016.29844 | 0.78645 | 24.4 |

##  3 Data Analysis

###  3.1 O-C Analysis

We segment each eclipse cycle of data into three parts, i.e. the primary
eclipse, the secondary eclipse, and the out-of-eclipse. The data points of the
primary and secondary eclipses are smoothed by using the
$\it{LSQUnivariateSpline}$ function in the interpolate module of $\it{Python}$
package $\it{Scipy}$, respectively (Fig. 2). Then, we take the time
corresponding to the minimum of the smoothed curve as the time of minimum. The
uncertainty of the time of minimum is estimated using Monte Carlo simulation
by repeating the above steps 100 times, randomly adding the observation
uncertainties of data points each time and finally calculating the standard
deviation. We derive the reference time of the primary minimum $T_{conj}=BJD\
2454956.737583\pm 0.033795$ d, the orbital period $P=4.3059177\pm 0.0000086$ d
, the interval between the phase of the primary and the secondary minimum
$sep=0.5034$, which are provided by the Villinova website. We fit the O-C
points of the time of the primary and the secondary minima, respectively, with
the linear function $T_{min}=T_{0}+PE$ (Sterken 2005), where $T_{min}$
represents the time of the minimum, $T_{0}$ is the reference time of the
minimum in the ephemeris, $P$ the orbital period, and $E$ the cycle number. So
the ephemeris for the time of the primary and the secondary minimum are
obtained as follows:

$\displaystyle T_{min1}$ $\displaystyle=BJD\
2454956.73835(4)+4.3059177(2)~{}E$ (1) $\displaystyle T_{min2}$
$\displaystyle=BJD\ 2454958.90556(9)+4.3059195(5)~{}E$ (2)

Compared with the ephemeris given by the Villinova website, the orbital period
$P$ obtained with the O-C analysis is consistent within its uncertainty, while
the difference between the reference time of the primary minimum is about and
0.00077 d (about 1.1 min) and the uncertainty is significantly reduced. Since
the uncertainty of the time of primary minimum is smaller, we adopt the new
ephemeris to fit the O-C points of the time of the primary minimum as the
basis for subsequent analysis (Fig. 3).

Figure 2: An example of smoothing one cycle (Cycle 3) of the long cadence
light curve. The upper left panel shows the smoothing and residuals of the
out-of-eclipse data points, the orange curve represents the smoothing result,
and the red and blue triangles indicate the positions of the primary and
secondary maxima. The upper right panel shows the smoothing and residuals of
the data points of the primary eclipse, while the lower right panel is the
secondary eclipse. The lower left panel is the overall smoothing result and
the residuals by connecting the three smoothed curves at the intersections.
Figure 3: O-C fitting. Upper panel: the red points are the time of primary
minima, and the cyan solid line represents the fitting for primary minima.
Lower panel: the blue points are the time of secondary minima, and the magenta
solid line is the fitting of secondary minima.

###  3.2 Spectra Analysis

The tomographic spectra disentangling technology can be applied to decompose
the time series composite spectrum of binary system into the spectrum of a
single sub-star, and at the same time give the orbital parameters and RVs. It
bases on the principle of Doppler shift, that is, the periodic revolving of
binary causes the periodic blue and red shift of the spectral lines. So no
template spectrum is required (Bagnuolo & Gies 1991; Simon & Sturm 1994;
Hadrava 1995). We use the $\it{fd3}$ tool333http://sail.zpf.fer.hr/fdbinary to
disentangle the LAMOST medium resolution spectra. This tool is the realization
of spectral tomography disentangling technology in frequency domain, which
needs to provide the respective flux factors of the primary and secondary
stars at different time of spectral observations (Ilijic et al. 2004;
Hensberge et al. 2008). We set the flux ratio of the secondary and the primary
stars $FR=F_{1}/F_{2}$ as a free parameter. Firstly, the coarse grid search is
carried out with the range of 0.2-1.0 and with the step size of 0.02. At each
step, the flux factors of the two component stars at each spectral observation
time are calculated, respectively, under the given flux ratio $FR$ according
to the phase folded and smoothed light curve, and the $\chi^{2}$ between the
observed and the modeled spectra is also calculated after spectral
disentangling. By fitting the variation of $\chi^{2}$ versus the flux ratio
$FR$ using a fifth-order polynomial, we obtain the $FR$ corresponding to the
minimum of $\chi^{2}$. Then, we carry out a fine grid search again with the
range of $FR$ $\in$ [FR-0.1, FR+0.1] with the step size of 0.005. The
uncertainty of $FR$ is estimated by the quadratic function fitting. After
searching, we get the best flux ratio $FR=0.446\pm 0.014$ and obtain the
spectrum of the primary and secondary star, respectively, with the fixed
orbital period $P$ in the spectra disentangling. At the same time, we also
obtain the RVs (Table 3) and orbital parameters of the two component stars.
Since the spectral disentangling technology only relies on the relative motion
of the two component stars, the binary system radial velocity $\gamma$ is
fixed to 0 km s-1. We use the Monte Carlo simulation method to estimate the
uncertainties of the orbital parameters and RVs by repeating the spectral
disentangling process 50 times, and randomly superimposing the observation
uncertainties of the observed spectra each time. Finally we calculate the
standard deviation of the 50 results as the uncertainty.

We use the IDL code
$\it{rvfit}$444http://www.cefca.es/people/~riglesias/rvfit.html (Iglesias-
Marzoa et al. 2015) to fit the RVs to obtain the orbit parameters (Table 4),
i.e. the time of periastron $T_{p}$, the eccentricity e, the angle of
periastron $\omega$, radial velocity amplitude of the primary and the
secondary star $K_{1}=78.945$ km s-1, $K_{2}=97.185$ km s-1, the mass ratio
$q=K_{1}/K_{2}=0.812$, and the orbital semi-major axis
$a\mathrm{\sin}i=14.984\ R_{\odot}$ (Table 4). Since the phase coverage of the
spectra is much less than that of the light curve, $K_{1}$ and $K_{2}$
obtained by spectral disentangling are relatively accurate, so the parameters
$q$ and $a\mathrm{\sin}i$ are more reliable. Therefore, we fix $q$ and
$a\mathrm{\sin}i$ in subsequent modeling with the light curves.

Table 3: RVs Obtained from LAMOST Medium Resolution Spectra Disentangling.

BJD-2400000 | $RV_{1}$ | Uncertainty | $RV_{2}$ | Uncertainty
---|---|---|---|---
(d) | (km s${}^{-1})$ | (km s${}^{-1})$ | (km s${}^{-1})$ | (km s${}^{-1})$
58270.260432 | 13.3 | 0.2 | -16.4 | 0.2
58270.276404 | 15.1 | 0.2 | -18.6 | 0.2
58270.293072 | 17.0 | 0.1 | -20.9 | 0.2
58270.309045 | 18.7 | 0.1 | -23.1 | 0.3
58625.278092 | 12.5 | 0.3 | -15.4 | 0.4
58625.294065 | 10.7 | 0.3 | -13.1 | 0.4
58625.310732 | 8.7 | 0.3 | -10.8 | 0.4
58644.232256 | -54.7 | 0.3 | 67.3 | 0.5
58644.248923 | -53.3 | 0.3 | 65.6 | 0.5
58644.264896 | -51.9 | 0.3 | 63.9 | 0.5
58644.281563 | -50.4 | 0.3 | 62.1 | 0.5
58646.251766 | 65.1 | 0.3 | -80.2 | 0.8
58646.268433 | 64.0 | 0.3 | -78.8 | 0.8
58646.290656 | 62.5 | 0.3 | -76.9 | 0.7
58649.238664 | 19.7 | 0.1 | -24.3 | 0.3
58649.270609 | 23.2 | 0.1 | -28.6 | 0.3
58649.286582 | 25.0 | 0.1 | -30.7 | 0.3
58649.302555 | 26.7 | 0.1 | -32.8 | 0.3
59001.232662 | -77.0 | 0.2 | 94.7 | 0.8
59001.260441 | -76.1 | 0.2 | 93.7 | 0.7
59001.278497 | -75.5 | 0.2 | 93.0 | 0.7
59001.294470 | -75.0 | 0.2 | 92.3 | 0.7
59001.311137 | -74.3 | 0.2 | 91.5 | 0.7
59003.226483 | 78.7 | 0.2 | -96.9 | 0.7
59003.242456 | 78.7 | 0.2 | -96.8 | 0.7
59003.258429 | 78.6 | 0.2 | -96.8 | 0.7
59003.275096 | 78.5 | 0.2 | -96.7 | 0.7
59003.291069 | 78.4 | 0.2 | -96.5 | 0.7
59003.307042 | 78.2 | 0.2 | -96.3 | 0.7
59004.244575 | 7.2 | 0.3 | -8.9 | 0.4
59004.260548 | 5.4 | 0.3 | -6.6 | 0.4
59004.276520 | 3.5 | 0.3 | -4.3 | 0.4
59004.293188 | 1.6 | 0.3 | -2.0 | 0.4
59004.309160 | -0.3 | 0.3 | 0.3 | 0.4
59006.298118 | -17.8 | 0.3 | 21.9 | 0.3
59011.262171 | 52.2 | 0.2 | -64.3 | 0.4
59011.278838 | 53.6 | 0.2 | -66.0 | 0.4
59011.294811 | 54.9 | 0.2 | -67.6 | 0.5
59011.311478 | 56.3 | 0.2 | -69.3 | 0.5
59015.206043 | 15.8 | 0.2 | -19.4 | 0.2
59015.222016 | 17.5 | 0.1 | -21.6 | 0.2
59015.238683 | 19.4 | 0.1 | -23.9 | 0.3
59015.261601 | 21.9 | 0.1 | -27.0 | 0.3
59015.278268 | 23.7 | 0.1 | -29.2 | 0.3
59015.294240 | 25.5 | 0.1 | -31.3 | 0.3
59016.249824 | 77.9 | 0.2 | -95.9 | 0.7
59016.265797 | 77.6 | 0.2 | -95.5 | 0.7
59016.281770 | 77.3 | 0.2 | -95.1 | 0.7
59016.298437 | 76.9 | 0.2 | -94.7 | 0.7

Figure 4: Fitting RVs obtained from LAMOST medium spectra disentangling. The
top panel shows the RVs of the primary (red points) and the secondary (blue
points) star, and the orbital fitting of the primary (red dashed curve) and
the secondary star (blue dashed curve). The middle and bottom panels are the
corresponding residuals with uncertainties of the RVs measurements plotted.

##  4 Binary Model Solution

We use PHOEBE 2.3 (PHysics Of Eclipsing BinariEs) package (Prša & Zwitter
2005; Prša et al. 2016; Jones et al. 2020; Conroy et al. 2020) to calculate
the forward model for each cycle of the Kepler light curves (Figs. 5 and 6).
The model is iteratively optimized using the least squares nonlinear fitting
function $\it{curve\\_fit}$ in the Python package $scipy.optimize$. The
objective function of optimization is
$\chi^{2}=\sum{1/\sigma^{2}{(f-f_{obs})}^{2}}$, where $f$ is the modeled flux,
$f_{obs}$ is the observed flux, and $\sigma$ is the flux uncertainty. The
optimal model solution is obtained corresponding to the minimum of $\chi^{2}$.
As there are a very large number of data points in the light curves,
especially for the short cadence data, running of the PHOEBE model calculation
is very time-consuming. In order to reduce the calculation time, we set the
grid step of the model calculation to be equivalent to the long cadence
sampling interval ($\sim$ 29.4 min), and then the model fluxes corresponding
to the observed points are calculated by linear interpolation. Since the
duration of eclipse is very short, in order to avoid abnormal light curves
caused by insufficient grid points in linear interpolation, we have increased
the dense of calculation grid for the eclipse parts, with the grid step of one
sixth of the long cadence sampling interval. The fixed parameters are the
orbital period $P=4.3059177$ d, the time of the primary minimum
$T_{conj}=\mathrm{BJD}\ 2454956.738351$ d, the mass ratio $q=0.812$,
$a\mathrm{\sin}i=14.984\ R_{\odot}$, and the temperature of the primary star,
$T_{eff1}=6425\pm 24$ K, the metalicity of the two sub-stars [Fe/H] = -0.146.
The value $T_{conj}$ is obtained from the O-C analysis in Section 3.1, the $q$
and $a\mathrm{\sin}i$ are obtained from the spectral disentangling in Section
3.2, where [Fe/H], $T_{eff1}$ are calculated as the average of corresponding
values given by the two low resolution spectra in LAMOST DR8 and DR9 catalogs
(Table 1). The parameters to be optimized are $e\mathrm{\cos}\omega$,
$e\mathrm{\sin}\omega$, the orbital inclination $i$, the temperature ratio of
the secondary star to the primary star $r_{teff}$, the radius of the primary
star $R_{1}$, and the radius of the secondary star $R_{2}$. Then, the masses
of the two sub-stars can be derived. We set the initial values of the
eccentricity e and the angle of periastron $\omega$ to the values obtained
from the spectral disentangling in Section 3.2. In order to determine the
appropriate initial values of other parameters, we use all the short cadence
and long cadence data, respectively, and take the $\it{emcee}$ tool to
randomly sample in a wide range of parameter space. The rough values of
parameters can be guessed using the solver of PHOEBE
$\it{estimator.lc\\_geometry}$ and $\it{estimator.ebai}$. So the sampling
range are set to a reasonable range based on the rough values and experience,
$e\mathrm{\cos}\omega\in$ (-0.1, 0.1), $e\mathrm{\sin}\omega\in$ (-0.1, 0.1),
$i\in$ (80.0, 90.0), $r_{teff}\in$ (0.8, 1.0), $R_{1}\in$ (0.8, 2.0),
$R_{2}\in$(0.8, 2.0). It converges after performing 500 times of sampling by
running 20 Monte Carlo Markov Chains (MCMC). Then, the average values and
uncertainties of the parameters are calculated according to the posterior
probability distribution to determine the initial values and the bounds of
parameters to be optimized. The binary parameters are hence optimized in a
narrower search range. In order to accelerate the convergence, we call the
eclipsing binary light curve fast calculation code $\it{ellc}$ (Maxted 2016)
as the model calculation backend, which produces almost the same result as
that of PHOEBE backend itself. After the optimization, we analyze the
residuals and investigate the lifetime and activity of the spot/facula (see
Sect. 5). Finally, the binary model and spot/facula model parameters are
globally optimized again in a narrower range at the same time (see Sect. 6.2)
for each cycle. We take the average of each parameter over all the 264 cycles
as the final parameter value, and estimate the uncertainty by calculating the
standard deviation of each parameter over all cycles. We compare the
parameters obtained from the short cadence data and the long cadence data, and
find that they are very consistent to each other within the uncertainties
(Table 4).

Figure 5: An example of the optimal model solutions and residuals of the long
cadence light curve of cycle 197. The red curve in each upper sub-panel
represents the model curve produced by PHOEBE. The black points in each lower
sub-panel are the corresponding residuals. Panels from left to right and top
to bottom, show the model solutions without spot/facula, a spot on the primary
star, a spot on the secondary star, a spot and a facula on the primary and the
secondary star, respectively.

Figure 6: The same as Figure 5 but with the short cadence data. Table 4: The
Average Parameters Obtained from Binary and Spot/facula Model Solutions.

Parameter | Short Cadence | Long Cadence
---|---|---
Primary | System | Secondary | Primary | System | Secondary
$T_{conj}$ (d) |  | $2454956.73835\pm 0.00004^{a}$ |  |  | $2454956.73835\pm 0.00004^{a}$ |
$P$ (d) |  | $4.3059177\pm 0.0000002^{a}$ |  |  | $4.3059177\pm 0.0000002^{a}$ |
$\gamma$ (km s-1) |  | $0.0^{b}$ |  |  | $0.0^{b}$ |
$q$ |  | $0.812\pm 0.007^{b}$ |  |  | $0.812\pm 0.007^{b}$ |
$a\mathrm{sin}i$ |  | $10.429\pm 0.048^{b}$ |  |  | $10.429\pm 0.048^{b}$ |
$i\ (^{\circ})$ |  | 87.71$\pm$0.04 |  |  | 87.71$\pm$0.05 |
$e\mathrm{cos}\omega$ |  | 0.00529$\pm$0.00004 |  |  | 0.00529 $\pm$0.00006 |
$e\mathrm{sin}\omega$ |  | -0.0211$\pm$0.0008 |  |  | -0.0209$\pm$0.0009 |
$e$ |  | 0.0217$\pm$0.0008 |  |  | 0.0215$\pm$0.0009 |
$\omega\ (^{\circ})$ |  | 284.1$\pm$0.5 |  |  | 284.2$\pm$0.6 |
$r_{teff}$ |  | 0.924$\pm$0.001 |  |  | 0.924$\pm$0.001 |
$R\ (R_{\odot})$ | 1.569$\pm$0.003 |  | 1.078$\pm$0.002 | 1.566$\pm$0.002 |  | 1.078$\pm$0.004
$M\ (M_{\odot})$ | 1.3467$\pm$0.0001 |  | 1.0940$\pm$0.0001 | 1.3467$\pm$0.0002 |  | 1.0940$\pm$0.0001
$T_{eff}$ (K) | $6425\pm 24^{c}$ |  | 5936$\pm$25 | $6425\pm 24^{c}$ |  | 5936$\pm$25
$L_{2}/L_{1}$ |  | $0.344\pm 0.009$ |  |  | $0.343\pm 0.004$ |
$colat1\ (^{\circ})$ |  |  | 70$\pm$32 |  |  | 65$\pm$34
$long1\ (^{\circ})$ |  |  | 144$\pm$72 |  |  | 103$\pm$61
$radius1\ (^{\circ})$ |  |  | 21$\pm$10 |  |  | 18$\pm$11
$relteff1$ |  |  | 0.727$\pm$0.559 |  |  | 0.911$\pm$0.553
$colat2\ (^{\circ})$ | 44$\pm$32 |  |  | 47$\pm$31 |  |
$long2\ (^{\circ})$ | 88$\pm$43 |  |  | 95$\pm$57 |  |
$radius2\ (^{\circ})$ | 34$\pm$27 |  |  | 30$\pm$26 |  |
$relteff2$ | 0.998$\pm$0.032 |  |  | 1.003$\pm$0.065 |  |

a: Derived by O-C fitting, fixed in binary modeling.
b: The RVs are only related to the relative motion of the two sub-stars with
the method of spectral disentangling technique, so the radial velocity of the
binary system is always fixed to 0 km s-1.
c: Obtained from LAMOST low resolution spectra.

##  5 Analysis of Out-of-eclipse Residuals

Ignoring the influence of spot/facula activity, the binary model parameters
obtained in the previous section, i.e. $e\mathrm{cos}\omega$,
$e\mathrm{sin}\omega$, the orbital inclination $i$, the temperature ratio of
the secondary star to the primary star $r_{teff}$, the radius of the primary
star $R_{1}$, and the radius of the secondary star $R_{2}$, are used to
calculate the theoretical fluxes and then the residuals for each cycle. It can
be seen that there is an obvious quasi-sinusoidal variation in the out-of-
eclipse. The primary and secondary maxima are not equal to each other (Fig.
7), which is the evidence of the O’Connell effect (O’Connell 1951). This is
generally considered to be modulated by the spot activity. Based on this
assumption, we analyze the out-of-eclipse residuals as follows.

Figure 7: Upper panel: the out-of-eclipse residuals (black points) for all
quarters (Q0-Q17). Lower panel: the residuals zoomed for Q6. The red dashed
vertical lines in the upper panel represent the bounds of different quarters
except for the first and last segments, each of which consists of two quarters
of data sets, i.e. Q0-Q1 and Q16-Q17, respectively. The red dashed vertical
lines in the lower panel indicate the positions of the primary minima.

###  5.1 Active Region Lifetime

Spots keep appearing and disappearing at different latitudes and longitudes on
the surface of the sub-stars, causing the modulation in the phase and
amplitude of the out-of-eclipse fluxes, so the active region shows a certain
lifetime. The auto-correlation function (ACF) method can be used to estimate
the lifetime of the dominant active region (McQuillan et al. 2013, 2014). The
ACF will have series of peaks at the lags close to integer multiples of the
rotation period of the dominant spot, where the signal has the highest degree
of self-similarity. The peak heights decrease with time delay due to spot
formation and decay. The decay rate of the side lobes can be used to
characterize the lifetime of the active region. The underdamped simple
harmonic oscillator (uSHO) function is used to fit the ACF for each year
respectively using the MCMC method (Giles et al. 2017):

$y(t)=e^{(-t/\tau_{AR})}(Acos(\frac{2\pi t}{P})+Bcos(\frac{4\pi
t}{P})+y_{0}),$ (3)

where $A$ and $B$ are the amplitudes of the cosine terms and $y0$ is the
offset to $y=0$. $P$ is the stellar rotation period corresponding to the time
lag at which the highest peak occurs. $\tau_{AR}$ represents the decay time-
scale of the ACF which indicates the lifetime of the dominant spot. The
results are shown in Figure 8 and listed in Table 5. It can be seen that the
lifetime of the active region is relatively long except for the first year.

Figure 8: Estimation of active region lifetime by fitting the ACF of out-of-
eclipsing residuals using uSHO function. Panels from left to right and top to
bottom, show the results from 1st to 4th year, respectively. In each panel,
the blue curve is the smoothed ACF, and the orange curve represents the
fitting using MCMC method. Table 5: Active Region Lifetime

Year | $\tau_{AR}$ | A | B | $y_{0}$ | $P_{rot}$
---|---|---|---|---|---
(d) | (d) | (d) | (d) | (d)
1st | 35.4 | 0.658 | 0.138 | -0.008 | 4.364
2nd | 106.8 | 0.820 | 0.099 | -0.008 | 4.295
3rd | 160.7 | 0.957 | 0.064 | -0.006 | 4.277
4th | 66.5 | 0.639 | 0.119 | -0.002 | 4.311

###  5.2 Rotation Period

Due to the differential rotation, the rotation period of the spot/facula at
different positions on the surface of a star would be different. For the
short-period eclipsing binary, the rotation period of the sub-stars is the
same as the orbital period due to the tidal synchronization. Compared with
ACF, the Discrete Fourier Transform (DFT) is more conducive to analyze the
rotation periods of spots/faculae. Therefore, the generalized Lomb-Scargle
(LS) module in the $\it{Astropy}$ package (Lomb 1976; Scargle 1982;
Zechmeister & Kürster 2009; Press & Rybicki 1989; VanderPlas 2018) is employed
to analyze the residuals of long cadence light curves for each quarter (Fig.
9). From Figures 9 and 10, it can be seen that the rotation period of
spots/faculae is close to the orbital period, with an average of
$P_{rot}=4.32$ d. The result is listed in Table 6. Most of the rotation
periods are slightly longer than the orbital period (4.3059177 d), indicating
that the rotation is slightly slower than that of the sub-star during most of
the time, which may be caused by either the migration of spots/faculae along
the longitude (Fig. 13) or the latitudinal differential rotation.

Figure 9: An example of the periodogram of the out-of-eclipse residuals for
Q6. The blue curve is the power spectrum. The orange curve is the spectrum
window. The red triangles mark the position of the significant peaks. From
left to right, vertical blue dashed lines represent the half and the full
orbital period, respectively. Figure 10: The rotation period of spot/facula is
shown for each quarter, and the horizontal dotted line represents the orbital
period. Table 6: The Rotation Periods of Spots/faculae for Each Quarter.

Quarter | Q0-Q1 | Q2 | Q3 | Q4 | Q5 | Q6 | Q7 | Q8 | Q9 | Q10 | Q11 | Q12 | Q13 | Q14 | Q15 | Q16-Q17
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$P_{rot}$ (d) | 4.403 | 4.298 | 4.521 | 4.283 | 4.318 | 4.304 | 4.317 | 4.331 | 4.324 | 4.306 | 4.326 | 4.233 | 4.425 | 4.365 | 4.332 | 4.333

###  5.3 Active Region Evolution

In order to quantitatively describe the flux modulation due to active region
during each eclipse cycle, two indicators $R_{var}$ and $rms$ are calculated
for each cycle. $R_{var}$ represents the variability range, which is defined
as the range between the 5th and 95th percentile of the sorted intensity, and
$rms$ is the root-mean-square scatter (Basri et al. 2011; Reinhold et al.
2017). Figure 11 shows the variations of $R_{var}$ and $rms$ versus cycle
number. It can be seen that the spot activity is enhanced around the cycle 90
and 160, but the four years of observation are not long enough to obtain an
explicit magnetic activity cycle.

Figure 11: The variability range and the root-mean-square scatter of the
active region versus cycle number.

##  6 Discussion

###  6.1 Periastron precession

After optimization of binary model with a spot and a facula for each cycle of
light curve, we finally get the parameters of KIC 8098300 that changed over
time. We find the periastron procession of the binary system. As can be seen
in Figure 12, the orbital parameters $T_{p}$, $e\mathrm{cos}\omega$ and
$\omega$, increase with the cycle number. There are no trends for the rest of
other parameters. This trend is the most evident for $e\mathrm{cos}\omega$.
However, it also superimposes a sinusoidal signal, which may be caused by the
modulation of spot/facula activity. $T_{p}$ and $\omega$ versus cycle number
are fitted with the linear functions with the periastron precession speeds is
of $0.000024\pm 0.000001$ d cycle-1 ($\sim$ 2.0736 min cycle-1) or $0.0020\pm
0.0004^{\circ}\ cycle^{-1}$, respectively. The $e\mathrm{cos}\omega$ values
are fitted with the combination of a linear and a sine function, and the
periastron precession is derived as $0.0015\pm 0.0003^{\circ}\ cycle^{-1}$. We
hence get a modulation time scale of 134 cycles (about 578 d), which might be
due to the magnetic activity cycle.

Figure 12: The periastron precession. The black open circles in each upper
sub-panel are the optimized binary parameters for long cadence data, while the
black filled squares represent that of short cadence data. The red curve in
each upper sub-panel represents the fitted curve. In the lower sub-panels, the
corresponding residuals are plotted.

###  6.2 Spot/facula model

From Figure 2, one can clearly see the O’Connell effect in this binary system,
i.e. the two out-of-eclipse maxima are unequally high. It can also be seen in
the residuals of the binary model solution without spot/facula for each cycle
of light curve (the upper left panels in Figs. 5 and 6). There are several
explanations for this effect, the most common of which is the spot modulation
(Wilsey & Beaky 2009). We have tested the single spot model, but it can not
well suite to the out-of-eclipse light curve whether on the primary or the
secondary star as some trend still exists in the residuals (see the upper
right and lower left panels in Figs. 5 and 6). We have tried to optimize the
other parameters like $q$, $a\mathrm{sin}i$, albedo and reflection
coefficients, but it hasn’t improved much. Thus we have constructed the model
with two spots, one for each component star, for which there is no trend and
with the smallest $rms$ scatter of the residuals. The facula can also be
called as hot spot, the temperature of which is higher than that of the
photosphere around it. We optimize the four parameters of each spot/facula,
i.e. the colatitude $colat$, the longitude $long$, the size $radius$, and the
contrast $relteff$, along with the binary parameters in Section 4 with the
narrower search range for each cycle light curve. The bounds of parameters for
optimization are $e\mathrm{cos}\omega\in$ (0.002, 0.008),
$e\mathrm{sin}\omega\in$ (-0.03, 0.001), $i\in$ (84.7, 89.7), $r_{teff}\in$
(0.85, 1.0), $R_{1}\in$ (1.37, 1.77), $R_{2}\in$ (0.88,1.28), $colat\in$ (0,
180), $long\in$ (0, 360), $radius\in$ (0, 90), and $relteff\in$ (0, 1.5). As
can be seen in the lower right panels in Figures 5 and 6, the optimized model
solution with two spots suite well to the light curves. The average values of
those parameters and uncertainties are listed in Table 4 for long and short
cadence data, respectively. Since the results based on the long and short
cadence data are very consistent to each other, we take the parameters from
the long cadence data as the final result, of which the uncertainties are
slightly smaller. The orbital eccentricity is $e=0.0217\pm 0.0008$. The
orbital inclination is $i=87.71\pm 0.04^{\circ}$. The angle of periastron is
$\omega=284.1\pm 0.5^{\circ}$. The masses and radii of the primary and
secondary star are $M_{1}=1.3467\pm 0.0001\ M_{\odot}$, $R_{1}=1.569\pm 0.003\
R_{\odot}$, and $M_{2}=1.0940\pm 0.0001\ M_{\odot}$, $R_{2}=1.078\pm 0.002\
R_{\odot}$, respectively. The temperature ratio is $r_{teff}=0.924\pm 0.001$,
so the temperature of the secondary star can be derived as $T_{eff2}=5936\pm
25$ K. Although affected by the degeneracy of spot parameters $colat$, size
$radius$ and contrast $relteff$, it hints that the temperature of spot on the
primary star is very close to or slightly higher than that of the photosphere,
and it is lower than that of the photosphere on the secondary star. So it may
be facula (hot spot) dominant for the primary star, and cold spot dominant for
the secondary star. As shown in Figure 13, it seems like that there is some
signature of the migration of spot longitudes from one hemisphere to the
opposite side on the secondary star, from about cycle 30 to 120, cycle 150 to
250 and cycle 260 to 330, respectively, which means that active regions are
formed and disappeared in different longitudes. Bahar et al. (2020) also found
the longitudinal spot migration on eclipsing binary KIC9821078, but the drift
direction was toward increasing longitudes. But we can not see this kind of
spot migration on the primary star.

Figure 13: Variations of the longitudes of spot/facula versus the cycle
number. The upper panel represents the secondary star, and the lower panel is
for the primary star. The color bars represent the contrast of spot/facula.

###  6.3 Comparison with spots of single stars

We compare the spot/facula activity on KIC 8098300 with that of 1774 single
main sequence (MS) stars analyzed by Giles et al. (2017) based on the
$\it{Kepler}$ light curves. As shown in the left panel of Figure 11, the rms
of the spot/facula on this binary is located at the bottom edge in the
corresponding temperature range, which means the spot/facula activity in this
binary is relatively weaker than that of most single MS stars. As shown in the
right panel of Figure 11, the lifetime of active region $\tau_{AR}$ on this
binary is located near the upper edge in the same temperature range, which
implies that the lifetime of the spots/faculae on this binary is longer than
that of most single MS stars. This is similar to the case of the eclipsing
binary KIC 5359678 (Wang et al. 2021). Montet et al. (2017) analyzed a sample
of 4000 Sun-like stars with measured rotation periods and found a transition
between spot-dominated and facula-dominated variability between rotation
periods of 15 and 25 days, which implied that the transition between the two
modes is complete for MS stars at the age of the Sun. For this short period
binary, it may be facula-dominated for the primary star, and spot-dominated
for the secondary star. So the binary may be undergoing the transition between
the two modes.

Figure 14: Comparison the spot/facula activity on KIC 8098300 with that of
spots on the single MS stars in the sample of Giles et al. (2017). The red
filled square represent the primary star, while the blue filled square is the
secondary star. The black open circles are the sample of single MS stars.

##  7 Summary

The LAMOST spectra and $\it{Kepler}$ light curves are combined to study the
detached eclipsing binary KIC 5359678, which shows the O’Connell effect caused
by spot modulation. The tomographic spectra disentangling technology is
applied to decompose the time series of spectra and derive the RVs. Thus the
parameters of $q=K_{1}/K_{2}=0.812\pm 0.007$ and $a\mathrm{\sin}i=14.984\pm
0.048\ R_{\odot}$ are obtained by orbital fitting of RVs. We construct and
optimize the binary model with the spot and facula using the PHOEBE code and
$\it{curve\\_fit}$ function in $\it{Scipy}$ package for each cycle of the
light curves. Each parameter for all 264 cycles is averaged as the final
value. The orbital eccentricity, the orbital inclination, and the angle of
periastron are obtained as $e=0.0217\pm 0.0008$, $i=87.71\pm 0.04^{\circ}$,
and $\omega=284.1\pm 0.5^{\circ}$, respectively. The masses and radii of the
primary and secondary star are $M_{1}=1.3467\pm 0.0001\ M_{\odot}$,
$R_{1}=1.569\pm 0.003\ R_{\odot}$, and $M_{2}=1.0940\pm 0.0001\ M_{\odot}$,
$R_{2}=1.078\pm 0.002\ R_{\odot}$, respectively. The temperature ratio of the
secondary and the primary star is $r_{teff}=0.924\pm 0.001$, so the
temperature of the secondary star can be derived as $T_{eff2}=5936\pm 25\ $ K.
We also find the periastron precession of $0.000024\pm 0.000001$ d cycle-1. It
may be spot-dominated for the secondary star and facula-dominated for the
primary star. This binary system may be undergoing the transition between the
two modes. It shows some signature of spot migration toward decreasing
longitudes on the secondary star, but it can not be seen on the primary star.

The residuals of out-of-eclipse are analyzed using ACF and DFT to determine
the lifetimes and the rotation periods of active regions. The lifetimes of
active regions are longer than that of most single MS stars in the same
temperature range. The average rotation period $P_{rot}=4.32$ d is slightly
longer than the orbital period (4.3059177 d), which may be caused by either
the migration of spots/faculae in longitude or the latitudinal differential
rotation. The spot/facula activity in this binary is relatively weaker (with
smaller $rms$) than that of most single MS stars.

###### Acknowledgements.

We would like to thank the referees for the valuable and constructive
comments. The authors acknowledge the support from the National Natural
Science Foundation of China (NSFC) through the grants 11833002, 12090040, and
12090042. 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. This paper includes data collected by the $\it{Kepler}$ mission.
Funding for the $\it{Kepler}$ mission is provided by the NASA Science Mission
directorate. Data presented in this paper were obtained from the Mikulski
Archive for Space Telescopes (MAST).

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