Hu-Biao Niu,Jian-Ning Fu,Jia-Xin Wang,and Chun-Qian Li

1 Department of Astronomy,Beijing Normal University,Beijing 100875,China; jnfu@bnu.edu.cn

2 Xinjiang Astronomical Observatory,Chinese Academy of Sciences,Urumqi 830011,China

3 Key Laboratory of Optical Astronomy,National Astronomical Observatories,Chinese Academy of Sciences,Beijing 100101,China

4 School of Astronomy and Space Science,University of Chinese Academy of Sciences,Beijing 100049,China

Abstract The LAMOST spectra and 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=K1/K2=0.812±0.007,and the orbital semimajor axis a sin i= 14.984 ±0.048 R⊙a(bǔ)re obtained by ftiting the RV curves.We optimize the binary modelconcerning the spot/facula activity with the code PHOEBE and obtain precise parameters of the orbit including the eccentricity e=0.0217±0.0008,the inclination i=87°.71±0°.04,and the angle of periastron ω=284°.1±0°.5.The masses and radii of the primary and secondary star are determined as M1=1.3467±0.0001 M⊙,R1=1.569±0.003 R⊙,and M2=1.0940±0.0001 M⊙,R2=1.078±0.002 R⊙,respectively.The ratio of temperatures of the two component stars is rteff=0.924±0.001.We also obtain the periastron precession speed of 0.000024±0.000001 day cycle-1.The residuals of out-of-eclipse are analyzed using the auto-correlation function and the discrete Fourier transform.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 Prot=4.32 days 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 obtain the precise parameters and investigate the difference of spot behavior between binaries and single stars.It is helpful for a deeper understanding of the stellar magnetic activity and dynamo theory.

Key words: (stars:) binaries: eclipsing–(stars:) starspots–stars: individual (KIC 8098300)

1.Introduction

It has been well known that the Sun has an 11 yr 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 quasisinusoidal 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 Kepler(Borucki et al.2010),TESS(Ricker et al.2015)and 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°,but distributed widely in longitude.They also estimated the surface differential rotation rate of the K star.?zavc? et al.(2018)reconstructed maps of the K1-type subgiant component of the eclipsing binary KIC 11560447 with lightcurve inversions(LCI)using 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 (Czesla et al.2019; Pi et al.2019).Czesla et al.(2019) modeled light curves of the eclipsing binary CoRoT 105895502,and found one short-lived (～40 days) but quasistationary spot,and one spot drifting at a rate of 2°.3 day-1.Pi et al.(2019) modeled light curves of the RS CVntype eclipsing binary DV Psc.They found two active regions at longitude belts of about 90°and 270°,with the activity cycles of 3.60±0.03 yr and 3.42±0.02 yr,respectively.They also found a correlation between spots and flares.

The LAMOST-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 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 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 Kepler light curves,respectively.KIC 8098300 is a detached eclipsing binary located in the Kepler field,and has been observed for 4 yr by Kepler.The magnitude is Kp=12.817 mag,and the orbital period is P=4.3059177 days.It has also been observed by the LAMOST-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-ofeclipse residuals are analyzed to study the characters of active regions in Section 5.The discussion and summary are given in Sections 6 and 7,respectively.

2.Observations

Kepler had carried out almost continuous photometric observations for 4 yr (from 2009 to 2013) and obtained highprecision light curves of KIC 8098300.There are a total of 18 quarters (Q0–Q17) of long cadence data with the sampling of 29.4 minutes,and four segments of short cadence data with the sampling of 58.8 s (Pr?a et al.2011; Slawson et al.2011;Kirk et al.2016).Those data can be accessed via the Mikulksi Archive for Space Telescopes (MAST) archive.5https://archive.stsci.edu/kepler/kepler_fov/search.phpThe MAST website provides the Simple Aperture Photometry (SAP) light curves obtained from the Kepler data processing pipeline and the Presearch Data Conditioning SAP 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 normalization6http://keplerebs.villanova.edu(Pr?a et al.2011;Slawson et al.2011;Kirk et al.2016).We use all of the 18 long cadence and four short cadence light curves accessed via the Villinova website(Figure 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 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 two 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～1800) spectrum is 370–900 nm (Cui et al.2012; Zhao et al.2012; De Cat et al.2015).The wavelength coverage of the medium resolution (R～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 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,Teff1=6425±24 K,[Fe/H]=-0.146±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(S/N).It can be better than 1 km s-1when the S/N is higher than 10,which is below the S/N of our spectra.

Table 1 LAMOST Low Resolution Spectra

Table 2 LAMOST Medium Resolution Spectra

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 LSQUnivariateSpline function in the interpolate module of Python package Scipy,respectively (Figure 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 Tconj=BJD 2,454,956.737583±0.033 795 days,the orbital period P=4.3059177±0.0000086 days,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 functionTmin=T0+PE(Sterken 2005),where Tminrepresents the time of the minimum,T0is the reference time of the minimum in the ephemeris,P is the orbital period,and E is the cycle number.So the ephemeris for the time of the primary and the secondary minimum are obtained as follows:

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 days (about 1.1 minutes) and the uncertainty issignificantly reduced.Because 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 (Figure 3).

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 redshift of the spectral lines.So no template spectrum is required (Bagnuolo &Gies 1991; Simon & Sturm 1994; Hadrava 1995).We use the fd3 tool7http://sail.zpf.fer.hr/fdbinaryto 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=F1/F2as a free parameter.First,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 χ2between the observed and the modeled spectra is also calculated after spectral disentangling.By fitting the variation of χ2versus the flux ratio FR using a fifth-order polynomial,we obtain the FR corresponding to the minimum of χ2.Then,we carry out a fine grid search again with the range of FR ∈[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±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.Because the spectral disentangling technology only relies on the relative motion of the two component stars,the binary system RV γ 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 rvfit8http://www.cefca.es/people/～riglesias/rvfit.html(Iglesias-Marzoa et al.2015) to fit the RVs to obtain the orbit parameters (Figure 4),i.e.,the time of periastron Tp,the eccentricity e,the angle of periastron ω,RV amplitude of the primary and the secondary star K1=78.945 km s-1,K2=97.185 km s-1,the mass ratio q=K1/K2=0.812,and the orbital semimajor axisasini=14.984R⊙(Table 4).Since the phase coverage of the spectra is much less than that of the light curve,K1and K2obtained by spectral disentangling are relatively accurate,so the parameters q andasiniare more reliable.Therefore,we fix q andasiniin subsequent modeling with the light curves.

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.

4.Binary Model Solution

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.

Table 3 RVs Obtained from LAMOST Medium Resolution Spectra Disentangling

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.,ecosω,esinω,the orbital inclination i,the temperature ratio of the secondary star to the primary star rteff,the radius of the primary star R1,and the radius of the secondary star R2,are used to calculate the theoretical fluxes and then the residuals for each cycle.It can be seen that there is an obvious quasisinusoidal variation in the out-of-eclipse.The primary and secondary maxima are not equal to each other (Figure 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 outof-eclipse residuals as follows.

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 autocorrelation 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):

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.τARrepresents the decay timescale 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.

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 Astropy package (Lomb 1976;Scargle 1982; Zechmeister & Kürster 2009; Press &Rybicki 1989; VanderPlas 2018) is employed to analyze theresiduals of long cadence light curves for each quarter(Figure 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 Prot=4.32days.The result is listed in Table 6.Most of the rotation periods are slightly longer than the orbital period(4.3059177 days),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 (Figure 13) or the latitudinal differential rotation.

Table 4 The Average Parameters Obtained from Binary and Spot/facula Model Solutions

Table 5 Active Region Lifetime

5.3.Active Region Evolution

In order to quantitatively describe the flux modulation due to active region during each eclipse cycle,two indicators Rvarand rms are calculated for each cycle.Rvarrepresents the variability range,which is defined as the range between the 5th and 95th percentile of the sorted intensity,and rms is the root-meansquare scatter (Basri et al.2011; Reinhold et al.2017).Figure 11 shows the variations of Rvarand 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 observations are not long enough to obtain an explicit magnetic activity cycle.

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 Tp,ecosωand ω,increase with the cycle number.There are no trends for the rest of other parameters.This trend is the most evident forecosω.However,it also superimposes a sinusoidal signal,which may be caused by the modulation of spot/facula activity.Tpand ω versus cycle number are fitted with the linear functions with the periastron precession speeds is of 0.000024±0.000001 daycycle-1(～2.0736 minutescycle-1)or 0°.0020±0°.0004 cycle-1,respectively.Theecosωvalues are fitted with the combination of a linear and a sine function,and the periastron precession is derived as 0°.0015±0°.0003 cycle-1.We hence obtain a modulation timescale of 134 cycles(about 578 days),which might be due to the magnetic activity cycle.

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 stars,respectively.

Figure 6.The same as Figure 5 but with the short cadence data.

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.

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 yr,respectively.In each panel,the blue curve is the smoothed ACF,and the orange curve represents the fitting using the MCMC method.

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.

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 Figures 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 cannot well suite to the out-ofeclipse 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 Figures 5 and 6).We have tried to optimize the other parameters like q,asini,albedo and reflection coefficients,but it has not 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 areecosω∈(0.002,0.008),esinω∈(-0.03,0.001),i ∈(84.7,89.7),rteff∈(0.85,1.0),R1∈(1.37,1.77),R2∈(0.88,1.28),colat ∈(0,180),long ∈(0,360),radius ∈(0,90),and relteff ∈(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.Because 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±0.0008.The orbital inclination is i=87°.71±0°.04.The angle of periastron is ω=284°.1±0°.5.The masses and radii of the primary and secondary stars are M1=1.3467±0.0001 M⊙,R1=1.569±0.003 R⊙,and M2=1.0940±0.0001 M⊙,R2=1.078±0.002 R⊙,respectively.The temperature ratio is rteff=0.924±0.001,so the temperature of the secondary star can be derived as Teff2=5936±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.However,we cannot see this kind of spot migration on the primary star.

Figure 10.The rotation period of spot/facula is shown for each quarter,and the horizontal dotted line represents the orbital period.

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

Table 6 The Rotation Periods of Spots/faculae for Each Quarter

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 Kepler light curves.As shown in the left panel of Figure 14,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 14,the lifetime of active region τARon 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 faculadominated 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 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.

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.

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 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=K1/K2=0.812±0.007 andasini= 14.984 ±0.048R⊙a(bǔ)re obtained by orbital fitting of RVs.We construct and optimize the binary model with the spot and facula using the PHOEBE code and curve_fit function in 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±0.0008,i=87°.71±0°.04,and ω=284°.1±0°.5,respectively.The masses and radii of the primary and secondary star are M1=1.3467±0.0001 M⊙,R1=1.569±0.003 R⊙,and M2=1.0940±0.0001 M⊙,R2=1.078±0.002 R⊙,respectively.The temperature ratio of the secondary and the primary star is rteff=0.924±0.001,so the temperature of the secondary star can be derived as Teff2=5936±25 K.We also find the periastron precession of 0.000 024±0.000001 daycycle-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 cannot 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 Prot=4.32days is slightly longer than the orbital period (4.3059177 days),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.

Acknowledgments

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.Guo Shou Jing 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 Kepler mission.Funding for the 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).