The nature of the eccentric doubled-lined eclipsing binary system KIC 2306740 with Kepler space photometry
D. Koçak, K. Yakut, J. Southworth, P. P. Eggleton, T. İçli, C. A. Tout, S. Bloemen
DD RAFT VERSION F EBRUARY
8, 2021Typeset using L A TEX twocolumn style in AASTeX62
The nature of the eccentric doubled-lined eclipsing binary system KIC 2306740 with
Kepler space photometry
D. K OC ¸ AK , K. Y
AKUT ,
1, 2
J. S
OUTHWORTH , P. P. E
GGLETON , T. ˙I C ¸ LI , C. A. T
OUT , AND
S. B
LOEMEN
5, 61
Department of Astronomy and Space Sciences, University of Ege, 35100, Bornova–˙Izmir, Turkey Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Astrophysics Group, Keele University, Staffordshire ST5 5BG, UK Lawrence Livermore National Laboratory, 7000 East Ave, Livermore, CA94551, USA Instituut voor Sterrenkunde, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium Department of Astrophysics, IMAPP, University of Nijmegen, PO Box 9010, 6500 GL Nijmegen,, the Netherlands (Received February 8, 2021; Revised February 8, 2021; Accepted February 8, 2021)
Submitted to ApJABSTRACTWe present a detailed study of KIC 2306740, an eccentric double-lined eclipsing binary system.
Kepler satellite data were combined with spectroscopic data obtained with the 4.2 m William Herschel Telescope(WHT). This allowed us to determine precise orbital and physical parameters of this relatively long period( P = 10 . d ) and slightly eccentric, ( e = 0 . ) binary system. The physical parameters have been determinedas M = 1 . ± . M (cid:12) , M = 1 . ± . M (cid:12) , R = 1 . ± . R (cid:12) , R = 1 . ± . R (cid:12) , L = 2 . ± . L (cid:12) , L = 1 . ± . L (cid:12) and orbital seperation a = 26 . ± . R (cid:12) through simultaneoussolutions of Kepler light curves and of the WHT radial velocity data. Binarity effects were extracted from thelight curve in order to study intrinsic variations in the residuals. Five significant and more than 100 combinationfrequencies were detected. We modeled the binary system assuming non-conservative evolution models withthe Cambridge
STARS ( TWIN ) code and we show evolutionary tracks of the components in the log L − log T plane, the log R − log M plane and the log P − age plane for both spin and orbital periods together with eccen-tricity e and log R . The model of the non-conservative processes in the code led the system to evolve to theobserved system parameters in roughly . Gyr.
Keywords: stars: evolution — stars: binaries: eclipsing — stars: binaries: spectroscopic — stars: oscillations— stars: individual: KIC 2306740 INTRODUCTIONDouble-lined eclipsing detached binary stars are an impor-tant source for accurately determining the physical parame-ters of the component stars (Torres, Andersen, & Gim´enez2010). Pulsations can be used for determining physical pa-rameters as well as understanding stellar structure. Pulsat-ing components in binary systems play an important rolein understanding stellar structure because they are effec-tively laboratories for investigating stellar interiors (Aerts2013).Therefore, having a pulsating component in a bi-nary provides an independed verification of stellar param-eters. Using continuous high precision observations from theCoRoT,
Kepler and TESS satellites’ data provide the oppor-tunity to study a variety of pulsating stars in binary systems.Recently many observational results for different type pulsat-ing stars, including those of binary components, have beenstudied in the literature (e.g. Wood, Olivier & Kawaler 2004; Yakut, Aerts & Morel 2007; Welsh, et al. 2011; Maceroni etal. 2014; Murphy, Moe, Kurtz, Bedding, Shibahashi & Bof-fin 2018; Qian, Li, He, Zhang, Zhu & Han 2018; Johnston, etal. 2019). Pulsating stars in close binary systems have beendiscussed in detail by many authors (e.g. Zahn 1975; Aerts& Harmanec 2004; Breger 2005; Reed, Brondel & Kawaler2005; Aerts, Christensen-Dalsgaard, & Kurtz 2010; Huber2015; Southworth et al. 2020).The
Kepler satellite observed more than 200,000 stars, in-cluding some with planetary companions, binary/multiplestellar systems, and pulsating stars, to obtain very high pre-cision photometry (Koch et al. 2010; Borucki et al. 2010;Gilliland et al. 2010; Brown et al. 2011).These photometricresults have found previously unknown variations, providingfurther constraints to current models. The precision of the
Kepler observations allow us to disentangle low-amplitudevariation in a binary star system. One such system is KIC2306740 which will be the focus of this work. a r X i v : . [ a s t r o - ph . S R ] F e b K OC ¸ AK ET AL . Table 1.
Basic parameters for KIC 2306740. B and V color val-ues are taken from Zacharias et al. (2005) and other parameters aretaken from the Kepler Input Catalogue, Gaia and Simbad.Parameter Value2MASS ID 19290475+3741535Gaia ID 2051885033280089216 α
19 29 04.75 δ +37 41 53.5B . m V . m R . m G (Gaia) . m J (2MASS) . m H (2MASS) . m K s (2MASS) . m K p ( Kepler ) . m E B − V . m Period 10.31 d π (mas) 0.6606 KIC 2306740 ( P = 10 . d , e = 0 . , V= . m , K p =13 . m ) is an eccentric double-lined detached eclipsing bi-nary system that was discovered by Kepler satellite. Somebasic parameters for the system given in Table 1. Thefirst preliminary binary solution of the system was foundby Prˇsa et al. (2011). They refined the orbital period as . d ± . d and temperature ratio ( T /T ) as 0.834.Kjurkchieva, Vasileva, & Atanasova (2017) subsequently es-timated the relative radii of the components as r = 0 . . and r = 0 . , mass ratio as 0.972, the orbital eccentricityas 0.299, and the argument of periastron as o . However,the parameters obtained in the current work are quite differ-ent from those two studies. This is because of the carefulinteractive analysis of the light curve (LC) and radial veloc-ity (RV) data made in this study, rather than the automatedLC modeling.In this paper we study the binary nature of the system aswell its and rotational behaviour using the Kepler data com-bined with a set of high precision RVs. KIC 2306740 wasobserved by
Kepler in quarters Q0 to Q16. The new spec-troscopic observations and data analysis of radial velocitiesare described in Section 2. We present the
Kepler data andlight curve solution of the system in Section 3. Using theradial velocity and light curve solution we obtained the phys-ical parameters of the system in Section 4. Light variationoutside eclipses is discussed in Section 5. Section 6 containsa discussion of the possible evolutionary state of the systemand our conclusions. SPECTROSCOPIC OBSERVATIONS
Table 2.
Radial velocity measurements for KIC 2306740.HJD Phase V ( O − C ) V ( O − C ) (2456000+) km s − km s − km s − km s − We obtained spectra of KIC 2306740 at 15 epochs with theIntermediate dispersion Spectrograph and Imaging System(ISIS) on the William Herschel Telescope (WHT), in July2012. These were timed to provide the best possible coverageof the orbital phases, given that the orbital period of 10.3 d issignificantly longer than the duration of the observing runof 7 nights. Two epochs occurred close to eclipse when thevelocity separation of the stars was small. These measuredradial velocities are strongly affected by line blending and sowere not used in our analysis.WHT equipped with the double-armed ISIS. Spectra weretaken simultaneously in the blue and red arms, covering theregions around H γ and H α . In the blue arm we used the grat-ing H2400B, with a wavelength coverage of 4200 to 4550 ˚A.In the red arm the R1200R grating was used and gave cover-age of 6100 to 6730 ˚A. The slit was set to 0.5 arcsec in orderto limit the effects of telescope pointing errors so that a re-solving power of R ≈
22 000 is achieved. We used exposuretimes of 1500 s for all spectra to give a signal-to-noise (S/N)of roughly 30 per resolution element in the blue and 80 inthe red. We bracketed each with spectra of CuAr+CuNe arclamps for wavelength calibration. The data were reduced us-ing the
PAMELA package (Marsh 1989).To measure the radial velocities (RVs) of the two stars fromthese spectra we used standard cross-correlation (e.g. Tonry& Davis 1979) and its two-dimensional extension
TODCOR (Zucker & Mazeh 1994). Synthetic template spectra werecalculated with the
UCLSYN code (Smith 1992; Smalley etal. 2001) and
ATLAS T eff of 5500 K and no rotational broadening. For ourfinal RVs we adopt those given by standard cross-correlation.These are very similar to those obtained with TODCOR . HE NATURE OF THE ECCENTRIC DOUBLED - LINED ECLIPSING BINARY SYSTEM
KIC 2306740 3
Table 3.
Spectroscopic orbital parameters of KIC 2306740. Thestandard errors σ are given in parentheses in the last digit quoted.Parameter e not fixed e fixed at 0.301T /d 2456399.19(24) 2456399.21(25)P/d 10.3069(78) 10.3075(81)e 0.322(22) 0.301 ω /rad 4.81(3) 4.80(3)K /km s − /km s − o /km s − m /m sin i / R (cid:12) sin i / R (cid:12) sin i / M (cid:12) sin i / M (cid:12) Figure 1.
The radial velocity observations of KIC 2306740 as afunction of phase. The filled and open circles represent the veloci-ties of the primary and the secondary component, respectively. Theresiduals are shown in the bottom panel. The data are listed in Table2 and the curve fitting corresponds to the elements given in Table 3.3.
KEPLER
OBSERVATIONS OF THE SYSTEM ANDMODELING OF THE LIGHT CURVEThe system was observed over approximately 1460 daysduring seventeen quarters (Q0 to Q16) with a long cadence(exposure time of about min) and a total of 64 370 datapoints were obtained using the satellite. The light curveshows deep eclipses with periods of totality, plus periodicvariations due to pulsations. Kepler satellite observationsshow some fluctuations due to common instrumental effects (Jenkins et al. 2010). Using the techniques outlined in (Jenk-ins et al. 2010), cotrending and detrending were applied toeliminate systematic variations. We studied each quarter sep-arately and, to de-trend the data, a third-order polynomial fitwas applied as we did in our earlier
Kepler study (Yakut etal. 2015; C¸ okluk, et al. 2019). The raw data of KIC 2306740is shown in Fig. 2 (upper panel). The de-trended normalizedlight variation is shown in Fig. 2 (lower panel). The quartersare shown in different colors.Using the
Kepler observations we derived the linearephemeris given in Eq.1.HJD Min I = 24 56399 d . d . . (1)During the calculation of the orbital phases in the Figures 2 -3 and Table 4 we used Eq.1.The Kepler light curves along with the WHT RV curveswere modeled simultaneously with the
JKTEBOP code (seeSouthworth, Maxted & Smalley 2004; Southworth 2013) andalso with the P HOEBE (Prˇsa & Zwitter 2005) program whichuses the W–D code (Wilson and Devinney 1971). The curvedependent weights were assigned as described by Wilson(1979). We ran the code assuming a detached configura-tion. During analysis we iteratively solved the LC and RVcurves: the LC gave a more accurate estimate of the eccen-tricity ( . ± . ) than did the RV curves ( . ± . ), andso we fixed e to this photometric solution when re-solving theRV curves. Even though Kepler data are very sensitive theyare all obtained in a single filter. This prevents us from deter-mining accurate temperatures from multiple color analyses.Since the spectral data obtain is not sufficient to determinea precise temperature, the temperature of the hotter star wasfixed to 6060 K found in Armstrong et al. (2014).To determine the uncertainties of the measured propertiesof the KIC 2306740 system, we turned to
JKTEBOP as itis much faster than
PHOEBE . JKTEBOP and derivatives ofthe Wilson-Devinney code have been found to yield resultsconsistent at the 0.2% level or better in well-separated sys-tems (Maxted, et al. 2020). For computational efficiency wephase-binned the data by sorting it into orbital phase andcombining each group of 20 consecutive datapoints, givinga total of 3219 datapoints.The phase-binned data and the radial velocities were mod-eled using
JKTEBOP . The fitted parameters were the frac-tional radii, orbital eccentricity, inclination, argument of pe-riastron, central surface brightness ratio of the stars, thirdlight, the linear limb darkening coefficient of the primarystar, and the velocity amplitude of each star and systemicvelocity of the system. Limb darkening was implementedusing the quadratic law for both stars. Numerical integra- K OC ¸ AK ET AL . Figure 2.
Kepler Q0 to Q16 observations of KIC 2306740 raw data(upper panel) and de-trended (lower panel) data. The quarters areshown in different colors. tion was used to account for the fact that the data were ob-tained in long cadence by the
Kepler satellite (see South-worth 2011). The fractional radii were fitted using their sumand ratio as these are less correlated. The orbital eccentric-ity e and argument of periastron ω , were fitted using thePoincar´e parameters e cos ω and e sin ω , for the same rea-son. Uncertainties were calculated for each fitted parameterand for each derived parameter in this study. This was donein two ways: using Monte Carlo and residual-permutationsimulations (Southworth 2008). The uncertainties from theresidual-permutation algorithm were found to be larger bytypically a factor of 1.5 than those from the Monte Carlo al-gorithm, so were adopted as the final errorbars.Simultaneous LC and RV solutions were made using thefull Q0–Q16 data and the analyses are summarized in Ta-ble 4. In Fig. 3 the computed light curves are shown by solidlines. Prˇsa et al. (2011) gave preliminary orbital parame-ters for 1879 Kepler binary systems, including KIC 2306740.They obtained a temperature ratio of 0.86, a sum of the frac-tional radii of 0.1374 and a sin i of 0.99919. Our analysisincludes RVs as well as much more extensive LCs, so the re-sults given in Table 4 differ from those found by Prˇsa et al.(2011). PHYSICAL PARAMETERS OF THE COMPONENTSThe physical parameters of a binary system can best bederived if it is a double-lined eclipsing binary system withan accurate light curve. Hence, the detached binary systemKIC 2306740, for which the photometric and spectroscopic
Table 4.
Fitted and parameters for KIC 2306740 from the
JKTEBOP analysis.Parameter Value
Fitted parameters:
Sum of the fractional radii 0.11102 ± ± ± ◦ ) 89.670 ± e cos ω ± e sin ω − ± ± − ) 64.00 ± − ) 70.87 ± − ) 18.70 ± − ) 18.57 ± Derived parameters:
Fractional radius of cool component 0.06421 ± ± ± e ± ω ( ◦ ) 274.44 ± ± Table 5.
Astrophysical parameters of KIC 2306740. The standard σ errors of the last digits are given in parentheses.Parameter Cool Component Hot ComponentMass ( M/M (cid:12) ) 1.194(8) 1.078(7)Radius (
R/R (cid:12) ) 1.682(4) 1.226(5)Temperature log ( T eff /K ) 3.764(18) 3.782(17)Luminosity log ( L/L (cid:12) ) 0.449(75) 0.260(72)Surface gravity log ( g/ cm s − ) 4.063(2) 4.294(4)Bolometric magnitude (M b ) 3.61(19) 4.08(18)Absolute magnitude (M V ) 3.69 4.09Semi-major axis ( a/R (cid:12) ) 26.201(44) data are both of high precision, is excellent for accurate de-termination of its parameters. The detailed LC solution of thesystem indicates that the stars are well detached from theirRoche lobes (see Section 3).In this study, RVs and LCs were analysed simultaneouslyand the orbital parameters of the system were obtained. Withthe measurements given in Tables 3 and 4 we can estimatethe physical parameters of the components given in Table 5. HE NATURE OF THE ECCENTRIC DOUBLED - LINED ECLIPSING BINARY SYSTEM
KIC 2306740 5
Figure 3. (a)
Kepler full-data set observation (blue dots) and com-puted (red line) light curve of the system. Zoomed secondary (b)and primary (c) minima are shown to emphasize the agreement andresiduals between the observed values and the corresponding LCmodel (d).
We have used the
JKTABSDIM code to estimate the physicalparameters of the components, with uncertainties propagatedfrom the LC and RV solution using a perturbation analysis.The nominal physical constants and solar properties recom-mended by the IAU were used (Prˇsa, et al. 2016). All thecalculated parameters of the binary system are summarizedin Table 5 with their estimated errors.The masses of the two stars are slightly greater than solarand the radii are significantly larger. The hotter star is the Table 6.
Computed genuine frequencies, amplitudes and phaseshifts of the solution. Frequencies with signal-to-noise ratios (
S/N )exceeding 4 are considered as significant.
Frequency Amplitude Phase S/N/d − /mmag φ less massive and has the smaller radius. This indicates thatthe larger star is near the end of its main-sequence lifetime.We discuss this in Section 6. LIGHT VARIATION OUTSIDE ECLIPSEInvestigating of the sinusoidal brightness variation in theLC requires the extraction of the effects of binarity fromthe LC. After making the simultaneous LC and RV analy-sis, we subtracted the binary model from the observations.The residuals from the phased light variation of the binaryare plotted in the Fig. 5. The oscillatory pattern can be seenclearly.Since programs using modern light curve modeling werenot perfect at representing
Kepler data, we performed a fre-quency analysis on all the long cadence data obtained out-of-eclipse using the S IG S PEC (Reegen 2007) and P
ERIOD
S/N > thresholdwas chosen as a criterion to consider a frequency as signifi-cant (Breger et al. 2011). We searched for significant peaksin the frequency interval from 0 to the Nyquist frequency of
25 d − but found no meaningful peak above − . Fig. 4ashows the amplitude spectrum before pre-whitening of anyfrequency between 0 and
25 d − . Higher amplitude peaksgather below a frequency of − . We continued to obtainpre-whitened frequencies until the signal amplitudes fell be-low four times the average noise. Fig. 4b represents the spec-tra after pre–whitening.Table 6 lists these genuine frequencies, their amplitudes,phases and S/Ns sorted by decreasing amplitude. Signal-to-noise ratios were computed over an interval of − . In thetop panel of Fig. 5 the agreement between 100 calculatedfrequencies and the observational data is plotted for almost4 yr. The bottom panel illustrates the zoomed data of 10 d forclarity. The analysis resulted in the detection of five genuineand more than 100 combination frequencies.What could be the mechanism that caused a change in themaximum amplitude of the KIC 2306740 system? Gener-ally, such changes may result from stellar pulsating and/orperiodic changes. In binary star systems, inhomogeneousstructures (e.g. stellar spots) on the surface of one or both K OC ¸ AK ET AL . Figure 4.
Amplitude A spectrum of the components before (a) andafter (b) pre-whitening of all frequencies ν . of the component stars can cause changes in the light curvein asynchronous situations, known as rotational variability.For the frequencies obtained from the Fourier analysis of thelight variation of the KIC 2306740 system (Table 6), peri-ods of approximately 0.7 d, 3.4 d and 10.3 d were obtained.The 10.3 d period is related to the orbital period and the 3.4 dperiod is related to the spin period of the stars (see Section6). The source of the 0.7 d periodicity may be γ Dor-typepulsation or spot modulations on one or both of the compo-nent(s). Besides, looking into out-of-eclipse of light variationof the system we analyzed minima phases of the light curves.There is a variation with an amplitude of 0.03 at the primaryminimum and a variation with an amplitude of 0.013 at thesecondary minima. However, residual data is not sufficientto estimate new frequencies. RESULTS AND CONCLUSIONWe have modeled the light and radial velocity curvesof the well-detached binary system KIC 2306740 and de-termined its orbital and physical parameters (Tables 4 and5). Solutions indicate that the more massive and coolercomponent of the system is more evolved than the hotterless massive component. Recently, Gaia gave a parallax of0.6606 ± m V − M V = 11 . m and magnitudes for the componentsand the system of V = 14 . m ± . , V = 15 . m ± . and V total = 14 . m ± . . The new stellar parameters andreddening, reveal the distance of the system to be 1.53 kpc,which is very close to the distance obtained by Gaia as 1.51kpc. With these results we can add the system to the listof well-determined binary stars. The maximum light showscyclical variations. Figure 5.
Part of data used in the frequency analysis (upper panel)is zoomed for different time intervals (lower panel).
Out-of-eclipse light variations have been obtained aftereliminating the effects of binarity for the established orbit. Afrequency analysis of these data revealed more than 100 fre-quencies of which 5 are significant non-combination frequen-cies. The results from this study indicate that the systemKIC 2306740 may contains a non-radial pulsating γ Dor typestar which pulsates in high-order gravity-modes. Consider-ing the other frequencies ( ∼ . d, half of the spin periods)obtained, we have found that this is related to the predictedspin period of ∼ . days (see Fig. 6). HE NATURE OF THE ECCENTRIC DOUBLED - LINED ECLIPSING BINARY SYSTEM
KIC 2306740 7we have found that the component stars are related to thespin periods ( ∼ . d) obtained (see Fig. 6c). This tells usthat at least one of the components has changed due to inho-mogeneous structures on its surface. This also tells us thatspot modulation is a possible explanation for the variation atthe maxima phases, instead of pulsations. Using the parame-ters we have obtained ( Table 5) show that the components ofbinary system KIC 2306740 is outside or on the edge of theinstability zone in the HR diagram. Therefore, the light vari-ation seen in this system is likely caused by spot modulation.We use this system to test the theory of modeling stel-lar interiors by comparing with its the observational prop-erties. Models were constructed with the EV code (Eggle-ton & Kiseleva-Eggleton 2002) and its much more power-ful TWIN variant (Yakut & Eggleton 2005; Eggleton 2006;Eggleton 2010), both of which are based on the CambridgeSTARS code (Eggleton 1971, 1972, 1973; Pols et al. 1995).In single-star evolution, as in the STARS code, the effectsof rotation and magnetic dynamo activity on mass loss andthe proximity of the companion are usually not considered.
TWIN allows various non-conservative processes to be ap-plied to the primary component of a binary system.
Both components are modeled simultaneously so that the effectsof tidal friction, magnetic dynamo activity, and hence massloss, can be incuded according to a self-consistent prescrip-tion. Mass loss carries off angular momentum by way ofmagnetic braking.In the case of non-conservative evolution it is hard to findinitial parameters that would lead to the current binary sys-tem (see Eggleton & Yakut 2017). However we can reason-ably assume that the initial masses were larger than now. Ta-ble 7 gives parameters for a model that is slightly metal-rich( Z = 0 . ) compared to the Sun. This metallicity gives asomewhat better fit than solar.After some experimentation, we evolved a pair of starswith initial masses of 1.21 and 1.08 M (cid:12) , each with spin pe-riods of 3.0 d, an eccentricity of 0.34 and an orbital periodof 10.30 d. We expect the orbital period to decrease as theorbit circularizes and as angular momentum is lost by windmass loss and magnetic braking. However the period also in-creases on account of the orbit acquiring some angular mo-mentum from the spins of the stars by tides. The particularmodel of the non-conservative processes in the TWIN codeled the system to evolve to roughly the observed masses andradii in about . Gyr.Fig. 6 shows evolutionary tracks in three planes, (a) the log L − log T plane or theoretical HR diagram, (b) the log R − log M plane and (c) the log P − age plane for both spinand orbital periods periods together with eccentricity e and log R . Our data for KIC 2306740 are represented by squaresin panels (a) and (b). Each square is surrounded by a cloud ofplusses generated by a Gaussian random number generator to Figure 6.
Evolutionary tracks for our best model of KIC 2306740.Our data for KIC 2306740 are shown as squares. Panel (a) shows theevolution in the log L − log T plane. The primary’s and secondary’stracks are shown with red and green lines, respectively. The extentof the red and green pluses represent roughly the observational un-certainties. Panel (b) shows the primary’s radii (red) and lobe radii(dark blue) and secondary’s radii (green) and lobe-radii (light blue)as a function of mass. Panel (c) shows the time-evolution of or-bital period (dark blue), spin period of the primary (red) and thesecondary (green), eccentricity (light blue) and radius of primary(black). K OC ¸ AK ET AL . Table 7.
An Evolutionary Model for KIC 2306740.Parameter Zero age Age 5.12 Gyr Observed P/ d e M /M (cid:12) log ( R /R (cid:12) ) log ( L /L (cid:12) ) log ( T / K) M /M (cid:12) log ( R /R (cid:12) ) –0.013 0.071 0.088 log ( L /L (cid:12) ) –0.005 0.194 0.260 log ( T / K) illustrate the extent of the standard errors in the basic obser-vational data ( K , K , T , . . . ). The evolutionary tracks forthe more massive star (*1) are red and for less massive star(*2) green. A blue circle approximately marks the best fit to ∗ and a blue asterisk the coeval point for ∗ . Panel (b) showsthe Roche lobe radii as well as the actual radii. Neither star iswithin a factor of 5 of its Roche radius. Panel (c) shows thespin periods in red and green. Both were started arbitrarilyat d and rather rapidly evolved to about d before reachinga plateau. During this fairly rapid initial spin-down the com-ponents lost about . and . M (cid:12) of their masses. Thelogarithm of the orbital period is dark blue, the eccentrcity is light blue and the radius of star 1 is black. Table 7 showsthe evolutionary changes in some major variables. Our over-all conclusion from Fig. 6 is that the fit of to the theoreticalmodel is acceptable at a 1 σ level but it would be better if themodel temperatures matched more closely those observed.We are very grateful to an anonymous referee for com-ments and helpful, constructive suggestions, which helpedus to improve the paper. The authors gratefully acknowledgethe numerous people who have helped the NASA Kepler mis-sion possible. This study was supported by the Turkish Sci-entific and Research Council (T ¨UB˙ITAK 117F188). DK isgrateful to the Astronomy Department of the University ofGeneva (Geneva Observatory) for the kind hospitality dur-ing her visit and gratefully acknowledge the support providedby the T ¨UB˙ITAK-B˙IDEB 2211-C and 2214-A scholarships.CAT thanks Churchill College for his Fellowship. KY wouldlike to acknowledge the contribution of COST (European Co-operation in Science and Technology) Action CA15117 andCA16104.
Facilities:
Kepler, William Herschel Telescope (WHT)
Software:
TODCOR (Zucker & Mazeh 1994) ,
PAMELA (Marsh 1989), S IG S PEC (Reegen 2007),
PERIOD
04 (Lenz& Breger 2005),
TWIN (Yakut & Eggleton 2005; Eggleton2006; Eggleton 2010), C
AMBRIDGE
STARS C
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HE NATURE OF THE ECCENTRIC DOUBLED - LINED ECLIPSING BINARY SYSTEM