2.1. Spectroscopy

Medium resolution optical spectra of KIC4832197 were obtained at TUB¨ ITAK National˙ Observatory (TNO) with the 1.5 m Russian-Turkish telescope and the Turkish Faint Object Spectrograph Camera (TFOSC¹). An Andor DW436-BV 2048 × 2048 pixels CCD camera with a pixel size of 13.5 × 13.5 µm ² was used, which allows recording optical spectra between 3900 - 9100 Å in 11 ´echelle orders. This instrumental set-up provided an average resolution of R = λ/∆λ = 2700 ± 500 around the λ = 5500 Å wavelength region. Observations were carried out on eight nights between July 2014 and April 2017. Ten spectra were recorded in total.

Conventional procedures for reducing ´echelle spectrum images were followed. These steps are applied under IRAF environment² and start with the removal of bias level and continue by division of biasremoved object and calibration lamp images by a normalized flat-field image. Then, scattered light correction and cosmic rays removal steps are applied to the bias and flat-field corrected images. Finally, object and calibration lamp spectra are extracted from the ´echelle orders. Wavelength calibration of the extracted object spectra is done by using a Fe-Ar calibration lamp spectrum recorded on the same observing night. After completing standard reduction steps, all object spectra were normalized to unity by applying 4th or 5th order cubic spline functions.

2.2. Kepler Photometry

Very high precision space photometry of KIC4832197 was obtained by Kepler spacecraft. The spacecraft recorded images in 6.02 second exposure times with 0.52 second read-out times in a broad wavelength range between 4100Å and 9100Å (^{Gilliland et al. 2010}). The broad wavelength range allows to record more photons in a single exposure, which increases precision, at the expense of losing colour information. From the recorded images, two data sets were created depending on two separate integration times; 58.9 second (short cadence data) and 29.4 minute (long cadence data). These integrations were grouped as separate data sets, where each set covered approximately three months, called quarters, except the first quarter, which covered ten days of commissioning phase and is called quarter zero (Q0). During operation of the spacecraft over 4 years (between 2009 and 2013) photometric data were collected for eighteen quarters in total (from Q0 to Q17). Continuous long cadence photometry obtained in each quarter is available for the majority of Kepler targets, including KIC4832197. Long cadence photometry of KIC4832197 for each quarter is obtained from Mikulski Archive for Space Telescopes (MAST). Prior to analyses, it is necessary to remove instrumental effects from the long cadence data. For each quarter, simple aperture photometry (SAP) fluxes are considered. These fluxes are de-trended as described in ^{Slawson et al. (2011)} and ^{Prša et al. (2011)}. De-trended fluxes are then normalized to unity. The analysis and modelling process is based on these normalized fluxes. We show the long cadence light curve of KIC4832197 in Figure 1.

Fig. 1 In panel a the long cadence light curve of KIC4832197 is shown. In panels b and c, different portions of the long cadence light curve are shown, where the shape of the light curve remarkably changes. 

3.1. Light Time Variations

The first step of the analysis is to determine mideclipse times of KIC4832197 for primary eclipses. In principle, mid-eclipse times can be determined straightforwardly by applying the Kwee - van Woerden method (^{Kwee & van Woerden 1956}) to observations close by to mid-eclipse time. An alternative way is to fit a function to the observational data around the estimated mid-eclipse time and determine the extremum point of this function, which corresponds to the mid-eclipse time. These methods may work flawlessly for a light curve of an ordinary eclipsing binary, which shows symmetric eclipse light curves with respect to the mid-eclipse time and exhibits flat or slightly distorted maxima at out-ofeclipse phases. Such light curves indicate the absence of intrinsic variability for any of the component. Looking at Figure 1, one may easily notice the variable nature of the light curve. Because of the relatively short orbital period and integration time of long cadence data, a few data points can be found around the expected mid-eclipse time for a given orbital cycle. Combination of these two effects gives a complex shape to the light curve and makes the methods mentioned above unsuited for the precise determination of mid-eclipse times. In such cases, one of the most reliable ways is to find a best-fitting light curve model and then only adjust the mid-eclipse time of the primary minimum in the model for each cycle. We follow exactly this way for a precise determination of mid-eclipse times from long cadence data.

Actually, this is an iterated process, which starts by preparing a phase-folded light curve with initial light elements, i.e. an ephemeris reference time (T ₀) and an orbital period (P), then continues by finding the best-fitting light curve model. In the case of KIC4832197, initial T ₀ and P values are adopted from the Kepler Eclipsing Binary Catalogue³ (^{Prša et al. 2011}; ^{Slawson et al. 2011}) given in equation 1:

After finding the best-fitting light curve model, T ₀ is adjusted for each orbital cycle separately, by keeping all other model parameters fixed. In this step, time-based long cadence data are considered, instead of phase-folded light curve. Thus, the mideclipse time of each orbital cycle can be determined precisely together with formal errors. Practical application of this method is done with the Wilson-Devinney code (see later sections for details). After determination of mid-eclipse times, a simple eclipse time variation (etv) diagram is constructed and linear corrections are determined and applied to both T ₀ and P. Then the whole process is repeated by using the corrected T ₀ and P. In most cases, two iterations are enough to obtain self consistent T ₀, P, and light curve model parameters. The converged solution leads to the corrected light elements given in equation 2:

The numbers in parentheses are statistical uncertainties for the last digit of the corresponding parameter. These uncertainties are computed from a linear least squares fit. T ₀ and P values given in equation 2 are adopted for further analyses and kept fixed. The resulting eclipse time variation diagram is shown in Figure 2. In the figure, an irregular undulating variation is noticeable among the scatter, with an approximate amplitude of 0^d .003. Since that pattern does not repeat itself strictly over four years, it is not likely to be due to a possible third body. A more likely explanation of this pattern might be the out-of-eclipse variability of the system, which can be noticed in the b and c panels of Figure 1.

Fig. 2 ETV diagram for KIC4832197. 

3.2. Radial Velocities and Spectroscopic Orbit

The next step of our analysis is to determine radial velocities of each component from each observed spectrum and to model the spectroscopic orbit of the system. In order to determine radial velocities of the components, we use the optical spectrum of HD184499 (T _eff = 5743, log g = 4.07; ^{Prugniel et al. 2011}) as a template spectrum, which was recorded with the same instrumental set-up on 20th August, 2014. Then, each observed spectrum of KIC4832197 was cross-correlated with the spectrum of HD184499 by following the method proposed by ^{Tonry & Davis (1979)}. The practical application of this method used the fxcor task (Fitzpatrick 1993) under IRAF environment. All clear absorption lines (except strongly blended or very broad spectral lines) were considered in the 5th and 6^th échelle orders, which cover a wavelength range between 4900 Å and 5700 Å. In this wavelength range, it was possible to detect strong signals of both components. We show cross-correlation functions of two spectra of KIC4832197, recorded at two separate orbital quadratures, in Figure 3. The heliocentric radial velocities are tabulated in Table 1.

Fig. 3 Cross-correlation functions of two observed spectra recorded around orbital quadratures. P and S denote the primary and the secondary component, ϕ the orbital phase. The colour figure can be viewed online. 

Preliminary inspection of the phase-folded light curve shows that the mid-primary and the midsecondary eclipses precisely occur at 0.0 and 0.5 orbital phases, respectively. This finding strongly suggests a circular orbit. Therefore, the spectroscopic orbit of the system is determined assuming zero eccentricity (i.e. e = 0). In this case, the longitude of the periastron (ω) is undefined; thus the T ₀ value found in the light time variation analysis step is adopted instead of the periastron passage time. T ₀ and P values are kept fixed during the modelling, while radial velocity semi-amplitudes of the components (K ₁ and K ₂), center-of-mass velocity of the system (V _γ ) are adjusted. Application of a linear least squares fitting method to the observed radial velocities results in the best-fitting spectroscopic orbit parameters tabulated in Table 2. The agreement between the observed radial velocities and their bestfitting model is shown in Figure 4.

Fig. 4 Phase-folded radial velocities of the primary (blue) and the secondary (red) components are shown as filled circles. The best-fitting spectroscopic orbit model is over-plotted for each component as a continuous line. The colour figure can be viewed online. 

3.3. Spectral Type

Medium resolution TFOSC spectra allow us to determine spectral types and global atmospheric parameters for each component of KIC4832197. A spectrum recorded around any of the orbital quadrature is suitable for the atmospheric parameter determination of individual components, since the spectral lines of both components are often sufficiently separated from each other and can be distinguished at those orbital phases. This is observed in the TFOSC spectrum recorded on 30th July 2016, corresponding to 0.7130 orbital phase (see lower panel of Figure 3). Spectral types and atmospheric parameters are determined from this spectrum.

Determination of spectral types and atmospheric parameters is an iterated process, as in the case of determination of mid-eclipse times described in § 3.1. During the analysis, we fix the micro-turbulence velocity of each component at 2 kms⁻¹. Actually, analysing high resolution optical spectra it is possible to determine the micro-turbulence velocity and the [Fe/H] abundance simultaneously, via the Blackwell diagram (^{Blackwell & Shallis 1979}). However, due to the insufficient resolution of TFOSC spectra, we were unable to do so. Instead, we implicitly assume a 2 km/s micro-turbulence velocity and fix this value during the analysis. Although there is no strict relation in the literature to estimate micro-turbulence velocities reliably, limited observational studies indicate that 2 kms⁻¹is appropriate for solar type stars (^{Landstreet et al. 2009}).

It is possible to apply a constraint on the logarithm of gravity (log g) of the components. To do this, spectroscopic orbit model parameters are combined with preliminary light curve solution parameters. This step allows computation of the masses and radii of the components. Then, log g values are computed via the computed masses and radii of the components. Computed log g values are fixed in the spectral type analysis and effective temperatures of the components (T _eff1 and T _eff2 ) are estimated together with overall metallicity ([Fe/H]). Then, the estimated temperature of the primary component and the overall metallicity are adopted as fixed parameters, and combined light curve and radial velocity modelling is repeated. Two or three iterations are usually enough to reach a self-consistent solution.

Effective temperatures and overall metallicity are estimated by the spectrum synthesis method. ATLAS9 (^{Castelli & Kurucz 2004}) model atmospheres, which adopt a plane-parallel atmosphere, are used for synthetic spectrum computation. A grid of synthetic spectra is computed for a temperature range of 6000 K to 7000 K with a step of 100 K. This computation is repeated for metallicities between solar ([Fe/H]=0) and sub-solar ([Fe/H]=−1.0), with a step of 0.25. The computation of synthetic spectra is done by a Python framework, iSpec software (^{Blanco-Cuaresma et al. 2014}). Among various radiative transfer codes provided in iSpec, the Spectrum code (^{Gray & Corbally 1994}) is adopted. iSpec also includes a comprehensive line list compiled from the third version of the Vienna atomic line database (VALD3, ^{Ryabchikova et al. 2015}). Each computed synthetic spectrum is convolved with a Gaussian line spread function to reduce the spectral resolution to the resolution of the TFOSC spectra. After that, a trial synthetic spectrum is chosen for each component among the computed spectra, and a composite spectrum of the system is computed. In order to compute the composite spectrum of the system, each individual spectrum is shifted in wavelength with respect to the radial velocity of the corresponding component and scaled with respect to the square of the ratio of the radii of the components.

Self-consistent effective temperatures and overall metallicities are determined in the third iteration, which gives T _eff1 = 6300 K, T _eff2 = 6100 K and [Fe/H] = −0.25. The estimated uncertainties are around 200 K for temperature and 0.25 for metallicity. The adopted ratio of radii is R ₁ /R ₂ = 1.27 and log g values are log g ₁ = 4.30 and log g ₂ = 4.47 (see next section for details). These results indicate F7V and F9V spectral types for the primary and the secondary components (^{Gray 2005}), respectively. The observed TFOSC spectrum and the best-fitting composite spectrum are shown in Figure 5.

Fig. 5 Observed (blue) and computed composite spectrum (red) for KIC 4832197 in three different portions of the optical wavelengths. Residuals from the best-fitting model are plotted as shifted upwards by 0.5 (blue) for a better viewing purpose. The colour figure can be viewed online. 

3.4. Light Curve Analysis and Evolutionary Status

Long cadence photometric data of KIC4832197 include 65190 individual data points in total, which means that a large amount of CPU time is required to find a best-fitting light curve model. Therefore, an average phase-folded light curve is prepared by computing an average in every 0.01 phase at out-of eclipse phases. Around mid-eclipse phases a phase step of 0.0005 is adopted in order to detect the ingress and egress phases of eclipses precisely. The average phase-folded light curve includes 556 data points in total, which enormously reduces the required CPU time for light curve modelling. Weighting of each normal data point in the phase-folded average light curve is done by considering the total number of data points, which produces the normal data point.

Light curve modelling is done with the 2015 version of the well-known Wilson-Devinney (WD) eclipsing binary light curve modelling code (^{Wilson & Devinney 1971}; ^{Wilson & Van Hamme 2014}). Its practical application is done by the user friendly Python GUI PyWD2015 (^{Güzel & Özdarcan 2020}). PyWD2015 allows the use of almost all features of the 2015 version of the WD code. In addition to the capabilities of the WD code, PyWD2015 includes many useful features and tools to speed up the modelling process and to trace successive iterations visually.

Two most critical parameters for an accurate light curve modelling are T _eff1 and q. These have already been determined in previous sections, and are kept fixed during modelling. Effective temperatures of the components clearly indicate that both components possess a convective outer envelope; thus, gravity darkening (g) and albedo (A) values of each component are set to 0.32 (^{Lucy 1967}) and 0.5 (^{Ruciński 1969}), respectively. The low resolution of TFOSC spectra does not allow to determine reliable rotational velocities of the components; so, considering the non-eccentric orbit of the system, synchronous rotation is implicitly assumed for both components by fixing the rotation parameter (F) of each component to unity. A logarithmic limb darkening law (^{Klinglesmith & Sobieski 1970}) is adopted for both components and limb darkening coefficients are taken from ^{van Hamme (1993)} for the Kepler passband. Adjustable parameters are: inclination of the orbit (i), effective temperature of the secondary component (T _eff2 ), dimensionless potentials of the components (Ω₁, Ω₂) and luminosity of the primary component (L ₁). Coarse and fine grid numbers for the stellar surface are set to 60. Modelling attempts with a possible third light contribution do not yield a reasonable model, which indicates extremely small or negative third light contribution. Therefore, the modelling process is carried out with no third light contribution. Convergence is not very quick due to the shallow eclipse depths. However, successive iterations converge slowly but firmly to a global minimum. Model parameters for the best-fitting light curve are tabulated in Table 3. In Figure 6, the phase-folded light curve and and the best-fitting light curve model are shown. A separate close-up view of each eclipse is shown in Figure 7.

Fig. 6 Phase-folded long cadence Kepler observations and the best-fitting light curve model of KIC4832197 are shown along with residuals from the model. The colour figure can be viewed online. 

Fig. 7 Close-up view of observations and the best-fitting light curve model around eclipse phases. The colour figure can be viewed online. 

Combination of the best-fitting spectroscopic orbit and light curve model parameters allows us to compute the absolute dimension of the system as well as its distance. Both the UBV colours (^{Everett et al. 2012}) and the optical spectrum of the system indicate an F7 spectral type, which means that reddening and interstellar extinction should be negligible. Trial plotting of the system on the UBV colour-colour diagram gives E(B − V ) = 0^m . 009, which is well below the observational error of B − V . Hence might be ignored for the distance computation. Considering P, K ₁, K ₂, i, ⟨r ₁⟩ and ⟨r ₂⟩ along with the published UBV magnitudes, the absolute dimension of the system is computed with the JKTABSDIM code⁴ (^{Southworth et al. 2005}) and tabulated in Table 4. Calibrations given in ^{Kervella et al. (2004)} are adopted for the bolometric correction and distance computation.

Comparing the locations of the components with stellar evolutionary tracks on the Log T _eff − Log L/L _⊙ plane, it is possible to estimate the age of the system. In Figure 8, the components of KIC4832197 are plotted along with stellar evolutionary tracks computed with the PAdova and TRieste Stellar Evolution Code (PARSEC, ^{Bressan et al. 2012}). The best-matching isochrone to the positions of the components has log(age)=9.45, which indicates an age of 2.8 Gyr for the system. The estimated uncertainty for this age is approximately 0.8 Gyr. The primary source of this uncertainty is the uncertainty of the estimated effective temperatures and its propagation to the computed luminosities. Both components are still on the main sequence. However, the primary component is almost half-way through its main sequence life.

Fig. 8 Positions of the primary and the secondary components on log T _eff −log L/L _⊙ plane. Dashed lines show evolutionary tracks for different masses. Each track is labelled with its corresponding mass. The continuous curve shows the isochrone for log(age)=9.45. The tracks are for a slightly sub-solar metallicity with Y = 0.273 and Z = 0.014. The colour figure can be viewed online. 

3.5. Out-of-Eclipse Variability

A scatter of ±0.01 is clearly noticeable in the phase-folded light curve (see Figure 7). This scatter is a natural result of dominant brightness variability at out-of-eclipse phases (see Figure 1, lower panels). For further investigation of this variability, the best-fitting light curve model is subtracted from observations, and residuals from the model are obtained. In principle, these residuals do not include variability due to eclipses nor proximity effects of the components, and give hints on intrinsic variability of one or both components. Residuals from the best-fitting light curve model are obtained with PyWD2015 by inputting all long cadence data into the WD code and making a single differential corrections iteration. The obtained residuals are plotted in Figure 9. Quick changes in the residual light curve shape and amplitude are remarkable.

Fig. 9 Residuals from the best-fitting light curve model are plotted against time (BJD). Panel a shows the whole long cadence residuals, while Panels b and c show different portions of the whole set. 

In order to see the reflection of this variability in the frequency-amplitude plane, a Lomb-Scargle periodogram (^{Lomb 1976}; ^{Scargle 1982}) is applied to the whole long cadence residuals via a Python script pysca (^{Herzberg & Glogowski 2014}). pysca allows quick and practical computation of the amplitude spectrum and can extract significant frequencies above a defined noise level. Adopting a 24.498 cycle/day (c/d) as Nyquist frequency, a single pysca run results in the amplitude spectrum given in Figure 10. The adopted Nyquist frequency corresponds to twice (≈ 59 min) the integration time of a single long cadence exposure. In the resulting amplitude spectrum, two dominant peaks are clearly seen around the orbital frequency, but no signal appears at the precise location of the orbital frequency. Frequency values of the two dominant peaks are tabulated in Table 5, together with the orbital frequency.

Fig. 10 Amplitude spectrum of light curve residuals. The uppermost panel shows the whole amplitude spectrum between 0 and 24.498 c/d. The middle panel focuses on 0 and 2 c/d, where dominant peaks appear. The lowermost panel shows a detailed view of the two most dominant peaks and the orbital frequency. In each panel, the orbital frequency is marked with a vertical dashed (red) line. The colour figure can be viewed online. 

It is also possible to trace the behaviour of the detected frequency peaks for each Kepler quarter separately. A close view of the f ₁ and f ₂ peaks is shown in Figure 11 for each Kepler quarter, separately. It is apparent that f ₁ strongly appears in most of quarters while f ₂ weakens or disappears in some quarters.

Fig. 11 Close view of the amplitude the spectrum of light curve residuals for each quarter of Kepler data. Panels are focused on the frequency region around f _orb , which is shown by a vertical (red) dashed line. Each panel is labelled with its corresponding quarter in the upper left corner. The colour figure can be viewed online. 

The corresponding frequencies of the first two dominant peaks in the amplitude spectrum give the phase folded residual light curves shown in Figure 12. Phase folding is done with respect to the reference ephemeris given in equation 2 and the corresponding period values of the frequencies listed in Table 5. A wave-like light curve pattern can be noticed easily in both panels of the figure. This picture appears to be the reflection of a double-humped structure of the light curve (see Figure 1, Panels b and c), which is the frequently observed through four years of long cadence data. Estimated spectral types of the components in § 3.3 suggest that these f ₁ and f ₂ frequencies are likely the result of cool spot activity on one or both components.

Fig. 12 Phase folded light curve residuals with respect to f ₁ and f ₂ frequencies. Each frequency is shown in the corresponding panel.