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Revista mexicana de astronomía y astrofísica

versión impresa ISSN 0185-1101

Rev. mex. astron. astrofis vol.54 no.1 Ciudad de México abr. 2018

 

Artículos

KIC 9451096: Magnetic activity, flares and differential rotation

O. Özdarcan1 

E. Yoldaş1 

H. A. Dal1 

1Ege University, Science Faculty, Department of Astronomy and Space Sciences, Izmir, Turkey.


Abstract

We present a spectroscopic and photometric analysis of KIC9451096. The combined spectroscopic and photometric modelling shows that the system is a detached eclipsing binary in a circular orbit and composed of F5V + K2V components. Subtracting the best-fitting light curve model from the whole long cadence data reveals additional low (mmag) amplitude light variations in time and occasional flares, suggesting a low, but still remarkable level of magnetic spot activity on the K2V component. Analyzing the rotational modulation of the light curve residuals enables us to estimate the differential rotation coefficient of the K2V component as k = 0.069 ± 0.008, which is 3 times weaker compared with the solar value of k = 0.19, assuming a solar type differential rotation. We find the stellar flare activity frequency for the K2V component as 0.000368411 h-1 indicating a low magnetic activity level.

Key Words: binaries: eclipsing; stars: activity; stars: flare; stars: fundamental parameters; stars: individual: KIC9451096

Resumen

Presentamos un análisis espectroscópico y fotométrico de KIC9451096. El modelo combinado muestra que el sistema es una binaria eclipsante separada, en órbita circular y compuesta de dos estrellas, F5V + K2V. Al sustraer la mejor curva de luz modelada de la cadencia completa de datos se revelan pequeñas variaciones en magnitud (mmag) así como ráfagas ocasionales, lo que sugiere una baja pero notable actividad de manchas magnéticas en la componente K2V. El análisis de la modulación rotacional de la curva de luz nos permite estimar que el coeficiente de la rotación diferencial de la componente K2V es k = 0.069 ± 0.008, tres veces más débil que l el valor solar, k = 0.19. Encontramos que la frecuencia de la actividad de ráfagas en la K2V es 0.000368411 h-1, lo que indica una baja actividad magnética.

1. Introduction

Although the primary aim of the Kepler mission is to detect transiting planets by obtaining very high precision photometric measurements, it provides further benefits, especially in terms of clear and reliable determination of very small amplitude light variations on eclipsing and intrinsic variable stars. About 150000 targets have been observed in the mission, and apart from the exoplanets, numerous variable stars have been discovered. The unprecedented precision of the Kepler photometry clearly revealed low amplitude (mmag) light variations, which were used in the analysis of stellar flares, spot activity and differential rotation (Balona 2015; Balona et al. 2016; Reinhold & Reiners 2013; Reinhold et al. 2013a). Among these variable stars, 2876 eclipsing binary stars have been discovered (Prša et al. 2011; Slawson et al. 2011). Careful light curve modelling of the binaries with cool components (Teff < 6500 K) revealed rotational modulation of the light curves and flares in model residuals. KIC 09641031 (Yolda ̧s & Dal 2016), KIC09761199 (Yolda ̧s & Dal 2017) and KIC2557430 (Kamil & Dal 2017), GJ1243, GJ 1245A and B (Hawley et al. 2014), KIC 2300039, KIC4671547 (Balona 2015) are examples of such stars.

The analyses of the patterns of magnetic activity exhibited by these stars reveal some clues about their evolutionary stages. Although there are several indicators found in these analyses, two of them are the energy spectra defined by Gershberg (1972) and the flare frequencies described by Ishida et al. (1991). Both of them have been computed, especially from the 1970’s to the 1980’s, in order to figure out the magnetic activity levels for the stars with detected flares. In 1990’s, Leto et al. (1997) examined the flare frequency variation of EV Lac, a well-known UV Ceti type star. There are a few studies on the activity levels of three magnetic active stars discovered in the Kepler Mission depending on their flare frequencies. Yolda ̧s & Dal (2016) detected 240 flares from KIC09641031, and Yolda ̧s & Dal (2017) detected 94 flares from KIC09761199. In addition, Kamil & Dal (2017) detected 69 flares from KIC2557430. Yolda ̧s & Dal (2016) derived the one phase exponential association (hereafter OPEA) model, and the flare frequency N1 was found to be 0.41632 h-1 for KIC09641031. Yolda ̧s & Dal (2017) computed N1 as 0.01351 h-1 for 69 flares for KIC 09761199. However, an interesting situation occurs in the case of KIC2557430. Kamil & Dal (2017) find that some of the flares detected from KIC 2557430 come from a third body; it is unclear whether it is a component in the system or an undetected background light source. Depending on the OPEA model derived from 69 flares, Kamil & Dal (2017) reveal that 40 (called Group 1) of them come from the secondary component, while 29 flares (called Group 2) come from a third body. They computed the flare frequency N1 as 0.02726 h-1 for Group 1 and 0.01977 h-1 for Group 2. As discussed by Yolda ̧s & Dal (2016) and Gershberg (2005), the flare frequency is one of the parameters indicating the nature of the flare mechanism in the stellar atmosphere. Apart from the classical parameters described by Gershberg (2005), Dal & Evren (2010, 2011) have also described some new parameters derived from the OPEA models in order to determine the flare process occurring on the stellar surface.

Continuous photometry of variable single stars discovered by Kepler enabled to trace photometric period variations as a proxy of differential rotation via Fourier transform (see, e.g. Reinhold et al. 2013b; Reinhold & Reiners 2013). However, the Fourier transform may not perfectly work in case of eclipsing binaries, where the amplitude of the rotational modulation of star spots is usually embedded into the relatively large amplitude light variations caused by eclipses and the lack of spherical symmetry of the binary components. Furthermore, insufficient representation of light curve models, especially around mid-eclipse phases, may require discarding data around those phases and may cause regular gaps in the light curve, which would lead to unwanted alias periods and harmonics. In this case, alternative methods should be adopted to trace photometric period variation, such as an O − C diagram based on minimum times of rotationally modulated light curves (see, e.g. özdarcan et al. 2010).

In the case of eclipsing binary stars, additional intrinsic variations may not be determined at first, due to the reasons explained above. KIC 9451096 is such an eclipsing binary in the Kepler eclipsing binary catalog1 (Prša et al. 2011; Slawson et al. 2011) with a short period, and with a confirmed third body (Borkovits et al. 2016). Beyond the properties provided by the catalog, such as morphology and eclipse depths, Armstrong et al. (2014) provided physical information, estimated from the spectral energy distribution based on photometric measurements. They estimated the effective temperature of the components of KIC 9451096 as 7166 K and 5729 K for the primary and the secondary component, respectively.

In this study, we carry out a photometric and spectroscopic analysis of KIC 9451096, based on Kepler photometry and optical spectroscopic observations with intermediate resolution described in § 2. § 3 describes the spectroscopic and photometric modelling of the system, and the analysis of the out-of-eclipse variations. In the final section, we summarize and discuss our findings.

2. Observations and data reductions

2.1. Kepler Photometry

Photometric data obtained by the Kepler spacecraft cover a broad wavelength range between 4100 ̊A and 9100 ̊A; this has the advantage of collecting many more photons in a single exposure and reaching sub-milli-mag precision, but also has the disadvantage of having no “true” photometric filter, hence no photometric color information. There are two types of photometric data having different exposure times. These are short cadence data (having an exposure time of 58.89 seconds) and long cadence data (having an exposure time of 29.4 minutes). In this study we use long cadence data of KIC 9451096 obtained from the Kepler eclipsing binary catalog. The catalog provides detrended and normalized intensities, which are obtained by application of procedures described by Slawson et al. (2011) and Prša et al. (2011). The whole data covers ≈4 years of time, with 65307 data points in total. The MAST archive reports 0.9% contamination level in the measurements, practically indicating no additional light contribution to the measured fluxes of KIC 9451096.

2.2. Spectroscopy

We obtained optical spectra of KIC 9451096 with the 1.5 m Russian - Turkish telescope equipped with the Turkish Faint Object Spectrograph Camera2 (TFOSC) at Tubitak National Observatory. TFOSC enables one to obtain intermediate resolution optical spectra inéchelle mode. In our case, the instrumental setup provides actual resolution of R = λ/∆λ ≈ 2800 around 6500 ̊A, and the observed spectra cover a usable wavelength range between 3900-9100 ̊A in 11 échelle orders. A back illuminated 2048 × 2048 pixels CCD camera, which has pixel size of 15 × 15 μm2, was used to record spectra.

We obtained ten optical spectra of KIC9451096 during the 2014 and 2016 observing seasons. In order to obtain enough signal, we used 3600 s of exposure time for each observation. The estimated signal-to-noise ratio (SNR) of observed spectra is mostly between 80-100, except for a few cases, where the SNR is around 50. SNR estimation is based on photon statistic. Together with the target star, we also obtained high SNR optical spectra of HD 225239 (G2V, vr = 4.80 km s-1) and ι Psc (HD 222368, F7V, vr = 5.656 km s-1), and adopted them as radial velocity and spectroscopic comparison templates.

We reduced all observations using standard IRAF3 packages and tasks. A typical reduction procedure starts with obtaining a master bias frame from several bias frames taken nightly, and subtracting the master bias frame from all object, calibration lamp (Fe-Ar spectra in our case) and halogen lamp frames. Then the bias corrected halogen frames are combined to form an average halogen frame and this average frame is normalized to unity to produce the normalized master flat frame. After that, all target and calibration lamp spectra are divided by the normalized flat field frame. Next, cosmic rays removal and scattered light corrections are applied to the bias and flat corrected frames. At the end of these steps, reduced frames are obtained and these frames are used for the extraction of spectra. In the final steps, Fe-Ar frames are used for wavelength calibration of the extracted spectra and the wavelength calibrated spectra are normalized to unity by using cubic spline functions.

3. Analysis

3.1. Radial Velocities and Spectroscopic Orbit

The first step of our analysis is to determine the radial velocities of the components and the spectroscopic orbit of the system. We cross-correlated each observed spectrum of KIC9451096 with spectra of template stars HD 225239 and ι Psc, as described in Tonry & Davis (1979). In practice we used the fxcor task in IRAF environment. We achieved better cross-correlation signals (especially for the weak secondary component) when we used HD 225239 as template; thus, we determined all radial velocities with respect to the HD225239 spectrum. We obtained acceptable cross-correlation signals of both components inéchelle orders 5 and 6, which cover a wavelength range between 4900-5700 ̊A. Figure 1 shows the cross-correlation functions of two spectra obtained around orbital quadratures.

Fig. 1 Cross-correlation functions of two spectra obtained around orbital quadratures. The letter ϕ denotes corresponding orbital phase. P and S indicate the primary component and the secondary component, respectively. 

We list the observation log and the measured radial velocities of the components in Table 1. Note that we use the ephemeris and period given by Borkovits et al. (2016) and listed in their Table 2 to calculate orbital phases and for further analysis.

Table 1 Log of spectroscopic observations* 

HJD Orbital Exposure Primary Secondary
(24 00000+) Phase time (s) Vr σ Vr σ
56842.5435 0.7794 3600 91.4 8.2 -152.5 36.9
56844.4052 0.2682 3600 -79.9 6.3 151.9 39.1
56844.4479 0.3024 3600 -74.4 6.6 155.0 37.2
56889.4315 0.2781 3600 -77.1 5.7 148.1 40.0
56890.2958 0.9693 3600 14.5 5.0 - -
57591.4532 0.7199 3600 88.5 7.2 -153.3 32.0
57601.4386 0.7058 3600 88.7 5.4 -149.8 32.1
57616.4778 0.7333 3600 86.0 4.3 -145.2 38.7
57617.5188 0.5659 3600 31.0 5.8 - -
57672.3009 0.3779 3600 -54.8 5.1 111.1 47.9

*together with measured radial velocities and their corresponding standard errors (σ) in kms−1.

Table 2 Spectroscopic orbital elements of KIC 9451096. * 

Parameter Value
Porb (day) 1.25039069 (fixed)
T0 (HJD24 00000+) 54954.72942 (fixed)
(kms−1) 2.8±0.5
K1 (kms−1) 84.1±2.3
K2 (kms−1) 153.2±14.6
e 0 (fixed)
a sin i (R⊙) 5.92±0.35
M sin3 i (M⊙) 1.79±0.25
Mass ratio (q = M2/M1) 0.55±0.05
rms1 (kms−1) 3.7
rms2 (kms−1) 4.9

*M1 and M2 denote the masses of the primary and the secondary component, respectively, while M shows the total mass of the system.

We achieved a reasonable solution for the spectroscopic orbit assuming zero eccentricity, where an undefined longitude of periastron is taken. We checked this assumption by inspecting the Kepler light curve of the system, where we observe deeper and shallower eclipses at 0.0 and 0.5 orbital phases, respectively, indicating a circular orbit (see § 3.3, Figure 4). In order to reach the final spectroscopic orbital solution, we prepared a simple script written in Python language, which applies Markov chain Monte Carlo simulations to the measured radial velocities, considering their measured errors. We list the final spectroscopic orbital elements in Table 2 and plot the measured radial velocities, their observational errors, the theoretical spectroscopic orbit and residuals from the solution in Figure 2.

Fig. 2 (a) Observed radial velocities of the primary and the secondary (blue and red filled circles, respectively), and their corresponding theoretical representations (blue and red curve). (b) Residuals from theoretical solution. The color figure can be viewed online. 

3.2. Spectral Type

We rely on our intermediate resolution TFOSC optical spectra to determine the spectral type of the components. Most of our spectra correspond to the phases around orbital quadratures, where we observe the signal of the two components separated. However, there are two spectra obtained at phases close to the eclipses, where the two components can not be resolved separately. One of these spectra corresponds to ≈ 0.56 orbital phase (see Table 1), where we cannot observe the radial velocity signal of the secondary component in cross-correlation. Even at the orbital quadratures, the cross-correlation signal of the secondary component is considerably weak compared to the primary component, indicating a very small light contribution from the secondary component to the total light of the system. Our preliminary light curve analysis shows that the contribution of the secondary component to the total light does not exceed ≈ 10%. In this case, the signal from the secondary component becomes almost negligible at the resolution of our observed spectrum at ≈ 0.56 orbital phase. Therefore, we assume that we only observe the spectrum of the primary component and adopt this spectrum as reference spectrum for the primary component. We confirm this assumption by calculating the composite spectrum of the binary via final parameters of the components (see § 3.3), where we observe that the contribution of the secondary component affects the theoretical composite spectrum less than 2% for the wavelength range of 4900-5700 ̊A. We refrain from performing a detailed analysis with spectral disentangling. Future studies could take advantage of this technique and derive atmospheric parameters of the secondary.

We first compare the reference spectrum with the template spectra of HD 225239 and ι Psc. We observe that ιPsc spectrum provides a closer match to the reference spectrum but also indicates earlier spectral type and slightly lower metal abundances for the primary component. At that point, we switch to the spectrum synthesizing method. We use the latest version of python framework iSpec (Blanco-Cuaresma et al. 2014) which enables practical and quick calculation of a synthetic spectrum with a given set of atmospheric parameters via different radiative transfer codes. Among these codes we adopt the SPECTRUM4 code (Gray & Corbally 1994), together with ATLAS-9 (Castelli & Kurucz 2004) model atmospheres and the actual line list from the third version of the Vienna atomic line database (V ALD3, Ryabchikova et al. 2015).

Considering the spectral type of ι Psc, we synthesize spectra for effective temperatures between 6000 K and 7000 K in steps of 250 K, and metallicity values ([Fe/H]) between −1.0 and 0.0 in steps of 0.5. For all synthetic spectra we fix the gravity (log g) to 4.15, which we precisely calculate by light curve modelling (see § 3.3). Since we do not have a high resolution spectrum, we fix the microturbulence velocity to 2 kms-1. We convolve all calculated spectra with a proper Gaussian line spread function in order to degrade their resolution to the resolution of the TFOSC spectra. Instrumental broadening in TFOSC spectra is 2.2 ̊A, corresponding 119 km s-1 for wavelengths around 5500 ̊A. The estimated projected rotational velocities of the components are 62 km s-1 and 36 km s-1 for the primary and the secondary component respectively (see § 3.3). Since instrumental broadening is the most dominant broadening source in the observed spectra, we do not consider rotational broadening and other line broadening mechanisms.

Among the calculated spectra we find that the model with 6500 K effective temperature and an [Fe/H] value of −0.5 provides the closest match to the reference spectrum. The final effective temperature indicates F5 spectral type (Gray 2005). Considering the effective temperature and metallicity steps in model atmospheres, and the resolution of TFOSC spectra, the final values and their estimated uncertainties are Teff = 6500±200 K and [Fe/H] = −0.5±0.5 dex, respectively. Note that even if we considered the neglected contribution of the secondary component in the reference spectrum, its effect would be within the estimated uncertainties above. The final Teff value is ≈ 670 K lower than the 7166 K value estimated in Armstrong et al. (2014). In Figure 3 we show portions of the reference spectrum and the model spectrum, calculated with the final parameters above.

Fig. 3 Representation of the observed (black), best matched (red) synthetic spectrum and residuals (blue) for three regions. Note that we shift the residuals upwards by 0.3 for the sake of simplicity. 

3.3. Light Curve Modelling and Physical Properties

Global visual inspection of KIC 9451096 Kepler photometry reflects properties of a typical close eclipsing binary. We start the light curve modelling by phasing the whole long cadence data with respect to the ephemeris and period given by Borkovits et al. (2016), and re-binning the phased data with a phase step of 0.002 via the freely the available fortran code lcbin5 written by John Southworth. We plot the binned and phased light curves of the system in Figure 4, panels a and aa. The light curve indicates a detached configuration for the system. Mid-eclipse phases are 0.0 and 0.5 phases, indicating a circular orbit. There is no conspicuous asymmetry in the light curve.

Fig. 4 (a) Phase binned light curve of KIC9451096 (black filled circles) together with best-fitting model (red curves). (b) Close up view of the light curve at light maxima. c) Residuals from the best-fitting model. Panels at right (aa, bb and cc) are the same as left panels but for phased long cadence data. The color figure can be viewed online. 

We model the light curve with the 2015 version of the Wilson-Devinney code (Wilson & Devinney 1971; Wilson & Van Hamme 2014). In the modelling, we first fix the most critical two parameters of the light curve modelling, i.e., the mass ratio (q) of the system and the effective temperature of the primary component (T1). Since we have reliably derived these parameters in previous sections as q = 0.55 and T1 = 6500 K, we adopt them as fixed parameters. The calculated atmospheric properties of the primary component reveal that both stars have convective envelopes. Therefore, we set albedo (A1 , A2 ) and gravity darkening (g1, g2) coefficients of the components to 0.5 and 0.32, respectively, which are typical values for stars with convective outer envelopes. We also consider a slight metal deficiency of the system, and thus adopt the internal stellar atmosphere formulation of the Wilson-Devinney code according to the determined [Fe/H] value of −0.5. We assume that the rotation of the components is synchronous with the orbital motion, and thus fix the rotation parameter of each component (F1 , F2 ) to 1.0. We adopt a square root law (Klinglesmith & Sobieski 1970) for limb darkening of each component; this is more appropriate for stars cooler than 9000 K. We take the limb darkening coefficients for the Kepler passband (x1, x2, y1, y2) and the bolometric coefficients (x1bol , x2bol , y1bol , y2bol ) from van Hamme (1993). In the modelling, we adjust inclination of the orbit (i), temperature of the secondary component (T2), dimensionless omega potentials of the components (Ω1, Ω2) and luminosity of the primary component (L1). We also include a phase shift parameter as adjustable in the modelling, since we expect a shift in the ephemeris due to the light-time effect of the third body (Borkovits et al. 2016). The model quickly converged to a steady solution in a few iterations. We list the model output in Table 3 and we plot the best-fitting model in Figure 4, panels a, b, and the residuals from the model in panel c.

Table 3 Light curve modelling results of KIC9451096.a 

Parameter Value
q 0.55*
T1(K) 6500*
g1, g2 0.32*, 0.32*
A1, A2 0.5*, 0.5*
F1 = F2 1.0*
phase shift 0.00108(2)
i (◦) 79.07(4)
T2(K) 5044(200)
Ω1 4.4942(49)
Ω2 4.8885(125)
L1/(L1+L2) 0.897(1)
x1bol, x2bol 0.136*, 0.293*
y1bol, y2bol 0.583*, 0.401*
x1, x2 0.106*, 0.482*
y1, y2 0.670*, 0.313*
(r1),(r2) 0.2557(3), 0.1506(5)
Model rms 3.0 × 10−4

a(r1),(r2) denote the mean fractional radii of the primary and the secondary components, respectively. Internal errors of the adjusted parameters are given in parentheses for the last digits. Asterisk symbols in the table denote fixed values for the corresponding parameter. Note that we adopt the uncertainty of T1 for T2 as well, since the internal error of T2 is unrealistically small (∼1 K).

In Figure 4, panel b, one can easily see the model inconsistency around 0.25 orbital phase. The inconsistency indicates an additional light variation, which is known as O´Connell effect, i.e. difference between light levels of subsequent maxima in an orbital cycle. Possible sources of the difference may be Doppler beaming, a hot spot or a cool spot on one of the component of the system. KIC9451096 is a detached eclipsing binary, thus we can safely exclude possibility of mass transfer between components, i.e., a hot spot possibility. Doppler beaming was detected observationally among some Kepler binaries (see, e.g. van Kerkwijk et al. 2010), which becomes important for systems with very low mass ratios, especially for systems with a compact component, such as a white dwarf or a hot sub-dwarf. In addition, if the effect is in progress, then it would change the light levels of each maxima. However, we observe inconsistency only for phase 0.25, while the model fairly represents the light level at phase 0.75. Thus, Doppler beaming should have a negligible effect in the case of KIC 9451096, if any. A remaining possibility is cool spots located preferably on the cooler component.

Here we do not chose to model this inconsistency alone, which would only show the cumulative effect of hundreds of light curves, but instead we subtract the best-fitting model from the whole long cadence data and inspect the residuals in order to investigate further light variations. We will focus on this in § 3.4.

We complete the light curve modelling section with a calculation of the absolute parameters of the system by combining the spectroscopic orbital solution and light curve model results. In Table 4, we give the physical properties of each component. Our analysis reveals that the system is formed by an F5V primary and a K2V secondary component.

Table 4 Absolute physical properties of KIC9451096.* 

Parameter Primary Secondary
Spectral Type F5V K2V
[Fe/H] −0.5 ± 0.5
Mass (M⊙) 1.18(26) 0.65(9)
Radius (R⊙) 1.53(10) 0.90(6)
Log L/L 0.574(76) −0.327(88)
log g (cgs) 4.14(4) 4.34(1)
Mbol (mag) 3.31(19) 5.57(22)

*The error of each parameter is given in parantheses for the last digits.

3.4. The Out-of-Eclipse Variations

In this section, we subtract the best-fitting light curve model from the whole long cadence data and obtain residuals. Here, we first divide the whole long cadence data into subsets, where each subset covers only a single orbital cycle, resulting in 1026 individual light curves. Then we apply the differential corrections routine of the Wilson-Devinney code and fix all parameters, except the ephemeris reference time. In this way, we find a precise ephemeris reference time for each individual subset, therefore eliminating any shift in the ephemeris time due to the third body reported by Borkovits et al. (2016), and obtain precise residuals. In Figure 5, we plot three different parts of the residuals. Note that we remove data points that correspond to the eclipse phases due to the insufficient representation of the model at those phases. This mainly arises from the inadequacy of radiative physics used in light curve modelling for a very high photometric precision and can clearly be seen in Figure 4 panel c.

Fig. 5 (a) Residuals from whole long cadence data. Remaining panels show different time ranges of residuals, where we observe different light curve shapes, and flares. 

Inspecting residual brightness, we immediately see a variation pattern which changes its shape from time to time. Furthermore, we observe a sudden increase and gradual decrease in the residual brightness which occasionally occurs over four years of time span and has short time scale of a few hours. These patterns are traces of magnetic spot activity, which is very possible for the K2V secondary component. Observational confirmation of this possibility can be done by inspecting magnetic activity sensitive spectral lines, such as the Hα and Ca II H & K lines. We inspected these lines in our TFOSC spectra and did not notice any emission features, which could be considered as the sign of the activity. However, one should consider that the contribution of the secondary component to the total light does not exceed 10% at optical wavelengths and will steeply decrease towards the ultraviolet region of the spectrum. Furthermore, the variation patterns observed in Figure 5 exhibit very small amplitudes. Therefore, the existence of magnetic spot activity cannot be confirmed or excluded via spectral line inspection in the case of KIC 9451096. Nevertheless, variation patterns and flares observed in the residuals indicate weak magnetic spot activity in the secondary component, which can still be detected with the very high precision of the Kepler photometry.

We analyze rotational modulation and flares of the secondary component via residuals by assuming that the source of all variation patterns is only the secondary component.

3.4.1. Photometric Period and Differential Rotation

Conventional periodogram methods for determining rotational period do not perfectly work in our case because the observed variation patterns exhibit quick changes in amplitude and mean brightness level over short time scales of a few days, which is comparable to the orbital period. Moreover, since we remove data points at eclipse phases, this causes regular gaps in the data which repeat each ≈ 0.625 day (i.e. half of the orbital period); thus, it causes an alias period and its harmonics, and disturbs the real periods. Furthermore, one can clearly see that the rotational modulation of residuals has an asymmetric shape. Considering an individual light curve with an asymmetric shape, it is not possible to find a single period to represent the whole light curve perfectly, and additional periods (i.e. harmonics) are required. Therefore we apply an alternative method based on tracing the time of a minimum light observed in an orbital cycle, which was previously applied to RS CVn system HD 208472 (özdarcan et al. 2010). For each orbital cycle, we find the time of the deepest minimum in the cycle by fitting a second or third order polynomial to the data points around the expected minimum time. The order of the polyno Subset mial depends on the light curve shape. After obtaining all minimum times, we construct an O − C diagram by adopting the first minimum time observed in the residuals as initial ephemeris reference time, and the orbital period as the initial period, and obtain O−CI values. Then we apply a linear fit to the O−CI values and calculate an average ephemeris reference time and period given in Equation 1, together 6 with statistical uncertainties given in parentheses for the last digits.

T0BJD=2,454,954.0224+1.d2454436×E. (1)

In the equation, T0(BJD) and E denote ephemeris reference time and integer cycle number, respectively. We plot O − CI values and linear fit in Figure 6, panel a. After obtaining an average ephemeris and period, we subtract the linear fit from O−CI data and obtain O−CII data, which in principle shows the real period variation for a given time range. Figure 6, panel b shows O − CII data. We divide O − CII data into 30 subsets by grouping data points that appear with a linear slope. The linear trend of a subset gives the difference between the best-fitting photometric period of the subset and the grand average photometric period given in Equation 1. Therefore we can calculate a mean photometric period for each subset. We plot the calculated mean photometric periods versus time in Figure 6, panel c, together with the statistical uncertainties. We list photometric periods for 30 subsets in Table 5, and tabulate our O − C analysis results in Table 8.

Fig. 6 (a) O−CI diagram of observed minimum times (blue filled circles) and linear fit (red line). 

Table 5 photometric periods found from O − C analysis 

Subset BJD (24 00000+) P (day) σ(P) (day)
1 54994.8107 1.2456 0.0004
2 55048.8731 1.2326 0.0008
3 55094.1598 1.2441 0.0004
4 55139.0644 1.2260 0.0019
5 55169.9192 1.2459 0.0008
6 55208.0721 1.2489 0.0006
7 55250.0831 1.2584 0.0011
8 55314.8252 1.2484 0.0004
9 55366.4562 1.2355 0.0006
10 55425.0957 1.2470 0.0006
11 55478.0779 1.2517 0.0010
12 55507.4240 1.2437 0.0006
13 55539.3828 1.2216 0.0025
14 55629.1787 1.2430 0.0004
15 55702.5236 1.2447 0.0004
16 55740.2684 1.2522 0.0007
17 55793.0150 1.2485 0.0004
18 55840.9410 1.2223 0.0022
19 55868.2947 1.2534 0.0005
20 55894.6874 1.2712 0.0022
21 55924.7567 1.2494 0.0006
22 55960.4676 1.2391 0.0011
23 55996.8636 1.2507 0.0005
24 56026.2172 1.2474 0.0009
25 56073.0738 1.2528 0.0005
26 56136.3924 1.2449 0.0005
27 56258.6328 1.2509 0.0004
28 56333.3104 1.2323 0.0019
29 56359.5423 1.2565 0.0008
30 56400.8932 1.2504 0.0004

The average period given in Equation 1 represents the average rotation period for magnetic activity features on the surface of the secondary component, which are typically cool and dark regions, i.e., star spots, and indicates a slightly (∼0.5% day) shorter period compared to the orbital period. This is clearly observed in Figure 6 panel c, where the mean photometric periods of subsets are mostly shorter than the orbital period. Assuming a solar type differential rotation, this means that the orbital period is slightly longer than the equatorial rotation period of the secondary component. Under the same assumption, the differential rotation coefficient can be estimated from (Pmax −Pmin)/Pequ = kf, where Pmax , Pmin , k and f denote observed maximum and minimum period, differential rotation coefficient and a constant that depends on the range of spot forming latitudes, respectively (Hall & Busby 1990). Considering the small amplitude of rotational modulation of residuals, we assume that the secondary component is not largely spotted and that the total latitudinal range of the spot distribution is 45 degrees, which causes the f constant to take values between 0.5 and 0.7 (Hall & Busby 1990). Using maximum and minimum photometric periods from the O − C analysis, and assuming that the shortest period corresponds to the equatorial rotation period of the star, we find k = 0.081 ± 0.011 and k = 0.058 ± 0.006 for f = 0.5 and f = 0.7, respectively. Since these k values are calculated via boundary values of f, the real differential rotation coefficient must lie in the range of k values calculated above. An average k is found as 0.069±0.008.

3.4.2. Flares

We detect 13 flares in the residuals from long cadence data. In the flare analysis, it is critical to determine the quiescent level, which denotes the brightness level in the absence of a flare. In our case, we determine the quiescent level by applying Fourier analysis to the single orbital cycle where the flare occurs. The Fourier analysis represents the rotational modulation of residuals in the cycle, and then we remove the Fourier representation from the data. The remaining residuals show only the quiescent level and the flare itself. We show such a flare light curve in Figure 7.

Fig. 7 An example of a flare light curve. The filled black circles represent the observations, while the red line represents the quiescent level derived from the data out-of-flare. The color figure can be viewed online. 

The energy (E) is a very important parameter for a flare. However, the energy parameter has the luminosity L of the star as a factor in equation E = P × L described by Gershberg (1972). Due to the disadvantages described in Dal & Evren (2010), we use the flare equivalent duration instead of the flare energy, which is more proper. We compute the equivalent durations of flares via the equation P = [(Iflare − I0)/I0]dt (Gershberg 1972), where P is the flare equivalent duration in seconds, I0 is the quiescent level intensity, and Iflare is the intensity observed at the moment of the flare. Considering the quiescent level, the times of flare beginning, flare maximum and flare end are determined, together with flare rise duration, flare decay duration and flare amplitude. We list all computed values in Table 6 for each of the 13 flares.

Table 6 The parameters calculated for each 

BJD (24 00000+) 55021.2171 P (s) 11.4 Tr (s) 1763 Td (s) 15889 Amp (mag) -0.001516
55043.1016 5.6 1763 5296 -0.002483
55310.6569 7.6 1763 8830 -0.002047
55326.5140 2.7 1771 1763 -0.001618
55412.0302 5.9 1763 7068 -0.001648
55416.9343 12.1 1771 14118 -0.002853
55824.2162 4.3 1763 5296 -0.001578
55931.1213 4.5 3534 3534 -0.001453
55971.7021 4.9 1763 5296 -0.002152
56142.9809 6.0 3534 7059 -0.001983
56284.8887 3.4 1771 3525 -0.001806
56286.5642 4.4 1771 3525 -0.001568
56375.4705 2.2 1763 1763 -0.001429

Dal & Evren (2010, 2011) suggest that the best function to represent the relation between flare equivalent duration and flare total duration is the OPEA, where the flare equivalent duration is considered on a logarithmic scale. The OPEA function is defined as y = y0 +(Plateau−y0)×(1−e−kx), where y is the flare equivalent duration on a logarithmic scale, x is the flare total duration, and y0 is the flare equivalent duration in the logarithmic scale for the least total duration, according to the definition of Dal & Evren (2010). It should be noted that the y0 does not depend only on the flare mechanism, but also depends on the sensitivity of the optical system used in the mission. The most important parameter in the model is the Plateau value, which defines the upper limit for the flare equivalent duration on a logarithmic scale and is defined as the saturation level for a star (Dal & Evren 2011). Using the least squares method, the OPEA model leads to the results in Table 7. We plot the resulting model in Figure 8 with its 95% statistical sensitivity limit.

Table 7 Parameters derived from the OPEA 

Parameter Value
Y0 −0.015961±0.13891
Plateau 1.2394±0.14441
K 0.00011438±0.000036715
Half-time 6060
R2 0.94535
P value ∼0.10

*Using the least squares method.

Fig. 8 The OPEA model obtained over 13 flares. The blue filled circles show each flare while the continuous red line shows the OPEA model and the dotted red lines show the sensitivity range of the model. The color figure can be viewed online.  

We tested the derived model by using method proposed by D’Agostino & Stephens (1986) to understand whether there are any other functions to model the distribution of flare equivalent durations on this plane. In this method, the probability value (P value), is found to be ≈ 0.10, which means that there is no other function to model the distributions (Motulsky 2007; Spanier & Oldham 1987).

Ishida et al. (1991) described a frequency for the stellar flare activity as N1 = Σnf /ΣTt, where Σnf is the total flare number detected in the observations, while ΣTt is the total observing duration from the beginning of the observing season to the end. In the case of KIC9451096 we find the N1 frequency as 0.000368411 h-1 adopting the total long cadence observing duration as 1470.2786 days from the times of the first and last long cadence data points.

4. Summary and discussion

Photometric and spectroscopic analysis of KIC9451096 reveals that the system is composed of an F5V primary and a K2V secondary star in a circular orbit with a detached binary configuration. Medium resolution TFOSC spectra suggest that the system has one third of the [Fe/H] of the Sun. Light curve modelling reasonably represents the observations. However, we are able to catch the signals of additional light variation, which is very weak compared to the variations due to the binarity and eclipses, but still observable due to the very high precision of the Kepler photometry.

We observe occasional flares and rotational modulation of the light curve residuals from the eclipsing binary model. Considering the physical and atmospheric properties of the components, we attribute these variations to the secondary component, which is a perfect candidate for magnetic star spot activity with its deep convective zone owing to its spectral type and very fast rotation caused by short orbital period. We inspect rotational modulations of the residuals to trace the photometric period of the secondary component, and analyze its flare characteristics.

Photometric period analysis via O − C diagrams shows that the average photometric period is shorter than the orbital period by ≈ 0.5% day. Under any type of differential rotation assumption (either solar like, or anti-solar like), this means that the orbital period does not correspond to the equatorial rotation period of the star. Following the method proposed by Hall & Busby (1990), we find an average differential rotation coefficient of k = 0.069 ± 0.008, suggesting ≈ 3 times weaker differential rotation compared to the solar value of 0.19. We note that the type of differential rotation cannot be determined from photometry alone and we implicitly assume a solar type differential rotation in the case of KIC 9451096. However, the k = 0.069 value, which is extracted from very high precision continuous photometry for a restricted time range (four years in our case), defines a lower limit for the strength of the differential rotation of the star. A quick comparison of k values for other stars can be done by looking at the 17 stars listed in Hall & Busby (1990), where k values are usually a few percent or less, except for BY Dra with k = 0.17.

A more reliable way of detecting differential rotation with its magnitude and type is Doppler imaging, which is based on high resolution time series spectroscopy. Considering other stars whose k values were determined by Doppler imaging, we see mostly weak differential rotation with a k value of a few percent, both among solar type differential rotators (HD 208472 k = 0.015 (özdarcan et al. 2016), XXTri k = 0.016 (Künstler et al. 2015), ζ And k = 0.055 (Kövári et al. 2012), KUPeg k = 0.04 (Kövári et al. 2016)) and among anti-solar type differential rotators (UZLib k = −0.004 (Vida et al. 2007), σ Gem k = −0.04 (Kövári et al. 2015), HU Vir k = −0.029 (Harutyunyan et al. 2016)). Due to the binary nature of KIC 9451096, a considerable effect of tidal forces on the redistribution of the angular momentum in the convective envelope of the components can be expected, which would alter the magnitude of differential rotation (Scharlemann 1982). Based on observational findings, Collier Cameron (2007) suggests suppression of differential rotation by tidal locking, which is possibly in progress for KIC 9451096.

We detect 13 flares in the residuals from long cadence data, which are attributed to the secondary component with a corresponding B − V value of 0m.92 (Gray 2005). We apply the OPEA model to analyze flare characteristic and find that the calculated flare parameters and resulting OPEA model parameters seem to be in agreement with parameters derived from stars analogous to the secondary component, except for the half-time value. A possible source of disagreement for the half-time value is that there are not enough sample flares at the beginning of the OPEA model.

We find an N1 value of 0.000368411 h-1 for KIC9451096. N1 was found to be 0.41632 h-1 for KIC09641031 (Yolda ̧s & Dal 2016), 0.01351 h-1 for KIC 09761199 (Yolda ̧s & Dal 2017), and 0.02726 h-1 for Group 1 and 0.01977 h-1 for Group 2 of KIC 2557430 (Kamil & Dal 2017). Among these systems, KIC9451096 has the lowest N1 value, which indicates that the magnetic activity level of the secondary component of KIC 9451096 is the lowest, according to Dal & Evren (2011).

We thank TüBI ̇TAK for partial support in using RTT150 (Russian-Turkish 1.5-m telescope in Antalya) with project number 14BRTT150-667. This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission Directorate.

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Appendix

A.1. O − C analysis results

We tabulate O − C analysis results in Table 8. N is the number of the minimum, beginning from the first observed minimum in the data set. E is the decimal cycle number and E rounded is the rounded E number to the nearest integer or half integer. Note that as time progress O − C differences approach a cycle. When this the occurs, one needs to add an additional increment of 0.5 to the E rounded value in order to see O − CI diagram on a trend without any discontinuity.

Table 8 O − C analysis results  

N BJD (24 00000+) E E rounded OCI (day) OCII (day) N BJD (24 00000+) E E rounded OCI (day) OCII (day)
1 54954.3397 -0.01 0.0 0.037559 0.378738 73 55058.7485 83.49 84.0 -0.587093 0.170127
2 54955.5840 0.98 1.0 0.031503 0.377636 74 55059.9785 84.47 85.0 -0.607518 0.154654
3 54956.8162 1.97 2.0 0.013253 0.364338 75 55061.1873 85.44 86.0 -0.649122 0.118004
4 54958.0797 2.98 3.0 0.026397 0.382435 76 55066.1529 89.41 90.0 -0.685052 0.101885
5 54959.3084 3.96 4.0 0.004684 0.365675 77 55067.4011 90.41 91.0 -0.687308 0.104582
6 54960.5831 4.98 5.0 0.028962 0.394906 78 55068.6338 91.39 92.0 -0.705006 0.091837
7 54961.8311 5.98 6.0 0.026601 0.397497 79 55069.8833 92.39 93.0 -0.705847 0.095948
8 54965.5429 8.95 9.0 -0.012834 0.372921 80 55071.1320 93.39 94.0 -0.707572 0.099177
9 54966.7917 9.94 10.0 -0.014400 0.376308 81 55072.3928 94.40 95.0 -0.697178 0.114523
10 54968.0426 10.94 11.0 -0.013919 0.381742 82 55073.6453 95.40 96.0 -0.695042 0.121612
11 54969.2711 11.93 12.0 -0.035832 0.364781 83 55074.9064 96.41 97.0 -0.684360 0.137247
12 54970.5301 12.93 13.0 -0.027244 0.378323 84 55076.1196 97.38 98.0 -0.721625 0.104935
13 54971.7622 13.92 14.0 -0.045532 0.364987 85 55077.3677 98.38 99.0 -0.723868 0.107644
14 54973.0157 14.92 15.0 -0.042376 0.373097 86 55078.6214 99.38 100.0 -0.720625 0.115841
15 54974.2711 15.93 16.0 -0.037394 0.383031 87 55079.8524 100.36 101.0 -0.739938 0.101480
16 54975.5424 16.94 17.0 -0.016495 0.408883 88 55081.1133 101.37 102.0 -0.729506 0.116866
17 54976.7582 17.92 18.0 -0.051121 0.379210 89 55082.3594 102.37 103.0 -0.733732 0.117592
18 54978.0108 18.92 19.0 -0.048923 0.386361 90 55083.6067 103.37 104.0 -0.736889 0.119388
19 54979.2462 19.90 20.0 -0.063908 0.376328 91 55084.8310 104.35 105.0 -0.762963 0.098267
20 54980.5036 20.91 21.0 -0.056872 0.388318 92 55086.1013 105.36 106.0 -0.743025 0.123158
21 54981.7375 21.90 22.0 -0.073396 0.376746 93 55094.7741 112.30 113.0 -0.823060 0.077792
22 54982.9940 22.90 23.0 -0.067271 0.387824 94 55096.0183 113.29 114.0 -0.829243 0.076563
23 54984.2387 23.90 24.0 -0.073003 0.387045 95 55097.2744 114.30 115.0 -0.823548 0.087210
24 54985.4801 24.89 25.0 -0.081962 0.383039 96 55098.4829 115.26 116.0 -0.865443 0.050269
25 54986.7385 25.90 26.0 -0.074023 0.395930 97 55102.2238 118.26 119.0 -0.875730 0.054840
26 54987.9747 26.89 27.0 -0.088237 0.386670 98 55103.4779 119.26 120.0 -0.872063 0.063460
27 54989.2224 27.88 28.0 -0.090861 0.388998 99 55104.7113 120.25 121.0 -0.889036 0.051439
28 54990.4801 28.89 29.0 -0.083562 0.401251 100 55105.9617 121.25 122.0 -0.889065 0.056364
29 54991.7160 29.88 30.0 -0.098088 0.391677 101 55107.1998 122.24 123.0 -0.901346 0.049035
30 54992.9545 30.87 31.0 -0.109983 0.384735 102 55108.4680 123.25 124.0 -0.883525 0.071809
31 54994.1980 31.86 32.0 -0.116895 0.382776 103 55109.7219 124.25 125.0 -0.880072 0.080215
32 54995.4493 32.86 33.0 -0.115976 0.388648 104 55110.9729 125.25 126.0 -0.879402 0.085838
33 54996.6932 33.86 34.0 -0.122437 0.387139 105 55112.2163 126.25 127.0 -0.886467 0.083725
34 55004.1716 39.84 40.0 -0.146475 0.392819 106 55115.9232 129.21 130.0 -0.930695 0.054357
35 55005.3983 40.82 41.0 -0.170123 0.374124 107 55117.1913 130.23 131.0 -0.913059 0.076945
36 55006.6502 41.82 42.0 -0.168696 0.380504 108 55118.4321 131.22 132.0 -0.922596 0.072361
37 55007.8905 42.81 43.0 -0.178787 0.375366 109 55119.6639 132.20 133.0 -0.941263 0.058647
38 55009.1251 43.80 44.0 -0.194516 0.364590 110 55120.9037 133.20 134.0 -0.951789 0.053074
39 55010.3779 44.80 45.0 -0.192162 0.371896 111 55122.1667 134.21 135.0 -0.939179 0.070636
40 55011.6258 45.80 46.0 -0.194622 0.374390 112 55125.8541 137.15 138.0 -1.003025 0.021649
41 55012.8550 46.78 47.0 -0.215858 0.358106 113 55127.0843 138.14 139.0 -1.023189 0.006438
42 55019.1204 51.79 52.0 -0.202426 0.396303 114 55128.3384 139.14 140.0 -1.019523 0.015057
43 55020.3475 52.78 53.0 -0.225773 0.377908 115 55129.5583 140.12 141.0 -1.050067 -0.010535
44 55021.6086 53.78 54.0 -0.215099 0.393535 116 55130.7947 141.11 142.0 -1.064060 -0.019574
45 55022.8554 54.78 55.0 -0.218692 0.394895 117 55132.0135 142.08 143.0 -1.095642 -0.046204
46 55024.1181 55.79 56.0 -0.206306 0.412234 118 55133.2565 143.07 144.0 -1.102987 -0.048595
47 55025.3559 56.78 57.0 -0.218929 0.404564 119 55134.4648 144.04 145.0 -1.145085 -0.085741
48 55026.5982 57.77 58.0 -0.227020 0.401426 120 55135.7251 145.05 146.0 -1.135194 -0.070897
49 55027.8428 58.77 59.0 -0.232822 0.400576 121 55136.9848 146.06 147.0 -1.125890 -0.056640
50 55029.0872 59.77 60.0 -0.238863 0.399489 122 55138.2260 147.05 148.0 -1.135135 -0.060932
51 55030.3152 60.75 61.0 -0.261241 0.382063 123 55139.3961 147.98 149.0 -1.215360 -0.136205
52 55031.5630 61.75 62.0 -0.263886 0.384371 124 55140.6664 149.00 150.0 -1.195475 -0.111366
53 55032.8374 62.76 63.0 -0.239869 0.413341 125 55141.9514 150.03 151.0 -1.160879 -0.071817
54 55034.0580 63.74 64.0 -0.269684 0.388479 126 55143.6087 151.35 152.5 -1.379206 -0.282715
55 55035.2816 64.72 65.0 -0.296409 0.366706 127 55144.7802 152.29 153.5 -1.458106 -0.356662
56 55036.5589 65.74 66.0 -0.269523 0.398546 129 55148.5363 155.29 156.5 -1.453207 -0.336905
57 55037.7875 66.72 67.0 -0.291324 0.381697 130 55151.0239 157.28 158.5 -1.466431 -0.340223
58 55039.0026 67.70 68.0 -0.326650 0.351324 131 55152.2748 158.28 159.5 -1.465906 -0.334745
59 55040.2237 68.67 69.0 -0.355947 0.326980 128 55146.0738 153.33 154.5 -1.414879 -0.308483
60 55041.4737 69.67 70.0 -0.356277 0.331603 132 55158.0603 162.91 164.0 -1.307165 -0.153716
61 55042.7426 70.69 71.0 -0.337835 0.354998 133 55159.3610 163.95 165.0 -1.256856 -0.098454
62 55043.9667 71.67 72.0 -0.364077 0.333709 134 55160.6064 164.95 166.0 -1.261887 -0.098533
63 55045.1588 72.62 73.0 -0.422461 0.280277 135 55161.8491 165.94 167.0 -1.269607 -0.101299
64 55046.3730 73.59 74.0 -0.458642 0.249050 136 55163.0822 166.93 168.0 -1.286891 -0.113631
65 55047.6337 74.60 75.0 -0.448283 0.264361 137 55164.3032 167.90 169.0 -1.316294 -0.138081
66 55048.8564 75.58 76.0 -0.475977 0.241620 138 55165.6046 168.95 170.0 -1.265253 -0.082087
67 55050.0805 76.55 77.0 -0.502283 0.220267 139 55166.8299 169.92 171.0 -1.290347 -0.102228
68 55051.3086 77.54 78.0 -0.524603 0.202900 140 55168.0530 170.90 172.0 -1.317624 -0.124552
69 55052.5696 78.55 79.0 -0.513982 0.218473 141 55169.3099 171.91 173.0 -1.311137 -0.113112
70 55053.7928 79.52 80.0 -0.541189 0.196220 142 55170.5385 172.89 174.0 -1.332958 -0.129981
71 55055.0220 80.51 81.0 -0.562386 0.179975 143 55171.8078 173.91 175.0 -1.314118 -0.106187
72 55057.5244 82.51 83.0 -0.560812 0.191455 144 55173.0292 174.88 176.0 -1.343042 -0.130159
145 55174.2930 175.89 177.0 -1.329707 -0.111871 220 55303.6152 279.32 280.5 -1.423718 0.306739
146 55175.5226 176.88 178.0 -1.350498 -0.127709 221 55304.8802 280.33 281.5 -1.409120 0.326290
147 55176.7983 177.90 179.0 -1.325210 -0.097468 222 55306.1357 281.33 282.5 -1.404018 0.336345
148 55178.0453 178.89 180.0 -1.328543 -0.095849 223 55311.1195 285.32 286.5 -1.421824 0.338350
149 55179.2763 179.88 181.0 -1.347928 -0.110280 224 55312.3630 286.32 287.5 -1.428739 0.336388
150 55180.5189 180.87 182.0 -1.355797 -0.113197 225 55313.6050 287.31 288.5 -1.437108 0.332972
151 55181.7781 181.88 183.0 -1.347004 -0.099451 226 55314.8625 288.31 289.5 -1.430004 0.345028
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775 56208.2411 1002.79 1007.0 -5.212331 0.116379 850 56338.2570 1106.77 1111.0 -5.237886 0.605923
776 56209.4810 1003.79 1008.0 -5.222828 0.110836 851 56339.4611 1107.74 1112.0 -5.284151 0.564610
777 56210.7714 1004.82 1009.0 -5.182829 0.155787 852 56340.6906 1108.72 1113.0 -5.305020 0.548694
778 56213.3022 1006.84 1011.0 -5.152775 0.195747 853 56341.9756 1109.75 1114.5 -5.895645 -0.034501
779 56214.5478 1007.84 1012.0 -5.157621 0.195854 854 56343.2492 1110.77 1115.5 -5.872415 -0.006319
780 56215.7983 1008.84 1013.0 -5.157546 0.200882 855 56344.4293 1111.71 1116.5 -5.942756 -0.071706
781 56217.0470 1009.84 1014.0 -5.159207 0.204174 856 56345.7230 1112.74 1117.5 -5.899509 -0.023507
782 56218.2981 1010.84 1015.0 -5.158470 0.209863 857 56346.9865 1113.76 1118.5 -5.886309 -0.005354
783 56219.5546 1011.84 1016.0 -5.152404 0.220883 858 56348.2050 1114.73 1119.5 -5.918239 -0.032331
784 56220.8009 1012.84 1017.0 -5.156531 0.221708 859 56349.4638 1115.74 1120.5 -5.909852 -0.018991
785 56222.0363 1013.83 1018.0 -5.171527 0.211665 860 56350.6891 1116.72 1121.5 -5.934905 -0.039092
786 56223.2783 1014.82 1019.0 -5.179907 0.208238 861 56352.0308 1117.79 1122.5 -5.843619 0.057148
787 56224.5537 1015.84 1020.0 -5.154868 0.238230 862 56353.3171 1118.82 1123.5 -5.807787 0.097932
788 56225.8164 1016.85 1021.0 -5.142631 0.255419 863 56354.5736 1119.82 1124.5 -5.801645 0.109027
789 56227.0506 1017.84 1022.0 -5.158845 0.244159 864 56355.8250 1120.82 1125.5 -5.800643 0.114982
790 56228.3015 1018.84 1023.0 -5.158301 0.249655 865 56357.0925 1121.84 1126.5 -5.783577 0.137001
791 56229.5273 1019.82 1024.0 -5.182880 0.230030 866 56360.7909 1124.80 1129.5 -5.836365 0.099071
792 56230.8226 1020.85 1025.0 -5.138000 0.279862 867 56362.0787 1125.83 1130.5 -5.798958 0.141432
793 56232.0699 1021.85 1026.0 -5.141082 0.281733 868 56363.3212 1126.82 1131.5 -5.806816 0.138526
794 56233.2894 1022.83 1027.0 -5.171953 0.255815 869 56364.5903 1127.83 1132.5 -5.788165 0.162130
795 56234.5424 1023.83 1028.0 -5.169397 0.263324 870 56365.8475 1128.84 1133.5 -5.781346 0.173902
796 56235.7786 1024.82 1029.0 -5.183627 0.254046 871 56367.0963 1129.84 1134.5 -5.782885 0.177316
797 56239.5391 1027.82 1032.0 -5.174262 0.278270 872 56368.3384 1130.83 1135.5 -5.791247 0.173906
798 56240.7773 1028.81 1033.0 -5.186457 0.271028 873 56369.6252 1131.86 1136.5 -5.754795 0.215312
799 56242.0276 1029.81 1034.0 -5.186631 0.275807 874 56370.8651 1132.85 1137.5 -5.765300 0.209759
800 56243.2577 1030.80 1035.0 -5.206922 0.260468 875 56372.1318 1133.87 1138.5 -5.749028 0.230984
801 56244.5006 1031.79 1036.0 -5.214427 0.257917 876 56373.3682 1134.85 1139.5 -5.763039 0.221926
802 56253.3022 1038.83 1043.0 -5.165545 0.341468 877 56374.6499 1135.88 1140.5 -5.731719 0.258199
803 56254.5576 1039.84 1044.0 -5.160566 0.351401 878 56375.8468 1136.84 1141.5 -5.785202 0.209668
804 56255.8080 1040.84 1045.0 -5.160585 0.356334 879 56377.1089 1137.85 1142.5 -5.773500 0.226324
805 56257.0728 1041.85 1046.0 -5.146118 0.375754 880 56378.3842 1138.87 1143.5 -5.748615 0.256161
806 56258.3140 1042.84 1047.0 -5.155379 0.371446 881 56379.6388 1139.87 1144.5 -5.744433 0.265297
807 56259.5516 1043.83 1048.0 -5.168190 0.363588 882 56380.8733 1140.86 1145.5 -5.760302 0.254380
808 56260.7955 1044.82 1049.0 -5.174669 0.362061 883 56382.1355 1141.87 1146.5 -5.748486 0.271149
809 56262.0533 1045.83 1050.0 -5.167215 0.374469 884 56383.3994 1142.88 1147.5 -5.735011 0.289577
810 56263.2775 1046.81 1051.0 -5.193508 0.353128 885 56384.6451 1143.87 1148.5 -5.739710 0.289831
811 56264.5440 1047.82 1052.0 -5.177382 0.374208 886 56385.8893 1144.87 1149.5 -5.745889 0.288604
812 56265.7843 1048.81 1053.0 -5.187410 0.369132 887 56387.1412 1145.87 1150.5 -5.744449 0.294998
813 56267.0202 1049.80 1054.0 -5.201944 0.359551 888 56388.3931 1146.87 1151.5 -5.742949 0.301450
814 56270.7982 1052.82 1057.0 -5.175126 0.401228 889 56389.6473 1147.87 1152.5 -5.739137 0.310215
815 56272.0692 1053.84 1058.0 -5.154509 0.426798 890 56394.6352 1151.86 1156.5 -5.752815 0.316349
816 56275.1726 1056.32 1060.5 -5.177166 0.416523 891 56395.8689 1152.85 1157.5 -5.769539 0.304577
817 56276.4370 1057.33 1061.5 -5.163184 0.435458 892 56397.1298 1153.86 1158.5 -5.758961 0.320109
818 56277.6912 1058.34 1062.5 -5.159377 0.444217 893 56398.3793 1154.86 1159.5 -5.759917 0.324105
819 56278.9163 1059.32 1063.5 -5.184624 0.423924 894 56399.6365 1155.86 1160.5 -5.753126 0.335849
895 56400.8789 1156.86 1161.5 -5.761061 0.332867 902 56409.6487 1163.87 1168.5 -5.744102 0.384496
896 56402.1367 1157.86 1162.5 -5.753714 0.345167 903 56410.8892 1164.86 1169.5 -5.753967 0.379583
897 56403.3823 1158.86 1163.5 -5.758478 0.345355 904 56412.1522 1165.87 1170.5 -5.741409 0.397095
898 56404.6358 1159.86 1164.5 -5.755425 0.353362 905 56413.4054 1166.87 1171.5 -5.738612 0.404844
899 56405.8873 1160.86 1165.5 -5.754252 0.359487 906 56420.9048 1172.87 1177.5 -5.741557 0.431616
900 56407.1374 1161.86 1166.5 -5.754610 0.364082 907 56422.1512 1173.87 1178.5 -5.745583 0.432544
901 56408.3847 1162.86 1167.5 -5.757700 0.365945 908 56423.4023 1174.87 1179.5 -5.744909 0.438170

Received: May 17, 2017; Accepted: September 05, 2017

H. A. Dal, O. Özdarcan, and E. Yoldaş:

Ege University, Science Faculty, Department of Astronomy and Space Sciences, 35100 Bornova, Izmir, Turkey (orkun.ozdarcan@ege.edu.tr).

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