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        KIC 2557430: A Triple System Containing One γ Dor and Two Flaring Components?
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        KIC 2557430: A Triple System Containing One γ Dor and Two Flaring Components?
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        KIC 2557430: A Triple System Containing One γ Dor and Two Flaring Components?
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Abstract

The results are presented for eclipsing binary KIC 2557430. The mass ratio was computed as 0.868 ± 0.002, while the inclination (i) was found as 69°.75 ± 0°.01 with T 2 = 6271±1 K. 50 frequencies were found in the period analysis. 48 frequencies of them are caused due to the primary component, a γ Doradus star, while two of them are caused by the cool spots. 69 flares were detected in the analyses. Two OPEA models were derived for flares, which indicates that the flares were come from two different sources. The Plateau value was found to be 1.4336 ± 0.1104 s for Source 1, which is seen as possible the secondary component and 0.7550 ± 0.0677 s for Source 2, which is seen as possible third body. The half-life value was computed as 2278.1 s for Group 1 and 1811.2 s for Group 2. The flare frequency N 1 was found to be 0.02726 h −1 and N 2 was computed as 0.00002 for Group 1, while N 1 was found to be 0.01977 h −1 and N 2 was computed as 0.00001 for Group 2. In a results, KIC 2557430 is a possible triple system consisting of a γ Doradus-type star, a chromospherically active star, and also a flaring third body.

1 INTRODUCTION

It is well known for about four decades that 65% of the stars in our Galaxy is consisted from red drawfs, whose 75% called UV Ceti type stars exhibits strong flare activity (Rodonó Reference Rodonó1986). There are lots of studies in the literature that UV Ceti type stars are very young stars and their population increases in the open cluster and especially in the associations (Mirzoyan Reference Mirzoyan1990; Pigatto Reference Pigatto, Mirzoyan, Pettersen and Tsvetkov1990). The population rate of the flaring red dwarfs is decreasing while the age of the cluster is increasing, as it is expected according to Skumanich’s law (Skumanich Reference Skumanich1972; Pettersen Reference Pettersen1991; Stauffer Reference Stauffer, Catalano and Stauffer1991; Marcy & Chen Reference Marcy and Chen1992). The mass loss is one of the important parameters, which affect the stellar evolution, especially for the low-mass stars like red dwarfs. In the case of magnetic active stars, the chromospheric activity of the stars are relatively very strong due to the their high rotational speed, when the stars have just came to the Zero Age Main Sequence (ZAMS), which led the rate of the mass loss to increase. The observations indicate that the mass loss rate of the Sun is about 2 × 10−14 M y−1 (Gershberg Reference Gershberg2005); however, it is about several 10−10 M y−1 in the case of young UV Ceti type stars (Schrijver and Zwaan Reference Schrijver and Zwaan2000; Boyarchuk Reference Boyarchuk and Habing1970). This indicates that the UV Ceti type stars lose the largest part of their angular momentum in the beginning of the main-sequence life (Marcy & Chen Reference Marcy and Chen1992).

However, we do not exactly know how the flare process causing mass loss with the high rate is working on a UV Ceti type star. The highest energy detected from two-ribbon flares that is the most powerful flares occurring on the sun is found to be 1030–1031 erg (Gershberg Reference Gershberg2005; Benz Reference Benz2008). Beside the Sun, RS VCn binaries, whose components are generally from the spectral types G or K giants, also exhibit the flare activity. Although their flare activity are generally observed in the radio or X-Ray bands (Pandey & Singh Reference Pandey and Singh2012), there are several white-light observations of the visible region, in which the white-light flare light curves have been obtained (Mathioudakis et al. Reference Mathioudakis, Doyle, Avgoloupis, Mavridis and Seiradakis1992). Examining these white-light light curves, the energies emitting in the flares detected from RS CVn binaries are about 1031 erg (Haisch, Strong, & Rodonó Reference Haisch, Strong and Rodonó1991). Apart from RS CVn binaries, the ground-based observations demonstrated that the events are more frequently occurring on the surfaces of the dMe stars. In the case of dMe stars, the flare energy can reach a level from 10 times to 1 000 times of the level reached in the solar case (Güdel & Nazé Reference Güdel and Nazé2009). In fact, the observations reveal that the flare energy level varies from 1028 to 1034 erg in the case of dMe stars (Haisch et al. Reference Haisch, Strong and Rodonó1991; Gershberg Reference Gershberg2005). In addition, some dMe stars in the young clusters such as the Pleiades cluster and Orion association exhibit some powerful flare events, whose energies reach 1036 erg (Gershberg & Shakhovskaya Reference Gershberg and Shakhovskaya1983).

As it has been summarised above, there are so serious differences between the flare patterns, such as the flare energies and mass loss, of different type stars, such as the solar and dMe cases, that their flare process should be different from each other. However, the flare process working in UV Ceti stars has been explained by the Standard Solar Dynamo in spite of all. Especially, the source of the flare energy in the case of dMe stars is generally assumed the magnetic reconnection processes (Gershberg Reference Gershberg2005; Hudson & Khan Reference Hudson, Khan, Bentley and Mariska1997). On the other hand, there are lots of the points waiting to be explained. At this point, comparing the flares obtained from the different type stars, all the similarities and differences should be demonstrated. For example, first of all, the sources of the differences in some parameters like the flare energy spectra or flare frequency should be found. In this aim, it should be analysed whether the differences in the flare parameters from a star to the next are caused by the some physical parameters of the source stars, such as stellar mass, age, etc. At this point, there is also one more question that it should be analysed whether being a binary system causes some effects on the flare activity occurring on the components.

To understand the real reasons of the differences in the flare parameters from a star to the next, it should be demonstrated the nature of these stars with all their internal layers. There are few indicators to reveal the internal layers of the stars. One of them is the stellar pulsation. Unfortunately, the pulsations cannot be observable for each star. On the other hand, the initial analysis and some studies in the literature, such as Uytterhoeven et al. (Reference Uytterhoeven2011), indicated that KIC 2557430 is one of the candidates for the eclipsing binaries with pulsating component. Moreover, the pulsation behaviour seen in the stars is very important pattern to understand the stellar itself and its evolution. According to the observations lasting as long as several decades indicate that there are several type pulsating stars such as Cepheid, γ Doradus, and δ Scuti type pulsating stars in the Instability Strip in the Hertzsprung–Russell diagram, especially on the main sequence. All these types are separated by their locations in the Instability Strip from each other. Analysing the pulsation frequencies, which is generally known as stellar seismology called asteroseismology, the physical processes behind both the pulsating nature and stellar interiors can be revealed. This is why the pulsating stars have an important role to understanding stellar evolution (Cunha et al. Reference Cunha2007; Aerts, Christensen-Dalsgaard, & Kurtz Reference Aerts, Christensen-Dalsgaard, Kurtz, Burton and Trimble2010).

In addition, a pulsating star could be a component in an eclipsing binary system. In this case, this pulsating star has more important role to figure out the stellar evolution process. This is because, in some cases, the pulsation features cannot be enough to reveal the entire stellar interior and its physical parameters. However, analysing the light curves of the eclipsing binaries, lots of parameters such as mass (M), radius (R), and logg, can be easily obtained (Wilson & Devinney Reference Wilson and Devinney1971; Wilson Reference Wilson1990). At this point, considering the results obtained from both pulsating and eclipsing behaviour, the physical natures of components can be easily identified (Lampens Reference Lampens, Sterken and Aerts2006; Pigulski Reference Pigulski, Sterken and Aerts2006). As it is well known, there are many pulsating single stars in the Instability Strip. However, the number of the pulsating stars being a component in an eclipsing binary is very small (Kim et al. Reference Kim, Lee, Lee and Youn2010).

In this study, we analysed the light variations of KIC 2557430, which is one of the first samples for its kind. One component of this system is seen as a pulsating star and another one is a chromospherically active star with strong flare activity. However, this chromospherically active component has some different physical properties from its analogue UV Ceti type stars due to being a component in a binary system. In this aim, first of all, the frequencies of the variations out-of-eclipses were analysed to figure out source of the variations. Then, we did complete light curve analyses of the system for the first time in the literature in order to find out the physical properties of the components, using the phoebe V.0.32 software (Prša & Zwitter Reference Prša and Zwitter2005), whose method depends on the 2003 version of Wilson–Devinney Code (Wilson & Devinney Reference Wilson and Devinney1971; Wilson Reference Wilson1990) . Finally, the flares occurring on the chromospherically active component were detected to model the magnetic activity nature of the system, comparing the active component with its analogue. Unfortunately, the physical parameters of internal layers for a star can be determined by analysing the multicolour light curves in the case of the photometric data (Soydugan et al. Reference Soydugan, Soydugan, Kanvermez and Liakos2013). At this point, the data used in this study were taken from the Kepler Mission Database, which are the monochromatic data (Borucki et al. Reference Borucki2010; Koch et al. Reference Koch2010; Caldwell, Kolodziejczak, & Van Cleve Reference Caldwell, Kolodziejczak and Van Cleve2010). Because of this, the pulsation modes were not determined from available data. In the second step of the study, we will try to obtain the multicolour photometric data, then we can complete the pulsation analysis. In this paper, we just presented the pulsation frequencies determined from the available monochromatic data.

KIC 2557430 is listed as an Algol type eclipsing binary with a brightness of V = 11 m .63, in the SIMBAD Database. Although the system was listed in the Tycho Inpute Catalogue by Egret et al. (Reference Egret, Didelon, McLean, Russell and Turon1992) for the first time, there are so few studies about this system in the literature. The system was listed as 2MASS J19072286+3748571 in the 2MASS All-Sky Survey Catalogue (Cutrie et al. Reference Cutri2003), in which its JHK brightness were given as J = 13 m .574, H = 12 m .926, K = 12 m .782 (Kharchenko Reference Kharchenko2001). In the literature, there are several approaches for the temperature of the system, though it has been observed very long time in the Kepler Mission (Borucki et al. Reference Borucki2010; Koch et al. Reference Koch2010; Caldwell et al. Reference Caldwell, Kolodziejczak and Van Cleve2010). Considering the data taken by the Kepler Satellite (Slawson et al. Reference Slawson2011) computed the inclination (i) of the system as 77°.17 and the colour excess as E(BV) = 0 m .077, while the temperature of the system was found to be 6 248 K with the temperature ratio of 0.951 for the components. Examining all the data in the literature of KIC 2557430, Pinsonneault et al. (Reference Pinsonneault, An, Molenda-Zakowicz, Chaplin, Metcalfe and Bruntt2012) found its metallicity as [Fe/H] = −0.34 dex, and stated that the temperature of the primary component has a value between 6 547–6 248 K. Huber et al. (Reference Huber2014) found the temperature of the system between 6 539–6 531 K, and computed the mass and radius of the primary component as M =1.117 M and R =1.524 R. Using some calibrations obtained from the data taken by the 2MASS All-Sky Survey and the Kepler Mission, Armstrong et al. (Reference Armstrong, Chew, Faedi and Pollacco2014) found the temperature of the system as 6 913 K. In the literature, there are so few studies, in which the variations out-of-eclipses were analysed. The period of the system was found to be 1.297743 d for the first time by Uytterhoeven et al. (Reference Uytterhoeven2011), who stated that KIC 2557430 is an eclipsing binary with a γ Doradus-type component. Coughlin et al. (Reference Coughlin2014) confirmed that the orbital period of the system is 1.297747 d. However, Balona (Reference Balona2015), who revealed the flare activity detected from the system, stated that the rotational period is 2.02 d for KIC 2557430 apart from the orbital period.

2 DATA AND ANALYSES

The data analysed in this study were taken from the Kepler Mission Database (Slawson et al. Reference Slawson2011; Matijevič et al. Reference Matijevič2012). The Kepler Mission is a space mission in the aim of finding out exoplanet. More than 150.000 targets have been observed in this mission from 2009 (Borucki et al. Reference Borucki2010; Koch et al. Reference Koch2010; Caldwell et al. Reference Caldwell, Kolodziejczak and Van Cleve2010). The quality and sensitivity of Kepler observations have the highest one ever reached in the photometry (Jenkins et al. Reference Jenkins2010a, Reference Jenkins2010b). Lots of variable stars, such as new eclipsing binaries or pulsating stars, etc., have been also discovered apart from the exoplanets in this mission (Slawson et al. Reference Slawson2011; Matijevič et al. Reference Matijevič2012). In addition, the observations indicate that an important part of single or double stars discovered among these newly discoveries, which some of them are the eclipsing binaries, exhibiting chromospheric activity (Balona Reference Balona2015).

Considering the analyses of the flare activity in the study, the data were taken in short cadence format from the Database. All the available data reveal that KIC 2557430 has been observed in two observing parts. One of them was lasting about one month between HJD 24 55002.5109509–24 55033.3041338, while the second one was again lasting about one month but between HJD 24 55093.2155524– 24 55123.5566582. Because of this, the analyses were sometimes done in two steps, and the first month observations were called as Part 1 data and the second month observations were called as Part 2 data in these analyses.

(1) $$\begin{eqnarray} JD (\text{Hel.}) & = & 24 54954.461036 \pm 0.017907 \nonumber\\ &&+\, 1^{d}.12977364 \pm 0^{d}.0000015 \times E. \end{eqnarray}$$

The data were phased by using both the epoch and the orbital period given by Equation (1), which were taken from the Kepler Mission database and the light curves versus phase are shown in Figure 1. The entire light curves were shown in the bottom panel, while the light curves out-of-eclipses are shown in the upper panel for better visibility of light variations. As it is seen from the upper panel of the figure, the light curves out-of-eclipses are changing from one cycle to the next. Three variations are seen from the figure. The one of them is the primary and secondary minima due to the eclipses; the second one is a sinusoidal variation, and finally, the last one is instant short-term flare events. In the analyses, we arranged the data in suitable format considering the light curve analysis, sinusoidal variation analysis, and the minimum time variation (OC).

Figure 1. All the light curves of KIC 2557430 obtained from the available short cadence data in the Kepler Mission Database. The full of the light curves are shown in the bottom panel, while the maxima of the curves are shown in the upper panel to reveal the variations out of eclipses.

2.1. Variability out-of-eclipses: Pulsation

To examine the sinusoidal variations out-of-eclipses, both all the minima due to the eclipses and all the flare events, sudden—rapid increasing in the light, were removed from the entire light curves. Thus, the remaining light curves were obtained, which is hereafter called as the pre-whitened light curves. For this purpose, the data of all primary minima observations between the phases of 0.955−0.045 and all secondary minima observations between 0.455 and 0.545 in phase were removed from the general light curve data. Comparing the consecutive light curve cycles in the pre-whitened light curves revealed that the consecutive cycles are absolutely different from each other. It is seen that the phases and levels of maxima and minima are rapidly changing from one cycle to the next.

In the first place, the source of these variations is seen as the rotational modulation due to the stellar cool spots. However, considering both the orbital period of 1.297747 d and the flare activity, there must be another source affected these variations out-of-eclipses. If the sinusoidal variations out-of-eclipses were caused due to just spot activity, the shape of the consecutive pre-whitened light curves should not been absolutely changed in 1.297747 d from one cycle to the next. Because of this, there must be one more source like stellar pulsation. In fact, Uytterhoeven et al. (Reference Uytterhoeven2011) indicated that KIC 2557430 is an eclipsing binary with a γ Doradus-type component.

In this purpose, the pre-whitened light curve data were analysed with the PERIOD04 program (Lenz & Breger Reference Lenz and Breger2005), which depends on the method of Discrete Fourier Transform (DFT) (Scargle Reference Scargle1982). The results obtained from DFT were tested by two other methods. One of them is CLEANest, which is another Fourier method (Foster Reference Foster1995), and the second method is the Phase Dispersion Minimization (PDM), which is a statistical method (Stellingwerf Reference Stellingwerf1978).

(2) $$\begin{equation} L(\theta )= A_{0} + \sum _{\mbox{\ i=1}}^N A_{i} \cos (i \theta ) + \sum _{\mbox{\ i=1}}^N B_{i} \sin (i \theta ), \end{equation}$$

where A 0 is the zero point, θ is the phase, while A i and B i are the amplitude parameters.

Considering both the errors of parameters and also the signal-to-noise ratios (S/N), the results of the frequency analysis by the PERIOD04 program (Lenz & Breger Reference Lenz and Breger2005), which is based on Equation (2) described by Scargle (Reference Scargle1982) and Lenz & Breger (Reference Lenz and Breger2005), indicate that there are 50 different frequencies. The normalised power-spectrums distribution obtained from the DFT (Scargle Reference Scargle1982) is shown in Figure 2, while the obtained parameters are listed in Table 1. In the table, the frequency numbers are listed in the first column, while the obtained frequency values are listed in the second column. The amplitudes are listed in the fourth column, while the phase values are listed in the sixth column. The error of each parameter is listed in just next column in the table. In addition, the S/N of each frequency is listed in the last column.

Figure 2. The normalised power-spectrums distribution obtained from the Discrete Fourier Transform (Scargle Reference Scargle1982).

Table 1. All the frequencies and their parameters obtained from the Discrete Fourier Transform (Scargle Reference Scargle1982), using the PERIOD04 program.

Examining each frequency of all 50 frequencies, it is seen that the frequencies F1 and F4 are relevant to the orbital period of system. The frequency F1 is relevant to the orbital period itself, while the frequency F4 is relevant to the half of this period. In this case, these two frequencies must be relevant to a variation caused any rotational modulation possibly due to the stellar cool spots. Therefore, 48 frequencies must be relevant to the stellar pulsation apart from frequencies F1 and F4.

Using the obtained frequencies, the synthetic light curve was derived for the variation out-of-eclipses by Equation (2) (Scargle Reference Scargle1982; Lenz & Breger Reference Lenz and Breger2005). This synthetic curve and the pre-whitened are shown in Figure 3. As it is seen from the figure, the synthetic curve perfectly modelled the pre-whitened curve, which indicates that the analysis correctly worked.

Figure 3. The variation out-of-eclipses and the synthetic model derived by the frequencies obtained from the Discrete Fourier Transform (Scargle Reference Scargle1982). The filled circles represent the observations, while the red line represents the model.

2.2. Variability out-of-eclipses: Stellar spot activity

After removing all the synthetic sinusoidal waves associated with just 48 frequencies of the pulsations from all the pre-whitened light curves, we got the residual variations, which are associated with the rest two frequencies listed as F1 and F4 in Table 1. However, as it was shown in Figure 4, these residual data show very striking variation because there is still a sinusoidal variation. However, this variation is not stable. As it is seen from Figure 3, the shape of the residual light curve obtained with 48 frequencies is not changing along several cycles of orbital period. This is because this variation is caused due to stellar pulsation. On the other hand, it is clearly seen that the shape of the residual light curve obtained with the frequencies of F1 and F4 is absolutely changing from the Part 1 data to the Part 2 data. As it is seen from Figure 4, the phase of the residual variation and its amplitude are absolutely changed from the Part 1 data to the Part 2 data. It seems to be that there are two minima, which the deeper one is located in the later phases, in the residual light curve of Part 1 data, while the residual light curve of the Part 2 data has just one minimum, which is located in earlier phases. However, the latter light curve is seen more asymmetric. There are two months from the observations of Part 1 to those of Part 2. In this time interval, the phase of the deeper minimum has migrated from the phase of 0.64 to 0.24.

Figure 4. The residual variations obtained removing all the pulsation variations from all the pre-whitened light curves of KIC 2557430. In each panel, the black circles represent the first 10-d observations, the blue circles represent the second 10-d observations, while the red circles represent the third 10-d observations for both Part 1 data taken between HJD 24 55002.51095–24 55033.30413 and Part 2 data taken between HJD 24 55093.21555– 24 55123.55666.

We wanted to check whether this shape change is a systematic slow variation or unsystematic sudden. For this purpose, the consecutive cycles of the residual light curve was examined for Part 1 and Part 2, separately. It was seen that the shape of the residual light curve is so slowly changing from one cycle to the next that the variation needs eight or nine cycles to be noticeable. Because of this, the residual data of each part observations were separated to three subsets that each of them contains about 10-d data. In the upper panel of Figure 4, the data of subsets were plotted in different colour for both part observations. To better view, each of the 10-d data was averaged phase by phase with interval of 0.01. The averaged light curves were plotted in the bottom panel of Figure 4. As it is seen from this panel, the shape of the residual light curve is clearly changing from the one 10-d data to the next. In Figure 4, the filled black points represent the first 10-d data, while the filled blue points represent the second 10-d data, and finally, the filled red points represent the last 10-d data. As it is seen in the figure, the minima of the residual light curves are migrating towards the earlier phases from the first 10-d data to the last 10-d data in both part observations. These small migrations in each part support that the shape of the residual light curve is slowly and systematically changing from the Part 1 data to the Part 2 data.

Apart from the pulsation, this variation can be caused by a third body or the stellar cool spot activity. In the first places, the reason of the variation cannot be third body. As it can be seen from the next section, although there are some light excess in the total light of the system due to a third body, but this is so small excess to show any sinusoidal variation seen in Figure 4. However, considering the existence of flare activity, it means that there are the spotted areas migrating towards the earlier longitudes on a component.

2.3. Light curve analysis

Examining the entire light curves of KIC 2557430 observing by Kepler Mission along two months cycle by cycle, it was seen that there are three different variations, such as eclipses, flare, and sinusoidal variation out-of-eclipses, in the light curves. However, the frequency analyses indicated that the sinusoidal variation is not just caused by the stellar pulsations, but also a stellar spot activity on a component. Stellar spot activity can be easily modelled in the light curve analyses, but the instant-short-term variation like flare activity or complex cyclic variation in the different phase like pulsation cannot model in the light curve analyses. Because of this, the variations caused by both the flare activity and pulsation waves from all the entire light curves were removed, before analysing the light curve. The flare activity as an instant short-term is clear to easily detect with their distinct light variation. Therefore, first of all, we removed all the variations due to the flare activity from the data. After then, the synthetic light curve, derived with 48 frequencies obtained from the frequency analyses, shown in Figure 3 was also removed from all the entire light curves. However, the residual sinusoidal variation caused by both the frequencies listed as F1 and F4 in Table 1 was leaved to be modelled in the light curve analysis. However, the shape of the residual sinusoidal variation is different in the Part 1 and 2 observations. Because of this, in the light curve analysis, the residual sinusoidal variation was modelled separately for the Part 1 and Part 2 data.

Using the phoebe V.0.32 software (Prša & Zwitter Reference Prša and Zwitter2005), which is employed in the 2003 version of the Wilson-Devinney Code (Wilson & Devinney Reference Wilson and Devinney1971; Wilson Reference Wilson1990), we analysed the light curves obtained from the averages of all the detrended short cadence data. We attempted to analyse the light curves in various modes, including the detached system mode (Mod2), semi-detached system with the primary component filling its Roche-Lobe mode (Mod4), and semi-detached system with the secondary component filling its Roche-Lobe mode (Mod5). If the obtained stellar absolute parameters and the stellar evolution models are considered together in the analysis, the initial test demonstrated that an astrophysically reasonable solution was obtainable only in the detached system mode; no results that were statistically consistent with reasonable solutions could be obtained in any of the other modes.

Although there is no much more detailed studies depending on the light curve analysis for the system in the literature, lots of temperature values were given for the system, but there is no any clear vision about the temperature of the system. Because of this, taking each temperature value given in the literature as the temperature of the primary component respectively in the initial tests, we tried to find out which temperature value is the correct one. These tests indicated that an astrophysically reasonable solution can be obtainable only taking the temperature value of 6 913 K given by Armstrong et al. (Reference Armstrong, Chew, Faedi and Pollacco2014) for the primary component. Thus, the temperature of the primary component was fixed to 6 913 K, while the temperature of the secondary component was taken as adjustable parameter. Considering the spectral type corresponding to this temperature, the albedos (A 1 and A 2) and the gravity-darkening coefficients (g 1 and g 1) of the components were adopted for the stars with the convective envelopes (Lucy Reference Lucy1967; Rucinski Reference Rucinski1969). The non-linear limb-darkening coefficients (x 1 and x 2) of the components were taken from van Hamme (Reference van Hamme1993). In the analyses, their dimensionless potentials (Ω1 and Ω2), the fractional luminosity (L 1) of the primary component, the inclination (i) of the system, the mass ratio of the system (q), and the semi-major axis (a) were taken as the adjustable free parameters. In addition, the fractional luminosity of the third body is also taken as the adjustable free parameter.

In addition, we modelled the residual sinusoidal variations out-of-eclipses by two cool spots on the secondary component in the phoebe V.0.32 software. Moreover, it was seen that the light curve analysis also gives a third light contribution of 0.339%. All the parameters obtained from the light curve analysis were listed in Table 2, while the synthetic light curve derived with these parameters is shown in Figure 5. It must be noted that the temperature of the secondary component was found to be 6271±1 K. The error of the temperature is not realistic value, this is because the temperature error is statistically found in the process of the used code.

Table 2. The parameters obtained from the light curve analysis of KIC 2557430.

Figure 5. The observational and the synthetic light curves obtained from the light curve analyses of KIC 2557430 for the observations of Part 1 data taken between HJD 24 55002.51095–24 55033.30413 (upper panel) and Part 2 data taken between HJD 24 55093.21555– 24 55123.55666 (bottom panel). The filled circles represent the observations, while the red line represents the model.

In the light curve analysis, the luminosity of the primary component was found to be 6.75791 L. Taking into count both its temperature and the luminosity, the primary component was plotted in the log(T eff)−log(L/L) plane in Figure 6. As it is seen from the figure, the primary component is located among the γ Doradus type stars in the Instability Strip derived with the parameters taken from Girardi et al. (Reference Girardi, Bressan, Bertelli and Chiosi2000) and Rolland et al. (Reference Rolland, Costa, Rodriguez, Amado, Garcia-Pelayo, de Coca and Olivares2002).

Figure 6. The place of the primary component of KIC 2557430 among γ Doradus type stars in the HR diagram. In the figure, the small filled black circles represent γ Doradus type stars listed in Henry et al. (Reference Henry, Fekel and Henry2005). The asterisk represents the primary component of the system. The dash dotted lines (red) represent the borders of the area, in which γ Doradus type stars take place. In addition, we plotted the hot (HB) and cold (CB) borders of the δ Scuti stars for comparison. In the figure, the small filled pink circles represent some semi- and un-detached binaries taken from Soydugan et al. (Reference Soydugan, Soydugan, Demircan and İbanoǧlu2006) and references there in. The ZAMS and TAMS were taken from Girardi et al. (Reference Girardi, Bressan, Bertelli and Chiosi2000), while the borders of the Instability Strip were computed from Rolland et al. (Reference Rolland, Costa, Rodriguez, Amado, Garcia-Pelayo, de Coca and Olivares2002).

2.4. Orbital period variation: OC analysis

Using the available short cadence detrended data of the system in the Kepler Mission Database (Slawson et al. Reference Slawson2011; Matijevič et al. Reference Matijevič2012), the minima times were computed without any extra correction on these detrended data. The minima times were computed with a script depending on Kwee and van Woerden method described by Kwee & van Woerden (Reference Kwee and van Woerden1956), which considers just minima with its branches to compute the time of the minima, using a theoretical fit derived by the Least Squares Method. For all the minima times, the differences between observations and calculations were computed to determine the residuals (OC) I . Some minima times have very large error, for which the light curves were examined again. It was seen that there is a flare activity in these minima, then, these minima times were removed from the list. Finally, 89 minima times were obtained from the available short cadence detrended data. Using the regression calculations, a linear correction was applied to the differences, and the (OC) II residuals were obtained. After the linear correction on (OC) I , new ephemerides were calculated as following:

(3) $$\begin{equation} JD (\text{Hel}.) = 24 54954.46175(2) + 1^{d}.1297728(3) \times E. \end{equation}$$

All the calculated minima times, (OC) II calculated from the differences between observations and calculations (OC) were listed in Table 3. The minima times, epoch, minimum type, and (OC) II residuals are listed in the table, respectively. It is seen as an interesting phenomenon in the variation of the (OC) II residuals versus time. The (OC) II residual variations are shown in Figure 7. A similar phenomenon has been recently demonstrated for chromospherically active other systems by Tran et al. (Reference Tran2013) and Balaji et al. (Reference Balaji, Croll, Levine and Rappaport2015).

Table 3. All the minima times and (OC) II residuals.

Figure 7. The variations of the minima times computed from the available short cadence data in the Kepler Database. All the (OC) II residuals for both the primary and the secondary minima and the parabola fit derived from all the minima are shown in the upper panel, while the (OC) II residuals for both the primary and the secondary minima are shown separately in the bottom panel. In both panels, the filled circles represent the (OC) II residuals, while the lines represent the parabola fits. In the bottom panel, the red circles represent the secondary minima, while the blue circles represents the primary minima.

2.5. Flare activity and the OPEA model

To understand the flare behaviour of the system, first of all, it needs to determine the flares from the available data. Then, it needs to determine and model the quiescent levels at the moment of the flares. For this purpose, all the primary minima between the phases of 0.955−0.045 and all the secondary minima between 0.455 and 0.545 in phase were removed from the entire light curves. Then, the observations with large error caused by some technical problems were also removed from the light curves.

To compute the parameters of a flare, it needs to determine where a flare is beginning and end. In this aim, we attempted to derive the quiescent levels in the light curves. At this point, the synthetic models obtained by the frequency analyses were used. All the sinusoidal variations, which are occurring due to the stellar pulsation and also the stellar spot activity, were derived for the light curves out-of-eclipses. At this point, the synthetic model was re-derived with the Fourier transform again, using all the found 50 frequencies for this time. In each point of the entire light curve, this synthetic model was assumed as the quiescent level of the light curve without any flare events. Some samples of the detected flares and the synthetic quiescent light curve at the moment are shown in Figure 8. The filled circles represent the observations, while the red lines represent the synthetic quiescent level of the light curve.

Figure 8. The flare light curve samples chosen from different parts of the short cadence data in the Kepler Database. In the figures, the filled circles represent the observations, while the red lines represent the synthetic curves assumed as the quiescent state of the star.

Using this synthetic model, the flare rise time (T r), the decay time (T d), amplitude of the flare maxima, and the flare equivalent duration (P) were computed for each flare, after defining both the flare beginning and the end for each flare. All these parameters are listed in Table 4.

Table 4. All the calculated parameters of flares detected from the short cadence observational data of KIC 2557430. As it is explained in the text, the flares were separated into two groups, such as Group 1 and Group 2.

Considering all the available short cadence data given in the Kepler Database, 69 flares were detected in total. In the analysis, the equivalent duration of each flare was computed using Equation (4) taken from Gershberg (Reference Gershberg1972):

(4) $$\begin{equation} P = \int [(I_{\text{flare}}-I_{0})/I_{0}] \text{d}t, \end{equation}$$

where P is the flare-equivalent duration in the observing band, while I 0 is the flux of the star in the observing band while in the quiet state. As it has just been described above, we computed the parameter I 0 using by the synthetic models derived with the Fourier transform. I flare is the intensity observed at the moment of the flare. Here it should be noted that the flare energies were not computed to be used in the following analyses due to the reasons described in detail by Dal & Evren (Reference Dal and Evren2010, Reference Dal and Evren2011). Instead of the flare energy, flare equivalent duration has been used in the analysis. This is because of the luminosity term in the equation of flare energy, given by Gershberg (Reference Gershberg1972). The luminosities of stars with different spectral types have large differences. Although the equivalent durations of two flares detected from two stars in different spectral types are the same, calculated energies of these flares are different due to different luminosities of these spectral types. Therefore, we could not use these flare energies in the same analysis. However, flare equivalent duration depends just on power of the flare. Moreover, the given distances of a star in different studies could be quite different. These differences cause the calculated luminosities become different.

In a result, obtained parameters, such as flare maximum times, equivalent durations, rise times, decay times, and amplitudes of flare maxima, are listed from the first column to the last in Table 4, respectively.

Examining the relationships between the flare parameters, it was seen that the distributions of flare equivalent durations on the logarithmic scale versus flare total durations are varying in a rule. The distributions of flare equivalent durations on the logarithmic scale cannot be higher than a specific value for the star, and it is no matter how long the flare total duration is. Using the SPSS V17.0 (Green, Salkind, & Akey Reference Green, Salkind and Akey1999) and GrahpPad Prism V5.02 (Dawson & Trapp Reference Dawson and Trapp2004) programs, Dal & Evren (Reference Dal and Evren2010, Reference Dal and Evren2011) demonstrated that the best function is the One Phase Exponential Association (hereafter OPEA) for the distributions of flare equivalent durations on the logarithmic scale versus flare total durations. The OPEA function (Motulsky Reference Motulsky2007; Spanier & Oldham Reference Spanier and Oldham1987) has a Plateau term, and this makes it a special function in the analyses. The OPEA function is defined by Equation (5):

(5) $$\begin{equation} y = y_{0} + (\text{Plateau} - y_{0}) \times (1 - e^{-k \times x}), \end{equation}$$

where the parameter y is the flare equivalent duration on a logarithmic scale, the parameter x is the flare total duration as a variable parameter, according to the definition of Dal & Evren (Reference Dal and Evren2010). In addition, the parameter y 0 is the flare-equivalent duration in on a logarithmic scale for the least total duration, which it means that the parameter y 0 is the least equivalent duration occurring in a flare for a star. Logically, the parameter y 0 does not depend on only flare mechanism occurring on the star, but also depends on the sensitivity of the optical system used for the observations. In this case, the optical system is optical systems of the Kepler Satellite. The parameter Plateau value is upper limit for the flare equivalent duration on a logarithmic scale. Dal & Evren (Reference Dal and Evren2011) defined Plateau value as a saturation level for a star in the observing band.

After the OPEA model was derived for all 69 flares detected from KIC 2557430, it was seen that the correlation coefficient squared (R 2) is very low, while the probability value (p-value) is found to be very high. It means that the model does not perfectly fit the distributions. In fact, it had been seen that the distributions of flare equivalent durations on the logarithmic scale (logP) versus flare total time (T t) split into two groups in the T t − log(P) plane. Especially, this dissociation gets much clearer to be seen for the flares, whose total flare time is longer than 1 400 s. Because of this, the flares with the total times longer than 1 400 s were split into two groups. Then, using the least-squares method, the OPEA models were derived separately for two groups. In addition, the confidence intervals of 95% were also derived in these models. In the second step, considering the derived two OPEA models with the confidence intervals of 95%, the flares with the total times shorter than 1 400 s were separated into these two groups. As a result, all the distributions of flare equivalent durations on the logarithmic scale (logP) versus flare total time (T t) and all the derived models with the confidence intervals of 95% are shown in Figure 9. In the figure, the filled red circles represent the flares called as Group 1 in this paper, while the filled blue circles represent the flares of Group 2. Using the least-squares method, the parameters of both models were computed and listed in Table 5. The span value listed in the table is difference between Plateau and y 0 values. The half-life value is equal to ln2/K, where K is a constant expressing in the same units as the x value, at the model reaches the Plateau value (Dawson & Trapp Reference Dawson and Trapp2004). In other words, the n × half-life parameter is half of the minimum flare total time, which is enough to the maximum flare energy occurring in the flare mechanism.

Figure 9. Using the least-squares method, the OPEA models derived from the detected 69 flares. In the figure, the filled circles represent the observations, while the lines represent the models.

Table 5. The parameters obtained from the OPEA models using the least-squares method.

The OPEA models derived for both groups was tested by using three different methods, such as the D’Agostino–Pearson normality test, the Shapiro–Wilk normality test, and also the Kolmogorov–Smirnov test, given by D’Agostino & Stephens (Reference D’Agostino, Stephens, D’Agostino and Stephens1986) to understand whether there are any other functions to model the distributions of flare equivalent durations on the logarithmic scale versus flare total durations. In these tests, as it is listed in Table 5, the probability value called as p-value was found to be p-value < 0.001, and this means that there is no other function to model the distributions of flare equivalent durations (Motulsky Reference Motulsky2007; Spanier & Oldham Reference Spanier and Oldham1987). Therefore, as it is seen from the correlation coefficient squared (R 2) obtained for both models, the separation of the flares as two groups are statistically real.

In the Kepler Mission program, KIC 2557430 was observed along 61.134289 d (1467.22299 h) in total, from HJD 24 55002.5109509 to 24 55033.3041338 and from 24 55093.2155524 to 55123.5566582. The significant 69 flares were detected in total from the available data. As it is listed in Table 6, 40 samples of all flares belong to Group 1, while the 29 of them belong to Group 2. Ishida et al. (Reference Ishida, Ichimura and Shimizu1991) described two frequencies for the stellar flare activity. These frequencies are defined as given by Equations (6) and (7):

(6) $$\begin{equation} N_{1} = \Sigma n_{\text{f}} / \Sigma T_{\text{t}}, \end{equation}$$
(7) $$\begin{equation} N_{2} = \Sigma P / \Sigma T_{\text{t}}, \end{equation}$$

where Σn f is the total flare number detected in the observations, and ΣT t is the total observing duration, while ΣP is the total equivalent duration obtained from all the flares. In this study, both N 1 and N 2 flare frequencies were computed for all flares and were also computed separately for both groups. All the results of the flare frequencies are listed in Table 6.

Table 6. Flare frequencies computed for all flares and grouped flares.

3 RESULTS AND DISCUSSION

In this study, we exerted remarkable effort to figure out the nature of the KIC 2557430, which is classified as Algol type binary in the SIMBAD Database, taking the observational data from the database of the Kepler Mission (Slawson et al. Reference Slawson2011; Matijevič et al. Reference Matijevič2012). The initial analyses indicate that there are three type variations, such as eclipses, sinusoidal variations, and flare activity, different from each other. The data were analysed in the suitable ways to figure out these sources.

First of all, examining the variations out-of-eclipses indicates that both the shapes and the phases of the wave minima are changing from one cycle to the next, whose cycles are computed with the orbital period. Considering the flare activity detected from the system, this variation seems to be caused by the stellar cool spots on the active component. However, the orbital period of the system is given as 1.297747 d in the literature (Coughlin et al. Reference Coughlin2014). In addition, the rotational period is given as 2.02 d in just one study for KIC 2557430 (Balona Reference Balona2015). In this case, the changing shape of the light curve out-of-eclipses cannot be explained by the stellar spot activity. There must be some other sources to rapid changes in the light curve out-of-eclipses. Because, it should not be expected that the configuration of the spotted areas on a star is changing to cause the radical changes in the light curve shape in the short time intervals such as 1.297747 or 2.02 d (Gershberg Reference Gershberg2005). As it is known that one of the components is a γ Doradus-type star (Uytterhoeven et al. Reference Uytterhoeven2011). Considering the available information about the system, we analysed the light variation out-of-eclipses to find its characteristic frequencies. In total, 50 frequencies with S/N of 5.0 or with larger ratio than 5.0 were found in the analysis of the PERIOD04 program. Two of the found frequencies listed in Table 1, F1 and F4, are relevant to the orbital period of the system given by Coughlin et al. (Reference Coughlin2014) and its half. Thus, the rest of them are relevant to the stellar pulsation. The later analyses indicated that there is no more astrophysically acceptable frequency. As it seen from Figure 3, the synthetic curve obtained from the found frequencies is really well fitted the observations.

Although the frequencies F1 and F4 are relevant to the orbital period, if the synthetic curve is obtained with the rest 48 frequencies, it is seen that this synthetic curve cannot perfectly fit the observations. If this synthetic curve is extracted from the observations, it is seen that a residual sinusoidal variation still remain in the light curve. It means that there is another source for the variation out-of-eclipses apart from the stellar pulsation. In addition, the analysis of this residual sinusoidal variation indicated an interesting result that the shape of this variation is dramatically changing from the Part 1 observations to the Part 2. Considering that these two frequencies are relevant to the orbital period, this residual sinusoidal variation out-of-eclipses must be caused by the rotational modulation due to the stellar cool spots probably. Kron (Reference Kron1952) demonstrated that the UV Ceti type stars also exhibit the stellar cool pot activity too. This phenomenon, called BY Dra syndrome in later years, has been studied for several decades (Bopp & Evans Reference Bopp and Evans1973; Kunkel Reference Kunkel1975; Vogt Reference Vogt1975). As it is in the case of Solar, it has been observed in lots of cases that a spotted area exhibits a cyclic migration on the surface of the star, while its location and size are varying at the moment, evolving on the surface by the time (Fekel et al. Reference Fekel, Henry, Eaton, Sperauskas and Hall2002; Berdyugina Reference Berdyugina2006; Oláh et al. Reference Oláh, Korhonen, Kövári, Forgcs-Dajka and Strassmeier2006; Korhonen & Elstner Reference Korhonen and Elstner2005; Korhonen and Järvinen Reference Korhonen, Järvinen, Hartkopf, Guinan and Harmanec2007; Gershberg Reference Gershberg2005). In the case of KIC 2557430, the residual sinusoidal variation must be caused by the spotted area, which is evolving while it is also migrating on the surface. This is explained why the shapes of the residual sinusoidal variation are different in Parts 1 and 2. Consequently, considering the available flare observations, the sinusoidal variation found as the residual of the stellar pulsation in the pre-whitened light curves must be caused the rotational modulation due to the stellar chromospheric activity.

Although there are several approaches about the physical parameters of the system found using some calibration with the available data in the literature, there is no any entire light curve analysis for KIC 2557430. For this reason, the entire light curve of the system was analysed for the first time in the literature, using the phoebe V.0.32 software (Prša & Zwitter Reference Prša and Zwitter2005), which uses the 2003 version of the Wilson-Devinney Code (Wilson & Devinney Reference Wilson and Devinney1971; Wilson Reference Wilson1990).

Although the temperatures given in the literature for the system vary from 5 544 K (Ammons et al. Reference Ammons, Robinson, Strader, Laughlin, Fischer and Wolf2006) to 6 913 K (Armstrong et al. Reference Armstrong, Chew, Faedi and Pollacco2014). Trying each temperature value given in the earlier studies, we found that an astrophysically reasonable solution was obtainable only taking the temperature value of 6 913 K for the primary component. Fixing this temperature for the primary component, the temperature of the secondary component was found to be 6 271±1 K. The mass ratio (q) of the system was found to be 0.868±0.002, while the inclination (i) of the system was computed as 69.75±0.01. The dimensionless potentials (Ω1 and Ω2) of the components were found to be 5.8362±0.0010 and 5.0301±0.0009, while the fractional radii of the components were calculated as 0.2029±0.0004 and 0.2218±0.0006. On the other hand, the light curve analysis indicated the existence of the luminosity of 0.0034 L for a third body.

According to these results, we revealed which component is located in the Instability Strip. In Figure 6, we plotted ZAMS and TAMS taken from Girardi et al. (Reference Girardi, Bressan, Bertelli and Chiosi2000) and the borders of the Instability Strip computed from Rolland et al. (Reference Rolland, Costa, Rodriguez, Amado, Garcia-Pelayo, de Coca and Olivares2002) in the log(T eff)−log(L/L) plane. Apart from this, we also plotted some pulsating stars, which are the components in the binary systems, from two different types. Some of them are some δ Scuti stars taken from Soydugan et al. (Reference Soydugan, Soydugan, Demircan and İbanoǧlu2006), while some of them are γ Doradus type stars taken from Henry, Fekel, & Henry (Reference Henry, Fekel and Henry2005). When the components of KIC 2557430 were plotted in this log(T eff)−log(L/L) plane, it was seen that the primary component is located among γ Doradus type stars. However, according to the light curve analysis, the temperature of the secondary component was computed as log(T eff) =3.797 and its luminosity was calculated as log(L/L) =0.729. In this case, the secondary component is located absolutely out of the Instability Strip in this plane. Therefore, as it was earlier stated by Uytterhoeven et al. (Reference Uytterhoeven2011), one of components is located in the Instability Strip and seems to be a γ Doradus type star.

The minima times were computed for both the primary and secondary minima from the available short cadence data in the Kepler Database. After determining the differences (OC) I between observations and calculations, the linear correction was applied on (OC) I and (OC) II residuals were obtained. At this point, the variation of (OC) II residuals versus time exhibits two characteristic behaviours. First of all, as it is seen from the upper panel of Figure 7, the least-squares method indicated that all the (OC) II residuals computed for both the primary and secondary minima show an inverse parabolic variation, which means normally that there is a mass transfer from the primary component to the secondary (Pringle & Wade Reference Pringle and Wade1985). However, in the case of this system, the secondary component is filling its Roche Lobe, not the primary. Possible interpretation of the situation is that, considering existence of the flare activity exhibiting by KIC 2557430, the inverse parabolic variation must be caused due to the mass loss from the system. Second, if the variations of (OC) II residuals obtained from the primary minima versus time and from the secondary minima are examined separately, it is clearly seen that the (OC) II residuals of the primary and secondary minima vary in the same way, but in the different phases, asynchronously. This behaviour was demonstrated by Tran et al. (Reference Tran2013) and Balaji et al. (Reference Balaji, Croll, Levine and Rappaport2015). According to their results, the synchronous variation but in opposite directions of the (OC) II residuals is explained by the stellar spot activity on the component. In fact, the light curve analysis revealed the existence of two cool spotted areas on the secondary component.

The relations between the parameters indicate that the distributions of flare equivalent durations on the logarithmic scale versus flare total durations are varying in a rule. The distributions of flare equivalent durations had been modelled with the OPEA function by Dal & Evren (Reference Dal and Evren2011). The authors also demonstrated that the OPEA models get a form depending on the (BV) colour indexes of the stars. According to their results, the OPEA models have the saturation levels, which are defined depending on the stellar (BV) colour indexes. In this study, available short cadence data given in the Kepler Database indicate that KIC 2557430 was observed along 61.134289 d (1467.22299 h) in total. In total, 69 flares were detected from these data, and their parameters were computed. we initially tried to derive the OPEA model using all the detected 69 flares. However, the correlation coefficient squared (R 2) obtained from this initial OPEA model is very low, while its probability value (p-value) is very high. In this case, the model cannot be statistically acceptable. Examination of the distributions of flare equivalent durations revealed that the dissociation gets start in the data around the flare total time of 1 400 s. Because of this, we split the data into two groups, depending on the flares with larger flare total time than 1 400 s. Then, two OPEA models were derived for both flare groups. When the parameters derived from the OPEA models were statistically compared, these statistical comparisons indicated that these two OPEA models are absolutely different from each other. For example, the Plateau value, which was defined as a saturation level for the model by Dal & Evren (Reference Dal and Evren2011), was found to be 1.4336±0.1104 s for the flares of Group 1, while it was found to be 0.7550±0.0677 s for the flares of Group 2. Considering the errors of the Plateau values, it is clearly seen that these Plateau values cannot be equal to each other, which is very interesting because there is not any other sample for this case. According to the results obtained by Dal & Evren (Reference Dal and Evren2011) and Dal, Sipahi, & Özdarcan (Reference Dal, Sipahi and Özdarcan2012), the flares detected from one star are always modelled with just one OPEA model. However, the flares detected from KIC 2557430 must be modelled with two different OPEA models. Dal & Evren (Reference Dal and Evren2011) and Dal et al. (Reference Dal, Sipahi and Özdarcan2012) demonstrated that the OPEA models were formed depending on the stellar (BV) colour indexes. Thus, two OPEA models derived for the flares detected from KIC 2557430 definitely indicate existence of two flare sources in this system. In brief, the flares of Group 1 are come from one star, while the flares of Group 2 are come from another star in the system. If the Plateau values of the models are considered, it will be understood that the flares of different groups must be come from two stars with different (BV) colour indexes. As a result, two components of KIC 2557430 must be separately flare stars.

On the other hand, here is a bit problematic case. The light curve analysis gave an astrophysically reasonable solution, if the temperature of the primary component was taken as 6 913 K. In this case, the temperature of the secondary component is found to be 6 271 K. Using the calibration given by Tokunaga (Reference Tokunaga and Cox2000), the colour index of the primary component was computed as BV = 0 m .372, it was found to be BV = 0 m .516 for the secondary component. In the study of Yoldaş & Dal (Reference Yoldaş and Dal2016, Reference Yoldaş and Dal2017), the Plateau value of FL Lyr, whose (BV) colour index is 0 m .74, was found to be 1.232 s, while it was computed as 1.951 s for KIC 9761199, whose (BV) colour index is 1 m .303. In addition, the Plateau value of KIC 7885570, whose (BV) colour index is 0 m .643, was found to be 1.9815 s by Kunt & Dal (Reference Kunt and Dal2017). Moreover, the Plateau value was obtained as 3.014 s for EV Lac (BV = 1 m .554), 2.935 s for EQ Peg (BV = 1 m .574), and also 2.637 s for V1005 Ori (BV = 1 m .307) by Dal & Evren (Reference Dal and Evren2011) and Dal et al. (Reference Dal, Sipahi and Özdarcan2012). According to the results given in this brief summary, in the case of KIC 2557430, the flares of Group 1 seem to be come from the secondary component, considering its (BV) colour index with the Plateau value. Thus, the flares of Group 2 should come from the primary component. However, it is not possible due to its (BV) colour index according to the results obtained by Dal & Evren (Reference Dal and Evren2011) and Dal et al. (Reference Dal, Sipahi and Özdarcan2012). At this point, our thought is that the flares of Group 2 could come from the third body the light curve analysis indicated. However, there is no data about what its (BV) colour index is. In the future, KIC 2557430 needs a spectral observation with high resolution.

From the analyses of the OPEA models derived for KIC 2557430, the half-life values were found to be 2278.1 s for the flares of Group 1 and 1811.2 s for the flare of Group 2. The half-life values found for KIC 2557430 are remarkably higher than those found for the UV Ceti type single stars. According to Dal & Evren (Reference Dal and Evren2011) and Dal et al. (Reference Dal, Sipahi and Özdarcan2012), it is 433.10 s for DO Cep (BV = 1 m .604), 334.30 s for EQ Peg, and 226.30 s for V1005 Ori. As it is seen from this brief summary, the flares can reach the maximum energy level at their Plateau value, when their total durations reach about n × 5 min for a UV Ceti type single star. In the case of KIC 2557430, it needs n × 38 min for the flares of Group 1, while it is n × 30 min for the flares of Group 2. Similarly, in the cases of FL Lyr and KIC 9761199, it needs n × 39 and n × 17 min for these systems (Yoldaş & Dal Reference Yoldaş and Dal2016, Reference Yoldaş and Dal2017). In addition, it was found to be n × 66 min for KIC 7885570 by Kunt & Dal (Reference Kunt and Dal2017). In a result, KIC 2557430 is similar to FL Lyr in the point of the half-life values.

The maximum flare total time (T t) was found to be 9827.78 s for the flares of Group 1, while it was 9827.92 s for the flares of Group 2. In addition, the maximum flare rise time (T r) was found to be 941.53 s for the flare Group 1, while it was 3059.99 s for the others. Yoldaş & Dal (Reference Yoldaş and Dal2016, Reference Yoldaş and Dal2017) found that the maximum flare rise time (T r) is 5 179 s for the flares of FL Lyr, while it is 1118.1 s for the flares of KIC 9761199. Moreover, the maximum flare total time (T t) was found to be 12770.62 s for the flares of FL Lyr, while it was 6767.72 s for the flares of KIC 9761199. At the moment, Kunt & Dal (Reference Kunt and Dal2017) found that the maximum flare rise time (T r) is 7768.210 s and also the maximum flare total time (T t) is 16890.30 s for KIC 7885570. On the other hand, for the single UV Ceti type stars, Dal & Evren (Reference Dal and Evren2011) found that the maximum flare rise time (T r) is 2 062 s for V1005 Ori, 1 967 s for CR Dra. Similarly, the maximum flare total time (T t) was found to be 5 236 s for V1005 Ori and 4 955 s for CR Dra. As it is seen from these results, the flare time scales indicate that the Group 2 flares detected from KIC 2557430 have the same character with KIC 9761199. However, the flares of Group 1 have the same character with the single UV Ceti type stars.

The Short Cadence data in the Kepler Mission Database reveal that KIC 2557430 was observed 1467.22299 h. In these observations, 69 flares were detected. Forty of them were classified as Group 1, the rest of them were classified as Group 2. The flare frequencies of KIC 2557430 were computed as N 1 =0.02726 h −1 and N 2 = 0.00002 for Group 1, while they were calculated as N 1 =0.01977 h −1 and N 2 =0.00001 for Group 2. Comparing these frequencies with those computed from single UV Cety type stars, it is seen that the flare energy level found for KIC 2557430 is remarkably lower than those found from them. For instance, the observed flare number per hour for UV Ceti type single stars was found to be N 1 =1.331 h −1 in the case of AD Leo, while it was found to be N 1 =1.056 h −1 for EV Lac. Moreover, N 2 frequency was found to be 0.088 for EQ Peg, while it was found to be N 2 =0.086 for AD Leo (Dal & Evren Reference Dal and Evren2011). However, according to Yoldaş & Dal (Reference Yoldaş and Dal2016, Reference Yoldaş and Dal2017), the flare frequencies were found as N 1 = 0.4163 h −1 and N 2 =0.0003 for FL Lyr (BV = 0 m .74), and N 1 = 0.0165 h −1 and N 2 =0.00001 for KIC 9761199 (BV = 1 m .303). In addition, N 1 and N 2 were computed as N 1 =0.00362 h −1 and N 2 =0.00001 for KIC 7885570 (BV = 0 m .643) by Kunt & Dal (Reference Kunt and Dal2017). It is clearly seen that the flare frequencies of KIC 2557430 for both groups are similar to the frequencies of KIC 9761199.

As a result, the frequency analyses and the light curve analysis indicate that the primary component of KIC 2557430 is most probably a γ Doradus-type pulsating star. However, they also indicate that the secondary component of the system exhibits the cool spot activity on its surface. The analysis of the (OC) II residuals reveals a mass loss from the whole system. In fact, there is a distinctive flare activity detected from the system. The analyses of the flare activity indicate the existence of two possible OPEA models, which means that the flares detected from KIC 2557430 come from two different targets with different (BV) colour indexes. On the other hand, the primary component of the system is a γ Doradus-type pulsating star. This component can certainly exhibit some flares, but it is too hot to exhibit often any huge flares like a UV Ceti star (Gershberg Reference Gershberg2005). However, the light curve analysis indicated that the second component exhibits the spot activity, so it is possible that the secondary component can also exhibit the flare activity. At this point, there should be one more target to exhibit the flare activity.

In our opinion, this second target for the flare activity is the third body, whose existence was found in the light curve analysis. On the other hand, there is a handicap in this case. If the third bode found in the light curve analysis is able to exhibits some flare activity, this source must be an M dwarf at least. In this case, it is expected that the third light contribution in the total light must be much more than 0.339%. However, the third body may be not a component in the system. This source can be a chromospherically active star located in the same direction with KIC 2557430 in the sky, but not in the same distance. If the case is real, the third body must be far away from the KIC 2557430 in the space. However, the stellar spot activity indicated by the residual frequencies of F1 and F4 should not relate to the third body, because the F1 and F4 frequencies are correlated with the orbital period of the system.

It is seen that the problem about the third body and the second source of the flare activity needs the spectral observations with high resolution in the future. For this reason, KIC 2557430 should be observed spectroscopically with high time resolution in order to check the existence of the spectral lines for the third body.

ACKNOWLEDGEMENTS

The authors acknowledge generous allotments of observing time at Ege University Observatory and TUBITAK National Observatory of Turkey. We also wish to thank the Turkish Scientific and Technical Research Council for supporting this work through grant No. 116F349. We also thank the referee for useful comments that have contributed to the improvement of the paper.

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