Activity Analyses for Solar-type Stars Observed with Kepler. II. Magnetic Feature versus Flare Activity

The light curves of solar-type stars present both periodic fluctuation and flare spikes. The gradual periodic fluctuation is interpreted as the rotational modulation of magnetic features on the stellar surface and is used to deduce magnetic feature activity properties. The flare spikes in light curves are used to derive flare activity properties. In this paper, we analyze the light curve data of three solar-type stars (KIC 6034120, KIC 3118883, and KIC 10528093) observed with Kepler space telescope and investigate the relationship between their magnetic feature activities and flare activities. The analysis shows that: (1) both the magnetic feature activity and the flare activity exhibit long-term variations as the Sun does; (2) unlike the Sun, the long-term variations of magnetic feature activity and flare activity are not in phase with each other; (3) the analysis of star KIC 6034120 suggests that the long-term variations of magnetic feature activity and flare activity have a similar cycle length. Our analysis and results indicate that the magnetic features that dominate rotational modulation and the flares possibly have different source regions, although they may be influenced by the magnetic field generated through a same dynamo process.


INTRODUCTION
High-precision and continuous light curves of solar-type stars observed with Kepler space telescope Koch et al. 2010;Borucki 2016) and other space missions like CoRoT (Auvergne et al. 2009) and MOST (Walker et al. 2003) present two significant ingredients: periodic fluctuation (rotational modulation) and flare spikes. The rotational modulation on the stellar brightness variation is interpreted as being caused by the corotating magnetic features (e.g., dark starspots and bright faculae) on the stellar surface (Vaughan et al. 1981;Baliunas et al. 1983;Debosscher et al. 2011;McQuillan et al. 2014;Suárez Mascareño et al. 2015;He et al. 2015;Hempelmann et al. 2016;Mehrabi et al. 2017), and the sudden spikes in the light curves is explained as flare phenomenon originated from the starspot regions associated with intense and concentrated magnetic field Maehara et al. 2012;Shibayama et al. 2013;Notsu et al. 2013;Kowalski et al. 2013;Hawley et al. 2014;Balona 2015;Karoff et al. 2016;Davenport 2016;Yun et al. 2016Yun et al. , 2017. In this paper, we analyze the relationship between these two aspects of stellar activity, that is, the magnetic feature activity properties manifested by the fluctuation characteristics of light curves versus the flare activity properties manifested by the flare spikes in light curves, for three solar-type stars observed with Kepler. (For a general review of stellar activity, please refer to, e.g., Choudhuri 2017.) He et al. (2015) (paper I of this series) suggested two quantitative measures, i AC and R eff , as the magnetic activity proxies of solar-type stars based on the Kepler light curve observations. The first proxy i AC (autocorrelation index) describes the degree of periodicity of a light curve, which indicates the stability of magnetic features. The second proxy R eff measures the effective fluctuation range of a light curve, which indicates the spatial size or coverage of magnetic features and hence the intensity of magnetic activity (Basri et al. , 2013García et al. 2010;Chaplin et al. 2011;McQuillan et al. 2012McQuillan et al. , 2013. The analysis by He et al. (2015) illustrates that both the two proxies can reflect the cyclic variation of magnetic activities of solar-type stars. Their result shows that, for a solar-type star, the time variations of the two magnetic proxies may be in the same phase (positive correlation) or in the opposite phase (negative correlation), which implies two different magnetic activity behaviors of solar-type stars. As demonstrated by He et al. (2015) using the solar light-curve data (Fröhlich et al. 1997), the Sun is a negative correlation star. Mehrabi et al. (2017) further analyzed the correlation between the two proxies for a large sample of G-type main sequence Kepler targets. They found that the number of positive correlation stars is much larger than the number of negative correlation stars in Kepler sample, and the positive correlation stars tend to have shorter rotation period and larger magnitude of light-curve variation than the negative correlation stars.
On the Sun, major flares occur in the sunspot regions holding a complex magnetic geometry (e.g., Priest & Forbes 2002;Leka & Barnes 2003, 2007Benz 2008;Wang et al. 2009;Shibata & Magara 2011;He et al. 2008He et al. , 2014Wang et al. 1996Wang et al. , 2015Yang et al. 2014Yang et al. , 2017. Statistical studies show that the occurrence frequency of solar flares has a clear cyclic variation in pace with the 11-year solar cycle (e.g., Aschwanden & Freeland 2012;Hathaway 2015), that is, the flare frequency is higher around solar maxima and is lower at solar minima. The solar cycle is characterized by sunspot number or sunspot coverage on the disk of the Sun. The variation of sunspot coverage along with the solar cycle can also be reflected by the cyclic variation of the measure R eff (effective fluctuation range) of the solar light curves, since it is sunspots that dominate the rotational modulation in the light curve of the Sun . Thus the 11-year cyclic variation of the occurrence frequency of solar flares is in phase with the cyclic variation of the measure R eff of solar light curves. On the other hand, Aschwanden & Freeland (2012) found that the power-law distribution of magnitudes of solar flares (defined as the background-subtracted soft X-ray peak flux) is invariant through the solar cycles. This means that even during a cycle minimum the high magnitude solar flares can also occur (Hathaway 2015), though with very low frequency. In short, the sunspot coverage on the solar disk affects the occurrence frequency of solar flares but not the proportionality of flares between different magnitudes.
For the solar-type stars, the relations between stellar flare activity and magnetic activity have also been investigated by many authors in literature. The Kepler dataset has large sample of stellar flares and was extensively used in these studies. For example, in the paper by Shibayama et al. (2013), they extended the pioneer work of Maehara et al. (2012) and compiled a superflare catalog of solar-type (G-type) stars based on the observational data of Kepler. By utilizing this catalog, Notsu et al. (2013) found that the energy of superflares is related to the amplitude of brightness variation of the host stars, that is, the stars with higher brightness variation amplitude tend to have more energetic flares. Similar results (flare magnitude vs. amplitude of stellar brightness variation) were also obtained for other kinds of stars with different spectral types or effective temperatures Balona 2015), but the stellar brightness variability is only weakly correlated with the flare occurrence frequency . Maehara et al. (2012) reported a relation between the flare occurrence frequency and the stellar brightness variation amplitude for solar-type stars in the Supplementary Information of their paper, but this relation is not confirmed in the follow-up studies using a larger sample of superflare stars Notsu et al. 2013). The amplitude of stellar brightness variation is commonly attributed to the coverage of starspots, yet the statistical result with stellar flare frequency implies a contradiction from the result of the Sun.
The stellar magnetic activity can also be reflected by chromospheric observations (Eberhard & Schwarzschild 1913;Wilson 1978;Hall 2008). By employing a chromospheric activity index (Vaughan et al. 1978) derived from the spectrum observations of the Ca ii H and K lines for Kepler objects (De Cat et al. 2015) using the ground-based Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST) (Cui et al. 2012), Karoff et al. (2016) demonstrated that the superflare stars in the catalog of Shibayama et al. (2013) generally possess higher chromospheric activity level than average stars including the Sun, and this higher activity level of superflare stars is consistent with the result of the stellar activity analysis performed by Notsu et al. (2015). Karoff et al. (2016) also found that their chromospheric activity index values are positively correlated with the stellar periodic photometric variability amplitudes measured by McQuillan et al. (2014) using the Kepler data down to 1000 ppm (parts per million) which is just the order of amplitude of the Sun (note that the concept of stellar brightness variation amplitude or periodic photometric variability amplitude is equivalent to the measure R eff in this work). The stellar photometric variability amplitude is attributed to the coverage of starspots as usual, yet the emission of Ca ii H and K lines employed to derive the chromospheric activity index mainly comes from the plage regions in chromosphere (e.g., Skumanich et al. 1975;Schrijver et al. 1989). Chromospheric plages correspond to enhanced network magnetic field and facula regions in photosphere (Solanki et al. 2006), which might surround but are not necessarily associated with starspots (Skumanich et al. 1975;Schrijver et al. 1989).
In previous studies, relations between stellar magnetic activity and flare activity for solar-type stars are analyzed mainly based on large sample of stars and lack time evolution information. In this work, we perform the analysis of magnetic feature activity versus flare activity for three individual solar-type stars. As the approaches for the Sun, we investigate the time variations of stellar flare activity and magnetic activity, with a particular interest in their phase relationship. The investigated superflare stars are selected from the catalog of Shibayama et al. (2013). These selected Kepler stars possess different rotation periods and have different flare frequencies, which are explained in Section 2. For each star, two time-series components are extracted from the original Kepler light curves: the rotational modulation component (caused by corotating magnetic features) and the flare component (constituted by flare spikes). The procedures for the Kepler light curve data processing are described in Section 3. The magnetic activity properties of the stars are evaluated based on the rotational modulation component by using the two magnetic proxies introduced in paper I . The flare activity properties are derived from the flare component via several flare indexes. The methods for the magnetic proxies and flare indexes evaluation are described in Section 4. The relationship between the magnetic feature activity and the flare activity are analyzed in section 5. Section 6 presents the summary and discussion.

SUPERFLARE STARS INVESTIGATED
The individual Kepler stars analyzed in this work are selected from the superflare star catalog of Shibayama et al. (2013). For studying the time evolution of stellar activity properties, the investigated stars should have continuously observed light-curve data, i.e., the Long Cadence data (Jenkins et al. 2010a) from quarter 2 to 16 (Q2-Q16). The data in Q0, Q1, and Q17 are not included because the three quarters are too short (see illustration in paper I) to yield a compatible result with the other full-length (about three months; Haas et al. 2010) quarters. We picked out all the stars with complete Q2-Q16 Long Cadence data from the catalog of Shibayama et al. (2013) and sorted them by their flare counting numbers given by Shibayama et al. (2013). The star at the top of the list (i.e., with the highest flare occurrence frequency) is KIC 6034120. In this paper, our analyses of stellar magnetic activity versus flare activity are mainly based on this flare-abundant star (notice that the same Kepler object has been quoted in the paper by Maehara et al. (2012) as a representative superflare star). For comparison, two other superflare stars (KIC 3118883 and KIC 10528093) with different rotation periods and flare occurrence frequencies are also employed in this study. The stellar parameters and basic Kepler data information of the three investigated superflare stars are listed in Table  1.

LIGHT-CURVE DATA PROCESSING
We adopt the PDC (Pre-search Data Conditioning) flux product (Stumpe et al. , 2014Smith et al. 2012) of the Long Cadence light-curve data in Kepler Data Release 25 (Thompson et al. 2016) for our study. The PDC data is named because it is the output of the PDC module of the Kepler science processing pipeline (Jenkins et al. 2010b;Jenkins 2017) in which the systematic errors in the raw data are corrected (Stumpe et al. , 2014Smith et al. 2012). The time resolution of the data (i.e., Long Cadence PDC flux) is 29.4 minutes per data point (Jenkins et al. 2010a).
The absolute values of Kepler light-curve data feature discontinuities across quarter boundaries owing to the 90 • -roll of the telescope between quarters and hence the shift of CCD modules (Borucki 2016;Van Cleve et al. 2016, notice that the Kepler mission pursues high differential photometric precision instead of absolute flux accuracy). Considering this issue of the data, we process the Kepler data of each quarter separately as done in paper I. Our aim is to extract two time series components from the original Kepler light curves: One is the rotational modulation component (associated with magnetic features), and another is the flare component (constituted by flare spikes). Then the relation of magnetic feature activity with flare activity for the selected stars can be studied. The procedures for Kepler light-curve data processing are described in the following subsections.

Extracting Rotational Modulation Component
We use the light-curve data of KIC 6034120 in Q7, as an example, to demonstrate the procedures for data processing. Firstly, we extract the gradual fluctuation component from the original light curve, which is illustrated in Figure 1. Figure 1(a) shows the curve of the original PDC flux (denoted by F ). To extract the gradual fluctuation component, we identify the prominent flare spikes in the light curve (by visual inspection with the aid of a semi-automatic software tool) and then remove all the data points that constitute the identified flare spikes from the light curve. The flareremoved flux curve is shown in Figure 1(b) in black color. The void data points left by the flare spikes are subsequently patched up through the linear interpolation. The patched flux curve is plotted in Figure 1(b) in gray. As done in paper I, we filter out the transient variation component (mainly noises, with some minor flares which are not identified in the previous step) in the patched light curve by using a Fourier-based low-pass filter. The upper cutoff frequency of the filter is set to 1/0.5 day −1 which is an empirical value and determined via visual inspection. The flux curve after filtering (i.e., the gradual component of the original light curve; denoted by F G where the letter 'G' refers to 'gradual') is plotted in Figure 1(c).
We calculate the relative fluxes f and f G for F and F G , respectively, to dispel the discontinuity issue at quarter boundaries since all the relative fluxes of different quarters fluctuate around zero (see paper I). The baseline value F 0 for deriving the relative fluxes of a given quarter is defined as (1) Then the relative fluxes f and f G can be calculated by the following equations: The derived f and f G flux curves of the example data are shown in Figure 2(a) (in gray and black, respectively). Some residual long length-scale trends are remained in the PDC data of the three investigated stars listed in Table  1. To extract the pure rotational modulation components from the light curves, we filter out this long length-scale trend (denoted by f L where the letter 'L' refers to 'long length-scale'; see the dotted curve in Figure 2(a)) through a high-pass filter acting on the f G flux data. The lower cutoff frequency of the high-pass filter is set to 1/1024 longcadence −1 (about 1/21 day −1 ) as adopted by the multiscale MAP (maximum a posteriori) algorithm of the Kepler science processing pipeline (Stumpe et al. 2014, note that 1 long-cadence ≈ 29.4 minutes). The resulting flux is the expected rotational modulation component of the original Kepler light curve and is denoted by f M (where the letter 'M' refers to 'modulation'). In short, The f M flux curve of the example data is shown in Figure 2(b). The magnetic activity proxies of the three investigated stars are evaluated based on the f M flux data (see Section 4 for details).

Extracting Flare Component
The flare component (denoted by f F , where the capital letter 'F' refers to 'flare') of the original Kepler light curve can be obtained by subtracting the gradual component f G from the total relative flux f , that is, The f F flux curve of the example data is shown in Figure 2 Figure 2.) It can be seen from equation (5) and Figure 2 that the noises in the original Kepler light curves are left in the f F flux data. Since the noise level of the Kepler data is low (Borucki 2016;Van Cleve et al. 2016) and the flare signal is high (see Figure 2(c)), it will not affect the evaluation of flare activity indexes in Section 4. We use the two magnetic proxies i AC and R eff suggested in paper I to quantitatively describe the magnetic activity properties of the three investigated stars. The first parameter i AC measures the degree of periodicity of a light curve, which reflects the stability of the magnetic features that dominate the rotational modulation. The second parameter R eff measures the effective range of the light-curve fluctuation, which reflects the spatial size or coverage of the magnetic features. In this paper, the values of i AC and R eff are evaluated based on the f M flux data (rotational modulation component of the total relative flux f ) derived in section 3.

Light-Curve Data After Processing
For a time series of f M data with N evenly spaced data points {x t , t = 0, 1, . . . , N − 1}, the parameter i AC is calculated by the following equations : wherex is the mean value of the time series {x t }, ρ(h) is the autocorrelation coefficient of {x t } at time lag h, and i AC (autocorrelation index) is defined as the average value of |ρ(h)| for the first half of the ρ(h) function. The possible values of i AC are in the interval between 0 and 3 2π (≈ 0.48). Larger i AC means stronger periodicity. (See paper I for more details.) The parameter R eff (effective fluctuation range) for the time series {x t } is calculated by equation ) where x rms is the rms value of {x t }, and the rms is multiplied by a scaling factor 2 √ 2 to obtain the effective range between crest and trough (see paper I for details). We calculated the values of i AC and R eff in each quarter of Q2-Q16 for the three investigated stars based on the f M data. The results are listed in Table 2. The mean values of i AC and R eff within the 15 quarters were also calculated and are given in the last line of Table 2.

Flare Indexes
The flare signals in Figures 3(c), 4(c) and 5(c) show variations of flare occurrence frequency as well as flare magnitude for the three investigated stars. To quantitatively describe flare activity properties, we introduce three flare indexes based on the f F flux data. The three flare indexes are: the time occupation ratio of flares R flare , the total relative power (energy rate) of flares P flare , and the averaged relative flux magnitude of flares M flare . Because the three flare indexes are defined as ratio or relative values (see definitions below), they are all dimensionless quantities. As done for the two magnetic proxies in Section 4.1, we intend to evaluate the three flare indexes quarter by quarter to exhibit time evolution information of flare activities.
In Figure 6, we use the the f F flux curve of KIC 6034120 in Q7 (as shown in Figure 2(c)) to explain the quantities employed in the flare index definitions. Firstly, the subset of flare spikes (denoted by f S , where the capital letter 'S' refers to 'spikes') in f F flux data are identified by the criterion f F 0.001. The threshold value 0.001 is empirically determined and just above the noise level of the original Kepler light-curve data (see diagram illustration in Figure  6). Then the total occupation time of the flare spikes (denoted by T flare ) can be counted accordingly (see illustration in Figure 6). The total valid observing time of the quarter (as recorded in Kepler data files) is denoted by T obs (see illustration in Figure 6). Notice that the time gaps between the three months of the quarter for spacecraft data downlink (Borucki 2016;Van Cleve et al. 2016) are not included in T obs .
The time occupation ratio of flares, R flare , is defined as the ratio of T flare to T obs , Note that the concept of R flare is equivalent to the concept of flare occurrence frequency. We adopt R flare in this work because it is more convenient for evaluation.    For defining the total relative power (energy rate) of flares, P flare , we first evaluate the total relative energy (denoted by U flare ) released by all the identified flares in the quarter via equation The integral in equation (10) is over the discrete time intervals occupied by the identified flare spikes, which are indicated by the plus symbols in Figure 6. Then the total relative power (energy rate) of flares, P flare , can be calculated by equation Notice that U flare has the dimension of time (and is also called equivalent duration of flare in literature, e.g., Hawley et al. 2014) and P flare is dimensionless. Since f S is evaluated relative to the background flux of the star (see section 3), the value of P flare is also relative to the background energy emission rate of the star. The averaged relative flux magnitude of flares, M flare , is defined by equation It can be seen from equations (9), (11), and (12) that M flare is connected with R flare and P flare by equation We calculated the values of the three flare indexes, R flare , P flare , and M flare , in each quarter of Q2-Q16 for the three investigated stars, and the results are listed in Table 3. The mean values of the flare indexes within the 15 quarters are given in the last line of Table 3. Variations of R eff (top row) and iAC (middle row) with time for the three investigated stars and correlation analysis between the two magnetic proxies (bottom row). Each column corresponds to one star, and each data point in the plots corresponds to one quarter in Q2-Q16. In the bottom row, the dotted lines are the linear fitting for the paired data of (iAC, R eff ) and 'c.c.' represents 'correlation coefficient'. Figure 7 corresponds to one quarter in Q2-Q16. It can be seen from the plots that both the two magnetic proxies i AC and R eff show long length-scale variations, and the long length-scale variations of i AC and R eff are in the same phase for all the three stars. To confirm this impression, we perform correlation analysis between the paired data of the two magnetic proxies. The scatter plots of (i AC , R eff ), the correlation coefficients, and the linear fitting for the paired data are given in the bottom row of Figure 7. The results of the correlation analysis affirm that the two magnetic proxies are positively correlated for the three stars with the correlation coefficients being 0.77, 0.72, and 0.78, respectively. That is, all the three stars investigated in this paper belong to the positive-correlation-star catalog described in the work by Mehrabi et al. (2017).

Flare Activity
The variations of the three flare indexes with time for the three investigated stars are plotted in Figure 8 (top row for R flare , middle row for P flare , bottom row for M flare , and each column corresponding to one star). The values of the three indexes are taken from Table 3. It can be seen from Figure 8(a)-(c) that the curves of the flare index R flare (time occupation ratio of flares) show apparent long-term trends of flare activity for the three stars, while the flare index M flare (averaged relative flux magnitude of flares) fluctuates randomly around its mean value for the three stars (see Figures 8(g)-(i)). The flare index P flare (total relative power of flares) displays similar long-term trends as the flare index R flare (see Figures 8(d)-(f)). Considering P flare is the product of R flare and M flare (see equation (13)), and M flare varies randomly, the long-term variation of P flare is in fact the reflection of the long-term variation of R flare .
Since the R flare values of KIC 6034120 is higher than KIC 3118883 and KIC 10528093 (see Table 3), i.e., there are more flares on KIC 6034120 than KIC 3118883 and KIC 10528093 (see Figures 3-5), the statistical significance of the R flare variation for KIC 6034120 is also higher, thus the R flare curve of KIC 6034120 is smoother (less random fluctuation) than the other two stars as presented in Figures 8(a)-(c).
Moreover, recall that the variations of the flare index M flare (averaged flare magnitude) are random (Figures 8(g)-(i))

Correlation Analysis Between Magnetic Feature and Flare Activities
Because the long length-scale variation of the two magnetic proxies i AC and R eff are in the same phase for the three investigated stars (see Section 5.1), we adopt R eff (effective range of light-curve fluctuation) as the representative parameter of magnetic feature activity for the three investigated stars. As demonstrated in Section 5.2, the long-term trends of the flare activities of the three stars are best represented by the flare index R flare (time occupation ratio of flares). Thus, the correlation between the magnetic feature activities and the flare activities of the three stars can be quantitatively described by the correlation between the values of R eff and R flare of the three stars. The results of the correlation analysis between R eff and R flare for the three stars are given in Figure 9.
To indicate the long-term trends of magnetic feature and flare activities more clearly, we performed polynomial fitting for the R eff and R flare curves of the three investigated stars. The results of the polynomial fitting are shown in the top row (for R eff ) and middle row (for R flare ) of Figure 9 (each column corresponding to one star). The polynomial fitting curves are shown in black and the original R eff and R flare curves are shown in gray. The orders of polynomial fitting were determined empirically, which are 5, 3, and 2 for KIC 6034120, KIC 3118883, and KIC 10528093, respectively. The scatter plots of the paired data (R eff , R flare ) are given in the bottom row of Figure 9. The correlation coefficients between R eff and R flare of the three stars are 0.06, −0.14, and −0.26, respectively.
It can be seen in the top and middle rows of Figure 9 that the long-term variations of R eff and R flare are not in phase with each other for all three stars. This impression is affirmed by the weak or negative correlations between R eff and R flare values as demonstrated in the bottom row of Figure 9. The fact that the magnetic feature activity and the flare activity are not positively related indicates that they may have different source regions.

Cycle Length of Magnetic Feature and Flare Activities
The polynomial fitting plots of the star KIC 6034120 in Figures 9(a) and (d) show apparent cyclic variations for both the parameters R eff and R flare , which may represent the magnetic feature activity cycle and flare activity cycle, respectively. Although the cyclic variations of two parameters are not in phase with each other as shown in Section 5.3, the rhythms of the variations are compatible (see Figures 9(a) and (d)), that is, they may have the similar cycle lengths. The fitting curves of the other two stars KIC 3118883 and KIC 10528093 also show clues of this impression (see Figures 9(b), (c), (e), and (f)), but not as apparent as KIC 6034120 since the R eff and R flare curves of the two stars do not exhibit a full cycle of variation.
We quantitatively evaluate the periods of the cyclic variations (i.e., cycle lengths; denoted by P cyc ) of R eff and R flare for the star KIC 6034120 by using the generalized Lomb-Scargle (GLS) periodogram method (Zechmeister & Kürster 2009). The GLS periodograms of the R eff and R flare variations for KIC 6034120 are shown in Figure 10 (bottom row); left column is for R eff (representing magnetic feature activity) and right column is for R flare (representing flare activity). In the top row of Figure 10 we show the first harmonic (fundamental) components of the R eff and R flare variations, and in the middle row the second harmonic components. The sine curves of the harmonic components are plotted in black and the original R eff and R flare curves are in gray. The values of the first harmonic period (denoted by P 1 ) and the second harmonic period (denoted by P 2 ) are also given in the corresponding panels in Figure 10. The first and second harmonic periods of R eff or R flare are evaluated through the two most prominent peaks in the associated periodograms (see bottom row of Figure 10). The two periods (as well as the two peaks) are indicated by two vertical dotted lines in each periodogram and are labeled P 1 and P 2 , respectively. The signals of the higher-order harmonics are too low to be utilized.
The value of P cyc can be evaluated either via the first harmonic period P 1 (P cyc = P 1 ) or via the second harmonic period P 2 (P cyc = 2P 2 ). The evaluated cycle length of the magnetic feature activity (denoted by P cyc−M ) through the parameter R eff and the cycle length of flare activity (denoted by P cyc−F ) through the parameter R flare for the star  Figure 10. Evaluating the periods of the cyclic variations of R eff and R flare for the star KIC 6034120 by using the GLS periodogram method. Left column is for R eff (magnetic feature activity) and right column is for R flare (flare activity). Top row shows the first harmonic (fundamental) components of the R eff and R flare variations, and middle row shows the second harmonic components. The harmonic components are plotted in black and the original R eff and R flare curves are in gray. The bottom row shows the GLS periodograms of the R eff and R flare curves. The two dotted lines in each periodogram indicate the first harmonic period P1 and the second harmonic period P2, which correspond to the two most prominent peaks in the periodogram.
KIC 6034120 are summarized in Table 4. It can been seen from Table 4 that the two values of P cyc−F (flare activity) via the first harmonic and the second harmonic are almost identical (897 and 892 days, respectively), while the discrepancy between the two values of P cyc−M (magnetic feature activity) are larger (1239 and 924 days, respectively). By comparing FWHM (full width at half maximum) of the two peaks associated with the first and second harmonics in the periodogram of magnetic feature activity (see Figure 10(e)), the P cyc−M value via the second harmonic (i.e., 924 days) is associated with a smaller FWHM and thus is more accurate (less uncertainty) than the value via the first harmonic. After all, the total time of Kepler observation from Q2 to Q16 (about 1388 days) only cover one full length of P 1 (see Figure 10(a)) but three times of P 2 (see Figure 10(c)). That is why the FWHM of the peak associated with P 1 is larger in the periodogram. The similar property also exists for the periodogram of flare activity (see right column of Figure 10).
Based on the above analysis, we adopt the P cyc−M and P cyc−F values evaluated via the second harmonics of the R eff and R flare curves as the representative cycle lengths for the magnetic feature and flare activities of the star KIC 6034120, i.e., 924 days for P cyc−M and 892 days for P cyc−F (bottom line of Table 4). The relative difference (see rightmost column of Table 4) between the two values is only 3.5%. This result suggests that the long-term variations of the magnetic feature activity and the flare activity of the star KIC 6034120 has a similar cycle length. Together with the fact that the two cyclic variations are not in phase with each other (see Section 5.3), this might indicate that two activities are controlled by two distinct aspects of the magnetic field generated through one same dynamo process. In this paper, we analyze the light-curve data of three solar-type stars observed with Kepler (KIC 6034120, KIC 3118883, and KIC 10528093) and investigate the relationship between their magnetic feature activity and flare activity. The information of magnetic feature activity is deduced from the rotational modulation signal in the light curves and is quantitatively described by two measures of the light curves. The first measure i AC describes the degree of periodicity of the light curves, which reflects the stability of the magnetic features that cause the rotational modulation. The second measures R eff describes the effective range of light-curve fluctuation, which reflects the spatial size or coverage of magnetic features. The analysis shows that the two measures are positively correlated for all three investigated stars. So we use R eff as the representative quantity of magnetic feature activities.
The information of flare activity is deduced from the flare spike signals in the light curves. We employ three flare indexes to describe the flare activity properties. The three flare indexes are the time occupation ratio of flares R flare , the total relative power (energy rate) of flares P flare , and the averaged relative flux magnitude of flares M flare . The three quantities are connected with each other by equation (13). The analysis shows that for the three investigated stars, M flare varies randomly with time, while the long-term variations of R flare and P flare are in the same phase. So we employ R flare as the representative quantity of flare activity. Note that R flare is equivalent to the concept of flare occurrence frequency.
By analyzing the relationship between the two parameters R eff and R flare quantitatively, we found that: (1) Both the magnetic feature activity and the flare activity exhibit long-term variations as the Sun does; (2) Unlike the Sun, the long-term variations of magnetic feature activity and flare activity are not in phase with each other; (3) The analysis of the star KIC 6034120, which has the highest flare occurrence frequency among the three stars, shows that the long-term variations of the magnetic feature activity and flare activity has a similar cycle length.
Our analysis and results indicate that the magnetic features that dominate the rotational modulation and flares possibly have different source regions, although they may be influenced by the magnetic field generated through the same dynamo process.
We speculate that the positive correlation between the R eff and i AC suggest that the magnetic field on these three stars, unlike the Sun, is an agent of stability more than activity. We further speculate that the poloidal component of the magnetic field on these stars are stronger than the toroidal component of the magnetic field (for a brief description of the poloidal and toroidal magnetic fields in the context of dynamo model, see, e.g., Choudhuri 2017). According to Zhang et al. (2006), the poloidal component of the magnetic field is an agent for confinement and stability and the torodial component of the magnetic field is an agent for magnetic free energy and eruption, the stronger component of poloidal field naturally explained our results why the R eff and i AC are positively correlated and why R eff and R flare are not or even negatively related, contrary to what happens on the Sun.