SUPERFLARES ON SOLAR-TYPE STARS OBSERVED WITH KEPLER. I. STATISTICAL PROPERTIES OF SUPERFLARES

Kepler carries an optical telescope with a 95 cm aperture and 105 deg² field-of-view (about 12° diameter), which is in Cygnus, Lyra, and Draco. This spacecraft was launched by NASA in Spring 2009 to search for exoplanets by finding planetary transit events, faint decreases in brightness due to a crossing planet. Since the orbiting planet is very small compared with the host star, the luminosity decrease from planetary transit is usually less than one-hundredth of the total brightness of the star. In addition, the planet passes in front of the star only when the orbit is nearly parallel with the line of sight. Kepler is, therefore, designed to obtain high-precision and long-period light curves of many stars. The typical precision is 0.1 mmag for a star of 12 mag and the number of observed stars is more than 160,000 (Koch et al. 2010). The time resolution is about 30 minutes and 1 minute. The resultant light curves are useful in detecting not only planetary transits but also other small brightness variations like stellar flares. In fact, there have been some previous studies of stellar flares using Kepler data (Walkowicz et al. 2011; Balona 2012), which focus on flares on early- or late-type stars.

The data we used were taken during the period from 2009 May to 2010 September. We retrieved the data from the Multimission Archive at the Space Telescope Science Institute and analyzed the long-cadence (the time resolution is about 30 minutes) corrected flux of 9511, 75598, 82811, 82586, 89188, 86248, and 82052 stars in quarters 0, 1, 2, 3, 4, 5, and 6, respectively (all public light curves of G-type dwarfs observed by Kepler). The number of G-type dwarfs and monitoring period of each quarter are shown in Table 1. In Table 1, we also show the beginning and the end dates of each quarter, which are shown in the Kepler Data Release Notes.⁵

Since the aim of this study is to detect superflares on G-type dwarfs, we selected G-type dwarfs in all of the observed stars using the Kepler Input Catalog (Brown et al. 2011). The condition is 5100 K ⩽T_eff < 6000 K and log g > 4.0 and the number of selected stars is about 80,000 in 160,000.

There are various flare detection algorithms which detect flares from light curves (Osten et al. 2012; Walkowicz et al. 2011). In these algorithms, light curves are analyzed after they have detrended, and are searched for cases where the relative flux becomes statistically larger than a certain threshold for two or more times consecutively. The time resolution of the light curve of Kepler is about 30 minutes and some stars exhibit short periodic brightness variation (several hours). This period is comparable with the time scale of flare duration so that a detrending light curve has the potential to detrend the superflares themselves. To avoid misdetection of short stellar brightness variation and not overlook large flares, we calculated the distributions of brightness variation between all pairs of consecutive data points after creating light curves of all selected stars using the corrected flux in the data. The threshold of the flare was determined to be three times the value at the top 1% of the distribution. This threshold was chosen as a result of a test run so as not to misdetect other brightness variation. We also calculated the standard deviation of the distribution (σ_diff) for defining the end time of a flare.

Figure 1 shows a schematic figure of our method. Figures 1(a) and (b) show a light curve of KIC 9603367 and the distribution of brightness variations between consecutive data points. Figures 1(c) and (d) are the same as (a) and (b) but for KIC 4830001. We show examples of the determination of threshold in the figures. Vertical lines in the right panels, written as "1%," correspond to the value of the top 1%, and the other vertical lines written as "1%×3" show the threshold of flare detection. Bars in the left panels at t = 2 days show the threshold determined by the right panels. Although the amplitudes of brightness variation of KIC 9603367 and KIC 4830001 are almost the same, the period of the brightness variation of KIC 9603367 (12.2 days) is longer than that of KIC 4830001 (1.0 days). Since the brightness variation between two consecutive observations in the stars with short-period variations (e.g., KIC 4830001) is larger than that of the stars with long-period variations (e.g., KIC 9603367), the distribution of brightness variations in short-period systems tend to spread wider than those of long-period systems, and the detection threshold of flares in short-period systems, therefore, becomes larger than that of flares in long-period systems. The detection threshold also depends on the amplitude of brightness variations. The thresholds in the stars showing large-amplitude variations tend to be larger than those in the small-amplitude systems.

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The detection limit of this algorithm is determined from the value of the threshold divided by the average flux of the light curve. These detection limits are between 0.001 to 0.01 for most of our targets; therefore, flares are detected almost completely when the amplitudes are more than 1% of the average flux. The typical total energy of flare of 1% amplitude is 5 × 10³⁴ erg.

The time at which the flux first exceeds the threshold is determined as the flare start time, and we removed long-term brightness variations around the flare by fitting with spline curves of three points. Here, three points are determined by the average of some data points just before the beginning of the flare and around 5 hr and 8 hr after the peak of the flare. After detrending, the flare end time was defined to be the time at which the flux residuals become smaller than 3σ_diff. We only analyzed flares with a duration longer than 0.05 days (at least two consecutive data points exceed the threshold) and excluded flare candidates with a shorter decline phase than the increase phase.

The pixel scale of the Kepler CCDs is about 4 arcsec and the typical photometric aperture for a 12 mag star contains about 30 pixels (van Cleve & Caldwell 2009). This suggests that the brightness variation of neighboring stars can affect the flux of the target star. If there is a flare star near the target and if large-amplitude flares occur on the flare star, Kepler can detect fake flares on the target star. In order to avoid these false events, we chose stars without neighboring stars within 12 arcsec for the analysis. The total number of samples that satisfy this condition was about 30%. The spatial distance between a pair of stars is calculated from the star properties in KIC. Superflares are rare events, therefore, the probability is very low that stars visually nearby exhibit superflares during the same 30 minute cadence. Such flare candidates are thought to be misdetected because of instrumental characteristics or contaminations of other sources. Although the reason for these instrumental characteristics is uncertain, light curves occasionally show discontinuity, which is thought to be caused by the variation of pointing accuracy or cosmic rays. Therefore, we excluded pairs of flares that occurred at the same time and whose spatial distance was less than 24 arcsec.

After removing the candidates of flares on neighboring stars, we checked light curves of flare candidates by eye, and examined the pixel level data of each G-type dwarf exhibiting flares by eye. Some contaminations of eclipsing binaries or transit events are found from light curves. If the spatial distribution of brightness on the CCDs is different between flares and at the quiescence, the flare is revealed to be a brightness variation of another source. A few percent and about 10% of flare candidates are removed by checking light curves and pixel level data, respectively.

We calculated periods of brightness variation from light curves, which correspond to the rotational period in the case of a single star with a dark spot. The variation period was calculated by the discrete Fourier transform method, and we chose the largest peak as the rotational period of the star when the peak was larger than the error level.

Figure 2 shows number distributions of all G-type dwarfs (solid line), flare stars (dash-dotted line), and flares (dotted line) observed by Kepler as a function of the rotational period. The periods of these stars are calculated from new data. A dashed line indicates the distribution of the period calculated from the old data, which is calibrated by the pipeline of the previous version. In this figure, we only selected flares whose total energy was more than 5 × 10³⁴ erg. We obtained public data of releases 14 and 16 in this study. The dashed line shows the distribution of the period calculated from light curves from releases 4 to 9, which were used in Maehara et al. (2012). In releases 4 to 9, the data are calibrated using the old pipeline, and long-term brightness variation longer than 15 days tends to be removed (van Cleve et al. 2010). Hence, the number of stars with a period of more than 20 days was small (Table 2 and Figure 2). The reduction algorithm for the new Kepler data in releases 14 and 16 is improved (Jenkins et al. 2012; Stumpe et al. 2012). As a result, long-term brightness variation has been detected much more accurately than before, and more stars with a period of longer than 20 days are, therefore, detected from the new data (release 14 and 16) than from the old data (release 4–9). We chose 10 days as the threshold of Sun-like stars. A rotational period of 10 days corresponds to a radial velocity of ∼5 km s⁻¹, and this threshold is enough to distinguish slowly rotating stars from rapidly rotating stars (e.g., v > 10 km s⁻¹). This threshold is consistent with the threshold of "ordinary G-type dwarfs" in Schaefer et al. (2000).

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We estimated the total energy of each flare from stellar luminosity, amplitude, and duration of flares by assuming that the spectrum of white-light flares can be described by a blackbody radiation with an effective temperature of 10,000 K (T_flare) (Mochnacki & Zirin 1980; Hawley & Fisher 1992). Assuming the effective temperature is 9000 K, the estimated energy becomes 66% of our result. This is the possible systematic uncertainty of our energy estimate.

Assuming that the star is a blackbody radiator, the bolometric flare luminosity (L_flare) is calculated from T_flare and the area of flare (A_flare) using the following equation,

$$\begin{equation} L_{\rm flare} = \sigma _{\rm SB} T_{\rm flare}^4 A_{\rm flare}, \end{equation} \tag{ 1 }$$

where σ_SB is the Stefan–Boltzmann constant. Since the star is not the blackbody radiator, this estimate is not accurate and may have an error of a few tens of percent. For the estimate of A_flare we use the observed⁶ luminosity of the star ($L^{\prime }_{\rm star}$), flare ($L^{\prime }_{\rm flare}$), and flare amplitude of the light curve of Kepler after detrending ($C^{\prime }_{\rm flare}$). These estimates are calculated from the following equations,

$$\begin{eqnarray} L^{\prime }_{\rm star} &=& \int R_{\lambda }B_{\lambda (T_{\rm eff})}d\lambda \cdot \pi R_{\rm star}^2, \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} L^{\prime }_{\rm flare} &=& \int R_{\lambda }B_{\lambda (T_{\rm flare})}d\lambda \cdot A_{\rm flare}, \ {\rm and} \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} C^{\prime }_{\rm flare} &=& L^{\prime }_{\rm flare}/L^{\prime }_{\rm star}, \end{eqnarray} \tag{ 4 }$$

where λ is the wavelength, B_λ(T) is the Plank function, and R_λ is the response function of the Kepler instrument. The instrument has a spectral bandpass from 400 nm to 850 nm (van Cleve & Caldwell 2009). We can estimate A_flare from these equations as

$$\begin{equation} A_{\rm flare} = C^{\prime }_{\rm flare} \pi R^2 \frac{\int R_{\lambda }B_{\lambda (T_{\rm eff})}d\lambda }{\int R_{\lambda }B_{\lambda (T_{\rm flare})}d\lambda }. \end{equation} \tag{ 5 }$$

L_flare can be estimated from Equations (1) and (5), and $C^{\prime }_{\rm flare}$ is a function of time; therefore, L_flare is also a function of time. Total bolometric energy of superflare (E_flare) is an integral of L_flare during the flare duration,

$$\begin{equation} E_{\rm flare} = \int _{\rm flare} L_{\rm flare}(t) dt. \end{equation} \tag{ 6 }$$

According to Brown et al. (2011), 1 − σ uncertainties of the stellar radius and the surface temperature in the Kepler Input Catalog are about ±0.2 dex and ±200 K, respectively. Hence, the uncertainty in the stellar luminosity can be estimated to be about ±60%. Although the time cadence of the Kepler data was not sufficiently short to determine the peak and duration of flares, the uncertainties of the amplitude and duration of flares are much smaller than the uncertainty in the stellar luminosity because, when we calculate the flare energy, we integrate the brightness variation due to the flare over time from the start to the end of the flare. Therefore, total uncertainty in the flare energy is about ±60%, and the occurrence rate of superflares in the unit energy range includes not only the counting error (the square root of the number of flares) but also the uncertainty of the flare energy.

We detected 1547 superflares on 279 G-type dwarfs, including 44 superflares on 19 Sun-like stars from the Kepler long-cadence data. The monitoring duration was about 500 days (quarters 0–6). Three hundred and sixty-five superflares were detected from 148 stars in a monitoring duration of about 120 days by Maehara et al. (2012). The number of superflares in this study is more than four times that of the previous study (Maehara et al. 2012), while the number of flare-generating stars is less than double. Kepler is monitoring the same stars and some active stars exhibit more than a few flares. One hundred and one superflare stars show only one superflare, while the number of superflare stars that exhibit more than one superflare is 178, and 8 Sun-like stars show more than one superflare in 19 Sun-like superflare stars. All flares are listed in Table 7 with parameters of stars and flares. BV Amp indicates the amplitude of stellar brightness variation in the quarter when the flare is detected. This value basically corresponds to the apparent fraction of the spot area to that of the disk. The definition of this value is as follows. We calculated ΔF(t)(= F(t) − F_av), where F(t) is flux of light (stellar brightness) at date t and F_av is average flux. Then, excluding data points whose absolute values of ΔF are included in the top 1% of them, we calculated (max (ΔF) − min (ΔF))/F_av, which is defined as BV Amp of the star in the quarter.

The numbers of detected superflares and stars that exhibit these flares are listed in Table 5. The fractions of superflare stars among all Sun-like stars, all G-type dwarfs, etc., are shown in Table 6 with numbers of flare stars and all observed stars in the category. In Figures 5 and 6, G-type dwarfs and Sun-like stars correspond to the lower right and middle left panels, respectively. Figure 3 shows four light curves of the most energetic superflares with Kepler ID, flare peak date, and total flare energy in the right panels (see Table 3 for detailed parameters of these stars.). Left panels indicate long-period light curves (30 days) and right panels show enlarged light curves of superflares during about a day. Vertical short lines in some left panels in Figure 3 show the time at which the flare is detected with our detection algorithm. Bars at t = 2 of left panels and t = 0.1 of right panels show the flare detection thresholds of the light curves.

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Various different periodic brightness variations are observed on many flare stars. These variations suggest the existence of large starspots on superflare stars (Rodono et al. 1986). There are, however, other mechanisms causing the brightness variation such as orbital motion of a binary system, eclipse by an accompanying star (Kopal 1959), or stellar pulsation. Here, stellar pulsation can be excluded because the pulsation period of the G-type dwarfs is shorter than a few hours (Unno et al. 1989). The possibility of the brightness variation being due to rotation must be carefully distinguished from its being a result of orbital motion, on the basis of the difference in the shape of the light curves (Debosscher et al. 2011). Figure 4 shows examples of superflares on Sun-like stars. The durations of the detected superflares are typically a few hours, and their amplitudes are generally of order 0.1 ∼ 1% of the stellar luminosity.

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A major concern is the contamination from unresolved sources such as a low-mass companion. We discussed this topic in the supplementary information of Maehara et al. (2012) and concluded that most of the flares we detected are thought to be produced by G-type dwarfs because of the positive correlation between the amplitude of stellar brightness modulations and the flare frequency in Figure 5(b) of Maehara et al.'s (2012) supplementary information. Furthermore, our results show a dependence of flare occurrence rate on the effective temperature of the G-dwarfs (See Table 6). This dependence cannot be explained by the contamination from a low-mass companion.

Figure 5 represents the occurrence rate distributions of superflares. The configuration of this figure is basically the same as Figure 2 of Maehara et al. (2012) but the number of flares is more than four times that of the flares analyzed by Maehara et al. (2012). In Figure 5(a) we show the number of flares as a function of observed flare amplitude. The number of flares (N) is 1547, and the errors are estimated from the square root of the event number in each bin. Error bars from small number statistics are given by binomial distribution (Gehrels 1986); the error bars in figures, therefore, are overestimated when the number of flares is not large enough.

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The average occurrence rate of superflares can be estimated from the number of observed superflares, the number of observed stars, and the length of the observational period. For example, in the case of Sun-like stars, 44 superflares were detected from the data of about 14,000 stars over 500 days. Hence, the occurrence rate of superflares is 2.3 × 10⁻³ flares per year per star, which corresponds to a superflare occurring on a star once every $440_{-60}^{+80}$ yr. We show the frequency distributions of flares as a function of flare energy in Figures 5(b)–(d), but the sets of superflare-generating stars for each curve are different. The vertical axis indicates the number of superflares per star per year per unit energy. We show the distribution of all superflares (solid line) and superflares on slowly rotating (longer than 10 days period) stars (dashed line) in panel (b), superflares on all G-type dwarfs whose effective temperature (T_eff) is 5100 K ⩽T_eff < 5600 K (solid line) and 5600K ⩽ T_eff < 6000K (dashed line) in panel (c), and slowly rotating G-type dwarfs of the same temperature range as (c) in panel (d). We show comparisons of this study and Maehara et al. (2012) in panels (e) and (f). Results of this study basically agree with the results in Maehara et al. (2012), and we can see reduced errors. It is seen from Figure 5(b) that the occurrence rate of superflares in all G-type dwarfs shows a power-law distribution

$$\begin{equation*} dN/dE \propto E^{-\alpha } \end{equation*}$$

with α ≃ 2.2 for all G-type dwarfs and α ≃ 2.0 for flares on slowly rotating G-type dwarfs. It is interesting to note that these distributions are quite similar to those for solar flares (e.g., Aschwanden et al. 2000) and stellar flares on red dwarfs (e.g., Shakhovskaia 1989).

We further found from Figures 5(b)–(d) that (1) the frequency of superflares on slowly rotating stars is smaller than that for all G-type dwarfs, and (2) the frequency of superflares on hot G-type dwarfs (5600 < T_eff < 6000 K) is smaller than that for cool G-type dwarfs (5100 < T_eff < 5500 K). These are basically the same results as those found in Maehara et al. (2012). The number of superflares on Sun-like stars is 36 on 13 stars, which is much more than the 14 superflares on 10 Sun-like stars found in Maehara et al. (2012). The occurrence rate distribution of superflares on these Sun-like stars is again similar to that for our Sun. The occurrence rate of superflares with energy of 10³⁴ erg is once in 800 yr and once in 5000 yr for superflares with energy of 10³⁵ erg.

According to the Kepler candidate planet data explorer (Batalha et al. 2013), 2321 planets have been found in 1790 stars among 156,453 stars. Hence, the probability of finding exoplanets orbiting stars is about 1%. Howard et al. (2012) showed that the probability of finding a hot Jupiter was 0.5%. However, none of our superflare stars (279 G-type dwarfs) have a hot Jupiter according to the data explorer.

For a G-type dwarf with a hot Jupiter, the probability of a transit of the planet across the star is about 10% averaged over all possible orbital inclinations (Kane & von Braun 2008). If all of our 279 superflare stars are caused by a hot Jupiter as suggested by Rubenstein & Schaefer (2000), Kepler should detect 28 of them from transits. However, we did not find any hot Jupiters around our superflare stars as mentioned previously. This suggests that hot Jupiters associated with superflares are rare.

Figure 6 represents the light curve of the most active superflare star (KIC10422252). This star exhibits 57 superflares in about 500 days, and hence the superflare occurrence rate is more than once in 10 days. The variation of the stellar brightness contains modulation of the frequency which is different from but near the main frequency. Light curves of that sort are typically shown by the starspot system with differential rotation (Frasca et al. 2011).

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Figure 7 is the same as Figure 6 but for KIC10471412. This star is an active Sun-like star, which produced four superflares over the course of the entire observation period (500 days). This occurrence rate of superflares corresponds to once in 100 days.

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Figure 8 shows the flare occurrence rate distribution of the most active G-type dwarfs and Sun-like stars. The solid line corresponds to the average frequency of superflares on the 9 most active G-type dwarfs, whose superflare occurrence rate is more than once in 10 days. The dashed line shows the distribution for Sun-like stars that showed more than one superflare over the course of the entire observing period (about 500 days). Error bars are estimated from the square root of the numbers of superflares in each bin.

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It is interesting to note that the frequency of superflares with ∼10³⁴ erg in the dM4.5e star YZ CMi is once a month (Lacy et al. 1976; Kowalski et al. 2010), which is comparable to those of the most active G-dwarfs and Sun-like stars (once in 10–100 days).

We shall compare the occurrence rate distribution of superflares with those of solar flares and stellar flares on active G-type dwarfs discussed by Schrijver et al. (2012) in Figure 9. The solid-line histogram shows the flare occurrence rate distribution of superflares on G-type dwarfs and the dashed histogram corresponds to the frequency distribution of superflares on the most active G-type dwarfs. The error bars in the histogram represent the square root of event number in each bin. Solid lines indicate the frequency distribution of nanoflares observed in EUV (Aschwanden et al. 2000), microflares in soft X-rays (Shimizu 1995), and solar flares in hard X-rays (Crosby et al. 1993), respectively. The thin dot--dashed line shows a power-law line with the distribution dN/dE∝E^−α with index α ∼ 1.8. The thick dotted line corresponds to the frequency distribution of stellar flare on active G-type dwarfs shown in Schrijver et al. (2012) (κ Cet, EK Dra, and 47 Cas). The frequency of flares on these G-type dwarfs is comparable to that of most active G-type dwarfs in our data. The maximum energy of these stellar flares is lower than that of superflares on the most active G-type dwarfs.

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It is quite interesting to see that solar flares and superflares on Sun-like stars are roughly on the same power-law line. This suggests that the mechanism of superflares is similar to that of solar flares and microflares, because white-light flare energy is roughly in proportion to EUV and X-ray flare energy (but see Figure 13 for the case $F_{\rm white-light} \propto F_{\rm X-ray}^{0.65}$).

In Figure 10, we compare the frequency distribution of superflares on the most active Sun-like stars (the same curve as the dashed line of Figure 8) with those of solar flares and average Sun-like stars. It is found that the frequency of superflare occurrence in the most active Sun-like stars is about 1000 times larger than that of average Sun-like stars. In Figure 10, we also added the flare frequency in solar maximum (in 2001) and minimum (in 2008) using the statistics of solar flares based on GOES X-ray flux classification (X-class, M-class, C-class flares; see Table 4; T. T. Ishii et al. 2012, private communication). Here we assumed that the bolometric flux of flare (F_bol) (70% white-light; Kretzschmar 2011) is in proportion to the GOES X-ray flux (F_GOES).⁷ It is seen that the flare frequency in the solar maximum is larger than that in the solar minimum by a factor of about 200. It is also found that the superflare frequency curve in the most active Sun-like stars is well above that of solar flares during the maximum solar cycle. We can now understand from this figure that the difference in the activity level of the Sun (or sunspot number or total magnetic flux) leads to the difference in solar flare frequency in the different phases of the solar activity cycle.

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This suggests that the total magnetic flux of the most active Sun-like stars is larger than that of the solar maximum. Since the total magnetic flux during the solar maximum and minimum are about 10²⁴ Mx and 10²³ Mx (e.g., Solanki et al. 2002; Vieira & Solanki 2010), the total magnetic flux in the most active Sun-like stars may be estimated to be 3 × 10²⁴ ∼ 10²⁵ Mx.

Furthermore, we show the frequency of stellar flares on κ Cet with a thick dotted line in Figure 10. κ Cet is a G5-type main sequence star (T_eff = 5520 K, vsin i = 3.9 km s⁻¹, P_rot = 9.4 days; Schrijver et al. 2012; Audard et al. 2000), and hence is slightly cooler than Sun-like stars. The frequency of flares on κ Cet is larger than that of the most active Sun-like stars by a factor of 100, although the maximum energy of flares on κ Cet (10³⁴ erg) is lower than that of flares on the most active Sun-like stars (5 × 10³⁵ erg).

Flare frequency of Sun-like stars is assumed to be determined by the average magnetic activity level of these stars. In Figure 10, it is interesting to see that the flare occurrence rate distribution of the average of all Sun-like stars is located between the solar maximum and minimum. This suggests that the average activity level of all Sun-like stars is comparable to that of the Sun. It is suggested from Figure 10 that Sun-like stars have a hyperactive cycle, when they exhibit 1000 times more flares than the average of Sun-like stars. The number of all Sun-like stars observed by Kepler is 14,410 and the number of hyperactive stars we detected was 6. The fraction of stars showing a hyperactive cycle is, therefore, of the order of 10⁻⁴.

Rapidly rotating stars (rotational period: P < 10 days) tend to exhibit superflares more frequently than slowly rotating stars (P > 10 days) (see Tables 5 and 6). As mentioned in Section 2.2, the flare selection criteria, and therefore the detection completeness of flares, depends on the rotation period of the star. Our results show, however, that rapidly rotating stars show higher flare occurrence rate than slowly rotating stars. This trend cannot be affected by the nonuniformity of the detection completeness. The rotation period is correlated with the chromospheric activity, which is known to be an indicator of the magnetic activity of the stars (Noyes et al. 1984), and the rapidly rotating stars have higher magnetic activity than slowly rotating stars (Pallavicini et al. 1981). According to the dynamo theory of magnetic field generation, magnetic activity results from the interaction between rotation and convection (Parker 1979), and rapid rotation can cause high magnetic activity. Our result implies that rapidly rotating stars with higher magnetic activity can cause more frequent superflares.

The rotation period of a star is also known to be related to the stellar age and younger stars show more rapid rotation (Skumanich 1972; Barnes 2003). Our findings suggest that superflares occur more frequently on the young G-type dwarfs (i.e., rapidly rotating stars). Moreover, on G-type dwarfs similar in age to the Sun, superflares occur less frequently, though the maximum energy of superflares is nearly independent of rotation period (Maehara et al. 2012).

The relation between properties of superflares and the rotation period of superflare stars will be analyzed in detail in a future paper (Notsu et al. 2013b).

It has been pointed out that there is no record of solar superflares over the past 2000 yr (Schaefer et al. 2000). According to the measurement of the impulsive nitrate events in polar ice, the largest proton flare event during the past 450 yr is the Carrington event (Shea et al. 2006), which occurred on 1859 September 1 (Carrington 1859). The total energy released in this flare was estimated to be on the order of 10³² erg (Tsurutani et al. 2003), which is only 1/1000 of the maximum energy of flares on the slowly rotating Sun-like stars we detected. Our criterion of rotation period for Sun-like stars is more than 10 days, and the average rotation period of Sun-like superflare stars is 12.7 days. This rotation period corresponds to a rotation velocity of ∼4 km s⁻¹. According to the rotation–age relation for single stars in Ayres (1997), this rotation velocity corresponds to an age of ∼1 Gyr. The rotational velocity of the Sun is ∼2 km s⁻¹ (rotational period is 25 days) and the age of our Sun is 4.6 Gyr. Do superflares really occur on Sun-like stars most similar to the Sun which have a rotational period longer than 25 days? Table 7 shows 4 superflares (5.0 × 10³⁴, 2.4 × 10³⁴, 9.9 × 10³³, and 4.4 × 10³⁴ erg) that occurred on Sun-like stars with surface temperatures of 5600 K < T_eff < 6000 K and rotational periods longer than 25 days. The number of such stars is about 5000 in our sample. This implies that superflares whose total energy is of the order of 10³⁴ erg might occur on our Sun once in ∼2000 yr, although more detailed analysis is needed.

It has also been proposed that hot Jupiters have an important effect on stellar magnetic activity (Rubenstein & Schaefer 2000; Cuntz et al. 2000; Ip et al. 2004) and that superflares occur only on G-type dwarfs with hot Jupiters. However, there is no hot Jupiter in our solar system. For these reasons, it was suggested that a superflare on the Sun was extremely unlikely (Schaefer et al. 2000; Rubenstein & Schaefer 2000).

We found, however, no hot Jupiters around the 279 G-type dwarfs with superflares (see Section 3.3). This suggests that a hot Jupiter is not a necessary condition for superflares. Recent theoretical research (Shibata et al. 2013) also suggested that G-type dwarfs can store enough magnetic energy to cause a superflare in the overshoot layer. Moreover, Miyake et al. (2012) reported an occurrence of an energetic cosmic-ray event in the eighth century recorded in a tree ring of Japanese cedar trees. There is a possibility that this event was produced by a superflare (with energy of ∼10³⁵ erg) on our Sun.

In this paper, we found 1547 superflares on 279 G-type dwarfs in 500 days of Kepler data. Using these superflares, we calculated superflare occurrence rate distributions as a function of total superflare energy. This shows that power-law distributions and the power-law index were comparable to those of solar flares (∼2), and the average occurrence rate of superflares with energy of 10³⁴–10³⁵ erg is once in 800–5000 yr. According to the Kepler candidate planet data explorer, there was no host star of a hot Jupiter in all superflare stars. We showed a comparison of flare occurrence distributions of solar maximum, minimum, and superflare stars, and pointed out the existence of a hyperactive cycle of G-type dwarfs. We found hyperactive G-type superflare stars, which exhibit superflares more than once in 10 days. We found 36 superflares on Sun-like stars, and this is much more than the 14 superflares on Sun-like stars discovered by Maehara et al. (2012). Nevertheless, this number is still not enough to derive accurate statistical properties of superflares on Sun-like stars. This is important for discussion of the possibility of superflares on our present Sun. Hence, it will be important to continue to observe Sun-like stars for a longer time to increase the potential observation of superflares on Sun-like stars. It will also be important to get short-time (1 minute) cadence data to obtain information on small superflares (with energy of 10³³–10³⁴ erg) which bridge the gap between superflares on Sun-like stars and the largest solar flares 3 × 10³² erg. Kepler only obtains photometric data. Rotational periods and areas of star spots calculated from them are an indirect estimate based on some presumptions. In order to get direct evidence of slow rotation, large spot, high activity, and binarity, high-dispersion spectroscopic observation is necessary. We have observed some superflare stars with the HDS on the Subaru telescope. The results from these observations will be reported elsewhere (Notsu et al. 2013a).