Statistical properties of superflares on solar-type stars based on 1-min cadence data

We searched for superflares on solar-type stars using Kepler data with 1 min sampling in order to detect superflares with short duration. We found 187 superflares on 23 solar-type stars whose bolometric energy ranges from the order of $10^{32}$ erg to $10^{36}$ erg. Some superflares show multiple peaks with the peak separation of the order of $100$-$1000$ seconds which is comparable to the periods of quasi-periodic pulsations in solar and stellar flares. Using these new data combined with the results from the data with 30 min sampling, we found the occurrence frequency ($dN/dE$) of superflares as a function of flare energy ($E$) shows the power-law distribution ($dN/dE \propto E^{-\alpha}$) with $\alpha \sim -1.5$ for $10^{33}<E<10^{36}$ erg which is consistent with the previous results. The average occurrence rate of superflares with the energy of $10^{33}$ erg which is equivalent to X100 solar flares is about once in 500-600 years. The upper limit of energy released by superflares is basically comparable to a fraction of the magnetic energy stored near starspots which is estimated from the photometry. We also found that the duration of superflares ($\tau$) increases with the flare energy ($E$) as $\tau \propto E^{0.39\pm 0.03}$. This can be explained if we assume the time-scale of flares is determined by the Alfv$\acute{\rm e}$n time.


Introduction
Solar flares are eruptive events in the solar atmosphere caused by the magnetic reconnection (e.g. Shibata & Magara 2011). The occurrence frequency of solar-flares decreases as the flare energy increases.
The frequency-energy distribution of solar flares can be fitted by a simple power-law function with an index of −1.5 -−1.9 in the flare energy range between 10 24 erg and 10 32 erg (e.g. Aschwanden et al. 2000;Shimizu 1995;Crosby et al. 1993). The total bolometric energy released by the largest solar flares is estimated to be the order of 10 32 erg (Emslie et al. 2012) and the occurrence frequency of such flare is about once in 10 years. On the other hand, much larger flares called "superflares" have been observed on a wide variety of stars including solar-type stars (Landini et al. 1986;Schaefer 1989;Schaefer et al. 2000).

Methods and Data
We searched for flares (sudden brightenings of the star) from the short-cadence data (∼ 1 min interval) observed with the Kepler space telescope between 2009 April (quarter 0: Q0) and 2013 May (Q17) (Koch et al. 2010;Gilliland et al. 2010). We selected solar-type (G-type main sequence) stars from the data set by using the surface temperature of the star (T eff ) and the surface gravity (log g) taken from Huber et al. (2014) instead of those from Brown et al. (2011) (initial Kepler Input Catalog). In previous papers (Maehara et al. 2012;Shibayama et al. 2013), we used stellar parameters taken from Brown et al. (2011) and the selection criteria of 5100K < T eff < 6000 K and log g > 4.0. However the temperatures in Brown et al. (2011) are systematically lower by ∼ 200 K than those in Huber et al. (2014). Since the occurrence frequency of superflares depends on the surface temperature (e.g. Candelaresi et al. 2014), this systematic difference in surface temperature of stars would affect the resultant flare occurrence frequency. Therefore, in order to reduce the difference in the occurrence frequency of superflares caused by the systematic difference in temperature, we used the selection criteria of 5300K < T eff < 6300K and 4.0 < log g < 4.8. The total number of solar-type stars observed with short-cadence mode is 1547.
We used the Presearch Data Conditioned (PDC) light curve Smith et al. 2012)for the detection of flares. The typical length of each continuous observation in short-cadence mode is about 30 days. We first calculated the distribution of the following parameter (∆F 2 ) for all consecutive data points from two pairs of the stellar brightness (F (t)) at the time of t i−n−1 , t i−n , t i , and t i+1 (i and n are integer numbers and t i means time of the i-th data point): In the case of typical 30-day short-cadence data blocks, i ranges from 1 to ∼ 44000. If the flare peak is around at t i and the stellar brightness at t i−n is close to the quiescent level, the value of ∆F 2 (t i ) becomes much larger than the mode of the distribution. Therefore t i − t i−n should be comparable to or longer than the rise time of superflares. According to Shibayama et al. (2013), the rise time of superflare on KIC 11610797 is about 2 min. In the case of the X17 1 solar flare (Kopp et al. 2005), the rise time is about 4 min. Thus we used the separation between two data point n = 3 (t i − t i−n ∼ 3 min) and n = 10 (∼ 10 min) for the analysis. In the case of larger n value, we can detect flares having longer rise-time.
The threshold of the flare was determined to be three times the value at the top 1% of the distribution.
We removed long-term brightness variations around the flare peak by fitting a quadratic function. We used the data from 0.05 to 0.01 (in the case of n = 3) or from 0.15 to 0.03 (n = 10) days before the flare peak and the data from 0.05 to 0.10 day (n = 3) or 0.15 to 0.25 day (n = 10) after the peak for the fitting. After removing the long-term light variations, the flare start time and flare end time are defined to be the time at which the residuals of stellar brightness become larger and smaller than three times of the photometric error (σ photo ; PDC flux errors given in the Kepler data) respectively. We analyzed only flares with the duration longer than 5 min (at least 5 consecutive data points exceed 3σ photo ) since it is difficult to distinguish between true flares and false events in the case of a single data point above the threshold (e.g. spikes in figure 1 (h)).
We excluded the flare candidates which have longer rise time than decay time. Since the number of stars observed in short-cadence mode is only ∼ 1500, the possibility that two or more stars exhibit flares at the same time is negligible. Thus we also removed the pairs of flare events on different stars which have the same flare peak time. After removing these false events, we checked all light curves of flare candidates by eye and removed non-flare events, since our flare detection method misidentified some eclipses or light curve jumps as flares. We also examined the pixel level data of flare stars in order to exclude the contamination of flares from nearby stars or other false events. If the spatial pixel-distributions of the brightness of a star on CCDs are different between during the flare and at the quiescence, the candidate of a flare is revealed to be a light variation of nearby source or a non-astrophysical event (e.g. Maehara et al. 2012). We removed 5 events in this step.
We checked the Kepler Eclipsing Binary Catalog Slawson et al. 2011;Kirk et al. 2013) and removed all flare candidates which occurred on the eclipsing binary (14 stars). We also removed all flare candidates on the stars which are listed as visual or spectroscopic binaries in Notsu et al. (2015a) (2 stars). In total, we eliminated 152 flare candidates on these 16 stars. Since the fraction of known binary systems would be small, there may be more unknown binary systems in the sample. According to Notsu et al. (2015a), the lower limit of binary fraction among superflare stars based on high-dispersion spectroscopy is about 30 %, which is not so different from the binary fraction of nearby solar-type stars (e.g. Duquennoy & Mayor 1991).
Bolometric energy emitted by each flare was estimated from the stellar luminosity, flare amplitude and flare duration with the same manner as Shibayama et al. (2013). The luminosity of each flare star were estimated from the effective temperature and the stellar radius taken from Huber et al. (2014). The error in bolometric energy was estimated from errors in the temperature and stellar radius from Huber et al. (2014). Since the error in the temperature is σ Tstar ∼ 200 K (σ Tstar /T star ∼ 3-4%) and that in the stellar radius is σ Rstar /R star ∼ 10 %, the typical error in bolometric energy would be σ E flare /E flare ∼ 25 %.
We estimated periods of long-term light variations from light curves of each star by using the discrete Occurrence frequency distribution Figure 2 represents the occurrence frequency distribution of superflares with energy > 10 33 erg as a function of the rotation period. The flare frequency strongly depends on the rotation period and the frequency rapidly decreases as the rotation period increases in the period above a few days. No flare was detected on the stars with P rot ≥ 20 day from short-cadence data and the upper limit of the frequency for this bin is < 1 in 80 years. The flare occurrence frequency on the stars with 2.5 ≤ P rot < 20 day roughly proportional to P −3 rot . The flare frequency on the stars with P rot ≥ 20 day expected from this decrease trend is ∼ 0.004 year −1 star −1 which is about 8 times lower than that on the stars with 10 ≤ P rot < 20 day. The similar relation between flare frequency and rotation period was reported by previous studies using the long-cadence data (e.g. Notsu et al. 2013b). The rotation period is thought to be related to the age of stars (e.g. Ayres 1997). This result suggests that the young solar-type stars would exhibit superflares more frequently. solar-type stars derived from short-cadence data and dashed histogram represents that from long-cadence data (Shibayama et al. 2013). The average flare frequencies in a given energy bin from short-cadence data are about 5 times higher than those from long-cadence data. This difference in the flare frequency is mainly caused by that in the rotation period distribution of the observed targets. The period distribution of the targets in short-cadence mode is biased to the shorter-period end. The fraction of stars with the rotation period (P rot ) < 10 days observed in long-cadence mode is 8.1 % (8312/102709). On the other hands, that of stars with P rot < 10 days in short-cadence mode is 32% (499/1547). As mentioned above, the stars with shorter rotation period exhibit more frequent superflares. Therefore, the average flare frequency derived from short-cadence data is higher than that from long-cadence data. Solid histogram in figure 3 (b) represents the corrected frequency distribution of superflares from short-cadence data taking into account the difference in the period distribution of the observed samples. Since the ratio of total observation time (sum of the length of observation time for each star) for stars with P rot < 10 day to that for stars with P rot ≥ 10 day in short-cadence mode (1:1.7) is about 6.5 times larger than the ratio in long-cadence mode (1:11), we calculated the corrected occurrence frequency by adjusting the ratio of total observation time in short-cadence mode to that in long-cadence mode. Both frequency distributions from short-and long-cadence data are almost the same for the flare energy between 10 34 erg and 10 36 erg. The frequency of flares derived from the long-cadence data is less than that from the short-cadence data for the flare energy below 10 34 erg. This difference is caused by the detection limit of flare search method. Since the time-resolution of long-cadence data is much lower than that of short-cadence data, we could not detect smaller flares from long-cadence data and observed flare frequency from long-cadence data becomes much lower than that from short-cadence data.
The frequency distribution of superflares from the short-cadence data can be fitted by a power-law function with the index of −1.4 ± 0.2 for the energy between 4 × 10 33 erg and 1 × 10 36 erg. This value is comparable to the power-law index for the frequency distribution of superflares derived from the longcadence data. Using the combined data set from both short-and long-cadence data, the power-law index is −1.5 ± 0.1 for the flare energy of 4 × 10 33 -1 × 10 36 erg. Shibata et al. (2013) pointed out that the frequency distribution of superflares on Sun-like stars (early-G dwarfs with P rot ≥ 10 day) and those of solar flares are roughly on the same power-law line. Figure 4 represents the comparison between the frequency distribution of superflares on Sun-like stars (5800 ≤ T eff < 6300 and P rot ≥ 10 day) derived from both short-(filled circles) and long-cadence (solid line) data and those of solar flares (dashed lines). The thin dotted line indicates the power-law function with an index of -1.8 taken from Shibata et al. (2013). The frequency distribution of superflares derived from short-cadence data is also on the same power-law line. This distribution suggests that average occurrence frequency of superflares on Sun-like stars with bolometric energy of 10 33 erg, 10 34 erg, and 10 35 erg are once in ∼ 70 years, ∼ 500 years and ∼ 4000 years respectively. As mentioned in previous paragraph, the frequency of superflares strongly depends on the rotation period. The flare frequency on the stars with the same rotation period as the Sun (P rot ∼ 25 day) would be ∼ 8 times smaller that that on the stars with 10 ≤ P rot < 20 day. Therefore the occurrence frequency of superflares with energy geo-effective CMEs is less than 10% (e.g. Gopalswamy et al. 2007). If the fraction is same for superflares on our Sun, the occurrence rate of extreme GIC events (10 times or more larger than the Carrington event) would be less than once in a few thousand years.

Relation between flare energy and area of starspots
Most of superflare stars show large-amplitude light variations with periods from a few days to ∼ 20 days.
If we assume these quasi-periodic variations are caused by the rotation of the star, the amplitude and period of light variations correspond to the area of starspots and the rotation period (Notsu et al. 2013b).
The large-amplitude light variations of superflare stars suggest that the surface of superflare stars are corved by large starspots.
where A star is the apparent area of the star, T spot and T star are the temperature of starspot and photosphere of the star (Notsu et al. 2013b). Here we define the amplitude as the brightness range normalized by the average stellar brightness, in which the lower 99% of the distribution of the brightness difference from average, except for the flares, are included. We estimated the amplitude from the longcadence data of the quarter at which each superflares detected from the short-cadence data occurred. Here the stored magnetic energy can be roughly estimated by where B and L correspond to the magnetic field strength of the starspots and the scale length of the starspot group respectively. By using the total area of starspot group A spot ∼ L 2 , the energy released by a flare can be written as where f is the fraction of magnetic energy released by the flare. This suggests that the flare energy is roughly proportional to the area of the starspot group to the power of 3/2. Since f = 0.1 (Aschwanden et al. 2014), the typical sunspot area for generating X10-class flares (∼ 10 32 erg) observed in 1989-1997 was 3 × 10 −4 of the half area of the solar surface (Sammis et al. 2000) and the typical magnetic field strength of sunspot is the order of 1000 G (e.g. Antia et al. 2003), equation (5) can be written as Solid and dashed lines in the figure 5 represent equation (6) for f = 0.1, B = 1000 and 3000 G respectively.
Majority of superflares detected from short-cadence data (filled-squares) and almost all solar flares (small dots) are below these lines. This suggests that the upper limit of the energy released by the flare is basically determined by the magnetic energy estimated from the area of starspots. However the bolometric energies of superflares detected from long-cadence data (small-crosses) are biased to largerenergy end. As mentioned above, due to the low time-resolution, only superflares with larger bolometric energies could be detected from long-cadence data. In addition, superflares with large energy but small occurrence frequency could be detected from the long-cadence data because of the large number of stars (∼ 10 5 ).
There are some superflares above the analytic relation. KIC 7093428 is one of superflare stars exhibiting superflares whose energy is much larger than that expected from equation (6). This object is located within the error circle of an X-ray source 1SXPS J184925.7+423901 (Evans et al. 2014). The X-ray luminosity of the object is ∼ 10 28 erg/s if the distance of the object is ∼ 200 pc. The empirical relation between the X-ray luminosity and the magnetic flux of the star (Pevtsov et al. 2003) suggests that the magnetic flux of KIC 7093428 is the order of 10 24 Mx. This value corresponds to the spot area of 3 × 10 20 cm 2 for B = 3000G, and the energy of the largest superflare on KIC 7093428 is comparable to the flare energy estimated from equation (6). The spot area of the object expected from the X-ray luminosity is more than one order of magnitude larger than that estimated from the amplitude of quasi-periodic modulations. The amplitude of light variations due to the rotation of star is affected by the inclination of rotation axis and the latitude of starspots (e.g. Notsu et al. 2013a,b). In the case of stars with lowinclination angle or stars with starspots around polar region, the light variation caused by the rotation become small and the area of starspots derived from the amplitude of light variation is smaller than the actual area. According to Notsu et al. (2015b), some superflare stars which show superflares with energy larger than that expected from the amplitude of light variation have low-inclination angles.
In addition, polar spots are often observed on rapidly-rotating cool stars (e.g. Strassmeier 2009). Another possibility is flares on the companion star. Kitze et al. (2014) found that the astrometric position of the Sun-like superflare star, KIC 7133671 which show a superflare with energy larger than expected from the photometric spot area, was shifted by 25 mas during the flare. This suggests that the flare on KIC 7133671 occurred on a faint companion star. In this case, the flare energy would not be related to the area of starspot group on the primary star.
Correlation between flare duration and flare energy where τ flare and E flare indicate the duration and bolometric energy of flares. The duration of superflares derived from long-cadence data (crosses) is longer than that from short-cadence data. This difference may be caused by the difference in time resolution of the data and selection bias. Since the time resolution of long-cadence data is 30 min and much longer than that of short cadence data (1 min), the flare detection method for long-cadence data used in Maehara et al. (2012) and Shibayama et al. (2013) can detect only superflares with long duration. The bolometric energy and duration in optical wavelength of the X17solar flare on 2003 October 28 are the order of 10 32 erg and ∼ 10 min (Kopp et al. 2005), which are roughly comparable to the correlation between the duration and energy of superflares on solar-type stars.
Similar correlation between the flare duration and energy was observed in solar flares. The power-law slope for the correlation between the duration of solar flares and X-ray fluence observed with the GOES is about 1/3 (Veronig et al. 2002). The correlation between the duration and peak flux observed with RESSI also shows similar power-law slope of ∼ 0.2 (Christe et al. 2008). These similarity between solar flares and superflares on solar-type stars suggests that solar flares and superflares are caused by the same physical process (i.e. reconnection).
As discussed in the previous subsection, the flare energy is related to the magnetic energy stored near the starspots as follows: where f , B, and L correspond to the fraction of energy released by a flare, the magnetic field strength of the starspots and the scale length of the starspot group respectively. On the other hand, the duration of flares, especially the duration of impulsive phase of flares, is comparable to the reconnection time (τ rec ) which can be written as where τ A = L/v A is the Alfvén time, v A is the Alfvén velocity, and M A is the non-dimensional reconnection rate which is comparable to 0.1-0.01 for the fast reconnection such as Petschek model (e.g. Shibata & Magara 2011). Since all G-type main sequence stars have similar stellar properties (e.g. T eff and log g), B and v A might not be so different among them. Therefore, from equation (8) and (9), the duration of flares can be written as This suggests that the power-law slope for the correlation between the flare duration and flare energy is about 1/3 and this is comparable to the observed value of 0.39 ± 0.03.

Conclusions
We found 187 superflares on 23 solar-type stars from the 1-min cadence data obtained with the Kepler space telescope. Because of the high temporal resolution of the data, we found the following new results.
(1) The power-law frequency distribution of superflares (e.g. Maehara et al. 2012;Shibayama et al. 2013) continues to the flare energy of 10 33 erg and the average occurrence rate of the X100 superflares on the stars with similar rotation period to the Sun is about once in 500-600 years.
(2) Some of superflares show multiple peaks during a flare with the peak separation of 100-1000 seconds.
The time scale of these modulation during the flare is comparable to the periods of quasi-periodic pulsations in solar and stellar flares.

Competing interests
The authors have no competing interests to declare.   The vertical axis indicates the number of superflares per star and per year in each period bin. Period bins are defined as follows: P < 2.5 day , 2.5 ≤ P < 5 day, 5 ≤ P < 10 day, 10 ≤ P < 20 day, and P ≥ 20 day. Solid-line histogram represents the occurrence frequency of superflares with an energy of    Superflares (Shibayama et al. 2013) dN/dE∝E −1.53 Solar flares (Crosby et al. 1993) dN/dE∝E −1.74 Microflares (Shimizu 1995) dN/dE∝E −1.79 Nanoflares (Aschwanden et al. 2000) Figure 4. Comparison between occurrence frequency superflares on Sun-like stars and those of solar flares. Filled-circles indicates the occurrence frequency distribution of superflares on Sun-like stars (G-type main sequence stars with P rot > 10 days and 5800 < T eff < 6300K) derived from short-cadence data. Horizontal error bars represent the range of each energy bin. The definition of the vertical error bars is the same as figure 3. Bold-solid line represents the power-law frequency distribution of superflares on Sun-like stars taken from Shibayama et al. (2013). Dashed lines indicate the power-law frequency distribution of solar flares observed in hard X-ray (Crosby et al. 1993), soft X-ray (Shimizu 1995), and EUV (Aschwanden et al. 2000). Occurrence frequency distributions of superflares on Sun-like stars and solar flares are roughly on the same power-law line with an index of −1.8 (thin-solid line) for the wide energy range between 10 24 erg and 10 35 erg. Flare energy (erg) Figure 6. Scatter plot of the duration of superflares as a function of the bolometric energy. Filled-squares and small-crosses indicate superflares on G-type main sequence stars detected from short-and long-cadence data respectively. We used e-folding decay time as the flare duration.
Dotted line indicates the linear regression for the data of superflares from short-cadence data. The power-law slope of the line is 0.39 ± 0.03.