A Comparative Study of the Magnetic Activities of Low-mass Stars from M-type to G-type

NASA launched the Kepler spacecraft, which is equipped with a 0.95 m aperture Schmidt telescope and forty-two 50 × 25 mm CCDs at 2200 × 1024 pixels each, on 2009 March 7. Kepler FOV covers 115 square degrees, around 0.25% of the sky. It provided an excellent photometric precision (only a few ppm for a star of V = 12) in visible light, and near-infrared region from 423 to 897 nm (Koch et al. 2010). Its primary science goal was to discover the Earth-size and super-Earth-size exoplanets in or near the habitable zones of their host stars by searching for the transit events from the light curves of target stars in the Cygnus–Lyra region (Koch et al. 2006; Borucki et al. 2010). There were four reaction wheels installed in the Kepler spacecraft for keeping the field of view focusing on the Cygnus–Lyra region. Unfortunately, two of the reaction wheels were out of control in 2012 July and 2013 May, respectively. Later, NASA started planning the new observation program (the K2) with the last two reaction wheels and the thrusters. Finally, it was decided to use the sunlight as the third wheel to control the spacecraft pointing. The K2 mission started in 2013 November with its FOV covering a large area in Earth-trailing heliocentric orbit and changing every three months in successive campaigns (Howell et al. 2014). Since 2013 until 2018 October, the K2 mission had collected 20 campaigns' (campaigns 0–19) data.

There are two types of time resolutions in the Kepler archive data, the LC in 29.4 minutes and the short cadence (SC) in 58.9 s (Gilliland et al. 2010). For the K2 mission, on average, there are between 10,000 and 20,000 LC targets and 50–100 SC targets in each campaign (Van Cleve & Caldwell 2016). Since the number of SC targets is much lower than the number of LC targets, we decided to use the LC targets for this study.

The photometric precision of the K2 data is about 80 ppm for a star of V = 12 on a timescale of 6 hr, and it is within a factor of 3–4 times the original Kepler's precision (Howell et al. 2014). The lengths of the time series of the K2 data in most of campaigns 10 and 11 were operationally separated into two segments (e.g., C101, C102, and C111, C112) due to the initial pointing error in C101 and the spacecraft roll-angle error in C111, respectively. Although these errors were corrected in C102 and C112, the divided campaigns' time-series lengths are inevitably shorter than others. Many of the target pixel apertures of C101 data are nullified which would make photometric precision degenerate, and we thus do not use C101 data for this study. Also, the loss of module 4 during C102 led to a 14-day data gap in time series. Fortunately, it did not affect the flux values of targets on either side of the data gap because the spacecraft was correctly at the relative location to the Earth without any significant shifts.

The specific-target Kepler light-curve data files in binary FITS format have been derived from the Target Pixel File (TPF). Each of the FITS files contains two pieces of primary flux information, the Simple Aperture Photometry (SAP) flux with 1σ statistical uncertainties and the SAP that includes artifact mitigation called Pre-search Data Conditioning SAP (PDCSAP) flux with uncertainties (Smith et al. 2012; Stumpe et al. 2012). Since the SAP is contaminated by pointing drift and the systematics focus artifacts, it is not the best option to explore subtle astrophysics phenomena such as flare events. On the other hand, the PDCSAP is the SAP with removal of the errors such as systematics while preserving any astrophysically interesting signals by using the PDC pipeline module (Smith et al. 2012). As an example, Figure 1 shows the difference between the SAP and PDCSAP light curve of the star, KIC 1569836. We chose the PDCSAP of the LC data to inspect the flare events on the dwarf stars.

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We collected 10,750 K2 LC light-curve data, including 3501 M- and 7249 K-type stars in several campaigns from the Mikulski Archive for Space Telescopes (MAST) data archive. The numbers of M dwarfs in the individual Campaign are 1094 (C1), 526 (C3), 941 (C4), 472 (C5), 4 (C12), 71 (C13), 80 (C14), 294 (C102), 9 (C111), and 10 (C112); for K dwarfs, they are 675 (C1), 799 (C3), 870 (C4), 742 (C5), 770 (C6), 1687 (C7), 720 (C8), 836 (C12), 674 (C13), 525 (C14), 1028 (C15), 675 (C16), 573 (C102), 196 (C111), and 176 (C112). The observation periods of these campaigns are 82 days (C1), 69 days (C3 and C102), 70 days (C4), 74 days (C5), 79 days (C6), 83 days (C7), 79 days (C8), 79 days (C12), 80 days (C13), 79 days (C14), 88 days (C15), 80 days (C16), 67 days (C102), 23 days (C111), and 48 days (C112), respectively. The selection criteria included the distance from the Earth, the mass, the effective temperature and all of these stellar parameters were obtained from Huber et al. (2016). The stars had to be within 100 pc (M dwarfs) and 200 pc (K dwarfs) from the Earth because we plan to study the exoplanets around the nearby flare stars in a future work. Because our intention was to compare the M, K, and G dwarfs' magnetic activities with each other, the mass and the effective temperature of stars we selected must be in the range of 0.08 M_⊙ to 0.81 M_⊙ and from 2300 K to 5230 K, respectively. Also, the surface gravity, log(g) must be in the range of 4.32 to 4.63.

We generated the detrended (or background) light curve of each dwarf in the data analysis. For the analysis of the brightness variation, we first produced a fake curve by using the four-point moving median for subtraction of the original curve (F_PDC) from the fake curve; then we computed the median absolute derivation (MAD) of the residuals. If the residual was greater than 4.5 times MAD, the counterpart data point in F_PDC would be excluded from the second four-point moving median. The gaps that we created during the detrending would be filled in by interpolation with quadratic fitting. After second-time detrending, we obtained the background curve (F_background). We subtracted the F_background from F_PDC and searched for the flare events in the residual curve (F_R) with MAD of residuals.

The flare's detection algorithm is given by the expression:

$$\begin{eqnarray}&&{F}_{{Ri}}\gt 0\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{{Ri}}}{\mathrm{MAD}({F}_{R})}\geqslant 3\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\mathrm{Cons}\geqslant 2.\end{eqnarray} \tag{ 3 }$$

F_Ri is the ith point in F_R. The Cons is the number of the consecutive points that have satisfied Equations (1) and (2). We also calculated the flare amplitude of each flare. The flare amplitude represents, in comparison with the stellar flux, how much brighter the flare peak is. The flare peak amplitude can be calculated with a flux normalization equation:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}F}{\tilde{F}}=\displaystyle \frac{{F}_{\mathrm{PDC}}-\tilde{F}}{\tilde{F}},\end{eqnarray} \tag{ 4 }$$

where F_PDC is the PDCSAP flux, $\tilde{F}$ is the median of the F_PDC. Figure 2 shows the F_PDC and F_background curves of EPIC 210564347 and Figure 3 illustrates the detection method of the corresponding flare candidates.

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We have detected 10,240, and 10,934 flare candidates on 850 M dwarfs and 3247 K dwarfs, respectively. The next step is to exclude the candidates that are not real flares but rather flare shaped noises or the burst signals caused by cosmic-ray hits, etc. Figure 4 demonstrates how to exclude fake flares and collect the real flares in reliable flare candidates by inspecting the F_R, F_PDC, and F_barkground. Please note that all of these flux curves have been normalized with Equation (4). The upper panel shows an example of EPIC 201303324. This flare candidate does not have typical flare shape in F_R. Also, in the F_PDC, the flare candidate is the end of the gap, which may be caused by the telescope motion during the observation. Consequently, the flare candidate has to be excluded. The middle panel in Figure 4 shows an example of EPIC 211392382. In F_R, the flare candidate is in the flare shape. In the light curve, however, the flare candidate is apparently one of the noise signal (or under the noise level) because of the low signal-to-noise ratio. Again, it has been excluded from the real flare events. The lower panel shows an example of EPIC 201102172, which is a real flare.

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How good is our flare detection method? To prove that our flare detection method is appropriate, we searched for the flares on a solar-type star KIC 10422252 and compare our result with that of Shibayama et al. (2013; see Figure 5). Shibayama et al. (2013) detected 57 flares on KIC 10422252 in about 500 days, and we found 114 flares in the same time interval. Although we missed 5 of the 57 flares detected by Shibayama et al. (2013), it demonstrates that our method is appropriate, and it is more sensitive to the small flares near the trough and at increasing phase.

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Stellar flares are caused by explosive energy release, and their energy estimates are necessary for appraising the habitability of the exoplanets around flare stars. We referred to Wu et al. (2015), Chang et al. (2017, 2018), and Shibayama et al. (2013) for details of flare energy estimates. Briefly speaking, we first estimate the bolometric energy of each flare with the following equation:

$$\begin{eqnarray}&&{E}_{\mathrm{flare},\mathrm{bol}}={L}_{* }\int \displaystyle \frac{{\rm{\Delta }}F(t)}{\tilde{F}}\,{dt}\ (\mathrm{erg}),\end{eqnarray} \tag{ 5 }$$

where ΔF(t) represents the flux differences between flares in the time profile, and L_* is the stellar luminosity calculated from the Stefan–Boltzmann law:

$$\begin{eqnarray}&&{L}_{* }=4\pi {R}_{* }^{2}{\sigma }_{\mathrm{sb}}{T}_{\mathrm{eff}}^{4}\ (\mathrm{erg}\,{{\rm{s}}}^{-1}),\end{eqnarray} \tag{ 6 }$$

where σ_sb is the Stefan–Boltzmann constant. The stellar radius (R_*) and the effective temperature (T_eff) are given in the K2 EPIC catalog. Because the bandpass from 423 nm to 897 nm of the CCDs on the Kepler telescope has a certain transmission curve (Van Cleve & Caldwell 2016), there will be a reduction to the actual stellar and flare fluxes that we refer to as the instrument response factor α(T). By convoluting the Planck function at temperature T with the Kepler wavelength-dependence instrument function, α(T) can be expressed as a polynoimal function (see Figure 6):

$$\begin{eqnarray}\begin{array}{rcl}\alpha (T) & = & 0.19970{x}^{6}-1.45856{x}^{5}+4.30523{x}^{4}\\ & & -\,6.44683{x}^{3}+4.93603{x}^{2}-1.64091x+0.19489,\end{array}\end{eqnarray} \tag{ 7 }$$

where $x=\tfrac{T}{{T}_{\odot }}$ for T_⊙ = 5800 K. The above equation is applied to the stellar effective temperature range from 2500 to 10,000 K. Then, we derived the following equation for the Kepler response flare energy estimation:

$$\begin{eqnarray}&&{E}_{\mathrm{flare}}=\displaystyle \frac{1}{4}\displaystyle \frac{\alpha ({T}_{* })}{\alpha ({T}_{\mathrm{flare}})}{E}_{\mathrm{flare},\mathrm{bol}}(\mathrm{erg}),\end{eqnarray} \tag{ 8 }$$

where T_* is the effective temperature of the stellar photosphere and T_flare is the flare temperature at maximum brightness. The white light flare emission was assumed to be characterized by a blackbody spectrum with T_flare = 9000 ± 500 K (Hawley & Fisher 1992; Hawley et al. 2003; Kretzschmar 2011), thus α(9000 K) = 0.073.

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Figure 7 shows the duration distribution. Obviously, the flares with durations ∼1.5 hr are the majority of the flares. The shortest duration of the flare is also about 1.5 hr because of the time resolution of the LC light curve and the flare detection algorithm. The longest flare duration is about 5 hr.

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Figure 8 shows the flare peak amplitude distributions of different spectral types. The maximum flare amplitude we found on the M dwarfs in the present data set is about 12.1, which means that at its brightest moment the flare is about 12.1 times brighter than the star (see also Chang et al. 2018 for an analysis of M-dwarf flares from the primary Kepler mission). For K dwarfs, the highest ${\rm{\Delta }}F/\tilde{F}$ is about 78% of the stellar luminosity. Therefore, the later type dwarfs tend to have flares with higher flare peak amplitudes.

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Figure 9 shows the histograms of the distributions of flare energies for different spectral types. The most intense flare, which exploded on a M dwarf released about 1 × 10³⁵ erg, is almost 1000 times larger than the most powerful solar flare. The most powerful flare on K dwarfs released about 4 × 10³⁵ erg. This shows that earlier type dwarfs tend to have more powerful flares even though their peak amplitudes are smaller than those of later type stars.

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Figure 10 displays the stellar flare occurrence percentage distribution. The flare percentage can be derived from the equation given by Walkowicz et al. (2011):

$$\begin{eqnarray}&&\mathrm{Flare}\quad \mathrm{occurrence}\quad \mathrm{percentage}=\displaystyle \frac{\sum {t}_{\mathrm{flare}}}{\sum {t}_{\mathrm{star}}}\times 100 \% ,\end{eqnarray} \tag{ 9 }$$

where $\sum {t}_{\mathrm{flare}}$ is the summation of the flare duration of each star, and $\sum {t}_{\mathrm{star}}$ is the observation period of each star. We found that the later type (M) stars tend to have higher flare percentages (2.5% at maximum) than G-type stars (1.0% at maximum).

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Because of the stellar magnetic dynamo mechanism, the rotating period is one of the critical parameters for producing the flare events (Pallavicini et al. 1981; Noyes et al. 1984). We estimated the rotation periods of detrended light curves of 548 flaring M dwarfs and 343 flaring K dwarfs by using the python package gatspy (VanderPlas & Ivezic 2015) from which the Lomb–Scargle method (Barning 1963; Lomb 1976; Scargle 1982) can be used to deduce the period or frequency that exists in the time series signal data.

The time interval of the K2 data is inevitably shorter than that of the Kepler (prime mission) data due to the difference between their survey strategies. It may affect the accuracy of the rotation period estimation, especially for the stars with long-term brightness variation. To clarify how much influence this has, we analyzed the LC data of the 1000 Kepler targets that have the rotation period published by McQuillan et al. (2014). Also, we compared our results with those from McQuillan et al. (2014) to demonstrate that the rotation period method in the present study is reliable.

In addition, we estimated the accuracy of rotation period determinations based on the relatively short time intervals (∼80 days in each campaign) obtained from the K2 observations. It was found that the success rate is 92% for P < 10 days and about 80% for P ∼ 10–20 days. For the rest, the success rate is below 50% for P > 20 days (see Table 5 in Appendix). Figure 11 shows the rotation period distributions of the three types of low-mass dwarfs in this study. An alternative presentation is to have only three categories of rotation periods, i.e., P < 10 days, P ∼ 10–20 days, and P > 20 days. The proportion of the slow-rotating (and hence old) flare stars is small (∼1%) but important to the issue of exoplanetary habitability and the spaceweather effect of the Sun itself. See the Appendix for details.

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In this study, the K2 light curves of 3501 M dwarfs and 7249 K dwarfs have been analyzed. We detected flare events on 548 M-type (≈15.62%) and 343 K-type stars (≈4.73%). On the other hand, Shibayama et al. (2013) have found only 279 flaring G dwarfs in 80,000 solar-type stars (≈0.34%). We have investigated the proportion of flare stars in several specific temperatures and mass ranges (see Figure 16). Since we cannot verify which 80,000 solar-type stars Shibayama et al. (2013) investigated, we only show the M- and K-type flaring stars-to-all stars ratio in Figure 12. In general, the lower mass or cooler stars tend to have high flare occurrence probability. However, we found that the proportion of flare stars do not decrease with increasing temperatures or mass all along. A plateau appears in the temperature range of 2500–4000 K, or in the mass range of 0.1–0.3 M_⊙. This means that stars with mass and temperature around 0.08–0.1 M_⊙ and 2300–2500 K, respectively, have the highest proportion of flaring stars.

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The flare peak amplitude, ${\rm{\Delta }}F/\tilde{F}$, can be used to classify the flares in different levels. In general, the flares with ${\rm{\Delta }}F/\tilde{F}\gt 1$ are called the hyperflares (Chang et al. 2018) with the corresponding flare peak flux reaching more than 100% stellar luminosity. We found 19 hyper-flare events in 15 M dwarfs (Table 1).

Table 1.  19 Hyperflares on 15 M Dwarfs

EPIC	${\rm{\Delta }}F/\tilde{F}$	Eflare	Duration	Tpositiona	Teff	Mass
		(Log(erg))	(hr)	(JD-2454833)	(K)	(M⊙)
211441334	12.11	34.86	2.0	2334.23	3384	0.147
210715010	5.33	32.92	2.0	2273.14	2376	0.082
211441334	3.64	34.86	2.5	2334.26	3384	0.147
206181579	2.74	33.68	2.0	2153.05	2991	0.121
210843552	2.64	35.08	2.0	2259.02	3711	0.378
210626041	2.26	35.11	2.9	2268.34	3724	0.376
211441334	2.12	34.10	2.0	2319.28	3384	0.147
211796220	1.88	33.64	2.0	2316.15	3037	0.121
201396752	1.87	33.50	2.0	2019.88	2981	0.120
210658027	1.60	34.55	2.5	2237.03	3572	0.284
211577353	1.47	34.19	2.5	2323.00	3458	0.200
211107439	1.44	34.91	4.4	2297.47	3611	0.289
211016654	1.37	34.85	2.9	2232.89	3622	0.270
210450012	1.37	34.06	2.0	2241.14	3375	0.196
201396752	1.32	33.32	2.0	1989.40	2981	0.120
210395925	1.27	34.26	2.0	2268.03	3512	0.214
210454684	1.24	34.56	2.9	2233.87	3597	0.292
211107439	1.15	34.88	3.9	2297.49	3611	0.289
211091536	1.10	34.18	2.0	2271.14	3446	0.196

Note.

^aThe flare time position in units of Julian day-2454833 day in the light curve.

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A flare with ${\rm{\Delta }}F/\tilde{F}\approx 12$ was detected on the M dwarf EPIC 211441334 (Figure 13). This flare released an energy of about 7 × 10³⁴ erg. EPIC 249394096 produced the maximum ${\rm{\Delta }}F/\tilde{F}$ flare of K-type stars with a peak flare amplitude of about 0.78 (78%) and an energy ≈2 × 10³⁵ erg (see Figure 14). From the work of Shibayama et al. (2013), the maximum ${\rm{\Delta }}F/\tilde{F}$ of G-type stars can be found to be about 0.32 (32%) with an energy ≈3 × 10³⁵ erg on KIC 12354328. No hyper-flare event has been discovered in K- or G- type stars. The reason is that the ${\rm{\Delta }}F/\tilde{F}$ value is based on the luminosity baseline of the stars. Because the K- and G- type stars are brighter than the M dwarfs, the ${\rm{\Delta }}F/\tilde{F}$ of the flares with the same energy on stars of K- and G-type would be smaller than those of M-type stars. Figure 15 provides a summary of the correlations between the ${\rm{\Delta }}F/\tilde{F}$ value and the stellar temperature or the mass. We found the power-law expressions: ${\rm{\Delta }}F/\tilde{F}\propto {T}_{\mathrm{eff}}^{-3.49}$ and ${\rm{\Delta }}F/\tilde{F}\,\propto {M}^{-1.18}$. Namely, a negative correlation. However, the correlations with the flare energy and the stellar parameters are positive, and the power-law expressions are ${E}_{\mathrm{flare}}\propto {T}_{\mathrm{eff}}^{7.63}$ and E_flare ∝ M^2.43. Note that Figures 8 and 9 also show that the later type dwarfs tend to have higher ${\rm{\Delta }}F/\tilde{F}$ but smaller flare energies, and the earlier type dwarfs tend to have the lower ${\rm{\Delta }}F/\tilde{F}$ but larger flare energies. Some of the most powerful flares in M dwarfs can reach the same order as the powerful flares in G-type dwarfs, which are hotter than the M dwarfs.

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The rotation period of a star is known to be related to the stellar age, and younger stars show more rapid rotation according to the gyrochronology relation (Skumanish 1972; Barnes 2003, 2007, 2010; Reinhold & Gizon 2015). Angus et al. (2015) correlated the rotation period measurements of 310 Kepler stars with asteroseismic ages, six field stars including the Sun and stars in the Hyades and Coma clusters with precise age determination to calibrate the gyrochronology relation. These authors confirmed the general trend for a power-law approximation with rotation period, P, increasing from about 7 days at t(age) ∼0.5 Gyr to about 20–40 days at t ∼ 10 Gyr (see Figure 6 in their paper), while cautioning the presence of deviations from the precise relation and the need for further investigations.

Since the discovery of the superflares by Maehara et al. (2012), the relationship between the superflare frequencies/magnetic activities and the rotation periods and hence the stellar age, by implication, has been investigated by Wu et al. (2015), Karoff et al. (2016), Chang et al. (2017, 2018), Maehara et al. (2017), Notsu et al. (2017), and He et al. (2018). Briefly speaking, from a statistical study of the chromospheric emissions of 5648 solar-type stars including 48 superflare stars culled from the Kepler data, Karoff et al. (2016) showed that the superflare stars have a larger S-index, which is a measure of the Ca ii and K line emission level. Wu et al. (2015) and Chang et al. (2017, 2018) examined the link between stellar rotation periods and superflare frequencies of different types of low-mass stars and showed that P = 10 days is likely to be a critical value dividing active and inactive stars. Maehara et al. (2017) compared the size frequency distributions of the star spots with the flare frequencies of the solar-type stars in detail. They found that generally not just the flare occurrence rates but also the spot sizes tend to decrease if the stellar rotation periods increase. Note that He et al. (2018) found that there exists no clear correlation between the timing of the flare occurrence and the locations of the magnetically active regions identified by the star spots according to their detailed analysis of the light curves of three solar-type stars (KIC 6034120, KIC 311833, and KIC 10528093). This result is in general agreement with the work of Hou (2018). Using high-resolution spectroscopic measurements with Subaru, Notsu et al. (2017) compared the superflare frequencies of some solar-type stars with their corresponding X-ray fluxes and the Li abundance, which can be used as a proxy of the stellar ages. They found that young stars tend to be fast-rotating and magnetically active even though some old and slowly rotating stars as old as the Sun can also produce superflares.

Following these recent studies, it is possible to explore the dependence of flare activity with the stellar age making use of the gyrochronology relation mentioned earlier. With the gyrochronology formulation published by Barnes (2007), we found old flaring stars (see Table 2), and three flaring solar-type stars (KIC 6633602, KIC 8212826, and KIC 11401109) are older than the Sun. The oldest one, KIC 11401109, is up to 6 Gyr old. However, the oldest M- (∼2.8 Gyr) and K-type (∼2.9 Gyr) flaring stars with P > 40 days are younger than the Sun.

We also found ${\rm{\Delta }}F/\tilde{F}$ and E_flare increased with decreasing rotation period (see Figure 16). An obvious transition region appears at P ≈ 10 days in the M dwarfs' flare-period correlation. The level of flare activity drops sharply for those stars with rotation periods longer than 10 days. The undetectable transition region in the K and G dwarfs' flares may be due to observational bias. A possible reason is that higher stellar luminosities of K and G dwarfs will impede the identification of flares with small ${\rm{\Delta }}F/\tilde{F}$ values in the K2 light curves, which are flux-limited. On the other hand, it is clear that fast-rotating stars exhibit more flare activities than stars with longer rotation periods.

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Figure 17 compares the flare energy variations with flare durations in M-, K-, and G-type dwarfs. Since we used the LC data whose time resolution is about 29.4 minutes and the flare detection algorithm is designed to exclude the flare shape signal without three consecutive data points, the minimum flare duration of the flares in this study is 1.5 hr. For the 3589 flares detected in M dwarfs, the flare energy increases with increasing flare duration, and there is a flare energy upper limit that is about 10³⁵ erg. This trend can also be found in K dwarfs. That is, 1647 flares have been detected in K dwarfs with an energy upper limit ≈4 × 10³⁵ erg. Wu et al. (2015) showed that the upper limit of G dwarf flare energy is about 2 × 10³⁷ erg, but they did not consider the Kepler response function for the flare energy calculation. The flare that released an energy of about 2 × 10³⁷ erg was from the star KIC 11551430 in Wu et al. (2015). With an effective temperature 5541 K, we recomputed the flare energy by considering the Kepler response function and obtained a value of 6 × 10³⁶ erg instead. Shibayama et al. (2013) detected 1547 superflares on 279 G-type dwarfs. The energy upper limit was found to be about 10³⁶ erg. As mentioned before, a superflare that exploded on the young M-dwarf binary system DG CVn released an energy of 4 ×10³⁵–9 × 10³⁵ erg larger than the M dwarfs' flare energy upper limit (∼10³⁵ erg) we found (Osten et al. 2016). This means that the magnetic energy storage mechanism of the binary system may be capable of yielding more powerful flares than single stars.

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The stars with the top 10 highest flare occurrence percentages are shown in Table 3. Figure 18 shows the correlation of the flare occurrence percentages with the stellar masses and rotation periods. The stars are divided into several groups according to the rotation period. With similar rotation periods, the lower mass stars tend to be more active than the more massive stars. The flare occurrence percentage of stars is also significantly affected by the rotation period. For the four M dwarfs with rotation periods >40 days, their flare occurrence percentages are all below 0.5%, which is lower than some more massive stars with faster rotation periods. A transition region appears near 0.6–0.7 M_⊙, forming a "hockey stick" pattern. The highest flare occurrence percentages in each rotation group are 3.22% (P ≤ 10 days); 1.84% (10 days < P ≤ 20 days); 1.98% (20 days < P ≤ 30 days); 0.95% (30 days < P ≤ 40 days); and 0.78% (P > 40 days).

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Table 3.  The Stars with the 10 Highest Flare Percentages

EPIC	Flare Percentage (%)	Mass (M⊙)	Teff (K)	Prot (day)	Max ${\rm{\Delta }}F/\tilde{F}$	Max Eflare[Log(erg)]
210564347	3.23	0.223	3496	5.121	0.065	33.19
206262336	2.72	0.222	3495	9.741	0.185	33.59
210909746	2.45	0.373	3748	4.039	0.439	34.50
206024567	2.39	0.278	3585	6.052	0.234	33.75
210434976	2.33	0.509	3954	2.553	0.215	34.48
211120664	2.32	0.320	3683	1.497	0.199	33.92
211006108	2.30	0.268	3582	0.959	0.470	34.34
201738857	2.16	0.206	3463	1.378	0.092	33.17
211724534	2.16	0.282	3604	2.454	0.141	33.72
206050032	2.07	0.090	2617	1.244	0.095	31.68

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In a previous study, the solar and stellar flare activities have been interpreted in terms of a magnetic reconnection-driven, self-organized critical system (Boffetta et al. 1999). The corresponding solar/stellar flare frequency distribution FFD can be expressed as the power law:

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{{dE}}\propto {E}^{-\gamma }.\end{eqnarray} \tag{ 10 }$$

We estimate the cumulative FFDs and derive the power-law index by using the maximum likelihood method (Clauset et al. 2009; Klaus et al. 2011) with a Python package "powerlaw" which is a toolbox for analysis of heavy-tailed distributions (Alstott et al. 2014). Figure 19 shows the cumulative FFDs of M, K, and G dwarfs. The power-law indices of M and K dwarfs are γ = 1.82 ± 0.02, γ = 1.86 ± 0.02, respectively, and these are quite close to each other. Shibayama et al. (2013) showed that the power-law index of G-type stars is γ ≈ 2.2 by binning the data set while our result is γ = 2.01 ± 0.03. The flare occurrence rate of M dwarfs is about once in 0.6 yr for E_flare > 1 × 10³³ erg and once in 6.5 yr and 350 yr for superflares with E_flare of 10³⁴ erg and 10³⁵ erg, respectively. In comparison, the K dwarfs' flare occurrence rate is about once in 4.7 yr with E_flare > 1 × 10³⁴ erg, namely, more frequent than the M dwarfs'. The G dwarfs' flare occurrence rate of the flares with E_flare > 1 × 10³⁵ erg is also higher than the M dwarfs', it is about once in 320 yr. For the solar-like stars (5600 K ≤ T_eff < 6000 K and P_rot > 10 days and log(g) > 4), the flare occurrence rates are about once in 800 yr with E_flare > 1 × 10³⁴ erg and once in 5000 yr with E_flare > 1 ×10³⁵ erg (Shibayama et al. 2013). If we only consider the flares from stars with rotation periods shorter than 10 days, the power-law indices of M- and K-type stars (1.78 versus 1.82) are nearly the same (see Figure 20), it may indicate they have similar flare generation mechanisms.

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