Stellar Superflares Observed Simultaneously with Kepler and XMM-Newton

Simultaneous Kepler and XMM-Newton observations of superflares on the young Pleiades stars during the Kepler/K2 mission have been studied earlier by Guarcello et al. (2019a, 2019b); in contrast, we consider here the primary Kepler observational campaign (2009 May–2013 May). Simultaneous Kepler and XMM-Newton observations during the primary Kepler campaign have been studied before by Pizzocaro et al. (2019); however, Pizzocaro et al. (2019) focused mainly on the general stellar activity indicators (average X-ray luminosity, flare occurrence rate, etc.) and did not analyze the individual flares in detail.

The Kepler data archive ³ provides nearly continuous light curves for hundreds of thousands of stars within its field of view, for the abovementioned time range. As the starting point for the X-ray data, we used the 3XMM-DR5 serendipitous source catalog (Rosen et al. 2016), which contains hundreds of thousands of sources detected by XMM-Newton; light curves are provided for the brightest sources (including the signals from the selected source regions and from nearby background regions, to filter out the background fluctuations). To find the flares that occurred simultaneously in both spectral ranges, we (a) selected the XMM-Newton (3XMM-DR5) detections within the Kepler primary campaign field of view and time range; (b) matched the Kepler Input Catalog (KIC; Brown et al. 2011) and the 3XMM-DR5 catalog to select the objects with the mutual positional differences of ≲5'' (this value is determined by the 3XMM-DR5 positional error, cf. Rosen et al. (2016), while the KIC positional error is negligible), accounting for the proper motion; (c) selected the XMM-Newton (3XMM-DR5) detections with $\gt {10}^{3}$ source counts in any EPIC detector ($0.2\mbox{--}12\,\,\mathrm{keV}$ range), for which reliable X-ray light curves with a sufficiently high time resolution are available (see Rosen et al. 2016); and (d) inspected the Kepler and XMM-Newton light curves of the selected objects visually, to search for simultaneous flares. As a result, we have identified three stars that exhibited well-defined correlated peaks in the optical and X-ray light curves ⁴ ; parameters of these stars are summarized in Table 1.

Table 1. Parameters of the Selected Stars: Spectral Types (SpT), Effective Temperatures (${T}_{\mathrm{eff}}$), Luminosities (L), Distances (d), Metallicities ([Fe/H]), Rotation Periods (${P}_{\mathrm{rot}}$), Optical Extinctions (A_V ), and Magnitudes in the Johnson (B and V), 2MASS (K_S ), and Gaia (G, ${G}_{\mathrm{BP}}$ and ${G}_{\mathrm{RP}}$) Bands

Star	KIC 8093473	KIC 8454353	KIC 9048551
SpT a	M3	M2	K7
${T}_{\mathrm{eff}}$, K b	${3357}_{3344}^{3528}$	${3541}_{3514}^{3923}$	${4124}_{4025}^{4138}$
L, L☉ b	${0.111}_{0.103}^{0.120}$	${0.051}_{0.050}^{0.052}$	${0.084}_{0.084}^{0.085}$
d, pc b	${205.9}_{198.4}^{213.9}$	${168.5}_{167.5}^{169.4}$	${125.9}_{125.6}^{126.2}$
[Fe/H], dex	$+{0.04}_{-0.06}^{+0.14}$ c	$-{0.16}_{-0.27}^{-0.05}$ c	$-{0.04}_{-0.05}^{-0.03}$ d
${P}_{\mathrm{rot}}$, days e	6.043	1.496	8.553
AV , mag f	0.171	0.109	0.080
B, mag g	17.273	17.297	15.345
V, mag g	15.883	16.005	14.085
KS , mag h	${11.160}_{11.149}^{11.171}$	${11.551}_{11.531}^{11.571}$	${10.466}_{10.448}^{10.484}$
G, mag b	${14.696}_{14.694}^{14.698}$	${14.941}_{14.940}^{14.943}$	${13.336}_{13.335}^{13.337}$
${G}_{\mathrm{BP}}$, mag b	${15.989}_{15.980}^{15.999}$	${16.094}_{16.086}^{16.101}$	${14.187}_{14.181}^{14.192}$
${G}_{\mathrm{RP}}$, mag b	${13.561}_{13.556}^{13.566}$	${13.863}_{13.859}^{13.867}$	${12.439}_{12.434}^{12.444}$
${M}_{\mathrm{star}}$, M☉ i	0.57	0.44	⋯
${R}_{\mathrm{star}}$, R☉ i	0.55	0.42	⋯
a, au i	0.068	0.024	⋯

Notes. We present also the estimated component masses (${M}_{\mathrm{star}}$) and radii (${R}_{\mathrm{star}}$) as well as orbital separations (a) for the supposed binaries (see Section 4.3).

^a Spectral types were estimated from the effective temperatures according to Pecaut & Mamajek (2013). ^b Gaia Collaboration (2018). ^c Gaidos et al. (2016). ^d Jönsson et al. (2020). ^e McQuillan et al. (2014). ^f Brown et al. (2011). ^g Pizzocaro et al. (2019). ^h Cutri et al. (2003). ⁱ This work.

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To check further the stellar parameters for our sample, we have analyzed the available catalogs and photometry data; we have compared the observations with theoretical PARSEC isochrones (Marigo et al. 2017) and gyrochronology relations by Barnes (2007), Mamajek & Hillenbrand (2008), and Angus et al. (2019). In particular, Figure 1 demonstrates locations of the selected stars on the Hertzsprung–Russell diagram; a more detailed analysis is presented in Appendix A. The conclusions can be summarized as follows:

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KIC 8093473 is located well above the main sequence, but is too faint to be a giant or subgiant. Most likely, it is an unresolved binary or multiple system consisting of several (from two to four) M dwarfs; the estimation of the number of components depends on the assumed age of the system (which is actually unknown). Determining the exact configuration of this system will require further observations.

KIC 8454353 is located slightly above the main sequence. Most likely, it is an unresolved binary consisting of two more-or-less similar M dwarfs, with an age of ≳100 Myr.

KIC 9048551 is located on the main sequence and can be identified as a single K dwarf, with an estimated age of ∼120–280 Myr.

Since about 30%–40% of K and M dwarfs in the solar neighborhood have been found to form binary or multiple systems (e.g., Raghavan et al. 2010; Winters et al. 2019), it is not surprising that two of the three stars in our sample belong to this category, too. We note that, for an unresolved binary or multiple system, the parameters ${T}_{\mathrm{eff}}$ and L in the Gaia catalog (see Table 1) represent, respectively, the average temperature and total luminosity of the system (see Andrae et al. 2018). We have analyzed long-term Kepler light curves of KIC 8093473 and KIC 8454353, and found no significant secondary rotation periods in both cases, which implies that either the components of these systems are tidally locked (for KIC 8093473, this favors a binary), or the secondary components are inactive. In Section 4.3, we demonstrate that an orbital interaction in the KIC 8093473 and KIC 8454353 systems is unlikely to affect the observed flares. All stars in our sample are relatively cool (K and M dwarfs) and rapidly rotating; i.e., they belong to the category of stars where superflares occur most frequently (e.g., Davenport 2016; Van Doorsselaere et al. 2017; Brasseur et al. 2019).

Figures 2–4 demonstrate the light curves of the selected stars versus time in Modified Julian Day (MJD). For the X-rays, we consider here and below the data from the XMM-Newton EPIC PN detector (Strüder et al. 2001) only, because it is the most sensitive one; the instrumental background is subtracted. The Kepler light curves have a cadence of about 29.5 minutes; the bin sizes of the XMM-Newton light curves vary from 270 to 1290 s for different objects, depending on the X-ray flux (Rosen et al. 2016). The gradual trends in the "raw" optical light curves (panels (a)) are caused by the rotational modulation (influence of starspots). The main features of the light curves are summarized below.

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KIC 8093473 (Figure 2): there are two prominent white-light flares at around MJD 55164.93 and 55165.37, which are accompanied by weak but noticeable X-ray counterparts. The brightest X-ray flare occurs at around MJD 55165.13, with several weaker quasi-periodic peaks occurring in its decay phase; both the first X-ray flaring peak and the subsequent weaker peaks are accompanied by nearly simultaneous white-light brightenings. Such a behavior is typical of large solar and stellar flares, with the multiple peaks either caused by the modulation of the flaring emissions by magnetohydrodynamic oscillations, or representing separate acts of magnetic reconnection (subflares) in the same active region (see, e.g., the recent review by Kupriyanova et al. 2020). Notably, the flares at MJD 55164.93 and 55165.37 are more pronounced in the optical range than in the X-rays, while the flare at MJD 55165.13 is more pronounced in the X-rays than in the optical range; we discuss this peculiarity in Section 4.1.

KIC 8454353 (Figure 3): there is a prominent sharp flare at around MJD 55829.48, occurring simultaneously in the optical and X-ray ranges. Another weaker but slightly longer X-ray flare occurs at around MJD 55829.75; it is accompanied by a very faint but noticeable white-light counterpart.

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KIC 9048551 (Figure 4): there are two prominent partially overlapping flares at around MJD 55735.54 and 55735.65, occurring nearly simultaneously in the optical and X-ray ranges (the X-ray flares are slightly shorter); several weaker peaks are visible at later times in both spectral ranges. The shaded region in the X-ray plots represents the time interval when significant background X-ray fluctuations (with the flux comparable to that from the target star) occurred, which makes the data less reliable; for this reason, we do not analyze here the flares that occurred during the mentioned background event.

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The observational light curve data behind Figures 2–4 are available as the supplementary material in the online version of the journal.

There was no noticeable correlation between the flares and the stellar rotational phase: the flares occurred both around the local minimum (for KIC 8454353) and near the local maxima (for KIC 8093473 and KIC 9048551) of the long-term Kepler light curves. Although the number of flares in our sample is rather small, the absence of a preferable rotational phase is consistent with the results of the large-scale statistical study by Doyle et al. (2018, 2019, 2020). Most likely, similarly to the Sun, the considered stars possess multiple active regions (and hence multiple starspots), and flares do not necessarily occur in the largest of them.

To analyze the identified stellar flares quantitatively, we best-fitted the observed light curves with model ones. The model that we used represents a flaring component superimposed on a variable quiescent background: $I(t)={I}_{\mathrm{back}}(t)+{I}_{\mathrm{flare}}(t)$. Each flare (in either the optical or X-ray range) was modeled as an asymmetric peak with a Gaussian rise phase and an exponential decay phase; i.e., the contribution of the flares had the form:

$$\begin{eqnarray}\begin{array}{rcl}{I}_{\mathrm{flare}}(t) & = & \sum _{i=1}^{N}{I}_{\mathrm{flare}}^{(i)}(t)\\ & = & \sum _{i=1}^{N}{g}_{i}\left\{\begin{array}{ll}\exp \left[-\displaystyle \frac{{\left(t-{t}_{i}^{0}\right)}^{2}}{{\left({\tau }_{i}^{\mathrm{rise}}\right)}^{2}}\right], & t\lt {t}_{i}^{0},\\ \exp \left(-\displaystyle \frac{t-{t}_{i}^{0}}{{\tau }_{i}^{\mathrm{decay}}}\right), & t\geqslant {t}_{i}^{0},\end{array}\right.\end{array}\end{eqnarray} \tag{ 1 }$$

where N is the number of flares revealed for each star. For the optical observations, the number N was determined by the number of local maxima in the original light curves. To account for different time resolutions and select the events that revealed themselves in both spectral ranges, searching for flares in the X-ray light curves included an additional step: we first smoothed a light curve by filtering out the high-frequency component (with timescales shorter than 30 minutes, which corresponds to the Kepler cadence), and then all local maxima exceeding a 1σ threshold in the residual signal were considered as potential flare candidates and their number determined the number N in Equation (1). This model implies that all significant peaks in a light curve (e.g., both the brightest X-ray flare on KIC 8093473 and the weaker peaks occurring in its decay phase, see Figure 2(c)–(d)) are considered as separate flares; this approach is consistent with the concept of "complex flares" used by Davenport et al. (2014), and may also be attributed to a "build-up and release" scenario for flares in the solar and stellar coronae (Hudson 2020).

In the optical range, the background variations are caused mainly by the stellar rotation. Therefore, for KIC 8093473 and KIC 9048551, we modeled the quiescent optical background with the following function:

$$\begin{eqnarray}\begin{array}{rcl}{I}_{\mathrm{back}}^{\mathrm{WL},\ 1}(t) & = & {A}_{1}\sin \left(\displaystyle \frac{2\pi t}{{P}_{\mathrm{rot}}}+{\varphi }_{1}\right)\\ & & +{A}_{2}\sin \left(\displaystyle \frac{4\pi t}{{P}_{\mathrm{rot}}}+{\varphi }_{2}\right)+C,\end{array}\end{eqnarray} \tag{ 2 }$$

where ${P}_{\mathrm{rot}}$ is the rotation period determined from earlier studies (see Table 1). Equation (2) represents a truncated Fourier expansion, including the first and second harmonics of the rotation frequency. For KIC 8454353, better agreement with the observations was achieved using a different background model function:

$$\begin{eqnarray}&&{I}_{\mathrm{back}}^{\mathrm{WL},\ 2}(t)=({A}_{0}{t}^{{A}_{1}}+{A}_{2})\sin \left(\displaystyle \frac{2\pi t}{{P}_{\mathrm{rot}}}+\varphi \right)+C.\end{eqnarray} \tag{ 3 }$$

In this case, the gradual variation (increase) of the modulation depth with time could be attributed to the evolution of the stellar active regions (starspots). The Kepler light curves were truncated to the time windows slightly extending those of the XMM-Newton observations, as shown in Figures 1–3. For the X-ray emission, we adopted a simple linear model of the quiescent background: ${I}_{\mathrm{back}}^{\mathrm{XMM}}(t)={At}+B$.

The model light curves were fitted to the observations using the Bayesian inference and MCMC sampling (see, e.g., Gregory 2010, and references therein). In this work, we used the MCMC sampling implementation by Pascoe et al. (2017) and Anfinogentov et al. (2021). All the model parameters, except the rotation period ${P}_{\mathrm{rot}}$ and the number of flares N (that was determined by the number of local maxima prior to the fitting procedure, as described above), were considered as free parameters. As an initial guess for the flare parameters in Equation (1), we used positions of the local maxima (for t_i ⁰), the corresponding flare fluxes (for g_i ), and the time intervals between the neighboring apparent flare peaks (for ${\tau }_{i}^{\mathrm{rise}}$ and ${\tau }_{i}^{\mathrm{decay}}$). Then, for the X-ray emission, the MCMC fitting procedure used the original (nonsmoothed) light curves.

To cope with the original long-cadence observations and sample the flare shapes properly, we used the method of supersampling, i.e., the model light curves were initially calculated on a fine time grid with 10 s cadence. After that, the model fine-resolution light curves were binned over the instrument exposure intervals, and then compared with the observations to evaluate the residuals and the likelihood function through the MCMC sampling algorithm.

The MCMC fitting provided the most probable values of the model parameters, as well as robust estimations of their confidence intervals. Parameters of the flares that occurred simultaneously in the optical and X-ray ranges are presented in Table 2; all significant peaks in the light curves revealed by the above-described analysis are listed in Table 3 in Appendix B. In Figures 2–4, panels (a) and (c), the best-fitted model light curves and the quiescent background components are overplotted on the observed data, while panels (b) and (d) demonstrate the flaring (background-subtracted) components.

Table 2. Parameters of the Flares That Occurred Simultaneously in the White-light (WL) and X-Ray (X) Ranges: Peak Times (t₀), Rise Times (${\tau }_{\mathrm{rise}}$), Decay Times (${\tau }_{\mathrm{decay}}$), Peak Luminosities (L_max), Emitted Energies (${E}_{\mathrm{flare}}$), and Peak Equivalent GOES X-Ray Fluxes (${I}_{\max }^{\mathrm{GOES}}$)

No.	${t}_{0}^{\mathrm{WL}}$,	${\tau }_{\mathrm{rise}}^{\mathrm{WL}}$,	${\tau }_{\mathrm{decay}}^{\mathrm{WL}}$,	${L}_{\max }^{\mathrm{WL}}$,	${E}_{\mathrm{flare}}^{\mathrm{WL}}$,	${t}_{0}^{{\rm{X}}}$,	${\tau }_{\mathrm{rise}}^{{\rm{X}}}$,	${\tau }_{\mathrm{decay}}^{{\rm{X}}}$,	${L}_{\max }^{{\rm{X}}}$,	${I}_{\max }^{\mathrm{GOES}}$,	${E}_{\mathrm{flare}}^{{\rm{X}}}$,
	(days)	(minutes)	(minutes)	(1028 erg ${{\rm{s}}}^{-1}$)	(1032 erg)	(days)	(minutes)	(minutes)	(1028 erg ${{\rm{s}}}^{-1}$)	(10−2 W ${{\rm{m}}}^{-2}$)	(1032 erg)
KIC 8093473
1	${0.926}_{0.917}^{0.935}$	${29.6}_{11.2}^{45.0}$	${73.3}_{46.1}^{104.0}$	${83.2}_{56.8}^{202.5}$	${46.0}_{33.4}^{111.1}$	${0.940}_{0.940}^{0.942}$	${16.7}_{9.1}^{37.6}$	${19.3}_{18.4}^{29.7}$	${63.2}_{21.7}^{89.5}$	${29.5}_{10.3}^{41.3}$	${10.6}_{6.5}^{17.7}$
2	${1.117}_{1.107}^{1.134}$	${21.9}_{7.3}^{57.2}$	${62.7}_{16.6}^{65.6}$	${36.6}_{15.0}^{97.0}$	${15.1}_{8.6}^{38.4}$	${1.127}_{1.127}^{1.129}$	${17.8}_{16.9}^{21.1}$	${36.8}_{31.0}^{39.7}$	${434.0}_{357.8}^{523.7}$	${147.1}_{127.1}^{170.7}$	${135.1}_{114.8}^{157.8}$
3	${1.182}_{1.166}^{1.199}$	${32.4}_{13.2}^{100.7}$	${50.2}_{13.9}^{108.5}$	${21.5}_{7.9}^{44.1}$	${12.9}_{6.8}^{34.7}$	${1.185}_{1.183}^{1.186}$	${9.0}_{6.9}^{23.1}$	${14.6}_{14.4}^{23.3}$	${45.4}_{37.8}^{93.3}$	${15.4}_{13.4}^{30.4}$	${8.9}_{6.7}^{13.3}$
4	${1.276}_{1.265}^{1.293}$	${44.2}_{19.8}^{96.7}$	${40.2}_{10.1}^{79.9}$	${41.8}_{22.0}^{92.7}$	${16.6}_{11.3}^{46.5}$	${1.278}_{1.277}^{1.279}$	${29.3}_{9.4}^{37.6}$	${18.5}_{16.4}^{26.6}$	${19.0}_{10.3}^{36.3}$	${6.4}_{3.6}^{11.8}$	${4.7}_{3.2}^{8.4}$
5	${1.370}_{1.364}^{1.379}$	${20.4}_{10.9}^{38.3}$	${91.4}_{68.9}^{108.7}$	${95.0}_{66.0}^{217.8}$	${61.6}_{45.1}^{138.1}$	${1.388}_{1.386}^{1.389}$	${16.0}_{11.8}^{31.7}$	${25.5}_{24.6}^{39.8}$	${68.2}_{25.4}^{92.5}$	${27.3}_{10.4}^{36.3}$	${15.1}_{9.5}^{21.9}$
KIC 8454353
1	${0.479}_{0.477}^{0.482}$	${16.9}_{16.7}^{22.5}$	${33.2}_{23.6}^{41.4}$	${61.2}_{46.0}^{146.8}$	${17.6}_{13.9}^{40.4}$	${0.484}_{0.477}^{0.498}$	${14.2}_{12.4}^{41.5}$	${25.9}_{26.3}^{70.5}$	${43.1}_{18.5}^{56.3}$	${16.2}_{5.7}^{23.5}$	${10.4}_{6.6}^{16.3}$
2	${0.747}_{0.736}^{0.750}$	${77.0}_{22.8}^{98.9}$	${63.0}_{15.0}^{72.6}$	${8.6}_{4.8}^{32.6}$	${6.2}_{3.7}^{18.1}$	${0.728}_{0.661}^{0.742}$	${47.7}_{13.2}^{64.7}$	${26.6}_{26.9}^{158.6}$	${9.5}_{8.3}^{18.0}$	${1.1}_{0.8}^{2.3}$	${5.3}_{3.9}^{9.3}$
KIC 9048551
1	${0.542}_{0.541}^{0.547}$	${10.7}_{10.6}^{20.1}$	${135.2}_{74.9}^{139.2}$	${25.6}_{20.0}^{52.1}$	${18.2}_{14.0}^{36.5}$	${0.548}_{0.543}^{0.559}$	${12.3}_{5.3}^{27.3}$	${50.2}_{29.1}^{66.6}$	${13.0}_{8.5}^{15.9}$	${1.1}_{0.6}^{1.4}$	${4.5}_{3.6}^{5.4}$
2	${0.645}_{0.643}^{0.655}$	${13.6}_{13.3}^{33.8}$	${138.1}_{85.7}^{164.9}$	${19.2}_{15.2}^{41.9}$	${17.3}_{13.1}^{35.4}$	${0.657}_{0.640}^{0.665}$	${42.6}_{14.0}^{56.2}$	${36.7}_{16.2}^{118.9}$	${8.1}_{5.8}^{11.8}$	${0.7}_{0.4}^{1.0}$	${4.0}_{3.0}^{5.9}$

Note. The flare peak times ${t}_{0}^{\mathrm{WL}}$ and ${t}_{0}^{{\rm{X}}}$ for KIC 8093473, KIC 8454353, and KIC 9048551 are relative to MJD 55164, MJD 55829, and MJD 55735, respectively.

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Table 3. Parameters of All Flares (with the Amplitude above 1σ Level and in the "Good" Time Intervals) Detected in the White-light (WL) and/or X-Ray (X) Ranges: Peak Times (t₀), Rise Times (${\tau }_{\mathrm{rise}}$), Decay Times (${\tau }_{\mathrm{decay}}$), Peak Luminosities (L_max), Emitted Energies (${E}_{\mathrm{flare}}$), and Peak Equivalent GOES X-Ray Fluxes (${I}_{\max }^{\mathrm{GOES}}$)

No.	${t}_{0}^{\mathrm{WL}}$,	${\tau }_{\mathrm{rise}}^{\mathrm{WL}}$,	${\tau }_{\mathrm{decay}}^{\mathrm{WL}}$,	${L}_{\max }^{\mathrm{WL}}$,	${E}_{\mathrm{flare}}^{\mathrm{WL}}$,	${t}_{0}^{{\rm{X}}}$,	${\tau }_{\mathrm{rise}}^{{\rm{X}}}$,	${\tau }_{\mathrm{decay}}^{{\rm{X}}}$,	${L}_{\max }^{{\rm{X}}}$,	${I}_{\max }^{\mathrm{GOES}}$,	${E}_{\mathrm{flare}}^{{\rm{X}}}$,
	(days)	(minutes)	(minutes)	(1028 erg ${{\rm{s}}}^{-1}$)	(1032 erg)	(days)	(minutes)	(minutes)	(1028 erg ${{\rm{s}}}^{-1}$)	(10−2 W ${{\rm{m}}}^{-2}$)	(1032 erg)
KIC 8093473
1	${0.926}_{0.917}^{0.935}$	${29.6}_{11.2}^{45.0}$	${73.3}_{46.1}^{104.0}$	${83.2}_{56.8}^{202.5}$	${46.0}_{33.4}^{111.1}$	${0.940}_{0.940}^{0.942}$	${16.7}_{9.1}^{37.6}$	${19.3}_{18.4}^{29.7}$	${63.2}_{21.7}^{89.5}$	${29.5}_{10.3}^{41.3}$	${10.6}_{6.5}^{17.7}$
⋯	${1.035}_{1.017}^{1.048}$	${47.3}_{12.7}^{78.6}$	${66.7}_{12.2}^{86.7}$	${25.7}_{10.6}^{54.0}$	${12.2}_{6.8}^{33.0}$	⋯	⋯	⋯	⋯	⋯	⋯
2	${1.117}_{1.107}^{1.134}$	${21.9}_{7.3}^{57.2}$	${62.7}_{16.6}^{65.6}$	${36.6}_{15.0}^{97.0}$	${15.1}_{8.6}^{38.4}$	${1.127}_{1.127}^{1.129}$	${17.8}_{16.9}^{21.1}$	${36.8}_{31.0}^{39.7}$	${434.0}_{357.8}^{523.7}$	${147.1}_{127.1}^{170.7}$	${135.1}_{114.8}^{157.8}$
3	${1.182}_{1.166}^{1.199}$	${32.4}_{13.2}^{100.7}$	${50.2}_{13.9}^{108.5}$	${21.5}_{7.9}^{44.1}$	${12.9}_{6.8}^{34.7}$	${1.185}_{1.183}^{1.186}$	${9.0}_{6.9}^{23.1}$	${14.6}_{14.4}^{23.3}$	${45.4}_{37.8}^{93.3}$	${15.4}_{13.4}^{30.4}$	${8.9}_{6.7}^{13.3}$
⋯	⋯	⋯	⋯	⋯	⋯	${1.219}_{1.218}^{1.220}$	${24.7}_{12.9}^{52.2}$	${38.6}_{26.8}^{43.3}$	${42.2}_{20.1}^{66.0}$	${14.3}_{7.1}^{21.5}$	${12.0}_{8.3}^{18.2}$
4	${1.276}_{1.265}^{1.293}$	${44.2}_{19.8}^{96.7}$	${40.2}_{10.1}^{79.9}$	${41.8}_{22.0}^{92.7}$	${16.6}_{11.3}^{46.5}$	${1.278}_{1.277}^{1.279}$	${29.3}_{9.4}^{37.6}$	${18.5}_{16.4}^{26.6}$	${19.0}_{10.3}^{36.3}$	${6.4}_{3.6}^{11.8}$	${4.7}_{3.2}^{8.4}$
5	${1.370}_{1.364}^{1.379}$	${20.4}_{10.9}^{38.3}$	${91.4}_{68.9}^{108.7}$	${95.0}_{66.0}^{217.8}$	${61.6}_{45.1}^{138.1}$	${1.388}_{1.386}^{1.389}$	${16.0}_{11.8}^{31.7}$	${25.5}_{24.6}^{39.8}$	${68.2}_{25.4}^{92.5}$	${27.3}_{10.4}^{36.3}$	${15.1}_{9.5}^{21.9}$
⋯	${1.491}_{1.468}^{1.512}$	${19.0}_{11.7}^{98.3}$	${62.8}_{12.5}^{107.8}$	${13.5}_{1.4}^{31.4}$	${4.4}_{2.1}^{17.6}$	⋯	⋯	⋯	⋯	⋯	⋯
KIC 8454353
1	${0.479}_{0.477}^{0.482}$	${16.9}_{16.7}^{22.5}$	${33.2}_{23.6}^{41.4}$	${61.2}_{46.0}^{146.8}$	${17.6}_{13.9}^{40.4}$	${0.484}_{0.477}^{0.498}$	${14.2}_{12.4}^{41.5}$	${25.9}_{26.3}^{70.5}$	${43.1}_{18.5}^{56.3}$	${16.2}_{5.7}^{23.5}$	${10.4}_{6.6}^{16.3}$
⋯	${0.642}_{0.634}^{0.647}$	${13.5}_{11.3}^{99.4}$	${11.1}_{8.5}^{71.5}$	${2.4}_{0.9}^{15.1}$	${0.8}_{0.5}^{3.6}$	⋯	⋯	⋯	⋯	⋯	⋯
2	${0.747}_{0.736}^{0.750}$	${77.0}_{22.8}^{98.9}$	${63.0}_{15.0}^{72.6}$	${8.6}_{4.8}^{32.6}$	${6.2}_{3.7}^{18.1}$	${0.728}_{0.661}^{0.742}$	${47.7}_{13.2}^{64.7}$	${26.6}_{26.9}^{158.6}$	${9.5}_{8.3}^{18.0}$	${1.1}_{0.8}^{2.3}$	${5.3}_{3.9}^{9.3}$
KIC 9048551
⋯	${0.321}_{0.318}^{0.325}$	${43.0}_{18.2}^{43.4}$	${33.1}_{7.9}^{85.4}$	${3.3}_{0.7}^{11.7}$	${1.0}_{0.6}^{3.4}$	⋯	⋯	⋯	⋯	⋯	⋯
⋯	${0.385}_{0.379}^{0.386}$	${40.0}_{9.4}^{49.2}$	${51.2}_{15.0}^{114.1}$	${4.9}_{1.3}^{13.8}$	${2.1}_{1.1}^{6.1}$	⋯	⋯	⋯	⋯	⋯	⋯
⋯	${0.465}_{0.461}^{0.469}$	${56.5}_{23.3}^{57.9}$	${91.6}_{9.8}^{111.1}$	${5.3}_{1.4}^{12.9}$	${2.2}_{1.3}^{6.6}$	⋯	⋯	⋯	⋯	⋯	⋯
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Note. Only the simultaneous flares in both wavelength ranges are numbered. The flare peak times ${t}_{0}\mathrm{WL}$ and ${t}_{0}^{{\rm{X}}}$ for KIC 8093473, KIC 8454353, and KIC 9048551 are relative to MJD 55164, MJD 55829, and MJD 55735, respectively.

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We estimated the white-light flare luminosities and energies following the approach of Shibayama et al. (2013) and Namekata et al. (2017). Namely, we assumed that the spectrum of a white-light flare can be described by a blackbody radiation with a temperature of ${T}_{\mathrm{flare}}$, the star itself is a blackbody source with a temperature of ${T}_{\mathrm{star}}$, and the normalized (by the average stellar flux) flare amplitude in the light curve is proportional to a fraction of the stellar disk covered by the white-light flare ribbons, accounting for the different spectral shapes of the quiescent and flaring emissions as well as for the instrumental bandpass. This gives

$$\begin{eqnarray}&&\displaystyle \frac{{I}_{\mathrm{flare}}^{\mathrm{WL}}(t)}{\left\langle {I}_{\mathrm{star}}^{\mathrm{WL}}\right\rangle }=\displaystyle \frac{{A}_{\mathrm{flare}}(t)}{\pi {R}_{\mathrm{star}}^{2}}\displaystyle \frac{\int {R}_{\lambda }(\lambda ){B}_{\lambda }(\lambda ,{T}_{\mathrm{flare}}){\rm{d}}\lambda }{\int {R}_{\lambda }(\lambda ){B}_{\lambda }(\lambda ,{T}_{\mathrm{star}}){\rm{d}}\lambda },\end{eqnarray} \tag{ 4 }$$

where ${I}_{\mathrm{flare}}^{\mathrm{WL}}(t)$ is the background-subtracted Kepler flare light curve, $\left\langle {I}_{\mathrm{star}}^{\mathrm{WL}}\right\rangle$ is the average Kepler stellar flux, ${A}_{\mathrm{flare}}(t)$ is the visible (projected) area of the flare ribbons, ${R}_{\mathrm{star}}$ is the stellar radius, λ is the wavelength, ${B}_{\lambda }(\lambda ,T)$ is the Planck function, and ${R}_{\lambda }(\lambda )$ is the Kepler response function (in the 350–950 nm spectral band; Van Cleve & Caldwell 2016). As a result, the bolometric luminosity of a white-light flare ${L}_{\mathrm{flare}}^{\mathrm{WL}}(t)$ can be expressed in the form

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{L}_{\mathrm{flare}}^{\mathrm{WL}}(t)}{\left\langle {L}_{\mathrm{star}}^{\mathrm{WL}}\right\rangle } & = & \displaystyle \frac{1}{4}\displaystyle \frac{{I}_{\mathrm{flare}}^{\mathrm{WL}}(t)}{\left\langle {I}_{\mathrm{star}}^{\mathrm{WL}}\right\rangle }\\ & & \times \displaystyle \frac{{T}_{\mathrm{flare}}^{4}}{{T}_{\mathrm{star}}^{4}}\displaystyle \frac{\displaystyle \int {R}_{\lambda }(\lambda ){B}_{\lambda }(\lambda ,{T}_{\mathrm{star}}){\rm{d}}\lambda }{\displaystyle \int {R}_{\lambda }(\lambda ){B}_{\lambda }(\lambda ,{T}_{\mathrm{flare}}){\rm{d}}\lambda },\end{array}\end{eqnarray} \tag{ 5 }$$

where $\left\langle {L}_{\mathrm{star}}^{\mathrm{WL}}\right\rangle$ is the average bolometric stellar luminosity; the factor 1/4 arises because we consider the total stellar luminosity (for a spherical source), while the sources of the flaring white-light emission look like nearly flat patches on the stellar surface. For an unresolved binary or multiple system, the temperature ${T}_{\mathrm{star}}$ in Equations (4)–(5) represents an average effective temperature of the system components, and the parameters $\pi {R}_{\mathrm{star}}^{2}$ and $\left\langle {L}_{\mathrm{star}}^{\mathrm{WL}}\right\rangle$ should be replaced by the total visible area of the stellar disks ($\pi {R}_{{\rm{A}}}^{2}+\pi {R}_{{\rm{B}}}^{2}+\ldots$) and the total luminosity of the system ($\left\langle {L}_{{\rm{A}}}^{\mathrm{WL}}+{L}_{{\rm{B}}}^{\mathrm{WL}}+\ldots \right\rangle$), respectively. We used the stellar temperature and luminosity values determined by Gaia (see Table 1), which can be applied to both single stars and unresolved systems, as described above; the average Kepler fluxes were determined by averaging the light curves over the entire respective quarters. Following Maehara et al. (2012), Shibayama et al. (2013), and Namekata et al. (2017), we adopted ${T}_{\mathrm{flare}}={\rm{10,000}}$ K as the typical effective temperature of the white-light flares, with possible variations in the range of 9000–14,000 K (Kowalski 2016); this uncertainty in the flare temperature is responsible for about half of the uncertainties in the estimated white-light flare energies and luminosities.

The total radiated flare energy (in either spectral band) is an integral of its luminosity:

$$\begin{eqnarray}&&{E}_{{\rm{flare}}}^{{\rm{WL}},{\rm{X}}}=\int {L}_{{\rm{flare}}}^{{\rm{WL}},{\rm{X}}}(t){\rm{d}}t.\end{eqnarray} \tag{ 6 }$$

To estimate the energies and peak luminosities of the individual flares, we used the fitted model light curves, i.e., ith components in Equation (1). The most probable values and confidence intervals for the integrals under the flare light curves were determined from the MCMC fitting procedure.

The X-ray luminosity of a stellar source (for the flaring or/and quiescent emissions, see below) can be estimated as

$$\begin{eqnarray}&&{L}^{{\rm{X}}}=2\pi {d}^{2}{\int }_{{E}_{\min }}^{{E}_{\max }}F({E}_{\mathrm{ph}}){E}_{\mathrm{ph}}\,{\rm{d}}{E}_{\mathrm{ph}},\end{eqnarray} \tag{ 7 }$$

where d is the distance to the star, ${E}_{\mathrm{ph}}$ is the photon energy, $F({E}_{\mathrm{ph}})$ is the model X-ray spectral flux density, and ${E}_{\min }\,\leqslant {E}_{\mathrm{ph}}\leqslant {E}_{\max }$ is the considered energy range ($0.2\mbox{--}12\,\,\mathrm{keV}$ in this work). We note that both the model X-ray spectrum $F({E}_{\mathrm{ph}})$ and the resulting luminosity ${L}^{{\rm{X}}}$ in Equation (7) are being averaged over a time interval where the spectral fitting is performed, i.e., ${L}^{{\rm{X}}}\equiv \left\langle {L}^{{\rm{X}}}\right\rangle$.

As mentioned above, we used the XMM-Newton EPIC PN detector data. We extracted the time-resolved spectral data for the selected stars from the XMM-Newton science archive ⁵ using the same source and background regions and "good" time intervals (presented in the 3XMM-DR5 database) that were used to produce the light curves. Then we analyzed the X-ray spectra in selected time intervals (see below) using the OSPEX package (Tolbert & Schwartz 2020); the spectra were fitted with a single-temperature optically thin thermal model (vth), accounting for interstellar absorption (computed using the model of Morrison & McCammon 1983). Following Guarcello et al. (2019a), Pizzocaro et al. (2019), etc., we estimated the absorption column density for each star as ${N}_{{\rm{H}}}=1.79\,\times {10}^{21}{A}_{V}$ ${{\rm{cm}}}^{2}$ ${{\rm{mag}}}^{-1}$, where A_V is the known optical extinction (see Table 1). We set the abundances of heavy elements to 0.2 of the solar ones—a typical value for the coronae of active stars (e.g., Robrade & Schmitt 2005; Pandey & Singh 2012). Considering the absorption column and/or abundances as free parameters has not significantly affected the obtained results.

Subtraction of the quiescent background is an important part of spectral analysis of flaring X-ray emission. However, for the considered observations, it was not possible to obtain reliable pre- or postflare background spectra. Instead, we assumed that the spectra of the flaring and quiescent emission components have similar shapes: e.g., for the optically thin thermal emission model, the plasma temperature remains nearly constant throughout time, and only the emission measure varies. This assumption can be justified for active K-M stars, where the quiescent X-ray emission is partially produced by the hot coronae (with the temperatures comparable to the temperatures during flares), and partially consists of multiple unresolved weaker flares (Güdel 2004). In this case, the time-dependent X-ray flare luminosity ${L}_{\mathrm{flare}}^{{\rm{X}}}(t)$ can be estimated as

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\mathrm{flare}}^{{\rm{X}}}(t)}{\left\langle {L}_{\mathrm{total}}^{{\rm{X}}}\right\rangle }\simeq \displaystyle \frac{{I}_{\mathrm{flare}}^{{\rm{X}}}(t)}{\left\langle {I}_{\mathrm{total}}^{{\rm{X}}}\right\rangle },\end{eqnarray} \tag{ 8 }$$

where ${I}_{\mathrm{flare}}^{{\rm{X}}}(t)$ is the background-subtracted XMM-Newton flare light curve, $\left\langle {I}_{\mathrm{total}}^{{\rm{X}}}\right\rangle$ is the average total (i.e., including both the flares and the quiescent background) XMM-Newton flux in the selected time interval, and $\left\langle {L}_{\mathrm{total}}^{{\rm{X}}}\right\rangle$ is the average total X-ray luminosity in the selected time interval (Equation (7)). This approach is certainly an approximate one; however, as demonstrated, e.g., by Flaccomio et al. (2018), the inaccuracy introduced due to neglecting the plasma temperature variations during flares is small with respect to other sources of uncertainties, such as the measurement and spectral fitting uncertainties.

We performed spectral fitting of the XMM-Newton spectra in the "flaring" time intervals, each selected to cover either one flare or several overlapping flares; these time intervals and the resulting best-fit spectral parameters are presented in Table 4 in Appendix B. We estimated the radiated energies and peak luminosities of the individual X-ray flares in the same way as for the white-light flares, i.e., using Equation (6) and the fitted model light curves.

Table 4. Parameters of the X-Ray Spectral Fits for the Selected Time Intervals: Emission Measures (EM), Temperatures (T), Average Luminosities ($\left\langle {L}_{\mathrm{total}}^{{\rm{X}}}\right\rangle$), and Average Equivalent GOES X-Ray Fluxes ($\left\langle {I}_{\mathrm{total}}^{\mathrm{GOES}}\right\rangle$)

Time Range, days	Flare Nos.	EM, 1052 ${{\rm{cm}}}^{-3}$	T, keV	$\left\langle {L}_{\mathrm{total}}^{{\rm{X}}}\right\rangle$, 1028 erg ${{\rm{s}}}^{-1}$	$\left\langle {I}_{\mathrm{total}}^{\mathrm{GOES}}\right\rangle$, 10−2 W ${{\rm{m}}}^{-2}$
KIC 8093473
0.911–0.961	1	${20.0}_{18.9}^{21.1}$	${5.30}_{4.08}^{6.52}$	${151.3}_{120.3}^{182.3}$	${70.7}_{57.3}^{84.0}$
1.102–1.305	2, 3, 4	${34.3}_{33.6}^{35.0}$	${2.31}_{2.17}^{2.45}$	${187.0}_{164.1}^{209.8}$	${63.4}_{58.3}^{68.4}$
1.355–1.442	5	${19.9}_{19.1}^{20.8}$	${3.29}_{2.84}^{3.74}$	${126.3}_{104.2}^{148.3}$	${50.5}_{42.7}^{58.3}$
KIC 8454353
0.472–0.562	1	${4.1}_{3.7}^{4.5}$	${2.84}_{1.91}^{3.78}$	${24.3}_{18.2}^{30.4}$	${9.1}_{5.6}^{12.7}$
0.666–0.771	2	${3.7}_{3.3}^{4.1}$	${0.76}_{0.69}^{0.82}$	${14.2}_{12.6}^{15.8}$	${1.6}_{1.1}^{2.0}$
KIC 9048551
0.528–0.725	1, 2	${2.7}_{2.5}^{2.8}$	${0.66}_{0.64}^{0.69}$	${10.3}_{9.6}^{10.9}$	${0.9}_{0.7}^{1.0}$

Note. The flare numbers correspond to those in Table 2. The time ranges for KIC 8093473, KIC 8454353, and KIC 9048551 are relative to MJD 55164, MJD 55829, and MJD 55735, respectively.

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We estimated also the equivalent GOES fluxes—i.e., the X-ray flare fluxes as if observed by the GOES satellite from a distance of 1 au. They are given by the expression ${I}_{\mathrm{flare}}^{\mathrm{GOES}}(t)\,={L}_{\mathrm{flare}}^{\mathrm{GOES}}(t)/(2\pi {{ \mathcal R }}^{2})$, where ${ \mathcal R }=1\,\mathrm{au}$, and the luminosities ${L}_{\mathrm{flare}}^{\mathrm{GOES}}(t)$ are computed as described above, but for the GOES energy range ($1.55\mbox{--}12.4\,\,\mathrm{keV}$ or 1–8 Å). Since the X-ray fluxes from the stellar flares were measured reliably only at the energies of up to a few keV, the estimations of the equivalent GOES fluxes are based largely on extrapolation.

Figure 5 demonstrates scatter plots of the radiated flare energies and peak flare luminosities in the optical and X-ray ranges. Most of the analyzed flares released more energy in the optical range than in the X-ray one: ${E}_{\mathrm{flare}}^{\mathrm{WL}}/{E}_{\mathrm{flare}}^{{\rm{X}}}\sim 3\mbox{--}4$ in five flares, and ${E}_{\mathrm{flare}}^{\mathrm{WL}}/{E}_{\mathrm{flare}}^{{\rm{X}}}\sim 1.5$ in three flares. Similarly, the peak white-light flare luminosities were usually higher than or comparable to the X-ray ones: ${L}_{\max }^{\mathrm{WL}}/{L}_{\max }^{{\rm{X}}}\sim 1\mbox{--}2$ in seven flares. This energy partition is typical of the solar flares, where the white-light continuum emission is responsible, on average, for about 70% of the total radiated flare energies (Kretzschmar 2011). Similar relations between the optical and X-ray emissions in stellar flares were reported, e.g., by Fuhrmeister et al. (2011), Flaccomio et al. (2018), Guarcello et al. (2019a, 2019b), and Schmitt et al. (2019). ⁶

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A prominent outlier is the powerful flare #2 on KIC 8093473, where the X-ray emission strongly dominated: ${E}_{\mathrm{flare}}^{\mathrm{WL}}/{E}_{\mathrm{flare}}^{{\rm{X}}}\simeq 1/9$ and ${L}_{\max }^{\mathrm{WL}}/{L}_{\max }^{{\rm{X}}}\simeq 1/12$. This difference from the other flares looks even more intriguing because the flare #2 together with subsequent weaker flares #3 and #4 were likely parts of one long complex event (see Figure 2); however, the flares #3 and #4 demonstrated more typical relations between the optical and X-ray emissions (with ${E}_{\mathrm{flare}}^{\mathrm{WL}}\gtrsim {E}_{\mathrm{flare}}^{{\rm{X}}}$ and ${L}_{\max }^{\mathrm{WL}}\sim {L}_{\max }^{{\rm{X}}}$). In fact, in the sequence of flares #2–4 on KIC 8093473, the ${E}_{\mathrm{flare}}^{\mathrm{WL}}/{E}_{\mathrm{flare}}^{{\rm{X}}}$ and ${L}_{\max }^{\mathrm{WL}}/{L}_{\max }^{{\rm{X}}}$ ratios tended to increase with time. We propose two explanations of this phenomenon:

(a) Soft electron spectrum. The white-light emission in the flare #2 on KIC 8093473 could be relatively weak, if the accelerated electrons in this flare, despite of a large total energy flux, had a relatively soft spectrum; therefore, these electrons heated a large amount of plasma in the lower corona (which produced the soft X-rays), but were unable to penetrate into the deeper layers of the stellar atmosphere where the white-light emission is produced. According to the gas-dynamic simulations of Katsova et al. (1980), the optical continuum emission should be negligible for the electron beams with the spectral indices of $\delta \gtrsim 4.5;$ due to the limited spectral coverage of the observations, we cannot estimate the parameters of nonthermal energetic particles in the analyzed flares independently. If the variations of the ${E}_{\mathrm{flare}}^{\mathrm{WL}}/{E}_{\mathrm{flare}}^{{\rm{X}}}$ ratio from flare to flare were indeed caused by the mentioned effect, the spectra of energetic electrons in the long complex event including the flares #2–4 on KIC 8093473 should have become increasingly harder with time—a behavior ("soft-hard-harder") that has also been observed in some solar flares (see, e.g., Fletcher et al. 2011).

(b) Limb flare. The observed white-light emission depends on the flare location on the stellar disk: the area ${A}_{\mathrm{flare}}$ in Equation (4) is the projected area of the optically thick emission source, which decreases with the distance from the disk center and approaches zero for the flares at the limb. In contrast, the optically thin soft X-ray emission is not affected by the source location (unless the emitting volume is partially occulted). Therefore, the observed ratios ${L}_{\mathrm{flare}}^{\mathrm{WL}}/{L}_{\mathrm{flare}}^{{\rm{X}}}$ and ${E}_{\mathrm{flare}}^{\mathrm{WL}}/{E}_{\mathrm{flare}}^{{\rm{X}}}$ are expected to decrease significantly for the flares near the limb; this effect is confirmed by observations of solar flares, where the flare location is known (e.g., Woods et al. 2006). Thus the flare #2 on KIC 8093473 could be an example of a flare that occurred near the stellar limb, so that only a small fraction (∼0.03) of the total white-light emission was observed. In this case, the subsequent flares #3 and #4 should have occurred at different locations, approaching the stellar disk center with time. We note that similar (albeit smaller) shifts of the flare ribbons have been observed in solar flares: as demonstrated, e.g., by Grigis & Benz (2005), Zimovets & Struminsky (2009), Zimovets et al. (2013), and Kuznetsov et al. (2016), the energy release sites responsible for different flaring peaks in long complex events (at least, in some of them) are not cospatial—they are located in different magnetic loops that are "ignited" successively by a propagating disturbance (e.g., a magnetohydrodynamic wave); consequently, the flaring loop footpoints (where the white-light emission is produced) move along the flaring arcade. The flares #2–4 on KIC 8093473 were produced in a large active region, with the size comparable to the stellar radius (see Sections 4.2–4.3); therefore, the distances between the different loop footpoints within the flaring arcade could be sufficient to provide a significant variation of the viewing angle.

Figure 6(a) shows the delays between the X-ray and optical flares, defined as ${\rm{\Delta }}t={t}_{0}^{{\rm{X}}}-{t}_{0}^{\mathrm{WL}}$. The obtained delays were always smaller than the associated uncertainties (caused mainly by the limited time resolution of Kepler). Nevertheless, the delays were mostly positive, which implies that the X-ray flares, as a rule, were delayed with respect to the optical ones; the only exception was a weak flare #2 on KIC 8454353 with a relatively slow rise phase, where the peak times were poorly determined. This result is consistent with the Neupert effect (Neupert 1968)—a delay of flaring thermal emissions relative to nonthermal ones, which is often (but not always) observed in solar and stellar flares (see, e.g., Benz & Güdel 2010, and references therein).

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Figure 6(b) compares the flare durations in the optical and X-ray ranges. The durations were defined as $\tau ={\tau }_{\mathrm{rise}}+{\tau }_{\mathrm{decay}}$, and their most probable values and confidence intervals were determined from the MCMC fitting procedure. The observed X-ray flares were mostly shorter than their optical counterparts (on average, ${\tau }^{\mathrm{WL}}/{\tau }^{{\rm{X}}}\sim 2$), which is inconsistent with the Neupert effect. This behavior is uncommon for solar and stellar flares, but not extraordinary: stellar flares with ${\tau }^{\mathrm{WL}}/{\tau }^{{\rm{X}}}\gtrsim 1$ have been reported earlier, e.g., by Guarcello et al. (2019a, 2019b). The relatively short durations of the X-ray flares can be attributed, e.g., to rapid radiative cooling of the emitting plasma, which, in turn, could be caused by a relatively high (in comparison with solar flares) plasma density in the coronal X-ray sources in the analyzed events (cf. Güdel 2004; Mullan et al. 2006).

We now compare the obtained results with the characteristics of solar flares that were also observed in both the optical and X-ray ranges; we used the sample of solar flares (50 events) presented in Namekata et al. (2017). Figure 7(a) shows the scatter plot of the radiated white-light flare energies versus the peak soft X-ray fluxes in the GOES range; we used the observed GOES fluxes for the solar flares and the estimated equivalent GOES fluxes for the stellar flares. We note that the distribution of stellar flares has a low-energy cutoff due to limited sensitivity of the used instruments. We fitted the relation between these parameters with a power-law dependence in the form of ${E}_{\mathrm{flare}}^{\mathrm{WL}}\propto {({I}_{\max }^{\mathrm{GOES}})}^{\alpha }$, which provided $\alpha =0.982\pm 0.024$. This result is consistent with conclusions and theoretical scaling laws by Namekata et al. (2017) and indicates that the flare energies, on average, are nearly proportional to the peak soft X-ray fluxes.

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In Figure 7(b), we present estimations of the total radiated flare energies. For the stellar flares, the total flare energy was estimated as the sum of the white-light and X-ray energies: ${E}_{\mathrm{flare}}\,\simeq {E}_{\mathrm{flare}}^{\mathrm{WL}}+{E}_{\mathrm{flare}}^{{\rm{X}}}$. Since Namekata et al. (2017) did not present the radiated X-ray energies for the flares in their sample, we used the abovementioned statistical conclusion by Kretzschmar (2011) that the white-light emission is responsible for about 70% of the total radiated energy of solar flares; therefore, the total energies of the solar flares were estimated as ${E}_{\mathrm{flare}}\simeq {E}_{\mathrm{flare}}^{\mathrm{WL}}/0.7$. Considering the total flare energy instead of the white-light one does not affect significantly the above conclusions: the fit in the form of ${E}_{\mathrm{flare}}\propto {({I}_{\max }^{\mathrm{GOES}})}^{\alpha }$ provided the power-law index of $\alpha =0.975\,\pm 0.017$. The most noticeable difference is that the flare #2 on KIC 8093473 now agrees much better with the power-law fit; i.e., despite of an anomalously low ${E}_{\mathrm{flare}}^{\mathrm{WL}}/{E}_{\mathrm{flare}}^{{\rm{X}}}$ ratio and the largest (among the considered flares) total energy, this flare was otherwise a quite typical one. Apart from the agreement in general, six out of nine detected flares in Figure 7(b) coincide with the obtained power-law fit within the estimated uncertainties, while three flares deviate slightly from the fit. However, in this work we do not elaborate on this discrepancy, because the three outlying events were characterized by relatively low plasma temperatures ($\lt 1\,\,\mathrm{keV}$, see Table 4 in Appendix B), and hence the extrapolated X-ray fluxes in the GOES range could be underestimated.

Maehara et al. (2015) and Namekata et al. (2017) derived the scaling laws describing the relations between the flare parameters, under the assumptions that (a) the flare energy is proportional to the magnetic energy in the flaring volume, and (b) the flare duration is proportional to the Alfvén travel time through the flaring region. For a constant coronal plasma density, these scaling laws have the form:

$$\begin{eqnarray}&&\begin{array}{c}\tau \propto {E}^{1/3}{B}^{-5/3},\\ \tau \propto {E}^{-1/2}{L}^{5/2},\end{array}\end{eqnarray} \tag{ 9 }$$

where τ is the flare duration, E is the released flare energy, B is the characteristic magnetic field strength in the active region, and L is the length scale of the active region. ⁷ For consistency with the results of Namekata et al. (2017), we consider here the white-light flare decay time as an estimation of the flare duration, i.e., $\tau \simeq {\tau }_{\mathrm{decay}}^{\mathrm{WL}}$.

Figure 8 demonstrates the scatter plots of the white-light flare decay times versus the radiated white-light or total flare energies for the stellar flares analyzed in this work and the solar flares reported by Namekata et al. (2017); theoretical lines corresponding to several constant values of B and L, according to Equations (9), are overplotted. If we consider the total radiated flare energies, the characteristic magnetic field strengths in the stellar active regions can be estimated as $B\sim 25\mbox{--}70$ G, which is very similar to those in the solar active regions. On the other hand, the estimated length scales of the stellar active regions ($L\sim {\rm{250,000}}\mbox{--}{\rm{500,000}}$ km) far exceed those of the solar active regions. Thus, according to the $E\mbox{--}\tau$ diagram, the analyzed stellar superflares look like "oversized" versions of solar flares, with nearly the same magnetic field strengths in the reconnection sites, but much larger sizes of the corresponding active regions—comparable to the stellar radii.

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Since two stars in our sample seem to be nonsingle (KIC 8454353 is likely a binary, and KIC 8093473 is either a binary or even a higher-order multiple system), we have analyzed how this multiplicity can affect the flaring processes. We have estimated the masses and radii of individual components of these systems using the empirical relations derived by Mann et al. (2015) that link the stellar mass and radius to absolute magnitude in the K_S band and metallicity; we have assumed that both KIC 8093473 and KIC 8454353 are binaries consisting of two identical components each, i.e., the magnitude of an individual component $K{{\prime} }_{S}$ is related to the observed magnitude of an unresolved system K_S as $K{{\prime} }_{S}\,={K}_{S}+2.5\mathrm{log}2$. The orbital parameters were estimated under the assumption of tidally locked binaries with circular orbits and the orbital periods equal to the rotation ones; the resulting estimations can be considered as lower limits for the orbital separations. The obtained results are presented in Table 1; according to them, both considered systems are sufficiently separated—with the distances between the components a of about 26.6 ${R}_{\mathrm{star}}$ for KIC 8093473 and 12.7 ${R}_{\mathrm{star}}$ for KIC 8454353.

As follows from Section 4.2, the estimated sizes of stellar active regions L (i.e., heights of the flaring loops above the photospheres) are comparable to the stellar radii ($L\lesssim {R}_{\mathrm{star}}$) and hence much smaller than the orbital separations ($L\ll a$) for both considered systems. This implies that (a) the flaring regions are confined entirely within the coronae of the respective stars, i.e., the flaring processes occur in closed magnetic loops that belong to one of the stars, rather than in long interbinary magnetic loops potentially connecting the system components; (b) consequently, the magnetic energy released during the flares comes from the dynamo processes in the stellar interiors rather than from a star–star interaction. Therefore, although the presence of a companion can potentially provide some triggering effect (i.e., control where and when the flares occur), the flaring processes themselves are expected not to be much different from those on single stars—at least for the flares analyzed in this work; this conclusion is supported by the correlations presented in the previous section.