KEPLER FLARES. IV. A COMPREHENSIVE ANALYSIS OF THE ACTIVITY OF THE dM4e STAR GJ 1243

As the most active M dwarf in the Kepler field, GJ 1243 was the primary subject of multiple Kepler "Guest Observer" campaigns (GO programs 20016, 20028, 20031, 30002, 30021). These campaigns yielded 11 months of short cadence data from the primary Kepler mission. Paper 2 used what was at the time the most recent reduction of the Kepler light curves, including the PDC-MAP Bayesian detrending analysis from Smith et al. (2012), to build its flare sample, which we use here.

The flare identification process we adopt here was thoroughly detailed in Paper 2. Briefly, flare candidates were first identified by automatically selecting events of two or more time steps in length with positive flux excursions greater than 2.5σ from the starspot-subtracted light curve. After this, all data were visually inspected using the FBEYE package⁷ to validate the initial computer classifications. Multiple users classified each month of data, and the final flare sample⁸ was selected from a composite of all user flare identifications. Flare start and end times were selected to include all observations identified as part of a flare by at least two users.

We note that since the publication of Paper 2, additional errors in the short cadence data processing have been uncovered, which impact the data on GJ 1243 from Kepler Q10 and Q12.⁹ The amplitude of these calibration errors is typically small. However, as the impact on this specific data set is as yet unknown, we note that some caution must be taken when interpreting the rates of the smallest energy flares. Future versions of this work will utilize Data Release 25 when it becomes available in mid- to late-2016.

To break the modeling degeneracy due to Kepler's single bandpass, we obtained ground-based spectroscopic observations of the star simultaneous with Kepler observations. Spectra for GJ 1243 were collected with the DIS on the Astrophysical Research Consortium (ARC) 3.5 m telescope at the Apache Point Observatory (APO) on 2012 May 18 (during Kepler Q13), following the procedure described in Kowalski et al. (2013). The low-resolution B400/R300 gratings were used, providing continuous wavelength coverage from $\lambda \sim 3400\mbox{--}9200\,\mathring{\rm A}$, except for a dichroic feature that affected flux calibration at $\lambda \sim 5200\mbox{--}5900\,\mathring{\rm A}$. Exposure times ranged from 45 to 60 s, with ∼10 s readout times. Short cadence spectra were interspersed to avoid nonlinearity and saturation in the red; this typically provided a signal-to-noise of ∼10 at 3600 Å. GJ 1243 was observed with the 5'' slit, which facilitates absolute flux calibration, mitigates the effects of slit loss, and allows for reduced exposure times. The slit was oriented perpendicular to the horizon to compensate for atmospheric differential refraction.

The spectra were reduced using standard IRAF procedures via a customized PyRAF wrapper, developed from the reduction software of Covey et al. (2008). Wavelengths were calibrated from HeNeArHg and HeNeAr lamps, resulting in dispersions of 1.83 Å pixel⁻¹ in the blue and 2.3 Å pixel⁻¹ in the red. Spectral resolutions were determined from the He I $\lambda 4471$ arc line recorded at the start of observations, resulting in a resolution of ∼18 Å ($R\sim 250$); however, for the 5'' slit, the spetral resolution was determined by the seeing, leading to a higher resolution on the observed objects ($R\sim 320$). To correct for the radial velocity uncertainty that results from the wide slit, we found it necessary to apply a wavelength shift to the blue and red spectra using the centers of the Hγ and Hα emission lines, respectively. The spectrophotometric standard star BD+28 4211 was observed and used to flux-calibrate the spectra, and an airmass correction was applied using the atmospheric extinction curve for APO published by the Sloan Digital Sky Survey. Despite this airmass correction, we observe a residual 5%–8% color-independent variation in spectrum brightness, which corresponds well to the change in airmass; spectra generally get brighter as airmass decreases through the night. With multiple standard star observations through the night, a residual airmass correction could be applied; however, to maximize the number of observations simultaneous with Kepler, we only observed a standard for flux-calibration purposes at the start of the night (at airmass 1.5).

We collected optical spectra of GJ 1243 and Gliese 406 (Wolf 359) on UT 2011 June 23, with the Astrophysical Research Consortium Echelle Spectrograph (ARCES) on the ARC 3.5 m telescope at APO. ARCES (Wang et al. 2003) is a high-resolution, cross-dispersed spectrograph, collecting $R\,\sim$ 31,500 spectra between 3600 and 10000 Å. In addition to these stars, we recorded bias, flat-field, and ThAr lamp exposures. These data were then reduced with standard IRAF procedures.¹⁰

FFDs, plots of the cumulative frequency of flares (log number of flares per day with energy greater than E), as a function of the energy (log E), were constructed for the entire data set and for each month of data, as well as quarterly; the FFD for the full data set is seen in Figure 3. Following common practice, we characterize the probability distribution for flare energies as a power law,

$$\begin{eqnarray}&&N(E){dE}\propto {E}^{-\alpha }{dE},\end{eqnarray} \tag{ 1 }$$

where $1-\alpha$ is the slope of a linear fit to a log–log representation of the cumulative FFD. We characterize the cumulative FFD as a power law for energies above a limiting energy ${E}_{\mathrm{lim}}$ determined primarily by the completeness of the sample for each segment in time. The power-law fits we obtain for our FFDs of GJ 1243 were determined with a least-squares fit to the flares above ${E}_{\mathrm{lim}}$. To mitigate the potential over-weighting of the highest-energy flares, where flare energies are much more uncertain (see Paper 3), Poisson uncertainties were used to weight the fit toward energy ranges with better sampling. We did not exclude flares for which peak fluxes are 95% or more of the quoted full depth of the chip, due to the lack of available data; using these flares with the calculated energies as lower limits still provides more information than excluding them entirely. We initially fit the data using the energy cutoff of $\mathrm{log}({E}_{\mathrm{lim}})=31$ found in Paper 1; values of α over time based on this cutoff are found in Figure 4. We also evaluated α by determining a cutoff for each data subset (monthly, quarterly, and the full set). To do this, we increased ${E}_{\mathrm{lim}}$ and re-fit until the change in slope did not exceed the uncertainty of the fit. To counter-act potential biases in the determination of ${E}_{\mathrm{lim}}$ (and thus α) due to the step size for the increase in ${E}_{\mathrm{lim}}$, we ran fits for a variety of step sizes, ranging from $\bigtriangleup \mathrm{log}({E}_{\mathrm{lim}})=0.025$ to 0.125. The values of α found for each step size often had comparable ${\chi }^{2}$ values, but the actual values could vary widely. Our adopted value for α in each segment of time (month or quarter) is instead based on a visual inspection of these final values, determining whether the fit line fit the distribution of flares adequately by eye, rather than being confounded by an additional outlier. The range of possible values of α depending on the choice of step size is represented by the shaded regions in Figure 5.

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Our values for α and limiting energy ${E}_{\mathrm{lim}}$ for each segment in time are listed in Table 1. We note that for Q6, we adopt the flare identifications from Paper 2, rather than the initial flare identifications made in Paper 1. This leads to a small difference between our reported value of α for this period and those reported in Paper 1.

Table 1.  Power-law Fits to Kepler FFDs for GJ 1243

Month	Quarter	α	$\mathrm{log}({E}_{\mathrm{lim}})$	$\alpha (\mathrm{log}({E}_{\mathrm{lim}})=31)$
1	6a	2.035 ± 0.018	31.3	2.042 ± 0.007
2	6b	1.774 ± 0.003	30.6	1.785 ± 0.008
3	10a	1.872 ± 0.037	31.7	1.940 ± 0.005
4	10b	1.766 ± 0.005	30.6	1.838 ± 0.013
5	10c	1.592 ± 0.003	30.3	1.847 ± 0.008
6	12a	1.815 ± 0.006	30.8	1.862 ± 0.011
7	12b	1.783 ± 0.005	30.7	1.904 ± 0.013
8	12c	2.389 ± 0.035	31.3	2.177 ± 0.017
9	13a	1.797 ± 0.004	30.8	1.845 ± 0.004
10	13b	1.825 ± 0.006	30.9	1.885 ± 0.008
11	13c	1.699 ± 0.004	30.6	1.788 ± 0.009
1−2	6	1.902 ± 0.003	31.0	1.913 ± 0.003
3−5	10	1.903 ± 0.004	31.1	1.889 ± 0.003
6−8	12	2.118 ± 0.005	31.3	1.971 ± 0.007
9−11	13	1.808 ± 0.003	31.0	1.823 ± 0.003
All data	⋯	2.008 ± 0.002	31.5	1.901 ± 0.001

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There is apparent variation in α over time. As seen in Figure 4, there is no clear pattern to this variation, even when holding a fixed energy cutoff of $\mathrm{log}({E}_{\mathrm{lim}})=31$ (the cutoff found in Paper 1). This suggests that the variation is likely due to the varying levels of completeness for each sample. Variation in time is more apparent in the adopted values for α when determining the cutoff via the iterative method described above, as seen in Figure 5. However, as the large range of the shaded regions in Figure 5 suggests, systematic error from the choice of step size is a large factor here, and thus makes the variation less apparent overall. Determining $\mathrm{log}({E}_{\mathrm{lim}})$ separately for each epoch allows each fit to probe to a much higher ${E}_{\mathrm{lim}}$ in the sample; this effect, combined with the larger sample size, is why α for the full data set in Figure 5 is higher than the mean of α for the individual epochs. The systematic error due to step size choice, however, makes this method of evaluation less effective for evaluating time evolution on monthly and quarterly timescales. The large systematic error is generally consistent with the conclusion that the completeness of each sample is a key factor; the systematic error is much lower for the full ($\sim 90 \%$ complete) data set than it is for each time segment. We discuss further evidence to support this conclusion in Section 4.2.

It is particularly interesting to examine the behavior of the highest energy flares, which show clear variation around the single-power-law fit and may contribute to the large range of values of α seen in certain time segments (months 1, 3, 8, and 11). Maehara et al. (2015) searched for a separate power law for the highest-energy flares on solar-type stars, with a fixed bolometric energy lower limit of $\mathrm{log}(E)=32$. However, for several reasons, this cutoff cannot simply be applied directly. The Maehara et al. (2015) energies are the inferred bolometric flare energies, rather than specifically those in the Kepler bandpass, making the two measurements incompatible. Additionally, a solar-type star can achieve a higher energy flare at lower relative amplitude, due to its higher intrinsic luminosity (using the equivalent duration method of calculating energy); this leads to far fewer saturated flares at these energies on solar-type stars than it does in our data set. As seen in Figure 3, the majority $(\sim 78 \% )$ of the 49 flares in which the peak fluxes equaled or exceeded 95% of Kepler's quoted full well depth occur above the $\mathrm{log}({E}_{\mathrm{Kp}})=32$ limit, but these saturated flares also occur down to energies of $\mathrm{log}({E}_{\mathrm{Kp}})\sim 31$.

Paper 3 found a deviation from a single power-law fit at high energies, but attributed this exclusively to the lower-limit effect of the nonlinearity of some high-energy flares. However, close inspection of the FFD at high energies (as in Figure 6) shows that this deviation occurs in unsaturated flares of high energy. This suggests that the break-away from the single power law seen in the FFDs here and in Paper 3 is not exclusively a factor of entering the saturated regime of the CCD, which could correspond to a different power law at high energies. The relatively small number of flares in question suggests that this difference could be due simply to small-number statistics. Nevertheless, a significant change in slope from the single-power-law model has many implications for our understanding of flare generation and behavior, and warrants further study. Future work will include the addition of high-energy flares found in Kepler long-cadence data, to improve the completeness of the sample at these energies (Davenport 2016).

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Figure 7 shows the relationship between flare amplitude, energy, and duration for classical and complex flares. As in Paper 1, the binning of flare durations at the low end is an artificial result of the 1 minute cadence of observations; with higher time resolution this would likely smooth out, as energy and amplitude do. This would agree with the morphological conclusions in Paper 2, which found an empirical model for classical flares that depended exclusively on relative flux and full-time at half-maximum (that is, the time between half-maximum flare-only flux during the rise and decay phases).

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Following Maehara et al. (2015), we quantify the relationship between flare duration and energy. Using our definitions of flare duration ${\tau }_{\mathrm{Kp}}$ and energy ${E}_{\mathrm{Kp}}$ (rather than those used by Maehara et al. 2015)¹¹ , we find that ${\tau }_{\mathrm{Kp}}\propto {E}_{\mathrm{Kp}}^{0.342\pm 0.003}$ for the full set of classical flares. For complex flares, we see ${\tau }_{\mathrm{Kp}}\propto {E}_{\mathrm{Kp}}^{0.363\pm 0.006}$. Both of these values are similar to, but slightly lower than, the value for "superflares" on solar-type stars found by Maehara et al. (2015); however, they fit well within the wide boundaries for values set by it and studies of solar flares (e.g., Veronig et al. 2002; Christe et al. 2008). This suggests that, as expected, all these flares are likely caused by the same basic physical process of magnetic reconnection.

Of note is the difference in spectral index and offset between the classical and complex relationships, which indicates that complex flares have longer durations than their classical counterparts of the same energy. The complex slope and classical slope show some difference, but not a clearly significant difference between the distributions. Paper 2 found that many multiple-peak events (the so-called "complex" flares) were well-described by multiple overlapping classical flare templates; this potentially significant difference in the relationship between duration and energy provides further support for this assessment. Furthermore, qualitatively the complex flares exhibit somewhat lower amplitudes than classical flares of the same energy, which would also be expected from a "superposition" model.

As values of α show some variation over time, the question of whether the correlations here and the scatter related to them also vary merits further investigation. To examine this, with the variation in α in mind, we show the relationships between flare amplitude, energy, and duration for the months with the highest and lowest values of α, and month 6, a month near the median of values, in Figure 8. This subset of the full data set demonstrates that for each month, the relationships between these three variables are qualitatively identical to the full data set, for both classical and complex flares. Furthermore, while the overall trend appears to be the same month-to-month, months 5 (lowest α) and 8 (highest α) demonstrate a clear gap, much larger than that observed in the full data set, between the highest-energy and next-highest-energy flares in all three attributes. This suggests that the values for α in these time segments (as well as the large range of possible values for α seen in Figure 5) are due to these outliers in how the underlying distribution was sampled.

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Long- and short-cadence Kepler observations of GJ 1243 show a regular, ∼0.59 day periodicity in its brightness, believed to be due to a persistent starspot or region of starspots (Davenport et al. 2015). As follow-up to the analysis of correlation with starspot modulation in Paper 1, we also examined the distribution of flares as a function of star phase month-by-month as well as for the full 11-month sample, to observe whether flares were more likely to occur when the largest fraction of the visible hemisphere was covered with starspots. Figure 9 shows the histogram for the full data set and demonstrates no clear trend, as was found in Paper 1. Similar results are seen for the individual months and quarters. Similarly, there is no apparent trend in flare energy as a function of phase, the same result as Paper 1.

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Davenport et al. (2015) identified a long-lived starspot at higher latitude on GJ 1243 (possibly a spot cap or group) which holds very stable in phase (standard deviation in longitude of only 16° over four years of data). They were able to weakly constrain the latitude of the "primary starspot" to a region that implies that the spot covers a significant fraction of the pole. Given the inclination derived in Section 3, part of the spot will always be in view, leading to a minimal correlation between flare occurrence and phase even if the flares are associated with the spot.

Kowalski et al. (2013) defined several observational parameters for spectra, which we adopt here. Briefly, we use the average flux measurements C3615 and C4170, their ratio ${\chi }_{\mathrm{flare}}$, and the flux measures BaC3615, BaC_tot, and PseudoC. We refer the reader to Kowalski et al. (2013) for definitions of these measures. In addition to these, we record line fluxes for the Balmer series through Hδ, as well as Ca ii K, using the same line and continuum windows used in Kowalski et al. (2013) for consistency.

All times recorded for both ground- and space-based observations are times at the midpoint of the observation. All times from the spectroscopy (generally given in MJD) have been converted to BJD–2454833, a standard time for the Kepler data that was used in Papers 1 and 2, to ensure synchronization.

The third flare began at Kepler date 1232.9472656 and lasted for 15.64 minutes. The recorded $\mathrm{log}{E}_{\mathrm{Kp}}$ of the flare was 30.66, nearly four times as strong as the first flare. As seen in Figure 11(a), this flare exhibits a delay of one minute between maximum C4170 and maximum Balmer emission. While there is a delay, the minimal size (the smallest detectable delay in our time resolution) suggests that these peaks were produced by a common heating mechanism, as with most classical impulsive events examined in Kowalski et al. (2013).

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The shape of the flares with simultaneous photometry and spectroscopy suggests that they are "low-amplitude," "classical" flares, using broad qualitative descriptors from Kowalski et al. (2013). Kowalski et al. (2013) defined a quantitative "impulsiveness index" ${ \mathcal I }$ based on the full width of the U-band light curve at half maximum and the quantity ${I}_{f,U}$ defined in Gershberg (1972). A similar impulsiveness index could be defined for the Kepler data, but would yield very little information without multiple flares with differing levels of impulsiveness. Instead, we calculate a rough filter-weighted count flux measurement for flare 3 in the U-band, based on our spectral data. To avoid extreme noise effects at the blue end of the spectrum and the portion of the bandpass missing from our spectra, we set the filter sensitivity to 0 blueward of 3400 Å. The results (along with synthetic B-band flux, actual Kepler flux, and a synthetic measurement of Kepler flux from spectra) are shown in Figure 11. Following Section 2.1 from Kowalski et al. (2013) to find values ${I}_{f,U}$ and ${t}_{1/2,U}$ for these data, we calculate ${{ \mathcal I }}_{U}\sim 0.4$ for this flare. To confirm this approximation of ${{ \mathcal I }}_{U}$, we look to other spectral diagnostics. Kowalski et al. (2013) showed an inverse relationship between impulsiveness of flare and ratio of BaC emission to total emission blueward of the Balmer jump (as measured by BaC3615/C3615); "gradual flare" events had much higher percentages ($55 \% \mbox{--}80 \%$) of C3615 due to BaC3615 at the peak than "impulsive" or "hybrid" events. All three "spectroscopic" flares exhibited BaC3615/C3615 ratios of $\sim 0.7\mbox{--}0.81$ at peak (Table 2), indicating that all were "gradual flare" events. The relationship between the U, B, and Kepler bands is discussed further in Section 5.6.

To constrain unexplained energy sources for model predictions (e.g., white-light continuum emission), we measure flux attributable to hydrogen Balmer (HB) emission relative to the total flare emission. We quantify this by summing the fluxes in Hδ, Hγ, Hβ, PseudoC, and BaC_tot. As in Kowalski et al. (2013), we measure the percentage of HB flux to total flux in spectra blueward of 5200 Å (%HB) at peak emission and the beginning of the gradual decay phase. These data provide direct comparison to radiative hydrodynamic (RHD) models. Data for each flare are listed in Table 2; here, we summarize some key findings for the third flare.

Like Kowalski et al. (2013), we find that in the third flare, the fraction of emission attributable to Balmer emission increases from peak continuum emission to the beginning of the gradual decay phase by $\sim 20 \%$. However, we also observe a dramatic ($\sim 30 \%$) decrease in %HB immediately afterward. Additionally, the third flare has $50 \% \mbox{--}60 \%$ of the total flux from HB emission at the beginning of the gradual decay phase, which also corresponds well to GF-type characteristics. At peak, each of the Hβ, Hγ, and Hδ lines produces $\sim 5 \%$ of the emission, in line with Kowalski et al. (2013).

Both Paper 2 and Kowalski et al. (2013) found specific changes in flare behavior between the impulsive and gradual decay phases. Paper 2 found this change in its two-component flare decay model, while Kowalski et al. (2013) found that the gradual decay phase was marked by an increase in HB flux (as a percentage of total flare flux) of $\sim 20 \% \mbox{--}30 \%$. Here, we use the results of spectral analysis of the third flare (Figure 11(a)) to explain the empirical model from Paper 2. The Kepler response function has minimal transmission below 4000 Ź² , so NUV components of the flare produce a minimal contribution to the overall shape. However, the lower-order Balmer lines (Hα, Hβ, Hγ, and Hδ) contribute to the Kepler flux, as does C4170.

The impulsive rise is most characterized by continuum emission (as represented by C4170), as expected. C4170 rapidly increases from its quiescent pre-flare level; the total (quiescent + flare) value of C4170 at peak is 1.15 times greater than the total value immediately pre-flare. C4170 drops back to near-quiescent levels fairly quickly (within five spectra of its peak), corresponding with the flare's impulsive decay phase, and then evolves similarly to the evolution of the Kepler light curve, gradually decaying until the end of the flare. This agrees well with the canonical $\sim {10}^{4}$ K "blackbody" emission component, as expected. The HB lines begin at a lower quiescent point than C4170 does, and peak one spectrum after C4170 does, in line with the Kepler peak. To evaluate the increase in line emission, we calculate the ratio of emission from each line to Hγ in total (quiescent + flare) spectra; to evaluate Hγ, we calculate the ratio of Hγ/C4170. At peak, values of the lines range from 2.1 (Hγ/C4170) to 7.1 (Hδ/Hγ) times their quiescent levels. Like C4170, these four lines show a rapid decay. The bluer lines (all but Hα) then exhibit a more gradual fade at approximately the same point that C4170 enters its gradual decay (approximately corresponding to the start of the Paper 2 gradual decay phase), while Hα mimics the Kepler light curve.

Paper 2 characterized their two-component empirical decay as representing two physically distinct regions, with independent cooling profiles, producing radiation throughout the flare decay. Our analysis of the third flare here suggests that the secondary component of this model connects the gradual decay phase blackbody (represented by C4170) and the increase in relative proportion of HB flux (Table 2), as was found in Kowalski et al. (2013). This implies that the blackbody and HB components of the gradual decay phase are physically connected to each other, originating from the same physical process, but are physically distinct from the impulsive phase of the flare, either by virtue of different spatial region, different atmospheric layer, or different cooling process. This result is critical for future modeling, as it suggests that the gradual decay phase and impulsive phase may potentially be modeled separately, but with a connected underlying physical origin. Current modeling efforts (e.g., Kowalski et al. 2015) focus on replicating the impulsive phase with one-dimensional RHD multithread modeling (e.g., Warren 2006); the gradual phase currently remains a separate challenge, due to the quite different timescales. Given the distinct characteristics of their evolution, it is unlikely that a single underlying fundamental heating/cooling mechanism can produce both the observed impulsive and gradual phases, unless the impulsive phase drastically alters the initial conditions of the flaring region for the gradual phase. As such, modeling the gradual phase as a separate, temporally overlapping problem will allow for more focused efforts to improve both models. For example, there is no known explanation for the observed evolution of Ca ii K, which peaks during the transition from impulsive to gradual decay for the third flare, though that evolution does indicate a less intense heating/cooling mechanism than the impulsive phase's electron-beam heating. A separate gradual-phase model with an independent heating and cooling mechanism would be able to directly address this, as well as the observed blackbody and HB evolution.

Connecting the results of studies of stellar flares from Kepler data to the literature is challenging, as most ground-based flare studies utilize U and B band optical data (see, e.g., Lacy et al. 1976; Kowalski et al. 2013), while Kepler data lack color information. To enhance comparison of Kepler flare data to canonical studies, it will be critical to find an appropriate conversion of measurements of ${L}_{\mathrm{fl}}$, the flare luminosity in a given bandpass, to ${L}_{\mathrm{fl}}/{L}_{\mathrm{bol}}$, the bolometric luminosity for the star.

As a preliminary step in this direction, we analyze the relationship between synthetic U, synthetic B, and synthetic and actual Kepler data, presented for flare 3 in Figure 11. Qualitatively, the synthetic Kepler data provides a satisfactory match to the observed data from Kepler, but with some additional noise due to observed fluctuations in the continuum level of the red end of the spectra. We therefore determine relationships between relative U, B, and Kepler fluxes based on these synthetic fluxes rather than the actual Kepler fluxes, as the spectral fluxes are synchronized.

We define ${\rm{\Delta }}{({\boldsymbol{f}})}_{\lambda }$ as the flux enhancement during a flare for each band. This is analogous to the ${I}_{f}+1$ quantity from Gershberg (1972), but utilizes the calibrated flux rather than the count flux. Figure 12 presents the flux enhancements ${\rm{\Delta }}{({\boldsymbol{f}})}_{\lambda }$ for the U and B bands for the third flare as a function of the Kepler flux enhancement. The synthetic Kepler flux values are only enhanced significantly ($\gt 3\sigma$) above the quiescent noise level during the two observations at the peak of this flare; as such, we focus on flux enhancement conversions during the peak.

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We find that for the peak of the third flare,

$$\begin{eqnarray}&&{\rm{\Delta }}{({\boldsymbol{f}})}_{B,\mathrm{peak}}=1.09{(\pm 0.01){\rm{\Delta }}({\boldsymbol{f}})}_{\mathrm{Kep},\mathrm{peak}}\end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray}&&{\rm{\Delta }}{(f)}_{U,\mathrm{peak}}=1.55{(\pm 0.02){\rm{\Delta }}(f)}_{\mathrm{Kep},\mathrm{peak}},\end{eqnarray} \tag{ 3 }$$

suggesting (as expected) that a small flux enhancement in the Kepler bandpass indicates a similarly sized enhancement in B, and a much larger enhancement in U. Future work (J. R. A. Davenport et al. 2016, in preparation) will expand on this exploration of conversion from the Kepler bandpass to the conventional U and B filters, using the flare light curve template for GJ 1243 from Paper 2 and robustly determined bolometric flare characteristics.

We demonstrated based on high-resolution echelle spectroscopy that GJ 1243 has a large $v\sin i$ of 25 $\mathrm{km}\,{{\rm{s}}}^{-1}$, as expected for its high activity level. We used the observed period in its Kepler light-curve and this measurement to find that the system's inclination angle i was 32^◦. Using data on its proper motion, distance, and radial velocity, we showed that GJ 1243 is likely (98.5% probability) a member of the Argus association, suggesting that its age is 30–50 Myr. This relatively young age agrees well with the high level of activity we observe on this star.

We find that generally a single power-law model fits each data set in monthly and quarterly increments, as well as the full data set. Over time, we show that α shows some uncorrelated variability over time, likely due to the varying sample of flares over each epoch studied. Interestingly, the values of α for each subset are generally lower than the value for the full data set. We believe this is in part due to the sample size of the full data set, which allowed us to probe the FFD at a higher energy resolution, compared to the subsets.

Additionally, we find that the highest energy flares, both saturated and unsaturated, deviate from a single power law in a way that suggests a steeper power law than the full sample. This would have significant implications for stellar modeling, as values of $\alpha \gt 2$ are a possible mechanism for the heating of the solar corona. However, these heating models require this larger value of α to extend to low energies, which our data do not. For the full data set $\alpha \approx 2$, though there is a turn-off at low energies where flares above the minimum energy threshold are not detected as expected, as was observed in Paper 1. Paper 2 showed that the 11 month sample used here is 90% complete at the highest energy levels, suggesting that there may be "missing" flares from the sample that could account for this. Additionally, some lower-energy "classical" events could have been incorporated into "complex" events. While Paper 2 found that this potential overlap could not account for all of the "missing" portion, it could be an additional contributing factor.

We find a strong, positive correlation between flare energy and flare duration, as well as between flare energy and amplitude, similar to the trends found in Paper 1. We find that complex flares have higher energies and longer durations than classical flares at the same amplitude. We quantify the relationship between duration and energy, and find that complex and classical flares show a distinct difference in this relationship, which agrees with the characterization of some complex flares as a superposition of classical flares, presented in Paper 2. As with Paper 1, we find no correlation of flare timing or energy with the phase caused by the persistent starspots on the surface, in monthly or quarterly time periods or in the full data set. This suggests that flaring is distributed across the stellar surface, rather than concentrated only in the regions where these starspots appear. Additionally, the high-latitude constraint for the persistent spot found in Davenport et al. (2015), combined with the inclination angle found in Section 3, indicates that there would be minimal correlation between flare occurrence and phase were the flares in fact associated with the spot.

We observed three low-energy flares with simultaneous ground-based spectroscopy and Kepler photometry. Using this simultaneous photometric and spectroscopic data, we were able to classify the third flare of this set as a "gradual-flare" type event, following the definition from Kowalski et al. (2013). We were also able to derive an impulsiveness index ${ \mathcal I }$ for this flare based on the Kepler data, finding that this value was two orders of magnitude lower than the same index in the U band (synthesized from spectra), due to the smaller flare-induced flux enhancement in the Kepler bandpass. We also presented a spectroscopy-based explanation for the two-component empirical flare model adopted in Paper 2, finding that the gradual phase HB and blackbody were tied to each other, but resulted from a cooling region physically distinct from that of the impulsive phase. This suggests that the impulsive and gradual phases are physically distinct phenomena potentially arising from the same initial conditions, which allows for the possibility of separately modeling the impulsive and gradual phases of the flare.

Using simultaneous photometric and spectroscopic data, we were able to develop a relationship between flux recorded in the Kepler bandpass and in the more traditional U and B bandpasses at the peak of a faint flare, directly connecting this result to the larger canon of previous flare surveys. This basic conversion will be invaluable for future studies of dMe flares recorded with Kepler, and will be improved by future work that will approach the problem beginning with the empirical Kepler flare model.