#  $K2$ Ultracool Dwarfs Survey. VI. White light superflares observed on an L5
dwarf and flare rates of L dwarfs

R. R. Paudel1,2,3, J. E. Gizis3, D. J. Mullan3, S. J. Schmidt4, A. J.
Burgasser5, P. K. G. Williams6,7
1NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
2University of Maryland, Baltimore County, 1000 Hilltop Cir, Baltimore, MD
21250, USA
3Department of Physics and Astronomy, University of Delaware, Newark, DE,
19716, USA
4Leibniz-Institute for Astrophysics Potsdam (AIP), An der Sternwarte 16,
14482, Potsdam, Germany
5Center for Astrophysics and Space Science, University of California San
Diego, La Jolla, CA 92093, USA
6Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, USA
7American Astronomical Society, 1667 K Street NW Ste. 800, Washington, DC
20006, USA Contact e-mail<EMAIL_ADDRESS>
###### Abstract

$Kepler$ $K2$ long cadence data are used to study white light flares in a
sample of 45 L dwarfs. We identified 11 flares on 9 L dwarfs with equivalent
durations of (1.3 - 198) hr and total (UV/optical/IR) energies of $\geq$0.9
$\times$ 1032 erg. Two superflares with energies of $>$1033 erg were detected
on an L5 dwarf: this is the coolest object so far on which flares have been
identified. The larger superflare on this L5 dwarf has an energy of
4.6$\times$ 1034 ergs and an amplitude of $>$300 times the photospheric level:
so far, this is the largest amplitude flare detected by the $Kepler/K2$
mission. The next coolest star on which we identified a flare was an L2 dwarf:
2MASS J08585891+1804463. Combining the energies of all the flares which we
have identified on 9 L dwarfs with the total observation time which was
dedicated by $Kepler$ to all 45 L dwarfs, we construct a composite flare
frequency distribution (FFD). The FFD slope is quite shallow (-0.51$\pm$0.17),
consistent with earlier results reported by Paudel et al. (2018) for one
particular L0 dwarf, for which the FFD slope was found to be -0.34. Using the
composite FFD, we predict that, in early and mid-L dwarfs, a superflare of
energy 1033 erg occurs every 2.4 years and a superflare of energy 1034 erg
occurs every 7.9 years. Analysis of our L dwarf flares suggests that magnetic
fields of $\geq$0.13-1.3 kG are present on the stellar surface: such fields
could suppress Type II radio bursts.

###### keywords:

stars: individual: VVV BD001 - stars: activity - stars: low-mass - stars:
flare

††pubyear: 2018††pagerange: $K2$ Ultracool Dwarfs Survey. VI. White light
superflares observed on an L5 dwarf and flare rates of L dwarfs–$K2$ Ultracool
Dwarfs Survey. VI. White light superflares observed on an L5 dwarf and flare
rates of L dwarfs

##  1 Introduction

L dwarfs are ultracool dwarfs (hereafter UCDs) with effective temperatures
$\lesssim$2300 K. Since the lowest mass hydrogen burning stars have spectral
types as late as $\sim$L4, the early L dwarfs can be either young brown dwarfs
or old hydrogen burning low mass stars. Objects with spectral types later than
L4 are all brown dwarfs (Chabrier et al., 2000; Saumon & Marley, 2008).
Empirically, L dwarfs have been found to have weak chromospheric emission
lines, but some have strong radio-emissions. Furthermore, the coronae are
usually not detected in these cool objects by currently available X-ray
satellites. L dwarfs are rapidly rotating objects with periods $<$10 hours,
but available data indicate that the L dwarfs do not follow the rotation-
activity correlations (RAC) which are typical for stars with spectral types G,
K, and M (Reiners & Basri, 2008; Berger et al., 2010; Williams et al., 2014;
Cook et al., 2014). In this paper, we use $K2$ long cadence data to study the
properties of white light flares (hereafter WLFs) of L dwarfs. In Section 1.1,
1.2 and 1.3, we give an overview of previous studies regarding the important
magnetic activity indicators: H$\alpha$, X-ray and radio emission, and flares
on L dwarfs. In Section 1.4, we discuss about flares on stellar and substellar
objects.

###  1.1 Previous studies: H$\alpha$ emission

Extensive studies of H$\alpha$ emission from L dwarfs have already been
reported by Schmidt et al. (2015) and Pineda et al. (2016). Schmidt et al.
(2015) studied 551 L dwarfs using data obtained from SDSS, 2MASS and the WISE
survey. Among the 551 targets, values of H$\alpha$ emission or upper limits
were reported for 181 stars. The authors classified chromospherically active L
dwarfs as those with H$\alpha$ equivalent width (EW) $>$0.75 Å, and
chromospherically inactive L dwarfs as those with upper limits (non-
detections) of H$\alpha$ EW $\leq$0.75 Å. They found that $\sim$90% of L0
dwarfs are chromospherically active, but this fraction decreases to $\sim$60%
in the case of L3 dwarfs. Likewise, the chromospheric activity fractions for
the L4 and L5 dwarfs were found to be 33% and 50% respectively. The L6-L8
dwarfs have either very weak or no chromospheric activity. Likewise, there is
a decline in the chromospheric activity (as measured by log $L_{\rm
H\alpha}$/$L_{\rm bol}$) from a value of $\sim$ -3.8 in M0-M4 dwarfs to a
value of -5.7 in the case of L3 dwarfs.

Pineda et al. (2016) studied H$\alpha$ emission from late-L and T dwarfs using
optical (6300-9700 Å) spectra obtained by the $Keck$ telescopes. In a sample
of 109 L4-T8 dwarfs, they reported a detection rate of 9.2$\pm^{3.5}_{2.1}$%.
In particular, they detected H$\alpha$ emission in 9.3$\pm^{4.5}_{2.4}$% of
mid-to-late L dwarfs (L4-L8) in a sample of 75 objects. The sample of L dwarfs
studied by Pineda et al. (2016) might include binary systems. In view of that,
it is possible that the overall detection rate of H$\alpha$ emission in single
L dwarfs might be lower than they have reported.

###  1.2 Previous studies: X-ray and radio emission

Although the L dwarfs are rapid rotators, their RACs depart from those which
have been obtained for stars earlier than M6.5 in the following sense. If the
L dwarfs were to obey the RAC for GKM stars, the L dwarfs would be expected to
have saturated X-ray emission, with $L_{\rm X}$/$L_{\rm bol}$ $\approx$ 10-3
(Cook et al., 2014). But this expectation is not observed. Instead, the L
dwarfs show significantly lower levels of X-ray emission.

Up to now, there have been only two detections of X-rays from L dwarfs. One
detection refers to Kelu-1AB which is a binary system of two L dwarfs: Kelu-1A
(L2$\pm$1) and Kelu-1B (L3.5$\pm$1) (Liu & Leggett, 2005). Audard et al.
(2007) reported the detection of quiescent X-rays from Kelu-1AB with an X-ray
luminosity of 2.9${}^{+1.8}_{-1.3}$ $\times$ 1025 erg s-1 . The second
detection refers to an L0-L1 star (J0331-27) which was found (in archival XMM
data) to undergo a flare (De Luca et al., 2020): at the peak of the flare,
this star had an X-ray luminosity $L_{X}$ = 1029.8 erg s-1 in the energy range
0.5-2 keV. This luminosity is comparable to, or slightly larger than, the
maximum luminosity observed in X-ray flares on M dwarfs with spectral types
M6.5-M9.5. De Luca et al. searched for quiescent X-rays from J0331-27 in XMM
and $Chandra$ data but found no significant detections of the source.

Prior to the detection by De Luca et al., Cook et al. (2014) had compiled the
largest catalog of L dwarfs with X-ray measurements: the catalog contains 10 L
dwarfs with spectral types in the range L0-L8. Of these 10 stars, one had an
X-ray detection, with log $L_{X}$/$L_{bol}$ = -4.7, while the remaining 9 had
only upper limits of X-rays in the range log $L_{X}$/$L_{bol}$ $<$ -5.0 to
-3.0. In addition to the L dwarfs reported by Cook et al. (2014), we have
independently searched for X-ray emission from another flaring L1 dwarf: WISEP
J190648.47+401106.8 using $Chandra$ data. No X-ray photons were unambiguously
detected from this L dwarf during an observation period of $\sim$50 ks (R. R.
Paudel et al., in prep.).

As regards the radio luminosity of the stars of interest to us, Berger et al.
(2010) report that L dwarfs exhibit a pronounced deviation from the Guedel-
Benz relation (GBR). The latter relation is based on observations of stars of
spectral types F, G, K, and early-M: for such stars, $L_{X}$ has been found to
be correlated linearly with the radio luminosity $L_{\nu,R}$ at frequency
$\nu$: log $L_{\nu,R}$/$L_{X}$ $\sim$ -15.5 (Guedel & Benz, 1993). However, at
a spectral type of M7-M9, the $L_{\nu,R}$/$L_{X}$ correlation is observed to
break down “sharply” (Berger et al. 2010, especially their Fig. 8). Though
there is observed to be a decline in X-ray emission as we go to later L
dwarfs, radio emission does not follow this decline. In some L dwarfs,
$L_{\nu,R}$/$L_{X}$ is found to be several orders of magnitude larger than
10-15.5 even when the stars are in the quiescent state (see also Williams et
al. 2014: their Fig. 6). A possible explanation for the breakdown in GBR in L
dwarfs has been proposed (Mullan, 2010) based on the physics of magnetic
reconnection. Two modes of reconnection are possible (slow and fast),
depending on the process which determines the electrical conductivity in the
ambient medium: if classical ohmic processes are dominant, reconnection occurs
at a relatively slow rate. But in conditions where the Hall effect dominates,
reconnection can occur orders of magnitude more rapidly. The latter case is
believed to be the case on solar-like coronal conditions. But in L dwarfs,
conditions may allow only slow reconnection to occur: in such cases, although
X-rays will not be efficiently generated, the electrons emerging from the
reconnection site can still have enough energy to be effective emitters of
electron cyclotron maser radio emission.

A compilation of the L dwarfs with radio emission can be found in Williams
(2018). The number of L dwarfs with known radio emission is currently 9, among
which 4 are confirmed to have radio emission that varies with periods of
$\lesssim$ 1 hr time scales. Radio emission has been observed on objects with
spectral type as late as T6.5 (Kao et al., 2016).

###  1.3 Previous studies: Flares on L dwarfs

L dwarfs exhibit lower rates of optical flares than M dwarfs do (see for e.g.
Paudel et al. 2018). As a result it is very difficult to study the flare rates
using ground-based telescopes with limited observation times. For example,
Koen (2013) and Ramsay et al. (2015) did not detect any flares on the L dwarfs
which they monitored using ground-based telescopes.The precise photometry and
continuous monitoring of the $K2$ mission make it relatively easy to study the
WLF rates of L dwarfs. Using this opportunity, Gizis et al. (2013) were able
to detect 21 white light flares on the L1 dwarf WISEP J190648.47+401106.8
(hereafter W1906+40) by monitoring this object for three months in short
cadence mode. The flares had (UV/visible/infrared) energies in the range
$\sim$(1029 \- 1032) erg. W1906+40 was the first L dwarf on which WLFs were
observed. Up to now, WLFs have been observed on 6 L dwarfs: W1906+40, SDSS
J053341.43 +001434.1 (hereafter S0533+00; L0), SDSSp J005406.55- 003101.8
(L1), 2M1221+0257 (L0), 2M1232-0951 (L0) and ULAS J224940.13-011236.9
(hereafter U22-011; L2.5) (Gizis et al., 2013; Schmidt et al., 2016; Gizis et
al., 2017a; Paudel et al., 2018; Jackman et al., 2019). All of these L dwarfs,
except S0533+00 and U22-011, were monitored by the $Kepler$/$K2$ mission.
Among all the WLFs observed on L dwarfs, the most powerful is the one observed
on S0533+00 by the ground-based All Sky Automated Survey for SuperNovae
(ASASSN). This flare had a total estimated bolometric energy of $>$6.2
$\times$ 1034 erg (Schmidt et al., 2016). Furthermore, U22-011 is the coolest
flaring object detected so far: with spectral type L2.5, this means that it
has $T_{\rm eff}$ $<$ 2000 K. It was observed by using the Next Generation
Transit Survey (NGTS) and the flare had a total estimated energy of 3.4
$\times$ 1033 erg (Jackman et al., 2019).

The noise level in the $K2$ long cadence data is $>$10% for the fainter
targets. As a result, small flares cannot be reliably detected. However, the
flares with large amplitudes (relative to the photospheric level) can be
easily detected even if the object is very faint and has very low photospheric
emission. One such large flare with total (UV/visible/infrared) energy
$\sim$1033 erg was observed on the L1 dwarf SDSSp J005406.55-003101.8 by Gizis
et al. (2017a) using long cadence data.

###  1.4 Flares on different classes of stars

It is well known that the Sun can produce flares in “active regions” where the
local magnetic field is strong and stressed: the stressing comes about because
of the motions of the gas in the photosphere due to convection. When the field
stresses exceed a certain limit, the field relaxes to a lower energy state and
some of the magnetic energy that was previously stored in the active region is
converted rapidly into heat, bulk motion of gas, and energetic particles: this
is a flare. The most common type of flare star has spectral class M: such
stars have been observed to have strong magnetic fields (Morin et al., 2010).
M stars also have extensive convective envelopes which can create stresses in
the magnetic field. For this reason, it is believed that flares on M dwarfs
also rely on the conversion of magnetic energy into heat, motion, and
particles (e.g. Mullan 1989). Our interest in this paper is stars which are of
spectral type L. The question is: does the internal structure of a star have
any fundamental difference on the properties of the flares which are produced
by different stars?

To address this, we first consider the internal structure of stars of interest
to us. The Sun has a radiative core with an outer envelope where convection
transports energy. The energy owes its origin ultimately to proton burning in
the core. Early-M dwarfs have internal structures such that proton burning
occurs in a radiative core, and there is an extensive convective envelope:
such a structure overlaps with the properties of the Sun. Stars with masses
which are less than (roughly) 0.3-0.35 $M_{\odot}$ are completely convective
on the main sequence (e.g. Mullan et al. 2015), but proton burning continues
to occur in such stars. However, for stars with masses $\lesssim$0.075
$M_{\odot}$ (corresponding to late-M on the main sequence), proton burning is
no longer possible. They are also known as brown dwarfs: the only nuclear
reactions in such objects (at least in their young stages) may involve
deuterium burning. The early-L dwarfs are a mixture of both evolved old stars
and young brown dwarfs. The internal structure of a star undoubtedly has an
effect on the dynamo processes which can occur in that star, i.e. on the
properties of the magnetic fields which can be generated (e.g. Mullan et al.
2015). In fact, observational properties of magnetic fields in M dwarfs show a
range of properties (Morin et al., 2010): some stars have strong fields which
are mainly dipolar, some have weak fields which are non-axisymmetric, and
others have toroidal fields. As regards the strength of the fields on M
dwarfs, the strongest surface fields are several kG (Kochukhov & Lavail, 2017;
Shulyak et al., 2017): these upper limits on the field strength at the surface
of a star may be constrained by equipartition with the photospheric gas
pressure (Saar 1996, especially his Fig. 3). Although there have been no
reports so far about magnetic detections in L dwarfs, it would not be
surprising to find that they also may contain a variety of magnetic
topologies, but with field strengths which are also no larger than several kG.

Turning now to a consideration of flares, although a star almost certainly
generates its field in a medium where magnetic pressure is overwhelmed by gas
pressure and/or gas kinetic energy, the processes of converting magnetic
energy into non-magnetic forms almost certainly rely on plasma processes which
occur in a medium where the magnetic field dominates the gas. Such a condition
typically requires a medium of low density and high temperature, i.e. a
stellar corona. In a region where magnetic flux tubes of opposite polarity can
be brought into close proximity, reconnection of the field is an effective
method of converting magnetic energy into the forms of energy which appear
during a flare (e.g. Petschek 1964). The process of reconnection releases
energy on a time-scale which is determined by the local physical conditions:
specifically, the thickness of the current sheet which exists when a plasma is
squeezed between two approaching magnetic flux tubes, and the electrical
conductivity of the plasma. If ohmic conductivity is at work, the rate of
energy release is rather slow. However, in certain conditions, e.g. if Hall
conductivity is relevant (i.e. ions decouple from the field lines but
electrons do not), the rate of reconnection can increase by factors of order
106 (Cassak et al., 2006). The ratio between the thickness of a current sheet
and certain basic length scales of the flare plasma helps to bring order to
the properties of a large sample of flares in the Sun and in flare stars
(Cassak et al., 2008).

In view of the fact that reconnection is controlled by local properties in the
corona, the flare process may be disconnected from the processes which
generate the field within the star. To the extent that this is true, we expect
that the properties of flares in stars of different internal structure need
not be all that different. In this regard, De Luca et al. (2020) have noted
the following aspects of the X-ray flare which they discovered recently on an
L dwarf: (i) the peak luminosity is similar to those observed on late-M
dwarfs; (ii) the short decay time-scale of the flare, indicating that the
flaring region on the star has a compact size, is consistent with flares on
late-type M dwarfs; (iii) the X-rays are emitted by a plasma which has a
temperature of $\sim$16 MK, within the range of values reported for late-M
dwarf flares. De Luca et al. conclude: “our observation shows that no
qualitative change takes place in the properties of X-ray flares at the bottom
of the main sequence down to $T_{\rm eff}$ $\sim$2100 K”. In fact, the flare
temperature reported by De Luca et al. for the L-dwarf flare (with a total
energy of 1033.3 ergs in X-rays) is only slightly higher than the most likely
temperature (12-13 MK) which occurs in a sample of 4500 smaller flares (with
total energies of 1029-31 ergs) on the Sun (Baylor et al., 2011).

These results suggest that flares in a range of stars from spectral type G to
L share a number of properties, despite the difference in their internal
structures.

###  1.5 Plan of the present paper

In this paper, we use photometry obtained by the $K2$ mission to study the
light curves of 45 L dwarfs. Only the L dwarfs which can be distinguished from
background in their target pixel level data are included in our sample. This
is the largest sample of L dwarfs for which the flare rates have been studied.
Our study will lead us to report for the first time the detection of flares on
an L5 dwarf which produced two superflares during $K2$ Campaign 11. One of
these flares is found to have the largest amplitude relative to the
photospheric level and also the largest energy among all the L dwarf flares
monitored by the $Kepler$/$K2$ mission.

###  1.6 Sample of L dwarfs observed by $K2$

We analyzed the light curves of 45 L dwarfs for which we obtained $K2$
photometry despite being very faint objects. In Table 2, we list the total
observation times of L dwarfs in each $K2$ campaign. The total observation
time of all L dwarfs in the sample is 4402.0 days and this is equal to 12.1
years.

In Table LABEL:table:list_of_L_dwarfs, we list the L dwarfs which were
observed in various $K2$ campaigns with good photometry. The first column is
the EPIC ID of each object and the second column is the 2MASS name of each
object except for EPIC 236324763. Likewise, the third and fourth column give
the information regarding the $K2$ campaign in which each object was observed
and its optical spectral type respectively. One of the L dwarfs does not have
an optical spectral type, so its near-infrared (NIR) spectral type is listed
instead. The fifth column is the reference for the spectral type and the sixth
column indicates whether the object was observed in short cadence mode or not
111Here, we don’t report any flares observed in short cadence mode only. We
report only those which were observed in both long and short cadence. The
results of short cadence data are either published in Paudel et al. (2018) or
will be published in Paudel et al. 2020, in prep.. The total number of L
dwarfs observed in short cadence mode is eleven. Figure 1 shows the
distribution of L dwarfs in each spectral type.

Figure 1: Distribution of L dwarfs by spectral type. Table 1: Total observation time by campaign Campaign # | # of L dwarfs | Time of Obs.
---|---|---
|  | (d)
2 | 1 | 77.2
3 | 0 | 0
4 | 2 | 144.5
5 | 16 | 1196.8
6 | 2 | 160.9
7 | 1 | 81.4
8 | 1 | 80.3
9 | 0 | 0
10 | 5 | 281.0
11 | 1 | 71.0
12 | 2 | 161
13 | 0 | 0
14 | 6 | 487.5
15 | 2 | 179.5
16 | 7 | 567.7
17 | 3 | 201.3
18 | 14 | 711.6
Total |  | 4402.0
Table 2: Total observation time by campaign

EPIC | 2MASS/other name | Cam. # | textbfSp. Type | ref | Short Cadence
---|---|---|---|---|---
204341806 | J16073799-2242468 | 2 | L0 | 1 |
210522262 | J04070752+1546457 | 4 | L3.5 | 2 |
210879793 | J04090950+2104393 | 4 | L3 | 3 |
211962038 | J08264262+1939224 | 5, 18 | L0 | 8 | Yes
211970944 | J08175266+1947279 | 5, 18 | L0 | 2 |
212102189 | J08302724+2203456 | 5, 16, 18 | L0 | 2 |
211328277 | J08433323+1024470 | 5, 18 | L1 | 2 | Yes
211357895 | J08503593+1057156 | 5, 18 | L6 | 4 |
211628806 | J0829066+145622 | 5, 18 | L2 | 3 | Yes
211680042 | J08312221+1538511 | 5, 18 | L1 | 2 |
211727819 | J08373282+1617380 | 5,18 | L0 | 2 |
211891128 | J08365239+1835455 | 5, 18 | L0 | 2 |
211963497 | J09094822+1940428 | 5 | L1 | 2 | Yes
211978512 | J09053102+1954334 | 5 | L0 | 2 |
212111554 | J08580549+2214582 | 5, 16, 18 | L1 | 2 |
212128548 | J08564793+2235182 | 5,18 | L3 | 5 |
229227169 | J13530778-0857119 | 6 | L0 | 1 |
217976219 | J19090821-1937479 | 7 | L1 | 1 | Yes
220186653 | J00540655-0031018 | 8 | L1 | 6 |
201181297 | 12130336-0432437 | 10 | L5 | 6 |
201299167 | J12025263-0227483 | 10 | L1 | 2 |
201482905 | J12035812+0015500 | 10 | L3 | 7 |
201658777 | J12212770+0257198 | 10 | L0 | 6 | Yes
228730045 | J12321827-0951502 | 10 | L0 | 8 | Yes
236324763 | VVV BD001 | 11 | L5 p | 12 |
246080803 | J2344062-073328 | 12 | L4.5 | 9 |
246303486 | J23255604-0259508 | 12 | L3 | 10 |
201528766 | J10501247+0058032 | 14 | L0 | 2 |
248442470 | J10484281+0111580 | 14 | L1 | 6 | Yes
248523311 | J10340564+0350164 | 14 | L0 | 2 |
248653486 | J10431944+0712326 | 14 | L0 | 2 |
248862470 | J10433508+1213149 | 14 | L8 (NIR) | 10 |
248891072 | J10345117+1258407 | 14 | L0 | 2 |
249343675 | J15230657-2347526 | 15 | L0 | 14 | Yes
249914869 | J1507476-162738 | 15 | L5 | 3 | Yes
211467731 | J08560211+1240150 | 16 | L0 | 2 |
211854467 | J08585891+1804463 | 16 | L2 | 2 |
212119590 | J08354537+2224310 | 16 | L0 | 2 |
212127137 | J08535917+2233363 | 5,16, 18 | L2 | 2 |
251355936 | J0918382+213406 | 16 | L2.5 | 4 |
229227143 | J13530778-0857119 | 6, 17 | L0 | 1 |
251551345 | J13433872-0220446 | 17 | L1 | 2 |
251555071 | J13334540-0215599 | 17 | L3 | 2 | Yes
211981633 | J08375977+1957279 | 5, 18 | L0 | 2 |
211602578 | J08403612+1434247 | 5, 18 | L1 | 2 |

References: 1) Best et al. (2018); 2) Schmidt et al. (2010); 3) Kirkpatrick et
al. (2000) 4) Kirkpatrick et al. (1999); 5) Cruz et al. (2003); 6) Schmidt et
al. (2015); 7) Schmidt et al. (2007); 8) Bardalez Gagliuffi et al. (2014); 9)
Kirkpatrick et al. (2008); 10) Burgasser et al. (2010); 11) Mace (2014); 12)
Beamín et al. (2013); 13) Griffith et al. (2012); 14) Koen et al. (2017)

##  2 Data reduction and computations

###  2.1 $K2$ photometry

We used the target pixel files (TPFs) of the objects and performed Point
Spread Function (PSF) photometry to extract the corresponding light curves.
The TPFs were obtained from the MAST archive. The PSF photometry is also known
as Point Response Function (PRF). Under this method, a parameterized model
developed by using the response of the pixels to stars of different brightness
is fitted to the data. The PSF method is preferred over aperture photometry
(AP) method for extracting the light curves of objects which lie in crowded
regions of the sky (Vinícius et al., 2018). We used the PSF fitting model
incorporated in the Python package ‘Lightkurve’ (Vinícius et al., 2018). The
light curves were then detrended against any systematics (such as the
instrumental noise) by using the K2 Systematics Correction (‘$K2SC$’, Aigrain
et al. 2016).

###  2.2 Flare detection

We used the false discovery rate (FDR) analysis described in Miller et al.
(2001) to identify possible flare candidates in the $K2SC$ detrended light
curve. The same method was used by Osten et al. (2012) to identify flares in
the light curves of their targets. For FDR analysis, we calculated the
relative flux $F_{\rm rel,i}$ for each data point of the detrended light curve
as follows:

$F_{rel,i}=\frac{F_{i}-F_{m}}{F_{m}}$ (1)

where Fi is the flux in ith epoch and $F_{\rm m}$ is the mean flux of the
entire light curve of each target. Using the values of $F_{rel,i}$, we then
calculated a statistic $\phi_{ij}$ for each consecutive observation epoch
($i,j$) as:

$\phi_{ij}=\Big{(}\frac{F_{rel,i}}{\sigma_{i}}\Big{)}\times\Big{(}\frac{F_{rel,j}}{\sigma_{j}}\Big{)},j=i+1$
(2)

Here $\sigma_{i}$ is the error in the flux which is associated with the ith
epoch. The possible flare candidates were first identified by estimating a
critical threshold value of $\phi_{ij}$ as described in Miller et al. (2001).
The detailed information regarding this method of identifying flare candidates
is given in Osten et al. (2012) and Paudel et al. (2018). The final flares
were confirmed by inspecting each candidate by eye. Among the sample of 45 L
dwarfs, 9 were observed to flare.

###  2.3 Flare energy estimation

To calculate the energies of our flares, we first measured the equivalent
duration (ED) of each flare using the $K2$ light curve. ED has units of time
and is the duration in which the flare produces the same amount of energy as
the (sub)stellar object when it is in its quiescent state (Gershberg, 1972).
We modeled each flare as a 10,000 K blackbody normalized to have the same
count rate through the $K2$ response curve as the photosphere of the flaring L
dwarf and estimated the energy of the flare emitted during an ED of 1s. This
energy is estimated for UV/optical/IR wavelengths. We then multiplied this
energy with the ED of each flare to get an estimate of the total energy
emitted by the flare. A similar procedure for flare energy estimation is
described in Gizis et al. (2017a, b). The values of flare energy for an ED of
1s for each flaring L dwarf are given in Table 3.

Table 3: Flare energy for ED = 1 s for each flaring L dwarf EPIC | energy
---|---
| (erg)
211891128 | 4.6 $\times$ 1028
220186653 | 5.3 $\times$ 1028
201658777 | 7.4$\times$ 1028
228730045 | 8.6$\times$ 1028
211854467 | 5.8$\times$ 1028
236324763 | 6.5$\times$ 1028
201528766 | 2.9$\times$ 1028
249343675 | 9.8$\times$ 1028
212102189 | 8.7$\times$ 1028

###  2.4 Flare on an L2 dwarf: 2MASS J08585891+1804463

2MASS J08585891+1804463 (hereafter 2M0858+1804) is an L2 dwarf (Schmidt et
al., 2010) and was observed by the $K2$ in Campaign 16 for 70.7 d. It has a
photospheric (continuum) level of 71 counts s-1 in the $K2$ light curve. A
flare was observed on 2M0858+1804 at $Kepler$ time 3286.7039 during which the
star brightened by a factor of $\sim$3 relative to the local photospheric
level measured 10 hours before the flare. 222There was a ‘thruster firing’ of
the spacecraft right before the peak flare time, but the centroid of the
target remained in the same pixel with no significant change in the brightness
level during that cadence. The flare has an ED of 2.0 hr and lasted for 0.20
d. It has a total estimated energy of 4.2 $\times$ 1032 erg. The flare light
curve is shown in Figure 2. The object 2M0858+1804 is the second L2 dwarf
known to produce a WLF.

Figure 2: The flare observed on the L2 dwarf 2MASS J08585891+1804463.

###  2.5 Flares on an L5 dwarf: VVV BD001

VVV BD001 (VVV J172640.2-273803, hereafter V01) is an L5 brown dwarf
discovered by Beamín et al. (2013) using the Vista Variables in the Vía Láctea
(VVV) survey (Minniti et al., 2010; Saito et al., 2012). It is an unusually
blue L dwarf (UBLD) that is located at a distance of 18.5$\pm$0.14 pc, towards
the Galactic bulge, and is a high proper motion object with $\mu_{\alpha}$ =
-544.5$\pm$0.6 mas yr-1 and $\mu_{\delta}$ = -326.4$\pm$0.4 mas yr-1. This
corresponds to a tangential velocity $V_{\rm tan}$ of 56.0$\pm$0.4 km s-1. We
note that, in a sample of 20 “known blue L dwarfs” which have unusually blue
$J$ \- $K_{s}$ colors, Kirkpatrick et al. (2010) listed values of $V_{\rm
tan}$ with a median value of 58$\pm$38 km s-1. Thus, in terms of $V_{\rm
tan}$, V01 does not appear at all exceptional when compared kinematically with
the known blue L dwarfs. Previously, a sub-sample of 10 UBLD’s in the sample
of Kirkpatrick et al. had been analyzed by Faherty et al. (2009) and had been
found to have a median $V_{\rm tan}$ of 99$\pm$47 km s-1: Kirkpatrick et al.
noted that although the median $V_{\rm tan}$ = 58 km s-1 of their expanded
UBLD sample is “somewhat lower” than the median value reported by Faherty et
al., still the result agrees with that of Faherty et al. within the 1$\sigma$
error bar. However, in a sample of 33 field L dwarfs, Vrba et al. (2004)
reported a median value of $V_{tan}$ = 24.5 km s-1 : this indicates that V01
(with $V_{\rm tan}$ = 56 km s-1) clearly has a larger tangential velocity than
the field L dwarfs. It therefore seems likely that V01 belongs to an older
population than the field L dwarfs.

V01 is the first brown dwarf to be discovered in the very crowded region of
sky where it is located. V01 was observed in long cadence mode by the $K2$
mission during Campaign 11 for $\sim$71 days. The median count rate of V01 in
the $K2$ light curve is 122 counts s-1. The photometric and physical
properties of V01 are summarized in Table 4.

Two superflares were detected by $K2$ on V01. The flare light curves are shown
in Figure 3. Following the detection of flares on the L2.5 dwarf U22-011 (in
2019), V01 is now the coolest object known to undergo flares. The larger of
the two superflares occurred at $Kepler$ mission time 2845.4432 during which
V01 brightened by more than 300 times above its photospheric level. The count
rate at the peak of this flare is 41874 counts s-1. This flare had an ED of
198 hr and lasted for 0.55 d. It is estimated that a 10,000 K flare with an ED
of 1s has an (UV/optical/IR) energy of 6.5 $\times$ 1028 erg. Using this
estimate, the total (UV/optical/IR) energy emitted during the largest flare is
estimated to be 4.6 $\times$ 1034 erg.

The weaker of the two superflares on V01 occurred at $Kepler$ mission time
2827.7696 during which V01 brightened by a factor of 15 times the photospheric
level. The count rate at the peak of this flare is 1832 counts s-1. It has an
ED of 8.0 hr and lasted for 0.06 d. The total estimated (UV/optical/IR) energy
of this flare is 2.0 $\times$ 1033 erg.

The fact that superflares have been detected on an unusually blue L5 star
raises a question: is there anything unique about V01 compared to other stars
in the UBLD class? Burgasser et al. (2008) suggested that UBLD’s differ from
“normal field L dwarfs” in the sense that the condensate clouds in the
photosphere of a UBLD may be either thinner, patchier, or contain larger
grains than the clouds in a normal L dwarf of the same effective temperature.
Burgasser et al. concluded that the UBLD’s may be “relatively old”, and they
suggested that the cloud properties might possibly be influenced by “magnetic
field strengths”. Magnetic fields are almost certainly at work in generating
superflares on V01, but if fields are present, they might have other physical
consequences as well, perhaps in determining the properties of the grains
which are permitted to condense. As regards the term “relatively old” used by
Burgasser et al. to describe UBLD’s, we note that Kirkpatrick et al. (2010)
also suggested, based on kinematics, that “many of the blue L dwarfs may be
old, though not as old as the halo population”. This still leaves a wide range
of possible ages, since the inner halo of our galaxy has an age of 11.4 Gyr
(Kalirai, 2012). The fact that UBLD’s may have ages of several Gyr is not
necessarily an argument against the possibility of magnetic activity: West et
al. (2008) have shown that the activity lifetimes of stars of spectral types
M0-M7 increases towards later types, reaching as old as 8 Gyr (or so) at type
M7. If the trend of multi-Gyr activity lifetimes continues beyond M7, we
expect that the L dwarfs could remain active even if they have ages of several
Gyr. In this regard, it is worth noting that the first L dwarf to be detected
as the site of an X-ray flare (J0331-27) has IR colors which indicate “that it
is likely an older source” with an age which may be $\geq$1 Gyr (De Luca et
al., 2020). Although J0331-27 was not present in the list of UBLD’s when
Kirkpatrick et al. compiled their list in 2010, De Luca et al. (2020) note
that J0331-27 is “bluer than field L1 dwarfs”: specifically, its $J$-$K_{s}$
color is 1.10$\pm$0.18, and this overlaps comfortably with the range of colors
listed by Kirkpatrick et al. (2010) for UBLD’s.

Thus, we now know of two UBLD’s which have been detected as flaring objects.
In this regard, we can answer the question posed at the start of the previous
paragraph in the negative: V01 is $not$ necessarily unique among UBLD’s.

The energies of the two flares in V01, i.e. 2.0 $\times$ 1033 ergs and 4.6
$\times$ 1034 ergs, are noteworthy in the context of the habitability of
possible planets orbiting such stars. In this regard, we note that Loyd et al.
(2018) have discussed the photolysis effects of the UV radiation from a flare
with total energy of 4 $\times$ 1033 ergs, i.e. within the range of the
energies of our two flares on V01. Loyd et al. conclude that the accumulated
erosion of the atmosphere of an exoplanet in the habitable zone (HZ) of an M
dwarf by flares with such energies “could be significant over timescales of
hundreds of megayears”. To be sure, the discussion of Loyd et al. applies to
stars with somewhat earlier spectral types (early M), but their results may
have applicability also to late L stars, where the HZ lies even closer to the
parent star. Moreover, another effect of large stellar flares is the
generation of energetic proton fluxes, which may also lead to significant
effects on the atmosphere of an exoplanet: Youngblood et al. (2017) have
suggested that energetic protons from M dwarf flares may lead to “complete
stripping of ozone from an Earth-like planet on timescales between 102 and 105
years”. Therefore, if the activity which we have discovered in V01 is typical
of late-L dwarfs, the habitability of exoplanets orbiting late-L dwarfs may
have to be called into question.

Table 4: Properties of VVV BD001

| Value | Units | Ref.
---|---|---|---
PHOTOMETRIC PROPERTIES |  |  |
Sp. Type | L5$\pm 1$ |  | 1
J | 13.27$\pm$0.02 | mag | 1
H | 12.67$\pm$0.02 | mag | 1
Ks | 12.20$\pm$0.02 | mag | 1
i | 17.90$\pm$0.02 | mag | 2
G | 18.23$\pm$0.00 | mag | 3
Kp | 14.73 | mag | 4
PHYSICAL PARAMETERS |  |  |
$\alpha$ | 261.7a ($\pm$0.3 mas) | deg | 3
$\delta$ | -27.6a ($\pm$0.2 mas) | deg | 3
parallax | 54.0 $\pm$ 0.4 | mas | 3
$\mu_{\alpha}$ | -544.5 $\pm$ 0.6 | mas yr-1 | 3
$\mu_{\delta}$ | -326.4 $\pm$0.4 | mas yr-1 | 3

aepoch J2015.5
References:
(1) Beamín et al. (2013); (2) Chambers et al. (2016);
(3) Gaia Collaboration et al. (2018a); (4) Huber et al. (2017)

Figure 3: Flares observed on V01. The time along the $X$-axis is centered at
peak flare time mentioned above each plot. The flux along the $Y$-axis is
normalized by the median flux which corresponds to photospheric level.

##  3 Results

We have identified a total of 11 flares on 45 L dwarfs observed in $K2$ long
cadence mode. Most of the flares were identified on the early L dwarfs. Those
flares have EDs in the range (1.3 - 198) hr and total estimated
(UV/optical/IR) energies in the range (0.9 - 460) $\times$ 1032 erg. More
information regarding the ED, peak flare time, flare energy and flare duration
of each flare is given in Table 5 333Flare energies of the targets which were
observed in both short and long cadence modes are reported by using the EDs
obtained from short cadence data to ensure more accuracy.. In Table 6, we list
the 2MASS, PS1, $Gaia$ properties of the flaring L dwarfs. The EW of H$\alpha$
emission is also listed whenever available.

We identified a strong flare on an L2 dwarf (2M0858+1804) and two superflares
on an L5 dwarf (V01). So far, 2M0858+1804 is the second L2 dwarf on which a
WLF has been observed. Likewise, V01 is the first L5 dwarf on which WLFs have
been reported. Hence, V01 is currently the coolest substellar object that is
known to produce WLFs. More interestingly, two superflares were identified on
the L5 dwarf in a single $K2$ campaign in an interval of $\sim$18 days. The
larger of these two superflares had an amplitude of $\sim$340 relative to the
photospheric level and an ED of 198 hr. An amplitude of $\sim$340 times
photospheric level is the largest amplitude among all the flares observed on
the targets we have studied using $K2$ data.

Assuming a similarity between the physical processes in solar flares and those
in flares on (sub)stellar objects, a rough estimation of the maximum magnetic
field strength $B$ associated with the flares can be obtained by using scaling
relations between flare energy and magnetic field which have been reported by
several authors: Aulanier et al. (2013), Shibata et al. (2013), and Notsu et
al. (2019). We consider these three in turn.

First, as regards flares on the Sun, Aulanier et al. (2013) make the following
assumptions: the flare occurs when a certain fraction of the total available
magnetic energy is released (e.g. as a result of reconnection) throughout an
active region. The active region is assumed to have a characteristic linear
dimension of $L_{bi}$ associated with the separation between bipolar spots of
dominant positive and negative polarity. The magnetic field strength in each
spot region on the solar surface is assumed to have a maximum value of
$B_{m}$. In the simplest case, the portion of the field which is reconnected
in order to power the flare is assumed to have a field strength which is a
fraction $f^{\prime}$ of $B_{m}$ throughout the flaring volume $V_{f}$ in the
atmosphere. Also in the simplest case, the volume $V_{f}$ is assumed to equal
some fraction $f^{\prime\prime}$ of the volume of a cube with sides of length
$L_{bi}$. With these assumptions, the magnetic energy $M_{E}$ released per cm3
is $M_{E}$ = $f^{\prime 2}$ $B_{m}^{2}$/8$\pi$ ergs cm-3 , while the magnetic
energy released in the flare as a whole is $E_{f}$ = $M_{E}$ $\times$ $V_{f}$
= 0.04 $f^{\prime 2}$ $f^{\prime\prime}$ $B_{m}^{2}$ $L_{bi}^{3}$ ergs. A
particular numerical example is provided by Aulanier et al. They find that, by
inserting values which may be (more or less) appropriate for conditions in the
Sun, i.e. $B_{m}$ = 103 G and $L_{bi}$ = 5 $\times$ 109 cm, the flare energy
is predicted (based on a model calculation which Aulanier et al. performed) to
be 5 $\times$ 1031 ergs. This is consistent with our expression for $E_{f}$
given above provided that the combination of terms 0.04 $f^{\prime 2}$
$f^{\prime\prime}$ has the numerical value of 4 $\times$ 10-4 , i.e.
$f^{\prime 2}$ $f^{\prime\prime}$ = 0.01. Provided that $f^{\prime}$ and
$f^{\prime\prime}$ do not differ by orders of magnitude, we can satisfy this
relation by assuming that $f^{\prime}$ and $f^{\prime\prime}$ are comparable,
in which case $f^{\prime}$ = $f^{\prime\prime}$ = 0.2. Thus, only about 20% of
the maximum available field strength needs to be reconnected, and only about
20% of the available volume needs to participate in the flare event. In other
words, we do not need to make extreme assumptions (e.g. by demanding that all
of the magnetic energy throughout the entire volume of the active region be
converted to flare energy) in order to account for the empirical properties of
solar flares.

Applying the scaling of Aulanier et al. (2013) to flare stars, we assume (in
the absence of any firm evidence to the contrary), that the numerical values
of the fractions $f^{\prime}$ and $f^{\prime\prime}$ can be assumed to be
essentially the same as in the Sun. In this context, the energies which are
expected to be released in flares on a flare star can be written in terms of
the star-spot magnetic field $B_{m}$ and the linear separation $L_{bi}$
between positive and negative spots on the stellar surface as follows: $E_{f}$
= 4 $\times$ 10-4 $B_{m}^{2}$ $L_{bi}^{3}$. According to the results in Table
5 above, the observed values of $E_{f}$ are mainly in the range 1032-34 ergs.
In order to set a lower limit on $B_{m}$, we consider the limiting case in
which the bipolar spots of an active region are placed as far apart as
possible on the surface of a star: this upper limit on $L_{bi}$ would occur if
the active region covered the entire surface of the star, with the positive
spot at one pole of the star, and the negative spot at the opposite pole. It
is physically impossible for the bipolar spots to be separated any farther
than this. In this limiting case, $L_{bi}$ $\approx$ $\pi$ $R_{\star}$ . With
brown dwarf radii somewhat larger than Jupiter’s radius, i.e. $R_{\star}$
$\approx$ 8 $\times$ $10^{9}$ cm, we find that, in the limiting case, we may
set $L_{bi}$ $\approx$ 2.5 $\times$ 1010 cm. Using this, in order to generate
flare energies in the range 1032-34 ergs (as reported in Table 5 of the
present paper), we find the following lower limits on $B_{m}$ on our stars:
$B_{m}$ $>$ 0.13$-$1.3 kiloGauss. In this case, because we have selected the
two bipoles to be situated at both poles of the star, the field $B_{m}$ is
truly to be considered as the global field of the star.

Second, Shibata et al. (2013) find that in a region of the Sun where the field
strength is $B_{\rm k}$ (in units of kG), and where a fraction $f$ of the
magnetic energy is released as flare energy, the amount of energy released in
a flare (in units of ergs) amounts to $E_{f}$ $\approx$ 7 $\times$ 1032
$B_{\rm k}^{2}$ ($f$/0.1) $L^{3}$. Here, the length $L$ is related to the area
$A_{\rm spot}$ (in units of cm2) of the sunspot in the active region where the
flare occurs according to $L$ = $\sqrt{A_{\rm spot}/10^{19}}$. The value of
$10^{19}$ cm2 which appears in this formula is the average area of a (large)
spot on the Sun capable of generating the largest X10 flares ($\sim$1032 erg):
such a spot occupies an area which is of order 3 $\times$ 10-4 times the area
of the visible hemisphere of the Sun. Applying the formula for $E_{f}$ to a
flare star, it is known that on such stars, spot sizes can occupy as much as
20% of the area of the visible hemisphere (e.g. Bopp & Evans 1973). Given that
the radius of a flare star $R_{\star}$ is of order 8 $\times$ 109 cm, $A_{\rm
spot}$ may be as large as 0.2 $\pi$ $R_{\star}^{2}$ $\approx$ 4 $\times$ 1019
cm2. This leads to $L$ = 2 in the above formula. Therefore, if $f$ retains a
value of order 0.1 in flare stars, we find that the formula of Shibata et al.
can be written as $E_{f}$ $\approx$ 5.6 $\times$ 1033 $B_{\rm k}^{2}$. In
order to replicate the flare energies listed in Table 5 of the present paper,
this formula suggests that we need to have field strengths $B_{\rm k}$ which
lie in the range 0.13-1.3 kG. These values are similar to the estimates we
have obtained from using the formula of Aulanier et al. (2013).

Third, Notsu et al. (2019) suggest that the formula for $E_{f}$ obtained by
Shibata et al. (2013) should be regarded as an upper limit. To test this, they
compare the Shibata et al. formula with the flares with maximum energy on each
star among a sample of 279 stars which were detected in $Kepler$ as flaring
objects. Visual inspection of their results (in their Figure 7) indicates that
the upper limit on field strength in stars with large spot areal coverage
($>$1% of the hemisphere) exceeds 3 kG in about 10% of the stellar sample.
However, when Notsu et al. (2019) limit their sample of stars for which $Gaia$
reported reliable distances in DR2 (Gaia Collaboration et al., 2018b), the
number of solar-like stars in their sample decreased to 106: for these stars,
only a few percent have upper limits as large as 3 kG. In most of the $Gaia$
stars, the mean value of the upper limit on surface field strength is of order
1 kG.

Other data sets are also beginning to generate data which are relevant to the
present discussion. E.g., $TESS$ data for Proxima Centauri (Vida et al., 2019)
indicate that with spot areal coverage of 5% or more, the observed flare
energies can be powered by fields in the range 1-2 kG. And data from EVRYSCOPE
(Howard et al., 2019) also indicate that flares on stars with spectral types
ranging from K7 to M4 have energies such that, with spot areal coverages of at
least 4-5%, the most energetic flares can be powered by magnetic fields with
strengths of 1.5 kG.

In summary, fields of order 1 kG appear to be adequate to provide the magnetic
energy which is needed to account for the upper limits of energies which are
released in stellar flares.

Is there any independent evidence that global fields of at least 0.13-1.3 kG
are present on the surface of low mass flare stars? In this regard, we may
cite magneto-convective models of low-mass flaring stars in which the magnetic
field impedes the onset of convection: in these models, global structural
changes occur in the star, including inflation of the stellar radius by
several percent (Mullan & MacDonald, 2001). In order to fit the empirical data
for any particular star, an appropriate value must be assigned to the vertical
magnetic field component which exists at the surface of the star.
Specifically, to fit the data for the flare stars UV Ceti and its companion
(with spectral types dM5.5 and dM6), it has been found that the global
vertical field on these two stars must be 1.85 and 2.25 kG (MacDonald et al.,
2018). And to fit the data for Trappist 1 (with spectral type M8), the models
indicate that the global vertical field must have values in the range
1.45-1.70 kG (Mullan et al., 2018). Thus, the interpretation we provided above
of the observed flare energies in our sample of L dwarfs in terms of fields
with strengths $B_{m}$ $\geq$ 0.13-1.3 kG is not inconsistent with the results
of magneto-convective modeling of low-mass flaring stars with spectral types
which reach almost as late as L0.

Table 5: Properties of flares on L dwarfs EPIC | Sp. Type | $T_{\rm peak}$ | ED | Energy | duration
---|---|---|---|---|---
|  | (BJD - 2454833) | (hr) | (erg) | (d)
211891128 | L0 | 2377.0389 | 11.4 | 2.0 $\times$ 1033 | 0.02
220186653 | L1 | 2595.7841 | 15.4 | 3.0 $\times$ 1033 | 0.2
201658777 | L0 | 2799.9338 | 0.30 | 8.8 $\times$ 1031 | 0.02
228730045 | L0 | 2811.7645 | 11.4 | 3.6 $\times$ 1033 | 0.18
228730045 | L0 | 2755.0599 | 1.3 | 3.9 $\times$ 1032 | 0.04
211854467 | L2 | 3286.7039 | 2.4 | 4.2 $\times$ 1032 | 0.06
236324763 | L5 | 2845.4432 | 198 | 4.6 $\times$ 1034 | 0.55
236324763 | L5 | 2827.7696 | 8.0 | 2.0 $\times$ 1033 | 0.06
201528766 | L0 | 3138.0292 | 19.7 | 2.0 $\times$ 1033 | 0.27
249343675 | L0 | 3238.3152 | 1.8 | 6.4 $\times$ 1032 | 0.18
212102189 | L0 | 3427.1071 | 3.5 | 1.1 $\times$ 1033 | 0.72
Table 6: Properties of flaring L dwarfs

EPIC | J | Ks | i | distance | H$\alpha$ EW
---|---|---|---|---|---
| (mag) | (mag) | (mag) | (pc) | (Å)
211891128 | 16.51$\pm$0.12 | 15.18$\pm$0.10 | 20.8$\pm$0.1 | 67.3$\pm$13.0a |
220186653 | 15.73$\pm$0.05 | 14.38$\pm$0.07 | 20.07$\pm$0.04 | 51.9$\pm$3.3b | 5.0e
201658777 | 13.17$\pm$0.02 | 11.95$\pm$0.03 | 17.44$\pm$0.01 | 18.5c | 6.1f
228730045 | 13.73$\pm$0.03 | 12.55$\pm$0.03 | 18.04$\pm$0.00 | 26.4$\pm$4.9d |
211854467 | 16.35$\pm$0.10 | 15.14$\pm$0.12 | 20.80$\pm$0.03 | 76.4$\pm$14.7a |
236324763 | 13.27$\pm$0.02 | 12.20$\pm$0.02 | 17.90$\pm$0.02 | 18.5$\pm$0.1b |
201528766 | 16.58$\pm$0.14 | 15.78 | 20.39$\pm$0.02 | 54.0$\pm$10.4a |
249343675 | 14.20$\pm$0.03 | 12.90$\pm$0.03 | 18.40$\pm$0.01 | 33.1$\pm$0.4b |
212102189 | 15.66$\pm$0.06 | 14.46$\pm$0.09 | 19.81$\pm$0.01 | 59.6$\pm$3.4b |

Notes:
i) $J$ and $K_{s}$ magnitudes are from 2MASS survey (Cutri et al., 2003).
ii) $i$ magnitudes are from Pan-STARRS survey (Chambers et al., 2016).
References:
aSchmidt et al. (2010); bGaia Collaboration et al. (2018b); cBailer-Jones et
al. (2018);
dPaudel et al. (2018); eGizis et al. (2017a); fSchmidt et al. (2015)

###  3.1 Flare frequency distribution of L dwarfs

In Figure 4, we plot the flare frequency distribution (FFD) of L dwarfs using
$K2$ long cadence data. This plot is useful in estimating the occurrence rate
of flares with certain energies. Here, the energies of all flares observed on
L0-L5 dwarfs are considered. In addition, the total time of observation (12.1
yrs) of all 45 L dwarfs is taken into account. The black dots represent the
observed flare energies and the red dashed line is a weighted least-squares
linear fit to power law distribution of flare energies. We used Poisson
uncertainties to determine the weights for least square fitting. In addition,
we did not include the minimum and maximum flare energies while fitting to
reduce any bias due to those energies. The kink seen around log $E$ $\sim$
33.3 erg is not real and is due to the closeness of the values of three flare
energies around this value. The slope of the fitted line is -(0.51$\pm$0.17)
and the cumulative frequency intercept at 1030 erg is log $\tilde{\nu}_{30}$ =
-1.42. Using these results, it is estimated that a superflare of energy 1033
erg occurs every 2.4 years and a superflare of energy 1034 erg occurs every
7.9 years on the L dwarfs.

The slope of -0.51 of the FFD for L dwarfs is shallower compared to the slopes
of FFD’s which have been observed in stars of earlier spectral types. For
example, Paudel et al. (2018), in their study of stars with spectral types
M6.5 - M8.5 reported FFD slopes which were on average equal to –(0.58 - 0.70)
regardless of stellar age. However, in the case of one of their L0 dwarfs,
they reported a shallower slope: -0.34. Mullan & Paudel (2018) have argued
that the shallowness of the slope for the FFD of an L0 star may be related to
the low values of electrical conductivity which are expected in the atmosphere
of such a cool star. This explanation could be helpful in understanding why
the FFD for the L dwarfs reported in the present paper is also shallower than
the averages reported for stars of earlier spectral types.

Figure 4: The FFD estimated by using the energies of all flares identified on
L dwarfs observed by $K2$ long cadence mode. The kink seen around log $E$
$\sim$ 33.3 is not real. The errors were estimated using Poisson
uncertainties.

## Acknowledgements

The material in this paper is based upon work supported by NASA under award
Nos. NNX15AV64G, NNX16AE55G and NNX16AJ22G. R.R.P acknowledges support from
the University of Delaware Dissertation Fellowship program. Some/all of the
data presented in this paper were obtained from the Mikulski Archive for Space
Telescopes (MAST). STScI is operated by the Association of Universities for
Research in Astronomy, Inc., under NASA contract NAS5-26555. This paper
includes data collected by the Kepler mission. Funding for the Kepler mission
is provided by the NASA Science Mission directorate. This work has made use of
data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This research has made use
of the VizieR catalogue access tool, CDS, Strasbourg, France. The original
description of the VizieR service was published in A&AS 143, 23. This work
made use of the http://gaia-kepler.fun crossmatch database created by Megan
Bedell.

Software: Python, IPython Perez & Granger (2007), Astropy (Astropy
Collaboration et al., 2013), Matplotlib (Hunter, 2007), Numpy (Oliphant,
2015), Lightkurve (Vinícius, Barentsen, Hedges, Gully-Santiago & Cody, 2018),
K2SC (Aigrain et al., 2016), Jupyter (Kluyver et al., 2016)

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