#  K2 Ultracool Dwarfs Survey. IV. Monster flares observed on the young brown
dwarf CFHT-BD-Tau 4.

Rishi R. Paudel, John E. Gizis, D. J. Mullan  Department of Physics and
Astronomy, University of Delaware, Newark, DE, 19716, USA  Sarah J. Schmidt
Leibniz-Institute for Astrophysics Potsdam (AIP), An der Sternwarte 16, 14482,
Potsdam, Germany  Adam J. Burgasser  Center for Astrophysics and Space
Science, University of California San Diego, La Jolla, CA 92093, USA  Peter K.
G. Williams  Harvard-Smithsonian Center for Astrophysics, 60 Garden Street,
Cambridge, MA 02138, USA  Edo Berger  Harvard-Smithsonian Center for
Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA

(Received XXXX; Revised XXXX; Accepted XXXX)

###### Abstract

We present photometric measurements of two superflares observed on a very
young brown dwarf CFHT-BD-Tau 4, observed during Campaign 13 of the Kepler K2
mission. The stronger of the two superflares brightened by a factor of
$\sim$48 relative to the quiescent photospheric level, with an increase in
Kepler magnitude $\Delta\tilde{K_{p}}$ = -4.20. It has an equivalent duration
of $\sim$107 hour, a flare duration of 1.7 day, and an estimated total
bolometric (ultraviolet/optical/infrared) energy up to 2.1 $\times$ 1038 erg.
The weaker of the two superflares is a complex (multipeaked) flare with an
estimated total bolometric (UV/optical/IR) energy up to 4.7 $\times$ 1036 erg.
They are the strongest flares observed on any brown dwarf so far. The flare
energies are strongly dependent on the value of visual extinction parameter
$A_{V}$ used for extinction correction. If we apply a solar flare-model to
interpret the two superflares, we find that the magnetic fields are required
to be stronger by as much as an order of magnitude than previous reports of
field measurements in CFHT-BD-Tau 4 by Reiners et al. (2009b). On the other
hand, if we interpret our data in terms of accretion, we find that the
requisite rate of accretion for the stronger superflare exceeds the rates
which have been reported for other young brown dwarfs.

brown dwarfs –stars: flare–stars: individual: CFHT-BD-Tau 4

††software: :

##  1 INTRODUCTION

Rapidly rotating young solar mass stars in the Orion Nebula Cluster are
capable of producing X-ray flares with energies in the range 1034-1036 erg
(Wolk et al., 2005). Similarly, low mass stars and young brown dwarfs
(hereafter BDs) in Orion Nebula region and Taurus molecular cloud also have
high X-ray emission. There is no significant difference in X-ray activity
levels of the low mass stars and the young BDs with similar spectral types
implying that X-ray activity levels are determined by effective temperatures
rather than masses and surace gravities of (sub)stellar objects (Preibisch et
al., 2005; Güdel et al., 2007; Grosso et al., 2007; Getman et al., 2008a, b).
Hence, young BDs can be magnetically active as the low mass stars, and produce
huge flares. The young stars and BDs may have accreting or non-accreting disks
which limit the X-ray flare loop sizes ($<$ Keplerian corotation radii)
(Getman et al., 2008b). In addition to the magnetic reconnection events which
occur in diskless stars, the interaction of disks of young stars with their
magnetospheres can also trigger large scale magnetic reconnection events which
manifest as flares (Zhu et al., 2009). Stelzer et al. (2000) report the
presence of an accretion disk makes no difference on X-ray flares on stars in
clusters with ages 1-3 Myr.

The continuous time coverage of Kepler K2 mission (Howell et al., 2014) over
months, has enabled us to study white light flare rates of several ultracool
dwarfs (hereafter UCDs) which have spectral types $\gtrsim$ M6 and various
ages (Gizis et al., 2017b, a; Paudel et al., 2018). Those flare rates will be
helpful to understand the evolution of magnetic dyanmo in UCDs. The results of
Gizis et al. (2017a) show that young brown dwarfs like 2MASS J03350208+2342356
(24 Myr old brown dwarf, hereafter 2M0335+2342) and CFHT-PL-17 (a brown dwarf
member of Pleiades) are capable of producing strong white light superflares
with energies $>$ 1033 erg. Superflares are thus ubiquitious in Kepler G, K, M
and L stars (Maehara et al., 2012; Shibayama et al., 2013; Notsu et al., 2013;
Candelaresi et al., 2014; Gizis et al., 2017a, b). The young brown dwarfs are
in the process of contraction and have radii $>$0.5$R_{\odot}$ if they are few
million years old. They are fully convective and have high luminosities. The
energy flux scaling law predicts that strong magnetic fields are produced in
young BDs and exoplanets (Reiners & Christensen, 2010; Christensen et al.,
2009). This is supported by detection of 5 kG magnetic field on a 22 Myr, M8.5
brown dwarf LSR J1835+3259 (Berdyugina et al., 2017). Reiners et al. (2009b)
however reported weak magnetic fields with strengths of few hundred gauss in
four young ($\lesssim$10 Myr) accreting BDs with $v\sin i$ $>$ 5 km s-1. The
weaker fields may be due to presence of disk around such objects or that they
do not follow scaling law (Reiners & Christensen, 2010).

Here, we present the photometric measurements of two superflares observed on a
very young brown dwarf CFHT-BD-Tau 4 (hereafter CT4). Discovered by Martín et
al. (2001), the presence of disk around CT4 makes it more valuable for studies
regarding planet formation around low mass stars and BDs (Ricci et al., 2014).
We present the photometric and physical properties of CT4 in Section 2. In
Section 3, we present the data reduction, flare photometry and flare energy
computation, and discuss the results in Section 4.

##  2 Target Characteristics

CFHT-BD-Tau 4 (2MASS J04394748+2601407) is a young M7 brown dwarf in the
Taurus star-forming region, with an estimated age $\sim$1 Myr old (Luhman et
al., 2017) and at a distance of 147.1$\pm$5.2 pc (Gaia Collaboration et al.,
2016, 2018). It has an effective temperature (Teff) equal to 2900 K and
bolometric luminosity ($L_{bol}$) equal to 0.03 $L_{\odot}$ (Reiners et al.,
2009a). It is a well studied BD in wavelengths ranging from X-rays to
millimeters (Martín et al., 2001; Liu et al., 2003; Pascucci et al., 2003;
Klein et al., 2003; Apai et al., 2004; Grosso et al., 2007). Using the
Submillimeter Common-User Bolometer Array (SCUBA) on James Clerk Maxwell
Telescope (JCMT) and the Max-Planck Millimeter Bolometer (MAMBO) array on the
IRAM 30 m telescope, Klein et al. (2003) reported the presence of
circumstellar cold dust around this young object. The spectral energy
distribution (SED) of this circumstellar dust fits a flat disk model better
than spherical dust distribution model and resembles to that of T Tauri disk
with a mass estimation of (0.3-1.5)MJ (Pascucci et al., 2003). The mid-
infrared observations done by using GEMINI/T-ReCS suggest the presence of
2$\mu$m silicon like grains is prominent in the disk (Apai et al., 2004). In
addition, Atacama Large Millimeter/submillimeter Array (ALMA) observations at
0.89 mm and 3.2 mm show that large grains of at least $\sim$ 1 mm are also
present in the outer disk region and the outer radius of disk is $>$80 AU
(Ricci et al., 2014).

Martín et al. (2001) detected a strong H$\alpha$ emission with an equivalent
width 340 Å, and Br$\gamma$ emission from CT4. Likewise, Grosso et al. (2007)
detected quiescent X-rays with luminosity equal to 24.3 $\times$ 1028 erg s-1
and X-ray activity, log(LX/Lbol) = -3 which is also the saturated X-ray
emission level of early M dwarfs. Hence the two magnetic activity indicators
H$\alpha$ emission and X-ray emission signify the presence of active
chromoshphere and corona in this substellar object. Jayawardhana et al. (2003)
classify CT4 as a non-accretor using the shape and width of H$\alpha$ emission
profile but Reiners et al. (2009b) mention that it may have magnetospheric
accretion. Different authors have reported different values of visual
extinction parameter $A_{V}$ for CT4. Martín et al. (2001) reported its value
equal to 3.0 mag in their discovery paper, using the I-J colors of field M
dwarfs and the interstellar extinction law of Rieke & Lebofsky (1985).
Likewise, Monin et al. (2010); Luhman et al. (2017); Alves de Oliveira et al.
(2012); Andrews et al. (2013) report its value equal to 2.6 mag, 5.0 mag, 5.4
mag and 5.67$\pm$0.89 mag respectively. Monin et al. (2010) do not specify
clearly how they estimated $A_{V}$ particularly for CT4. Alves de Oliveira et
al. (2012) estimated $A_{V}$ from the  J-H vs. H-Ks diagram. Luhman et al.
(2017) used SpeX spectrum of CT4 and estimated the value of $A_{V}$ by
comparing the spectral slopes at 1$\mu$m, of various young M dwarfs. Likewise,
Andrews et al. (2013) estimated $A_{V}$ by fitting the stellar photosphere
models to the optical and near-infrared spectral energy distribution (SED) of
the object. In addition to all the above values of $A_{V}$, Zhang et al.
(2018) estimated it to be 6.37$\pm$0.85 mag. Their estimation is based on the
intrinsic optical–near-infrared color as a function of spectral type. They
used spectral type of M7.2$\pm$0.9 for CT4, which was estimated by using their
own reddening-free spectral classification system. As this value of $A_{V}$ is
based on more precise photometry measured by Pan-STARRS1 3$\pi$ survey, we use
it for estimation of flare energies in this paper111Zhang et al. (2018) also
estimated the value of $A_{V}$ based on color-color diagrams using H2O indices
but suggest that the value of $A_{V}$ based on intrinsic optical–near-infrared
color sequences is more precise whenever the reddening-free spectral type of
the object is well-defined, and $\approx$M5-L2.. The photometric and physical
properties of CT4 are summarized in Table 1.

Table 1: Properties of CFHT-BD-Tau 4

Parameter | Value | Unit | Ref.
---|---|---|---
PHOTOMETRIC PARAMETERS |  |  |
Sp. Type | M7 |  | 1
V | 21.556$\pm$0.008 | mag | 2
J | 12.17$\pm$0.02 | mag | 3
H | 11.01$\pm$0.02 | mag | 3
Ks | 10.33$\pm$0.02 | mag | 3
r | 20.20$\pm$0.04 | mag | 4
i | 17.54$\pm$0.02 | mag | 4
z | 15.79$\pm$0.01 | mag | 4
G | 17.780$\pm$0.006 | mag | 5
PHYSICAL PARAMETERS |  |  |
Teff | 2900 | K | 6
M | 0.064222Ricci et al. (2014) mention the mass of CT4 to be 0.095$M_{\odot}$, based on the parameters derived by Andrews et al. (2013). | $M_{\odot}$ | 6
R | 0.65 | $R_{\odot}$ | 6
log L/L⊙ | -1.57 |  | 6
$v\sin i$ | 6${}^{+2}_{-4}$ | km s-1 | 6
Age |  $\sim$1 | Myr | 7
parallax | 6.80$\pm$0.24 | mas | 5
$\alpha$ | 069.94787355129b ($\pm$0.2 mas) | deg | 5
$\delta$ | +26.02787631047b ($\pm$0.1 mas) | deg | 5

bepoch J2015.5, ICRS
References:
(1) Martín et al. (2001); (2) Kraus et al. (2006);
(3) Skrutskie et al. (2006); (4) Chambers et al. (2016);
(5) Gaia Collaboration et al. (2018); (6) Reiners et al. (2009b);
(7) Luhman et al. (2017)

##  3 Data reduction and Computations

###  3.1 Kepler K2 photometry

CFHT-BD-Tau 4 was observed as EPIC 248029954 by Kepler K2 mission during
Campaign 13 (08 March, 2017 - 27 May, 2017) in long cadence ($\sim$30 minute)
mode (Jenkins et al., 2010). The total observation time is 80.52 day, and
there are 3651 good (Q = 0) data points. Average flux measured during 29.4
minute interval is obtained for each data point. We used astropy affiliated
photutils package to measure the photometry of flare from available Target
Pixel File (TPF), by following a similar process described in Gizis et al.
(2017a, b). To get the best estimate of the position of target in each pixel
image, we first estimated the median of centroids of all images. We then used
the information recorded as POS_CORR1 and POS_CORR2 in the FITS file headers
for each observation, to correct any off-positioning of the target centroid
due to spacecraft motion. Based on experience in our previous works Gizis et
al. (2017b, a), this method is preferable for UCDs.

We used Lund et al. (2015) relation: $\tilde{K_{p}}$ $\equiv$ 25.3 -
2.5log(count rate) to estimate the Kepler brightness magnitude $\tilde{K_{p}}$
which describes brightness level of the faint stars better than magnitude
$K_{p}$ provided in Kepler Input Catalog (KIC) (Gizis et al., 2017b, a). Here
flux is the count rate measured through a 3-pixel radius aperture. In case of
CFHT-BD-Tau 4, the median counts through 3-pixel radius aperture is less than
through 2-pixel radius aperture due to the negative counts in the outer pixels
surrounding the target pixel. The median count rate through 2-pixel radius
aperture is 478 count s-1 which corresponds to photoshperic continuum level.
This gives $\tilde{K_{p}}$ = 18.6. There is no significant difference in
$\tilde{K_{p}}$ calculated using either 2-pixel or 3-pixel radius aperture in
case of UCDs. We used 2-pixel radius aperture to measure the photometry of
flares discussed in this paper. The K2 light curve of CT4 is shown in Figure 1
in which we can see periodic nature of the curve. Using Lomb-Scargle
periodogram, we find that the period of this periodic feature is 2.98 days,
and is comparable to rotation period of 2.93$\pm^{0.9}_{2.4}$ day reported by
Scholz et al. (2018) for CT4. We compared the light curve obtained by using
aperture photometry with that obtained by using psf photometry and EVEREST
detrending code (Luger et al., 2016). We find no significant differences
between the light curves.

Figure 1: The K2 light curve of CT4. The two superflares are not shown in full
scale to focus the periodic nature of the curve. Using Lomb-Scargle
periodogram, the dominant period of the light curve is 2.98 day. The two
dashed vertical lines mark the peak flux times of the superflares.

###  3.2 Computation of Flare Energies

To compute energy of a flare, we first estimated its equivalent duration (ED).
It is the equivalent time during which the (sub)stellar object (in its
quiescent state) would have emited the same amount of energy as the flare
actually emitted (Gershberg, 1972). ED has units of time, and is area under
the flare light curve. It depends on the filter used but is independent of the
distance to the flaring object. However, it is affected by reddening. The
quiescent bolometric luminosity of flare is calculated by approximating the
flare to be a 10,000 K blackbody which produces the same count rate through
the Kepler response curve as CT4. We estimated the photospheric spectrum of
CT4 by using an active M7 template spectrum (Bochanski et al., 2007)
normalized to match the Pan-STARRS i-band photometry (Tonry et al., 2012;
Chambers et al., 2016; Magnier et al., 2016). As Kepler flux of CT4 is
affected by extinction, we first reddened the blackbody spectrum using astropy
application specutil.extinction for $A_{V}$ = 6.37, and $R_{V}$ = 3.1. Here,
the value of $R_{V}$(=$A_{V}/E(B-V))$ is the typical value used for Milky Way
Galaxy. We used the mean extinction law as mentioned in Cardelli et al.
(1989). Then, we computed the photospheric flux of the reddened 10,000 K
blackbody, which is normalized to have the same count rate through the Kepler
filter as the photosphere of CT4. The bolometric (UV/optical/IR) energy of CT4
flare having equivalent duration of 1 s is 4.9 $\times$ 1030 erg and 5.4
$\times$ 1032 erg corresponding to $A_{V}$ = 0.0 and 6.37 respectively. These
energies were used to estimate bolometric flare energy by multiplying with ED
of the flare. We adopted a distance of 147.1 pc to calculate the bolometric
luminosity. More details of flare energy calibration can be found in Gizis et
al. (2017b, a). Figure 2 shows optical spectral energy distribution (SED) of
CT4. For comparison, we also plot SEDs of 10,000 K and 6,500 K blackbody
flares that produce the same counts as CT4 through Kepler filter. The upper
plot shows the reddened SEDs, and the lower plot shows the dereddened SEDs for
two values of $A_{V}$: 6.37 and 3.0.

Figure 2: Optical spectral energy distribution (SED) of CT4 (black solid
curve). The three dashed lines represent the SEDs of hypothetical blackbody
flares of temperature 10,000 K, 6,500 K and 10,000 K which produce the same
counts through Kepler filter, corresponding to AV = 6.37, 6.37 and 3.0
respectively. The upper plot is the reddened version of SEDs, and the lower
plot is the dereddened version of SEDs.

###  3.3 Flare photometry

Figure 3: The strongest superflare observed on CFHT-BD-Tau 4. The blue dots
represent the observed data. The dashed vertical lines represent the start and
end times of flare. Figure 4: Next superflare observed on CFHT-BD-Tau 4. The
blue dots represent the observed data. The dashed vertical lines represent the
start and end times of flare.

The strongest superflare detected on CT4 is shown in Figure 3. At first, the
flux increased to 1753 count s-1 and 20006 count s-1 at Kepler mission day
2998.1243 and 2998.1447 respectively. It increased to peak value of 22788
count s-1($\tilde{K_{p}}$ = 14.4, $\Delta\tilde{K_{p}}$ = -4.20) on Kelper
mission day 2998.1651, during which the target brightened by $\sim$ 48 times
the quiescent photospheric level. Then, the flare continued to decay for over
the next several hours as seen in Figure 3. This flare has an equivalent
duration of $\sim$107 hour, and an estimated total bolometric (UV/optical/IR)
energy equal to 2.1 $\times$ 1038 erg, for $A_{V}$ = 6.37. The total flare
duration is 1.7 day. The rise time of the flare is longer than that of
superflares observed on some ultracool dwarfs. Further discussion is given in
Section 4.

We also detected another superflare on CT4, as shown in Figure 4. It is a
complex flare with two peaks. The first peak is at Kepler mission day
3058.4182, and the second peak is at Kepler mission day 3058.4999. The flux
counts at those times are 934 count s-1 and 665 count s-1 respectively. This
flare has an ED of $\sim$2.4 hour, and flare duration of 0.41 day. The
estimated total bolometric (UV/optical/IR) energy of this flare is 4.7
$\times$ 1036 erg, for $A_{V}$ = 6.37. The properties of both flares are
summarized in Table 2. For comparison, we also list the flare energies for
zero extinction i.e. $A_{V}$ = 0.0.

Table 2: Flare properties | ED (hour) |  $A_{V}$ = 0.0 |  $A_{V}$ = 6.37 | flare duration (day)
---|---|---|---|---
Flare 1 | 107 | 1.9 $\times$ 1036 erg | 2.1 $\times$ 1038 erg | 1.7
Flare 2 | 2.4 | 4.2 $\times$ 1034 erg | 4.7 $\times$ 1036 erg | 0.41

##  4 Discussion

We observed two superflares on CT4 using Kepler K2 Campaign 13 long cadence
data. We estimated total bolometric (UV/optical/IR) energies of those flares
to be 2.1 $\times$ 1038 erg and 4.7 $\times$ 1036 erg, for $A_{V}$ = 6.37. The
stronger of the two superflares has ED of $\sim$107 hour and the weaker has ED
of $\sim$2.4 hour. The energies of these flares are larger than the strongest
flares reported on other young brown dwarfs 2M0335+2342 and CFHT-PL-17 in
Gizis et al. (2017b). While the energy partition may not be same even for the
flares that occur on same star (Osten et al., 2016), we may get an approximate
estimation of soft X-ray (0.01-10 keV) energy (EX) radiated during the
stronger superflare by using the conversion factors: EX/Ebol = 0.3, and
EKp/Ebol = 0.16 (assuming 9,000 K blackbody temperature) for active stars,
listed in Table 2 of Osten & Wolk (2015). Here, Ebol is the total bolometric
flare radiated energy, and is related to the coronal radiated flare energy
Ecor and the optical flare energy Eopt, by Ebol = Ecor \+ Eopt. Likewise, EKp
is the Kepler flare energy. Our estimation of EKp for the stronger superflare
is 6.8 $\times$ 1037 erg (assuming 10,000 K blackbody temperature). Using this
value of EKp and the conversion factors mentioned above, we get EX $\approx$
1.3 $\times$ 1038 erg. The occurence of those two superflares during the
observed time of 78.31 days implies that CT4 is highly active which is also
supported by its high H$\alpha$ and X-ray emission and it is very young. We
observed only two flares on CT4 because it was observed in long cadence mode.
If we had short cadence ($\sim$1 min, Gilliland et al. 2010) data for CT4, it
would be possible to detect more weaker flares. Using the results of flare
frequency distribution of 24 Myr brown dwarf 2M0335+2342 over the range 1031
to 1033 erg as reported by Gizis et al. (2017a), the expected rate of flare
with energy equal to that of stronger superflare observed on CT4 is 0.004 per
year, much less than the observed results. It is interesting to note that the
energies of superflares observed on CT4 are comparable to energy of the most
powerful flare reported by Gahm (1990). He studied the energies of 13 flares
on seven T Tauri stars which were observed by using Str$\ddot{o}$mgren filters
and reported an upper limit of flare energy $\sim$7.0 $\times$ 1036 erg in
optical spectral region. His results are based on ground based observations
over several nights with as few as 3 observations per night.

We emphasize that the bolometric flare energies reported in this paper have
huge uncertainties for several reasons. The flare energies strongly depend on
the value of $A_{V}$, as can be seen in Table 2. The flares contribute more
flux in the shorter wavelengths in Kepler band pass, and the shorter
wavelengths are more affected by extinction in interstellar medium.
Consequently, fewer photons are recorded giving rise to shorter ED and less
flare energy. Furthermore, we have not taken into account the contribution of
atomic emission lines to total energy budget. The atomic emission lines and
blackbody components contribute in different proportions during the impulsive
and gradual phase of the flares (Hawley & Pettersen, 1991).

Gahm (1990) pointed out that the light curves of strong flares on T Tauri
stars are characterized by a slow rise and then a slow decline in brightness.
This is different than that is observed in flares of ordinary flare stars,
most of which are characterized by a rapid rise followed by a gradual decline.
While it would be more clear if we had short cadence data of CT4, the long
cadence data somehow suggests similar characteristics as pointed by Gahm
(1990). It took one hour for the strongest flare to reach the peak flux level.
The full width half maximum (FWHM) time scale of the flare is between 1.5 and
2.0 hours. This time scale is very different than that observed on typical
superflares on older, diskless targets. The FWHM time scale of strongest
flares on some UCDs is of the order of few minutes. For example, the two
superflares observed on the L0 dwarf 2MASS J12321827-0951502 and the M7 dwarf
2MASS J08352366+1029318 had very short FWHM time scales of order $\sim$2
minutes (Paudel et al., 2018). We should note that 2MASS J12321827-0951502 and
2MASS J08352366+1029318 were observed in short cadence ($\sim$1 min, Gilliland
et al. 2010) mode. It is not clear whether the presence of disk changes the
nature of flare light curves of young objects like CT4.

Assuming a similarity between the solar flares and the flares on BDs, we may
get a rough estimation of maximum magnetic field strength B${}^{max}_{z}$
associated with the stronger superflare observed on CT4 using the scaling
relation in Aulanier et al. (2013):

$E=0.5\times
10^{32}\Big{(}\frac{B^{max}_{z}}{1000G}\Big{)}^{2}\Big{(}\frac{L^{bipole}}{50Mm}\Big{)}^{3}erg.$
(1)

where E is the bolometric flare energy, and Lbipole is linear separation
between bipoles. If we take Lbipole to be equal to $\pi$$R$ as the maximum
distance between a pair of magnetic poles on the surface of CT4, a strong
magnetic field of 13.5 kG is required to produce the stronger superflare with
energy 2.1 $\times$ 1038 erg. In general, it is possible for fully convective
M dwarfs to have strong average magnetic fields with the highest observed
value to be 7.0 kG in the case of WX Ursae Majoris (Gliese 412 B) (Shulyak et
al., 2017). Likewise, a $\sim$22 Myr old M8.5 brown dwarf LSR J1835+3259 is
reported to have a strong magnetic field of strength 5 kG (Berdyugina et al.,
2017). However, the above value of magnetic field strength estimated in case
of CT4 using solar flare model is higher compared to results of Reiners et al.
(2009b) who reported weak magnetic fields (few hundred gauss) with an upper
limit $\sim$1 kG on four accreting brown dwarfs including CT4. Using
magnetospheric accretion model to young accreting brown dwarfs in combination
with observed data, Scholz & Jayawardhana (2006) and Stelzer et al. (2007)
also predicted weak magnetic fields of strength $\sim$kilogauss on the surface
of those young objects. The longer FWHM time scale of the superflare observed
on CT4 might be possible due to the reason that it occured as a result of
reconnection between the magnetic loops on CT4 and its disk. In such case,
even the weak magnetic fields might be capable of producing superflares with
huge energies because of the larger volumes associated with longer magnetic
loops.

It’s possible that an outbrust from magnetospheric accretion could mimic a
flare. Using magnetospheric model to a star-disk system, the rate of energy
released $L_{acc}$ due to accretion of gas onto the star can be calculated by
using

$L_{acc}=\frac{GM_{*}\dot{M}}{R_{*}}\Bigg{(}1-\frac{R_{*}}{R_{in}}\Bigg{)}$
(2)

where $M_{*}$ and $R_{*}$ are mass and radius of the star; $\dot{M}$ is the
accretion rate, and $R_{in}$ is the inner radius of accretion disk (Gullbring
et al., 1998). We do not have accurate estimation of inner disk radius and
accretion rate of CT4. Liu et al. (2003) suggest $R_{in}$ $\sim$ (2-3)$R_{*}$
for young brown dwarfs. Their results are based on 38 cool objects in IC348
and Taurus which have spectral types M6-M9 and ages $\leq$5 Myr. Using
$R_{in}$ = 2.5$R_{*}$, an accretion rate log$\dot{M}$ = -6.70 $M_{\odot}$yr-1
is required for the stronger superflare to emit the energy of 2.1 $\times$
1038 erg in the observed flare duration. This accretion rate is higher than
those reported for young accreting BDs GY11 and 2MASS J053825.4-024241.
Rigliaco et al. (2011) reported an accretion rate of log$\dot{M}$ $\sim$
-9.86$\pm$0.45 $M_{\odot}$yr-1 for 2MASS J053825.4-024241, with comparable
mass, age and spectral type as CT4. Likewise, Comerón et al. (2010) reported
log$\dot{M}$ $\sim$-9.02 $M_{\odot}$yr-1 for another deuterium burning brown
dwarf GY 11. The higher value of requisite rate of accretion suggests that it
is unlikely for the stronger superflare to occur due to magnetospheric
accretion process.

The TRAPPIST-1 planetary system (Gillon et al., 2016, 2017; Luger et al.,
2017) demonstrates the existence of planets around low-mass stars and BDs,
increasing the importance of the study of planet formation around low-mass
stars and BDs. The superflares observed on CT4 will be very helpful to
understand how such flares impact the dynamical and chemical evolution of disk
around it. Such flares result to enhanced UV and X-ray emission which can
create innerhole in the disk through photoevaporation (Owen et al., 2011). The
high energy X-rays also increase the ionization of the disk, which might
trigger magnetorotational instability (MRI) that is supposed to drive magneto-
hydrodynamical turbulence in the protoplanetary disk. This has several
consequences on planet formation and depends on the energy of X-rays
(Feigelson, 2010; Cleeves et al., 2015). In addition, the study of superflares
on young objects like CT4 might be helpful in explaining the mystries
regarding the formation of the chondrules and the calcium-aluminium-rich
inclusions (CAIs), which need transient heat sources to melt the precursor
material. It is impossible to explain the formation of these materials in the
context of thermodynamic equilibrium between the PMS stars and the disks
around them. Some possible proposed transient heat sources are nebular
lightning, protoplanetary induced shocks, activity associated with the young
star having disk, and nearby Gamma Ray Burst (GRB) (McBreen & Hanlon, 1999;
Desch & Cuzzi, 2000; Duggan et al., 2001; Feigelson et al., 2002). If strong
flares acted as heating sources for formation of chondrules/CAIs, it is
unclear how they were transported to the Asteroid belt (Feigelson, 2005).

Acknowledgements
We are thankful to Zhoujian Zhang for providing us the results of his paper.
This paper makes use of data collected by the Kepler mission. Funding for the
Kepler mission is provided by the NASA Science Mission derectorate. The
material is based upon work supported by NASA under award Nos. NNX15AV64G,
NNX16AE55G and NNX16AJ22G. P.K.G.W. and E.B. acknowledge support for this work
from the National Science Foundation through Grant AST-1614770. 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. Support for
MAST for non-HST data is provided by the NASA Office of Space Science via
grant NNX09AF08G and by other grants and contracts. This paper includes data
collected by the Kepler mission. Funding for the Kepler mission is provided by
the NASA Science Mission directorate. This publication also makes use of data
from the Sloan Digital Sky Survey. Funding for the Sloan Digital Sky Survey IV
has been provided by the Alfred P. Sloan Foundation, the U.S. Department of
Energy Office of Science, and the Participating Institutions. SDSS-IV
acknowledges support and resources from the Center for HighPerformance
Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS-
IV is managed by the Astrophysical Research Consortium for the Participating
Institutions of the SDSS Collaboration, which can be found at
http://www.sdss.org/collaboration/affiliations/. 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 work made use of the
http://gaia-kepler.fun crossmatch database created by Megan Bedell. Astropy
(Astropy Collaboration et al., 2013), Matplotlib (Hunter, 2007), Numpy
(Oliphant 2006) : Kepler

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# The smooth Hom-stack of an orbifold

David Michael Roberts and Raymond F. Vozzo111This research was supported under
the Australian Research Council’s Discovery Projects funding scheme (project
numbers DP120100106 and DP130102578). The authors thank Alexander Schmeding
and Eugene Lerman for helpful discussion and suggestions. This document is
released under a CC0 license http://creativecommons.org/publicdomain/zero/1.0/

###### Abstract

For a compact manifold $M$ and a differentiable stack $\mathcal{X}$ presented
by a Lie groupoid $X$, we show the Hom-stack
$\operatorname{\underline{\mathcal{H}om}}(M,\mathcal{X})$ is presented by a
Fréchet–Lie groupoid $\operatorname{Map}(M,X)$ and so is an infinite-
dimensional differentiable stack. We further show that if $\mathcal{X}$ is an
orbifold, presented by a proper étale Lie groupoid, then
$\operatorname{Map}(M,X)$ is proper étale and so presents an infinite-
dimensional orbifold.

Mathematics Subject Classification (2010). Primary 22A22; Secondary 57R18,
58B25, 58D15, 14A20, 18F99.

This note serves to announce a generalisation of the authors’ work [RV16],
which showed that the smooth loop stack of a differentiable stack is an
infinite-dimensional differentiable stack, to more general mapping stacks
where the source stack is a compact manifold (or more generally a compact
manifold with corners). We apply this construction to differentiable stacks
that are smooth orbifolds, that is, they can be presented by proper étale Lie
groupoids (see Definition 9).

Existing work on mapping spaces of orbifolds has been considered in the purely
topological case [CPR+15, GH06, Hae07, PS], the case of $C^{k}$ maps in
[Che06], Sobolev maps in [Wei07] and smooth maps in [BB13]. In all these
cases, some sort of orbifold structure has been found (for instance, Banach or
Fréchet orbifolds). However [Sch15] gives counterexamples showing that some of
the constructions in [BB13] are not well-defined.

Noohi [Noo10] solved the problem of constructing a topological mapping stack
between more general _topological_ stacks, when the source stack has a
presentation by a compact topological groupoid. See [RV16] for further
references and discussion.

We take as given the definition of Lie groupoid in what follows, using finite-
dimensional manifolds unless otherwise specified. Manifolds will be considered
as trivial groupoids without comment. We pause only to note that in the
infinite-dimensional setting, the source and target maps of Fréchet–Lie
groupoids must be submersions between Fréchet manifolds, which is a stronger
hypothesis than asking the derivative is surjective (or even split)
everywhere, as in the finite-dimensional or Banach case.

We will also consider groupoids in diffeological spaces. Diffeological spaces
(see for instance [BH11]) contain Fréchet manifolds as a full subcategory and
admit all pullbacks (in fact all finite limits) and form a cartesian closed
category such that for $K$ and $M$ smooth manifolds with $K$ compact, the
diffeological mapping space $M^{K}$ is isomorphic to the Fréchet manifold of
smooth maps $K\to M$.

Differentiable stacks are, for us, stacks of groupoids on the site
$\mathcal{M}$ of finite-dimensional smooth manifolds with the open cover
topology that admit a presentation by a Lie groupoid [BX11]. We can also
consider the more general notion of stacks that admit a presentation by a
diffeological or Fréchet–Lie groupoid.

######  Definition 1.

Let $\mathcal{X},\mathcal{Y}$ be stacks on $\mathcal{M}$. The _Hom-stack_
$\operatorname{\underline{\mathcal{H}om}}(\mathcal{Y},\mathcal{X})$ is defined
by taking the value on the manifold $N$ to be
$\operatorname{\bf{Stack}}_{\mathcal{M}}(\mathcal{Y}\times N,\mathcal{X})$.

Thus we have a Hom-stack for any pair of stacks on $\mathcal{M}$. The case we
are interested in is where we have a differentiable stack $\mathcal{X}$
associated to a Lie groupoid $X$, e.g. an orbifold, and the resulting Hom-
stack $\operatorname{\underline{\mathcal{H}om}}(M,\mathcal{X})$ for $M$ a
compact manifold.

We define a _minimal cover_ of a manifold $M$ to be a cover by regular closed
sets $V_{i}$ such that the interiors ${\kern 0.0ptV_{i}}^{\mathrm{o}}$ form an
open cover of $M$, and every ${\kern 0.0ptV_{i}}^{\mathrm{o}}$ contains a
point not in any other ${\kern 0.0ptV_{j}}^{\mathrm{o}}$. We also ask that
finite intersections $V_{i}\cap\ldots\cap V_{k}$ are also regular closed.
Denote the collection of minimal covers of a manifold $M$ by $C(M)_{\min}$,
and note that such covers are cofinal in open covers. Recall that a cover $V$
of a manifold defines a diffeological groupoid $\check{C}(V)$ with objects
$\coprod_{i}V_{i}$ and arrows $\coprod_{i,j}V_{i}\cap V_{j}$.222We can in what
follows safely ignore the issue of intersections of boundaries. We are
particularly interested in the case when we take the closure
$\\{\overline{U_{i}}\\}$ of $\\{U_{i}\\}$, a good open cover, minimal in the
above sense.

We denote the arrow groupoid of a Lie groupoid $X$ by $X^{\mathbbm{2}}$—it is
again a Lie groupoid and comes with functors $S,T\colon X^{\mathbbm{2}}\to X$,
with object components source and target, resp. Let $M$ be a compact manifold
with corners and $X$ a Lie groupoid. Define the _mapping groupoid_
$\operatorname{Map}(M,X)$ to be the following diffeological groupoid. The
object space $\operatorname{Map}(M,X)_{0}$ is the disjoint union over minimal
covers $V$ of the spaces ${X}^{\check{C}(V)}$ of functors $\check{C}(V)\to X$.
The arrow space $\operatorname{Map}(M,X)_{1}$ is

$\coprod_{V_{1},V_{2}\in
C(M)_{\text{min}}}{X}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}$

where the chosen minimal refinement $V_{12}\subset V_{1}\times_{M}V_{2}$ is
defined using the boolean product on the algebra of regular closed sets. The
maps

$S,T\colon{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\to{X}^{\check{C}(V_{12})}\qquad\text{and}\qquad{X}^{\check{C}(V_{i})}\to{X}^{\check{C}(V_{12})}\quad(i=1,2)$
(1)

give us a pullback and the two projections

${X}^{\check{C}(V_{1})}\times_{{X}^{\check{C}(V_{12})}}{(X^{\mathbbm{2}})}^{\check{C}(V_{12})}\times_{{X}^{\check{C}(V_{12})}}{X}^{\check{C}(V_{2})}\longrightarrow{X}^{\check{C}(V_{i})},$
(2)

induce, for $i=1,2$, the source and target maps for our groupoid resp.
Composition in the groupoid is subtle, but is an adaptation of the composition
of transformations of anafunctors given in [Rob12]. The proof of the following
theorem works exactly as in Theorem 4.2 in [RV16].

######  Theorem 2.

For $X$ a Lie groupoid and $M$ a compact manifold the Hom-stack
$\operatorname{\underline{\mathcal{H}om}}(M,X)$ is presented by the
diffeological groupoid $\operatorname{Map}(M,X)$. ∎

We need some results that ensure the above constructions give Fréchet
manifolds.

######  Proposition 3.

For $M$ a compact smooth Riemannian manifold (possibly with corners), $K$ a
compact regular closed Lipschitz subset of $M$ and $N$ a smooth manifold, the
induced restriction map $N^{M}\to N^{K}$ is a submersion of Fréchet manifolds.

###### Proof.

Recall that a submersion of Fréchet manifolds is a smooth map that is locally,
for suitable choices of charts, a projection out of a direct summand. This
means we have to work locally in charts and show that we have a split
surjection of Fréchet spaces. We can reduce this to the case that
$N=\mathbb{R}^{n}$, since the charts are given by spaces of sections of
certain vector bundles, and we can consider these spaces locally and patch
them together, and thence to $N=\mathbb{R}$. The proof then uses [Fre07,
Theorem 3.15], as we can work in charts bi-Lipschitz to flat $\mathbb{R}^{n}$,
hence reduce to the case of $K\subset\overline{B}\subset\mathbb{R}^{n}$, for
$B$ some large open ball. ∎

In particular this is true for sets $K$ that are closures of open geodesically
convex sets, and even more specifically such open sets that are the finite
intersections of geodesically convex charts in a good open cover. We also use
a special case of Stacey’s theorem [Sta13, Corollary 5.2]; smooth manifolds
with corners are smoothly $\mathfrak{T}$-compact spaces in Stacey’s sense.
Another proof, not using the technology of generalised smooth spaces, is given
in [AS17, Lemma 2.4].

######  Theorem 4 (Stacey).

Let $N_{1}\to N_{2}$ be a submersion of finite-dimensional manifolds and $K$ a
compact manifold, possibly with corners. Then the induced map of Fréchet
manifolds $N_{1}^{K}\to N_{2}^{K}$ is a submersion. ∎

The following proposition is the main technical tool in proving the mapping
stack is an infinite-dimensional differentiable stack.

######  Proposition 5.

The diffeological space ${X}^{\check{C}(V)}$ is a Fréchet manifold.

###### Proof.

First, the diffeological space of functors is isomorphic to the space of
simplicial maps $N\check{C}(V)\to NX$ between the nerves of the groupoids.
Then, since the subspaces of degenerate simplicies in $N\check{C}(V)$ are
disjoint summands, we can remove those, and consider semi-simplicial maps
between semisimplicial diffeological spaces instead. Then, since inverses in
$\check{C}(V)$ are also disjoint, we can remove those as well, and consider
the diffeological space of semisimplicial maps from the ‘nerve’ of the smooth
irreflexive partial order $\check{C}^{<}(V)$ to the nerve of $X$, considered
as a semisimplicial space (where we have chosen an arbitrary total ordering on
the finite minimal cover $V$). _This_ diffeological space is what we show is a
Fréchet manifold, by carefully writing the limit as an iterated pullback of
diagrams involving maps that are guaranteed to be submersions by Proposition 3
and Theorem 4, and using the fact that $X$ is appropriately coskeletal, i.e.
$(NX)_{n}=X_{1}\times_{X_{0}}\ldots\times_{X_{0}}X_{1}$. The original space of
functors is then a diffeological space isomorphic to this Fréchet manifold,
hence is a Fréchet manifold. ∎

######  Lemma 6.

Let $X\to Y$ be a functor between Lie groupoids with object and arrow
components submersions, and $V_{1}\to V_{2}$ a refinement of minimal covers.
Then the induced map ${X}^{\check{C}(V_{2})}\to{Y}^{\check{C}(V_{1})}$ is a
submersion between Fréchet manifolds. ∎

We will consider the special cases that the functor $X\to Y$ is the identity,
and also that the refinement $V_{1}\to V_{2}$ is the identity.

######  Theorem 7.

For a Lie groupoid $X$ and compact manifold $M$, $\operatorname{Map}(M,X)$ is
a Fréchet–Lie groupoid.

###### Proof.

The object space $\operatorname{Map}(M,X)_{0}$ is a manifold by Proposition 5.
The arrow space $\operatorname{Map}(M,X)_{1}$ is a manifold since it is given
by a pullback diagram built with the maps (1), which are submersions by Lemma
6. The identity map is smooth, as it is a smooth map between diffeological
spaces that happen to be manifolds, and so is composition. ∎

The following theorem is the first main result of the paper. The proof uses
the technique of [Noo10, Theorem 4.2] as adapted in [RV16], where it is shown
that all of the constructions remain smooth.

######  Theorem 8.

For a Lie groupoid $X$ and compact manifold $M$, the stack
$\operatorname{\underline{\mathcal{H}om}}(M,\mathcal{X})$ is weakly presented
by the Fréchet–Lie groupoid $\operatorname{Map}(M,X)$. ∎

A _weak_ presentation means that the pullback of the map
$\operatorname{Map}(M,X)_{0}\to\operatorname{\underline{\mathcal{H}om}}(M,\mathcal{X})$
against itself gives a stack representable by the Fréchet manifold
$\operatorname{Map}(M,X)_{1}$, and the two projections are submersions. For
the site of diffeological spaces, a weak presentation is an ordinary
presentation. This is also the case if we allow non-Hausdorff manifolds [BX11,
Proposition 2.2], so we either have to pay the price of a weak presentation or
working over a site of non-Hausdorff manifolds. If the groupoid
$\operatorname{Map}(M,X)$ is _proper_ , as in Theorem 10 below, then we can
upgrade this weak presentation to an ordinary one over Hausdorff manifolds.

######  Definition 9.

A (Fréchet–)Lie groupoid $Z$ is _proper_ if the map $(s,t)\colon Z_{1}\to
Z_{0}\times Z_{0}$ is a proper map (i.e. closed with compact fibres), _étale_
if the source and target maps are local diffeomorphisms, and an _orbifold
groupoid_ if it is a proper and étale.

It is a theorem of Moerdijk–Pronk that _effective_ orbifold groupoids are
equivalent to _reduced_ orbifolds defined in terms of orbifold charts—in
finite dimensions (see e.g. [Poh17]). For infinite dimensional orbifolds this
is not yet known, but may be possible for Fréchet–Lie groupoids with local
additions on their object and arrow manifolds,333Alexander Schmeding, Private
communication. of which our construction is an example.

Our second main result is then:

######  Theorem 10.

If $X$ is an étale Lie groupoid, then $\operatorname{Map}(M,X)$ is étale. If
$X$ is an orbifold groupoid, then $\operatorname{Map}(M,X)$ is an orbifold
groupoid.

###### Proof.

Stability of local diffeomorphisms under pullback mean that we only need to
show that the smooth maps
${S}^{\check{C}(V)},{T}^{\check{C}(V)}\colon{(X^{\mathbbm{2}})}^{\check{C}(V)}\to{X}^{\check{C}(V)}$,
for any minimal cover $V$, are local diffeomorphisms. If $X$ is an étale Lie
groupoid then the fibres of its source and target maps are discrete, and one
can show that ${S}^{\check{C}(V)},{T}^{\check{C}(V)}$ have discrete
diffeological spaces as fibres. But these maps are submersions of Fréchet
manifolds, hence are local diffeomorphisms.

Properness follows if we can show that $(s,t)$ for the mapping groupoid is
closed and every object has a finite automorphism group. This reduces to
showing that ${(S,T)}^{\check{C}(V)}$ is closed and its fibres are finite. We
can show the latter by again working in the diffeological category and showing
that the fibres of ${(S,T)}^{\check{C}(V)}$ are discrete, and also a subspace
of a finite diffeological space. As all the spaces involved are metrisable, we
use a sequential characterisation of closedness together with the local
structure of the proper étale groupoid $X$, and find an appropriate convergent
subsequence in the required space of natural transformations. ∎

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