# The white-light superflares from cool stars in GWAC triggers

Guang-Wei Li (李广伟) Key laboratory of Space Astronomy and Technology, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
Liang Wang (王靓) Nanjing Institute of Astronomical Optics & Technology, Chinese
Academy of Sciences, Nanjing 210042, China  CAS Key Laboratory of Astronomical
Optics & Technology, Nanjing Institute of Astronomical Optics & Technology,
Nanjing 210042, China  University of Chinese Academy of Sciences, Beijing
100049, China  Hai-Long Yuan (袁海龙) Key Laboratory of Optical Astronomy,
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, China  Li-Ping Xin (辛立平) Key laboratory of Space Astronomy and
Technology, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100101, China  Jing Wang (王竞)  Guangxi Key Laboratory for Relativistic
Astrophysics, School of Physical Science and Technology, Guangxi University,
Nanning 530004, China  Chao Wu (吴潮)  Key laboratory of Space Astronomy and
Technology, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100101, China  Hua-Li Li (黎华丽)  Key laboratory of Space Astronomy and
Technology, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100101, China  Hasitieer Haerken (哈斯铁尔·哈尔肯)  School of Artificial
Intelligence of Beijing Normal University, No.19, Xinjiekouwai St, Haidian
District, Beijing, 100875, China  Key laboratory of Space Astronomy and
Technology, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100101, China  Wei-Hua Wang (王伟华)  Changzhou Institute of Technology,
Changzhou, China  Hong-Bo Cai (蔡洪波)  Key laboratory of Space Astronomy and
Technology, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100101, China  Xu-Hui Han (韩旭辉)  Key laboratory of Space Astronomy and
Technology, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100101, China  Yang Xu (徐洋)  Key laboratory of Space Astronomy and
Technology, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100101, China  Lei Huang (黄垒)  Key laboratory of Space Astronomy and
Technology, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100101, China  Xiao-Meng Lu (卢晓猛)  Key laboratory of Space Astronomy
and Technology, National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100101, China  Jian-Ying Bai (白建迎)  Key laboratory of Space
Astronomy and Technology, National Astronomical Observatories, Chinese Academy
of Sciences, Beijing 100101, China  Xiang-Yu Wang (王祥玉)  School of Astronomy
and Space Science, Nanjing University, Nanjing 210093, China  Key Laboratory
of Modern Astronomy and Astrophysics (Nanjing University), Ministry of
Education, Nanjing 210093, China  Zi-Gao Dai (戴子高)  School of Astronomy and
Space Science, Nanjing University, Nanjing 210093, China  Key Laboratory of
Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education,
Nanjing 210093, China  En-Wei Liang (梁恩维)  Guangxi Key Laboratory for
Relativistic Astrophysics, School of Physical Science and Technology, Guangxi
University, Nanning 530004, China  Jian-Yan Wei (魏建彦)  Key laboratory of Space
Astronomy and Technology, National Astronomical Observatories, Chinese Academy
of Sciences, Beijing 100101, China

(Received June 1, 2019; Revised January 10, 2019)

###### Abstract

M-type stars are the ones that flare most frequently, but how big their
maximum flare energy can reach is still unknown. We present 163 flares from
162 individual M2 through L1-type stars that triggered the GWAC, with flare
energies ranging from $10^{32.2}$ to $10^{36.4}$ erg . The flare amplitudes
range from $\triangle G=0.84$ to $\sim 10$ mag. Flare energy increases with
stellar surface temperature ($T_{\rm eff}$) but both $\triangle G$ and
equivalent duration $\log_{10}(ED)$ seem to be independent of $T_{\rm eff}$.
Combining periods detected from light curves of TESS and K2, spectra from
LAMOST, SDSS and the 2.16 m Telescope, and the Gaia DR3 data, we found that
these GWAC flare stars are young. For the stars that have spectra, we found
that these stars are in or very near to the saturation region, and
$\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$ is lower for M7-L1 stars than for
M2-M6 stars. We also studied the relation between GWAC flare bolometric energy
$E_{\rm bol}$ and stellar hemispherical area $S$, and found that
$\log_{10}E_{\rm bol}$ (in erg) increases with increasing $S$ (in cm2), and
the maximum flare energy $\log_{10}E_{\rm bol,max}\geqslant\log_{10}S+14.25$.
For M7-L1 stars, there seem to be other factors limiting their maximum flare
energies in addition to stellar hemispherical area.

flares — M-type stars

††journal: ApJ††software: astropy (Astropy Collaboration et al., 2013),
lightkurve (Lightkurve Collaboration et al., 2018)

##  1 Introduction

Solar flares originated from the release of magnetic energy by the magnetic
reconnection in the corona (Haisch et al., 1991). The released energy ranges
from $10^{28}$ to $10^{32}$ erg (Shibata & Magara, 2011). The biggest solar
flare ever detected is the Carrington flare that occurred in 1859 (Carrington,
1859; Hodgson, 1859), which released energy of about $4\times 10^{32}$ erg
(Hudson, 2021). The Carrington Event caused a geomagnetic storm (Hudson,
2021), and interrupted telegraph services (Boteler, 2006). The aurorae can be
seen even near the Equator (Moreno Cárdenas et al., 2016).

Like the solar flares, stellar flares also originated from magnetic
reconnections (Yan et al., 2021; Yang & Liu, 2019). Most stellar flare
energies detected are between $10^{32}and10^{35}$ erg and for giants, their
flare energies can even be as large as $10^{38}$ erg (Yang & Liu, 2019;
Pietras et al., 2022). The white-light flare energies are often thought to be
from blackbody radiations with a temperature of 9000 K - 14,000 K (Kowalski et
al., 2013), and even as high as 42000 K (Howard et al., 2020), which is much
higher than the photospheric temperature of M dwarfs, so flares with energies
as low as of $10^{30}$ erg can be detected in near M-type dwarfs in optical
bands (Yang et al., 2017).

The stellar flare activity is related to rotation and also the spectral type.
The faster the rotation the stronger the activity, which is the stellar
activity-rotation relationship. The relationship has been found by X-ray
(Wright et al., 2011; Wright & Drake, 2016), Ca II H & K lines (Boudreaux et
al., 2022; Lehtinen et al., 2021), H$\alpha$ (Newton et al., 2017; Li et al.,
2023a). M dwarfs are more active than earlier type stars (Althukair &
Tsiklauri, 2023) and from M0 to M6 the fraction of flare stars increases from
about 10% to over 40% (Günther et al., 2020). M-type stars tend to produce
more frequent and powerful flares than the Sun, and the habitable zones of M
dwarfs are very near the hosts (Kane, 2018). Therefore, considering their
flare energies can be thousands of times higher than the Carrington Event of
the Sun and proximities of their habitable zones, the impacts of flares on
habitable planets can be several magnitudes higher than those of solar flares
on the Earth. The electromagnetic radiation from X-ray to radio (Osten et al.,
2005) during flares and coronal mass ejections (CMEs) can be released by
powerful flares: the more powerful the flare, the more likely the CME is
released (Li et al., 2021). The intense flares can release tremendous UV
fluxes and CME which can destroy O3 (Tilley et al., 2019), and the ultraviolet
fluxes can sterilize lives on the planet’s surface (Estrela & Valio, 2018). At
the same time, the atmosphere of the planet would be heated, expanded, eroded
(Linsky, 2019) and finally even disappeare (Atri & Mogan, 2021). On the other
hand, the intense flare can trigger the prebiotic chemistry and then life
(Rimmer et al., 2018; Xu et al., 2018; Günther et al., 2020; Chen et al.,
2021).

The biggest flares are very rare, especially the flares with energy of
$\geqslant 10^{34}$ erg from M dwarfs (Howard & MacGregor, 2022; Jackman et
al., 2023), which would have important impacts on the planet’s atmosphere and
life (e.g. Miranda-Rosete et al., 2023; Konings et al., 2022; Tilley et al.,
2019; Hu et al., 2022). Some biggest flares have been detected by the GWAC,
EvryFlare (Howard et al., 2019), ASAS-SN (Schmidt et al., 2019), and NGTS
(Jackman et al., 2021). The amplitudes can be $\bigtriangleup R\sim 9.5$ or
$\bigtriangleup V\sim 11.2$ (Xin et al., 2023b), $\bigtriangleup R\sim 9.5$
(Xin et al., 2021), $\bigtriangleup V>11$ mag (Schmidt et al., 2016),
$\bigtriangleup V\sim 10$ mag (Jackman et al., 2019), and so on.

As the ground instrument of Space-based multi-band astronomical Variable
Objects Monitor (SVOM; Wei et al., 2016), the Ground-based Wide Angle Cameras
(GWAC) system aims to monitor afterglows of gamma-ray bursts (Xin et al.,
2023a), and thus superstellar flares were also detected by the GWAC software
(Han et al., 2021). In this work, we collected 163 big flares that occurred
between 2017 November and 2023 March from 162 individual stars, and try to
explore the mechanism behind them. These flares and properties of host stars
are given in Table 1. Several of them have been carefully studied in Xin et
al. (2021), Li et al. (2023b), Wang et al. (2022), Wang et al. (2021), and Bai
et al. (2023). In Section 2 we will introduce the observations and data; in
Section 3, we will present the properties of GWAC flares and their host stars;
in Section 4, the rotation-age-activity relationship will be studied; in
Section 5 the lower limit of the maximum flare energy that a star can produce
is presented; the star age and activity pattern are discussed in Section 6;
and at last, the conclusion is given in Section 7.

Table 1: GWAC flares and properties of stars

Star No. | GWAC Name | TIC/K2 | Simbad Name | DR3 Name | RAJ2000 | DEJ2000 | Gmag | …
---|---|---|---|---|---|---|---|---
|  |  |  |  | (degree) | (degree) | (mag) | …
1 | GWAC181208A | TIC 456482672 | ATO J000.1128+13.6255 | Gaia DR3 2767338005380427264 | 0.112725 | 13.62563 | 14.740 | …
2 | GWAC220106A | TIC 432551405 | 2MASS J00013265+3841525 | Gaia DR3 2880981530065870720 | 0.386113 | 38.69797 | 18.193 | …
3 | GWAC180116A | TIC 357411008 |  | Gaia DR3 2769578225960672512 | 2.798485 | 15.87654 | 16.542 | …
4 | GWAC181206A | TIC 405305098 | 2MASS J00113451+0659388 | Gaia DR3 2742685889532913024 | 2.893805 | 6.99415 | 16.826 | …
5 | GWAC211124A | TIC 51940383 | V* BI Psc | Gaia DR3 2767679884775564928 | 3.176556 | 13.13609 | 15.637 | …
… |  |  |  |  |  |  |  |

Note. —

$a$ ’EB’ means eclipse binary, and ’WDMS’ means white dwarf-main sequence
binaries.

$b$ The spectral type is from their $G_{\rm BP}-G_{\rm RP}$.

(The full machine-readable form is available at https://nadc.china-
vo.org/res/r101350/)

##  2 Observations and Data

###  2.1 GWAC

GWAC has 36 cameras and can cover about $\sim 5000$ deg2 of sky (Xin et al.,
2021; Wang et al., 2020). There is a 4K $\times$ 4K CCD for each camera and
the pixel size is about $11^{\prime\prime}.7$. The cadence of GWAC is 15 s: 10
s for exposure and 5 s for readout. There is no filter on each camera with a
limit on magnitude about $G\sim 15$ mag, but in bad weather the limit
magnitude would be much brighter. For a transit that suddenly apparent on a
CCD, the GWAC software (Han et al., 2021) triggers one of the two Guangxi-NAOC
60 cm optical telescopes (F60A and F60B) to follow up immediately to check the
transit.

We obtained 163 big flares from 162 stars in the GWAC trigger database and all
were confirmed by an F60. The flare amplitude $\bigtriangleup G$ ranges from
0.83 to $\sim 10$ mag. Some GWAC images were not saved due to the overload of
the master computer and some unknown bugs in the communication program, which
is responsible for communications between more than 100 control and
calculation computers for GWAC cameras. As a result, we obtained images for
147 GWAC flares and made 147 movies for them. These GWAC light curve data,
their plots, and movies are available at https://nadc.china-
vo.org/res/r101350/.

Figure 1: Two GWAC flares: GWAC181208B and GWAC190101A. The upper two panels
are for GWAC181208B (from Star #18), and the bottom two panels are for
GWAC190101A (from Star #39). The left two panels are their preflare statuses
and the right two panels are the statuses at peaks. In each panel, the upper
is the GWAC image, and the bottom is the GWAC light curve with time in minutes
on the x-axis and flare amplitude on the y-axis. The black filled circle on
the light curve indicates the flare amplitude in the image.

In Figure 1, the flare GWAC181208B from Star #18 and the flare GWAC190101A
from Star #39 are shown, and the entire flare process can been seen in their
movies. The upper two panels are for GWAC181208B, and the bottom two panels
are for GWAC190101A. The left two panels show their pre-flare statuses and the
right two panels show flare peaks. Star #18 and Star #39 cannot be seen in the
upper left and bottom left panels, respectively, in their preflare status,
while they appeared at peaks in the upper right and bottom right panels,
respectively. The flare GWAC181208B has a common flare profile: a rapid rise
(lasting only about 1 minute) followed by a slow decay, with a flare amplitude
of $\sim 3.3$ mag, but for the GWAC190101A, its impulsive phase is much
slower, which lasted for about 10 minutes.

###  2.2 TESS and K2 Light Curves

We searched TESS light curves using the TESS-point Web Tool and found that
except Star #147 (GWAC191226A), all other stars have been observed by TESS. We
searched TESS and K2 light curves by the Python package, _Lightkurve_
(Lightkurve Collaboration et al., 2018), and found available light curves for
109 stars. For the remaining 52 stars, we extracted light curves from their
_TargetPixelFiles_. For all stars but Star #147 (GWAC191226A), we inspected
their Full Frame Images (FFIs) of TESS in MAST (STScI, 2022b) and also Aladin
(Bonnarel et al., 2000) by eye to ensure all light curves are not contaminated
by nearby bright stars. For all TESS light curves of 161 stars, the bad parts
of light curves were removed by hand. The K2 light curves were also obtained
if available. Finally, we obtained good light curves from 276 Sectors for 124
stars. From these light curves, we tried to obtain their periods and flares
using the algorithm from Li et al. (2023a). In short, the algorithm tries to
fit the light curve by a B-spline by iteratively removing flares, then obtains
the period by the Lomb-Scargle method (Scargle, 1982; Lomb, 1976) using
_LombScargle_ in _Astropy_ (Astropy Collaboration et al., 2013, 2018). After
minus the fitted B-spline from the original light curve, flares with 3
successive points higher than 5$\sigma$ were detected. Finally, 1478 flares
from 117 stars and periods for 105 stars were obtained. An example can be seen
in Figure 1 in Li et al. (2023a). All TESS and K2 light curves used in this
work, with flares and periods detected from these light curves are available
at https://nadc.china-vo.org/res/r101350/.

###  2.3 Flare Energy

We calculated the equivalent duration (ED; Gershberg, 1972) of each flare:

$ED=\int^{t1}_{t0}\frac{f(t)-f_{0}}{f_{0}}dt$

. Here, $t_{0}$ and $t_{1}$ are the flare start and end times in second
respectively, $f(t)$ is the flare flux in erg s-1 at the time $t$, and $f_{0}$
is the stellar quiescent flux in erg s-1. Then the flare energy $E=ED\times
f_{0}$.

GWAC has no filter and we used Gaia $G$ to calibrate its photometry. The
accuracy is better than 0.1 mag. We used the method in Li et al. (2023a) to
calculate the quiescent flux of a star, the zero point flux of $G$ and $T$
passbands are from Li et al. (2023a) and Sullivan et al. (2015), respectively.
The parallaxes of stars are from Gaia DR3 (Gaia Collaboration, 2022). Star #9
(GWAC171207A), #95 (GWAC210112A), #124 (GWAC210217A), and #104 (GWAC210202A),
have no available parallaxes in Gaia DR3, so their distances are estimated by
$M_{Ks}=1.844+1.116(V-J)$ (Raetz et al., 2020), where $Ks$ and $J$ are from
2MASS (Skrutskie et al., 2006) and $V$ is from TIC 8.2 (Paegert et al., 2022).
For Star #11 (GWAC181227A), #44 (GWAC190204A), #49 (GWAC211203A), #86
(GWAC210117A), #117 (GWAC190116B) and #120 (GWAC190117A), their flare energies
were calculated from their K2 light curves, so the method in Shibayama et al.
(2013) was used, with stellar radii and surface temperatures from Huber et al.
(2016). Finally, a blackbody with $T=9000$ K was used to estimate the
bolometric energy that a flare released. These flares are listed in Table 2.

Table 2: Flares from TESS or K2 light curves

TIC/K2 | BeginTime | EndTime | $\log_{10}E_{bol}$
---|---|---|---
| (days) | (days) | erg
TIC 156817806 | 2611.52955 | 2611.55733 | 31.78
TIC 156817806 | 2628.24472 | 2628.30028 | 32.35
TIC 156817806 | 2634.60568 | 2634.68901 | 32.26
TIC 156817806 | 2635.53621 | 2635.58482 | 32.05
TIC 318929976 | 1436.53726 | 1436.68309 | 33.38
TIC 13936933 | 1784.47441 | 1784.62024 | 33.03
TIC 118768009 | 2475.85775 | 2475.86748 | 33.17
TIC 118768009 | 2495.11225 | 2495.14420 | 33.88
TIC 118768009 | 2497.71940 | 2497.73468 | 33.63
TIC 303471889 | 2466.06797 | 2466.17215 | 33.85
TIC 303471889 | 2466.94999 | 2466.99861 | 33.55
… |  |  |

Note. —

’BeginTime’ and ’EndTime’ are Julian days (JD 2547000 days) when a flare
starts and ends.

(The full machine-readable form is available at https://nadc.china-
vo.org/res/r101350/)

###  2.4 Spectra

We searched low-resolution spectra ($R\sim 1800$) in LAMOST DR10
111http://www.lamost.org/dr10/ (Deng et al., 2012; Cui et al., 2012) and SDSS
DR17 222https://www.sdss4.org/dr17/, and found 68 low-resolution spectra for
46 stars and 17 spectra for 12 stars, respectively. We also obtained 69
spectra for 67 stars with a resolution of 2.34 Å pixel-1 with the instrument
G5 on the 2.16 m telescope (Zhao et al., 2018). There are five stars observed
by LAMOST and also by the 2.16 m Telescope, so 154 spectra for 120 spectra
were obtained. Because the resolution of spectra obtained with the 2.16 m
Telescope is too low, there is no available radial velocity can be calculated
from them.

Except for LAMOST spectra, all other spectra are flux calibrated. Then the
photometries of Pan-STARRS1 $g$ and $r$ (Chambers et al., 2016) were,
respectively, used to calibrate LAMOST spectra for stars with $r<15$ mag and
$r>15$ mag. The zero point fluxes and the filter transmission curves of Pan-
STARRS1 $g$ and $r$ are obtained from the Filter Profile
Service333http://svo2.cab.inta-csic.es/theory/fps/ of the Spanish Virtual
Observatory. All spectra cover H$\alpha$, and the equation

$F(\lambda)=A_{0}\exp(-|(\lambda-
A_{1})/A_{2}|^{A_{3}}/A_{3})+A_{4}\lambda+A_{5}$ (1)

is used to fit the H$\alpha$ emission, where $\lambda$ is the wavelength in Å,
$F(\lambda)$ is the spectral flux, and $A_{i},i=0,1,2,3,4,5$ is the
coefficients to be fitted. Then the apparent H$\alpha$ luminosity $l_{{\rm
H}\alpha}=\int_{w_{1}}^{w_{2}}A_{0}\exp(-|(\lambda-
A_{1})/A_{2}|^{A_{3}}/A_{3})d\lambda$, where $w_{1}=6544.61$ Å and
$w_{2}=6584.61$ Å. Figure 2 shows one LAMOST spectrum of TIC 336800108 (Star
#12; GWAC181129A), with the fitted H$\alpha$ emission shown by the red curve
in the insert panel. The function consists of an exponential function
$A_{0}\exp(-|(\lambda-A_{1})/A_{2}|^{A_{3}}/A_{3})$ and a linear function
$A_{4}\lambda+A_{5}$. The linear function is used to fit the spectral
continuum and the exponential function is used to fit the H$\alpha$ emission.
The LAMOST instrumental profile is not perfectly gaussian, and this
exponential function can fit LAMOST spectral line profiles very well (see
Figure 2). In fact, if $A_{3}$ is set to 2, then the exponential function
would be the gaussian function.

To obtain the apparent bolometric flux of a star $m_{\rm bol}$, we firstly
calculated the bolometric correction of $BC_{J}$ using the equation in
Cifuentes et al. (2020):

$BC_{J}=0.0115(G-J)^{3}-0.132(G-J)^{2}+0.735(G-J)+0.576$ (2)

, where $J$ is from 2MASS and $G$ from Gaia DR3. Then $m_{\rm bol}=J+BC_{J}$.
The apparent luminosity $l_{\rm bol}$ in erg cm-2 s-1 of a star can be
obtained by the definition of apparent bolometric magnitude in Mamajek et al.
(2015) by the equation

$\log_{10}(l_{\rm bol})=0.4\times(-18.997351-m_{\rm bol})+3$ (3)

That is,

$\log_{10}(l_{\rm bol})=0.4\times(-18.997351-J-BC_{J})+3$ (4)

Then the fraction of H$\alpha$ luminosity to bolometric luminosity $L_{\rm
H\alpha}/L_{\rm bol}=l_{\rm H\alpha}/l_{\rm bol}$. Thus, $\log_{10}(L_{\rm
H\alpha}/L_{\rm bol})=\log_{10}(l_{\rm H\alpha})-\log_{10}(l_{\rm bol})$.
$l_{\rm H\alpha}$, $l_{\rm bol}$, $\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$ and
radial velocities obtained from spectra with errors are given in Table 3.

For stars with available spectra, there are 3 stars: Star #94 (GWAC180218A),
#107 (GWAC200321A), and #125 (GWAC200317A), have no available H$\alpha$
emission data because of low signal-to-noise fluxes around H$\alpha$
emissions.

Table 3: $l_{\rm H\alpha}$, $l_{\rm bol}$, $\log_{10}(L_{\rm H\alpha}/L_{\rm
bol})$ and radial velocities obtained from spectra

Star No. | Telescope | $\log_{10}(l_{\rm bol})$ |  $\log_{10}(l_{\rm bol})$_err | $\log_{10}(l_{\rm H\alpha})$ |  $\log_{10}(l_{\rm H\alpha})$_err | $\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$ |  $\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$_err | rv | rv_err
---|---|---|---|---|---|---|---|---|---
|  | [erg cm-2 s-1] | [erg cm-2 s-1] | [erg cm-2 s-1] | [erg cm-2 s-1] |  |  | km s-1 | km s-1
1 | LAMOST | -10.10 | 0.0256 | -13.76 | 0.0065 | -3.65 | 0.0321 | 11.9 | 3.2
1 | LAMOST | -10.10 | 0.0256 | -13.72 | 0.0067 | -3.62 | 0.0323 | 15.2 | 3.7
3 | 216 | -10.58 | 0.0348 | -14.07 | 0.0052 | -3.49 | 0.0400 |  |
4 | 216 | -10.81 | 0.0252 | -14.22 | 0.0108 | -3.41 | 0.0360 |  |
6 | 216 | -10.46 | 0.0260 | -14.30 | 0.0122 | -3.83 | 0.0382 |  |
… |  |  |  |  |  |  |  |  |

Note. — The full machine-readable form is available at https://nadc.china-
vo.org/res/r101350/

##  3 Results

###  3.1 GWAC flares

The $\bigtriangleup G$ vs. $G_{\rm BP}-G_{\rm RP}$ and $\log_{10}(ED)$ vs.
$G_{\rm BP}-G_{\rm RP}$ diagrams are shown in the upper and bottom panels,
respectively, in Figure 3. Star #19 (GWAC181229A) has the biggest
$\bigtriangleup G\sim 10$ mag, but it is too faint to have available Gaia
photometry and had been well studied in Xin et al. (2021), so it is not shown
in this figure. From the upper two panels one can see that most $\triangle G$
are between 1 and 2.5 mag, and $\bigtriangleup G$ seems to be independent of
$G_{\rm BP}-G_{\rm RP}$. From two lower panels, $\log_{10}(ED)$ also seems to
be independent of $G_{\rm BP}-G_{\rm RP}$, or the stellar surface effective
temperature.

The bolometric energies of GWAC flares are shown by blue circles in Figure 4.
Stars that have available stellar surface effective temperatures in TIC 8.2
(Paegert et al., 2022) are shown in the right panel. From Figure 4 we can see
that the flare energy ranges from $10^{32.2}$ to $10^{36.4}$ erg, and
decreases with $G_{\rm BP}-G_{\rm RP}$ (increases with the stellar effective
temperature).

For comparison, the maximum flare energy of each star in Yang & Liu (2019)
from Kepler DR25 is shown in the right panel in gray in Figure 4, from which
we can see that the flare energies of GWAC triggers are higher than the
maximum ones recorded by Kepler. Yang & Liu (2019) estimated flare energies
from the Kepler band, while we estimated flare energies from the $G$ band.
However, both assumed that bolometric flare energies are from a blackbody with
a temperature of 9000 K. There are 402 Kepler stars with effective
temperatures in range of 2700-3700 K for comparison, and each star was
monitored by Kepler for 4 yr, which implies that GWAC superflares are at most
once every 4 yr in Kepler cool stars. We also compared GWAC superflares with
those detected by NGTS shown in Figure 4 in Jackman et al. (2023), and found
that GWAC superflares have similar energies as those in Jackman et al. (2023),
and significantly higher than those in Jackman et al. (2021), which implies
that these superflares may be from the same category of top energetic flares.

Figure 2: A LAMOST spectrum of TIC 336800108. The red line in the insert panel
is the fitted H$\alpha$ emission.  Figure 3: The distributions of
$\bigtriangleup G$ and $\log_{10}(ED)$ of GWAC flares.  Figure 4: The
bolometric energies of GWAC flares. The blue circles are energies from GWAC
flares, and the gray circles are the maximum flare energies of individual
stars from Yang & Liu (2019).

###  3.2 Spectral Type

The spectral types were assigned by comparing the standard spectra in
Kirkpatrick et al. (1991). Using the parallaxes and photometries from Gaia DR3
(Gaia Collaboration, 2022), the H-R diagram is given in Figure 5.

Both Star #40 (GWAC201218A) and #60 (GWAC191030A) have a spectral type of
M3.5, but their $G_{\rm BP}-G_{\rm RP}$ are in the range of M2 as shown in
Figure 5. Their spectra were obtained with the 2.16m Telescope and are plotted
in Figure 6 with the comparing spectrum of the normal M3.5V Star #152
(GWAC180212A), which was also obtained with the 2.16m Telescope. From their
spectra we can see that spectra of these 3 stars are similar, but the Na I
doublet line around 8190 Å of Star #40 (GWAC201218A) is significantly weaker
than those of Star #60 (GWAC191030A) and #152 (GWAC180212A), which means that
Star #60 (GWAC191030A) is a dwarf, while Star #40 (GWAC201218A) is a giant or
young star. We checked Star #40 (GWAC201218A) and noticed that its
$ruwe=6.958$ in Gaia
DR3444https://gea.esac.esa.int/archive/documentation/GDR2/Gaia_archive/chap_datamodel/sec_dm_main_tables/ssec_dm_ruwe.html,
which implies its data may be unreliable. As for Star #60 (GWAC191030A), it is
still unclear why its $G_{\rm BP}-G_{\rm RP}$ is significantly bluer than
other M3.5V stars in Figure 5.

Star #59 (GWAC210109A) is an M5.5 young star in the Taurus star-forming region
(Esplin & Luhman, 2019). Star #155 (GWAC211016A) is contaminated by a nearby
star with a distance of $<1.^{\prime\prime}2$, so its Gaia data may be
unreliable.

Star #143 (GWAC210518A) was reported to be a white dwarf + main sequence
(WDMS) binary (Rebassa-Mansergas et al., 2012). To find more WDMS stars, for
Star #34 (GWAC190901A), #43 (GWAC190121B) and #75 (GWAC190324A), #76
(GWAC201113A), #98 (GWAC180117A), #102 (GWAC190206A), #106 (GWAC201216A), #117
(GWAC190116B), #133 (GWAC200514A) and #135 (GWAC200319A) with abnormally
higher FUV and NUV, we fitted their SEDs of GALEX FUV, NUV, PS1 g, r, i, z, y,
2MASS J, H, K and WISE (Wright et al., 2010) W1, W2 bands by a white dwarf and
a main sequence star templates (Yuan et al., 2023). Finally, only Star #43
(GWAC190121B) and #76 (GWAC201113A) seem to be well fitted as shown in Figure
7. Certainly, UV excesses of these stars may be from their active
chromospheres.

For stars that have no available spectra, their spectral types were assigned
by their $G_{\rm BP}-G_{\rm RP}$ in Figure 5. From Figure 5, we can see that
spectral types of GWAC flare stars range from M2 to L1, and most stars are M4,
which implies that these stars around the convective boundary tend to produce
big flares.

The PARSEC isochrones (Chen et al., 2014) of 0.01 Gyr, 0.1 Gyr and 1 Gyr with
[Fe/H] $=$ 0 are overplotted in Figure 5, from which we can see that most
stars should be younger than 1 Gyr.

Figure 5: The H-R diagram and spectra type distribution of GWAC flare stars.
Figure 6: The spectra of Star #40 (GWAC201113A), #60 (GWAC191030A) and #152
(GWAC180212A). The spectra are normalized by dividing the fluxes at 7500 Å.
Figure 7: The SED fittings of Star #43 (GWAC190121B) and #76 (GWAC201113A). In
each panel, the blue spectrum is the template for a main sequence star (MS),
the yellow spectrum is a template for a white dwarf star (WD), the green
spectrum is the WD+MS spectrum, and the red circles are the photometries of
GALEX FUV, NUV, PS1 g, r, i, z, y, 2MASS J, H, K and WISE W1, W2 bands.

###  3.3 Flare Frequency Distribution

There are 16 GWAC flare stars with more than 20 flares in their TESS or K2
light curves, and their cumulative flare frequency distributions (FFDs) are
shown in Figure 8, where the cumulative flare frequencies of each star are
shown in gray circles with the bolometric flare energies from a blackbody of
$T=9000$ K. Each FFD is fitted by the linear function

$\log_{10}({\rm\nu})=\alpha\log_{10}(E_{\rm bol})+\beta$ (5)

and shown by a black line. Here, $\nu$ is the cumulative flare frequency in
day-1, and $\alpha$ and $\beta$ are the parameters to be fitted. These 16
stars with their $\alpha$ and $\beta$ and the predicted frequencies of GWAC
flares by Equation 5 are shown in Table 4. The GWAC flare energies on the
fitted FFD lines are shown by red pentagons in Figure 8. We found that most
(13/16) GWAC flares can happen more than once every year, and only three GWAC
flares happen once every several or even 20 yr.

Figure 8: The cumulative flare frequency distributions (FFDs) for 16 stars
with more than 20 flares detected in TESS or K2 light curves. The cumulative
flare frequencies are denoted by gray circles, and black lines are the fitted
functions; The bolometric energies of GWAC flares are shown by red pentagons.
Table 4: Stars with More than 20 Flares

Star No. | TIC/K2 | SpT | $\alpha$ |  $\alpha$_err | $\beta$ |  $\beta$_err | $\log_{10}\nu_{\rm GWAC}$ |  $\log_{10}\nu_{\rm GWAC}$_err
---|---|---|---|---|---|---|---|---
|  |  | $\left[{\rm day}^{-1}\right]$ | $\left[{\rm day}^{-1}\right]$ | $\left[{\rm day}^{-1}\right]$ | $\left[{\rm day}^{-1}\right]$ | $\left[{\rm day}^{-1}\right]$ | $\left[{\rm day}^{-1}\right]$
11 | EPIC 220404032 | M4V | -0.6705 | 0.0294 | 21.57 | 0.98 | -2.03 | 0.06
30 | TIC 22944327 | M4.5V | -0.4635 | 0.0246 | 14.00 | 0.81 | -1.38 | 0.05
33 | TIC 328254412 | M5.5V | -0.5478 | 0.0262 | 17.12 | 0.86 | -0.96 | 0.05
44 | EPIC 210878635 | M4V | -0.8935 | 0.0530 | 28.55 | 1.74 | -2.29 | 0.07
49 | EPIC 210417498 | M4V | -1.1381 | 0.1041 | 36.01 | 3.38 | -3.37 | 0.07
52 | TIC 283866910 | M4.5V | -0.9140 | 0.0745 | 29.08 | 2.39 | -1.97 | 0.06
56 | TIC 245936201 | M3.5V | -0.9368 | 0.0993 | 30.56 | 3.32 | -2.98 | 0.08
77 | TIC 461654150 | M8V | -0.6492 | 0.0423 | 19.80 | 1.36 | -1.89 | 0.05
86 | EPIC 212002525 | M4.5V | -0.9269 | 0.0563 | 29.61 | 1.86 | -2.79 | 0.07
87 | TIC 175241416 | M7V | -0.9632 | 0.0563 | 30.13 | 1.80 | -1.99 | 0.08
93 | TIC 471012520 | M6V | -1.0777 | 0.0588 | 33.61 | 1.87 | -1.66 | 0.09
96 | TIC 251079483 | M4.5V | -0.6915 | 0.0545 | 21.75 | 1.78 | -1.46 | 0.09
113 | TIC 289534997 | M6V | -0.7162 | 0.0219 | 22.10 | 0.72 | -2.40 | 0.04
117 | EPIC 248853090 | M4.5V | -0.9209 | 0.0426 | 29.69 | 1.42 | -2.33 | 0.06
119 | TIC 156151200 | M5V | -0.6080 | 0.0332 | 18.71 | 1.08 | -2.05 | 0.06
120 | EPIC 201664337 | M4V | -1.4847 | 0.1240 | 47.52 | 4.04 | -3.95 | 0.13

Note. — $\alpha$ and $\beta$ are the parameters in Equation 5, and
$\alpha$_err and $\beta$_err are respectively their errors. $\log_{10}\nu_{\rm
GWAC}$ is the logarithm of the predicted frequency of the GWAC flare energy
with the error of $\log_{10}\nu_{\rm GWAC}$_err.

###  3.4 Kinematics

To explore kinematic properties of these flare stars, we calculated their
tangential velocities ($V_{\rm T}$) for stars with available proper motions
and parallaxes in Gaia DR3, and for stars that are not proven to be binaries
in this work and have reliable radial velocities in SDSS and LAMOST data,
their velocities in the local standard of rest (LSR) ($U_{\rm LSR},V_{\rm
LSR},W_{\rm LSR}$) were also calculated by the python package
_astropy.coordinates.SkyCoord_ with the velocity of the Sun relative to the
LSR of $(U_{\rm LSR},V_{\rm LSR},W_{\rm LSR})_{\odot}=(11.1,12.24,7.25)$ km
s-1 (Schönrich et al., 2010). We notice that there are some stars have several
available radial velocities, so the medium velocity was used during
calculation. Here, $U_{\rm LSR}$ is directed to the Galactic center, $V_{\rm
LSR}$ is in the Galactic rotational direction, and $W_{\rm LSR}$ points to the
Galactic North Pole. The $V_{\rm T}$ was calculated by the proper motion $\mu$
and parallax $\varpi$ in Gaia DR3: $V_{\rm T}=4.74\mu/\varpi$, and the total
velocity was calculated by $V_{tot}=(U_{\rm LSR}^{2}+V_{\rm LSR}^{2}+W_{\rm
LSR}^{2})^{1/2}$. As a result, there are 157 stars have $V_{\rm T}$. After
removing spectroscopic binaries, there are 52 stars have velocities relative
to the LSR. The results are shown in Figure 9.

We used the certiria in Gaia Collaboration et al. (2018) to select thin disk
($V_{\rm T}<40$ km s-1 or $V_{\rm tot}<50$ km s-1), thick disk ($60<V_{T}<150$
km s-1 or $70<V_{\rm tot}<180$ km s-1) and halo stars ( $V_{\rm T}>200$ km s-1
or $V_{\rm tot}>200$ km s-1 ). From Figure 9 we can see that there is no halo
star, and only 8/157 ($V_{\rm T}>60$ km s-2) or 3/52 ($V_{\rm tot}>70$ km s-2)
are thick disk stars, but speeds of these stars are far from the upper limit
of thick disk stars. Therefore, GWAC flare stars are not old and most of them
should be young.

Figure 9: Left panel: the distributions of tangential velocities of 157 GWAC
flare stars; Right panel: Toomre diagram for 52 GWAC flare stars with
available radial velocities. The 1$\sigma$ errors are shown in gray lines.

##  4 Rotation-Age-Activity Relation

The periods of 105/162 stars were obtained from their TESS or K2 light curves
by the Lomb-Scargle method. We inspected all of the light curves by eye and
found there are no EA and EB eclipses, so the periods should be rotational
periods. We also inspected all folded light curves by eye and found that most
of them are sinusoidal, so if there are some EW binaries in GWAC flare stars,
their real periods would be halved.

The periods are shown in Panel A of Figure 10. Most stars (80/105) have
periods shorter than 2 days, and only 4 stars have periods longer than 10
days. In Panel B, GWAC flare stars are shown in red circles in the color-
period diagram. The gray and blue circles are, respectively, field and
Praesepe stars from Popinchalk et al. (2021). The gray circles show that the
most field stars converge to the upper belt with periods greater than several
10 days. Praesepe has an age of $\sim$670 Myr, and its upper periods are about
10-20 days. There is a fast reservoir of $1.2<G-G_{\rm RP}<1.4$ and $0.2<P<2$
days (Popinchalk et al., 2021), where GWAC flare stars with available periods
are also clustered. From this panel we can see that GWAC flare stars are still
far from the upper period belt where the field stars cluster, and rotate even
faster than Praesepe stars, so all these 105/162 GWAC stars should be younger
than 670 Myr.

In the works of Wright et al. (2018) and Wright et al. (2011), the convective
turnover time $\tau$ is a function of $V-Ks$, but the GWAC flare stars are
very red and faint, and then some of them have no available $V$ data. As a
result, we firstly obtained $V-Ks$ of some GWAC flare stars from Paegert et
al. (2022), then fitted the relation of $V-Ks$ vs. $G_{\rm BP}-G_{\rm RP}$,
and finally obtained their predicted $V-Ks$ from $G_{\rm BP}-G_{\rm RP}$:

$V-Ks=0.15226\times(G_{\rm BP}-G_{\rm RP})^{2}+0.67908\times(G_{\rm BP}-G_{\rm
RP})+2.1097$ (6)

which is shown in Panel C. Then we used the function given by Wright et al.
(2018) to calculate the convective turnover time $\tau$:

$\log_{10}(\tau)=0.64+0.25\times(V-Ks)$

.

There are 108 spectra with available $\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$
for 82 GWAC flare stars, which also have available periods. Their
$\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$ vs. Rossby number ($Ro=P/\tau$)
diagram is shown in Panel D. For stars that have more than one available
$\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$, all $\log_{10}(L_{\rm
H\alpha}/L_{\rm bol})$ are shown. We also used the equation presented by
Reiners et al. (2014) to calculate the saturation period $P_{\rm sat}$ in
days:

$P_{\rm sat}=(1.1\times 10^{34}/L_{\rm bol})^{1/2}$ (7)

We found that only Star #51 and #56 are around the saturation $Ro$ and also
$P_{\rm sat}$ which are shown in Panel D and E of Figure 10 by black
horizontal lines, and all other stars are in the saturation region. There are
117 GWAC flare stars have available $L_{\rm H\alpha}$ in total, and their
$\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$ vs. $G_{\rm BP}-G_{\rm RP}$ diagram
is shown in Panel F. In Panel D, E, and F, the blue triangles are stars of
M7-L1 and the red circles are stars of M2-M6. For red circles, the filled ones
are stars with $P<2$ days and the empty ones are stars with $P>2$ days. The
central gray line is the saturation line ($L_{\rm H\alpha}/L=1.49\times
10^{-4}$) and gray dotted lines are the 1$\sigma$ ($\sigma=0.26$) positions
given in Newton et al. (2017). From Panel D, E and F, we can see that:

  * •

Stars in Panel D and E are all in or very near to the saturation region, so
these stars should be very active.

  * •

Though in the saturation region, $\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$ are
significantly lower for M7 - L1 stars (blue triangles) than for M2 - M6 stars
(red circles).

  * •

For M2 - M6 stars (red circles), it seems that $L_{\rm H\alpha}/L_{\rm bol}$
slightly decreases with increasing $Ro$ (as seen in Panel D) and also
increasing period (as seen in Panel E), but is independent of $G_{\rm
BP}-G_{\rm RP}$ (as seen in Panel F). Especially, the $L_{\rm H\alpha}/L_{\rm
bol}$ are lower for stars with $P>2$ days (red empty circles) than for those
with $P<2$ days (red filled circles). This declination of $\log_{10}(L_{\rm
H\alpha}/L_{\rm bol})$ with increasing period may imply the increasing stellar
age (Kiman et al., 2021).

  * •

As shown in Panel F, $\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$ seems to
decrease with increasing $G_{\rm BP}-G_{\rm RP}$ (i.e. spectral type), which
was also presented by Berger et al. (2010) and Schmidt et al. (2015).

In summary, for M2 - M6 stars, their $\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$
decreases with increasing period and is independent of spectral type, but for
M7 - L1 stars, though they are rapid rotators, their $\log_{10}(L_{\rm
H\alpha}/L_{\rm bol})$ are significantly lower than those of M2 - M6 stars.

The functions to calculate the convective overturn time given by Wright et al.
(2018, 2011) and Reiners et al. (2014) are frequently used in literature, but
these equations were obtained from the stars with masses larger than 0.1 M⊙ or
spectral types no later than M6 (Wright et al., 2018, 2011; Jao et al., 2022).
Therefore, we do not know if Ro and saturation periods derived from these
functions can apply to stars later than M6. However, in any case, it is true
that the $\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$ is lower for M7 - L0 stars
than for M2 - M6 stars as shown in Panel D, E, F of Figure 10, because it is
also seen in literature (e.g. Berger et al., 2010; Schmidt et al., 2015; Kiman
et al., 2021).

Figure 10: The rotation-age-activity relation. Panel A: The distribution of
periods of 105 GWAC flare stars. Panel B: The color-period diagram. The gray
and blue circles are, respectively, field stars and Praesepe stars from
Popinchalk et al. (2021), and the red circles are GWAC flare stars. Panel C:
$V-Ks$ vs. $G_{\rm BP}-G_{\rm RP}$. The fitted relationship (Equation 6) is
shown in the red curve. Panel D: $\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$ vs.
Rossby number ($Ro$) for stars with available spectra and $Ro$. Panel E:
$\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$ vs. period for stars with available
spectra and periods. Panel F: The $\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$ vs.
$G_{\rm BP}-G_{\rm RP}$ diagram for 117 GWAC stars that have available
spectra. The dotted yellow vertical line is $G_{\rm BP}-G_{\rm RP}=4.05$,
which is roughly the demarcation for M6 and M7 stars. In Panel D, E, and F,
the blue triangles are M7-L1 stars and the red circles are stars M2-M6 stars.
For red circles, the filled ones are stars with $P<2$ days and the empty ones
are stars with $P>2$ days. For stars that have more than one available
$\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$, all $\log_{10}(L_{\rm
H\alpha}/L_{\rm bol})$ are shown. The central gray line is the saturation line
($L_{\rm H\alpha}/L=1.49\times 10^{-4}$) and gray dotted lines are the
1$\sigma$ ($\sigma=0.26$) positions given in Newton et al. (2017). The black
horizontal short lines are Star #51 and #56, which are around the saturation
threshold.

##  5 Maximum Flare Energy

Some works (e.g. Notsu et al., 2019; Shibata et al., 2013) suggested that for
a solar-like star, the upper limit of flare energy is usually determined by
the spot size, and thus the stellar hemisphere. Inspired by this, we inspected
the relation of the flare bolometric energy $E_{\rm bol}$ vs. the area of
stellar hemisphere $S=2\pi R^{2}$ ($R$ is the stellar radius from TIC 8.2 ),
as shown in the left panel of Figure 11. From this panel we can see that
$E_{\rm bol}$ increases with increasing $S$. The relation of flare energy
divided by stellar hemisphere area $\log_{10}(E_{\rm bol}/S)$ vs. stellar
surface effective temperature $T_{\rm eff}$ ($T_{\rm eff}$ from TIC 8.2), is
shown in the right panel of Figure 11, where the vertical dashed line is
$G_{\rm BP}-G_{\rm RP}=3.95$ and roughly the demarcation between M2 - M6 and
M7-L1 stars, and the red horizontal line is the median value of
$\log_{10}(E_{\rm bol}/S)$ of M2-M6 stars. From this panel we can see that for
M2-M6 stars, the distribution of $\log_{10}(E_{\rm bol}/S)$ is roughly same
and independent of $T_{\rm eff}$, and significantly higher than those of M7-L1
stars. We found that the maximum $\log_{10}(E_{\rm bol}/S)$ is about 14.25 for
GWAC flares in the right panel, which means that for M2-M6 stars, the lower
limit of the maximum flare energy is

$\log_{10}E_{\rm bol}=\log_{10}S+14.25$ (8)

which is also shown by the yellow dashed line in the left panel of Figure 11.

Figure 11:  The $E_{\rm bol}$-$S$-$T_{\rm eff}$ relation. The red circles are
M2-M6 stars, and the blue triangles are M7-L1 stars. The yellow dashed line in
the right panel is the lower limit of the maximum flare energy, which is
$\log_{10}E_{\rm bol}=\log_{10}S+14.25$. In the right panel, the vertical
dotted line is $G_{\rm BP}-G_{\rm RP}=3.95$ and roughly the demarcation
between M2-M6 and M7-L1 stars, and the red horizontal line is the median value
of $\log_{10}(E_{\rm bol}/S)$ of M2-M6 stars.

##  6 Discussion

###  6.1 Stellar Age

There are 105/162 stars having periods as shown in Panel A of Figure 10.
Medina et al. (2022) suggests that for stars with $P_{\rm rot}<10$ days, their
ages are from about 600 Myr for 0.2 - 0.3 M⊙ to about 2 - 3 Gyr for 0.1-0.2
M⊙. Pass et al. (2022) also suggested that for stars with 0.2 - 0.3 M⊙ and
$P_{\rm rot}<2$ days, their ages are less than 600 Myr, and $2<P_{\rm rot}<10$
days means their ages between 1-3 Gyr. Therefore, these GWAC stars should be
younger than 600 Myr - 3 Gyr. By comparing with Praesepe stars in Panel B of
Figure 10, we refined their ages as younger than 670 Myr.

There are 38 stars having available $L_{\rm H\alpha}/L_{\rm bol}$ but without
available periods. Kiman et al. (2021) proposed (see their Figure 8) that the
activity of M-type stars decreases with age. In their Figure 6, for M2 - M6
stars younger than 1 Gyr, their $L_{\rm H\alpha}/L_{\rm bol}$ would
significantly greater than $10^{-4}$, and then decreases abruptly to around
$10^{-4.25}\approx 5.6\times 10^{-5}$ at 1 - 3 Gyr, while for stars of M7 -
M9, $L_{\rm H\alpha}/L_{\rm bol}$ decreases abruptly to significantly
$<10^{-4.5}\approx 3.2\times 10^{-5}$ during $\sim$ 30 Myr through 3 Gyr. As a
result, for stars in Panel F of Figure 10, M2-M6 and M7-L1 stars should be
younger than 1 and 3 Gyr, respectively.

Among the 157 stars with $V_{\rm T}>60$ km s-1 in the left panel of Figure 9,
Star #40, #132, and #145 have neither an available period nor $L_{\rm
H\alpha}/L_{\rm bol}$. For the remaining five stars without available $V_{\rm
T}$, only Star #19 has neither available period nor $L_{\rm H\alpha}/L_{\rm
bol}$. Therefore, there are only 4/162 GWAC flare stars that cannot be
determined if they belong to the thin disk. Therefore, except for these four
stars, all other 158/162 stars should be young.

In summary, if we think these GWAC flare stars are from the same category,
then their periods show they are younger than 670 Myr, $L_{\rm H\alpha}/L_{\rm
bol}$ show they are younger than 3 Gyr, and $V_{\rm T}$ shows they are the
thin disk stars.

###  6.2 The Lower Activity Level of stars later than M6

For M and L stars, $L_{\rm H\alpha}/L_{\rm bol}$ decreases with increasing
spectral type (Mohanty & Basri, 2003; Berger et al., 2010; Schmidt et al.,
2015), which is also shown in Panel F of Figure 10, and the $L_{\rm
H\alpha}/L_{\rm bol}$ level is significantly lower for stars later than M6
than for earlier type stars. Kiman et al. (2021) suggested that for M7-M9
stars, they have the same rotation-activity relation as earlier type stars
only in their first $\sim$30 Myr, and after that their activity decreases
abruptly and keeps at the low level for more than 3-6 Gyr.

Besides $L_{\rm H\alpha}/L_{\rm bol}$, Berger et al. (2010) also found that
for the stellar activity there is a breakout around M7 in the quiescent X-ray
and radio: for stars ranging from F to M6, they have the similar $L_{\rm
X}/L_{\rm bol}$ pattern and $L_{\rm X}/L_{\rm bol}$ is higher than that of
later type stars; $L_{\rm radio}/L_{\rm bol}$ and $L_{\rm radio}/L_{\rm X}$
are lower for M1-M6 stars than for later type stars.

In the GWAC sample, for M7-L1 stars, the distributions of their
$\bigtriangleup G$ and $ED$ are similar to those of M2-M6 stars as shown in
Figure 3, but their $\log_{10}(E_{\rm bol}/S)$ are lower than the median
$\log_{10}(E_{\rm bol}/S)$ of M2 - M6 stars as shown in the right panel of
Figure 11. Therefore, if the stellar radii given in TIC 8.2 are reliable, then
the $\log_{10}(E_{\rm bol}/S)$ are lower for M7 - L1 stars than for M2-M6
stars. However, the number of M7-L1 stars is too small to draw a firm
conclusion.

The lower activity level of later type stars has been noticed for a long time.
Mohanty et al. (2002) suggested that as the surface temperature decreases, the
resistivity of the stellar atmosphere increases, then the charged particles in
the magnetic field are carried away due to collisions with neutral gases and
thus the magnetic free energy used to produce the flare decreases with the
spectral type. However, this theory still needs to be examined by further
observations.

##  7 Conclusion

In this work, we presented 163 big flares in GWAC triggers from 162 individual
stars with the spectral type from M2 to L1. The flare amplitude
$\bigtriangleup G$ ranges from 0.83 to $\sim$10 mag, and the flare energy
ranges from $10^{32.2}$ to $10^{36.4}$ erg. From TESS or K2 light curves, we
found 1478 flares from 117 stars and calculated periods for 105 stars.
Besides, we obtained 154 low-resolution spectra for 120 individual stars with
the 2.16 m Telescope in the Xinglong Station, LAMOST and SDSS. Among them,
there are 108 available $L_{\rm H\alpha}/L_{\rm bol}$ for 82 individual stars
with available periods. We also obtained tangent velocities $V_{\rm T}$ for
157 stars , and velocities relative to LSR for 52 stars. From these data we
found that:

  * •

The energy of GWAC flares decreases with increasing $G_{\rm BP}-G_{\rm RP}$
(decreasing stellar surface effective temperature $T_{\rm eff}$) (see Figure
4), but both distributions of flare amplitude $\bigtriangleup G$ and flare
equivalent duration $\log_{10}(ED)$ seem to be independent of $T_{\rm eff}$
(see Figure 3).

  * •

If these GWAC flare stars are from the same category, then their periods show
they are younger than 670 Myr, $L_{\rm H\alpha}/L_{\rm bol}$ show they are
younger than 3 Gyr, and $V_{\rm T}$ show they are the thin disk stars (see
Figure 9 and 10). Therefore, they are young stars.

  * •

There are 16 stars with flare numbers greater than 20. From their FFDs we
found that most (13/16) GWAC flares can happen more than once every year, and
only three GWAC flares happen once every several or even 20 years (see Figure
8).

  * •

Stars with spectra and periods are in or very near to the saturation region,
so these stars should be very active.

  * •

$\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$ is higher for M2-M6 stars than for
M7-L1 stars. $\log_{10}(L_{\rm H\alpha}/L_{\rm bol})$ of M2-M6 stars is around
the saturation level, and decreases with increasing period, while for M7-L1
stars, though they are all fast rotators, their $L_{\rm H\alpha}/L_{\rm bol}$
are significantly lower (see Figure 10).

  * •

The flare energy divided by the stellar hemispherical area $\log_{10}(E_{\rm
bol}/S)$ seems to be higher for M2 - M6 stars than for M7-L1 stars (see Figure
11). However, the number of M7-L1 stars is too small to draw a firm
conclusion.

  * •

The maximum flare energy of M2 - M6 stars should be larger than
$\log_{10}E_{\rm bol}=\log_{10}S+14.25$ (see Figure 11).

The different activity levels between M7-L1 and M2-M6 stars had been reported
by Berger et al. (2010) in X-ray, radio and H$\alpha$, but we still do not
know if the different activity levels exist in $\log_{10}(E_{\rm bol}/S)$,
which should be explored by the sample including a lot of stars later than M6
that release big white-light flare energies. The study would be very
interesting because the origin of the activity pattern of cool stars is still
unclear.

The authors thank the anonymous referee very much for the valuable report that
inspired us to improve this work. This work is supported by the National
Natural Science Foundation of China (NSFC) with grant No. 12073038. Liang Wang
acknowledges the National Natural Science Foundation of China under Grant No.
U2031144. Hai-Long Yuan acknowledges the support from the Youth Innovation
Promotion Association of the CAS (Id. 2020060). We acknowledge the science
research grants from the China Manned Space Project. Guoshoujing Telescope
(the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a
National Major Scientific Project built by the Chinese Academy of Sciences,
Funding for the project has been provided by the National Development and
Reform Commission. LAMOST is operated and managed by the National Astronomical
Observatories, the Chinese Academy of Sciences. We acknowledge the support of
the staff of the Xinglong 2.16m telescope. This work was partially supported
by the Open Project Program of the Key Laboratory of Optical Astronomy,
National Astronomical Observatories, Chinese Academy of Sciences. This
research has made use of ”Aladin sky atlas” developed at CDS, Strasbourg
Observatory, France. 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. Some/all of the data
presented in this paper were obtained from the Mikulski Archive for Space
Telescopes (MAST) at the Space Telescope Science Institute. The specific
observations analyzed can be accessed via https://doi.org/10.17909/55e7-5x63
(catalog https://doi.org/10.17909/55e7-5x63) (STScI, 2022a),
https://doi.org/10.17909/T9H59D (catalog https://doi.org/10.17909/T9H59D)
(STScI, 2013), https://doi.org/10.17909/fwdt-2x66 (catalog
https://doi.org/10.17909/fwdt-2x66) (STScI, 2018),
https://doi.org/10.17909/T93W28 (catalog https://doi.org/10.17909/T93W28)
(STScI, 2016). STScI is operated by the Association of Universities for
Research in Astronomy, Inc., under NASA contract NAS5–26555. Support to MAST
for these data is provided by the NASA Office of Space Science via grant
NAG5–7584 and by other grants and contracts. This publication makes use of
data products from the Wide-field Infrared Survey Explorer, which is a joint
project of the University of California, Los Angeles, and the Jet Propulsion
Laboratory/California Institute of Technology, funded by the National
Aeronautics and Space Administration. This publication makes use of data
products from the Two Micron All Sky Survey, which is a joint project of the
University of Massachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by the National Aeronautics
and Space Administration and the National Science Foundation.

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