Vol.0 (20xx) No.0, 000–000

11institutetext: Department of Physics and Astronomy, School of Physical and
Chemical Sciences, Queen Mary University of London, Mile End Road, London, E1
4NS, UK<EMAIL_ADDRESS><EMAIL_ADDRESS>
22institutetext: Physics Department, College of Sciences, Princess Nourah Bint
Abdulrahman University, Riyadh, PO Box 84428, Saudi Arabia
Received 20xx month day; accepted 20xx month day

# New cases of super-flares on slowly rotating solar-type stars and large
amplitude super-flares in G- and M-type main-sequence stars.

A. k. Althukair  ID 1122 D. Tsiklauri  ID 11

###### Abstract

In our previous work, we searched for super-flares on different types of stars
while focusing on G-type dwarfs using entire Kepler data to study statistical
properties of the occurrence rate of super-flares. Using these new data, as a
by-product, we found fourteen cases of super-flare detection on thirteen
slowly rotating Sun-like stars with rotation periods of 24.5 to 44 days. This
result supports earlier conclusion by others that the Sun may possibly have a
surprise super-flare. Moreover, we found twelve and seven new cases of
detection of exceptionally large amplitude super-flares on six and four main-
sequence stars of G- and M-type, respectively. No large-amplitude flares were
detected in A, F, or K main-sequence stars. Here we present preliminary
analysis of these cases. The super-flare detection, i.e. an estimation of
flare energy, is based on a more accurate method compared to previous studies.
We fit an exponential decay function to flare light curves and study the
relation between e-folding decay time, $\tau$, vs. flare amplitude and flare
energy. We find that for slowly rotating Sun-like stars, large values of
$\tau$ correspond to small flare energies and small values of $\tau$
correspond to high flare energies considered. Similarly, $\tau$ is large for
small flare amplitudes and $\tau$ is small for large amplitudes considered.
However, there is no clear relation between these parameters for large
amplitude super-flares in the main sequence G- and M-type stars, as we could
not establish clear functional dependence between the parameters via standard
fitting algorithms.

###### keywords:

stars: activity — stars: flare — stars: rotation — stars: solar-type — stars:
statistics — Sun: flares

It is believed the Solar and stellar flares are powered by a physical process
called magnetic reconnection, in which connectivity of magnetic field lines in
the atmospheres of stars changes rapidly (Masuda et al. 1994; Shibata et al.
1995). This is accompanied by acceleration of plasma particles and release of
heat. The source of this kinetic and thermal energy is the energy stored in
the magnetic field. Thus, the magnetic dynamo process which is one of possible
means to generate magnetic field via bulk plasma flows is of great importance
for understanding of what can power and therefore be source for a flare or a
super-flare. The energies of observed stellar flares lie in the wide range
from $10^{28}$ to $10^{37}$ erg, while the highest energy of any observed
solar flare is approximately few times $10^{32}$ erg. Thus, generally agreed
terminology is that a super-flare should have an energy in excess of $10^{34}$
erg. A plausible dynamo model capable to explain the generation of magnetic
energy sufficient to support super-flares has been recently suggested in
Kitchatinov & Olemskoy (2016) and then further investigated in Katsova et al.
(2018). In this scenario, rather than producing stellar cycles similar to the
solar 11 year cycle, the dynamos in superflaring stars excite some quasi-
stationary magnetic configuration with a much higher magnetic energy. Further,
Kitchatinov et al. (2018) used a flux-transport model for the solar dynamo
with fluctuations of the Babcock-Leighton type $\alpha$-effect to generate
statistics of magnetic cycles. As a result, they concluded that the statistics
of the computed energies of the cycles suggest that super-flares with energies
in excess of $10^{34}$ erg are not possible on the Sun.

Historical records suggest that no super-flares have occurred on the Sun in
the last two millennia. In the past there were notable examples detection of
super-flares on Sun-like stars. There are two references which support such
clam Schaefer et al. (2000) and Nogami et al. (2014).

As claimed by Schaefer et al. (2000) they identified nine cases of super-
flares involving $10^{33}$ to $10^{38}$ ergs on main sequence Sun-like stars.
Sun-like means that stars are on or near the main-sequence, have spectral
class from F8 to G8, are single (or have a very distant binary companion). The
super-flare energy estimation by Schaefer et al. (2000) was based on
photometric methods.

Nogami et al. (2014) reported the results of high dispersion spectroscopy of
two ’super-flare stars’, KIC 9766237, and KIC 9944137 using Subaru/HDS
telescope. These two stars are G-type main sequence stars, and have rotation
periods of 21.8 days, and 25.3 days, respectively. Their spectroscopic results
confirmed that these stars have stellar parameters similar to those of the Sun
in terms of the effective temperature, surface gravity, and metallicity. By
using the absorption line of Ca II 8542, the average strength of the magnetic
field on the surface of these stars was estimated to be 1-20 G. The super-
flare energy estimation by Nogami et al. (2014) was based semi-empirical
method based on magnetic energy density times volume of flare. These results
claim that the spectroscopic properties of these super-flare stars are very
close to those of the Sun, and support the hypothesis that the Sun may have a
super-flare.

Statistical study of super-flares on different stellar types has been an
active area of research (Maehara et al. 2012; Shibayama et al. 2013; Wu et al.
2015; He et al. 2015, 2018; Yang et al. 2017; Van Doorsselaere et al. 2017; Lu
et al. 2019; Yang & Liu 2019; Günther et al. 2020; Tu et al. 2020; Gao et al.
2022). Shibayama et al. (2013) studied statistics of stellar super-flares.
These authors discovered that for Sun-like stars (with surface temperature
5600-6000 K and slowly rotating with periods longer than 10 days), the
occurrence rate of super-flares with an energy of $10^{34}-10^{35}$ erg is
once in 800-5000 yr. Shibayama et al. (2013) confirmed the previous results of
Maehara et al. (2012) in that the occurrence rate ($dN/dE$) of super-flares
versus flare energy $E$ shows a power-law distribution with $dN/dE\propto
E^{-\alpha}$, where $\alpha\sim 2$. Such occurrence rate distribution versus
flare energy is roughly similar to that for solar flares. Tu et al. (2020)
identified and verified 1216 super-flares on 400 solar-type stars by analyzing
2-minute cadence data from 25,734 stars observed during the first year of the
TESS mission. The results indicate a higher frequency distribution of super-
flares compared to the findings from the Kepler mission. This difference may
be due to a significant portion of the TESS solar-type stars in the dataset
are rapidly rotating stars. The power-law index $\gamma$ of the super-flare
frequency distribution was determined to be $\gamma=2.16\pm 0.10$, which is
consistent with the results obtained from the Kepler mission. The study
highlights an extraordinary star, TIC43472154, which exhibits approximately
200 super-flares per year. Tu et al. (2020) analyzed the correlation between
the energy and duration of super-flares, represented as $T_{duration}\propto
E^{\beta}$. They derived a power-law index $\beta=0.42\pm 0.01$ for this
correlation, which is slightly larger than the value of $\beta=1/3$ predicted
by magnetic reconnection theory. We believe that method for flare energy
estimation used in (Shibayama et al. 2013; Yang et al. 2017) is more accurate
than one used by Schaefer et al. (2000) and Nogami et al. (2014). We therefore
base of flare energy estimate on method of Shibayama et al. (2013) and
Althukair & Tsiklauri (2022) referred thereafter as Paper I.

In Paper I we searched for super-flares on different spectral class stars,
while focusing on G-type dwarfs (solar-type stars) using Kepler data using
quarters $0-17$ with the purpose of study the statistical properties of the
occurrence rate of super-flares. Shibayama et al. (2013) studied statistics of
stellar super-flares based on Kepler data in quarters $0-6$ ($Q0-Q6$). In
Paper I we investigated how the results are modified by adding more quarters,
i.e. what is $\alpha$ power-law for data quarters $0-17$ and $7-17$. Here
using the more extended Kepler data, we also found 14 cases of detection of
Super-flares on 13 Slowly Rotating Sun-like starts in each of KIC 3124010, KIC
3968932, KIC 7459381, KIC 7459381, KIC 7821531, KIC 9142489, KIC 9528212, KIC
9963105, KIC 10275962, KIC 11086906, KIC 11199277, KIC 11350663 and KIC
11971032. Thus the main purpose of the present study is to present analysis of
these new 14 cases. The main novelty here is that the detection is based on a
Shibayama et al. (2013) method, which is more accurate, as compared to
Schaefer et al. (2000) and Nogami et al. (2014) for the flare energy
estimation. Our results support earlier conclusion by others (Schaefer et al.
(2000) and Nogami et al. (2014)) that the Sun may have a surprise super-flare.
We stress that Paper I has conducted a more comprehensive study of
determination of stellar rotation periods based on a robust method such as
used by McQuillan et al. (2014) in comparison to Shibayama et al. (2013). We
believe that more accurate period determination used in Paper I has led to the
current new results, presented in this paper. In addition to the 14 cases of
of Super-flares on Slowly Rotating Sun-like starts, we detected 12 and 7
super-flares with a large amplitude on five G-type and four M-type main
sequence stars, respectively.

Solar flares emit energy at all wavelengths, but their spectral distribution
is still unknown. When white-light continuum emission is observed, the flares
are referred to as ”white-light flares” (WLF). Kretzschmar (2011) identified
and examined visible light emitted by solar flares and found that the white
light is present on average during all flares and must be regarded as a
continuum emission. Kretzschmar (2011) also demonstrated that this emission is
consistent with a black body spectrum with temperature of 9000 K and that the
energy of the continuum contains roughly 70% of the total energy emitted by
the flares. WLFs are among the most intense solar flares, and it has been
demonstrated that an optical continuum appears anytime the flare’s EUV or soft
X-ray luminosity reaches a reasonably large threshold (McIntosh & Donnelly
1972; Neidig & Cliver 1983). Thus, optical continuum is presumably present in
all flares but only in a few cases it reaches a measurable degree of
brightness. This conclusion implies that WLFs are not fundamentally different
from conventional flares Neidig (1989). Nonetheless, WLFs are important in
flare studies because they are similar to stellar flares ways Worden (1983)
and because they represent the most extreme conditions encountered in solar
optical flares Neidig (1989). Using observations from the Transiting Exoplanet
Survey Satellite (TESS), Ilin et al. (2021) present four fully convective
stars that exhibited white light flares of large size and long duration. The
underlying flare amplitude as a fraction of the quiescent flux of two flares
is greater than two. After the discovery of the largest amplitude flares ever
recorded on the L0 dwarf, which reached $\Delta V\approx-11$ magnitude Schmidt
et al. (2016), Jackman et al. (2019) detected a large amplitude white-light
super-flare on the L2.5 dwarf ULAS J224940.13011236.9 with the flux $\Delta
V\approx-10$ magnitude which corresponds to a relative brightness ratio of
10000. This can be demonstrated as follows:

$\Delta{V}=V_{\rm max}-V_{\rm min}\approx-10,$ (1)

where $V_{\rm max}$ is the apparent magnitude in the visible band
corresponding to the flux at the maximum amplitude ($F_{\rm max}$), and
$V_{\rm min}$ is the apparent magnitude in the visible band corresponding to
the flux at the minimum state ($F_{\rm min}$). Using the magnitude difference
calculation :

$V_{\rm max}-V_{\rm min}=-10=2.5\log_{10}(F_{\rm min}/F_{\rm max}),$ (2)

it is clear that $F_{\rm max}/F_{\rm min}={\Delta{F}}/{F}=10000$.

Here we present the detection of 14 super-flares on 13 slowly-rotating Sun-
like stars, 12 and 7 cases of large amplitude super-flares on five G-type
dwarfs and four M-type dwarfs respectively. Section 1 presents the method used
including the flare detection, the flare energy estimation, and rotation
period determination. Section 2 provides the main results of this study.
Section 3 closes this work by providing our main conclusions.

##  1 Methods

###  1.1 Flare Detection

We conducted an automated search for super-flares on main-sequence stars type
(A, F, G, K, M) based on entire Kepler data, using our Python script on long
cadence data from Data Release 25 (DR 25) following (Maehara et al. 2012;
Shibayama et al. 2013) method. The parameters for all targets observed by
Kepler have been taken from the Kepler Stellar interactive table in NASA
Exoplanet Archive. The study was carried out on a sample of main sequence
stars, comprising 2222, 10307, 25442, 10898, and 2653 stars for the spectral
types of M, K, G, F, and A, respectively. The following is a brief description
of this method. We generate light curves of the stars using the PDCSAP flux.
Then, in order to be statistically precise, we computed the distributions of
brightness variation by calculating the flux difference in adjacent time
intervals between every two neighboring data points in the light curve. Then,
we determine the flux difference value at which the area under the
distribution equals 1% of the total area. In order to increase the threshold,
the 1% value of the area was multiplied by a factor of three. The start time
of a flare was defined as the time at which the flux difference between two
consecutive points exceeded the threshold for the first time. To determine the
end time of the flare, we computed the three standard deviations 3$\sigma$ of
the distribution of brightness variation. Figure 1 displays a typical results
for this method for KIC 9963105. The light curve of KIC 9963105 are shown in
Figure 1(a). Figure 1(b) shows the distribution of the brightness difference
between every two adjacent data points of KIC 9963105 light curve. 1% of the
total area under the distribution curve is represented by the green vertical
line. The red vertical line represents the flare detection threshold value,
which is equal to three times 1% value of the area under the distribution
curve. The blue vertical line is 3$\sigma$ of the distribution of the
brightness variation. To determine the flare end time, we fit a B-spline curve
through three points on the relative flux ($\Delta F/F_{\rm avg}$) distributed
around the flare. One point just before the flare, and the other two points
five and eight hours after the flare peak, respectively. Then we subtract the
B-spline curve from the relative flux in order to remove long-term brightness
variations around the flare Shibayama et al. (2013). We define the flare end
time as the time when the relative flux produced by the subtraction drops
below the value of 3$\sigma$ for the first time. After detecting flare events,
conditions were applied to all flare candidates. These conditions are: the
flare duration must exceed 0.05 days, corresponding to at least three data
points two of them after the flare peak, and the flare’s decline phase must be
longer than its rising phase Shibayama et al. (2013). Only flare incidents
meeting these criteria were analysed.

Figure 1: Illustrations of flares detection method used by Shibayama et al.
(2013). (a) The light curve of KIC 9963105. (b) The distribution of brightness
variation between each pair of adjacent data points in the light curves of KIC
9963105. The green vertical line represents the value of 1% of the total area
under the curve, the red vertical lines represents the flare detection
threshold and the blue vertical represents $3\sigma$ of the brightness
variation distribution.

###  1.2 Energy Calculation

Schaefer et al. (2000) identified nine cases of super-fares involving
$10^{33}-10^{38}$ ergs on normal solar-type stars. Their super-flare energy
estimate has a large uncertainty, e.g. Groombridge 1830 (HR 4550) total flare
energy (in the blue band alone) is $10^{35}$ ergs with an uncertainty of a
factor of a few due to having only four points on the light curve.

The possibility that super-flares can be explained by magnetic energy stored
on the star’s surface was considered by Nogami et al. (2014) Using the Ca ii
8542 absorption line, they estimate that the average magnetic field strength
($B$) of KIC 9766237 and KIC 9944137 is 1-20 Gauss, and the super-flare of
these targets has a total energy of $10^{34}$ erg. Under the assumption that
the energy released during the flare represents a fraction ($f$) of the
magnetic energy stored around the spot area. Their flare energy ($E_{\rm
flare}$) was calculated as follows:

$E_{\rm flare}\sim f\dfrac{B^{2}}{8\pi}L^{3}.$ (3)

The length of the magnetic structure causing the flare ($L$), has been
considered to be the same size as the spotted region, i.e. $L=\sqrt{a\pi
R_{\ast}^{2}}$ where $a$ is the spot’s area, which giving that

$E_{\rm flare}\sim f\dfrac{B^{2}}{8\pi}(a\pi R_{\ast}^{2})^{3/2}.$ (4)

Our energy estimation for each flare depends on the star’s luminosity ($L_{\rm
star}$), flare amplitude ($f_{\rm amp}$), and flare duration (Shibayama et al.
2013; Yang et al. 2017). $L_{\rm star}$, the amount of energy that the star
emits in one second, is proportional to the star’s radius $R$ squared and its
surface temperature $T_{\rm eff}$ to the fourth power, and is obtained from
the following equation:

$L_{\rm star}=\sigma_{\rm SB}T_{\rm eff}^{4}4\pi R^{2},$ (5)

where $\sigma_{\rm SB}$ is the Stefan-Boltzmann constant, $4\pi R^{2}$ is the
entire surface area of the star. The continuum emission from a white-light
flare is consistent with blackbody radiation at around 9000 K, as suggested by
(Hawley & Fisher 1992; Kretzschmar 2011). Based on (Shibayama et al. 2013;
Yang et al. 2017; Günther et al. 2020), we set $T_{\rm flare}$ = 9000 K and
derive the luminosity of a blackbody-emitting star as follows:

$L_{\rm flare}(t)=\sigma_{\rm SB}T^{4}_{\rm flare}A_{\rm flare},$ (6)

where $A_{\rm flare}$ is the flare’s area, as determined by the formula:

$A_{\rm flare}(t)=f_{\rm amp}(t)\pi R^{2}\frac{\int
R_{\lambda}B_{\lambda}(T_{\rm eff})d\lambda}{\int
R_{\lambda}B_{\lambda}(T_{\rm flare})d\lambda},$ (7)

where $f_{\rm amp}$ represents the flare amplitude for the relative flux and
$R_{\lambda}$ represents the Kepler instrument’s response function Caldwell et
al. (2010). The Kepler photometer covers various wavelengths, from 420 to 900
nm. The Plank function at a specific wavelength, denoted by $B_{\lambda}(T)$,
is given by:

$B_{\lambda}(T)=\frac{{2hc^{2}}/{\lambda^{5}}}{e^{{hc}/{\lambda kT}}-1},$ (8)

where $h$ represents Planck’s constant, $c$ the speed of light, $T$ the black
body temperature, and $k$ Boltzmann’s constant. By substituting Eq.(7) into
(6), we calculate the total flare energy by the integral of $L_{\rm flare}$
over the flare duration :

$E_{\rm flare}=\int_{t_{\rm start}}^{t_{\rm end}}L_{\rm flare}(t)dt.$ (9)

###  1.3 Rotational Period Determination

Light curve periods were calculated using the Lomb-Scargle periodogram, a
common statistical approach for detecting and characterising periodic signals
in sparsely sampled data. We used an oversampling factor of five VanderPlas
(2018), and use PDCSAP flux to generate a Lomb-Scargle periodogram for each
light curve from Q2 to Q16. Furthermore, the period corresponding to the
maximum power of the periodogram was allocated as the rotation period for the
Kepler ID in a specific quarter. This value was estimated with an accuracy of
a day without the decimal component because fractions of a day would not
significantly alter the results, allowing us to automate the selection of the
star’s rotation period rather than manually. We set 0.5 days for periods
shorter than a day and eliminated periods less than 0.1 days. Finally, for
each Kepler ID, we choose the most frequent period across all quarters from Q2
to Q16. Following the McQuillan et al. (2014) technique, we required that the
period chosen for all quarters be identified in at least two unique segments,
with the segment defined as three consecutive Kepler quarters. (Q2,Q3,Q4),
(Q5, Q6, Q7), (Q8, Q9, Q10), (Q11, Q12, Q13) and (Q14,Q15,Q16).

##  2 Results

By performing an automated search for super-flares on G-type main-sequence
stars during 1442 days of Kepler observation in all of (DR 25) long-cadence
data from Q0 to Q17, we found 14 super-flares on 13 slowly rotating Sun-like
stars in each of (KIC 3124010, KIC 3968932, KIC 7459381, KIC 7459381, KIC
7821531, KIC 9142489, KIC 9528212, KIC 9963105, KIC 10275962, KIC 11086906,
KIC 11199277, KIC 11350663 and KIC 11971032), with a surface temperature of
$5600K\leqslant T_{\rm eff}<6000K$, a surface gravity of $\log\ g>4.0$, and a
rotational period $P_{\rm rot}$ range between 24.5 to 44 days. Figure 3 shows
seven light curves of these events. The left panels display light curves over
a 90-day of observation. The blue arrow on the left panel indicates the
observed super-flare, which met all conditions. The right panels show zoomed
in time light curves of super-flares. The blue squares represent the data
points for a super-flare. We fitted an exponential decay function to the flare
light curve to characterise the flares, shown by a red dashed curve. This
exponential decay function is given by:

$f(t)=a\ e^{-t/\tau}+b$ (10)

where $f(t)$ is the relative flux as a function of time, $a$ is the flare peak
height, which is approximately equal to the relative flux at the flare peak,
$b$ is the relative flux in the quiescent state and $\tau$ is the decay time
of the flare, which is the time at which the relative flux is decreased to
$1/e\simeq$ 0.3679 of its initial value. The value of $\tau$, $a$ and $b$ of
the exponential decay function for each flare are shown in the right panel.

Figure 2: The left panel displays the light curves of super-flares. The x-axis
in the left panel is the time in (BJD) and the y-axis is the normalized flux.
The blue arrows indicates the occurrence of super-flares. The right panels
show zoom in time of these super-flares. The x-axis in the right panel is the
time from the flare peak in (day) and the y-axis is the relative flux
($\rm\Delta{F}/F_{avg}$). Each data points for a super-flare is represented by
blue squares in the right panel. The dashed red curve indicates an exponential
fit of the decay phase. $\tau$ in the equation refers to the best fit of
exponential decay time, $a$ refers to the final value of the amplitude fit and
$b$ refers to the fitted relative flux in the quiescent state.

Figure 3: Figure 3 continued

Flare parameters and their duration, amplitudes, energies and $\tau$ values
are listed in Table 1. The rotation periods for these slowly rotating Sun-like
stars were taken from McQuillan et al. (2014). Their flare energies range from
$(1.9-9.0)\times 10^{34}$ erg. The flare amplitude of the slowly rotating Sun-
like stars is relatively small, ranging between 0.002 and 0.018. We also find
that the duration of flares in all of these cases is the same 0.061 days. This
is probably due to the fact that flare duration of all small amplitude super-
flares is two data points in time or less. Therefore, because one of the flare
detection conditions in our code is that there must be at least two data
points between the flare’s peak and the end, cases with flare duration less
than two points were not detected, and we end up with the same duration super-
flares with the two data points. Because it appears that there are no small
amplitude super-flares with duration greater than two data points, all flare
duration end up the same. This selection effect is because we use long cadence
light curve data, which has a 29.4-minute interval between each data point in
time. The value of $\tau$ varies from 0.012 to 0.036 days.

Table 1: super-flares on slowly rotating Sun-like stars.

Kepler ID $\rm T_{eff}$ log g Radius $P_{\rm rot}$a $\rm t_{start}$ $\rm
t_{end}$ $\rm t_{peak}$ amp Flare Duration $\rm\tau$ Flare Energy ($K$)
($R_{\odot}$) (day) (BJD) (BJD) (BJD) (day) (day) (erg) 3124010 5688 4.46 1.01
25.90 913.24 913.30 913.26 0.008 0.061 0.017 $4.57\times 10^{34}$ 3968932 5716
4.39 0.96 24.56 868.88 868.94 868.90 0.004 0.061 0.026 $3.89\times 10^{34}$
7459381 5635 4.27 1.11 26.19 312.31 312.37 312.33 0.005 0.061 0.017
$4.89\times 10^{34}$ 7668358 5668 4.36 0.98 41.83 841.13 841.19 841.15 0.003
0.061 0.030 $1.94\times 10^{34}$ 7821531 5681 4.52 0.92 32.66 891.81 891.87
891.83 0.018 0.061 0.015 $8.63\times 10^{34}$ 9142489 5878 4.51 0.95 25.20
1561.13 1561.19 1561.15 0.004 0.061 0.028 $2.61\times 10^{34}$ 9528212 5872
4.42 0.97 61.43 1332.34 1332.40 1332.36 0.003 0.061 0.036 $2.31\times 10^{34}$
9963105 5751 4.39 1.01 28.09 883.80 883.86 883.82 0.016 0.061 0.012
$9.00\times 10^{34}$ 10275962 5782 4.51 0.91 26.12 213.21 213.27 213.23 0.007
0.061 0.035 $3.71\times 10^{34}$ 10275962 5782 4.51 0.91 26.12 599.85 599.91
599.87 0.004 0.061 0.029 $2.04\times 10^{34}$ 11086906 5758 4.38 1.11 29.18
1206.01 1206.07 1206.03 0.002 0.061 0.018 $1.90\times 10^{34}$ 11199277 5638
4.49 0.92 29.00 325.43 325.49 325.45 0.008 0.061 0.029 $3.72\times 10^{34}$
11350663 5966 4.49 0.96 36.92 1232.31 1232.37 1232.33 0.010 0.061 0.014
$5.21\times 10^{34}$ 11971032 5942 4.51 0.94 44.00 1231.92 1231.98 1231.94
0.006 0.061 0.030 $3.80\times 10^{34}$

  * •

a Rotation period from McQuillan et al. (2014).

In addition to the 14 cases of of super-flares on slowly rotating Sun-like
starts, we detected 12 super-flares with a large amplitude on 6 G-type dwarfs
in each of KIC 5865248, KIC 6783223, KIC 7505473, KIC 10053146, KIC 10057002
and KIC 11709752. Figure 5 shows eight light curves of these events same as
Figure 3. Table 2 shows the duration, amplitude, energy, and $\tau$ values for
these super-flares with their parameters. The energy of their flares range
from $1.67\times 10^{36}$ to $1.42\times 10^{38}$ erg. The rotation period of
KIC 1170952 was obtained by this work. As for the rotation period for the
other five stars, no such data is available. Even applying the method
described in Paper I does not allow period determination in these five cases.
According to Yang et al. (2017), there are three possible reasons: (i) due to
the inclination angle and low activity level, the light curve has a small
amplitude at the accuracy level of Kepler; (ii) fast-rotating stars have spots
in the poles Schüssler & Solanki (1992), making it hard to detect light
variation through rotation; and (iii) the rotation period is longer than 90
days (a quarter), making it difficult (or impossible) to detect them in the
frequency spectrum of the star. The flare amplitude for these cases range
between 4.05 and 35.60. These flares tend to last longer than flares with
smaller amplitude of slowly rotating Sun-like stars as their duration varies
between 0.061 and 0.143 day. The $\tau$ values of flares exhibiting large
amplitude on G-type main-sequence stars are observed to be higher than those
of flares occurring on slowly rotating Sun-like stars, as their values range
between 0.014 and 0.058 days.

Figure 4: Same as Figure 3 but for large amplitude super-flares on G-type
main-sequence stars.

Figure 5: Same as Figure 3 but for large amplitude super-flares on G-type
main-sequence stars Table 2: Large amplitude super-flares on G-type main-
sequence stars.

Kepler ID $\rm T_{eff}$ log g Radius $P_{\rm rot}$a $\rm t_{start}$ $\rm
t_{end}$ $\rm t_{peak}$ amp Flare Duration $\rm\tau$ Flare Energy ($K$)
($R_{\odot}$) (day) (BJD) (BJD) (BJD) (day) (day) (erg) 5865248 5780 4.44 1 NA
1476.22 1476.33 1476.24 13.16 0.102 0.036 $7.29\times 10^{37}$ 5865248 5780
4.44 1 NA 1496.09 1496.21 1496.11 8.78 0.123 0.041 $9.56\times 10^{37}$
5865248 5780 4.44 1 NA 1561.78 1561.86 1561.80 8.14 0.082 0.034 $6.75\times
10^{37}$ 6783223 5780 4.44 1 NA 1510.76 1510.86 1510.78 35.60 0.102 0.030
$8.6\times 10^{37}$ 7505473 5780 4.44 1 NA 1385.81 1385.95 1385.85 4.05 0.143
0.058 $1.67\times 10^{36}$ 10053146 5780 4.44 1 NA 1411.27 1411.33 1411.29
22.75 0.061 0.040 $1.42\times 10^{38}$ 10057002 5780 4.44 1 NA 1284.52 1284.58
1284.54 12.03 0.061 0.049 $6.24\times 10^{37}$ 11709752 5780 4.44 1 0.5
1501.56 1501.62 1501.58 4.21 0.061 0.031 $2.85\times 10^{37}$ 11709752 5780
4.44 1 0.5 1544.06 1544.13 1544.08 7.65 0.061 0.014 $2.85\times 10^{37}$
11709752 5780 4.44 1 0.5 1571.49 1571.55 1571.51 4.48 0.061 0.033 $3.94\times
10^{37}$ 11709752 5780 4.44 1 0.5 1575.35 1575.43 1575.37 5.11 0.082 0.046
$5.44\times 10^{37}$ 11709752 5780 4.44 1 0.5 1578.39 1578.45 1578.41 9.40
0.061 0.030 $3.94\times 10^{37}$

  * •

a Rotation period from (Althukair & Tsiklauri 2023a).

For stars of other spectral classes, no significant flares with large
amplitudes were detected on main-sequence stars of type A, F, and K. Only
M-type main-sequence stars manifested seven super-flares with large amplitude
on each of KIC 6580019, KIC 7123391, KIC 7341517 and KIC 9201463. Similar to
Figures 3 and 5, Figure 7 displays the seven light curves for these events.
The parameters of these super-flares, including their duration, amplitude,
energy, and $\tau$ values, are displayed in Table 3. These flares have an
energy between $3.16\times 10^{33}$ and $1.59\times 10^{35}$ erg, amplitude
ranges between 3.91 and 15.14 and their duration lasts between 0.018 and 0.044
day. $\tau$ values for super-flares with large amplitude on M-type main
sequence stars vary from 0.030 to 0.049 days.

Table 3: Large amplitude super-flares on M-type main-sequence stars.

Kepler ID $\rm T_{eff}$ log g Radius $P_{\rm rot}$ $\rm t_{start}$ $\rm
t_{end}$ $\rm t_{peak}$ amp Flare Duration $\rm\tau$ Flare Energy ($K$)
($R_{\odot}$) (day) (BJD) (BJD) (BJD) (day) (day) (erg) 6580019 2661 5.28 0.12
NA 609.97 610.09 609.99 3.91 0.123 0.044 7.04$\times 10^{33}$ 6580019 2661
5.28 0.12 NA 674.45 674.60 674.47 10.30 0.143 0.041 1.25$\times 10^{34}$
7123391 3326 5.12 0.19 NA 638.53 638.59 638.55 8.07 0.061 0.038 4.76$\times
10^{34}$ 7123391 3326 5.12 0.19 NA 692.09 692.19 692.13 12.38 0.102 0.018
1.23$\times 10^{35}$ 7123391 3326 5.12 0.19 NA 794.54 794.72 794.56 15.14
0.184 0.049 1.59$\times 10^{35}$ 7341517 2661 5.28 0.12 NA 877.95 878.12
877.99 5.27 0.163 0.041 4.82$\times 10^{33}$ 9201463 3319 5.14 0.18 NA 215.11
215.27 215.15 5.51 0.163 0.030 3.16$\times 10^{33}$

Figure 6: Same as Figures 3 and 5 but for large amplitude super-flares on
M-type main-sequence stars.

Figure 7: Same as Figures 3 and 5 but for large amplitude super-flares on
M-type main-sequence stars.

We examined whether there is a dependence between $\tau$ vs. flare amplitude
($f_{\rm amp}$) and $\tau$ vs. flare energy ($E_{\rm flare}$). Therefore, we
graphically display six panels in Figure 8 showing the relationship between
$\tau$ and the amplitude of flares and $\tau$ and the energy of flares in
slow-rotating Sun-like stars 8(a, b), G-type stars 8(c, d) and M-type stars
8(e, f) respectively. In 8(a) for slowly-rotating Sun-like stars, we find that
for small amplitude, $\tau$ is large, and when the amplitude is large, $\tau$
is consistently small in the range considered. The same applies to the
relation between $\tau$ and energy in Figure 8(b), we see that large $\tau$
values correspond to small energies and small values of $\tau$ correspond to
large energies considered. On the contrary, there is no clear relation between
$\tau$ vs. $f_{\rm amp}$ and $\tau$ vs. $E_{\rm flare}$ in G-type and M-type
main sequence stars in Figure 8(c-f). We tried various functions of fit using
Python’s curve_fit and Excel’s trendline, referred to as a (line of best fit),
to visualize the general trend for the data. We could not find any reliable,
functional fit dependence between those parameters because the coefficient of
determination, $\rm R^{2}$, which shows how well the data fit the regression
model, is less than 0.5 for all those cases in Figure 8(a to f). Hence any
attemted fit has been unreliable, as only fit with $\rm R^{2}>0.5$ can be
deemed acceptable.

Figure 8: The left panels display a scatter plot showing the relation between
tau values on the y-axis with the flare amplitude $f_{\rm amp}$ on the x-axis.
While the right panels display a scatter plot showing the relation between
$\tau$ values on the y-axis with the flare energy $E_{\rm flare}$ on the
x-axis. For flares on slowly rotating Sun-like stars,(a) demonstrates that
$\tau$ values are large for low flare amplitudes but consistently small for
high flare amplitudes. Likewise,(b) demonstrates that high $\tau$ values
correspond to low flare energies, whereas low $\tau$ values correspond to high
flare energies. For large amplitude super-flares on G-type dwarfs (c,d) and
M-type dwarfs (e,f), $\tau$ has no clear connection to the $f_{\rm amp}$ or
$E_{\rm flare}$.

##  3 Conclusions

Using our Python script on long cadence data from Data Release 25 (DR 25), we
searched for super-flares on main-sequence stars of types (A, F, G, K, and M)
based on the entire Kepler data following the method of (Maehara et al. 2012;
Shibayama et al. 2013). The Kepler targets’ parameters were retrieved from the
Kepler Stellar interactive table in the NASA Exoplanet Archive. Using these
data, we detected 14 super-flare on 13 Sun-like stars with a surface
temperature of $5600K\leqslant T_{\rm eff}<6000K$, and $P_{\rm rot}$ range
from 24.5 to 44 days. In addition, we found 12 and 7 cases of large amplitude
super-flares on six and four main-sequence G and M type stars, respectively.
Main-sequence stars of other spectral types A, F, and K showed no signs of
large-amplitude super-flares. To characterise the flares, we fit an
exponential decay function to the flare light curve given by $f(t)=a\
e^{-t/\tau}+b$. We study the relation between the decay time of the flare
after its peak $\tau$ vs. $f_{\rm amp}$ and $\tau$ vs. $E_{\rm flare}$. For
slowly rotating Sun-like stars, we find that $\tau$ is large for small flare
amplitudes and $\tau$ is small for large flare amplitudes considered.
Similarly, we find that large $\tau$ values correspond to small flare energies
and small $\tau$ values correspond to high flare energies considered. However,
for the main sequence stars of the G and M types, $\tau$ has no apparent
relation to the $f_{\rm amp}$ or $E_{\rm flare}$. We experimented with several
different fit functions between $\tau$ vs. $f_{\rm amp}$ and $\tau$ vs.
$E_{\rm flare}$ to better see the underlying pattern in the data. Since the
$\rm R^{2}$ is less than 0.5 in these cases, we could not identify a reliable
fit functional dependence between these parameters.

In conclusion, we believe that:
(i) the thirteen peculiar Kepler IDs that are Sun-like, slowly rotating with
rotation periods of 24.5 to 44 days, and yet can produce a super-flare with
energies in the range of $(2\textendash 9)\times 10^{34}$ erg; and
(ii) six G-type and four M-type Kepler IDs with exceptionally large amplitude
super-flares, with the relative flux in the range $\Delta F/F_{\rm avg}=4-35$;
defy our current understanding of stars and hence are worthy of further
investigation.

## Acknowledgements

Some 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 NNX13AC07G and by other grants and contracts.
Authors would like to thank Deborah Kenny of STScI for kind assistance in
obtaining the data, Cozmin Timis and Alex Owen of Queen Mary University of
London for the assistance in data handling at the Astronomy Unit.
A. K. Althukair wishes to thank Princess Nourah Bint Abdulrahman University,
Riyadh, Saudi Arabia and Royal Embassy of Saudi Arabia Cultural Bureau in
London, UK for the financial support of her PhD scholarship, held at Queen
Mary University of London.

## Data Availability

All data used in this study was generated by our bespoke Python script that
can be found at https://github.com/akthukair/AFD under the filename AFD.py and
other files in the same GitHub repository. The data underlying this article
were accessed from Mikulski Archive for Space Telescopes (MAST)
https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html. The long-
cadence Kepler light curves analyzed in this paper can be accessed via MAST
STScI (2016). The Kepler Stellar parameters table for all targets can be found
at Kepler Mission (2019). The derived data generated in this research will be
shared on reasonable request to the corresponding author.

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