# Finding the particularity of the active episode of SGR J1935+2154 during
which FRB 20200428 occurred: Implication from a statistics of Fermi/GBM X-ray
bursts

Sheng-Lun Xie Institute of Astrophysics, Central China Normal University,
Wuhan 430079, China  Key Laboratory of Particle Astrophysics, Institute of
High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing
100049, China  Yun-Wei Yu Institute of Astrophysics, Central China Normal
University, Wuhan 430079, China  Key Laboratory of Quark and Lepton Physics
(Central China Normal University), Ministry of Education, Wuhan 430079, China
Yun-Wei Yu<EMAIL_ADDRESS>Shao-Lin Xiong  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
19B Yuquan Road, Beijing 100049, China Shao-Lin Xiong<EMAIL_ADDRESS>Lin
Lin  Department of Astronomy, Beijing Normal University, Beijing 100088, China
Ping Wang  Key Laboratory of Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China
Yi Zhao  School of Computer and Information, Dezhou University, Dezhou 253023,
China  Yue Wang  Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049,
China  Wen-Long Zhang  School of Physics and Physical Engineering, Qufu Normal
University, Qufu, Shandong 273165, China  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
19B Yuquan Road, Beijing 100049, China

###### Abstract

By using the Fermi/GBM data of the X-ray bursts (XRBs) of SGR J1935+2154, we
investigate the temporal clustering of the bursts and the cumulative
distribution of the waiting time and fluence/flux. It is found that the bursts
occurring in the episode hosting FRB 20200428 have obviously shorter waiting
times than those in the other episodes. The general statistical properties of
the XRBs further indicate they could belong to a self-organized critical (SOC)
system (e.g., starquakes), making them very similar to the earthquake
phenomena. Then, according to a unified scaling law between the waiting time
and energy of the earthquakes as well as their aftershocks, we implement an
analogy analysis on the XRBs and find that the FRB episode owns more dependent
burst events than the other episodes. It is indicated that the FRB emission
could be produced by the interaction between different burst events, which
could correspond to a collision between different seismic/Alfven waves or
different explosion outflows. Such a situation could appear when the magnetar
enters into a global intensive activity period.

magnetars - soft gamma-ray repeaters: general - methods: data analysis -
techniques

##  1 Introduction

Generally, magnetars are a class of neutron stars with an ultrahigh magnetic
field ($\sim 10^{14}-10^{15}$ G), which usually exhibit in the Galaxy as
pulsars of a long spin period ($2\sim 12$ s) and a fast spin-down rate ($\sim
10^{-13}-10^{-11}$ s s-1, Kouveliotou et al., 1998; Olausen & Kaspi, 2014).
The marked feature of Galactic magnetars is their X-ray or soft $\gamma$-ray
burst emission (Thompson & Duncan, 1995), which are observed in more than 2/3
of the magnetar population (Olausen & Kaspi, 2014).

On 28th April 2020, from the direction of a Galactic magnetar, SGR J1935+2154,
the Canadian Hydrogen Intensity Mapping Experiment (CHIME) captured a fast
radio burst (FRB) emission (Bochenek et al., 2020; CHIME/FRB Collaboration et
al., 2020), which was precisely temporally associated with an X-ray burst
(XRB; Li et al., 2021, 2022a; Ge et al., 2022). This magnetar had just entered
an active state of XRBs about half a month before FRB 20200428 occurred (Li et
al., 2021; Cai et al., 2022a, b; Mereghetti et al., 2020; Younes et al., 2020;
Tavani et al., 2020; Veres et al., 2020; Ridnaia et al., 2020; Barthelmy et
al., 2020; Lin et al., 2020a; Ridnaia et al., 2021; Kaneko et al., 2021), and
remained active at other times as well (Israel et al., 2016; Lin et al.,
2020b; Borghese et al., 2022; Xie et al., 2022, 2023; Rehan & Ibrahim, 2023).
Meanwhile, the dispersion measure (DM) of FRB 20200428 ($\sim 332.7$ pc cm -3)
is well consistent with the distance of SGR J1935+2154 in the range of
$6.6-12.5$ kpc (Kothes et al., 2018; Zhou et al., 2020; Zhong et al., 2020).
These discoveries indicated that magnetars could be sources of the mysterious
FRB phenomena (Lorimer et al., 2007; Thornton et al., 2013).

In view of their millisecond duration and high energy releases, FRBs are
widely suggested to originate from the violent activities of compact objects,
in particular, magnetars (Popov & Postnov, 2010; Kulkarni et al., 2014; Katz,
2016; Connor et al., 2016; Cordes & Wasserman, 2016; Lyutikov, 2017a), which
was firstly favored by the discovery of the repeating FRB 20121102 as well as
its persistent radio counterpart (Kashiyama & Murase, 2017; Metzger et al.,
2017; Cao et al., 2017; Dai & Lu, 2017; Michilli et al., 2018). Then, the
direct connection of FRB 20200428 with SGR J1935+2154 undoubtedly offered a
smoking-gun evidence for one of the FRB/FRB-like origins, the magnetar origin,
although FRB 20200428 is still distinctive due to its obviously low luminosity
than its cosmological cousins (CHIME/FRB Collaboration et al., 2020; Bochenek
et al., 2020).

Generally, the amounts of the energy releases of the bursts and outbursts of
Galactic magnetars, from both soft gamma-ray repeaters (SGRs) and anomalous
X-ray pulsars (AXPs), indicate that XRBs probably correspond to sudden and
local (not global) catastrophic events. Since the energy is considered to
eventually come from magnetic fields, the XRBs are most likely to be caused by
sudden reordering of the magnetic field structure, which either deforms the
crust drastically or/and leads to a major reconfiguration of the
magnetosphere. As a result, an optically thick $e^{\pm}$ fireball can be
expected to form, radiating X-rays (Thompson & Duncan, 1995, 2001;
Beloborodov, 2021; Lyutikov, 2022). In the framework of such a starquake
model, regardless of the specific X-ray radiation mechanism, some scale
independent behaviors can be expected to appear in the statistics of the
energy and waiting times of XRBs, just as found from earthquakes (Cheng et
al., 1996). Many systems such as starquakes and earthquakes, consisting of a
large number of entities that interact with each other in a complex way, can
exhibit nonlinear behavior, which can be described by a self-organized
criticality (SOC) theory. The complex interactions in such systems can
occasionally lead to instabilities with subsequent small events. The largest
events, the subsequent events, and the myriads of smaller events can finally
share the same statistical property (Sornette, 2004; Tsallis, 2009;
Aschwanden, 2011).

Despite a basic consensus that could appear on the XRB origin, the mechanism
responsible for the FRB emission is still under great debate. In the
literature, two types of models have been suggested frequently, including the
curvature radiation in the magnetosphere (Lu & Kumar, 2018; Yang & Zhang,
2018, 2020; Long & Pe’er, 2018; Wang et al., 2019; Kumar & Lu, 2020) and the
shock synchrotron maser radiation (Lyubarsky, 2014; Beloborodov, 2017;
Ghisellini, 2017; Kumar et al., 2017; Lyutikov, 2017b; Waxman, 2017; Metzger
et al., 2019; Plotnikov & Sironi, 2019; Yu et al., 2021). In the former case,
the curvature radiation could happen in the magnetosphere because the crust
fracturing forms a charge-starved region and leads to the plasma discharges.
In the latter case, the shock responsible for the synchrotron maser can be
driven by the collisions between different ejecta of different burst
activities. In observation, although FRB 20200428 was found to be in temporal
coincidence with an XRB, the deep observations implemented by the Five Hundred
Meter Aperture Telescope (FAST) on some other XRBs of SGR J1935+2154 ruled out
the possibility of their association with FRB emission (Lin et al., 2020c). On
the one hand, this fact could just be due to the narrow beaming of FRB
emission, while the XRB emission is generally isotropic. In this case, there
is in principle no intrinsic difference between the different XRBs. On the
other hand, reversely, the un-association with FRB emission of other XRBs may
indicate that the XRB associated with FRB 20200428 is intrinsically different
from the others.

Therefore, in this paper, we first investigate the statistical distributions
of the energy and waiting times of the XRBs of SGR J1935+2154 in order to
reveal their possible SOC behaviors. Furthermore, these XRBs are separated
into several different active episodes and the possible particularity of the
active episode where FRB 20200428 occurred is further investigated, in order
to test whether the FRB-associated XRB is intrinsically different from the
other XRBs or the non-detection of FRB emission is just because of the off-
axis observational direction. This could help us to uncover the FRB emission
mystery.

Figure 1:  The XRBs (triangle markers) of SGR J1935+2154 detected by GBM
within different observation windows (horizontal lines). The length of the
lines represents the duration of the windows. The first burst in each window
is labelled by a black triangle, which is dropped in our statistics since its
waiting time cannot be defined. Then, the windows that only have one burst are
excluded (i.e, the grey lines). The remaining windows having sufficiently
large number of bursts can be basically separated into four active episodes,
as labelled by different colors (see Section 2.2 for active episode
definition).

Figure 2: The corner plot of the shape parameter $k$ and the event rate $\lambda$ for the Weibull distribution (see Section 2.1). Table 1: The start time and duration $T$ of the observation windows of the Fermi/GBM on SGR J1935+2154 Start time | $T$ | Start time | $T$ | Start time | $T$ | Start time | $T$
---|---|---|---|---|---|---|---
(UTC) | (hour) | (UTC) | (hour) | (UTC) | (hour) | (UTC) | (hour)
2014-07-05T09:11:45.34 | 0.74 | 2016-05-20T20:48:13.14 | 0.99 | 2019-11-04T07:04:07.94 | 0.99 | 2021-09-10T23:13:31.74 | 0.66
2015-02-22T06:50:37.34 | 0.65 | 2016-05-21T03:09:08.14 | 0.99 | 2019-11-04T08:39:19.94 | 0.99 | 2021-09-11T02:03:41.74 | 1.00
2015-02-22T08:30:36.34 | 0.57 | 2016-05-21T11:05:17.14 | 0.71 | 2019-11-04T10:14:31.94 | 0.99 | 2021-09-11T05:14:02.74 | 1.00
2015-02-22T13:28:02.34 | 0.38 | 2016-05-21T20:36:38.14 | 0.99 | 2019-11-04T11:53:16.94 | 0.93 | 2021-09-11T09:59:32.74 | 0.99
2015-02-22T14:22:13.34 | 0.44 | 2016-05-21T23:47:05.14 | 0.99 | 2019-11-04T15:00:43.94 | 0.98 | 2021-09-11T11:34:42.74 | 0.99
2015-02-22T17:32:56.34 | 1.07 | 2016-05-22T02:57:32.14 | 0.99 | 2019-11-04T18:10:29.94 | 0.99 | 2021-09-11T13:14:31.74 | 0.92
2015-02-22T19:08:17.34 | 1.07 | 2016-05-29T23:48:59.34 | 1.02 | 2019-11-04T19:45:40.94 | 0.99 | 2021-09-11T14:53:49.74 | 0.85
2015-02-22T20:43:38.34 | 1.07 | 2016-06-02T11:56:06.74 | 0.82 | 2019-11-04T22:56:04.94 | 0.99 | 2021-09-11T16:32:48.74 | 0.78
2015-02-22T23:54:20.34 | 1.07 | 2016-06-06T20:42:44.14 | 1.09 | 2019-11-05T00:31:15.94 | 0.84 | 2021-09-11T18:11:38.74 | 0.72
2015-02-23T01:29:41.34 | 1.07 | 2016-06-18T01:30:28.54 | 0.59 | 2019-11-05T05:16:50.94 | 0.99 | 2021-09-11T19:50:23.74 | 0.66
2015-02-23T04:58:26.34 | 0.76 | 2016-06-18T20:19:38.54 | 0.82 | 2019-11-05T06:52:02.94 | 0.99 | 2021-09-11T22:40:53.74 | 0.25
2015-02-23T06:39:32.34 | 0.67 | 2016-06-20T14:59:09.54 | 1.01 | 2019-11-14T00:16:54.94 | 0.72 | 2021-09-12T00:16:03.74 | 0.99
2015-02-23T14:12:29.34 | 0.42 | 2016-06-22T13:01:00.14 | 1.00 | 2019-11-14T19:19:05.94 | 0.58 | 2021-09-12T05:01:33.74 | 0.99
2015-02-23T15:47:50.34 | 0.51 | 2016-06-23T14:24:43.14 | 1.00 | 2019-11-15T20:41:57.94 | 0.59 | 2021-09-12T06:36:43.74 | 0.99
2015-02-24T06:28:23.34 | 0.68 | 2016-06-23T15:59:56.14 | 1.00 | 2020-04-10T08:45:15.54 | 0.99 | 2021-09-12T09:47:04.74 | 0.99
2015-02-24T21:59:21.34 | 1.05 | 2016-06-23T17:38:11.14 | 0.95 | 2020-04-27T17:58:04.14 | 1.03 | 2021-09-12T11:22:14.74 | 0.99
2015-02-25T10:42:03.34 | 0.19 | 2016-06-23T19:18:16.14 | 0.87 | 2020-04-27T19:33:15.14 | 1.03 | 2021-09-12T13:01:13.74 | 0.93
2015-02-28T10:40:49.34 | 0.54 | 2016-06-23T20:57:40.14 | 0.80 | 2020-04-27T21:08:26.14 | 1.03 | 2021-09-12T14:40:26.74 | 0.86
2015-03-01T07:11:29.34 | 0.67 | 2016-06-23T22:36:44.14 | 0.73 | 2020-04-27T22:43:37.14 | 1.03 | 2021-09-12T19:36:56.74 | 0.68
2015-03-05T18:51:25.34 | 1.00 | 2016-06-24T00:15:38.14 | 0.67 | 2020-04-28T00:18:47.14 | 1.03 | 2021-09-12T20:53:14.74 | 0.10
2016-02-01T09:32:31.14 | 0.99 | 2016-06-25T07:40:41.14 | 1.00 | 2020-04-28T01:53:58.14 | 1.03 | 2021-09-12T22:28:24.74 | 0.99
2016-05-14T07:40:05.54 | 0.99 | 2016-06-26T04:18:39.54 | 1.00 | 2020-04-28T03:29:09.14 | 1.03 | 2021-09-13T00:03:34.74 | 0.99
2016-05-14T15:36:16.54 | 0.94 | 2016-06-26T05:53:53.54 | 0.99 | 2020-04-28T05:45:39.14 | 0.34 | 2021-09-13T03:13:55.74 | 0.99
2016-05-14T21:57:24.54 | 0.99 | 2016-06-26T09:04:20.54 | 0.99 | 2020-04-28T09:49:53.14 | 0.52 | 2021-09-13T04:49:05.74 | 0.99
2016-05-16T19:59:04.54 | 0.99 | 2016-06-26T10:39:34.54 | 0.99 | 2020-04-29T11:12:49.14 | 0.61 | 2021-09-13T07:59:25.74 | 0.99
2016-05-18T06:54:13.54 | 0.99 | 2016-06-26T13:50:01.54 | 0.99 | 2020-04-29T20:43:56.14 | 1.05 | 2021-09-13T19:23:29.74 | 0.69
2016-05-18T08:29:27.54 | 0.99 | 2016-06-26T17:01:30.54 | 0.98 | 2020-05-03T23:06:22.14 | 1.08 | 2021-09-14T06:11:47.74 | 0.99
2016-05-18T10:04:41.54 | 0.74 | 2016-06-26T18:41:08.54 | 0.90 | 2020-05-05T02:36:16.14 | 0.57 | 2021-09-14T10:57:17.74 | 0.99
2016-05-18T14:50:23.54 | 0.89 | 2016-06-26T20:20:20.54 | 0.84 | 2020-05-09T00:02:16.14 | 0.78 | 2021-09-14T14:13:39.74 | 0.89
2016-05-18T16:25:37.54 | 0.99 | 2016-06-27T01:16:45.54 | 0.66 | 2020-05-10T21:46:13.14 | 1.07 | 2021-09-15T18:56:06.74 | 0.74
2016-05-18T18:00:51.54 | 0.99 | 2016-06-27T08:52:46.54 | 0.99 | 2020-05-19T18:24:38.14 | 1.00 | 2021-09-16T17:04:25.74 | 0.80
2016-05-18T19:36:05.54 | 0.99 | 2016-06-29T18:03:53.54 | 0.95 | 2020-05-20T13:26:54.14 | 1.00 | 2021-09-16T20:13:34.74 | 0.82
2016-05-18T21:11:19.54 | 0.99 | 2016-06-30T09:53:19.54 | 0.99 | 2020-05-20T21:30:58.94 | 0.86 | 2021-09-18T10:07:32.74 | 0.99
2016-05-19T00:21:46.54 | 0.99 | 2016-07-04T13:52:59.54 | 0.99 | 2021-09-09T18:38:30.74 | 0.69 | 2021-09-22T01:22:04.74 | 0.99
2016-05-19T05:07:28.54 | 0.99 | 2016-07-05T13:41:30.54 | 0.99 | 2021-09-09T20:17:16.74 | 0.63 | 2021-09-25T02:20:01.74 | 0.77
2016-05-19T11:28:24.54 | 0.75 | 2016-07-07T06:57:36.54 | 0.82 | 2021-09-10T00:40:59.74 | 1.00 | 2021-09-27T19:21:56.74 | 0.99
2016-05-19T19:24:33.14 | 0.99 | 2019-10-04T08:25:32.34 | 1.07 | 2021-09-10T02:16:09.74 | 1.00 | 2021-09-30T01:17:31.74 | 0.67
2016-05-20T03:20:42.14 | 0.99 | 2019-11-04T00:43:21.94 | 0.87 | 2021-09-10T05:26:30.74 | 1.00 | 2021-09-30T20:19:27.74 | 1.00
2016-05-20T04:55:56.14 | 0.99 | 2019-11-04T02:18:32.94 | 0.90 | 2021-09-10T08:36:50.74 | 1.00 |  |
2016-05-20T16:02:32.14 | 0.99 | 2019-11-04T03:53:44.94 | 0.94 | 2021-09-10T15:07:12.74 | 0.83 |  |

##  2 Temporal and energy distributions

We retrieve the burst start time, flux and fluence of SGR J1935+2154 which
were detected by the Fermi/Gamma-ray Burst Monitor (GBM) from the published
burst history (2014-2021, Table 4 in Lin et al. (2020b), Table 1 in Lin et al.
(2020a), Table 1 in Kaneko et al. (2021), Table 1 in Rehan & Ibrahim (2023)).
Then, we analyze the position history of Fermi/GBM observations using POSHIST
111https://heasarc.gsfc.nasa.gov/FTP/fermi/data/gbm/daily/ and calculate the
uninterrupted visible time (referred to as observation window hereafter) of
the GBM to the SGR J1935+2154, excluding the Earth occultation and SAA (South
Atlantic Anomaly) area, using the GBM Data Tools (Goldstein et al., 2022). The
start times of the observation windows and hundreds of bursts are listed in
Table 1 and shown in Fig. 1. To be specific, the burst samples of SGR
J1935+2154 consist of 158 observation windows, with each window having an
approximate duration of an hour because GBM orbits the Earth every about 90
minutes. However some observation windows are noticeably shorter than the 90
minutes because GBM occasionally shuts down when the satellite crosses the SAA
area.

###  2.1 The temporal clustering of XRBs

The waiting time between the successive bursts in the same observation window
is defined as

$\Delta{t}=t_{\mathrm{i}}-t_{\mathrm{i-1}}$ (1)

where $t_{i}$ is the burst start time of the $i-$th burst. Due to the unknown
waiting time for the first burst in each observation window, only 171 out of a
total of 329 bursts could be analyzed (see Fig. 1). A one-sample Kolmogorov-
Smirnov test is performed to examine whether the waiting times follow an
exponential distribution. The test yields a distance of $D=0.28$ and $p=0.03$,
indicating that a non-homogeneous model is necessary to describe the temporal
behavior of the bursts. Therefore, we use the Weibull function to model the
waiting times of the non-stationary Poisson process. Specifically, for an
observation window of a duration $T$ which owes a number of $N$ bursts, the
likelihood of this window can be calculated by (Oppermann et al., 2018),

$\displaystyle\mathcal{L}(N,t_{1},\ldots,t_{\mathrm{N}}|k,\lambda)$
$\displaystyle=\lambda\operatorname{F}(t_{1}|k,\lambda)\operatorname{F}(T-t_{\mathrm{N}}|k,\lambda)$
(2)
$\displaystyle\times\prod_{i=1}^{N-1}\mathcal{W}(t_{\mathrm{i}+1}-t_{\mathrm{i}}|k,\lambda)$

with

$\mathrm{F}(\Delta{t}|k,\lambda)=\mathrm{e}^{-[\Delta{t}\lambda\Gamma(1+1/k)]^{k}}$
(3)

where $\Gamma$ is the Gamma function and $\mathcal{W}(\Delta{t}\mid
k,\lambda)$ is the Weibull function. Two free parameters $k$ and $\lambda$ are
involved, which correspond to the shape of the Weibull function and the mean
burst rate, respectively. The meaning of the Weibull function represents the
probability density of the waiting time $\Delta t$, which can be expressed as

$\mathcal{W}(\Delta{t}\mid
k,\lambda)\quad=k\Delta{t}^{-1}[\Delta{t}\lambda\Gamma(1+1/k)]^{k}e^{-[\Delta{t}\lambda\Gamma(1+1/k)]^{k}}$
(4)

When we use the above expressions to describe a series of burst events, we
would conclude that the burst events can be clustered in the timeline if a
value of $k<1$ is obtained by maximizing the sum of the likelihoods of all
observation windows. Alternatively, if a value of $k=1$ is obtained, then it
means the Weibull distribution has reduced to the Poissonian case. By making
such a calculation, we finally obtain the corner diagram for the $k-\lambda$
parameter space as shown in Fig. 2, which is derived from a Markov Chain Monte
Carlo (MCMC) survey in the $k-\lambda$ space with 100,000 steps. To be
specific, the parameters are constrained to be $\lambda=58.04_{-4.05}^{+4.28}$
day-1 and $k=0.70\pm 0.03$, which demonstrates that the bursts indeed have
time clustering and do not occur randomly.

Figure 3: The logarithm waiting times, $\log(\Delta{t})$, of the XRBs v.s
their occurring times, where the time of FRB 20200428 is represented by the
green dashed line. The number distribution of the bursts on the waiting times
are exhibited in the right panel with the histograms, while the smooth lines
give the kernel density of these distributions. The histogram distributions
can be approximately fitted by a Gaussian function and its mean value and
variance are represented by the orange dot and the short black line in the
left panel.

###  2.2 The cumulative distribution of the waiting times

In Fig. 3, we plot the waiting times of the XRBs of SGR J1935+2154 v.s their
burst times, where an apparent clustering of bursts appear. Therefore, we can
divide these bursts into 4 separated active episodes as labelled by different
colors, except several bursts (grey data) outside of these episodes. Here, an
active episode can be defined as a period during which more than two bursts
are produced within 10 days, as previously adopted in Lin et al. (2020b, a).
The starting and ending times of the four active episodes are listed in Table
2. By according to the shape of the kernel of the histogram distributions of
$\log_{10}(\Delta{t})$ shown in the right panel of Fig. 3, we tentatively use
a Gaussian function to fit these histogram distributions. The obtained values
of the mean and variance of the Gaussians are listed in Table 2. It can be
found that the mean value of the waiting times in the 3rd episode (Apr.-May
2020) during which FRB 200428 occurred is significantly shorter than those in
the other episodes, indicating the particularity of the FRB episode. In
addition, these near-Gaussian distributions of the waiting times make these
burst activities of SGR J1935+2154 similar to the previous burst activities of
magnetars (Cheng et al., 1996; GöǧüŞ et al., 1999; Göǧüş et al., 2000; Younes
et al., 2020; Cai et al., 2022a).

Table 2: Parameters for the Gaussian fits of the histogram distribution of $\log_{10}(\Delta{t}/s)$ Episodes | UTC | MJD | $\mu$ | $\sigma$
---|---|---|---|---
1 | 2016-05-18 - 2016-05-22 | 57526 - 57530 | 3.11 $\pm$ 0.07 | 0.20 $\pm$ 0.09
2 | 2016-06-23 - 2016-06-27 | 57562 - 57566 | 2.67 $\pm$ 0.07 | 0.49 $\pm$ 0.07
3 | 2020-04-27 - 2020-05-20 | 58966 - 58989 | 1.80 $\pm$ 0.06 | 0.71 $\pm$ 0.06
4 | 2021-09-10 - 2021-09-13 | 59467 - 59470 | 2.63 $\pm$ 0.06 | 0.38 $\pm$ 0.06
Table 3: Parameters for the fittings of the CCD of burst fluence/flux | Model | $\alpha_{1}$ | $\alpha_{2}$ | $\delta$ |  $S_{\rm b}$ (or $F_{\rm b}$)
---|---|---|---|---|---
P( $>S$ ) | BPL | -0.02 $\pm$ 0.01 | 0.66 $\pm$ 0.06 | 0.61 $\pm$ 0.04 | 6.87 $\pm$ 0.31 $\times$ $10^{-7}$ erg cm-2 s-1
| SPL |  | 0.69 $\pm$ 0.01 |  |
P( $>F$ ) | BPL | -0.16 $\pm$ 0.07 | 0.92 $\pm$ 0.19 | 1.04 $\pm$ 0.14 | 6.25 $\pm$ 0.73 $\times$ $10^{-8}$ erg cm-2
| SPL |  | 0.95 $\pm$ 0.02 |  |

For a more detailed description for the distribution property of the waiting
times, we plot their complementary cumulative distribution (CCD)222The
cumulative distribution (CD) and CCD of the variable $x$ are defined in this
paper as follows:  $\displaystyle CD(x)$ $\displaystyle=P(\leq x)=\frac{n(\leq
x)}{N}$ (5) $\displaystyle CCD(x)$ $\displaystyle=P(>x)=1-CD(x)$ where $N$ is
the total number of the events. , $P(>\Delta{t})$, in the left panel of Fig.
4, where the distributions for different flux ranges are also displayed for
comparison. According to the heavy-tail feature of these CCDs and following
Reed & Jorgensen (2004), we try to use the double Pareto-Lognormal (dPLn)
distribution to describe their general properties. The dPLn distribution,
which consists of two Pareto tail distributions with a transition point in
between, arises from a geometric Browning motion process, that will be killed
when the killing rate comes to constant and the state becomes equilibrium
(Mitzenmacher, 2004). Specifically, the dPLn description for the CCD of the
waiting times can be expressed as (Reed & Jorgensen, 2004)

$\displaystyle P(\geq\Delta{t})$ $\displaystyle=$
$\displaystyle\Phi^{c}\Big{(}\frac{\log(\Delta{t})-\nu}{\tau}\Big{)}+\frac{1}{\alpha_{1}+\alpha_{2}}\Big{[}\alpha_{2}\Delta{t}^{-\alpha_{1}}$
(6) $\displaystyle
A(\alpha_{1},\nu,\tau)\Phi\Big{(}\frac{\log(\Delta{t})-\nu-\alpha_{1}\tau^{2}}{\tau}\Big{)}+\alpha_{1}\Delta{t}^{\alpha_{2}}$
$\displaystyle
A(-\alpha_{2},\nu,\tau)\Phi^{c}\Big{(}\frac{\log(\Delta{t})-\nu+\alpha_{2}\tau^{2}}{\tau}\Big{)}\Big{]},$

where $\Phi$/$\Phi^{c}$ is the CD/CCD of standard normal distribution, $\nu$
and $\tau^{2}$ represent the mean and variance of the log-normal for the
initial geometric Browning motion process, and

$A(\alpha,\nu,\tau)=\exp(\alpha\nu+\alpha^{2}\tau^{2}/2).$ (7)

The parameters $\alpha_{1}$ and $\alpha_{2}$ determine the slope of the two
Pareto tails. By using the package distribuisonsrd of Comprehensive R Archive
Network (CRAN, Mora-Cantallops et al., 2020), we perform the dPLn fitting to
the CCDs of the waiting times, as shown by the dashed line in Figure 4, and
obtain $\alpha_{1}$=3.85 and $\alpha_{2}$=0.73. This indicates that, in the
range of a relatively long waiting time, the CCD can be approximated by a
power law with a slope of $\alpha_{2}\sim 0.73$, which could further be
independent of the selection of the flux threshold and thus of the burst
energy.

Figure 4: The CCDs of the waiting time (Left) and fluence/flux (Right) of the
XRBs. In the left panel, the dashed line gives the dPLn fitting of the CCD,
while the dot and dashed lines in the right panel present the smoothly BPL and
SPL fits, respectively.

###  2.3 The cumulative distribution of the fluence/flux

In the right panel of Fig. 4, we further show the CCD of the fluence of the
XRBs, which can be fitted by a smoothly broken-power law (BPL) as

$P(>S)\propto{\left(\frac{S}{S_{b}}\right)^{-\alpha_{1}}}\left\\{\frac{1}{2}\left[1+\left(\frac{S}{S_{b}}\right)^{1/\delta}\right]\right\\}^{(\alpha_{1}-\alpha_{2})\delta}$
(8)

with the parameter values listed in Table 3. According to the above-obtained
value of $S_{b}$ (or $F_{b}$), we further ignore the CCD in the low
fluence/flux range and fit the high-fluence/flux distribution by a single
power law (SPL), which are also shown in the right panel of Fig. 4 by the
dashed lines. This result indicates that the CCD of the burst fluence/flux can
basically be described by a SPL, which is consistent with the results reported
in previous literature (van der Horst et al., 2012; Collazzi et al., 2015; Lin
et al., 2020b).

##  3 Unified scaling law for magnetar bursts

The temporal clustering of the XRBs of SGR J1935+2154 and the distributions of
their waiting time and fluence/flux make these burst activity very similar to
the earthquake phenomena. The earthquakes also satisfy two power-law
frequency–size statistics, i.e., the Omori law and Gutenberg-Richter law as

$p(\Delta{t})\propto\Delta{t}^{-\alpha_{\rm E}}$ (9)

with $\alpha_{\rm E}\sim(0.34-1.83)$ and

$P(>S_{\mathrm{E}})\propto S_{\mathrm{E}}^{-\beta_{\rm E}}$ (10)

with $\beta_{\rm E}\sim(0.6-1.0)$, where $S_{\mathrm{E}}$ is the maximum
seismic wave amplitude which determines the magnitude of the aftershocks of
the earthquake as $m=\mathrm{log}_{10}(S_{\mathrm{E}})$ (Aschwanden, 2011).
Earthquakes are known to be SOC phenomena. Under the long-term action of
external forces, such a SOC system can always enter into a statistically
stationary state characterized by spatial and temporal scale independence,
without fine-tuning the system parameters. In this state, a minor change can
lead to an avalanche and a collapse of the whole system. Therefore, it can be
naturally understood why the aftershocks of earthquakes can have temporal
clustering and power-law features of CD. Moreover, the connection between
these two characteristics can in principle be understood according to the
analysis by Aschwanden & McTiernan (2010), who demonstrated that the waiting
time can be distributed as

$\displaystyle p(\Delta t)$ $\displaystyle=$
$\displaystyle\frac{\lambda}{(1+\lambda\Delta t)^{2}}$ (11)
$\displaystyle\approx$ $\displaystyle(1/\lambda)\Delta
t^{-2},{\rm~{}for~{}}\Delta{t}\gg 1/\lambda,$

if the variability of the event rate shows a spike-like $\delta$-function with
$\lambda$ being the mean event rate. For the burst activity of SGR J1935+2154,
we have $\lambda=58.04_{-4.05}^{+4.28}$ day-1 as obtained in Section 2.

Figure 5: The $x-y$ USL of XRBs. The dashed lines give the fits of the data by
Eq. (12) for parameter values listed in Table 4.

As a SOC system, the earthquake phenomena had further been found to satisfy a
unified scaling law (USL) between the quantities of
$\Delta{t}S_{\mathrm{E}}^{-\beta_{\rm E}}L^{d_{f}}$ and
$\Delta{t}^{-\alpha_{\rm E}}p(\Delta{t})_{S,L}$, where $L$ is the fault
lengths, $d_{f}$ is the fractal dimension, and $p(\Delta{t})_{S,L}$ is the
distribution of waiting times $\Delta{t}$ between earthquakes occurring within
range $L$ whose amplitude greater than $S_{\mathrm{E}}$ (Bak et al., 2002).
Here, the scaling variable $\Delta{t}S^{-\beta_{\rm E}}L^{d_{f}}$ can be
interpreted as a measure of the average number of events occurring within a
time interval $\Delta{t}$ and a range $L$ with an amplitude greater than
$S_{\mathrm{E}}$. Then, in view of the similarity between the earthquakes and
the magnetar burst activities, it is supposed that, in principle, a similar
USL could also be found from the XRBs of magnetars, which reads

$y=f(x)\approx\left\\{\begin{array}[]{cc}{\rm const.},&{\rm for~{}}x<1,\\\
x^{-\gamma},&{\rm for~{}}x\geq 1,\end{array}\right.$ (12)

where $x\equiv c\Delta{t}S^{\beta}$ and
$y\equiv\Delta{t}^{-\alpha}P(\leq\Delta{t})$, $c$ is coefficient, and $S$
represents fluence which can be replaced with $F$ if flux is used. The scaling
variable $\Delta{t}S^{\beta}$ (or $\Delta{t}F^{\beta}$) can be interpreted as
a measure of the average number of bursts occurring within a time interval
$\Delta{t}$ with a fluence/flux less than $S$ (or $F$). By using the parameter
values of $\alpha=0.73$ derived from the CCD of waiting times and $\beta=0.69$
(or 0.95) from the CCD of fluence/flux, we plot the distribution of the XRBs
of SGR J1935+2154 in the $x-y$ plane in Fig. 5, where an appropriate value $c$
is adopted to define $x$ in order to set the break value of $x$ at $1$. As
shown, the distribution of the observational data indeed all behave as
described in Eq. (12), which have a constant-$y$ and decaying-$y$ part with
parameters listed in Table 4. This result is insensitive to the selection of
the fluence/flux limitation.

Table 4: Parameters for the USL presented in Eq. (12) | Selection | $c/10^{3}$ | $\gamma$
---|---|---|---
| $>10^{-7}$ |  | -0.30 $\pm$ 0.02
Fluence $S$ | $>10^{-6.5}$ | 1.68 | -0.33 $\pm$ 0.02
(erg cm-2) | $>10^{-6}$ |  | -0.28 $\pm$ 0.02
| $>10^{-7}$ |  | -0.27 $\pm$ 0.02
Flux $F$ | $>10^{-6.5}$ | 19.67 | -0.24 $\pm$ 0.01
(erg cm-2 s-1) | $>10^{-6}$ |  | -0.30 $\pm$ 0.02

In more detail, the USL indicates that the bursts locating the constant part
of $f(x)$, which corresponds to the Omori law in the earthquake case, are
likely to be dependent afterbursts of some main bursts. Meanwhile, the events
deviating from the Omori law can be classified as independent burst events.
This classification can be basically in agreement with the physical intuition
that XRBs that are separated by large energy difference or long waiting times
could be independent. According to this statistical classification, we plot
the inferred independent and dependent bursts in Figs. 6 and 7 by using
different symbols, the specific properties of which are listed in Table 6-9.
In Table 5, we list the numbers of the dependent and independent bursts for
the four different episodes. It can be found that the number fraction of
dependent bursts in the total bursts in the FRB episode (Apr.-May 2020) is
overwhelmingly higher than those in the other three episodes. Specifically,
the faction of the dependent bursts (i.e., the number fraction of the data of
$x<1$) can be calculated to be $\sim$33% (or $\sim$36%) for the fluence (or
flux) definition of $x$. The observation windows owing these dependent bursts
are plotted in Fig. 7. According to this result, it can be naturally
understood why the FRB episode has the smallest mean value of the waiting
times, as found from Fig. 3. The large fraction of the dependent bursts
increase the density of bursts in timeline and thus shorten the apparent
waiting time between the successive bursts. Following this consideration, it
is suspected that FRBs or slow radio bursts could exist in the last episode
(Sept. 2021), because several dependent bursts appear there.

In addition, it needs to be noticed that, due to the limited detection
efficiency of the instrument and the criteria used to distinguish the bursts,
a bright long burst may contain an independent burst and multiple dependent
bursts. The number of dependent bursts is likely underestimated. The energy of
a burst may be the sum of multiple dependent ones and may be misclassified as
dependent (see left panel of Fig. 6). Therefore, using the peak flux is
recommended for statistical analysis of magnetar bursts.

Table 5: Numbers of the inferred dependent/independent bursts Episodes | Total | Classified with flux | Classified with fluence
---|---|---|---
|  | Dependent | Independent | Dependent | Independent
1 | 7 | 0 | 7 | 0 | 7
2 | 18 | 1 | 17 | 2 | 16
3 | 107 | 39 | 68 | 35 | 72
4 | 31 | 3 | 28 | 5 | 26

Figure 6: The fluence/flux vs. the occurring time of XRBs, where the bursts
have been separated into the dependent and independent types by according the
USL distribution presented in Fig. 5.

Figure 7: The observation windows owing dependent bursts. The upper and lower
panels are plotted corresponding to results presented in the left and right
panels of Fig. 6, respectively.

##  4 Conclusion and discussion

The distribution of the waiting time and fluence/flux of the XRBs of SGR
J1935+21054 are investigated statistically in this paper, which firstly show
the temporal clustering of the bursts and the power-law behavior of the
distributions. These results indicate that the XRBs can belong to a SOC
phenomenon, which could correspond to sudden fractures and displacements of
the crust of the magnetar (i.e., starquakes; Thompson & Duncan, 1995; Cheng et
al., 1996). Our statistics further show that, different from the other
episodes, the episode during which FRB 20200428 occurred has an obviously
smaller mean value of the waiting time, which directly indicates the
particularity of this episode. In more detail, by applying the USL discovered
by Bak et al. (2002) for earthquakes, it is found that the regular episodes
could mainly consist of independent bursts, where as much more dependent
bursts can happen in the FRB episode. Nevertheless, it should be noted that we
actually cannot exactly specify the individual bursts to be dependent or
independent, only by according to the statistical investigation.

In any case, our statistical results indicate that the starquakes occurring in
regular episodes could be generally without casual link (i.e., independent).
On the contrary, in the FRB episode, independent starquakes could take place
more frequently and intensively and, thus, many secondary bursts could be
induced by the interactions among these primary independent bursts. This
further hints that FRB emission could result from such burst interactions
(e.g., the collision between different seismic/Alfven waves or different
explosion outflows) and be associated with a dependent burst. Such a situation
could appear in a global activity of a magnetar due to a conversion of the
magnetic field from a dipolar to a multi-polar configuration (e.g., the
situation considered in Yang & Zhang, 2021). Here, it is interesting to
mention that a giant glitch occurred approximately 3 days before FRB 20200428,
as reported recently by Ge et al. (2022). The occurrence of the glitch means
the magnetar crust had just obtained a sudden increase of the angular
momentum, which could be transferred from the core of the magnetar. In such a
case, due to the increase of the differential rotation between the crust and
the core, the stress forced on the whole crust by the internal magnetic field
could be consequently strengthened to exceed the elastic yield limit.
Therefore, it can be imaged that crust cracking events could take place nearly
globally and, simultaneously, a multi-polar field can be formed.

After the FRB 20200428 event, it has further been reported that the
Gravitational Wave High-energy Electromagnetic Counterpart All-sky Monitor
(GECAM) detected an XRB from SGR J1935+2154 at an arrival time consistent with
a radio burst detected by CHIME (Wang et al., 2022) on 2020 October 14 UTC
19:21:39.100. The Insight–Hard X-ray Modulation Telescope (HXMT) also detected
an XRB associated with a radio burst detected by the Yunnan 40 m telescope, on
2022 October 21 (Li et al., 2022b). Future works on these more XRB-FRB
association events would help us to test the discovery in this paper.

This work is supported by the National Key R&D Program of China
(2021YFA0718500), the National SKA program of China (2020SKA0120300), the
National Natural Science Foundation of China (Grant No. 12393811, 12173038),
and the Strategic Priority Research Program on Space Science (Grant No.
XDA15360102, XDA15360300, XDA15052700) of the Chinese Academy of Sciences.

##  Appendix A The X-ray bursts classification of SGR J1935+2154

Table 6: The list of the inferred dependent bursts classified by flux Episode | Time (UTC) | Flux | Fluence
---|---|---|---
2nd episode | 2016-06-23T21:31:12.600 | 9.27 | 0.89
3rd episode | 2020-04-27T18:31:25.234 | 11.39 | 1.89
| 2020-04-27T18:34:47.296 | 1.97 | 1.05
| 2020-04-27T18:35:57.633 | 18.00 | 0.45
| 2020-04-27T18:36:46.007 | 56.13 | 19.42
| 2020-04-27T18:39:09.331 | 5.43 | 0.19
| 2020-04-27T18:40:15.043 | 2.52 | 1.15
| 2020-04-27T18:40:32.031 | 4.55 | 6.16
| 2020-04-27T18:42:50.652 | 25.85 | 8.17
| 2020-04-27T18:46:39.414 | 5.88 | 9.70
| 2020-04-27T18:50:49.460 | 7.43 | 0.26
| 2020-04-27T20:17:27.317 | 9.06 | 0.58
| 2020-04-27T20:17:50.343 | 1.55 | 2.20
| 2020-04-27T20:17:58.442 | 14.74 | 1.15
| 2020-04-27T20:18:09.130 | 0.43 | 0.36
| 2020-04-27T20:19:47.631 | 1.37 | 1.16
| 2020-04-27T20:19:49.430 | 17.63 | 4.09
| 2020-04-27T20:20:44.640 | 2.21 | 0.74
| 2020-04-27T20:21:51.841 | 2.82 | 1.63
| 2020-04-27T20:21:55.136 | 17.01 | 9.34
| 2020-04-27T21:20:58.670 | 4.12 | 0.77
| 2020-04-28T00:46:00.034 | 7.58 | 6.05
| 2020-04-28T00:46:06.394 | 4.94 | 0.87
| 2020-04-28T00:46:23.528 | 3.21 | 2.71
| 2020-04-28T00:46:43.072 | 7.59 | 3.82
| 2020-04-28T00:47:24.957 | 3.51 | 0.84
| 2020-04-28T00:47:57.536 | 5.77 | 1.50
| 2020-04-28T00:48:44.833 | 4.24 | 1.88
| 2020-04-28T00:48:49.098 | 13.69 | 8.17
| 2020-04-28T00:49:00.270 | 1.09 | 2.85
| 2020-04-28T00:49:06.479 | 7.78 | 0.21
| 2020-04-28T00:49:16.609 | 12.59 | 3.94
| 2020-04-28T00:49:22.392 | 7.03 | 0.64
| 2020-04-28T00:49:27.008 | 1.84 | 0.64
| 2020-04-28T00:49:45.895 | 2.47 | 2.88
| 2020-04-28T00:50:01.012 | 7.29 | 3.64
| 2020-04-28T00:50:21.993 | 8.57 | 0.18
| 2020-04-28T00:50:41.835 | 1.56 | 0.63
| 2020-04-28T00:51:55.444 | 6.52 | 0.86
| 2020-04-28T00:52:06.141 | 1.97 | 0.75
| 2021-09-11T15:15:26.000 | 216.73 | 188.99
| 2021-09-12T12:19:21.000 | 44.75 | 1.79
4th episode | 2021-09-12T20:16:20.000 | 14.29 | 1.60
Table 7: The list of the inferred dependent bursts classified by fluence Episode | Time (UTC) | Flux | Fluence
---|---|---|---
2nd episode | 2016-06-23T15:16:26.836 | 17.27 | 3.04
| 2016-06-23T21:31:12.600 | 9.27 | 0.89
3rd episode | 2020-04-27T18:31:25.234 | 11.39 | 1.89
| 2020-04-27T18:34:47.296 | 1.97 | 1.05
| 2020-04-27T18:35:57.633 | 18.00 | 0.45
| 2020-04-27T18:36:45.376 | 20.00 | 0.28
| 2020-04-27T18:36:46.007 | 56.13 | 19.42
| 2020-04-27T18:39:09.331 | 5.43 | 0.19
| 2020-04-27T18:44:08.209 | 4.42 | 0.34
| 2020-04-27T18:50:28.665 | 10.80 | 0.27
| 2020-04-27T18:50:49.460 | 7.43 | 0.26
| 2020-04-27T20:16:15.285 | 10.67 | 0.32
| 2020-04-27T20:17:27.317 | 9.06 | 0.58
| 2020-04-27T20:17:58.442 | 14.74 | 1.15
| 2020-04-27T20:18:09.130 | 0.43 | 0.36
| 2020-04-27T20:19:23.068 | 10.33 | 0.31
| 2020-04-27T20:19:47.631 | 1.37 | 1.16
| 2020-04-27T20:19:49.430 | 17.63 | 4.09
| 2020-04-27T20:21:55.136 | 17.01 | 9.34
| 2020-04-27T21:20:58.670 | 4.12 | 0.77
| 2020-04-27T21:25:01.037 | 6.50 | 0.39
| 2020-04-28T00:25:43.946 | 5.95 | 0.25
| 2020-04-28T00:46:06.394 | 4.94 | 0.87
| 2020-04-28T00:46:23.528 | 3.21 | 2.71
| 2020-04-28T00:47:24.957 | 3.51 | 0.84
| 2020-04-28T00:48:49.098 | 13.69 | 8.17
| 2020-04-28T00:49:00.270 | 1.09 | 2.85
| 2020-04-28T00:49:06.479 | 7.78 | 0.21
| 2020-04-28T00:49:16.609 | 12.59 | 3.94
| 2020-04-28T00:49:22.392 | 7.03 | 0.64
| 2020-04-28T00:49:27.008 | 1.84 | 0.64
| 2020-04-28T00:49:45.895 | 2.47 | 2.88
| 2020-04-28T00:50:01.012 | 7.29 | 3.64
| 2020-04-28T00:50:21.993 | 8.57 | 0.18
| 2020-04-28T00:50:41.835 | 1.56 | 0.63
| 2020-04-28T00:51:55.444 | 6.52 | 0.86
| 2020-04-28T00:52:06.141 | 1.97 | 0.75
4th episode | 2021-09-10T00:46:21.000 | 17.50 | 0.63
| 2021-09-11T15:15:26.000 | 216.73 | 188.99
| 2021-09-12T06:51:14.000 | 300.37 | 88.91
| 2021-09-12T12:19:21.000 | 44.75 | 1.79
| 2021-09-12T20:16:20.000 | 14.29 | 1.60

Table 8: The list of the inferred independent bursts classified by flux

Episode | Time (UTC) | Flux | Fluence
---|---|---|---
outside-episodesaathe bursts outside of the 4 active episodes that are defined in this study (see Table 2).  | 2014-07-05T09:37:34.484 | 14.55 | 0.32
| 2015-02-22T19:44:16.881 | 44.00 | 1.76
| 2015-02-22T19:58:16.658 | 5.96 | 0.62
| 2015-02-23T05:24:53.876 | 6.78 | 2.17
1st episode | 2016-05-18T09:09:23.800 | 116.29 | 28.84
| 2016-05-18T10:28:02.817 | 22.79 | 2.37
| 2016-05-18T15:33:47.010 | 10.71 | 0.60
| 2016-05-18T17:05:12.813 | 2.73 | 0.24
| 2016-05-19T11:59:32.632 | 9.11 | 0.51
| 2016-05-19T12:07:46.594 | 32.03 | 2.05
| 2016-05-21T21:13:16.242 | 20.25 | 0.81
2nd episode | 2016-06-23T15:16:26.836 | 17.27 | 3.04
| 2016-06-23T16:49:57.493 | 26.49 | 4.45
| 2016-06-23T17:47:06.896 | 7.50 | 0.24
| 2016-06-23T17:55:48.768 | 4.62 | 0.37
| 2016-06-23T17:58:55.696 | 4.31 | 0.31
| 2016-06-23T18:22:43.425 | 5.38 | 0.56
| 2016-06-23T19:36:27.341 | 4.38 | 0.77
| 2016-06-23T19:37:59.997 | 6.84 | 0.93
| 2016-06-23T19:51:16.261 | 3.67 | 0.44
| 2016-06-23T20:06:37.158 | 28.02 | 2.69
| 2016-06-23T21:15:31.444 | 6.13 | 0.98
| 2016-06-23T21:16:24.612 | 19.04 | 7.77
| 2016-06-23T21:20:46.404 | 101.19 | 49.38
| 2016-06-23T21:23:36.780 | 54.86 | 7.90
| 2016-06-23T21:30:47.032 | 4.52 | 0.94
| 2016-06-23T21:38:13.432 | 5.36 | 0.30
| 2016-06-26T17:50:03.251 | 12.19 | 0.39
outside-episodesaathe bursts outside of the 4 active episodes that are defined in this study (see Table 2).  | 2019-11-04T09:17:53.492 | 18.35 | 5.89
| 2019-11-04T20:13:42.537 | 4.36 | 0.61
| 2019-11-04T20:29:39.804 | 10.70 | 1.37
| 2019-11-04T23:48:01.336 | 13.64 | 3.07
3rd episode | 2020-04-27T18:31:05.770 | 3.11 | 0.76
| 2020-04-27T18:33:53.116 | 7.18 | 0.51
| 2020-04-27T18:34:05.700 | 47.65 | 20.11
| 2020-04-27T18:34:46.047 | 16.95 | 3.83
| 2020-04-27T18:35:05.320 | 49.71 | 5.12
| 2020-04-27T18:35:46.623 | 21.64 | 1.32
| 2020-04-27T18:36:45.376 | 20.00 | 0.28
| 2020-04-27T18:38:20.206 | 8.86 | 0.93
| 2020-04-27T18:38:53.689 | 7.24 | 1.81
| 2020-04-27T18:42:40.816 | 11.29 | 0.35
| 2020-04-27T18:44:08.209 | 4.42 | 0.34
| 2020-04-27T18:46:08.767 | 18.74 | 3.86
| 2020-04-27T18:47:05.754 | 76.00 | 11.78
| 2020-04-27T18:48:38.675 | 6.95 | 1.69
| 2020-04-27T18:49:28.034 | 18.56 | 6.83
| 2020-04-27T18:50:28.665 | 10.80 | 0.27
| 2020-04-27T18:55:44.155 | 36.81 | 2.65
| 2020-04-27T18:57:35.574 | 8.12 | 0.78
| 2020-04-27T18:58:45.533 | 4.04 | 0.78
| 2020-04-27T19:37:39.328 | 5.37 | 3.89
| 2020-04-27T19:43:44.537 | 88.12 | 38.42
| 2020-04-27T19:45:00.478 | 5.05 | 0.51
| 2020-04-27T19:55:32.325 | 5.61 | 0.23
| 2020-04-27T20:01:45.681 | 10.52 | 5.08
| 2020-04-27T20:07:20.319 | 2.07 | 0.79
| 2020-04-27T20:13:38.263 | 9.45 | 0.52
| 2020-04-27T20:14:51.396 | 27.06 | 1.38
| 2020-04-27T20:15:20.583 | 148.03 | 189.77
| 2020-04-27T20:16:15.285 | 10.67 | 0.32
| 2020-04-27T20:17:09.139 | 6.45 | 0.71
| 2020-04-27T20:19:23.068 | 10.33 | 0.31
| 2020-04-27T20:25:53.415 | 3.88 | 1.55
| 2020-04-27T21:15:36.398 | 16.89 | 6.47
| 2020-04-27T21:20:55.561 | 10.79 | 0.96
| 2020-04-27T21:24:05.936 | 8.40 | 0.42
| 2020-04-27T21:25:01.037 | 6.50 | 0.39
| 2020-04-27T21:27:25.367 | 1.91 | 0.47
| 2020-04-27T21:43:06.346 | 10.18 | 1.66
| 2020-04-27T21:48:44.062 | 23.29 | 6.59
| 2020-04-27T21:57:03.989 | 7.93 | 0.23
| 2020-04-27T21:59:22.528 | 62.26 | 14.88
| 2020-04-27T22:55:19.911 | 7.41 | 1.97
| 2020-04-27T23:02:53.488 | 27.20 | 7.10
| 2020-04-27T23:06:06.135 | 12.17 | 2.02
| 2020-04-27T23:25:04.349 | 3.49 | 1.75
| 2020-04-27T23:27:46.293 | 47.50 | 3.23
| 2020-04-27T23:42:41.143 | 36.42 | 1.93
| 2020-04-27T23:44:31.818 | 45.00 | 14.49
| 2020-04-28T00:23:04.763 | 6.11 | 0.69
| 2020-04-28T00:24:30.311 | 146.14 | 34.49
| 2020-04-28T00:25:43.946 | 5.95 | 0.25
| 2020-04-28T00:37:36.160 | 5.48 | 0.63
| 2020-04-28T00:39:39.565 | 3.47 | 2.29
| 2020-04-28T00:40:33.077 | 6.21 | 4.28
| 2020-04-28T00:41:32.148 | 31.49 | 13.76
| 2020-04-28T00:43:24.784 | 7.71 | 6.52
| 2020-04-28T00:44:08.210 | 55.55 | 70.82
| 2020-04-28T00:45:31.098 | 8.04 | 0.86
| 2020-04-28T00:46:20.179 | 55.15 | 46.93
| 2020-04-28T00:51:35.912 | 11.16 | 0.77
| 2020-04-28T00:54:57.448 | 25.87 | 4.45
| 2020-04-28T00:56:49.646 | 4.39 | 1.44
| 2020-04-28T01:04:03.146 | 12.42 | 0.77
| 2020-04-28T02:27:24.905 | 7.69 | 0.20
| 2020-04-28T03:47:52.140 | 13.99 | 2.00
| 2020-04-28T04:09:47.317 | 17.18 | 1.89
| 2020-05-05T03:02:56.033 | 2.58 | 0.42
| 2020-05-19T18:57:36.305 | 5.15 | 0.17
4th episode | 2021-09-10T00:46:21.000 | 17.50 | 0.63
| 2021-09-10T01:00:44.000 | 34.71 | 2.36
| 2021-09-10T01:13:17.000 | 17.34 | 1.11
| 2021-09-10T01:14:37.000 | 27.50 | 3.63
| 2021-09-10T01:17:19.000 | 45.96 | 4.78
| 2021-09-10T01:18:54.000 | 19.35 | 1.78
| 2021-09-10T01:20:48.000 | 14.41 | 0.98
| 2021-09-10T01:27:05.000 | 6.25 | 0.90
| 2021-09-10T02:36:38.000 | 30.00 | 1.32
| 2021-09-10T02:44:34.000 | 35.00 | 2.80
| 2021-09-10T02:55:10.000 | 2.79 | 1.07
| 2021-09-11T03:02:28.000 | 7.78 | 0.56
| 2021-09-11T13:28:55.000 | 119.64 | 20.10
| 2021-09-11T15:06:43.000 | 199.73 | 81.49
| 2021-09-11T15:15:25.000 | 16.59 | 1.46
| 2021-09-11T15:17:45.000 | 141.90 | 49.38
| 2021-09-11T15:26:12.000 | 116.36 | 10.24
| 2021-09-11T15:32:33.000 | 36.21 | 4.49
| 2021-09-11T15:38:26.000 | 18.06 | 2.60
| 2021-09-11T16:50:04.000 | 99.77 | 4.39
| 2021-09-11T17:01:09.675 | 1.18 | 1.80
| 2021-09-11T17:04:29.740 | 14.59 | 4.30
| 2021-09-11T17:10:48.619 | 361.19 | 162.51
| 2021-09-11T20:13:40.000 | 27.33 | 4.70
| 2021-09-11T20:22:59.000 | 39.75 | 7.95
| 2021-09-12T00:45:49.000 | 36.25 | 1.45
| 2021-09-12T06:51:14.000 | 300.37 | 88.91
| 2021-09-12T07:28:07.000 | 40.80 | 3.59

Note. — The Flux in units of $\times 10^{7}$ erg $\cdot$ cm-2 s-1 and Fluence
in units of $\times 10^{7}$ erg $\cdot$ cm-2. The time range of the 1st, 2nd,
3rd and 4th episodes is May 2016, June 2016, Apr.-May 2020 and Sept. 2021
respectively (see Table 2).

Note. — The Flux in units of $\times 10^{7}$ erg $\cdot$ cm-2 s-1 and Fluence
in units of $\times 10^{7}$ erg $\cdot$ cm-2. The time range of the 1st, 2nd,
3rd and 4th episodes is May 2016, June 2016, Apr.-May 2020 and Sept. 2021
respectively (see Table 2).

Note. — The Flux in units of $\times 10^{7}$ erg $\cdot$ cm-2 s-1 and Fluence
in units of $\times 10^{7}$ erg $\cdot$ cm-2. The time range of the 1st, 2nd,
3rd and 4th episodes is May 2016, June 2016, Apr.-May 2020 and Sept. 2021
respectively (see Table 2).

Table 9: The list of the inferred independent bursts classified by fluence

Episode | Time (UTC) | Flux | Fluence
---|---|---|---
outside-episodesaathe bursts outside of the 4 active episodes that are defined in this study (see Table 2).  | 2014-07-05T09:37:34.484 | 14.55 | 0.32
| 2015-02-22T19:44:16.881 | 44.00 | 1.76
| 2015-02-22T19:58:16.658 | 5.96 | 0.62
| 2015-02-23T05:24:53.876 | 6.78 | 2.17
1st episode | 2016-05-18T09:09:23.800 | 116.29 | 28.84
| 2016-05-18T10:28:02.817 | 22.79 | 2.37
| 2016-05-18T15:33:47.010 | 10.71 | 0.60
| 2016-05-18T17:05:12.813 | 2.73 | 0.24
| 2016-05-19T11:59:32.632 | 9.11 | 0.51
| 2016-05-19T12:07:46.594 | 32.03 | 2.05
| 2016-05-21T21:13:16.242 | 20.25 | 0.81
2nd episode | 2016-06-23T16:49:57.493 | 26.49 | 4.45
| 2016-06-23T17:47:06.896 | 7.50 | 0.24
| 2016-06-23T17:55:48.768 | 4.62 | 0.37
| 2016-06-23T17:58:55.696 | 4.31 | 0.31
| 2016-06-23T18:22:43.425 | 5.38 | 0.56
| 2016-06-23T19:36:27.341 | 4.38 | 0.77
| 2016-06-23T19:37:59.997 | 6.84 | 0.93
| 2016-06-23T19:51:16.261 | 3.67 | 0.44
| 2016-06-23T20:06:37.158 | 28.02 | 2.69
| 2016-06-23T21:15:31.444 | 6.13 | 0.98
| 2016-06-23T21:16:24.612 | 19.04 | 7.77
| 2016-06-23T21:20:46.404 | 101.19 | 49.38
| 2016-06-23T21:23:36.780 | 54.86 | 7.90
| 2016-06-23T21:30:47.032 | 4.52 | 0.94
| 2016-06-23T21:38:13.432 | 5.36 | 0.30
| 2016-06-26T17:50:03.251 | 12.19 | 0.39
outside-episodesaathe bursts outside of the 4 active episodes that are defined in this study (see Table 2).  | 2019-11-04T09:17:53.492 | 18.35 | 5.89
| 2019-11-04T20:13:42.537 | 4.36 | 0.61
| 2019-11-04T20:29:39.804 | 10.70 | 1.37
| 2019-11-04T23:48:01.336 | 13.64 | 3.07
3rd episode | 2020-04-27T18:31:05.770 | 3.11 | 0.76
| 2020-04-27T18:33:53.116 | 7.18 | 0.51
| 2020-04-27T18:34:05.700 | 47.65 | 20.11
| 2020-04-27T18:34:46.047 | 16.95 | 3.83
| 2020-04-27T18:35:05.320 | 49.71 | 5.12
| 2020-04-27T18:35:46.623 | 21.64 | 1.32
| 2020-04-27T18:38:20.206 | 8.86 | 0.93
| 2020-04-27T18:38:53.689 | 7.24 | 1.81
| 2020-04-27T18:40:15.043 | 2.52 | 1.15
| 2020-04-27T18:40:32.031 | 4.55 | 6.16
| 2020-04-27T18:42:40.816 | 11.29 | 0.35
| 2020-04-27T18:42:50.652 | 25.85 | 8.17
| 2020-04-27T18:46:08.767 | 18.74 | 3.86
| 2020-04-27T18:46:39.414 | 5.88 | 9.70
| 2020-04-27T18:47:05.754 | 76.00 | 11.78
| 2020-04-27T18:48:38.675 | 6.95 | 1.69
| 2020-04-27T18:49:28.034 | 18.56 | 6.83
| 2020-04-27T18:55:44.155 | 36.81 | 2.65
| 2020-04-27T18:57:35.574 | 8.12 | 0.78
| 2020-04-27T18:58:45.533 | 4.04 | 0.78
| 2020-04-27T19:37:39.328 | 5.37 | 3.89
| 2020-04-27T19:43:44.537 | 88.12 | 38.42
| 2020-04-27T19:45:00.478 | 5.05 | 0.51
| 2020-04-27T19:55:32.325 | 5.61 | 0.23
| 2020-04-27T20:01:45.681 | 10.52 | 5.08
| 2020-04-27T20:07:20.319 | 2.07 | 0.79
| 2020-04-27T20:13:38.263 | 9.45 | 0.52
| 2020-04-27T20:14:51.396 | 27.06 | 1.38
| 2020-04-27T20:15:20.583 | 148.03 | 189.77
| 2020-04-27T20:17:09.139 | 6.45 | 0.71
| 2020-04-27T20:17:50.343 | 1.55 | 2.20
| 2020-04-27T20:20:44.640 | 2.21 | 0.74
| 2020-04-27T20:21:51.841 | 2.82 | 1.63
| 2020-04-27T20:25:53.415 | 3.88 | 1.55
| 2020-04-27T21:15:36.398 | 16.89 | 6.47
| 2020-04-27T21:20:55.561 | 10.79 | 0.96
| 2020-04-27T21:24:05.936 | 8.40 | 0.42
| 2020-04-27T21:27:25.367 | 1.91 | 0.47
| 2020-04-27T21:43:06.346 | 10.18 | 1.66
| 2020-04-27T21:48:44.062 | 23.29 | 6.59
| 2020-04-27T21:57:03.989 | 7.93 | 0.23
| 2020-04-27T21:59:22.528 | 62.26 | 14.88
| 2020-04-27T22:55:19.911 | 7.41 | 1.97
| 2020-04-27T23:02:53.488 | 27.20 | 7.10
| 2020-04-27T23:06:06.135 | 12.17 | 2.02
| 2020-04-27T23:25:04.349 | 3.49 | 1.75
| 2020-04-27T23:27:46.293 | 47.50 | 3.23
| 2020-04-27T23:42:41.143 | 36.42 | 1.93
| 2020-04-27T23:44:31.818 | 45.00 | 14.49
| 2020-04-28T00:23:04.763 | 6.11 | 0.69
| 2020-04-28T00:24:30.311 | 146.14 | 34.49
| 2020-04-28T00:37:36.160 | 5.48 | 0.63
| 2020-04-28T00:39:39.565 | 3.47 | 2.29
| 2020-04-28T00:40:33.077 | 6.21 | 4.28
| 2020-04-28T00:41:32.148 | 31.49 | 13.76
| 2020-04-28T00:43:24.784 | 7.71 | 6.52
| 2020-04-28T00:44:08.210 | 55.55 | 70.82
| 2020-04-28T00:45:31.098 | 8.04 | 0.86
| 2020-04-28T00:46:00.034 | 7.58 | 6.05
| 2020-04-28T00:46:20.179 | 55.15 | 46.93
| 2020-04-28T00:46:43.072 | 7.59 | 3.82
| 2020-04-28T00:47:57.536 | 5.77 | 1.50
| 2020-04-28T00:48:44.833 | 4.24 | 1.88
| 2020-04-28T00:51:35.912 | 11.16 | 0.77
| 2020-04-28T00:54:57.448 | 25.87 | 4.45
| 2020-04-28T00:56:49.646 | 4.39 | 1.44
| 2020-04-28T01:04:03.146 | 12.42 | 0.77
| 2020-04-28T02:27:24.905 | 7.69 | 0.20
| 2020-04-28T03:47:52.140 | 13.99 | 2.00
| 2020-04-28T04:09:47.317 | 17.18 | 1.89
| 2020-05-05T03:02:56.033 | 2.58 | 0.42
| 2020-05-19T18:57:36.305 | 5.15 | 0.17
4th episode | 2021-09-10T01:00:44.000 | 34.71 | 2.36
| 2021-09-10T01:13:17.000 | 17.34 | 1.11
| 2021-09-10T01:14:37.000 | 27.50 | 3.63
| 2021-09-10T01:17:19.000 | 45.96 | 4.78
| 2021-09-10T01:18:54.000 | 19.35 | 1.78
| 2021-09-10T01:20:48.000 | 14.41 | 0.98
| 2021-09-10T01:27:05.000 | 6.25 | 0.90
| 2021-09-10T02:36:38.000 | 30.00 | 1.32
| 2021-09-10T02:44:34.000 | 35.00 | 2.80
| 2021-09-10T02:55:10.000 | 2.79 | 1.07
| 2021-09-11T03:02:28.000 | 7.78 | 0.56
| 2021-09-11T13:28:55.000 | 119.64 | 20.10
| 2021-09-11T15:06:43.000 | 199.73 | 81.49
| 2021-09-11T15:15:25.000 | 16.59 | 1.46
| 2021-09-11T15:17:45.000 | 141.90 | 49.38
| 2021-09-11T15:26:12.000 | 116.36 | 10.24
| 2021-09-11T15:32:33.000 | 36.21 | 4.49
| 2021-09-11T15:38:26.000 | 18.06 | 2.60
| 2021-09-11T16:50:04.000 | 99.77 | 4.39
| 2021-09-11T17:01:09.675 | 1.18 | 1.80
| 2021-09-11T17:04:29.740 | 14.59 | 4.30
| 2021-09-11T17:10:48.619 | 361.19 | 162.51
| 2021-09-11T20:13:40.000 | 27.33 | 4.70
| 2021-09-11T20:22:59.000 | 39.75 | 7.95
| 2021-09-12T00:45:49.000 | 36.25 | 1.45
| 2021-09-12T07:28:07.000 | 40.80 | 3.59

Note. — The Flux in units of $\times 10^{7}$ erg $\cdot$ cm-2 s-1 and Fluence
in units of $\times 10^{7}$ erg $\cdot$ cm-2. The time range of the 1st, 2nd,
3rd and 4th episodes is May 2016, June 2016, Apr.-May 2020 and Sept. 2021
respectively (see Table 2).

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