# Analysis of the $\gamma n\to K^{+}\Sigma^{-}(1385)$ photoproduction

Ai-Chao Wang  School of Nuclear Science and Technology, University of Chinese
Academy of Sciences, Beijing 101408, China  Neng-Chang Wei  School of Nuclear
Science and Technology, University of Chinese Academy of Sciences, Beijing
101408, China  Fei Huang<EMAIL_ADDRESS>School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing 101408, China

###### Abstract

In our previous work [Phys. Rev. D $\bf{101}$, 074025 (2020)], the
photoproduction $\gamma p\to K^{+}\Sigma^{0}(1385)$ has been investigated
within an effective Lagrangian approach. There, the reaction amplitudes were
constructed by considering the $t$-channel $K$ and $K^{\ast}(892)$ exchanges,
$s$-channel $N$ contribution, $u$-channel $\Lambda$ exchange, generalized
contact term, and a minimum number of $s$-channel $N$ and $\Delta$ resonance
diagrams. It was found that the inclusion of one of the $N(1895){1/2}^{-}$,
$\Delta(1900){1/2}^{-}$, and $\Delta(1930){5/2}^{-}$ resonances is essential
to reproduce the available differential and total cross-section data for
$\gamma p\to K^{+}\Sigma^{0}(1385)$. In the present work, we employ the same
model to study the photoproduction $\gamma n\to K^{+}\Sigma^{-}(1385)$, with
the purpose being to understand the reaction mechanism and, in particular, to
figure out which one of the $N(1895){1/2}^{-}$, $\Delta(1900){1/2}^{-}$, and
$\Delta(1930){5/2}^{-}$ resonances is really capable for a simultaneous
description of the data for both $K^{+}\Sigma^{0}(1385)$ and
$K^{+}\Sigma^{-}(1385)$ photoproduction reactions. The results show that the
available data on differential and total cross sections and photo-beam
asymmetries for $\gamma n\to K^{+}\Sigma^{-}(1385)$ can be reproduced only
with the inclusion of the $\Delta(1930){5/2}^{-}$ resonance rather than the
other two. The generalized contact term and the $t$-channel $K$ exchange are
found to dominate the background contributions. The resonance
$\Delta(1930){5/2}^{-}$ provides the most important contributions in the whole
energy region considered, and it is responsible for the bump structure
exhibited in the total cross sections.

$K\Sigma(1385)$ photoproduction, effective Lagrangian approach, nucleon
resonance

###### pacs:

25.20.Lj, 13.60.Le, 14.20.Gk

##  I Introduction

The study of nucleon and $\Delta$ resonances ($N^{\ast}$s and
$\Delta^{\ast}$s) has been of great interest in hadron physics, since a deeper
understanding of $N^{\ast}$s and $\Delta^{\ast}$s can help us get insight into
the nonperturbative regime of quantum chromodynamics (QCD). It is well known
that most of our current knowledge about $N^{\ast}$s and $\Delta^{\ast}$s is
coming from $\pi N$ scattering or $\pi$ photoproduction reactions.
Nevertheless, there are lots of resonances predicted by quark model
calculations Isgur:1977ef ; Koniuk:1979vy  or lattice QCD simulations
Edwards:2013 ; Lang:2017 ; Kiratidis:2017 ; Andersen:2018  but not detected in
the $\pi N$ production experiments. This situation forces us to study
$N^{\ast}$s and $\Delta^{\ast}$s in other reaction channels, to which some of
the $N^{\ast}$s and $\Delta^{\ast}$s might have much stronger coupling
strengths than to the $\pi N$ channel. In this regard, the production
processes of heavier mesons such as $\eta$, $\eta^{\prime}$, $K$, $K^{\ast}$,
$\omega$, and $\phi$ provide us rather effective tools to investigate the
high-mass resonances, as these channels have much higher threshold energies
than that of $\pi N$. In the present work, we concentrate on the
photoproduction reaction of $\gamma n\to K^{+}\Sigma^{-}(1385)$.

Experimentally, the high-statistics data on differential cross sections and
photo-beam asymmetries for the reaction $\gamma n\to K^{+}\Sigma^{-}(1385)$
became available in recent years Paul:2014 ; Hicks:2009 . In 2009, the LEPS
Collaboration released the first high-statistics differential cross-section
data in the photon laboratory incident energy range $E_{\gamma}=1.5-2.4$ GeV
and the first photo-beam asymmetry data at three energy points Hicks:2009 . In
2014, the CLAS Collaboration at the Thomas Jefferson National Accelerator
Facility (JLab) reported the preliminary differential cross-section data in
the photon laboratory incident energy range $E_{\gamma}=1.6-2.4$ GeV Paul:2014
. The CLAS data are located in a larger scattering angle range than the LEPS
data.

Theoretically, three works based on effective Lagrangian approaches have
already been devoted to the study of $\gamma n\to K^{+}\Sigma^{-}(1385)$
reaction Zou:2010 ; Xiaoyun:2016 ; Byung:2017 . In Ref. Zou:2010 , the LEPS
differential and total cross-section data and beam asymmetry data for $\gamma
n\to K^{+}\Sigma^{-}(1385)$ together with the CLAS total cross-section data
for $\gamma p\to K^{+}\Sigma^{0}(1385)$ were analyzed, and it was reported
that a newly proposed $\Sigma$ $(J^{p}={1/2}^{-})$ state with a mass around
$1380$ MeV is helpful to reproduce the negative beam asymmetry data for
$\gamma n\to K^{+}\Sigma^{-}(1385)$. We mention that at that time the high-
precision differential cross-section data for $\gamma p\to
K^{+}\Sigma^{0}(1385)$ from the CLAS Collaboration Moriya2013  were not yet
being available, thus they were not included in the analysis of Ref. Zou:2010
to constrain their model. In Ref. Xiaoyun:2016 , the cross-section data for
$\gamma n\to K^{+}\Sigma^{-}(1385)$ from both the LEPS and CLAS Collaborations
were analyzed within an interpolated Regge model, and it was reported that the
data can be described without inclusion of any $s$-channel baryon resonances.
Note that the LEPS photo-beam asymmetry data for $\gamma n\to
K^{+}\Sigma^{-}(1385)$ were not considered in Ref. Xiaoyun:2016 , and it is
also not clear whether this model can describe the CLAS high-precision
differential cross-section data for $\gamma p\to K^{+}\Sigma^{0}(1385)$ or
not. In Ref. Byung:2017 , the available data on differential cross sections
for both $\gamma p\to K^{+}\Sigma^{0}(1385)$ and $\gamma n\to
K^{+}\Sigma^{-}(1385)$ and on beam asymmetries for $\gamma n\to
K^{+}\Sigma^{-}(1385)$ were investigated in a Regge model. The data were
qualitatively described with some room left for improvements as no resonance
was considered in this model.

In our previous work Wang:2020 , we have performed a detailed investigation of
the $\gamma p\to K^{+}\Sigma^{0}(1385)$ reaction within an effective
Lagrangian approach.111As mentioned in Ref. Wang:2020 , the $\gamma p\to
K^{+}\Sigma^{0}(1385)$ reaction has also been investigated in Refs. Yong:2008
; hejun:2014  in effective Lagrangian approaches. We considered the
$s$-channel $N$ contribution, $t$-channel $K$ and $K^{\ast}$ exchange,
$u$-channel $\Lambda$ exchange, and generalized contact term. Besides, the
contributions from a minimum number of $N$ and $\Delta$ resonances in the $s$
channel was also introduced in constructing the reaction amplitudes. The CLAS
high-precision data on differential cross sections were well described, and it
was found that one of the $N(1895){1/2}^{-}$, $\Delta(1900){1/2}^{-}$, and
$\Delta(1930){5/2}^{-}$ resonances was necessarily needed to reproduce the
data.

In the present work, we employ the same model as in our earlier work of Ref.
Wang:2020  that works quite well for the $\gamma p\to K^{+}\Sigma^{0}(1385)$
reaction to study the photoproduction reaction $\gamma n\to
K^{+}\Sigma^{-}(1385)$. The purpose is to get a unified description of both
$\gamma p\to K^{+}\Sigma^{0}(1385)$ and $\gamma n\to K^{+}\Sigma^{-}(1385)$
within the same theoretical model. Particularly, we want to figure it out
which one of the $N(1895){1/2}^{-}$, $\Delta(1900){1/2}^{-}$, and
$\Delta(1930){5/2}^{-}$ resonances can results in a simultaneous description
of the data for both $K^{+}\Sigma^{0}(1385)$ and $K^{+}\Sigma^{-}(1385)$
photoproduction reactions.

The paper is organized as follows. In Sec. II, we briefly introduce the
framework of our theoretical model. In Sec. III, we present our theoretical
results and discussions. Finally, a brief summary and conclusions are given in
Sec. IV.

##  II Formalism

The full photoproduction amplitudes for $\gamma N\to K\Sigma(1385)$ can be
expressed as Haberzettl:1997 ; Haberzettl:2006

$M^{\nu\mu}=M^{\nu\mu}_{s}+M^{\nu\mu}_{t}+M^{\nu\mu}_{u}+M^{\nu\mu}_{\rm
int},$ (1)

with $\nu$ and $\mu$ being the Lorentz indices of $\Sigma(1385)$ and the
photon $\gamma$, respectively. The first three terms $M^{\nu\mu}_{s}$,
$M^{\nu\mu}_{t}$, and $M^{\nu\mu}_{u}$ stand for the $s$-, $t$-, and
$u$-channel pole diagrams, respectively, with $s$, $t$, and $u$ being the
Mandelstam variables of the internally exchanged particles. They arise from
the photon attaching to the external particles in the $KN\Sigma(1385)$
interaction vertex. The last term, $M^{\nu\mu}_{\rm int}$, stands for the
interaction current that arises from the photon attaching to the internal
structure of the $KN\Sigma(1385)$ interaction vertex. All four terms in Eq.
(1) are diagrammatically depicted in Fig. 1.

(a) $s$ channel (b) $t$ channel

(c) $u$ channel (d) Interaction current

Figure 1: Generic structure of the amplitude for $\gamma n\to
K^{+}\Sigma^{-}(1385)$. Time proceeds from left to right. The symbols
$\Sigma^{\ast}$ and $K^{\ast}$ denote $\Sigma(1385)$ and $K^{\ast}(892)$,
respectively.

The interaction mechanisms considered in the present work for $\gamma n\to
K^{+}\Sigma^{-}(1385)$ are the same as those in Ref. Wang:2020  that works for
$\gamma p\to K^{+}\Sigma^{0}(1385)$, except that the $u$-channel $\Lambda$
exchange in $\gamma p\to K^{+}\Sigma^{0}(1385)$ is now replaced by the
$\Sigma$ exchange in $\gamma n\to K^{+}\Sigma^{-}(1385)$ due to the charges of
$\Sigma(1385)$ baryons. In brief, the following contributions (as shown in
Fig. 1) are considered in constructing the $s$-, $t$-, and $u$-channel
amplitudes for $\gamma n\to K^{+}\Sigma^{-}(1385)$: (i) $N$, $N^{\ast}$, and
$\Delta^{\ast}$ contributions in the $s$ channel, (ii) $K$ and $K^{\ast}(892)$
exchanges in the $t$ channel, and (iii) $\Sigma^{\ast}$ hyperon exchange in
the $u$ channel.

Using an effective Lagrangian approach, one can, in principle, obtain explicit
expressions for these amplitudes by evaluating the corresponding Feynman
diagrams. However, the exact calculation of the interaction current
$M^{\nu\mu}_{\rm int}$ is impractical, as it obeys a highly nonlinear equation
and contains diagrams with very complicated interaction dynamics. Following
Refs. Haberzettl:1997 ; Haberzettl:2006 ; Huang:2012 ; Huang:2013 , we model
the interaction current by a generalized contact current, which effectively
accounts for the interaction current arising from the unknown parts of the
underlying microscopic model:

$M^{\nu\mu}_{\rm int}=\Gamma^{\nu}_{\Sigma^{\ast}NK}(q)C^{\mu}+M^{\nu\mu}_{\rm
KR}f_{t}.$ (2)

Here $\nu$ and $\mu$ are Lorentz indices for $\Sigma(1385)$ and $\gamma$,
respectively; $\Gamma^{\nu}_{\Sigma^{\ast}NK}(q)$ is the vertex function of
$\Sigma(1385)NK$ coupling,

$\Gamma^{\nu}_{\Sigma^{\ast}NK}(q)=-\frac{g_{\Sigma^{\ast}NK}}{M_{K}}q^{\nu},$
(3)

with $q$ being the four-momentum of the outgoing $K$ meson; $M_{\rm
KR}^{\nu\mu}$ is the Kroll-Ruderman term,

$M^{\nu\mu}_{\rm KR}=\frac{g_{\Sigma^{\ast}NK}}{M_{K}}g^{\nu\mu}TQ_{K},$ (4)

with $T$ denoting the isospin factor of the $\Sigma(1385)NK$ coupling and
$Q_{K}$ being the electric charge of the outgoing $K$ meson; $f_{t}$ is the
phenomenological form factor attached to the amplitude of $t$-channel $K$
exchange, whose explicit form has been given in Ref. Wang:2020 ; $C^{\mu}$ is
an auxiliary current, which is nonsingular and is introduced to ensure that
the full photoproduction amplitudes of Eq. (1) are fully gauge invariant.
Following Refs. Haberzettl:2006 ; Huang:2012 , we choose $C^{\mu}$ for $\gamma
n\to K^{+}\Sigma^{-}(1385)$ as

$\displaystyle C^{\mu}=$
$\displaystyle-\tau_{t}(2q-k)^{\mu}\frac{f_{t}-1}{t-q^{2}}\left[f_{u}+\hat{A}\left(1-f_{u}\right)\right]$
$\displaystyle+\tau_{u}(2p^{\prime}-k)^{\mu}\frac{f_{u}-1}{u-p^{\prime
2}}\left[f_{t}+\hat{A}\left(1-f_{t}\right)\right].$ (5)

Here $p^{\prime}$, $q$, and $k$ are the four-momenta for the outgoing
$\Sigma(1385)$, outgoing $K$, and incoming photon, respectively;
$\tau_{t\left(u\right)}$ denotes the isospin factor in the corresponding
$t\left(u\right)$-channel hadronic vertex; $f_{u}$ and $f_{t}$ are the
phenomenological form factors for $u$-channel $\Sigma(1385)$ exchange and
$t$-channel $K$ exchange, respectively; $\hat{A}$ is a Lorentz-covariant,
crossing-symmetric phenomenological function introduced to prevent the
“violation of scaling behavior” as noted in Ref. Drell:1972 . The prescription
of Eq. (5) for the auxiliary current $C^{\mu}$ corresponds to set
$\hat{h}=1-\hat{A}$ in Ref. Wang:2020 . Following Ref. Xiaoyun:2016 ,
$\hat{A}$ is taken as

$\hat{A}(t,u)=A_{0}\frac{\Lambda_{c}^{4}}{\Lambda_{c}^{4}+\left(s-s_{\rm
th}\right)^{2}},$ (6)

with

$s_{\rm th}=\left(m_{\Sigma^{\ast}}+m_{K}\right)^{2}.$ (7)

Here the cutoff $\Lambda_{c}$ is fixed to be $2.5$ GeV to make $\hat{A}$ not
fall off too rapidly for the energy range considered, and the strength $A_{0}$
is taken as a fit parameter.

In Ref. Wang:2020 , we have presented the effective Lagrangians, resonance
propagators, and phenomenological form factors for the $\gamma p\to
K^{+}\Sigma^{0}(1385)$ reaction. The same formulas also apply for the $\gamma
n\to K^{+}\Sigma^{-}(1385)$ reaction. For the brevity of the present paper, we
do not repeat them here. For the newly introduced $u$-channel $\Sigma(1385)$
exchange, the Lagrangian for the $\gamma\Sigma^{\ast}\Sigma^{\ast}$
electromagnetic coupling reads

$\mathcal{L}_{\gamma\Sigma^{\ast}\Sigma^{\ast}}=e{\bar{\Sigma}}^{\ast}_{\mu}A_{\alpha}\Gamma_{\gamma\Sigma^{\ast}}^{\alpha,\mu\nu}\Sigma^{\ast}_{\nu},$
(8)

with

$\displaystyle A_{\alpha}\Gamma_{\gamma\Sigma^{\ast}}^{\alpha,\mu\nu}=$
$\displaystyle\;Q_{\Sigma^{\ast}}A_{\alpha}\left[g^{\mu\nu}\gamma^{\alpha}-\frac{1}{2}\left(\gamma^{\mu}\gamma^{\nu}\gamma^{\alpha}+\gamma^{\alpha}\gamma^{\mu}\gamma^{\nu}\right)\right]$
$\displaystyle-\frac{\kappa_{\Sigma^{\ast}}}{2M_{N}}\sigma^{\alpha\beta}\left(\partial_{\beta}A_{\alpha}\right)g^{\mu\nu},$
(9)

where $Q_{\Sigma^{\ast}}$ is the electric charge of $\Sigma(1385)$, and
$\kappa_{\Sigma^{\ast}}$ denotes the anomalous magnetic moment of
$\Sigma(1385)$ taken as $\kappa_{\Sigma^{\ast-}}=-2.43$ from a quark model
prediction Lichtenberg:1977 .

##  III Results and discussion

Table 1:  Fitted values of adjustable model parameters. The other parameters not shown in this table are taken from model III of Ref. Wang:2020 . $R$ denotes the $\Delta(1930){5/2}^{-}$ resonance. The cutoff parameters $\Lambda_{K}$ and $\Lambda_{\Sigma^{\ast}}$ are in MeV. $A_{0}$ | $\Lambda_{K}$ | $\Lambda_{\Sigma^{\ast}}$ | $g_{RN\gamma}^{(1)}g_{R\Sigma^{\ast}K}^{(1)}$ | $g_{RN\gamma}^{(2)}g_{R\Sigma^{\ast}K}^{(1)}$
---|---|---|---|---
$-0.056\pm 0.009$ | $805\pm 7$ | $915\pm 97$ | $-8.7\pm 0.3$ | $-46.5\pm 1.8$
Figure 2: Differential cross sections for $\gamma n\to K^{+}\Sigma^{-}(1385)$
as a function of $\cos\theta$ in model III of Ref. Wang:2020  with the
$\Delta(1930){5/2}^{-}$ resonance. The red squares and blue circles denote the
data from the LEPS Collaboration Hicks:2009  and CLAS Collaboration Paul:2014
, respectively. The numbers in parentheses denote the centroid value of the
photon laboratory incident energy (left number) and the corresponding total
center-of-mass energy of the system (right number), in MeV. Figure 3: Photo-
beam asymmetries for $\gamma n\to K^{+}\Sigma^{-}(1385)$ plotted against the
photon laboratory energy $E_{\gamma}$ in model III of Ref. Wang:2020  with the
$\Delta(1930){5/2}^{-}$ resonance. The red full squares denote the data from
the LEPS Collaboration Hicks:2009 .  Figure 4: Differential cross sections for
$\gamma n\to K^{+}\Sigma^{-}(1385)$ as a function of $\cos\theta$ (black solid
lines). The red squares and blue circles denote the data from the LEPS
Collaboration Hicks:2009  and CLAS Collaboration Paul:2014 , respectively. The
cyan dash-dotted, blue dashed, and red dot-double-dashed lines represent the
individual contributions from the $s$-channel $\Delta(1930){5/2}^{-}$
resonance, generalized contact term, and $t$-channel $K$ exchange,
respectively. The numbers in parentheses denote the centroid value of the
photon laboratory incident energy (left number) and the corresponding total
center-of-mass energy of the system (right number), in MeV. Figure 5:
Differential cross sections for $\gamma n\to K^{+}\Sigma^{-}(1385)$ as a
function of photon incident energy at four $\cos\theta$ intervals (black solid
lines). The notations are the same as in Fig. 4. The red full squares denote
the data from the LEPS Collaboration Hicks:2009 . Figure 6: The beam
asymmetries for $\gamma n\to K^{+}\Sigma^{-}(1385)$ plotted against photon
incident energy $E_{\gamma}$. The red full squares denote the data from the
LEPS Collaboration Hicks:2009 . Figure 7: Total cross sections for $\gamma
n\to K^{+}\Sigma^{-}(1385)$ with dominant individual contributions as a
function of photon incident energy. Data (red full squares) are taken from the
LEPS Collaboration Hicks:2009  but not included in the fit.

A combined analysis of the data for both the $\gamma n\to
K^{+}\Sigma^{-}(1385)$ and $\gamma p\to K^{+}\Sigma^{0}(1385)$ reactions will
put more constraints on theoretical models and thus result in more reliable
results. As mentioned in Sec. I, in literature, the photoproduction reaction
$\gamma n\to K^{+}\Sigma^{-}(1385)$ has been investigated by three theoretical
works in Refs. Zou:2010 ; Xiaoyun:2016 ; Byung:2017 . But only in a Regge
model of Ref. Byung:2017 , the differential cross-section data for $\gamma
p\to K^{+}\Sigma^{0}(1385)$ were considered. There, due to the lack of
resonance contributions, the cross-section data for $\gamma n\to
K^{+}\Sigma^{-}(1385)$ and $\gamma p\to K^{+}\Sigma^{0}(1385)$ were only
qualitatively reproduced.

In our previous work of Ref. Wang:2020 , we have investigated the
photoproduction reaction $\gamma p\to K^{+}\Sigma^{0}(1385)$ within an
effective Lagrangian approach. There, apart from the $t$-channel $K$ and
$K^{\ast}(892)$ exchanges, $s$-channel $N$ contribution, $u$-channel $\Lambda$
exchange, and generalized contact term, the contributions from a minimum
number of $N$ and $\Delta$ resonances were also considered to construct the
reaction amplitudes to describe the data. It was found that the high-precision
differential cross-section data from the CLAS Collaboration for $\gamma p\to
K^{+}\Sigma^{0}(1385)$ can be well reproduced by including the contribution
from one of the $N(1895){1/2}^{-}$, $\Delta(1900){1/2}^{-}$, and
$\Delta(1930){5/2}^{-}$ resonances in the $s$ channel.

In the present work, we plan to analyze all the available data on differential
cross sections from the LEPS Collaboration and CLAS Collaboration and on
photo-beam asymmetries from the LEPS Collaboration for the reaction $\gamma
n\to K^{+}\Sigma^{-}(1385)$ within the same model as in our previous work of
Ref. Wang:2020  that works quite well for $\gamma p\to K^{+}\Sigma^{0}(1385)$.
The purpose is to understand the reaction mechanism of $\gamma n\to
K^{+}\Sigma^{-}(1385)$ and, in particular, to figure out which one of the
$N(1895){1/2}^{-}$, $\Delta(1900){1/2}^{-}$, and $\Delta(1930){5/2}^{-}$
resonances really contributes in both the $\gamma n\to K^{+}\Sigma^{-}(1385)$
and $\gamma p\to K^{+}\Sigma^{0}(1385)$ reactions.

Most of the interaction mechanisms for $\gamma p\to K^{+}\Sigma^{0}(1385)$ and
$\gamma n\to K^{+}\Sigma^{-}(1385)$ are the same except for the $u$-channel
interaction and the generalized contact term. In $u$-channel amplitudes, the
$\Lambda$ exchange contributes only in $\gamma p\to K^{+}\Sigma^{0}(1385)$
while the $\Sigma^{-}(1385)$ exchange contributes only in $\gamma n\to
K^{+}\Sigma^{-}(1385)$ due to the charges of the outgoing $\Sigma(1385)$
particles. For the generalized contact term, one has different auxiliary
current $C^{\mu}$ [c.f. Eq. (5)] for $\gamma p\to K^{+}\Sigma^{0}(1385)$ and
$\gamma n\to K^{+}\Sigma^{-}(1385)$ due to the fact that apart from the
$t$-channel $K^{+}$ exchange which contributes to both the $\gamma p$ and
$\gamma n$ channels, the $s$-channel $N$ exchange (longitudinal part)
contributes only in the $\gamma p$ channel and the $u$-channel $\Sigma(1385)$
exchange contributes only in the $\gamma n$ channel.

In Ref. Wang:2020 , we have reported three models for $\gamma p\to
K^{+}\Sigma^{0}(1385)$, each with one of the $N(1895){1/2}^{-}$,
$\Delta(1900){1/2}^{-}$, and $\Delta(1930){5/2}^{-}$ resonances, respectively.
Here for $\gamma n\to K^{+}\Sigma^{-}(1385)$, we test these three models one
by one by comparing the theoretical results from each model with the data. In
all these three models, one has $A_{0}$ in Eq. (6) and the cutoff parameter
$\Lambda_{\Sigma^{\ast}}$ for $u$-channel $\Sigma(1385)$ exchange as fit
parameters, and in model I with the $N(1895){1/2}^{-}$ resonance, one has the
resonance electromagnetic coupling constant as an additional fit parameter.

The results from these three models show that although the differential cross-
section data for $\gamma n\to K^{+}\Sigma^{-}(1385)$ can be qualitatively
described, the negative photo-beam asymmetries cannot be reasonably
reproduced. As an illustration, we show in Fig. 2 the differential cross
sections and in Fig. 3 the photo-beam asymmetries resulted from model III of
Ref. Wang:2020  with the $\Delta(1930){5/2}^{-}$ resonance. One sees that the
differential cross-section data are satisfactorily described, however, in the
energy region of photon energy $E_{\gamma}>1.9$ GeV, the LEPS beam-asymmetry
data show negative values while the theoretical results are positive.

We then release the resonance electromagnetic coupling constants for $\Delta$
resonances and the cutoff value for $t$-channel $K$ exchange as fit parameters
instead of taking their fixed values from Ref. Wang:2020 . By doing this, it
is found that both the differential cross-section data and the photo-beam
asymmetry data for $\gamma n\to K^{+}\Sigma^{-}(1385)$ can be reasonably
reproduced in the model where the $\Delta(1930){5/2}^{-}$ resonance is
included, while in other two models with inclusion of either the
$N(1895){1/2}^{-}$ or the $\Delta(1900){1/2}^{-}$ resonance, the data cannot
be satisfactorily described. In the following parts of the paper, we show and
discuss the results from the model with the $\Delta(1930){5/2}^{-}$ resonance.

In Table 1, we present the fitted values of the adjustable parameters in the
model with the $\Delta(1930){5/2}^{-}$ resonance. The values of the parameters
that are not shown in this table are taken from model III of Ref. Wang:2020 .
One sees that the cutoff parameter of $t$-channel $K$ exchange is reduced from
$950$ MeV in model III of Ref. Wang:2020  to $805$ MeV. Although the
$t$-channel $K$ exchange itself results in negative beam asymmetries, this
diagram also contributes to the generalized contact term [c.f. Eq. (2)] which
results in positive beam asymmetries. This explains why the beam asymmetry
data for $\gamma n\to K^{+}\Sigma^{-}(1385)$ prefers a smaller cutoff value
for $t$-channel $K$ exchange. The fitted value of the cutoff parameter for
$u$-channel $\Sigma(1385)$ exchange has large uncertainties. This is because
that we don’t have data at backward angles where the $u$-channel
$\Sigma(1385)$ exchange is expected to contribute significantly, and thus this
parameter is not that well constrained by the available data.

The theoretical results for differential cross sections and photo-beam
asymmetries are shown in Figs. 4-6. There, the black solid lines represent the
results from the full calculation. The cyan dash-dotted, blue dashed, and red
dot-double-dashed lines represent the individual contributions from the
$s$-channel $\Delta(1930){5/2}^{-}$ resonance, generalized contact term, and
$t$-channel $K$ exchange, respectively. The contributions from other terms are
too small to be clearly seen with the scale used, and thus they are not
plotted. In Fig. 4, the numbers in parentheses denote the centroid value of
the photon laboratory incident energy (left number) and the corresponding
total center-of-mass energy of the system (right number), in MeV. One sees
from Figs. 4-6 that all our theoretical differential cross sections and photo-
beam asymmetries for $\gamma n\to K^{+}\Sigma^{-}(1385)$ agree with the
corresponding data quite well, except that at the lowest energy in Fig. 4, the
CLAS differential cross-section data Paul:2014  are somehow underestimated.
Note that the CLAS differential cross-section data are measured at a relative
large energy bin of $200$ MeV. In the first subfigure of Fig. 4, we show the
theoretical results are $E_{\gamma}=1600$ MeV, while the CLAS data were
measured at $E_{\gamma}=1500-1700$ MeV. We know that near threshold the phase
space is rather sensitive to the energy, which may partially explain the
deviation of our theoretical differential cross sections from the data at
$E_{\gamma}=1600$ MeV.

It is seen from Figs. 4 and 5 that the $\Delta(1930){5/2}^{-}$ resonance
contribution dominates the cross sections of $\gamma n\to
K^{+}\Sigma^{-}(1385)$ in the relative low-energy region ($E_{\gamma}\leq
2000$ MeV), the contact term contribute significantly in the whole energy
region considered, and the $K$ exchange makes considerable contributions at
forward angles in the high energy region.

In Fig. 7, we show the total cross sections for $\gamma n\to
K^{+}\Sigma^{-}(1385)$ with dominant individual contributions as a function of
photon incident energy. Note that the LEPS data are measured at the angular
interval $\cos\theta=0.6-1$, and correspondingly, our theoretical total cross
sections are obtained by an integral of the differential cross sections at the
angular interval $\cos\theta=0.6-1$. Although only the differential cross
sections at forward angles are considered, consistent observations can be made
from Fig. 7 compared with those obtained from the differential cross sections
of Figs. 4 and 5. The $\Delta(1930){5/2}^{-}$ resonance contribution dominates
the cross sections below $E_{\gamma}=2000$ MeV and is responsible for the bump
structure exhibited by the data near the $K\Sigma(1385)$ threshold. The
contact term provides significant contributions in the whole energy region
considered, and actually, it occupies the main body of the background
contributions. Considerable contributions can also be seen from the
$t$-channel $K$ exchange at higher energies.

##  IV Summary and conclusion

A combined analysis of the data for $\gamma n\to K^{+}\Sigma^{-}(1385)$ and
$\gamma p\to K^{+}\Sigma^{0}(1385)$ will put on more constraints on
theoretical models and thus result in more reliable results. In literature,
the data for the $\gamma n\to K^{+}\Sigma^{-}(1385)$ reaction have been
investigated in three theoretical works Zou:2010 ; Xiaoyun:2016 ; Byung:2017 .
But only in Ref. Byung:2017 , the data for the $\gamma p\to
K^{+}\Sigma^{0}(1385)$ reaction have been considered simultaneously. There,
the data for both $\gamma n\to K^{+}\Sigma^{-}(1385)$ and $\gamma p\to
K^{+}\Sigma^{0}(1385)$ are roughly described in a Regge model with some room
left for improvements due to the lack of contributions from nucleon resonances
in the Regge model.

In our previous work Wang:2020 , the high-precision differential cross section
data for $\gamma p\to K^{+}\Sigma^{0}(1385)$ from the CLAS Collaboration
Moriya2013  have been studied in an effective Lagrangian approach. There, we
constructed the reaction amplitudes by considering the contributions from a
minimum number of $N$ and $\Delta$ resonances in the $s$ channel besides the
$K$ and $K^{\ast}(892)$ exchanges in the $t$ channel, $\Lambda$ exchange in
the $u$ channel, $N$ contribution in the $s$ channel, and the generalized
contact term. It was found that the data for $\gamma p\to
K^{+}\Sigma^{0}(1385)$ can be well reproduced by considering one of the
$N(1895){1/2}^{-}$, $\Delta(1900){1/2}^{-}$, and $\Delta(1930){5/2}^{-}$
resonances.

In the present work, we use the same model developed in our previous work of
Ref. Wang:2020  for the $\gamma p\to K^{+}\Sigma^{0}(1385)$ reaction to
analyze the currently available differential cross-section data from the LEPS
Collaboration and CLAS Collaboration and the photo-beam asymmetry data from
the LEPS Collaboration for the reaction $\gamma n\to K^{+}\Sigma^{-}(1385)$,
with the purpose being to understand the reaction mechanism of $\gamma n\to
K^{+}\Sigma^{-}(1385)$ and, in particular, to figure out which one of the
$N(1895){1/2}^{-}$, $\Delta(1900){1/2}^{-}$, and $\Delta(1930){5/2}^{-}$
resonances is capable for a description of the data for $\gamma n\to
K^{+}\Sigma^{-}(1385)$. The interaction diagrams for these two reactions are
the same except that the $u$-channel $\Lambda$ exchange in $\gamma p\to
K^{+}\Sigma^{0}(1385)$ is replaced by the $\Sigma(1385)$ exchange in $\gamma
n\to K^{+}\Sigma^{-}(1385)$, and the generalized contact term in these two
reactions differ consequently as the $u$-channel $\Sigma(1385)$ exchange
contributes only in $\gamma n\to K^{+}\Sigma^{-}(1385)$ while the $s$-channel
$N$ exchange (longitudinal part) contributes only in $\gamma p\to
K^{+}\Sigma^{0}(1385)$. The values of most of the model parameters are taken
from Ref. Wang:2020  with the exception of the cutoff of $t$-channel $K$
exchange and the resonance electromagnetic couplings which are adjusted to fit
the data.

It was found that the available differential cross-section data the photo-beam
asymmetry data for $\gamma n\to K^{+}\Sigma^{-}(1385)$ can be satisfactorily
described by including the contribution from the $\Delta(1930){5/2}^{-}$
resonance in the $s$ channel. The $\Delta(1930){5/2}^{-}$ resonance
contribution dominates the cross sections of $\gamma n\to
K^{+}\Sigma^{-}(1385)$ below $E_{\gamma}=2000$ MeV and is responsible for the
near-threshold bump structure exhibited by the total cross-section data. The
contact term contributes significantly in the whole energy considered, and it
occupies the main body of the background contributions for $\gamma n\to
K^{+}\Sigma^{-}(1385)$. The $t$-channel $K$ exchange contributes considerably
at forward angles in the high energy region. The contributions from other
terms are found to be rather small.

Combining the results of the present work for $\gamma n\to
K^{+}\Sigma^{-}(1385)$ and of our previous work Wang:2020  for $\gamma p\to
K^{+}\Sigma^{0}(1385)$, a conclusion can be made that all the available data
for $\Sigma(1385)$ photoproduction on both proton and neutron targets can be
well described by including the contributions from the $\Delta(1930){5/2}^{-}$
resonance in $s$ channel in addition to the background contributions from the
$s$-, $t$-, and $u$-channel hadron exchanges and the interaction current in
our effective Lagrangian approach.

Seriously speaking, the cutoff parameter for $t$-channel $K$ exchange,
$\Lambda_{K}$, and the $\Delta(1930){5/2}^{-}$ resonance couplings for $\gamma
p\to K^{+}\Sigma^{0}(1385)$ should be the same as those for $\gamma n\to
K^{+}\Sigma^{-}(1385)$. However, as shown in Table I of the present work and
Table II of our previous work Wang:2020 , the values of $\Lambda_{K}$,
$g^{(1)}_{RN\gamma}g^{(1)}_{R\Sigma^{\ast}K}$, and
$g^{(2)}_{RN\gamma}g^{(1)}_{R\Sigma^{\ast}K}$ from the fit of the neutron
target data differ from those from the fit to the proton target data. The
possible reason is that in both our previous work and the present work, the
resonance hadronic coupling constant $g^{(2)}_{R\Sigma^{\ast}K}$ is set to be
$0$ for simplicity. With this parameter left free, the model is expected to
provide a self-consistent description of all the available data for both
$\gamma p\to K^{+}\Sigma^{0}(1385)$ and $\gamma n\to K^{+}\Sigma^{-}(1385)$ by
considering the $\Delta(1930){5/2}^{-}$ resonance contributions in $s$ channel
in addition to the background contributions. We leave such a work to the
future when more data for $\Sigma(1385)$ photoproduction become available.

###### Acknowledgements.

This work is partially supported by the National Natural Science Foundation of
China under Grants No. 12175240, No. 12147153, and No. 11635009, the
Fundamental Research Funds for the Central Universities, and the China
Postdoctoral Science Foundation under Grants No. 2021M693141 and No.
2021M693142.

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# Prevalence of non-stationarity in quasi-periodic pulsations (QPPs)
associated with M- and X- class solar flares

T. Mehta,1 A.-M. Broomhall,1,2 and L. A. Hayes, 3
1 Centre for Fusion, Space and Astrophysics, Department of Physics, University
of Warwick, Coventry, CV4 7AL, UK
2 Centre for Exoplanets and Habitability, University of Warwick, Coventry CV4
7AL, UK
3 European Space Agency (ESA), European Space Research and Technology Centre
(ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands E-mail:
<EMAIL_ADDRESS>
(Accepted 2023 May 26. Received 2023 May 26; in original form 2022 Oct 18)

###### Abstract

Quasi-periodic pulsations (QPPs) are frequently observed in solar and stellar
flare emission, with recent studies suggesting that an increasing
instantaneous period is a common characteristic of QPPs. Determining the
prevalence of non-stationarity in QPPs contributes to a better understanding
of which mechanism(s) is (are) responsible in QPP generation. We obtain the
rate of period evolution from QPPs in 98 M- and X- class flares from Solar
Cycle 24 with average periods between 8-130 s and investigate the prevalence
of QPP non-stationarity. We also investigate whether the presence of a Coronal
Mass Ejection (CME) impacts the period evolution of QPPs. We analyse soft
X-ray lightcurves obtained from GOES’ X-Ray Sensor (XRS) and assess the
dominant periods in the impulsive and decay phases of the flares using the
Fast Fourier Transform. We relate the rate of period evolution to flare
duration, peak flare energy, and average QPP period. We find evidence of non-
stationarity in 81$\%$ of the flares assessed, with most QPPs exhibiting a
period evolution of $\leq$10 s between the impulsive and decay phases, of
which 66$\%$ exhibited an apparent period growth and 14$\%$ showed an apparent
period shrinkage. We find a positive correlation between the absolute
magnitude of period evolution and the duration of the flare and no correlation
between the period evolution of the QPPs and flare energy or CME presence.
Furthermore, we conclude that non-stationarity is common in solar QPPs and
must be accounted for in flare analysis.

###### keywords:

Sun: flares, Sun: oscillations, Sun: particle emission, Sun: X-rays, Sun:
coronal mass ejections (CMEs)

††pubyear: 2023††pagerange: Prevalence of non-stationarity in quasi-periodic
pulsations (QPPs) associated with M- and X- class solar flares–10

##  1 Introduction

The emission from a solar flare often demonstrates fluctuations in intensity
as a function of time. These fluctuations are known as Quasi-Periodic
Pulsations (QPPs) and are characterised as repetitive bursts with similar
time-scales that can range from seconds to several tens of seconds (Nakariakov
& Melnikov, 2009; Van Doorsselaere et al., 2016; Kupriyanova et al., 2020).
QPPs are identified across the entire electromagnetic spectrum of flare
emissions, meaning that they are typically a multi-wavelength phenomenon (e.g.
see Clarke et al., 2021). While non-thermal hard X-ray and microwave
observations clearly demonstrate the most prominent pulsations during a flare,
measurements from the past Solar Cycle with Sun-as-a-star soft X-ray (SXR) and
extreme ultraviolet (EUV) observations have shown that small-amplitude QPPs
are a very common feature of solar flares (Simões et al., 2015; Dominique et
al., 2018; Hayes et al., 2020).

The study of solar flare emission fluctuations extends beyond our solar system
as stellar flare QPPs have been extensively observed (Zhilyaev et al., 2000;
Pugh et al., 2016; Broomhall et al., 2019a). These QPPs observed in stellar
flares are largely similar in characteristics to those observed in solar QPPs,
which strengthens the case for a solar-stellar analogy for QPPs (see Zimovets
et al., 2021, for an overview on recent advances in observations of stellar
QPPs). Therefore a better understanding of the mechanism(s) driving QPPs in
solar flares is likely to lead to advances in stellar QPPs.

The question as to what causes these repetitive flare emissions has been the
topic of significant discussion (McLaughlin et al., 2018), with over fourteen
different mechanisms suggested to date (See Kupriyanova et al., 2020; Zimovets
et al., 2021, and references therein for an overview on generation
mechanisms). The proposed generation mechanisms can be sorted into three
groups; 1. Mechanisms that modulate the direct release of plasma emissions as
the result of MHD oscillations; 2. Mechanisms where MHD waves modulate the
efficiency of energy release; 3. Mechanisms based on spontaneous quasi-
periodic energy release. Despite the growing number of mechanisms proposed to
underpin the generation of QPPs, we are not yet in a position to confidently
identify which mechanism is responsible and it seems likely that there are
multiple mechanisms at play in generating QPPs.

There is an expanding catalogue of QPPs which exhibit non-stationary
properties, with both the phase, period, and amplitude varying in time (see
Nakariakov et al., 2019, for review). For example, period drifts have been
identified in several flares (Kupriyanova et al., 2010; Simões et al., 2015;
Kolotkov et al., 2018), with it common to find the decay phase periods longer
than the associated impulsive phase periods (e.g. Hayes et al., 2016, 2020).
Notably, in some cases, QPPs can be observed to extend late into the decay
phase of solar flares and illustrate systematic increases in periods (Dennis
et al., 2017; Hayes et al., 2019). There is a growing need to understand how
the periods evolve over flares, whether period drifts are a common feature of
flare QPPs, and whether the period drifts are systematic based on flare class,
duration, or whether they are eruptive or not. We also need to address the
prevalence of non-stationarity in solar QPPs, as the majority of detection
methods used currently rely on a periodogram-based approach. As discussed in
Broomhall et al. (2019b), periodogram-based approaches tend to be less
successful when detecting a non-stationary QPP. It is likely that we are
missing, or at best, poorly characterising, the presence and behaviour of many
QPPs by assuming their dominant periods are stationary. In quantifying the
proportion of QPPs that exhibit non-stationarity we can better discern which
analysis methods are the most appropriate to use when searching and
categorising QPPs.

In this work we explore the nature of QPP period drifts by investigating
whether non-stationarity is an inherent feature of QPPs. To achieve this we
build upon the work of Hayes et al. (2020) and we present a comparison of the
dominant periods (the periodicity that corresponds to the largest peak
relative to the confidence level in a power spectrum) in the impulsive phase
of the flare (characterised as the time from the start of the flare to the
time corresponding to flare maximum) and the decay phase (after the flare
peak) in QPPs from M- and X- class flares from Solar Cycle 24. By examining
the prevalence of QPPs that show evidence of non-stationarity we can
potentially classify the different types of QPPs present in solar flare
emission, and help constrain which mechanisms can drive QPPs.

##  2 Observations and Analysis Methods

###  2.1 Data

To select a list of flares for which to perform this study, we utilise a list
of M- and X- GOES class flares from 1st February 2011 to 31st December 2018
(i.e. Solar Cycle 24) that demonstrated strong evidence of QPP signatures in
their emission from the study of Hayes et al. (2020). This list consists of
205 flare events that showed enhanced Fourier power in the periodograms of the
GOES-XRS 1–8 Å channel observations. We further analyse this list of flares by
focusing on the same 1–8 Å channel from the GOES-15 satellite which has a
cadence of 2.047 s, and focus on analysing the impulsive and decay phases of
the flares independently to identify features of non-stationarity and period
drift.

To determine the duration of the impulsive phase, we use the flare start and
peak times defined within the GOES flare catalogue produced by the National
Oceanic and Atmospheric Administration (NOAA). NOAA define the flare start
time as the first minute in a sequence of four minutes wherein there is a
steep monotonic increase in the 1–8 Å channel and the final flux value is
greater than the first by a factor of 1.4. The flare peak time is the time at
which the flares soft X-ray emission reaches its flare peak energy, which is
its maximal value as measured in the 1–8 Å channel. For our analysis, we limit
the time window of the decay phase to the same duration as the impulsive
phase. We use this method of choosing the end times rather than using the end
times defined within the GOES flare catalogue. This is because of our
implementation of criteria (ii) (discussed in Section 2.2) which requires 5 or
more full cycles in each phase of the flare. This means that for a flare with
impulsive/ decay phases of unequal length, each phase has a different upper
limit on the maximal periodicity that can be obtained. This discrepancy in the
upper limit threatens to artificially induce artefacts in the data. Therefore
for consistency we limit the time window of the decay phase to the same
duration as the impulsive phase, as can be seen in Fig. 1. However for most
events the end times we chose and those defined by the GOES catalogue were
similar.

To examine whether the presence of a Coronal Mass Ejection (CME) correlates
with the appearance or magnitude of a period evolution of the QPPs, we use the
publicly available SOHO/LASCO CME catalogue, to determine which flares had
associated CMEs.

###  2.2 Method

We separate the flare into the impulsive and decay phases, we perform a Fast
Fourier Transform (FFT) on each phase and test whether a periodic signature is
present above a 95$\%$ confidence level. We obtain the confidence levels by
making use of the technique outlined in Pugh et al. (2017) which is based upon
the work in Vaughan (2005). This method involves fitting the power spectrum
with a broken power law, which accounts for both the presence of red and white
noise in the signal and avoids the problems that can arise in assessing the
significance of an identified periodic signature when detrending data. Using
this fitting we determined the 95$\%$ confidence level. Any peaks in the power
spectra above these confidence levels were deemed to be statistically
significant.

We make use of this method as it was determined to be highly effective in
robustly detecting the period of QPPs in a Hare and Hound exercise (see Table
5 in Broomhall et al., 2019a). However we note that periodogram-based methods
do fail in the detection of non-stationary QPPs, (as discussed in Section 5.4
of Broomhall et al., 2019a), whereas EMD and other methods that allow for
varying time scales were more effective in detecting these QPPs. We chose not
to use EMD as it struggles with non-detrended data and can be a user intensive
process. Instead we opted to use the Fourier based method on a windowed
signal. This constrained our study to periodicities that are relatively
stationary within their shortened durations. This is a clear limitation in our
work as we are unlikely to detect periodicities that evolve rapidly in either
flare phase due to spectral leakage in the resulting power spectra. In theory,
we may be able to detect some of the more rapidly evolving periodicities in
the data using shorter or overlapping windows, should they exist, however
preliminary studies showed that reducing the duration of the signals resulted
in fewer overall detections which we attribute to the decreased number of
oscillatory cycles in the data. The flare database that this study uses
originates from a periodogram-based approach (Hayes et al., 2020), and so we
find it likely that the FFT will produce statistically meaningful results in
both phases. This technique allows for a statistically sound analysis that can
be applied to a large sample of flares.

We note that recent literature suggests that the significance of peaks in
periodograms can be overestimated for non-stationary QPPs if segments are
poorly selected. We follow a suggested mitigation strategy put forward in
Hübner et al. (2022) by splitting the flare event into two phases and only
assessing events in which there is similar statistically significant QPP-like
behaviour in both segments, as outlined below.

After performing a FFT on both phases of a given flare and obtaining the
dominant periods, we discard the data if it does not fulfil the following
criteria; (i) the periods obtained for both phases must be statistically
significant above a 95$\%$ confidence level, (ii) the periods for both phases
must be less than one tenth of the full duration of the flare, (iii) the
periods for both phases must be greater than four times the cadence of the
data (i.e. both periods must be greater that 8.19 s), and (iv) the impulsive
phase period must not be greater or smaller than the decay phase period by
more than a factor of eight. Criteria (ii) aims at targeting QPPs with at
least five full oscillatory cycles in both the impulsive and decay phase. We
also restrict our periods to be greater than four times the cadence of the
dataset (criteria (iii)). This is because we believe detections of periods
smaller than this are unreliable when detected by GOES alone and must be
accompanied by other data sources with better time resolution. Finally we
believe QPPs that exhibit a change in period by a factor larger than eight
(criteria (iv)) implies that the QPP in the impulsive phase does not
correspond to the QPP in the decay phase. This could, for example, be caused
by two periodicities present in the signal but one not reaching the 95$\%$
confidence level due to a change in the signal to noise ratio. It is important
to state that the absence of the above criteria being met for a given flare
event does not necessarily imply that no QPPs were present. Rather there may
have been QPPs that were not statistically significant in both phases, or one
whose period evolution was outside of the criteria we put forward. However we
restrict our study to these criteria in the interest of reliability and
consistency of results. This resulted in 98 flares which fulfilled all the
criteria, which are discussed in Section 3.

We define the term period drift to measure the change in period from the
impulsive phase to that in the decay phase, equal to PeriodDecay \-
PeriodImpulsive. A positive period drift implies an increase in dominant
period from the impulsive phase to the decay phase and vice versa. We
emphasise that there may be multiple processes present in generating the QPPs
and a positive period drift does not imply the growth in period of a singular
QPP process- for example such an effect could similarly be produced by a
process producing shorter period QPPs decaying in amplitude in tandem with a
secondary longer period process growing in amplitude. This would result in a
growth in dominant period across the two phases, i.e. a positive period drift.

We determine the average period of the flare by taking the mean of the
dominant periods in the impulsive and decay phases. As we are examining the
prevalence of non-stationarity in QPPs we avoid taking an FFT of the entire
duration of the flare to obtain the average period, as a non-stationary signal
that has significant period evolution is not well suited to the FFT which
assumes a stationary input. It is possible that a non-stationary signal which
evolves over several frequencies will show evidence of spectral leakage in its
associated power spectrum, leading to any dominant peaks being smeared out,
and presenting no statistically significant peaks. This is naturally still an
issue to be considered when assessing only the impulsive or decay phase and
any quickly evolving periodicity is likely to be obscured in the same manner,
which may lead to a number of false negatives in our results when
statistically significant periods are not found in our analysis. However by
splitting the flare into sections we still should be able to observe some
periods with sufficiently slow evolution and still pick up on their long-term
non-stationarity.

We determine the errors on the periods from the impulsive and decay phases by
use of the standard approach, and propagate these errors to obtain the errors
on period drift and the average period (See Section 4.2.1 in Hughes & Hase,
2010, for a detailed discussion on error propagation).

Figure 1: Profile of Flare 40 in GOES-XRS 1–8 Å, where the impulsive phase is
shaded in red and the decay phase is unshaded. The analysed impulsive and
decay phases are equal in duration and are delineated by flare maximum which
occurs at approximately 11:10 UT.

Figure 2: Fourier spectra of Flare 40. Top: Fourier spectrum of the impulsive
phase. Lower: Fourier spectrum of the decay phase. Fits of the spectra by
broken power laws are shown by solid red lines, and the 95$\%$ confidence
levels are indicated with dashed red lines. Statistically significant peaks
(indicated by vertical orange lines) can be seen corresponding to periods of
43.3 s in the impulsive phase and 54.9 s in the decay phase.

Fig 1 shows the 1–8 Å lightcurve for Flare 40 where the duration of the flare
has been symmetrically split into the impulsive phase until flare maximum, and
the decay phase. Fig. 2 shows the Fourier spectra of Flare 40’s impulsive and
decay phases, which show significant periods of
43.3$\genfrac{}{}{0.0pt}{2}{+1.7}{-1.6}$ and
54.9$\genfrac{}{}{0.0pt}{2}{+2.8}{-2.5}$ s respectively, corresponding to a
period drift of 11.6$\genfrac{}{}{0.0pt}{2}{+3.3}{-3.0}$ s.

##  3 Results

We examine 205 solar flares from M- and X-class flares over Solar Cycle 24,
resulting in 98 flares that show statistically significant periods in both the
impulsive and decay phases of the flare that have both periods greater than
four times the cadence of the dataset, less than one tenth of the full
duration of the flare, and separated in period by no more than a factor of
eight. We consider a period drift to be statistically significant if its
absolute magnitude is greater than 4.09 s, which is twice the cadence of the
data. This is a cautious approach as we see that the errors on periods are
generally smaller than the cadence. Of these 98 flares, 19 (equivalent to
19$\%$) showed no significant period drift. Of the remaining 79 QPPs, 65
(66$\%$ of the sample) exhibited a positive period drift where the dominant
period appears to increase from the impulsive to the decay phase. 14 flares
(14$\%$) exhibited a negative period drift where the dominant period appears
to shrink between the phases.

Figure 3: QPP impulsive phase periods against decay phase periods. A 1:1 ratio
line (which indicates no period drift) is shown as a solid black line. The
impulsive phase periods are between 8 – 75 s, and are approximately similar
across the all flares, whereas the decay phase periods have a larger spread
between 8 – 110 s. The line of best fit for QPP periods that grew between the
impulsive and decay phases is shown as a dashed blue line, and the line of
best fit for period that shrunk is shown as a dot dashed blue line. This
figure uses new data to recreate Fig. 10 from Hayes et al. (2020).

Fig. 3 shows the relationship between the impulsive and decay phase periods of
the 98 flares examined. It can be seen that the majority of results appear
above the 1:1 ratio line shown in solid black, which indicates more QPPs have
a larger decay phase period than impulsive phase period. For the QPPs showing
an apparent period growth, the decay phase periods are loosely correlated to
the impulsive phase periods by a factor of approximately $\sim$1.4, although
there is significant scatter for events with decay phase periods greater than
40 s. This correlation agrees well with the factor of $\sim$1.6 that was found
in a similar analysis, shown in Fig. 10 of Hayes et al. (2020), which shows
the difference in periods detected during the impulsive and decay phases of 28
flaring events (20 of which overlap with the study presented in this paper).
We note that the authors found that 26 of these events (92$\%$) showed a
larger decay phase period than impulsive phase period and their factor is
based on the fitting of all 28 events, not just those that show period growth.
For the 65 QPPs exhibiting positive period drift, the median period drift is
13$\genfrac{}{}{0.0pt}{2}{+13}{-6}$ s where the errors correspond to the
periods in the upper and lower 25th percentile. Similarly the median negative
period drift for the 14 flaring events is -10$\genfrac{}{}{0.0pt}{2}{+3}{-24}$
s.

We examine whether the presence of a CME associated with the flare impacts the
distribution of period drifts in QPPs. Of the 98 QPPs, 69 were associated with
a CME and 29 were not. Fig. 4 shows the histogram of period drifts in QPPs
from flares associated with CMEs (red) and those from flares not associated
with CMEs (black). The distributions of the two sets are reasonably similar
with median period drifts of 10$\genfrac{}{}{0.0pt}{2}{+13}{-9}$ s for the CME
associated flares and 5$\genfrac{}{}{0.0pt}{2}{+4}{-6}$ for the non-CME
associated flares. The maximal and minimal period drifts across both groups
are also similar with the CME-associated group having maximal and minimal
period drifts of 98 and -126 s, and the non-CME associated group with 121 and
-76 s.

Figure 4:  Histogram of period drifts of QPPs, separated by CME association.
The QPPs seen in flares associated with CMEs are given in red, and those not
associated with a CME are shown in black.

Fig. 5 shows the relationship between absolute period drift and average QPP
period. Positive period drifts are shown in blue, and the absolute values of
negative period drifts are shown in orange. QPPs associated with a CME are
shown with a triangle and non-CME associated events are marked with a circle.
The meanings of the colours and symbols used in Fig. 5 are consistent for the
remainder of this paper. A positive correlation, with a Pearson correlation
coefficient of 0.76, can be seen between the average period of the QPPs and
the magnitude of the period drift. However we emphasise that this artificial
correlation is largely induced by the selection criterion (iv) of the flares.

Figure 5:  Average QPP period plotted against the absolute magnitude of the
QPP period drift. Positive period drifts, indicating a growth in dominant
period, are shown in blue, and negative period drifts are shown in orange.
QPPs from flares associated with CMEs are indicated by a triangle marker
whereas those not associated with QPPs are shown with bullet points. The
Pearson correlation coefficient is 0.76 indicating a positive correlation. A
linear fit of the data is shown as a black dashed line.

Maximal flare energy, which is taken to be the maximal emission as measured in
the 1–8 Å channel, and QPP period drift are seen to have no correlation in
Fig. 6. As expected the flares not associated with CMEs are more commonly
found at lower energies but this distinction has no significant effect on the
magnitude or direction of the period drifts observed.

Figure 6: Peak flare energy as measured in GOES 1 – 8 Å plotted against QPP
period drift with no correlation. The meanings of colours and symbols are as
given in Fig. 5.

Fig. 7 shows a positive correlation between the absolute value of the period
drift of the QPPs and the duration of the flare, with a Pearson correlation
coefficient of 0.82. This relationship can likely be attributed to the fact
that longer duration flares allow more time for any non-stationary QPP periods
to evolve which leads to greater magnitude period drifts, in addition to the
artificial correlation between average period and absolute period drift, seen
in Fig. 5. There is no noticeable difference between the relationship of flare
duration to period drift magnitude for positive or negative period drifts.

Figure 7: Flare duration plotted against the absolute magnitude of the QPP
period drift. The Pearson correlation coefficient is 0.82, indicating a
positive correlation. A linear fit of the data is shown as a black dashed
line. The meanings of colours and symbols are as given in Fig. 5.

The period drift of all QPPs in the 98 flares may be visualised in Fig. 8 (or
explored in Table B1 found in the Appendix). The periods of the QPPs are given
in the horizontal axis, with bullet points indicating the period in the
impulsive phase, and arrow heads indicating the period at the decay phase.
Therefore arrows pointing right and coloured red indicate a positive period
drift. Conversely blue arrows, pointing left, indicate a negative period
drift. The period drift from a given flare is plotted against the
corresponding flare’s duration. The inset axes shows an enlarged region of the
plot for flares with durations less than 2500 s. Flares with longer durations
naturally allow for more time to evolve, leading to larger magnitude period
drifts as discussed previously. The majority of results are clustered for
flare durations less than 2500 s ($\sim$ 40 minutes), with impulsive and decay
phase periods of 40 s or less.

Figure 8: Arrows show evolution of statistically significant periods in the
impulsive and decay phases of 98 flares, with the arrow pointing from
impulsive phase (indicated with a bullet point) to decay phase (arrow head). A
period growth, i.e. a positive period drift, is shown in red and a negative
period drift is given in blue. The period drift in the QPP is plotted against
the flare’s duration, both given in seconds.

We control for the duration of the flares and now examine the rate at which
the QPP periods evolve. The rate of period drift is defined as the period
drift divided by half the duration of the flare, and is therefore a unitless
quantity. Fig. 9 shows the distribution of the magnitude of the rate of period
drift against average QPP period as a scatter plot (Top) and histogram
(Lower). As can be seen, the rates of period drift have considerable scatter,
although the absolute rate of period drift appears to cluster around
$\sim$0.01 for average periods greater than 40 s, an effect that cannot be
attributed to the selection criteria. Due to the selection criteria discussed
in the methods section, the maximal possible absolute rate of period drift for
the data used in this study is 1.4. The maximal rate of positive period drift
seen in these results is 0.06 and the maximal rate of negative period drift is
-0.1, although the majority of the rates of period drift are between0.02 and
0.03. There is no apparent correlation between the presence of a CME and the
rate at which the QPP in the associated flare evolves. We also find that there
is no correlation between the rate of period change and the flare energy,
which suggests that QPP periods evolve at a rate independent of the peak flare
energy. We also see the rate of period change to be uncorrelated with flare
duration. This can be seen in Figs. A1, A2 in the Appendix.

b

(a)

(b)

Figure 9: Top: Scatter plot of the absolute magnitude of the rate of period
drift, plotted against the average period of the QPP. Lower: Histogram of rate
of period drift. The meanings of colours and symbols in the Top panel are as
given in Fig. 5.

##  4 Discussion

Firstly, we remind the reader of the biases and limitations of our study. All
of the flaring events we examined had evidence of QPPs in the first place,
detected by Fourier analysis. This biases the dataset towards QPPs that were
stationary or slowly-evolving in periodicity, meaning that the results in this
paper are likely to underestimate the population of QPPs undergoing rapid
period evolution. We have chosen to split the flare into two phases, a choice
which is ultimately arbitrary and done for convenience. This again biases the
data and forces QPPs to be represented as stationary within an individual
phase. It also neglects the possibility of QPPs which exist in e.g. only the
impulsive or decay phase, or a shorter duration, which may be driven by
entirely different generation mechanisms to the QPPs examined here. A more
comprehensive study should look at QPP period evolution as a continuous
process. It may be that any apparent period evolution is non-linear and
follows some different schema. By repeating this analysis with some method
that has time resolution, such as a continuous wavelet-transform (CWT) or
Empirical Mode Decomposition (EMD) we may be able to uncover valuable
information about the time evolution of the apparent period drifts. This may
also be useful in discerning the generation mechanism(s) that are active in
the appearance of these QPPs.

As discussed earlier, a reader may be misled by these results into thinking
that a single process is occurring in which the period is growing or
shrinking. Instead it is possible that several periodicities exist at once,
each generated by a separate QPP mechanism. A limitation of this work is that
we only extract the period associated with the dominant peak from the FFT
spectrum, ignoring additional potentially statistically significant peaks. In
this paper, we associate the dominant periods in the FFT spectrum of each
phase to produce a period drift, however this may not always be the most
appropriate way to examine the change in instantaneous period of a QPP. For
example it is possible for a given stationary periodicity to be present
throughout the duration of the flare, and appear as the dominant peak in the
FFT spectrum of the impulsive phase but as a secondary peak in the FFT
spectrum of the decay phase due to an emergence of a secondary periodic
process with greater amplitude. This may produce the appearance of a large
magnitude period drift when both processes may in fact be stationary. However
for the majority of the events assessed here (77/98, 79$\%$), both the FFT
spectra of the impulsive and decay phases either resulted in dominant periods
that were similar in magnitude (suggesting the direct evolution of a singular
process) or produced only one peak in each phase that fulfilled the criteria
discussed in Section 2.2 and appeared above the 95$\%$ confidence level.
Therefore for these results the risks of drawing incorrect conclusions due to
erroneously associated periodicities is low.

We have shown that the majority (81$\%$) of flaring events which have evidence
of QPPs in both the impulsive and decay phases exhibit non-stationary
behaviour. Although this sample is not strictly representative of the
behaviour of QPPs en masse, due to the aforementioned biases in the data, the
results discussed here are a strong indicator that we must consider non-
stationarity to be a common property of QPPs and account for it in our
methodology. If we search for QPPs by utilising methods that assume a
stationary output, such as the FFT, we risk false-negative results where the
non-stationarity of QPPs may cause spectral leakage. We also risk poorly
categorising the behaviour of QPPs by assigning a single value for QPP period.
This is important because different QPP mechanisms allow for the presence of
non-stationarity in different ways and we must not omit the valuable data by
treating the QPP periods as a fixed value if we are to determine what causes
QPPs.

We also note the disparity in the proportion of flaring events showing a
positive period drift (66$\%$) compared to those showing a negative period
drift (14$\%$). This suggests an apparent growth in QPP period is more common
than an apparent shrinkage, as previously reported in single event studies
(e.g. Hayes et al., 2016; Dennis et al., 2017; Hayes et al., 2019), and for a
smaller statistical study (Simões et al., 2015; Hayes et al., 2020). We also
note that most of the period drift that we observe is of small magnitude- most
commonly between $\pm$10 s.

The rates at which the QPPs evolved in period exist over the same ranges and
in roughly the same populations for both growing and shrinking QPP periods,
without any dependence on QPP average period or maximum flare energy. We note
that the presence of CMEs or peak flare energy seem to have no effect on
whether the QPP periods grow or shrink, or the magnitudes of the period
drifts. We see that longer duration flares are correlated with greater
magnitude period drift. It is possible that other properties, such as CME
speed or the magnetic configuration of the Active Region could play a role in
determining if and how the QPP periods evolve.

##  5 Conclusions

There is clear evidence that non-stationarity is a common phenomenon in QPPs
observed in M- and X- class solar flares, with period growth appearing more
common than period shrinkage. We must consider this when investigating flaring
events for QPPs and be wary about how we assign values to QPP periodicities.
It appears that most QPP that show non-stationarity evolve in period at
similar rates. It is unlikely that the presence alone of CMEs, or the peak
flare energy impacts the presence or magnitude of QPP period evolution. As
seen in Table 1 of Zimovets et al. (2021) there are many generation mechanisms
(from all of the previously mentioned groupings) that have the potential to
produce QPPs with non-stationary properties. In building a catalogue of QPPs
that exhibit non-stationarity (see Table B1 in the appendix) future work may
determine commonalities, such as the magnetic configuration of the flare site,
which could be used to narrow down which mechanisms are responsible for
driving non-stationary behaviour. Further work with spatial resolution of the
flare site may be valuable in investigating the cause of QPP period evolution.

## Acknowledgements

L.A.H is supported by an ESA Research Fellowship. A-MB acknowledges support
from the Science and Technology Facilities Council (STFC) consolidated grant
ST/T000252/1. The CME catalogue is generated and maintained at the CDAW Data
Center by NASA and The Catholic University of America in cooperation with the
Naval Research Laboratory. SOHO is a project of international cooperation
between ESA and NASA. This research was supported by the International Space
Science Institute (ISSI) in Bern, through ISSI International Team project 527:
Bridging New X-ray Observations and Advanced Models of Flare Variability: A
Key to Understanding the Fundamentals of Flare Energy Release. This research
made use of sunpy (SunPy Community et al., 2020), matplotlib (Hunter, 2007),
numpy (Oliphant, 2007), pandas (Wes McKinney, 2010), and scipy (Virtanen et
al., 2020).

## Data availability

The data used here are all publicly available. The GOES-XRS data is available
online from NOAA (ngdc.noaa.gov/stp/satellite/goes/index.html) and the
SOHO/LASCO CME catalogue can be found here (cdaw.gsfc.nasa.gov/CME_list/). The
procedures described in Section 2.2 can be found in the following repository:
github.com/chloepugh/QPP-confidence-levels.

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##  6 Appendix A

We include Figs. A1, A2, which show the absolute values of the rate of period
drift against peak flare energy and flare duration respectively. No
correlation is observed in either figure. This is expected for Fig. A2 as we
obtain the rate of period drift by dividing the period drift by the flare
duration. Therefore we remove the duration dependence seen in Fig. 7.

Figure A1: Rate of period drift plotted against peak flare energy. The
meanings of colours and symbols are as given in Fig. 5. Figure A2: Rate of
period drift plotted against flare durations. The meanings of colours and
symbols are as given in Fig. 5.

##  7 Appendix B

Table B1 shows a table of the flares examined in this study that fulfilled the
three criteria outlined in Section 2.2 and the associated periods detected in
the impulsive and decay phases. We also include the flare’s GOES class,
duration, and period drift. Table B1 can be downloaded in comma-separated
values format (csv) by accessing the following repository:
github.com/TaraAthem/Non-stationarity-in-QPPs.

Table B1: Num gives the unique ID number of each flaring event. Date refers to the date that the flaring event began and Tstart, Tend, and Tpeak refer to the times used in this study corresponding to the start, end and peak flux of the flaring event, as measured in the long channel of GOES-15. GOES class gives the flaring class of the event. The CME column is ticked with a check mark if the flaring event is associated with a Coronal Mass Ejection. PeriodAvg is the mean period of the flaring event, found by taking the average of the impulsive and decay phase periods (given in columns PeriodImpulsive, PeriodDecay). Period drift is given in the final column of the table.  Num | Date | Tstart | Tend | Tpeak | Duration | GOES | CME | PeriodAvg | PeriodImpulsive | PeriodDecay | Period drift
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  | (seconds) | class |  | (seconds) | (seconds) | (seconds) | (seconds)
01 | 2011-02-14 | 17:20:00 | 17:31:55 | 17:26:08 | 715 | M2.2 | $\checkmark$ | 12.2$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 10.3$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 14.1$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$ | 3.7$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.6}$ $\dagger$
02 | 2011-02-15 | 01:44:00 | 02:07:56 | 01:56:44 | 1436 | X2.2 | $\checkmark$ | 22.5$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$ | 18.8$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 26.2$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$ | 7.4$\genfrac{}{}{0.0pt}{2}{+1.1}{-1.0}$
03 | 2011-02-18 | 20:56:00 | 21:11:53 | 21:03:58 | 953 | M1.3 |  | 13.1$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.3}$ | 8.4$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.1}$ | 17.8$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.6}$ | 9.4$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.7}$
04 | 2011-04-22 | 04:35:00 | 05:18:56 | 04:56:43 | 2636 | M1.8 |  | 19$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 15.6$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 22.4$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 6.8$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$
05 | 2011-05-29 | 10:08:00 | 10:57:54 | 10:33:15 | 2994 | M1.4 | $\checkmark$ | 54.7$\genfrac{}{}{0.0pt}{2}{+2.1}{-1.9}$ | 33.8$\genfrac{}{}{0.0pt}{2}{+0.8}{-0.7}$ | 75.7$\genfrac{}{}{0.0pt}{2}{+4.0}{-3.6}$ | 41.9$\genfrac{}{}{0.0pt}{2}{+4.1}{-3.7}$
06 | 2011-08-03 | 13:17:00 | 14:18:57 | 13:47:56 | 3717 | M6.0 | $\checkmark$ | 57.4$\genfrac{}{}{0.0pt}{2}{+1.5}{-1.4}$ | 42.4$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$ | 72.4$\genfrac{}{}{0.0pt}{2}{+2.9}{-2.7}$ | 30$\genfrac{}{}{0.0pt}{2}{+3.1}{-2.9}$
07 | 2011-09-07 | 22:32:00 | 22:43:52 | 22:38:44 | 712 | X1.8 | $\checkmark$ | 14.9$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$ | 11$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.3}$ | 18.8$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$ | 7.8$\genfrac{}{}{0.0pt}{2}{+1.1}{-1.0}$
08 | 2011-09-10 | 07:18:00 | 08:01:56 | 07:40:43 | 2636 | M1.1 | $\checkmark$ | 28.7$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 23.9$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 33.5$\genfrac{}{}{0.0pt}{2}{+0.9}{-0.8}$ | 9.6$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$
09 | 2011-09-23 | 23:48:00 | 00:03:54 * | 23:55:41 | 954 | M1.9 | $\checkmark$ | 12.3$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 11.6$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 13$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.3}$ | 1.4$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.4}$ $\dagger$
10 | 2011-09-24 | 17:19:00 | 17:30:55 | 17:25:33 | 715 | M3.1 |  | 13.2$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 10.7$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 15.7$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.7}$ | 5$\genfrac{}{}{0.0pt}{2}{+0.8}{-0.7}$
11 | 2011-09-25 | 09:25:00 | 09:44:54 | 09:35:56 | 1194 | M1.5 | $\checkmark$ | 16.1$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 18.4$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.6}$ | 13.8$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | -4.6$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.6}$
12 | 2011-10-01 | 08:56:00 | 11:01:52 | 09:59:21 | 7552 | M1.2 | $\checkmark$ | 48.4$\genfrac{}{}{0.0pt}{2}{+0.9}{-0.9}$ | 13.9$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | 82.9$\genfrac{}{}{0.0pt}{2}{+1.9}{-1.8}$ | 68.9$\genfrac{}{}{0.0pt}{2}{+1.9}{-1.8}$
13 | 2011-11-05 | 20:31:00 | 20:44:53 | 20:38:34 | 833 | M1.8 |  | 12.6$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 12.4$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 12.8$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 0.4$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$ $\dagger$
14 | 2012-01-17 | 04:41:00 | 05:04:56 | 04:53:40 | 1436 | M1.0 |  | 12.1$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 14.6$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 9.5$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | -5.1$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$
15 | 2012-01-19 | 13:44:00 | 18:25:54 | 16:03:17 | 16914 | M3.2 | $\checkmark$ | 127.9$\genfrac{}{}{0.0pt}{2}{+2.2}{-2.1}$ | 190.7$\genfrac{}{}{0.0pt}{2}{+4.4}{-4.2}$ | 65$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | -125.8$\genfrac{}{}{0.0pt}{2}{+4.4}{-4.2}$
16 | 2012-03-02 | 17:29:00 | 18:02:55 | 17:46:26 | 2035 | M3.3 | $\checkmark$ | 33.4$\genfrac{}{}{0.0pt}{2}{+0.8}{-0.8}$ | 31.1$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$ | 35.6$\genfrac{}{}{0.0pt}{2}{+1.3}{-1.2}$ | 4.5$\genfrac{}{}{0.0pt}{2}{+1.6}{-1.5}$
17 | 2012-03-05 | 02:30:00 | 05:47:56 | 04:08:34 | 11876 | X1.1 |  | 105.3$\genfrac{}{}{0.0pt}{2}{+2.4}{-2.3}$ | 45$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 165.6$\genfrac{}{}{0.0pt}{2}{+4.8}{-4.5}$ | 120.6$\genfrac{}{}{0.0pt}{2}{+4.8}{-4.5}$
18 | 2012-03-07 | 01:05:00 | 01:22:47 | 01:15:25 | 1067 | X1.3 | $\checkmark$ | 27.1$\genfrac{}{}{0.0pt}{2}{+1.8}{-1.6}$ | 12$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 42.2$\genfrac{}{}{0.0pt}{2}{+3.6}{-3.1}$ | 30.2$\genfrac{}{}{0.0pt}{2}{+3.6}{-3.1}$
19 | 2012-03-09 | 03:22:00 | 04:23:53 | 03:53:19 | 3713 | M6.3 | $\checkmark$ | 34.3$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$ | 9.1$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | 59.5$\genfrac{}{}{0.0pt}{2}{+2.0}{-1.8}$ | 50.4$\genfrac{}{}{0.0pt}{2}{+2.0}{-1.8}$
20 | 2012-03-10 | 17:15:00 | 18:12:52 | 17:43:55 | 3472 | M8.4 | $\checkmark$ | 47.7$\genfrac{}{}{0.0pt}{2}{+1.8}{-1.6}$ | 18.5$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 76.8$\genfrac{}{}{0.0pt}{2}{+3.6}{-3.3}$ | 58.3$\genfrac{}{}{0.0pt}{2}{+3.6}{-3.3}$
21 | 2012-05-06 | 01:12:00 | 01:23:55 | 01:18:05 | 715 | M1.1 |  | 13.3$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.3}$ | 14.3$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.6}$ | 12.2$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | -2.1$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.7}$ $\dagger$
22 | 2012-05-07 | 14:03:00 | 14:58:53 | 14:31:18 | 3353 | M1.9 | $\checkmark$ | 76.8$\genfrac{}{}{0.0pt}{2}{+2.8}{-2.6}$ | 64.1$\genfrac{}{}{0.0pt}{2}{+2.5}{-2.4}$ | 89.4$\genfrac{}{}{0.0pt}{2}{+5.0}{-4.5}$ | 25.4$\genfrac{}{}{0.0pt}{2}{+5.6}{-5.1}$
23 | 2012-05-09 | 21:01:00 | 21:08:53 | 21:05:22 | 473 | M4.1 | $\checkmark$ | 12.3$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 14.3$\genfrac{}{}{0.0pt}{2}{+0.9}{-0.8}$ | 10.3$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.4}$ | -4$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$ $\dagger$
24 | 2012-05-10 | 04:11:00 | 04:24:54 | 04:17:50 | 834 | M5.7 | $\checkmark$ | 16.3$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.4}$ | 16.1$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.6}$ | 16.6$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.6}$ | 0.5$\genfrac{}{}{0.0pt}{2}{+0.9}{-0.9}$
25 | 2012-07-19 | 04:17:00 | 07:38:55 | 05:57:51 | 12115 | M7.7 | $\checkmark$ | 102.3$\genfrac{}{}{0.0pt}{2}{+1.9}{-1.9}$ | 53.5$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 151.1$\genfrac{}{}{0.0pt}{2}{+3.9}{-3.7}$ | 97.6$\genfrac{}{}{0.0pt}{2}{+3.9}{-3.7}$
26 | 2012-07-30 | 15:39:00 | 15:56:55 | 15:48:28 | 1075 | M1.1 |  | 16.9$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 10.3$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 23.5$\genfrac{}{}{0.0pt}{2}{+1.1}{-1.0}$ | 13.2$\genfrac{}{}{0.0pt}{2}{+1.1}{-1.0}$
27 | 2012-08-11 | 11:55:00 | 12:44:54 | 12:19:52 | 2994 | M1.0 | $\checkmark$ | 12.3$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | 15.7$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 8.9$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | -6.8$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$
28 | 2012-08-30 | 12:02:00 | 12:19:55 | 12:11:36 | 1075 | M1.3 |  | 14.2$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 15.3$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 13.1$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | -2.2$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$ $\dagger$
29 | 2012-09-30 | 04:27:00 | 04:38:52 | 04:33:01 | 712 | M1.3 | $\checkmark$ | 10.9$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 8.6$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 13.2$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 4.5$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$
30 | 2012-10-08 | 11:05:00 | 11:28:53 | 11:16:56 | 1433 | M2.3 |  | 12.8$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 14.6$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 10.9$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | -3.7$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ $\dagger$
31 | 2012-11-27 | 15:52:00 | 16:01:53 | 15:57:35 | 593 | M1.6 |  | 11.1$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 10.3$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.3}$ | 12$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 1.8$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.6}$ $\dagger$
32 | 2013-05-03 | 17:24:00 | 17:39:53 | 17:32:13 | 953 | M5.7 | $\checkmark$ | 23.1$\genfrac{}{}{0.0pt}{2}{+1.2}{-1.0}$ | 14.2$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 32$\genfrac{}{}{0.0pt}{2}{+2.3}{-2.0}$ | 17.8$\genfrac{}{}{0.0pt}{2}{+2.3}{-2.1}$
33 | 2013-06-05 | 08:14:00 | 09:39:55 | 08:57:28 | 5155 | M1.3 | $\checkmark$ | 72.9$\genfrac{}{}{0.0pt}{2}{+2.6}{-2.4}$ | 33.2$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 112.5$\genfrac{}{}{0.0pt}{2}{+5.1}{-4.7}$ | 79.3$\genfrac{}{}{0.0pt}{2}{+5.2}{-4.7}$
34 | 2013-08-17 | 18:49:00 | 20:16:55 | 19:33:47 | 5275 | M1.4 | $\checkmark$ | 44.7$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$ | 48.8$\genfrac{}{}{0.0pt}{2}{+0.9}{-0.9}$ | 40.6$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.6}$ | -8.2$\genfrac{}{}{0.0pt}{2}{+1.1}{-1.1}$
35 | 2013-10-13 | 00:12:00 | 01:13:54 | 00:43:36 | 3714 | M1.7 | $\checkmark$ | 20.8$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 20$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 21.6$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.2}$ | 1.6$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ $\dagger$
36 | 2013-10-17 | 15:09:00 | 16:12:53 | 15:41:00 | 3833 | M1.2 | $\checkmark$ | 36.9$\genfrac{}{}{0.0pt}{2}{+0.8}{-0.8}$ | 55.4$\genfrac{}{}{0.0pt}{2}{+1.6}{-1.6}$ | 18.4$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | -37$\genfrac{}{}{0.0pt}{2}{+1.7}{-1.6}$
37 | 2013-10-28 | 01:41:00 | 02:24:53 | 02:02:57 | 2633 | X1.0 | $\checkmark$ | 16.7$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.1}$ | 17.5$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 16$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | -1.5$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ $\dagger$
38 | 2013-10-29 | 21:42:00 | 22:05:53 | 21:54:30 | 1433 | X2.3 | $\checkmark$ | 32.8$\genfrac{}{}{0.0pt}{2}{+1.1}{-1.0}$ | 30.7$\genfrac{}{}{0.0pt}{2}{+1.4}{-1.3}$ | 34.9$\genfrac{}{}{0.0pt}{2}{+1.8}{-1.6}$ | 4.3$\genfrac{}{}{0.0pt}{2}{+2.3}{-2.1}$
39 | 2013-11-05 | 18:08:00 | 18:17:56 | 18:12:57 | 596 | M1.0 | $\checkmark$ | 12.5$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.4}$ | 8.8$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 16.2$\genfrac{}{}{0.0pt}{2}{+0.9}{-0.8}$ | 7.4$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$
40 | 2013-11-21 | 10:52:00 | 11:29:55 | 11:11:07 | 2275 | M1.2 | $\checkmark$ | 49.1$\genfrac{}{}{0.0pt}{2}{+1.6}{-1.5}$ | 43.3$\genfrac{}{}{0.0pt}{2}{+1.7}{-1.6}$ | 54.9$\genfrac{}{}{0.0pt}{2}{+2.8}{-2.5}$ | 11.6$\genfrac{}{}{0.0pt}{2}{+3.3}{-3.0}$
41 | 2013-12-07 | 07:17:00 | 07:40:53 | 07:29:41 | 1433 | M1.2 | $\checkmark$ | 13.8$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 9.9$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | 17.7$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 7.7$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.4}$
42 | 2013-12-31 | 21:45:00 | 22:10:56 | 21:58:07 | 1556 | M6.4 | $\checkmark$ | 26.3$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.6}$ | 24.9$\genfrac{}{}{0.0pt}{2}{+0.8}{-0.8}$ | 27.7$\genfrac{}{}{0.0pt}{2}{+1.0}{-1.0}$ | 2.8$\genfrac{}{}{0.0pt}{2}{+1.3}{-1.2}$ $\dagger$
43 | 2014-01-01 | 18:40:00 | 19:04:10 | 18:52:03 | 1450 | M9.9 | $\checkmark$ | 14.1$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 19.7$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$ | 8.5$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | -11.3$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$
44 | 2014-01-08 | 03:39:00 | 03:54:54 | 03:47:45 | 954 | M3.6 | $\checkmark$ | 18$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 15.1$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 20.9$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$ | 5.8$\genfrac{}{}{0.0pt}{2}{+1.1}{-1.0}$
45 | 2014-01-30 | 07:54:00 | 08:28:17 | 08:10:51 | 2057 | M1.1 | $\checkmark$ | 20.6$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 9.8$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | 31.3$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$ | 21.5$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$
46 | 2014-02-11 | 16:34:00 | 17:07:54 | 16:51:43 | 2034 | M1.8 | $\checkmark$ | 31.2$\genfrac{}{}{0.0pt}{2}{+0.8}{-0.7}$ | 26.1$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.7}$ | 36.3$\genfrac{}{}{0.0pt}{2}{+1.3}{-1.3}$ | 10.2$\genfrac{}{}{0.0pt}{2}{+1.5}{-1.4}$
47 | 2014-02-24 | 11:03:00 | 11:31:20 | 11:17:07 | 1700 | M1.2 | $\checkmark$ | 32.2$\genfrac{}{}{0.0pt}{2}{+1.9}{-1.7}$ | 55.5$\genfrac{}{}{0.0pt}{2}{+3.9}{-3.4}$ | 8.8$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | -46.6$\genfrac{}{}{0.0pt}{2}{+3.9}{-3.4}$
48 | 2014-03-10 | 04:02:00 | 04:13:55 | 04:08:17 | 715 | M1.0 |  | 11.7$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 8.4$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 15$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.6}$ | 6.6$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.6}$
49 | 2014-03-12 | 10:55:00 | 11:14:54 | 11:05:09 | 1194 | M2.5 |  | 12.7$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 9.2$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | 16.3$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.4}$ | 7.1$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$
50 | 2014-04-18 | 12:31:00 | 13:34:52 | 13:02:58 | 3832 | M7.3 | $\checkmark$ | 42.9$\genfrac{}{}{0.0pt}{2}{+1.1}{-1.0}$ | 23.8$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 62.1$\genfrac{}{}{0.0pt}{2}{+2.1}{-1.9}$ | 38.2$\genfrac{}{}{0.0pt}{2}{+2.1}{-2.0}$
51 | 2014-05-07 | 16:07:00 | 16:50:56 | 16:29:08 | 2636 | M1.2 | $\checkmark$ | 41.1$\genfrac{}{}{0.0pt}{2}{+1.8}{-1.6}$ | 15.2$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 66.9$\genfrac{}{}{0.0pt}{2}{+3.6}{-3.2}$ | 51.8$\genfrac{}{}{0.0pt}{2}{+3.6}{-3.2}$
52 | 2014-06-12 | 04:14:00 | 04:27:55 | 04:21:17 | 835 | M2.0 | $\checkmark$ | 11.8$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 11.6$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 12.1$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.3}$ | 0.6$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ $\dagger$
53 | 2014-06-12 | 18:03:00 | 18:22:54 | 18:13:54 | 1194 | M1.3 |  | 14.7$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 12.3$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.2}$ | 17$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 4.6$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$
54 | 2014-06-15 | 11:10:00 | 12:07:53 | 11:39:34 | 3473 | M1.1 | $\checkmark$ | 28.7$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.6}$ | 10.2$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | 47.2$\genfrac{}{}{0.0pt}{2}{+1.3}{-1.2}$ | 37$\genfrac{}{}{0.0pt}{2}{+1.3}{-1.3}$
55 | 2014-07-10 | 22:29:00 | 22:38:56 | 22:34:15 | 596 | M1.5 | $\checkmark$ | 10.4$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.2}$ | 10.3$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.3}$ | 10.5$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 0.2$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ $\dagger$
56 | 2014-08-21 | 13:19:00 | 13:42:53 | 13:31:41 | 1433 | M3.4 | $\checkmark$ | 37.2$\genfrac{}{}{0.0pt}{2}{+1.7}{-1.5}$ | 27.6$\genfrac{}{}{0.0pt}{2}{+1.1}{-1.0}$ | 46.7$\genfrac{}{}{0.0pt}{2}{+3.3}{-2.9}$ | 19.1$\genfrac{}{}{0.0pt}{2}{+3.4}{-3.0}$
57 | 2014-08-25 | 20:06:00 | 20:35:56 | 20:20:50 | 1796 | M3.9 | $\checkmark$ | 28$\genfrac{}{}{0.0pt}{2}{+0.8}{-0.7}$ | 20.8$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 35.3$\genfrac{}{}{0.0pt}{2}{+1.4}{-1.3}$ | 14.5$\genfrac{}{}{0.0pt}{2}{+1.5}{-1.4}$
58 | 2014-09-03 | 13:20:00 | 14:27:55 | 13:54:11 | 4075 | M2.5 | $\checkmark$ | 37.2$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.6}$ | 49.1$\genfrac{}{}{0.0pt}{2}{+1.2}{-1.2}$ | 25.2$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | -23.8$\genfrac{}{}{0.0pt}{2}{+1.3}{-1.2}$
Table B2: continued

Num | Date | Tstart | Tend | Tpeak | Duration | GOES | CME | PeriodAvg | PeriodImpulsive | PeriodDecay | Period drift
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  | (seconds) | class |  | (seconds) | (seconds) | (seconds) | (seconds)
59 | 2014-09-10 | 17:21:00 | 18:08:55 | 17:45:10 | 2875 | X1.6 | $\checkmark$ | 56.7$\genfrac{}{}{0.0pt}{2}{+2.2}{-2.0}$ | 36.5$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$ | 77$\genfrac{}{}{0.0pt}{2}{+4.4}{-3.9}$ | 40.5$\genfrac{}{}{0.0pt}{2}{+4.5}{-4.0}$
60 | 2014-10-09 | 01:30:00 | 01:55:55 | 01:43:23 | 1555 | M1.3 |  | 59.7$\genfrac{}{}{0.0pt}{2}{+7.2}{-5.5}$ | 98.5$\genfrac{}{}{0.0pt}{2}{+14.3}{-11.1}$ | 20.9$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$ | -77.6$\genfrac{}{}{0.0pt}{2}{+14.3}{-11.1}$
61 | 2014-11-03 | 11:23:00 | 12:22:54 | 11:53:30 | 3594 | M2.2 | $\checkmark$ | 44.1$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$ | 31$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 57.2$\genfrac{}{}{0.0pt}{2}{+1.9}{-1.8}$ | 26.2$\genfrac{}{}{0.0pt}{2}{+2.0}{-1.8}$
62 | 2014-11-04 | 07:59:00 | 09:16:53 | 08:38:41 | 4673 | M2.6 | $\checkmark$ | 28.7$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 23.6$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 33.8$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 10.2$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$
63 | 2014-11-05 | 18:50:00 | 20:37:54 | 19:44:38 | 6474 | M2.9 | $\checkmark$ | 56.1$\genfrac{}{}{0.0pt}{2}{+1.1}{-1.0}$ | 30.3$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 81.8$\genfrac{}{}{0.0pt}{2}{+2.1}{-2.0}$ | 51.4$\genfrac{}{}{0.0pt}{2}{+2.1}{-2.0}$
64 | 2014-11-06 | 01:29:00 | 01:48:54 | 01:39:21 | 1194 | M3.2 | $\checkmark$ | 19$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.4}$ | 17.4$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 20.7$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.7}$ | 3.2$\genfrac{}{}{0.0pt}{2}{+0.9}{-0.9}$ $\dagger$
65 | 2014-11-06 | 21:53:00 | 22:38:49 | 22:16:01 | 2749 | M2.5 |  | 23.6$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 18.3$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 28.8$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.6}$ | 10.5$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.6}$
66 | 2014-11-07 | 10:13:00 | 10:30:56 | 10:22:15 | 1076 | M1.0 |  | 17.1$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 14.7$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 19.5$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.7}$ | 4.8$\genfrac{}{}{0.0pt}{2}{+0.8}{-0.8}$
67 | 2014-11-15 | 11:40:00 | 12:25:55 | 12:03:21 | 2755 | M3.2 | $\checkmark$ | 28.7$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 25.9$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 31.4$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.7}$ | 5.5$\genfrac{}{}{0.0pt}{2}{+0.9}{-0.8}$
68 | 2014-12-17 | 04:25:00 | 05:16:55 | 04:50:06 | 3115 | M8.7 | $\checkmark$ | 64$\genfrac{}{}{0.0pt}{2}{+2.0}{-1.8}$ | 56.8$\genfrac{}{}{0.0pt}{2}{+2.1}{-2.0}$ | 71.2$\genfrac{}{}{0.0pt}{2}{+3.4}{-3.1}$ | 14.3$\genfrac{}{}{0.0pt}{2}{+4.0}{-3.7}$
69 | 2014-12-18 | 21:41:00 | 22:14:55 | 21:58:03 | 2035 | M6.9 | $\checkmark$ | 29$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.6}$ | 22.8$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 35.2$\genfrac{}{}{0.0pt}{2}{+1.3}{-1.2}$ | 12.4$\genfrac{}{}{0.0pt}{2}{+1.4}{-1.3}$
70 | 2014-12-20 | 00:11:00 | 00:44:55 | 00:28:00 | 2035 | X1.8 | $\checkmark$ | 21.3$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 10.6$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | 31.9$\genfrac{}{}{0.0pt}{2}{+1.0}{-1.0}$ | 21.2$\genfrac{}{}{0.0pt}{2}{+1.0}{-1.0}$
71 | 2015-03-02 | 15:10:00 | 15:45:56 | 15:28:16 | 2156 | M3.7 | $\checkmark$ | 41.1$\genfrac{}{}{0.0pt}{2}{+1.3}{-1.2}$ | 48.9$\genfrac{}{}{0.0pt}{2}{+2.3}{-2.1}$ | 33.3$\genfrac{}{}{0.0pt}{2}{+1.1}{-1.0}$ | -15.6$\genfrac{}{}{0.0pt}{2}{+2.6}{-2.3}$
72 | 2015-03-06 | 04:14:00 | 05:39:54 | 04:57:36 | 5154 | M3.0 | $\checkmark$ | 82.3$\genfrac{}{}{0.0pt}{2}{+2.1}{-2.0}$ | 65.8$\genfrac{}{}{0.0pt}{2}{+1.7}{-1.6}$ | 98.7$\genfrac{}{}{0.0pt}{2}{+3.9}{-3.6}$ | 32.9$\genfrac{}{}{0.0pt}{2}{+4.3}{-4.0}$
73 | 2015-03-09 | 14:22:00 | 14:43:57 | 14:33:45 | 1317 | M4.5 | $\checkmark$ | 11.9$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 14.1$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 9.8$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | -4.4$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$
74 | 2015-03-09 | 23:29:00 | 00:16:54 * | 23:54:17 | 2874 | M5.8 | $\checkmark$ | 51.1$\genfrac{}{}{0.0pt}{2}{+1.7}{-1.5}$ | 36.3$\genfrac{}{}{0.0pt}{2}{+0.9}{-0.9}$ | 66$\genfrac{}{}{0.0pt}{2}{+3.2}{-2.9}$ | 29.7$\genfrac{}{}{0.0pt}{2}{+3.3}{-3.0}$
75 | 2015-03-11 | 16:11:00 | 16:32:54 | 16:21:38 | 1314 | X2.1 | $\checkmark$ | 22.3$\genfrac{}{}{0.0pt}{2}{+0.8}{-0.7}$ | 13.3$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 31.3$\genfrac{}{}{0.0pt}{2}{+1.6}{-1.4}$ | 18$\genfrac{}{}{0.0pt}{2}{+1.6}{-1.4}$
76 | 2015-03-12 | 04:41:00 | 04:50:53 | 04:46:00 | 593 | M3.2 |  | 11.4$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.3}$ | 8.4$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 14.3$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.7}$ | 5.9$\genfrac{}{}{0.0pt}{2}{+0.8}{-0.7}$
77 | 2015-03-12 | 11:38:00 | 12:01:53 | 11:50:26 | 1433 | M1.6 |  | 21.3$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 15.9$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.3}$ | 26.7$\genfrac{}{}{0.0pt}{2}{+1.0}{-1.0}$ | 10.8$\genfrac{}{}{0.0pt}{2}{+1.1}{-1.0}$
78 | 2015-03-12 | 12:09:00 | 12:18:53 | 12:14:23 | 593 | M1.4 |  | 17.1$\genfrac{}{}{0.0pt}{2}{+0.9}{-0.8}$ | 21.1$\genfrac{}{}{0.0pt}{2}{+1.6}{-1.4}$ | 13.2$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.6}$ | -7.9$\genfrac{}{}{0.0pt}{2}{+1.7}{-1.5}$
79 | 2015-03-12 | 13:50:00 | 14:25:53 | 14:08:39 | 2153 | M4.2 |  | 14.5$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | 13.1$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 16$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 2.9$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ $\dagger$
80 | 2015-03-13 | 03:47:00 | 04:14:54 | 04:01:49 | 1674 | M1.2 |  | 33.7$\genfrac{}{}{0.0pt}{2}{+1.3}{-1.2}$ | 22.4$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.6}$ | 44.9$\genfrac{}{}{0.0pt}{2}{+2.5}{-2.3}$ | 22.6$\genfrac{}{}{0.0pt}{2}{+2.6}{-2.4}$
81 | 2015-03-16 | 10:39:00 | 11:16:53 | 10:57:59 | 2273 | M1.6 |  | 27$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.6}$ | 18.3$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 35.6$\genfrac{}{}{0.0pt}{2}{+1.2}{-1.1}$ | 17.3$\genfrac{}{}{0.0pt}{2}{+1.2}{-1.1}$
82 | 2015-03-17 | 22:49:00 | 23:59:55 | 23:34:48 | 4255 | M1.0 | $\checkmark$ | 14.8$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | 8.7$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | 20.9$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 12.1$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$
83 | 2015-04-21 | 07:08:00 | 07:33:54 | 07:20:48 | 1554 | M1.0 |  | 19.1$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 15.7$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 22.5$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.6}$ | 6.8$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.7}$
84 | 2015-06-21 | 02:04:00 | 03:03:32 | 02:36:26 | 3572 | M2.7 | $\checkmark$ | 86.9$\genfrac{}{}{0.0pt}{2}{+3.3}{-3.0}$ | 76.4$\genfrac{}{}{0.0pt}{2}{+3.4}{-3.1}$ | 97.4$\genfrac{}{}{0.0pt}{2}{+5.6}{-5.0}$ | 21$\genfrac{}{}{0.0pt}{2}{+6.6}{-5.9}$
85 | 2015-06-21 | 02:06:00 | 03:05:51 | 02:36:45 | 3591 | M2.6 | $\checkmark$ | 84.8$\genfrac{}{}{0.0pt}{2}{+3.1}{-2.8}$ | 75.4$\genfrac{}{}{0.0pt}{2}{+3.3}{-3.0}$ | 94.1$\genfrac{}{}{0.0pt}{2}{+5.2}{-4.7}$ | 18.7$\genfrac{}{}{0.0pt}{2}{+6.2}{-5.6}$
86 | 2015-06-22 | 17:39:00 | 19:06:55 | 18:24:42 | 5275 | M6.5 | $\checkmark$ | 20.2$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | 21.2$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 19.2$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | -1.9$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ $\dagger$
87 | 2015-09-20 | 17:32:00 | 18:33:53 | 18:01:50 | 3713 | M2.1 | $\checkmark$ | 81.9$\genfrac{}{}{0.0pt}{2}{+2.8}{-2.6}$ | 70.2$\genfrac{}{}{0.0pt}{2}{+2.8}{-2.6}$ | 93.6$\genfrac{}{}{0.0pt}{2}{+5.0}{-4.5}$ | 23.4$\genfrac{}{}{0.0pt}{2}{+5.7}{-5.2}$
88 | 2015-09-28 | 07:27:00 | 07:42:53 | 07:34:42 | 953 | M1.1 |  | 13.5$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 13.8$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 13.3$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | -0.5$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$ $\dagger$
89 | 2015-10-02 | 17:08:00 | 17:27:54 | 17:18:27 | 1194 | M1.0 |  | 13.1$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.2}$ | 9.5$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.1}$ | 16.7$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 7.3$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$
90 | 2015-10-15 | 23:27:00 | 23:34:55 | 23:31:49 | 475 | M1.1 |  | 11.2$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.4}$ | 11.8$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.6}$ | 10.7$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | -1$\genfrac{}{}{0.0pt}{2}{+0.8}{-0.7}$ $\dagger$
91 | 2015-10-16 | 06:11:00 | 06:20:53 | 06:16:31 | 593 | M1.1 |  | 18.7$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$ | 23.1$\genfrac{}{}{0.0pt}{2}{+2.0}{-1.7}$ | 14.2$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.6}$ | -8.9$\genfrac{}{}{0.0pt}{2}{+2.1}{-1.8}$
92 | 2015-10-17 | 20:09:00 | 20:36:54 | 20:22:58 | 1674 | M1.1 |  | 18.7$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 9.3$\genfrac{}{}{0.0pt}{2}{+0.1}{-0.1}$ | 28.1$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$ | 18.7$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$
93 | 2015-11-04 | 11:55:00 | 12:10:56 | 12:03:17 | 956 | M2.5 | $\checkmark$ | 13.1$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 10.8$\genfrac{}{}{0.0pt}{2}{+0.2}{-0.2}$ | 15.4$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 4.6$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.5}$
94 | 2015-12-21 | 00:52:00 | 01:13:57 | 01:03:04 | 1317 | M2.8 | $\checkmark$ | 21.2$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.6}$ | 14.8$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 27.7$\genfrac{}{}{0.0pt}{2}{+1.2}{-1.1}$ | 12.8$\genfrac{}{}{0.0pt}{2}{+1.3}{-1.2}$
95 | 2015-12-22 | 03:15:00 | 03:52:55 | 03:34:19 | 2275 | M1.6 | $\checkmark$ | 32.3$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.7}$ | 27.1$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.6}$ | 37.4$\genfrac{}{}{0.0pt}{2}{+1.3}{-1.2}$ | 10.4$\genfrac{}{}{0.0pt}{2}{+1.4}{-1.3}$
96 | 2015-12-23 | 00:23:00 | 00:56:54 | 00:40:46 | 2034 | M4.7 | $\checkmark$ | 38.1$\genfrac{}{}{0.0pt}{2}{+1.2}{-1.1}$ | 29.1$\genfrac{}{}{0.0pt}{2}{+0.9}{-0.8}$ | 47.1$\genfrac{}{}{0.0pt}{2}{+2.3}{-2.1}$ | 18$\genfrac{}{}{0.0pt}{2}{+2.4}{-2.2}$
97 | 2016-07-23 | 05:00:00 | 05:31:52 | 05:16:43 | 1912 | M7.6 | $\checkmark$ | 20.9$\genfrac{}{}{0.0pt}{2}{+0.4}{-0.3}$ | 17.3$\genfrac{}{}{0.0pt}{2}{+0.3}{-0.3}$ | 24.6$\genfrac{}{}{0.0pt}{2}{+0.6}{-0.6}$ | 7.3$\genfrac{}{}{0.0pt}{2}{+0.7}{-0.7}$
98 | 2017-04-02 | 07:50:00 | 08:13:56 | 08:02:56 | 1436 | M5.3 | $\checkmark$ | 22.6$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 19.5$\genfrac{}{}{0.0pt}{2}{+0.5}{-0.5}$ | 25.7$\genfrac{}{}{0.0pt}{2}{+1.0}{-0.9}$ | 6.1$\genfrac{}{}{0.0pt}{2}{+1.1}{-1.0}$

* indicates that the flaring event took place over midnight, so the end time of the flare occurs on the subsequent day to the date indicated.
---
$\dagger$ indicates that the period drift is smaller in magnitude than 4.09 s
(twice the data cadence) and therefore the QPP is deemed to exhibit no period
drift
in this study.