# Relative yield of thermal and nonthermal emission during weak flares
observed by STIX during September 20–25, 2021

Arun Kumar Awasthi Space Research Centre, Polish Academy of Sciences, Bartycka
18A, 00-716 Warsaw, Poland  Tomasz Mrozek Space Research Centre, Polish
Academy of Sciences, Bartycka 18A, 00-716 Warsaw, Poland  Sylwester Kołomański
Astronomical Institute, University of Wroclaw, Kopernika 11, 51-622 Wroclaw,
Poland  Michalina Litwicka  University of Wrocław, Centre of Scientific
Excellence – Solar and Stellar Activity, Joliot-Curie 12, PL-50383 Wroclaw,
Poland  Space Research Centre, Polish Academy of Sciences, Bartycka 18A,
00-716 Warsaw, Poland  Marek Stęślicki  Space Research Centre, Polish Academy
of Sciences, Bartycka 18A, 00-716 Warsaw, Poland  Karol Kułaga  Astronomical
Institute, University of Wroclaw, Kopernika 11, 51-622 Wroclaw, Poland

###### Abstract

The disparate nature of thermal-nonthermal energy partition during flares,
particularly during weak flares, is still an open issue. Following the Neupert
effect, quantifying the relative yield of X-ray emission in different energy
bands can enable inferring the underlying energy release mechanism. During
September 20-25, 2021, the Solar Orbiter mission - being closer to the Sun
($\sim$0.6 AU) and having a moderate separation angle (<40∘) from the Sun-
Earth line provided a unique opportunity to analyze multi-wavelength emission
from $\sim$200 (mostly weak) flares, commonly observed by the Spectrometer
Telescope for Imaging X-rays (STIX), STEREO-A, GOES, and SDO observatories.
Associating the quotient (qf) of hard X-ray fluence (12-20 keV) and soft X-ray
flux (4-10 keV) with the peak SXR flux enabled us to identify strongly non-
thermal flares. Multi-wavelength investigation of spectral and imaging mode
observations of the 20 strongly non-thermal weak flares reveals an inverse
relationship of qf with the emission measure (EM) (and density), and a
positive relationship with the flare plasma temperature. This indicates that
plasma in tenuous loops attains higher temperatures compared to that in the
denser loops, in response to nonthermal energy deposition. This is in
agreement with the plasma parameters of the coronal loops, as derived by
applying the one-dimensional Palermo Harvard (PH) hydrodynamical code to the
coronal loop plasma having different initial coronal loop base pressures when
subjected to similar heating input. Our investigation, therefore, indicates
that the plasma parameters of the flaring loop in the initial phase have a
decisive role in thermal-nonthermal energy partitioning.

The Sun (1693) — Solar atmosphere (1477) — Solar flares (1496) — Solar flare
spectra (1982) — Solar X-ray flares (1816) — Solar coronal loops (1485) —
Solar extreme ultraviolet emission (1493)

††facilities: STIX (Solar Orbiter), GOES, SDO, STEREO-A††software:
SolarSoftWare (Freeland & Handy 1998, 2012), Sparse DEM inversion technique
(Cheng et al. 2012; Su et al. 2018b), Solar-MACH (Gieseler et al. 2023)

revtex4-1.clsrevtex4-2.cls

##  1 Introduction

Solar flares and accompanying events are one of the most energetic and dynamic
phenomena in the Solar System emitting a substantial amount of energy in the
X-ray and Extreme Ultraviolet (EUV) wavelength range. Being one of the first
observational signatures of solar eruption, flares not only enable probing the
plasma processes at play in the million-kelvin solar atmosphere but also
improve our capability of predicting the magnitude and timing of the impact of
solar eruptions on the Earth and beyond. Therefore, an in-depth analysis of
multi-wavelength emission from the Sun can enable us to infer the plasma
processes in the solar atmosphere and observationally unexplored
characteristics of stellar flares and coronal mass ejection events (Argiroffi
et al. 2019; Veronig et al. 2021).

Flare emission covers the entire electromagnetic spectrum with the high-
temperature (million–kelvin) plasma emitting in the X-ray and extreme ultra-
violet (EUV) wavelengths while the low-temperature plasma is best seen in the
H$\alpha$ and optical wavelengths. According to the standard model (CSHKP:
Carmichael 1964; Sturrock 1966; Hirayama 1974; Kopp & Pneuman 1976) of energy
released during solar flares (unified 2D & 3D models in Shibata & Magara 2011;
Aulanier et al. 2012, respectively), the energy produced following the
magnetic reconnection is predominately utilized in heating the ambient plasma
and accelerating particles. Such a non-thermal pool of particles (generally
electrons) deposits their energy in the denser layers of the solar atmosphere
and produces hard X-rays (HXRs). Thermal distribution of particles emitting
soft X-rays (SXRs) is subsequently generated from the chromospheric
evaporation-driven plasma. Due to this causal relationship, flares often
exhibit an ‘empirical’ temporal relationship between SXR and HXR emissions,
also termed the empirical Neupert effect (ENE; Neupert 1968). However, a
detailed quantification of the plasma parameters in the context of the Neupert
effect enables shedding light on the plasma processes occurring in various
layers of the solar atmosphere. For example, by investigating the differential
emission measure (DEM) of 80 flares, recorded by YOHKOH/SXT and BCS
instruments, McTiernan et al. (1999) found that primarily the flares with
high-temperature plasma ($\geq$16.5MK) exhibited the Neupert effect.

The integral form of the Neupert effect, which states that time-integrated HXR
emission should mimic SXR emission, enables assessing the thermal-nonthermal
energy partition during flares. However, Lee et al. (1995) and Veronig et al.
(2002a) questioned this simple form of relationship between the SXR and HXR
emission by demonstrating a discrepancy between the slopes of the HXR fluence
distributions and soft X-ray flux distribution. It has been further argued
that HXR and SXR emissions may not be a comprehensive indicator of thermal and
nonthermal energies, and it is the respective energy contents that should
exhibit the Neupert effect. Further, the proportionality between HXR fluence
and SXR flux should also additionally depend on the flare plasma parameters,
as shown by McTiernan et al. (1999). Therefore, Veronig et al. (2005)
formulated a theoretical Neupert effect (TNE) which associates the beam power
supply (inferred from HXR emission), and power required for observed SXR
emission. However, for simple approximations of loop geometry along with the
application of particle and energy transport schemes, the results of the TNE
relationship were found to be similar to ENE.

Based on a critical overview of thermal, nonthermal, and bolometric energy
content, Warmuth & Mann (2020), inferred the dependence of thermal-nonthermal
energy partition on the flare strength. They indicated that weak flares
exhibit a deficit of energetic electrons in contrast to the strong flare for
which the injected nonthermal energy has been found sufficient to account for
the thermal energy content. Several case studies of weak flares revealed the
extraordinary nature of HXR emission during microflares, for example, some
flares show SXR emission without distinctive enhancement in the HXR flux
(Awasthi et al. 2014, 2018b) while other cases exhibit non-thermal electrons
flux with a spectral slope as hard as 6 (Wright et al. 2017; Awasthi et al.
2018a). Evidently, weak flares often exhibit a departure from the Neupert
effect, as clearly seen in the correlation plot of GOES peak flux and HXR
fluence (from Hard X-Ray Burst Spectrometer (HXRBS) onboard Solar Maximum
Mission (SMM) observations) in Veronig et al. (2002b) where weaker events
exhibit a larger scatter. Conventionally, solar flares that release a huge
amount of X-ray emission (e.g. X-class flares) are naturally believed to be
extreme events in the context of space-weather impact. However, an extremely
broad range of variability within thermal-nonthermal emissions during flares
(e.g. strongly nonthermal flares (Awasthi et al. 2018a; Lysenko et al. 2018,
2023), thermal flares (Battaglia et al. 2009; Altyntsev et al. 2012; Awasthi
et al. 2014) reveals a gap in our understanding of the standard flare energy
release scheme. More recently, Fleishman et al. (2022) found the magnetic
reconnection to efficiently accelerate almost all the ambient electrons
leaving the region depleted of thermal electrons. Such episodes of energy
release and associated energy release processes should have an imprint on the
emission recorded in X-rays, extreme ultra-violet (EUV), and H$\alpha$
wavelengths.

Flare plasma parameters are also believed to affect the yield of HXR and SXR
emission, and therefore play a decisive role in the thermal-nonthermal energy
partition. According to Lee et al. (1995), the proportionality coefficient
between the HXR fluence and SXR flux may depend on the magnetic field
configuration and viewing angle and thus may vary from flare to flare. Early
impulsive (cold) flares, investigated by Lysenko et al. (2023) and Lysenko et
al. (2018), have been found to originate from relatively dense, and shorter
loops with a stronger magnetic field compared to the plasma charactersitics
and geometrical properties of average flares investigated. Saqri et al. (2023)
found conclusive evidence of microflares with strong nonthermal emission to
originate from the loops which have one of their footpoints rooted in the
umbral or penumbral region of sunspots. Motorina et al. (2020) investigated
the thermal-nonthermal energy partitioning during a flare that originated from
a two-loop system. Their investigation has revealed that although the
nonthermal energy is partitioned in the loops in a comparable amount, one of
the loops contained plasma with a higher temperature and lesser density
compared to that in the other loop. Such disparate thermal response of the
flare loops, even after being subjected to similar energy input from
nonthermal electrons, is attributed to different initial plasma parameters
(e.g. density) of the loops. On the contrary, Fleishman et al. (2021), based
on the investigation of a three-loop solar flare, found an uneven distribution
of nonthermal energy in the loops and suggested the initial distribution of
thermal plasma to guide the proportioning of thermal and nonthermal energies.
It is evident that initial flare plasma parameters and their subsequent
evolution characteristics may be the key to better understanding the thermal-
nonthermal energy partition during flares. Therefore, in this work, we first
conduct a statistical investigation of the yield of SXR and HXR emissions for
all the flares that occurred during September 20-25, 2021, as presented in
section 3.1. Next, an in-depth multi-wavelength investigation of 20 weak
flares, that exhibit strong HXR emission, is presented. We then employ one-
dimensional Palermo-Harvard (PH) hydrodynamical code to explore the response
of the flare plasma for different initial atmospheric conditions and present
the results in section 3.3. The discussion and conclusions are offered in
section 4.

##  2 Observations and Instruments

During September 2021, observing lines-of-sight of Spectrometer Telescope for
Imaging X-rays (Krucker et al. 2020, STIX; at $\sim$0.6 AU) onboard Solar
Orbiter mission (Müller et al. 2020), and STEREO-A (Kaiser et al. 2008, at
$\sim$0.95 AU) remained in an overlapping line-of-sight (Figure 1a), thus
offering a unique opportunity for a detailed multi-instrument investigation.
Although STEREO-A and Solar-Orbiter have been in a perfect alignment on 18
September 2021, the STIX observations have become available only since
$\sim$18:30 UT on 20 September 2021. Nevertheless, a very small separation
angle between the Solar Orbiter and STEREO-A (between 2.5–9∘ during September
20–25, 2021) provided a unique opportunity to investigate the long-term
evolution of the source active regions producing flares of different intensity
class. Besides, a relatively moderate (<40∘) separation angle between the
observing line-of-sight of STIX and the Sun-Earth line (Figure 1a) allowed a
very good coverage of STIX observed flares with the _Geostationary
Observational Environmental Satellite (GOES)_ and Solar Dynamics Observatory
(SDO; Pesnell et al. 2012). Therefore, we have conducted a multi-wavelength
investigation of solar flares that occurred between September 20–25, 2021.

Figure 1: Overview of X-ray and EUV emission recorded by various instruments
during September 20–25, 2021. (a) Positioning of various observatories on
September 20, 2021. A sequence of images in 195 Å from STEREO-A and 193 Å from
AIA on September 20, 2021, at 18:00 UT (b–c) and on September 25, 2021, at
23:57 UT (d–e) reveal overlap of field-of-view of STEREO-A and SDO. Active
regions 12871, 12872, and 12873 have been marked. (e–f) X-ray intensity
evolution in 4-10 keV, and 12-20 keV, from the quick-look mode observations
from STIX instrument, and in 1–8 Å from GOES mission, during 18:00–23:59 UT on
September 20, 2021 (e), and during 18:00–23:59 UT on September 25, 2021 (f),
demonstrating overlapping observations of flare events recorded by instruments
positioned at different vantage points.

Near-perfect alignment of Solar Orbiter and STEREO-A provided a co-temporal
imaging perspective. Full-disk observations in 195 Å, acquired by STEREO-A
showed the presence of three active regions 12871, 12872, and 12873 on
September 20, 2022 (figure 1b), which remained visible from the STIX
perspective until September 25, 2021 (figure 1d). Further, a detailed
investigation of EUV emission during flares is made by analyzing the
observations from Atmospheric Imaging Assembly (AIA; Lemen et al. 2012)
instrument onboard SDO. AIA provides the observations of full-disk Sun in
seven EUV channels (94, 131, 171, 193, 211, 304, and 335 Å) and two UV
wavelengths (1600 and 1700 Å) with a spatial and temporal cadence of 1.5
arcsecond and 12 s (24 s for UV channels), respectively. This enabled us to
identify the source region of the flares observed by STIX and GOES and derive
the thermal characteristics of the flare plasma. The active regions observed
by STEREO-A have also been observed by SDO for the analysis duration (figure
1c&e).

STIX onboard the Solar Orbiter observes the Sun uninterruptedly in X-ray
wavelength in a 4-150 keV energy range with a time cadence as high as 0.1 s.
The incoming X-ray emission, when passed through 30 pairs of X-ray opaque
Tungsten grids of varying pitch placed at different orientations before 30
pixelated (8 large pixels and 4 small pixels) detectors, forms a moiré pattern
which can be further analyzed to identify the location, shape, and brightness
of the source (Giordano et al. 2015; Hayes et al. 2022; Massa et al. 2022).
STIX observations are provided in two formats, pixel data (containing
observations from all the individual detectors and pixels), and spectrograms
(pixel-integrated observations). Besides, observations are also available in
the form of quick-look plots, background measurements, flare locations,
calibration spectra, etc. We have made use of the spectrogram mode
observations for the present investigation since these offer better coverage
of flares.

A moderate separation angle of STIX with respect to the Sun-Earth line
(between 30∘–38∘) allowed the investigation of flares observed by STIX from
the GOES as well. GOES provides the disk-integrated X-ray intensity profile in
1–8 Å and 0.5–4 Å with a time cadence as high as 1 second. From the X-ray
intensity profile in 4-10 keV, and 12-20 keV, from the quick-look mode
observations made available from the STIX instrument, and in 1–8 Å from GOES
mission, on September 20, 2021, and September 25, 2021 (Figure 1f–g), it is
evident that a majority of flares have been commonly recorded by both the STIX
and GOES observatories. The application of the solar flare identification
technique (Xiao et al. 2023) based on the STIX quick-look light curve in 4–10
keV, which also estimates the equivalent GOES intensity class, has revealed
that a total of 217 flares occurred in the investigated duration, out of which
only 13 flares are $\geq$ C-class. This indicates that the selected time
duration contains predominately weak flares.

##  3 Results

###  3.1 Relative yield of soft X-ray and hard X-ray emission during flares

According to the Empirical Neupert effect (ENE), the time-derivative of the
soft X-ray flux (FSXR) often mimics the hard X-ray (FHXR) emission profile. In
the integral form, this relation can be expressed in the following form (c.f.
section 13.5.5 of Aschwanden (2005),Veronig et al. (2005)).

$F_{SXR}(t_{p})\propto\mathcal{F}_{HXR}$ (1)

Here, $F_{SXR}(t_{p})$ corresponds to the soft X-ray (SXR) flux at the flare
peak time (tp) whereas $\mathcal{F}_{HXR}$ denotes the HXR fluence, derived by
integrating the hard X-ray (HXR) flux (FHXR) during the start (t0) and the
peak time (tp) of the flare, as expressed following.

$\mathcal{F}_{HXR}=\int_{t_{0}}^{t_{p}}F_{HXR}(t)dt$ (2)

This formulation of ENE allows quantifying the yield of HXR emission relative
to the SXR emission during a flare, defined as the quotient (qf) of hard X-ray
fluence and FSXR at the flare peak (tp) as presented in equation 3.

$q_{f}(t_{p})=\frac{\mathcal{F}_{HXR}}{F_{SXR}(t_{p})}$ (3)

We consider 4–10 keV and 12–20 keV energy bands to characterize the soft X-ray
(SXR) and hard X-ray (HXR) emissions, respectively. Temporal characteristics
of the flare X-ray intensity profile (t0, tp, and rise-time (tr), etc.),
including the peak and background fluxes in the SXR and HXR energy bands, have
been derived using a semi-automated scheme. This scheme starts with visually
identifying the time duration enveloping a flare event (tfl), as well as the
time duration for estimating background flux (tb). Next, the background fluxes
in both energy bands are estimated by averaging the respective fluxes during
tb, and their standard deviations ($\sigma$) are also calculated. Start time
(t0) in each energy band corresponds to the time instance when the flux values
exceed the background fluxes by 1$\sigma$ for the first time since the start
of tfl. Visually selected time durations, covering the flare maximum in both
the energy bands, enable deriving the peak time (tp) when the fluxes reach
their peak values. In such a way, we have characterized the temporal evolution
of a total of 203 flares, identified during the analysis period (September
20–25, 2021).

Figure 2a shows the X-ray intensity profile of two weak flares that occurred
on 21-Sep-2021 during 09:40–10:00 UT. Noticeably, despite similar peak soft
X-ray flux values, hard X-ray emission during the flares is different. This
disparate nature of relative X-ray emission in different energy bands can be
quantified in terms of the quotient (qf) (Figure 2b), as estimated to be 11.68
and 0.83, respectively for these flares. A higher value of qf indicates a
relatively stronger hard X-ray emission. Such a disparate nature is indicative
of a non-linear response of plasma environments toward the input nonthermal
energy. In a similar fashion, qf has been determined for all the investigated
flares by employing the background-subtracted flux in equation 3. We find the
qf values to be ranging between 0 (for 14 flares) and 11.68. According to the
standard flare energy release model for long cooling times, qf should attain a
value $\sim$1, which is indeed the case for 88 flares (corresponding to
0.5$\leq$qf$\leq$2.5; 43%).

We further categorize the relative yield of HXR emission with respect to the
FSXR(tp) (Figure 2c). This representation enabled outlining that a significant
fraction of the weak flares (FSXR(tp) <20 count/(s-cm2-keV)) (54 flare cases)
exhibit relatively large HXR emission for qf>2\. These flare cases are
hereafter termed as the HXR-rich weak flares. On the other hand, peak soft
X-ray flux FSXR(tp) exhibits a good correlation with hard X-ray fluence
($\mathcal{F}_{HXR}$) (Figure 2d), except for the weak flares as evident by a
cloud of scattered points below the unity line (grey diagonal line for
FSXR(tp)=$\mathcal{F}_{HXR}$) for FSXR(tp) <20 count/s-cm2-keV. This indicates
that a majority of weak flares have a tendency to exhibit predominant thermal
emissions. In agreement, Veronig et al. (2002b) also found a large scatter for
weak events in the correlation of HXR fluences (derived from HXRBS/SMM
observations) and GOES peak fluxes. The correlation between FSXR(tp) and
FHXR(tp) is found to be well explained by a power-law function with index 1.13
(Figure 2e). For more than 4000 RHESSI observed microflares, Hannah et al.
(2008) found the correlation of thermal and nonthermal flux to follow a power-
law relationship with an index of 1.11.

Figure 2: Statistical overview of hard X-ray flux yield (FHXR; 12-20 keV)
relative to soft X-ray Flux (FSXR; 4-10 keV). (a) Two weak flares during
21-Sep-2021 09:40–10:00 UT in Soft and Hard X-ray energy bands, which exhibit
the disproportionate nature of HXR emission irrespective of similar SXR
emission. Estimated background levels and time of the FSXR maximum (tp) have
also been plotted. (b) Background-subtracted FSXR, HXR-fluence, and quotient
(qf) of HXR-fluence and FSXR at tp. (c) Variation of qf with respect to
FSXR(tp). (d) FSXR(tp) versus FHXR(tp) exhibiting a power-law relationship.

###  3.2 Temporal and spectral properties of plasma emission during HXR-rich
weak flares

The variation of the relative yield of HXR emission (qf) with respect to the
peak SXR flux enabled us to select 20 HXR-rich weak flare cases (plotted in
blue in figure 2c) for a further in-depth multi-wavelength investigation, as
listed in Table 1. Flare intensity profiles in SXR (4-10 keV), and HXR (12-20
keV) energy bands, as determined from the STIX spectrograms, are plotted in
figure 3, and arranged in the descending order of qf value.

Table 1: Temporal characteristics of the SXR and HXR emission from HXR-rich weak flares and plasma properties during the SXR emission peak (tp) as derived from the analysis of X-ray and EUV emissions S.N. | Flare | FSXR111flux including pre-flare background, respective background values are provided in brackets | qf | EM (1046cm-3) | T (MK)
---|---|---|---|---|---
X-ray | EUV222averaged over the flaring region | X-ray | EUV
1 | SOL2021-09-21T09:46:22 | 19.5(10.5) | 11.68 | 1.593 | 0.002 | 26.0 | 8.5
2 | SOL2021-09-25T21:16:23 | 15.3(10.8) | 7.55 | 0.919 | 0.026 | 11.8 | 11.1
3 | SOL2021-09-23T09:02:20 | 15.9(11.1) | 7.22 | 0.273 | 0.090 | 11.2 | 11.1
4 | SOL2021-09-25T00:31:55 | 23.3(10.4) | 4.59 | 0.859 | 0.033 | 10.9 | 8.5
5 | SOL2021-09-21T04:00:28 | 16.9(10.9) | 4.50 | 0.021 | 0.004 | 12.6 | 9.4
6 | SOL2021-09-21T23:19:49 | 13.4(10.8) | 4.31 | 1.103 | 0.009 | 10.5 | 8.5
7 | SOL2021-09-24T18:02:42 | 17.4(10.8) | 4.13 | 0.856 | 0.006 | 11.6 | 10.6
8 | SOL2021-09-23T08:53:19 | 18.7(10.7) | 4.08 | 0.035 | 0.015 | 11.5 | 11.0
9 | SOL2021-09-21T06:25:24 | 24.4(10.9) | 3.97 | 1.996 | 0.019 | 11.4 | 8.5
10 | SOL2021-09-22T20:11:26 | 30.9(11.5) | 3.29 | 1.867 | 0.227 | 12.8 | 11.2
11 | SOL2021-09-23T06:00:50 | 21.3(11.6) | 2.68 | 2.414 | 0.045 | 10.5 | 11.0
12 | SOL2021-09-22T12:08:42 | 14.0(10.5) | 2.66 | 9.479 | 0.022 | 16.2 | 10.6
13 | SOL2021-09-22T21:50:57 | 17.8(10.4) | 2.48 | 1.782 | 0.025 | 10.0 | 9.3
14 | SOL2021-09-20T20:19:59 | 20.7(10.6) | 2.09 | 3.711 | 0.006 | 10.3 | 8.9
15 | SOL2021-09-22T13:51:59 | 24.7(10.5) | 2.04 | 1.246 | 0.019 | 11.5 | 9.4
16 | SOL2021-09-22T08:02:29 | 20.4(11.1) | 1.98 | 0.616 | 0.019 | 11.7 | 11.1
17 | SOL2021-09-23T14:11:38 | 26.9(10.2) | 1.83 | 2.326 | 0.019 | 11.2 | 8.0
18 | SOL2021-09-23T05:58:14 | 21.5(16.2) | 1.52 | 1.097 | 0.117 | 11.2 | 11.2
19 | SOL2021-09-20T21:21:23 | 19.4(10.4) | 1.49 | 2.075 | 0.004 | 11.7 | 11.2
20 | SOL2021-09-23T04:18:45 | 16.1(13.6) | 1.12 | 0.615 | 0.004 | 8.4 | 8.4
Figure 3: X-ray intensity profile of HXR-rich weak flares (qf>1;
FSXR(tp)$\sim$20 counts s-1 cm-2 keV-1 ), selected for further multi-
wavelength investigation and arranged in the descending order of qf value.
FSXR, FHXR, and $\mathcal{F}_{HXR}$ are plotted in red, blue, and green,
respectively. The time of FSXR peak emission (tp), when qf is determined, is
marked with vertical brown lines.

Next, we investigate the thermal and nonthermal characteristics of the plasma
during all the identified HXR-rich flares from the X-ray and EUV emission
recorded by STIX and AIA instruments, respectively. Although the time
durations of the flares range between 2 to 10 minutes, we have analyzed one
representative X-ray spectrum per flare, which has been prepared with a time-
integration ranging between 40–120 seconds covering the flare maximum phase.
We fit the X-ray spectrum in 4–20 keV employing the isothermal and broken-
power photon models available in the Object Spectral Executive (OSPEX) package
of SolarSoftWare (SSW). OSPEX enables forward fitting of the observed X-ray
spectrum by generating the theoretical spectrum from the selected photon model
functions and convolving it with the instrument-specific spectral response
matrix (SRM). The break energy in the broken power-law photon function is
allowed to vary in the range of 10–15 keV, while the power-law index below the
break energy is kept fixed to a very low value (equal to 1). This results in
effectively providing fit results for a single power-law function with a
cutoff at around 12 keV. In an iterative manner, the fitting aims to minimize
the $\chi^{2}$ value, a measure of the difference between the observed and
calculated count flux spectrum. We plot the spectral fit results for two
investigated flare cases SOL2021-09-21T09:46:22UT and SOL2021-09-22T20:11:26UT
(flare # 1 & 10 from table 1) in Figure 4 (a & e). The best-fit spectral model
enables deriving the thermal (temperature, emission measure) and non-thermal
(photon spectral slope; $\gamma$) characteristics of the flare plasma. From
the spectral fit, it is evident that the X-ray emission above 12 keV is best
represented primarily with the nonthermal fit in our investigation. This
supports our selection of energy bands that represent the thermal (SXR; 4–10
keV) and nonthermal (HXR: 12–20 keV) emission. Derived temperature and
emission measure of the flare plasma for all the analyzed flares ranges
between 8.4–26 MK, and 0.02–9.5 $\times$ 1046cm-3, respectively.

Figure 4: Thermal characteristics of the flare plasma as determined for HXR-
rich weak flare case#1 and #10 from Table 1. (a) Forward fit of the observed
X-ray spectrum (black) during the peak of SXR emission employing the
isothermal (red) and broken power-law (blue) photon models. Flare plasma
parameters that best fit the observations are annotated. 94 Å image of the
corresponding flare region (b), Emission measure map in 8–20 MK (c), and EM
distribution (EM[T]) corresponding to the flaring region (average of EM[T]
distribution over the pixels with EM > 1026cm-5 (yellow contours in (b) and
(c)) in various temperature bins), and the non-flaring region (grey) is
plotted in panel (d). Similarly, the bottom panels show the determination of
flare plasma parameters from the X-ray (e) and EUV (f–h) emissions during
flare case#10. EM-weighted temperature, and average EM in the range logT=[6.6,
7.5]K for the respective flares are annotated in panels (d) and (h).

The unavailability of pixel mode observations for a few flares restricted
performing image synthesis from STIX observations. However, EUV images from
AIA allow us to determine the morphological properties of the flaring region
as well as the thermal characteristics of the flare plasma. Identification of
the source region of weak flares is difficult, since several source regions of
apparently similar brightness may be simultaneously present. To locate the
source region of the flares, we first make use of the full-disk image of the
Sun in 94 Å at the time of maximum FSXR emission. For all the visually
identified locations of enhanced brightening, we have then derived the EUV
intensity profile from a sequence of 94 Å cut-out images by making an average
of the pixel intensities in all the pixels within the selected region that
have an intensity larger than 20 DN s-1. Subsequently, the temporal
association between the X-ray and EUV intensity profiles has been made to
distinctively identify the source region of the flares. From the 94 Å images
for flare#1 and 10 from table 1 (Figure 4 (b & f)), recorded at the flare
maximum, we note discrete kernel-like brightenings in addition to the extended
loop-like emissions. Such discrete brightenings are believed to have resulted
from the energy deposition from the nonthermal electron beams (Awasthi et al.
2018b).

To determine the thermal characteristics of the flare plasma from the EUV
observations, we apply a modified sparse differential emission measure (DEM)
inversion technique (Cheung et al. 2015; Su et al. 2018a). We employ this
scheme on a sequence of EUV images in six wavelengths (94 Å, 131 Å, 171 Å, 193
Å, 211 Å, and 335 Å), acquired close-in-time around the SXR emission peak
(tp). This enables determining the emission measure distribution (EM[T]) in
logT=[5.5, 7.5]K temperature range with binning of $\Delta$log$T$=0.05K.
Synthesized EM maps corresponding to the 8-20 MK temperature bin reveal hot
plasma of up to $\sim$20 MK temperature, co-spatial to the EUV brightening
seen in 94 Å images (Figure 4 b–c & f–g). This is in agreement with the flare
plasma temperature as derived from the X-ray spectral fit.

From the 8–20 MK EM map, the region with EM values higher than 1026cm-5 is
considered as the flare region (yellow contours in figure 4b–c & f–g)). The
area of the pixels that satisfy the aforesaid condition enabled determining
the area (A) of the flare region. Further, assuming the flare region to be
spherically symmetric and the estimated area (A) to represent its circular
projection, its radius ((A/$\pi$)1/2) may be regarded as the length equivalent
(s) of the flare region (Awasthi et al. 2016). Subsequently, the volume of the
flare plasma can be determined according to the relation, $V=\frac{4}{3}\pi
s^{3}$.

The EM[T] distribution of flare and quiet (EM < 1026cm-5) regions is
determined by averaging respective values of EM[T] in all the temperature bins
as plotted in red and grey, respectively, in figure 4d & h. We note that the
EM[T] distribution corresponding to the flare region exhibits a clear
enhancement in the temperature range logT=[6.6, 7.5] in comparison to that
corresponding to the quiet region. Therefore, we further estimate the EM-
weighted temperature ($<T_{EM}>$) in the aforesaid temperature range
(logT=[6.6, 7.5]) using the following relation.

$T_{EM}=\frac{\sum_{j}T_{j}\times{\it{EM}}(T_{j})}{\sum_{j}{EM}(T_{j})}$ (4)

Here, EM[Tj] is the emission measure value corresponding to temperature Tj,
and the averaging is performed over the temperature range logT=[6.6, 7.5]. A
representative value of emission measure for each flare case (EM_EUV) has been
determined by averaging the EM[T] distribution over the aforementioned
temperature range, and subsequently multiplying with the respective area of
the flare region (A). We have provided the values of TEM and EM_EUV in table
1. Temperature and EM values of the flare plasma from the EUV emission for all
the investigated weak flare cases, determined as explained above, vary in the
range of 8–11.2 MK, and 0.002–0.227 $\times$ 1046cm-3, respectively (see table
1).

To determine the role of flare plasma parameters in the relative yield of HXR
emission (qf) in the weak flares, we have determined the correlation of
temperature (T), emission measure (EM), and density (ne) as determined from
the X-ray as well as EUV observations (figure 5). The density of the flare
plasma is estimated from the following relationship.

$n_{e}=\sqrt{\frac{EM_{Xray}(cm^{-3})}{V_{EUV}(cm^{3})}}$ (5)

Here, $V_{EUV}$ is the volume of the flare region, estimated as discussed
above. Since the image synthesis from X-ray observations has not been possible
owing to the lack of ‘pixel mode’ of observations for several flare cases
investigated in this work, volumes estimated from the EUV images and EMXray
are used for the determination of respective density values.

Figure 5: Correlation of qf with flare plasma parameters, such as emission
measure (a), density (b), temperature from analyzing X-ray emission (c), and
EM derived from the EUV emission (d). The volumes estimated from the EUV
images and EMXray have been used for the density determination.

We find the derived EM and ne to be inversely related with qf, i.e., the
emission measure of the flare plasma is lower in the case of flares exhibiting
relatively larger HXR yield and vice-versa (figure 5a). This trend is evident
in the density versus qf correlation plot as well (figure 5b). Further, plasma
temperature is found to be positively correlated with the HXR-richness of the
flares (figure 5c). These correlations indicate that the stronger nonthermal
emission heats the flare plasma more efficiently without significantly
increasing the plasma density. This implication is in agreement with the case
studies performed by Motorina et al. (2020) and Fleishman et al. (2021).

Further, while we have derived the EM[T] distribution from the EUV emission
corresponding to the flare plasma having a temperature larger than 4 MK (logT
> 6.6 K), the EM (and temperature) determined from the EUV observations do not
exhibit any clear correlation with qf (figure 5d).

###  3.3 Hydrodynamical simulation of flare plasma

Although the analysis of X-ray and EUV observations in the present
investigation indicate that flare plasma parameters systematically affect the
partitioning of thermal and nonthermal emissions, it is not possible to
determine the effect of initial plasma conditions on the energy partitioning,
as indicated by Motorina et al. (2020) and Fleishman et al. (2021), primarily
due to low count statistics. Hydrodynamical models, on the other hand, allow
probing the response of the coronal loop plasma for different sets of initial
conditions. Therefore, we perform the hydrodynamical simulations using the
one-dimensional Palermo-Harvard (P-H) code (Peres et al. 1982; Betta et al.
1997). The PH code implements the strategy of solving the conservation
equations of the mass, momentum, and energy numerically for the case of a
compressible fluid. Further, the adoption of the adaptive grids enabled
resolving complex conditions like extremely large temperature gradients in the
transition region. A typical PH model run requires inputs such as loop length
and a heating function varying along the loop and in time. This enables
deducing the spatial and temporal evolution of temperature and density in the
loop.

To determine the impact of varying plasma conditions on the response of
coronal loop plasma that has been subjected to heating, we perform the PH
simulation for 25 sets of models comprised of five different initial base
pressures in the loop (10, 50, 100, 150, and 200 dyne cm-2) and all are
subjected to five values of heating inputs (5, 10, 15, 20, and 25 erg
s-1cm-3). Other input parameters for the simulations have been the loop semi-
length (L) = 109 cm, and the flare heating (Eh; spatial profile of the heating
- Gaussian centered at 2 $\times$ 108 cm above foot point, and width (FWHM)
L/5) has been applied during 100-130 sec of the simulation run, with a heating
decay time = 10 s. The first step of the PH run requires creating a
hydrostatic stable state loop plasma condition by subjecting the loop to a
background heating (Eh0), which varies in the range of 0.0485 – 1.16 erg
s-1cm-3, depending on the desired initial base pressure. PH model is allowed
to run for 1500 seconds. We have also tabulated the input parameters and
heating function in Table 2.

Table 2: Input parameters for PH model Loop semi-length (L) |  1 $\times$ 109 cm
---|---
Non-flare background heating (Eh0) |  0.0485 – 1.16 erg s-1cm-3
Flare heating (Eh) |  [5, 10, 15, 20, 25] erg s-1cm-3
Spatial Profile of Eh |  Gaussian centered at 2 $\times$ 108 cm above foot point,
|  and width (FWHM) L/5
heating decay time ($\tau$) |  10 s
ton–toff for Eh |  100-130 s
Base Pressure |  [10, 50, 100, 150, 200] dyne cm-2
Figure 6: Evolution of plasma parameters at the loop apex as resulting from
the application of the Palermo-Harvard hydrodynamical code. For two different
heating inputs (15 and 20 erg s-1 cm-3) that are applied for 30 seconds
starting at 100 seconds into the model run. The resulting temporal evolution
profiles of temperature, density, and simulated GOES flux in 1–8 Å wavelength
band are shown in panels (a)–(d), & panels (e)–(h), respectively.

To assess the response of the loop plasma with different initial conditions
when subjected to the same heating input, we plot the resulting plasma
parameters at the loop apex for three different cases of initial base pressure
when subjected to heating input of 15 and 20 erg s-1 cm-2 in the left and
right panels of figure 6, respectively. This analysis revealed that despite
applying the same heating input to the loops with the same geometrical
parameters, the model run for the case of lower base pressure resulted in a
higher peak temperature (and lower density) compared to that of a higher base
pressure. Similarly, the PH model run corresponding to the heating input of 20
erg s-1 cm-3 also revealed disparate temperature and density in the loops with
different initial base pressure, even though subjected to the same heating
input. Assuming the width of the loop as 5 arcseconds (w=2r; radius (r)), we
have derived the emission measure (EM) using the plasma density (ne) that has
been obtained from the hydrodynamical simulation using the relationship EM =
ne2V, where V is the volume and approximated to be equal to $\pi$r2L
considering the cylindrical geometry of the loop of length ‘L’ and radius ‘r’.
From the GOES fluxes in the long (1–8 Å) and short (0.5–4 Å) wavelength
ranges, estimated using the EM and PH model-derived temperature, we find that
the resulting soft X-ray flux in 1–8 Å also strongly depends on the initial
base pressure of the loop. This analysis asserts the role of initial loop
plasma parameters in the observed X-ray emission during flares and hence in
the thermal-nonthermal energy partitioning.

##  4 Discussion and Conclusions

While it is well known that non-potential energy available in the magnetic
system is released during solar flares governed by the magnetic reconnection
process, its partitioning between the thermal and nonthermal energies is not
well understood. This is evident from – 1) dissimilar SXR and HXR emissions
during a flare (e.g. figure 2a), and 2) observed disproportionate thermal and
nonthermal emissions within the flare loops that originate from the same flare
source region (Motorina et al. 2020; Fleishman et al. 2021). In addition to
exposing our lack of knowledge of the energy release mechanism during flares,
this poses questions if the flare intensity class, which is derived from the
peak flux recorded by GOES in 1–8 Å wavelength range, is a comprehensive
quantifier of energy released during the flare. In this regard, this work
characterizes the temporal evolution of SXR and HXR emissions during more than
200 flares, recorded by STIX during September 20-25, 2021. Using an integral
formulation of the Neupert effect, we have quantified the relationship between
SXR and HXR emission in the form of a quotient factor (qf). qf represents the
HXR fluence relative to the observed SXR emission at the flare maximum. This
quantification enables identifying the HXR-rich (strongly nonthermal) flares
from a large set of flares. By performing an in-depth multi-wavelength
investigation of 20 HXR-rich weak flare cases, our analysis revealed that the
relative HXR yield (qf) is inversely correlated with the flare plasma density
while exhibiting a positive correlation with the temperature. This indicates
that in comparison to denser loops, flare loops containing plasma of lower
density result in exhibiting relatively higher temperatures in response to
similar nonthermal energy input. We further explore the response of heating
the flare loop with different initial plasma conditions by the application of
a one-dimensional Palermo-Harvard code. We have performed hydrodynamical
simulations for various cases comprising five different initial base pressures
in the coronal loops when subjected to five different values of heating
inputs. This distinctively revealed that, for the same heating input, the
loops with lower base pressure result in exhibiting plasma of higher
temperature and lower density. This is in agreement with the implications made
from the X-ray spectral analysis of HXR-rich flare cases.

Motorina et al. (2020) conducted an in-depth analysis of the thermal response
of a two-loop solar flare. They found that, despite the deposition of a
comparable amount of nonthermal energy in both loops, the tenuous loop
contained hotter plasma than the denser loop. Only one of a complex three-loop
solar flare, investigated in Fleishman et al. (2021), contained nonthermal
electrons, while the other two loops contained only thermal plasma. More
recently, Fleishman et al. (2022) found the magnetic reconnection to
efficiently accelerate almost all the ambient electrons leaving the region to
be depleted of thermal electrons. These investigations emphasize the
importance of loop plasma conditions in thermal-nonthermal energy
partitioning, which is in agreement with the inferences derived from our
analysis. This further suggests that, in addition to investigating the
temporal association of the SXR and HXR emission made in accordance with the
Neupert effect, the density and temperature of the loop plasma should be
quantitatively included for a comprehensive understanding of the energy
release mechanism.

In addition to the thermal characteristics of the flare plasma, we also
derived the spectral slope ($\gamma$) of the HXR emission by forward fitting
the observed X-ray spectra with combined isothermal and broken power-law
photon models. We find the slope of nonthermal emission to inversely correlate
with the emission measure (figure 7a). This implies that the nonthermal
electrons with a harder spectrum result in an efficient chromospheric
evaporation, as evidenced by higher EM values. While this result is in
consensus with that revealed in the investigation of Motorina et al. (2020),
it is in disagreement with the simulation results of the chromospheric
evaporation in response to the beam heating (Fisher et al. 1985; Reep et al.
2015), in which nonthermal electron with a softer spectrum results in an
efficient evaporation. Despite this fact, the simulation results of Reep et
al. (2015) also indicated that, for weak flares, an explosive evaporation
threshold can be achieved with very little total nonthermal energy, and the
thermal response of the atmosphere may not strongly depend on the electron
energy in this regime. This may provide an explanation for a positive
correlation of spectral hardness with the emission measure, obtained in the
present work.

For the analyzed flares, we further deduce the correlation of temperature
versus emission measure, as shown in figure 7b. We may infer that relatively
strong HXR-rich flares (blue) exhibit a trend of having a lower EM and a
higher temperature. Flare plasma parameters corresponding to two strongly
nonthermal weak flare cases (nonthermal electron spectral index $\delta$ =
4–6) analyzed in Awasthi et al. (2018a) and Battaglia et al. (2023) are
overplotted which evidently follow the general behavior of T versus EM,
obtained in the present work. Despite the fact that our work considers flares
with similar peak SXR emission (< 20 counts s-1cm-2keV-1), from the ISO-flux
lines for GOES A, B, and C classes, drawn in figure 7b, the intensity classes
of the analyzed flares is found to vary from sub-A to B-class. This
representation further reveals the strong dependence of flare intensity class
on the thermal-nonthermal energy partition.

Figure 7: Variation of emission measure with HXR spectral slope (a) and
temperature (b) for the HXR-rich weak flare cases. Flare cases with qf>3 have
been shown in blue, while the rest are plotted in red. ISO-flux lines for GOES
A, B, and C classes are also drawn in grey. Relatively strong HXR-rich flares
(blue) exhibit a trend of having a lower EM and a higher temperature. Flare
plasma parameters for two strongly nonthermal weak flare cases analyzed in
Awasthi et al. (2018a) and Battaglia et al. (2023) are also plotted.

A comparative overview of temperature values derived from the X-ray (TX-ray)
and EUV (TEUV) observations reveals a significant difference for a few cases
while results in similar values for others (Table 1). This behavior may be
attributed to the non-equilibrium ionization (NEI). For heat pulses of less
than a minute duration, Reale & Orlando (2008) found that NEI may result in
much lower plasma temperature as determined from the observed spectra despite
the source electrons of more than 10 MK temperature. Lee et al. (2019)
investigated the effect of NEI on the thermal characteristics of the plasma,
which was initially in the state of ionization equilibrium (EI), and subjected
to rapid heating due to shock or magnetic reconnection. They found that the
temperature of the plasma in NEI is underestimated, compared to that in EI.
Therefore, a short heating of a coronal loop containing plasma of lower
initial density and temperature may lead to different responses as seen in the
AIA’s EUV channels owing to the fact that the highly ionized atoms, which are
needed to produce emission lines seen by AIA, will be in lower abundances.
Further, the time needed to attain a higher degree of ionization may be as
long as a few hundred seconds.

Summarily, this work emphasizes the crucial role of plasma density in the
coronal loops in the context of the thermal-nonthermal energy partition. While
it is difficult to determine initial plasma conditions solely from the X-ray
spectral observations due to the limited signal-to-noise ratio in the rising
phase of the flare, conducting case studies through multi-wavelength images
will help unravel the exact nature of initial plasma conditions on the thermal
response of the flare plasma and energy partition. Further, characterizing the
onset temperature (e.g., da Silva et al. (2023)) and emission measure for
flares of different intensity in a statistical sense may enable assessing
their quantitative impact on thermal-nonthermal energy partition of the flare
and is planned to be conducted in the future.

This research is supported by the PASIFIC project that received funding from
the European Union’s Horizon 2020 research and innovation programme under the
Maria Skłodowska-Curie grant agreement No. 847639. This research was funded in
part by the National Science Centre, Poland grant No. 2020/39/B/ST9/01591. For
the purpose of Open Access, the author has applied a CC-BY public copyright
licence to any Author Accepted Manuscript (AAM) version arising from this
submission. We acknowledge the insights provided by Janusz Sylwester and
Barbara Sylwester, Space Research Centre, Polish Academy of Sciences, Poland.
We also thank Ewan Dickson and Hualin Xiao for providing prompt support with
the STIX data processing. Open data policy of various space-based and ground-
based observatories namely Solar Orbiter, SDO, STEREO, and GOES is
acknowledged. The use of the Helioviewer project, an open-source project for
the visualization of solar and heliospheric data, is also acknowledged.

## References

  * Altyntsev et al. (2012) Altyntsev, A. A., Fleishman, G. D., Lesovoi, S. V., & Meshalkina, N. S. 2012, ApJ, 758, 138, doi: 10.1088/0004-637X/758/2/138
  * Argiroffi et al. (2019) Argiroffi, C., Reale, F., Drake, J. J., et al. 2019, Nature Astronomy, 3, 742, doi: 10.1038/s41550-019-0781-4
  * Aschwanden (2005) Aschwanden, M. J. 2005, Physics of the Solar Corona. An Introduction with Problems and Solutions (2nd edition)
  * Aulanier et al. (2012) Aulanier, G., Janvier, M., & Schmieder, B. 2012, A&A, 543, A110, doi: 10.1051/0004-6361/201219311
  * Awasthi et al. (2018a) Awasthi, A. K., Liu, R., Wang, H., Wang, Y., & Shen, C. 2018a, ApJ, 857, 124, doi: 10.3847/1538-4357/aab7fb
  * Awasthi et al. (2018b) Awasthi, A. K., Rudawy, P., Falewicz, R., Berlicki, A., & Liu, R. 2018b, ApJ, 858, 98, doi: 10.3847/1538-4357/aabd76
  * Awasthi et al. (2016) Awasthi, A. K., Sylwester, B., Sylwester, J., & Jain, R. 2016, ApJ, 823, 126, doi: 10.3847/0004-637X/823/2/126
  * Awasthi et al. (2014) Awasthi, A. K., Jain, R., Gadhiya, P. D., et al. 2014, MNRAS, 437, 2249, doi: 10.1093/mnras/stt2032
  * Battaglia et al. (2023) Battaglia, A. F., Wang, W., Saqri, J., et al. 2023, A&A, 670, A56, doi: 10.1051/0004-6361/202244996
  * Battaglia et al. (2009) Battaglia, M., Fletcher, L., & Benz, A. O. 2009, A&A, 498, 891, doi: 10.1051/0004-6361/200811196
  * Betta et al. (1997) Betta, R., Peres, G., Reale, F., & Serio, S. 1997, A&AS, 122, 585, doi: 10.1051/aas:1997157
  * Carmichael (1964) Carmichael, H. 1964, in NASA Special Publication, Vol. 50, 451
  * Cheng et al. (2012) Cheng, X., Zhang, J., Saar, S. H., & Ding, M. D. 2012, ApJ, 761, 62, doi: 10.1088/0004-637X/761/1/62
  * Cheung et al. (2015) Cheung, M. C. M., Boerner, P., Schrijver, C. J., et al. 2015, ApJ, 807, 143, doi: 10.1088/0004-637X/807/2/143
  * da Silva et al. (2023) da Silva, D. F., Hui, L., Simões, P. J. A., et al. 2023, MNRAS, doi: 10.1093/mnras/stad2244
  * Fisher et al. (1985) Fisher, G. H., Canfield, R. C., & McClymont, A. N. 1985, ApJ, 289, 414, doi: 10.1086/162901
  * Fleishman et al. (2021) Fleishman, G. D., Kleint, L., Motorina, G. G., Nita, G. M., & Kontar, E. P. 2021, ApJ, 913, 97, doi: 10.3847/1538-4357/abf495
  * Fleishman et al. (2022) Fleishman, G. D., Nita, G. M., Chen, B., Yu, S., & Gary, D. E. 2022, Nature, 606, 674, doi: 10.1038/s41586-022-04728-8
  * Freeland & Handy (1998) Freeland, S. L., & Handy, B. N. 1998, Sol. Phys., 182, 497, doi: 10.1023/A:1005038224881
  * Freeland & Handy (2012) —. 2012, SolarSoft: Programming and data analysis environment for solar physics, Astrophysics Source Code Library, record ascl:1208.013.  http://ascl.net/1208.013
  * Gieseler et al. (2023) Gieseler, J., Dresing, N., Palmroos, C., et al. 2023, Frontiers in Astronomy and Space Sciences, 9, 384, doi: 10.3389/fspas.2022.1058810
  * Giordano et al. (2015) Giordano, S., Pinamonti, N., Piana, M., & Massone, A. M. 2015, SIAM Journal on Imaging Sciences, 8, 1315, doi: 10.1137/141001111
  * Hannah et al. (2008) Hannah, I. G., Krucker, S., Hudson, H. S., Christe, S., & Lin, R. P. 2008, A&A, 481, L45, doi: 10.1051/0004-6361:20079019
  * Hayes et al. (2022) Hayes, L. A., Musset, S., üller, D. M., & Krucker, S. ä. 2022, arXiv e-prints, arXiv:2207.02079.  https://arxiv.org/abs/2207.02079
  * Hirayama (1974) Hirayama, T. 1974, Sol. Phys., 34, 323, doi: 10.1007/BF00153671
  * Kaiser et al. (2008) Kaiser, M. L., Kucera, T. A., Davila, J. M., et al. 2008, Space Sci. Rev., 136, 5, doi: 10.1007/s11214-007-9277-0
  * Kopp & Pneuman (1976) Kopp, R. A., & Pneuman, G. W. 1976, Sol. Phys., 50, 85, doi: 10.1007/BF00206193
  * Krucker et al. (2020) Krucker, S., Hurford, G. J., Grimm, O., et al. 2020, A&A, 642, A15, doi: 10.1051/0004-6361/201937362
  * Lee et al. (2019) Lee, J.-Y., Raymond, J. C., Reeves, K. K., et al. 2019, ApJ, 879, 111, doi: 10.3847/1538-4357/ab24bb
  * Lee et al. (1995) Lee, T. T., Petrosian, V., & McTiernan, J. M. 1995, ApJ, 448, 915, doi: 10.1086/176020
  * Lemen et al. (2012) Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, Sol. Phys., 275, 17, doi: 10.1007/s11207-011-9776-8
  * Lysenko et al. (2018) Lysenko, A. L., Altyntsev, A. T., Meshalkina, N. S., Zhdanov, D., & Fleishman, G. D. 2018, ApJ, 856, 111, doi: 10.3847/1538-4357/aab271
  * Lysenko et al. (2023) Lysenko, A. L., White, S. M., Zhdanov, D. A., et al. 2023, ApJ, 954, 122, doi: 10.3847/1538-4357/acea20
  * Massa et al. (2022) Massa, P., Battaglia, A. F., Volpara, A., et al. 2022, Sol. Phys., 297, 93, doi: 10.1007/s11207-022-02029-x
  * McTiernan et al. (1999) McTiernan, J. M., Fisher, G. H., & Li, P. 1999, ApJ, 514, 472, doi: 10.1086/306924
  * Motorina et al. (2020) Motorina, G. G., Fleishman, G. D., & Kontar, E. P. 2020, ApJ, 890, 75, doi: 10.3847/1538-4357/ab67d1
  * Müller et al. (2020) Müller, D., St. Cyr, O. C., Zouganelis, I., et al. 2020, A&A, 642, A1, doi: 10.1051/0004-6361/202038467
  * Neupert (1968) Neupert, W. M. 1968, ApJ, 153, L59, doi: 10.1086/180220
  * Peres et al. (1982) Peres, G., Serio, S., Vaiana, G. S., & Rosner, R. 1982, ApJ, 252, 791, doi: 10.1086/159601
  * Pesnell et al. (2012) Pesnell, W. D., Thompson, B. J., & Chamberlin, P. C. 2012, Sol. Phys., 275, 3, doi: 10.1007/s11207-011-9841-3
  * Reale & Orlando (2008) Reale, F., & Orlando, S. 2008, ApJ, 684, 715, doi: 10.1086/590338
  * Reep et al. (2015) Reep, J. W., Bradshaw, S. J., & Alexander, D. 2015, ApJ, 808, 177, doi: 10.1088/0004-637X/808/2/177
  * Saqri et al. (2023) Saqri, J., Veronig, A. M., Battaglia, A. F., et al. 2023, arXiv e-prints, arXiv:2312.06856, doi: 10.48550/arXiv.2312.06856
  * Shibata & Magara (2011) Shibata, K., & Magara, T. 2011, Living Reviews in Solar Physics, 8, 6, doi: 10.12942/lrsp-2011-6
  * Sturrock (1966) Sturrock, P. A. 1966, Nature, 211, 695, doi: 10.1038/211695a0
  * Su et al. (2018a) Su, Y., Veronig, A. M., Hannah, I. G., et al. 2018a, ApJ, 856, L17, doi: 10.3847/2041-8213/aab436
  * Su et al. (2018b) —. 2018b, ApJ, 856, L17, doi: 10.3847/2041-8213/aab436
  * Veronig et al. (2002a) Veronig, A., Temmer, M., Hanslmeier, A., Otruba, W., & Messerotti, M. 2002a, A&A, 382, 1070, doi: 10.1051/0004-6361:20011694
  * Veronig et al. (2002b) Veronig, A., Vršnak, B., Dennis, B. R., et al. 2002b, A&A, 392, 699, doi: 10.1051/0004-6361:20020947
  * Veronig et al. (2005) Veronig, A. M., Brown, J. C., Dennis, B. R., et al. 2005, ApJ, 621, 482, doi: 10.1086/427274
  * Veronig et al. (2021) Veronig, A. M., Odert, P., Leitzinger, M., et al. 2021, Nature Astronomy, 5, 697, doi: 10.1038/s41550-021-01345-9
  * Warmuth & Mann (2020) Warmuth, A., & Mann, G. 2020, A&A, 644, A172, doi: 10.1051/0004-6361/202039529
  * Wright et al. (2017) Wright, P. J., Hannah, I. G., Grefenstette, B. W., et al. 2017, ApJ, 844, 132, doi: 10.3847/1538-4357/aa7a59
  * Xiao et al. (2023) Xiao, H., Maloney, S., Krucker, S., et al. 2023, arXiv e-prints, arXiv:2302.00497, doi: 10.48550/arXiv.2302.00497