# Understanding The Duration of Solar and Stellar Flares at Various
Wavelengths

Jeffrey W. Reep Space Science Division, Naval Research Laboratory, Washington,
DC 20375<EMAIL_ADDRESS>Vladimir S. Airapetian American
University, 4400 Massachusetts Avenue NW, Washington, DC 20016 USA
NASA/GSFC/SEEC, Greenbelt, MD, 20771, USA

###### Abstract

Recent irradiance measurements from numerous heliophysics and astrophysics
missions including SDO, GOES, Kepler, TESS, Chandra, XMM-Newton, and NICER
have provided critical input in understanding the physics of the most powerful
transient events on the Sun and magnetically active stars, solar and stellar
flares. The light curves of flare events from the Sun and stars show
remarkably similar shapes, typically with a sharp rise and protracted decay
phase. The duration of solar and stellar flares has been found to be
correlated with the intensity of the event in some wavelengths, such as white
light, but not in other wavelengths, such as soft X-rays, but it is not
evident why this is the case. In this study, we use a radiative hydrodynamics
code to examine factors affecting the duration of flare emission at various
wavelengths. The duration of a light curve depends on the temperature of the
plasma, the height in the atmosphere at which the emission forms, and the
relative importance of cooling due to radiation, thermal conduction, and
enthalpy flux. We find that there is a clear distinction between emission that
forms low in the atmosphere and responds directly to heating, and emission
that forms in the corona, indirectly responding to heating-induced
chromospheric evaporation, a facet of the Neupert effect. We discuss the
implications of our results to a wide range of flare energies.

Sun: atmosphere; Sun: corona; Sun: flares; Sun: transition region; stars:
flares; stars: solar-type

##  1 Introduction

The Sun and all other cool dwarf stars produce powerful eruptive processes via
magnetic reconnection in the form of flares and coronal mass ejections (CMEs).
Flares cause the dissipation of magnetic energy, generating emission across
the electromagnetic spectrum, from radio to $\gamma$-rays. The relationship
between various emission bands can be used to understand the mechanism of
magnetic energy conversion into thermal and kinetic energy, and thus provide
insights into the physics of flares. Studies of the energy partition in solar
flares provide an estimate of the total solar irradiance in the ultraviolet-
optical bands and X-ray bands.

Observations of flares with Kepler (Borucki et al., 2010) and the Transiting
Exoplanet Survey Satellite (TESS; Ricker et al. 2014) on other solar-like
stars opened a new avenue in the study of flare physics by discovering
frequent superflares, events with energy above $10^{33}$ erg and often over
$10^{36}$ erg (Maehara et al., 2012), whereas the largest solar events are
around a few $\times 10^{33}$ erg (Emslie et al., 2012). Younger solar
analogues in the first 100 Myr of their life produce superflares at a rate as
high as 1 event/day (e.g. Colombo et al. 2022). Eruptions from the Sun and
other active stars can generate high fluxes of ionizing radiation in the X-ray
and extreme ultraviolet bands that can affect the magnetosphere, upper
atmosphere, and climate of our planet as well as impact atmospheric evolution
of exoplanetary atmospheres via photoionization driven heating that ignites
atmospheric escape and affects atmospheric chemistry (Airapetian et al., 2017,
2020; Siskind et al., 2022). These effects can influence the planetary climate
and produce detrimental conditions for habitability. The D-region of the
ionosphere, in particular, responds to even small scale changes in the X-rays
(Hayes et al., 2017) and in proportion to their brightness (Hayes et al.,
2021). Thus, studies of stellar variability hold the key to understanding the
potential extremes of solar/stellar flare effects and their impacts on space
weather in the solar system and other exoplanetary systems. The peak intensity
of the flare and the duration for which ionizing radiation remains elevated
both are important to understand the impact on planetary atmospheres. However,
the mechanisms that drive the energy release and transport in solar and
stellar flares are not fully understood.

In this paper, we focus on one aspect of flare physics, the flare duration.
Solar and stellar flares occur over a range of durations spanning two to three
orders of magnitude, from minutes to days. The measured duration depends on
the wavelength in which the flare is observed, and the duration varies for the
same event at different wavelengths. In some wavelengths, there appears to be
a correlation between the flare intensity and the duration (e.g. white light,
Maehara et al. 2015), while the same relation is not apparent in other
wavelengths (e.g. X-rays). Why does flare intensity correlate with duration in
some wavelengths, but not others?

###  1.1 Observations

Solar flares are often categorized by their properties as seen by the
Geostationary Orbital Environmental Satellites (GOES), a series of satellites
operated by NOAA that has provided continuous coverage of the Sun’s X-ray
activity since the 1970s. Specifically, the X-ray Sensors (XRS) onboard GOES
are two instruments measuring the solar irradiance in soft X-ray (SXR) bands
between 0.5 – 4 Å (3 – 24 keV) in XRS-A and between 1 – 8 Å (1.5 – 12 keV) in
XRS-B. The peak flux in 1-minute XRS-B data is used to classify solar flares,
where X-class flares have flux $\geq 10^{-4}$ W m-2, and M-, C-, B-, and
A-flares have flux decreasing in order of magnitude, respectively. Reep &
Knizhnik (2019) showed that the duration of solar flares, as measured by the
full-width-at-half-maximum (FWHM) in either XRS channel, is not correlated
with the flare’s class, and is consistent with a log-normal distribution. The
temperature and emission measure as inferred from XRS also do not appear to be
correlated with the duration. However, Toriumi et al. (2017) showed that there
is in fact a relation between the magnetic energy in a flare and the duration
in XRS ($\tau_{\text{FWHM}}\propto E_{\text{mag}}^{0.45\pm 0.05}$), but they
note that the magnetic energy is not clearly correlated with the peak
intensity of the SXR emission, so there is some disconnect between the energy
release and resultant intensity.

The duration of flares observed in white light is clearly correlated with the
released energy of those events. For example, Maehara et al. (2015) found that
there is a correlation between the energy release and duration of superflares
observed on solar-like stars with the Kepler satellite in white light
emission. Those authors derived a scaling law relating the duration of a flare
to its energy, $\tau_{\text{flare}}\propto E_{\text{flare}}^{1/3}$, while
their observations found $\tau_{\text{obs}}\propto E_{\text{obs}}^{0.39\pm
0.03}$. Namekata et al. (2017) found a similar relation between the duration
and intensity of white light emissions in M- and X-class solar flares,
although the solar events were systematically of shorter duration than the
stellar events, which those authors suggest is evidence that the stellar
flares have larger field strengths. Results from the TESS satellite by Pietras
et al. (2022) found a similar correlation between duration and intensity, and
also examined relations with stellar spectral type and magnetic field
strength. Getman et al. (2008) also examined X-ray flares in pre-main sequence
stars, finding a similar relationship between the duration and luminosity
(Getman 2023, private communication).

In a set of well-observed large solar flares, (Toriumi et al., 2017) found
that the SXR duration was uncorrelated with active region parameters (sunspot
area, total magnetic flux, average magnetic field strength), but that the
duration was linearly correlated with properties of the flare ribbons
(distance between ribbons, ribbon area, and ribbon magnetic flux). The authors
argue that magnetic reconnection continues for a longer time when the magnetic
flux is larger, and thus the energy release is related to the flare duration.
This was explored with modeling of a flare arcade in Reep & Toriumi (2017),
who showed that SXR duration depends directly on the duration of magnetic
reconnection as new loops are energized in turn, as well as on the loop length
of the longest loops in the arcade, which dictates the approximate cooling
time. Additionally, the occurrence of an associated CME does not correlate
with SXR duration (Harra et al., 2016); that is, eruptive and non-eruptive
flares are not distinguishable by duration. Finally, Hayes et al. (2020) found
a correlation between the SXR duration and the period of quasi-periodic
pulsations (QPPs) in the same energy band, and that the QPP period is also
correlated with the ribbon properties. Furthermore, the QPP period is
independent from the peak GOES/XRS flux, and the relation between QPP period
and flare duration scales differently in the presence or absence of a CME.
However, QPPs are not observed in all flares, but their detection is more
likely in larger class flares (Simões et al., 2015b; Hayes et al., 2020).

The relation of QPP period to flare duration raises another interesting
consideration. While the mechanisms that produce them are not definitively
known (Zimovets et al., 2021), QPPs are clearly related to the energy release.
The analysis of an X-class flare by Hayes et al. (2019) shows a clear example
where they begin to occur during the impulsive phase of the flare, that the
hard X-ray (HXR) bursts correspond to peaks in the SXR derivatives (a facet of
the Neupert effect), and that the HXR intensity peaks are modulated
significantly more than the SXR derivative peaks. A recent study by Lörinčík
et al. (2022) additionally showed that the QPPs seen in GOES/XRS derivatives
and EOVSA radio emission from sources above the looptop corresponded to strong
redshifts of Si IV emission observed with IRIS, which was likely a direct
signature of outflows from the reconnection region.

The Neupert effect (Neupert, 1968) describes a relation between the SXRs and
HXRs, namely that the intensity of the SXRs is proportional to the fluence of
the HXRs:

$I_{\text{SXR}}\propto\int I_{\text{HXR}}\ dt$ (1)

or, equivalently,

$I_{\text{HXR}}\propto\frac{d}{dt}\Big{(}I_{\text{SXR}}\Big{)}$ (2)

This relation occurs because the non-thermal HXRs are a direct signature of
heating from the electron beam, which induces chromospheric evaporation that
more slowly heats and fills the corona, causing the brightening of SXR
emission. Dennis & Zarro (1993) found that the Neupert effect is apparent in
around 80% of flares. This effect is more general than just a relation between
SXR and HXR emission, in that emission that forms lower in the atmosphere
brightens in direct response to the heating, and emission in the corona
primarily brightens in response to the evaporation process.

Bursts in HXR emission, due to bremsstrahlung from non-thermal electron beams,
are strongly correlated spatially and temporally with transition region (TR)
line emission. This was first discovered with the Solar Maximum Mission (Cheng
et al., 1981; Woodgate et al., 1983; Poland et al., 1984), and has also been
verified with more recent comparisons between SDO/AIA and RHESSI (Cheng et
al., 2012; Simões et al., 2015a) or IRIS and RHESSI (Warren et al., 2016).
This correlation occurs because the heating effectively pushes the TR
downwards in height, so that the TR lines form in regions of higher density,
immediately causing a jump in intensity ($\propto n^{2}$).

The production of white light emission in flares occurs when there is a
significant energy deposition deep in the chromosphere. The emission is
cospatial with the HXR emission in on-disk observations (Krucker et al.,
2011). Additionally, in limb flares, the HXR and white light emission has been
found to form at the same heights in the atmosphere (Martínez Oliveros et al.,
2012; Krucker et al., 2015), suggesting that the non-thermal electrons of
energy $\gtrsim 30$ keV cause the white light emission (Krucker et al., 2015).
Watanabe et al. (2017) found that there is a statistical difference in the
energy deposition rates due to non-thermal electrons between flares with and
without white light enhancements, specifically that white light flares have
larger deposition rates over a smaller range of time (i.e. much more impulsive
heating). Similarly, it is well known that the energy deposition from non-
thermal electrons is well correlated with X-ray intensity, as well as the rise
in temperature and emission measure (e.g. Warmuth & Mann 2016). Additionally,
using the data set of solar flares from Warmuth & Mann (2016), we find that
there is a correlation between SXR duration and non-thermal electron beam
parameters (the electron flux, non-thermal power, and low-energy cutoff),
which we show in Appendix A.

Additionally, while the acceleration of electrons is relatively well
understood, the transport of energy via other mechanisms like
magnetohydrodynamic (MHD) waves or ion beams must also play an important role.
In the largest events, particularly superflares with energy above $10^{33}$
erg, the expected energy fluxes of an electron beam become unstable to
Langmuir and ion-acoustic waves (Drummond et al., 1970; Zharkova & Siversky,
2011), so that ion beams likely become significantly more important to the
energy balance of the chromosphere. In this work, we only examine heating due
to an electron beam, but other heating mechanisms need to be further
constrained in the future.

###  1.2 Loop Cooling Time

The duration of an observed flare depends directly on the cooling time of the
loops composing that flare. Coronal loops cool primarily through thermal
conduction, radiation, and enthalpy flux (draining). Early numerical
simulations of coronal loops suggested that there is a direct relation between
the cooling and draining time of a loop: $T\propto n^{2}$ for the radiative
cooling phase of a flare loop (Serio et al., 1991; Jakimiec et al., 1992),
where $T$ is the temperature and $n$ the number density. More generally, the
scaling can be written as $T\propto n^{\delta}$, (Bradshaw & Cargill, 2005),
where $\delta=\gamma-1+\frac{\tau_{v}}{\tau_{R}}$ relates the timescales of
draining $\tau_{v}$ to radiative cooling $\tau_{R}$, and $\gamma$ is the ratio
of specific heats, usually taken to be $\frac{5}{3}$. For adiabatic cooling,
$\delta=\frac{2}{3}$, and for radiation-dominated cooling,
$\delta\rightarrow\infty$ (Bradshaw & Cargill, 2010). Additionally, a non-
uniform cross-sectional area fundamentally alters the rate of cooling due to
both thermal conduction and draining, and thus the typical $T\propto n^{2}$
relation does not hold in general (Cargill et al., 2022; Reep et al., 2022).
Rather, draining is suppressed from the loop ($\tau_{v}\rightarrow\infty$),
and radiation dominates the cooling until the onset of catastrophic collapse.
The total cooling time is considerably longer as a result.

Cargill et al. (1995) derived a simple expression to approximate the total
(evaporative) cooling time of a loop:

$\displaystyle\tau_{cool}$
$\displaystyle\approx\frac{5}{3}\tau_{c}^{7/12}\tau_{r}^{5/12}$
$\displaystyle\tau_{cool}$
$\displaystyle\approx\frac{5k_{B}L^{5/6}}{\chi^{7/12}\kappa_{0}^{5/12}(nT)^{1/6}}$
$\displaystyle\tau_{cool}$ $\displaystyle\propto\frac{L^{5/6}}{(nT)^{1/6}}$
(3)

where $\tau_{c}$ and $\tau_{r}$ are the conductive and radiative timescales,
$k_{B}$ the Boltzmann constant, $\kappa_{0}$ the coefficient of thermal
conductivity, and $L$ the loop half length. The radiative losses are commonly
taken to have a piece-wise power law form, so that the emissivity can be
written $\epsilon(T)=\chi T^{\alpha}$, with $\alpha=-\frac{1}{2}$ in the
temperature range of interest. Equation 3 was derived using the Spitzer-Härm
approximation to thermal conduction in the absence of electric fields (Spitzer
& Härm, 1953; Spitzer, 1962; Braginskii, 1965), but it can be modified to
account for turbulence. Bian et al. (2016a) derived a treatment for the
suppression of thermal conduction in the presence of turbulence, and its
impact on scaling laws for the cooling of coronal loops was examined by
Bradshaw et al. (2019). Bian et al. (2016b) derived a turbulent conductive
timescale:

$\displaystyle\tau_{\text{Turb}}$
$\displaystyle=\Big{(}\frac{c_{R}}{\gamma-1}\Big{)}\frac{L^{2}}{\kappa_{0}\lambda_{T}T^{1/2}}$
$\displaystyle\tau_{\text{Turb}}$
$\displaystyle\propto\frac{L^{2}}{\lambda_{T}T^{1/2}}$ (4)

Importantly, this has the same dependence on loop length as the Spitzer-Härm
case, has a weaker dependence on temperature, and is independent of density.
This can then be substituted to find the approximate total cooling time of a
turbulent loop

$\displaystyle\tau_{cool}$
$\displaystyle\approx\frac{5}{2}\Big{(}\frac{2k_{B}}{\chi}\Big{)}^{7/12}\Big{(}\frac{c_{R}}{\kappa_{0}\lambda_{T}}\Big{)}^{5/12}\frac{T^{2/3}L^{5/6}}{n^{7/12}}$
$\displaystyle\tau_{cool}$
$\displaystyle\propto\frac{T^{2/3}L^{5/6}}{\lambda_{T}\ n^{7/12}}$ (5)

where $c_{R}$ is a constant and $\lambda_{T}$ is the turbulent mean-free path
(see Bradshaw et al. 2019). The same dependence on loop length is recovered,
but the cooling time is more strongly dependent on both the temperature and
density, and it increases with temperature. In the presence of strong
turbulence, hotter loops take significantly longer to cool since energy
transport out of the corona by thermal conduction is reduced, particularly in
less dense loops. We will show an example of this in Section 3.

In this work, we examine how various cooling and heating parameters affect the
evolution of Sun-as-a-star time series. We have run hydrodynamic simulations
and forward modeled emission forming across a wide range of temperatures to
better understand how durations can be impacted by different cooling
processes, loop geometry, and heating parameters. We find that different
spectral lines evolve in two separate manners: those that respond directly to
heating, and those that respond indirectly via chromospheric evaporation. In
other words, there is a general Neupert effect for emission that forms in the
corona versus emission in the transition region or chromosphere.

##  2 Model

In order to explore how flare duration depends on wavelength as well as
physical parameters, we employ field-aligned hydrodynamic simulations. We use
the HYDrodynamics and RADiation code (HYDRAD111https://github.com/rice-solar-
physics/HYDRAD), which describes the evolution of plasma constrained to a
magnetic flux tube (Bradshaw & Mason, 2003). The code includes the physics
relevant to coronal loops and flares, including radiative losses from the
chromosphere through corona, thermal conduction with either the Spitzer-Härm
approximation or turbulent suppression of conduction, and heating functions
due to direct heating or from an electron beam (Bradshaw & Cargill, 2013; Reep
et al., 2019). The loop geometry is very general, and can include open or
closed loops (e.g. Scott et al. 2022), with uniform or expanding cross-
sectional area of any specified form. In this paper, we examine closed loops
both with and without area expansion.

From the simulations, we obtain the temperatures, densities, velocities, and
ion fractions as a function of both space and time, so we can synthesize
emission that can be used to facilitate comparisons to observations. In this
paper, we synthesize thermal emission using the CHIANTI atomic database (Dere
et al., 1997), version 10 (Del Zanna et al., 2021). The thermal, optically-
thin calculation with CHIANTI includes free-free, free-bound, bound-bound, and
two-photon continuum emission.

We assume solar photospheric abundances (Asplund et al., 2009) in both the
radiative losses and forward model, which is consistent with observations of
large solar flares (Warren, 2014). Recent observations have additionally found
that abundances of some elements (with low first ionization potential) vary
with time during a flare, from coronal abundances towards photospheric in the
impulsive phase, and then back towards coronal during the gradual phase
(Mondal et al., 2021; Sylwester et al., 2023; Nama et al., 2023).
Additionally, abundances can vary spatially (Doschek & Warren, 2016) and from
flare to flare (Sylwester et al., 1998). In this work, we do not attempt to
model variable elemental abundances, which would modify the radiative losses
from a given loop and thus directly impact the duration of emission. This
assumption also implies that flares from stars with metallicities different to
the Sun would have different durations as loops would cool at different rates.
That is, the cooling rates depend directly on the metallicity of the host star
(Sutherland & Dopita, 1993; Washinoue & Suzuki, 2023). For instance, low
metallicity stars such as $\epsilon$-Eridani (HD 22049, Santos et al. 2004)
show flares with longer duration than the Sun (Coffaro et al., 2020).
Additionally, metallicities of such stars can be up to an order of magnitude
lower than solar values (Santos et al., 2004; Testa, 2010), and so we would
expect the radiative cooling rate to be similarly reduced, by a factor of
$\approx 4$ from $\log{T}=4-6$ and a factor of $\approx 2$ around $\log{T}=7$.
We emphasize that the qualitative behavior of the emission is our focus in
this paper, rather than exact durations.

For the synthesis of optically thick lines (H-$\alpha$, Lyman-$\alpha$, He II
304), we have run a few simulations of HYDRAD using its NLTE chromosphere
(Reep et al., 2019) and synthesized the emission with the
RH1.5D222https://github.com/ITA-Solar/rh radiative transfer code (Pereira &
Uitenbroek, 2015). In the calculations of X-ray spectra, we additionally
synthesize the non-thermal free-free emission directly, using the assumed
electron beam parameters and the full Bethe-Heitler cross-section (Bethe &
Heitler, 1934) with the Elwert correction factor (Elwert, 1939) that
accurately captures the cross-section across both non-relativistic and
relativistic energies (Koch & Motz, 1959). We additionally synthesize non-
thermal free-bound emission for beam electrons recombining onto Fe XXII
through Fe XXVI ions, based on the derivation in Brown & Mallik (2008) and
corrections in Reep & Brown (2016). We synthesize all of the emission as
irradiance, that is, spatially unresolved or Sun-as-a-star spectra. See Kerr
(2022, 2023) for an in-depth review of flare models and forward modeling of
radiation in recent years.

##  3 Single Loop Evolution

Let us consider first the commonly assumed case of a magnetic loop of uniform
cross-sectional area, with the Spitzer-Härm approximation to thermal
conduction. Figure 1 shows an example of the cooling behavior of a 100 Mm
loop. The plasma was heated with an electron beam with a triangular pulse,
with $10^{11}$ erg s-1 cm-2 peak flux, lasting 100 seconds, low-energy cutoff
$E_{c}=25$ keV, and spectral index $\delta=5$. The corona is rapidly heated to
a temperature above 40 MK, while the density rises more slowly to a peak
nearing $10^{11}$ cm-3 due to chromospheric evaporation. The top plots show
the evolution of the electron temperature (left) and electron density (right)
at the apex of the loop, which takes about 50 minutes in total to cool, and 80
minutes to drain after the onset of heating. The vertical lines mark the times
when the corona hits the peak formation temperature of six spectral lines: Fe
XXIII 133 Å ($\log{T}=7.15$), Fe XVIII 94 Å ($\log{T}=6.85$), Fe XVI 335 Å
($\log{T}=6.45$), Fe XIII 202 Å ($\log{T}=6.25$), Fe IX 171 Å
($\log{T}=5.90$), and Ne VII 465 Å ($\log{T}=5.70$). The bottom six plots show
the evolution of the irradiance of these six spectral lines, respectively.

Figure 1: The cooling behavior of a uniform 100 Mm loop heated by a strong
electron beam. The top plots show the time evolution of the electron
temperature (left) and electron density (right) at the loop apex. The bottom
six plots show the evolution of synthetic irradiance of 6 spectral lines. The
vertical lines on each plot mark the time at which the coronal temperature
reaches the peak formation of each spectral line. Once the corona cools beyond
the formation temperature for each ion, the ion recombines and the irradiance
falls off rapidly. The corona catastrophically collapses after about 50
minutes, and the lines fade entirely after this point.

There is a clear pattern in the time series of these spectral lines in Figure
1. The irradiance of each line rises sharply with the onset of heating, and
for the duration of that heating, there is a local peak in the intensity. This
initial spike is most noticeable in the coolest lines, where it is the
brightest emission during the entirety of the loop’s evolution. This is
because the initial heating pulse pushes the transition region downwards in
height, so that the emission of the lines occurs deeper in the atmosphere,
where the density is higher. After that initial spike, there is a gradual
increase in intensity that tracks the increase in the density due to
chromospheric evaporation. Then, as the corona cools, the temperature falls
below the peak formation temperature of each line in turn (marked by the
vertical lines in the plots). After this occurs, the irradiance of each line
falls sharply as the ion population falls due to collisional recombination.
For example, the corona cools to the peak formation temperature of Fe XXIII at
about 14 minutes after heating onset, after which the irradiance of that line
falls off rapidly. In the case of the cooler lines, like Fe IX 171 Å, the
corona is too hot for Fe IX ions to be present before about 40 minutes, so the
emission of that line is coming from the transition region prior to then. As
the corona cools below 1 MK, though, the hotter Fe ions recombine to Fe IX,
and the line begins to form in the corona itself, causing a spike in the 171 Å
line. Finally, after about 50 minutes, the corona begins to catastrophically
collapse, and the emission from all of these lines disappears.

Figure 2: The cooling behavior of a 100 Mm loop with expanding cross section
(factor of 11 from footpoint to apex) heated by a strong electron beam.
Similar to Figure 1. The density remains constant through the radiative
cooling phase, so the intensities track the change in ionization fractions.

Figure 2 shows a similar set of plots for a loop with gradual area expansion
across the length of the loop (by a factor of 11 from photosphere to apex).
Because of the area expansion, the flow of mass from the corona is
significantly reduced (see detailed comparisons in Reep et al. 2022), so that
the cooling is dominated by radiation. That is, for the scaling relation
$T\propto n^{\delta}$, $\delta\rightarrow\infty$ and the cooling is driven by
radiation. The loop does not significantly drain until the onset of
catastrophic cooling, so the coronal density remains constant, and energy is
not lost through an enthalpy flux. The cooling time is therefore longer than
for a loop with uniform area.

In Figure 2, the spectral lines evolve in two phases. The first corresponds to
the initial heating pulse, where the transition region is pushed downwards in
height, raising the densities and causing intense brightenings. This is
particularly noticeable in the cool lines (e.g. Fe IX 171 Å). The coronal
density then rises from evaporation, and remains approximately constant
through the entire cooling period. As a result, the intensities of the lines
change only due to changes in the ionization fraction. Consider the evolution
of the Fe XVIII 94 Å case, where the intensity increases from about 20 to 40
minutes after the onset of heating, and then dims until about 70 minutes. The
coronal density is constant over this whole time period, so this change in
intensity corresponds directly to a change in ionization fraction (recombining
from Fe XIX initially as it rises, then recombining into Fe XVII as it fades).
This same behavior can be seen in all 6 lines. Additionally, for the cooler
lines, there is a strong component of emission from the transition region,
that is roughly constant in density and ionization fraction after the initial
heating pulse. This can be seen in the roughly constant intensity in e.g. Fe
IX 171 Å, until the coronal portion of the loop cools enough to begin forming
this ion.

Next, we consider the case of a uniform loop with suppression of thermal
conduction due to turbulence in Figure 3. Following the formalism of Bradshaw
et al. (2019), we set the turbulent mean-free path $\lambda_{T}=10^{7}$ cm,
which causes significant suppression of conduction. We see from the apex
temperature that the loop is heated to well over $10^{8}$ K as the energy is
bottled up in the corona, gradually falling over time as radiation and flows
carry away that energy. Comparing to the case with Spitzer-Härm conduction, at
the start of the cooling period, the temperature is significantly higher, and
the density lower, so from Equation 5, we can see that the cooling time should
be significantly lengthened. The loop slowly drains during the course of the
cooling, additionally, which gradually reduces the radiative losses.

Figure 3: The cooling behavior of a uniform 100 Mm loop heated by a strong
electron beam, with strong turbulent suppression of thermal conduction.
Similar to Figure 1. The loop becomes much hotter and less dense than the
laminar case, which causes the cooling and draining times to be significantly
longer.

The time series of spectral lines in Figure 3 show a rather different
evolution than the previous cases. There is still a spike in intensity
corresponding to the onset of heating when the transition region is pushed
downwards, and as before, it is much more noticeable in the cooler lines.
After that heating pulse, however, the intensities of the lines closely trace
the evolution of the density. The intensity of each rises as evaporation fills
the loop, peaking at about 10 minutes after heating. Then, as the density
begins to drain, the intensity of each line begins to decrease. There is an
inflection point in each line, corresponding again to the time when the corona
cools to the peak formation temperature of each ion. So, for example, after
about 40 minutes, the corona has cooled to $\log{T}=7.15$, when Fe XXIII is
most abundant, and there is an inflection point in the time series of the 133
Å line where the intensity falls sharply as the corona recombines to cooler
ions. The same behavior can be seen in all six lines.

Finally, in Figure 4, we consider the case of a loop with expanding cross
section ($A_{max}=11$) as well as turbulent suppression of thermal conduction
($\lambda_{T}=10^{7}$ cm). The behavior now shows an amalgamation of the
previous two cases. The coronal temperature rises to approximately $10^{8}$ K.
The coronal density rises with chromospheric evaporation to about $10^{10}$
cm-3, but, due to the area expansion, does not drain significantly during the
course of the cooling period and remains approximately constant until the
onset of catastrophic cooling. Since cooling due to conduction and enthalpy
flux have both been suppressed in this case, radiation is the only effective
cooling mechanism, and this takes significantly longer than in any previous
case.

Figure 4: The cooling behavior of a 100 Mm loop with expanding cross section
heated by a strong electron beam, with strong turbulent suppression of thermal
conduction. Similar to Figure 1. Both thermal conduction and draining of the
loop have been suppressed, so cooling proceeds only through radiation, and
takes significantly longer than the previous cases.

The evolution of the spectral lines in Figure 4 shows behavior similar to
those in Figure 2. There is a sharp spike in intensity due to the initial
heating pulse, causing the transition region to be pushed to lower heights and
higher densities. Then, the corona is heated and filled, but is too hot to
strongly emit in any of the six lines. In this case, all six lines initially
form in the transition region as the corona is far too hot for any of the ions
to be abundant. As the corona slowly cools, however, each ion begins to form
as hotter ions recombine. The coronal density remains constant for the cooling
period, so once again the change in intensity closely corresponds to the
change in ionization fractions. As a result, each has a roughly constant
component emanating from the transition region that produces a weak but
constant intensity. The coronal component dominates when the corona reaches
the peak formation temperature of each ion, and then rapidly falls as the
corona cools further.

We briefly turn our attention to the X-rays, which are used to classify solar
flares and used for diagnosing particle acceleration. Figure 5 shows synthetic
X-ray emission from each of the four simulations in Figures 1 to 4, from top
to bottom respectively. The plots in the left column show the X-ray spectra at
three different times corresponding to the peak of the heating event (50 s),
decay of the heating event (75 s), and near the start of the cooling period
(125 s). The total spectrum (solid) is calculated by combining the non-thermal
emission (dotted) from the electron beam with the thermal line and continuum
emission (dashed). The spectra were calculated at 0.01 keV resolution. The
emission is dominated by thermal bremsstrahlung at low energies, while the
non-thermal bremsstrahlung dominates at high energies during the heating
period. We then compare time series using energy ranges from three X-ray
instruments, the Neutron Interior Composition Explorer (NICER; Gendreau et al.
2016), X-ray Multi-Mirror Mission (XMM-Newton; Jansen et al. 2001), and
GOES/XRS. The right column shows the time series of the X-rays in physical
units corresponding to five energy ranges observed by NICER SXR (0.2 to 2
keV), NICER HXR (2 to 12 keV), XMM (0.2 to 20 keV), GOES XRS/A (3 to 24 keV),
and GOES XRS/B (1.5 to 12 keV). In each case, the softer energy ranges for XMM
and NICER SXR remain bright for longer than for the others. The high energy
emission, dominated by thermal bremsstrahlung, decays rapidly as the
temperature of the loop decreases. That is, time series of higher energy X-ray
bands should be shorter in duration than lower energy bands because of the
strong dependence of bremsstrahlung on temperature.

Figure 5: The evolution of X-ray emission in the four loop simulations
described in Figures 1 through 4, from top to bottom, respectively. The left-
hand column shows the X-ray spectrum at the time of peak heating (50 s), as
the heating decays (75 s), and at the onset of the cooling phase (125 s). The
right-hand column shows the irradiance time series in the energy ranges
observed by NICER, Newton/XMM, or GOES/XRS (assuming a distance of 1 AU).

The time evolution of emission, and thus the duration, depends on the height
at which the emission forms, the relative importance of the three cooling
mechanisms, and the details of the heating. We summarize the cooling evolution
and resultant emission with a cartoon description in Figure 6, including a
cool spectral line, a coronal line, and thermal free-free emission. There are
generally three phases of evolution. In the first phase, the onset of heating
causes a rise in temperature in the corona through the upper chromosphere. The
transition region is pushed to lower heights, and chromospheric evaporation
begins. Lines forming in the transition region or upper chromosphere, as well
as non-thermal emission from accelerated particles, spikes sharply during this
phase. Hotter emission begins to rise with the temperature, and grows slowly
as the coronal density increases (the Neupert effect). During the second
phase, the onset of cooling, evaporation slows and the coronal density reaches
its highest value. The plasma initially cools from thermal conduction, but
transitions to a radiative cooling phase. If the cross-sectional area of the
loop is close to uniform, it will slowly drain during this period, while large
area expansions will inhibit draining. Thermal bremsstrahlung peaks near the
onset of cooling, and rapidly falls off as the plasma cools. Hot lines grow in
intensity as the coronal plasma approaches their peak formation temperature,
and then begin to fall off as the ions recombine to lower ionization states.
Cooler lines, forming in the transition region at this time or below, remain
approximately constant in intensity. Finally, at the onset of catastrophic
collapse, the temperature plummets rapidly in the corona. The hot lines, which
may have some residual emission, may have a small spike here before
disappearing. As the coronal portion of the loop cools, cooler ions,
previously only present in the transition region, begin to form in the corona,
and there is a large spike in emission at this point. As the temperature
continues to plummet and material rapidly drains from the loop, all emission
fades away rapidly.

Figure 6: A cartoon description of the phases of evolution of emission in a
coronal loop. There are generally three phases: a heating phase, a radiative
cooling phase, and catastrophic collapse. The example light curves show a cool
line (blue), forming near the lower transition region or upper chromosphere, a
coronal line (gold), forming in the corona and upper transition region, and
thermal free-free emission (red), forming in the corona. The evolution depends
strongly on the height at which emission forms.

##  4 Parameter Survey

We briefly turn to examine how various parameters can impact the duration of
light curves. In Figure 7, we show a comparison of the time series for the
same six spectral lines as before with five loop lengths, [25, 50, 100, 150,
200] Mm. We use a uniform cross-sectional area, and Spitzer-Härm conduction in
these cases. As derived by Cargill et al. (1995), the total cooling time of a
loop scales as $L^{5/6}$, so the duration of each time series grows with loop
length. As pointed out in Figure 1 and accompanying text, the duration also
differs for each spectral line, with hotter lines fading away sooner.
Additionally, in longer loops, the density increase from evaporation is
relatively smaller than for shorter loops, so the intensity spike during the
heating period is more prominent in longer loops.

Figure 7: Irradiance time series for uniform, laminar loops of total lengths
between 25 and 200 Mm. The duration depends strongly on loop length as well as
the formation temperature of each spectral line. The intensity spike during
the heating period is relatively stronger in longer loops.

We next examine the low-energy cutoff $E_{c}$, which is the minimum value of
the accelerated electron distribution (assuming a power law distribution of
electrons, see discussions by Holman et al. 2011 and Xia et al. 2021). This
value determines the average depth at which the electrons collide with the
ambient plasma, and thus impacts the strength of chromospheric evaporation
(Nagai & Emslie, 1984). For smaller values of $E_{c}$, smaller energy fluxes
can cause explosive evaporation, and vice versa (Reep et al., 2015). In Figure
8, we show a comparison of five uniform, laminar loops heated by electron
beams carrying the same total energy flux (peak of $10^{11}$ erg s-1 cm-2),
with cutoffs $E_{c}$ of 10, 25, 30, 40, and 50 keV. When the cutoff is
relatively small, evaporation proceeds explosively, and thus the hottest lines
are not greatly impacted by the value of $E_{c}$. For large values of $E_{c}$,
however, the evaporation is significantly weaker, and thus the corona is much
lower density so that the intensities fade away much more quickly.
Additionally, for larger values of $E_{c}$, the transition region is not
pushed significantly downwards in height, and thus the densities of the TR
lines are not as high, reducing their intensities generally. Finally, for
large values of $E_{c}$, the energy deposition occurs too low to cause strong
evaporation, but does still produce a spike in intensity during the heating
period, which becomes relatively more prominent in cooler lines. Compare the
observational measurements in Appendix A, which shows the duration of SXR
emission is weakly correlated with the low-energy cutoff.

Figure 8: Irradiance time series for uniform, laminar 100 Mm loops, heated by
an electron beam of $10^{11}$ erg s-1 cm-2 with various low-energy cutoffs
$E_{c}=[10,25,30,40,50]$ keV. The durations and intensities are strongly
affected by the value of $E_{c}$, which affects the strength of evaporation.

We finally examine the behavior a few lines that form deeper in the
atmosphere, in the lower transition region and chromosphere. In Figure 9, we
show the evolution of H-$\alpha$, Lyman-$\alpha$, and He II 304 Å
(Lyman-$\alpha$ for He II). The simulations were run on 100 Mm loops with low
energy cut-off $E_{c}=15$ keV and spectral index $\delta=5$, with peak heating
rates of $F=[10^{9.5},10^{9.7},10^{10.0},10^{10.2},10^{10.3}]$ erg s-1 cm-2.
The plots at left show the evolution of the line profiles for the strongest
heating case, at a 100 s cadence from 0 to 5000 s from blue to yellow in time,
where the dotted vertical line marks the rest wavelength (in vacuum). The
plots at right show the time series of the total irradiance for all five cases
(normalized to the time zero value) at a 5 second cadence. H-$\alpha$, forming
in the mid-chromosphere (Carlsson & Stein, 2002), is initially in absorption,
quickly jumps into emission an order of magnitude above the continuum, and
then gradually decays over time. Stronger heating produces a longer duration
time series in H-$\alpha$. The behavior of Lyman-$\alpha$ and He II 304, both
forming in the upper chromosphere and lower transition region, behave somewhat
differently. In both cases, the initial central inversion disappears at the
onset of heating, causing a strong brightening that decays over similar
timescales as H-$\alpha$. In both lines, there is a strong redshifted
component during the gradual phase stemming from draining of material through
the transition region. In all three lines, the time series shows similar
behavior to the optically thin transition region lines examine in previous
Figures, with a long and gradual decay after a spike during the heating phase.

Figure 9: The evolution of H-$\alpha$, Lyman-$\alpha$, and He II 304 Å. The
plots at left show the evolution of the line profiles from a simulation with a
peak heating rate of $F=10^{10.3}$ erg s-1 cm-2, going from blue to yellow in
time at a 100 s cadence, while the plots at right show the time series from
five simulations with various heating rates.

##  5 Discussion

The evolution of a time series at a given wavelength depends intimately on the
details of the heating, cooling, emission mechanism, and loop and arcade
geometry. Whether there is a correlation between the intensity and duration at
a given wavelength depends primarily on the height in the atmosphere where the
emission forms. Cooler emission responds immediately to heating as the
transition region is effectively pushed to lower heights in the atmosphere,
which causes an immediate increase in the density and thus intensity. On the
other hand, hotter emission forming in the corona only sees a rise in density
through chromospheric evaporation, which is delayed from the heating as it
takes time to accelerate the plasma. The strength of evaporation (speed or
mass flux) depends on the exact heating parameters (Fisher et al., 1985; Reep
et al., 2015), and therefore the rise in intensity is only an indirect
response to the heating. This is effectively the same as the Neupert effect
(Neupert, 1968), which is more traditionally stated as a relation between the
HXR time series and time derivative of the SXRs. More generally, it relates
the heating to the evaporative response of the chromosphere, which causes that
observed relation between HXRs and SXRs (e.g. Li et al. 1993; Lee et al. 1995;
Veronig et al. 2002). The Neupert effect is detected routinely in many
observations in many different bandpasses of solar (e.g. Dennis & Zarro 1993;
Kundu et al. 2001; Lee et al. 2017) and stellar flares (Osten et al. 2004;
Fuhrmeister et al. 2011; Caballero-García et al. 2015).

The Neupert effect points to a clear relation between the heating and observed
emission. The heating of individual loops within an arcade cannot be directly
measured in general, and so must be inferred from other properties. Qiu et al.
(2013) pointed out that there is a two-phase evolution in AIA’s 1600 Å
bandpass (dominated by C IV emission) in a solar flare: a first phase that
shows a direct response to heating, and a second phase that tracks the
evaporation of material into the corona. Observations of a two-phase evolution
of time series have similarly been found in stellar flares (e.g. Hawley et al.
2003) and solar flares (e.g. Zeng et al. 2014). These kinds of observations
have also been used to attempt to constrain the heating properties of
individual loops in a flaring arcade (Liu et al., 2013; Zhu et al., 2018; Qiu,
2021), with the underlying assumption that brightenings were directly related
to the heating. Rubio da Costa et al. (2016) similarly used individual peaks
of the time derivative of GOES/XRS emission to inform heating functions in a
flare arcade model.

We summarize a few key points:

  * •

The Neupert effect implies different behavior of time series for hot and cold
emission. Heating causes the transition region to be pushed to lower heights,
raising the density, which directly causes emission from cool plasma to
brighten. This is evidenced by the direct correlation between non-thermal HXR
emission and transition region lines (Cheng et al., 1981; Warren et al.,
2016). Hot plasma brightens slowly as chromospheric evaporation raises the
density of the coronal segment of loops in the flare arcade, which occurs more
gradually.

  * •

The duration of emission in a given spectral line depends on the height in the
atmosphere where it forms. For hotter lines that form in the corona, the
intensity in a loop rises with evaporation, peaks when the ionization fraction
is maximized, and rapidly falls off below its formation temperature. For
cooler lines forming in the transition region or chromosphere, the emission
peaks during the heating period when the density jumps up, stays roughly
constant or slowly declines in the radiative cooling phase, and has a smaller
secondary peak when the coronal portion of the loop cools to its formation
temperature.

  * •

Cooling processes are vitally important for the duration of emission. It is
well known that longer loops take longer to both cool ($\propto L^{5/6}$) and
drain ($\propto L$). Expansion of the cross-sectional area suppresses draining
of the loop, while turbulence suppresses cooling by thermal conduction, so
these processes effectively lengthen the duration of light curves. Both the
area and the turbulence are vitally important, therefore, to correctly model
loop evolution and time series, but are currently poorly constrained
observationally.

  * •

All of the parameters of the heating determine the temperatures and strength
of evaporation. Cooling due to radiation and conduction both depend intimately
on the temperature and density, and thus the heating parameters indirectly
impact the cooling durations. Appendix A shows the relation of electron beam
parameters to SXR duration, for example.

It is clear that the light curves of hot and cold emission evolve differently.
There is a test of the cartoon description in Figure 6 in so-called “cold”
solar flares, where there is ample evidence for particle acceleration in HXR
or microwave emission emanating from the chromosphere, but the coronal
temperature and emission measure are not raised significantly (e.g. Fleishman
et al. 2011, 2016). A statistical study of 27 cold flares by Lysenko et al.
(2018) found that these events are typically shorter duration than the
reference group in both HXRs and microwaves. We have briefly examined the SXR
emission for these 27 events in Appendix A. We find that in cold flares, as
with “hot flares,” there is no relation between SXR flux and duration, but the
median duration ($\approx$ 90 s) is almost an order of magnitude shorter than
the median duration of hot events ($\approx$ 630 s). This suggests that there
is only a very limited radiative cooling phase in cold flares, and the
duration of the event corresponds roughly to the heating, but a more detailed
study is warranted.

In this work, we have only considered heating during the impulsive phase since
we were considering single loops. However, it is well known that there is
significant energy release well into the gradual phase of flares (Czaykowska
et al., 2001; Isobe et al., 2002; Qiu et al., 2013; Kuhar et al., 2017), which
would, of course, change the duration of emission. Additionally, we have only
considered heating due to an electron beam, known to deliver significant
amounts of heating to the chromosphere (Emslie et al., 2012), but by no means
the only mechanism of energy transport. Other potential mechanisms are less
well constrained, but likely include direct energy transport via thermal
conduction (e.g. López et al. 2022), wave generation of MHD waves (e.g. Emslie
& Sturrock 1982; Reep & Russell 2016), and acceleration of ion beams (e.g.
Canfield & Chang 1985; Kerr et al. 2023). Future work with a multi-threaded
model of flares should attempt to better constrain these mechanisms and their
direct role in energy transport over the course of an event.

In summary, the duration of solar and stellar flares depends intimately on the
details of the energy release and transport mechanisms. The parameters of both
the heating and cooling processes impact the duration of a flare. In this
paper, we have examined only flare heating mediated by an electron beam as HXR
emission constraints its parameters, but there is certainly energy transport
by ion beams and magnetohydrodynamic waves, which remains poorly constrained.
This is particularly important for the energy transport by larger electron
energy fluxes greater than 1012 erg cm-2 s-1, when electron beams become
unstable due to excitation of Langmuir and ion-acoustic waves (Drummond et
al., 1970; Zharkova & Siversky, 2011; Yao et al., 2022). Powerful stellar
superflares at energies over 1034 erg occur in larger active regions, with
stronger magnetic flux, and thus more efficiently accelerate particles,
including both electron and ion beams. As shown by Zharkova & Siversky (2011),
intense electron beams are expected to be damped in higher layers of the
chromosphere, causing excitation of plasma waves. Meanwhile, intense proton
beams are stable with respect to excitation of collective modes in plasma and
their mean free path is mostly driven by the collisional dissipation of their
energy, and thus can reach deeper, denser layers of the stellar chromosphere,
causing significant line broadening due to the Stark effect (Kowalski et al.,
2017). All of these factors play a role on the energy transport through the
lower atmosphere and the resultant impacts on X-ray spectra (Alaoui & Holman,
2017; Allred et al., 2020; Alaoui et al., 2021). In order to better understand
the impact of stellar flares on (exo)planets, more work is needed to build a
full thermodynamic model of flare irradiance across the X-ray, extreme, far,
and near ultraviolet bands to simulate the response of their magnetosphere,
upper and lower atmospheres.

JWR was supported by the Office of Naval Research 6.1 Support Program. VSA was
supported by the GSFC Sellers Exoplanet Environments Collaboration (SEEC),
which is funded by the NASA Planetary Science Division’s Internal Scientist
Funding Model (ISFM) and NASA NNH21ZDA001N-XRP F.3 Exoplanets Research Program
grants. We thank Dr. Konstantin Getman for helpful discussions. The authors
also thank the two anonymous referees for suggestions which have improved this
manuscript.

##  Appendix A Statistical Relations

We can check whether the SXR duration is related to the electron beam
parameters. Warmuth & Mann (2016) measured these parameters using spectral
inversion with RHESSI for a sample of 24 flares ranging from C to X-class. We
plot their measured values against the peak flux in both GOES/XRS channels as
well as the FWHM of both channels for each of these events. The scatter plots
in Figure 10 show this comparison. The plots also note the Pearson correlation
coefficient $r_{p}$ and Spearman coefficient $r_{s}$ for both XRS channels.
The lines show a linear regression in log-log space done with a non-parametric
Theil-Sen estimator, which is resistant to outliers. Their measured slopes are
also indicated. As noted by Warmuth & Mann (2016), there is a clear
correlation between the XRS-B flux and each of the electron beam parameters
(low-energy cut-off, spectral index, electron rate, and non-thermal power).
This holds in both XRS channels. Additionally, there is a weaker correlation
between the duration in each XRS channel with the low-energy cut-off, electron
rate, and non-thermal power, but perhaps not with the spectral index.

Figure 10: Scatter plots comparing the electron beam parameters in 24 flares
measured by Warmuth & Mann (2016) to the XRS peak flux and the XRS FWHM (in
each channel, XRS-A in blue, XRS-B in red). The Pearson $r_{p}$ (correlation)
and Spearman $r_{s}$ (monotonicity) coefficients are indicated. As reported by
those authors, the parameters are strongly correlated with the peak SXR flux.
Additionally, there may be a weaker correlation between the low-energy cut-
off, electron rate, and non-thermal power with the duration of the flare in
SXRs. Theil-Sen linear fits in log-log space have been overplotted and their
slopes labeled.

We have also examined the durations of so-called cold flares, where there is
noticeable emission from non-thermal bremsstrahlung (in either HXRs or radio),
but little increase in temperature or emission measure. In these events, we
expect that the SXR emission should be impulsive, responding to the heating
and not significantly to the more gradual evaporation. Lysenko et al. (2018)
presented a statistical study of 27 such events in depth. Using the event list
in that paper, in Figure 11 we have plotted the background-subtracted SXR flux
against SXR duration, finding little correlation (small Pearson $r_{p}$ and
small Spearman $r_{s}$). The Theil-Sen linear fits have a slope consistent
with 0 in both channels. Additionally, we show the distributions of FWHM in
each channel, with median values of 63 and 90 s in XRS-A and XRS-B,
respectively. The statistics are too limited to make any conclusions about the
distribution. Reep & Knizhnik (2019) showed similar distributions for around
3,000 “hot” flares observed with GOES-15, finding median values about 360 and
630 s in XRS-A and XRS-B. In both hot and cold flares, there is no relation
between duration and GOES class, but cold flares are of notably shorter
duration, indicating that the radiative cooling phase is negligible in cold
flares.

Figure 11: A scatter plot of peak flux and duration in each XRS channel for
the set of 27 cold flares from Lysenko et al. (2018). The correlation
coefficients are small, and the slopes are consistent with 0. The
distributions of durations are also shown for each XRS channel. The average
durations are about an order of magnitude shorter than for “hot” flares.

## References

  * Airapetian et al. (2017) Airapetian, V. S., Glocer, A., Khazanov, G. V., et al. 2017, ApJ, 836, L3
  * Airapetian et al. (2020) Airapetian, V. S., Barnes, R., Cohen, O., et al. 2020, International Journal of Astrobiology, 19, 136
  * Alaoui & Holman (2017) Alaoui, M., & Holman, G. D. 2017, ApJ, 851, 78
  * Alaoui et al. (2021) Alaoui, M., Holman, G. D., Allred, J. C., & Eufrasio, R. T. 2021, ApJ, 917, 74
  * Allred et al. (2020) Allred, J. C., Alaoui, M., Kowalski, A. F., & Kerr, G. S. 2020, ApJ, 902, 16
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481
  * Bethe & Heitler (1934) Bethe, H., & Heitler, W. 1934, Proceedings of the Royal Society of London Series A, 146, 83
  * Bian et al. (2016a) Bian, N. H., Kontar, E. P., & Emslie, A. G. 2016a, ApJ, 824, 78
  * Bian et al. (2016b) Bian, N. H., Watters, J. M., Kontar, E. P., & Emslie, A. G. 2016b, ApJ, 833, 76
  * Borucki et al. (2010) Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977
  * Bradshaw & Cargill (2005) Bradshaw, S. J., & Cargill, P. J. 2005, A&A, 437, 311
  * Bradshaw & Cargill (2010) —. 2010, ApJ, 717, 163
  * Bradshaw & Cargill (2013) —. 2013, ApJ, 770, 12
  * Bradshaw et al. (2019) Bradshaw, S. J., Emslie, A. G., Bian, N. H., & Kontar, E. P. 2019, ApJ, 880, 80
  * Bradshaw & Mason (2003) Bradshaw, S. J., & Mason, H. E. 2003, A&A, 401, 699
  * Braginskii (1965) Braginskii, S. I. 1965, Reviews of Plasma Physics, 1, 205
  * Brown & Mallik (2008) Brown, J. C., & Mallik, P. C. V. 2008, A&A, 481, 507
  * Caballero-García et al. (2015) Caballero-García, M. D., Šimon, V., Jelínek, M., et al. 2015, MNRAS, 452, 4195
  * Canfield & Chang (1985) Canfield, R. C., & Chang, C. R. 1985, ApJ, 295, 275
  * Cargill et al. (2022) Cargill, P. J., Bradshaw, S. J., Klimchuk, J. A., & Barnes, W. T. 2022, MNRAS, 509, 4420
  * Cargill et al. (1995) Cargill, P. J., Mariska, J. T., & Antiochos, S. K. 1995, ApJ, 439, 1034
  * Carlsson & Stein (2002) Carlsson, M., & Stein, R. F. 2002, ApJ, 572, 626
  * Cheng et al. (1981) Cheng, C. C., Tandberg-Hanssen, E., Bruner, E. C., et al. 1981, ApJ, 248, L39
  * Cheng et al. (2012) Cheng, J. X., Kerr, G., & Qiu, J. 2012, ApJ, 744, 48
  * Coffaro et al. (2020) Coffaro, M., Stelzer, B., Orlando, S., et al. 2020, A&A, 636, A49
  * Colombo et al. (2022) Colombo, S., Petralia, A., & Micela, G. 2022, A&A, 661, A148
  * Czaykowska et al. (2001) Czaykowska, A., Alexander, D., & De Pontieu, B. 2001, ApJ, 552, 849
  * Del Zanna et al. (2021) Del Zanna, G., Dere, K. P., Young, P. R., & Landi, E. 2021, ApJ, 909, 38
  * Dennis & Zarro (1993) Dennis, B. R., & Zarro, D. M. 1993, Sol. Phys., 146, 177
  * Dere et al. (1997) Dere, K. P., Landi, E., Mason, H. E., Monsignori Fossi, B. C., & Young, P. R. 1997, A&AS, 125, 149
  * Doschek & Warren (2016) Doschek, G. A., & Warren, H. P. 2016, ApJ, 825, 36
  * Drummond et al. (1970) Drummond, W. E., Malmberg, J. H., O’Neil, T. M., & Thompson, J. R. 1970, Physics of Fluids, 13, 2422
  * Elwert (1939) Elwert, G. 1939, Annalen der Physik, 426, 178
  * Emslie & Sturrock (1982) Emslie, A. G., & Sturrock, P. A. 1982, Sol. Phys., 80, 99
  * Emslie et al. (2012) Emslie, A. G., Dennis, B. R., Shih, A. Y., et al. 2012, ApJ, 759, 71
  * Fisher et al. (1985) Fisher, G. H., Canfield, R. C., & McClymont, A. N. 1985, ApJ, 289, 425
  * Fleishman et al. (2011) Fleishman, G. D., Kontar, E. P., Nita, G. M., & Gary, D. E. 2011, ApJ, 731, L19
  * Fleishman et al. (2016) Fleishman, G. D., Pal’shin, V. D., Meshalkina, N., et al. 2016, ApJ, 822, 71
  * Fuhrmeister et al. (2011) Fuhrmeister, B., Lalitha, S., Poppenhaeger, K., et al. 2011, A&A, 534, A133
  * Gendreau et al. (2016) Gendreau, K. C., Arzoumanian, Z., Adkins, P. W., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9905, Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, ed. J.-W. A. den Herder, T. Takahashi, & M. Bautz, 99051H
  * Getman et al. (2008) Getman, K. V., Feigelson, E. D., Broos, P. S., Micela, G., & Garmire, G. P. 2008, ApJ, 688, 418
  * Harra et al. (2016) Harra, L. K., Schrijver, C. J., Janvier, M., et al. 2016, Sol. Phys., 291, 1761
  * Hawley et al. (2003) Hawley, S. L., Allred, J. C., Johns-Krull, C. M., et al. 2003, ApJ, 597, 535
  * Hayes et al. (2019) Hayes, L. A., Gallagher, P. T., Dennis, B. R., et al. 2019, ApJ, 875, 33
  * Hayes et al. (2017) Hayes, L. A., Gallagher, P. T., McCauley, J., et al. 2017, Journal of Geophysical Research (Space Physics), 122, 9841
  * Hayes et al. (2020) Hayes, L. A., Inglis, A. R., Christe, S., Dennis, B., & Gallagher, P. T. 2020, ApJ, 895, 50
  * Hayes et al. (2021) Hayes, L. A., O’Hara, O. S. D., Murray, S. A., & Gallagher, P. T. 2021, Sol. Phys., 296, 157
  * Holman et al. (2011) Holman, G. D., Aschwanden, M. J., Aurass, H., et al. 2011, Space Sci. Rev., 159, 107
  * Isobe et al. (2002) Isobe, H., Yokoyama, T., Shimojo, M., et al. 2002, ApJ, 566, 528
  * Jakimiec et al. (1992) Jakimiec, J., Sylwester, B., Sylwester, J., et al. 1992, A&A, 253, 269
  * Jansen et al. (2001) Jansen, F., Lumb, D., Altieri, B., et al. 2001, A&A, 365, L1
  * Kerr (2022) Kerr, G. S. 2022, Frontiers in Astronomy and Space Sciences, 9, 1060856
  * Kerr (2023) —. 2023, Frontiers in Astronomy and Space Sciences, 9, 425
  * Kerr et al. (2023) Kerr, G. S., Allred, J. C., Kowalski, A. F., et al. 2023, ApJ, 945, 118
  * Koch & Motz (1959) Koch, H. W., & Motz, J. W. 1959, Reviews of Modern Physics, 31, 920
  * Kowalski et al. (2017) Kowalski, A. F., Allred, J. C., Uitenbroek, H., et al. 2017, ApJ, 837, 125
  * Krucker et al. (2011) Krucker, S., Hudson, H. S., Jeffrey, N. L. S., et al. 2011, ApJ, 739, 96
  * Krucker et al. (2015) Krucker, S., Saint-Hilaire, P., Hudson, H. S., et al. 2015, ApJ, 802, 19
  * Kuhar et al. (2017) Kuhar, M., Krucker, S., Hannah, I. G., et al. 2017, ApJ, 835, 6
  * Kundu et al. (2001) Kundu, M. R., Nindos, A., White, S. M., & Grechnev, V. V. 2001, ApJ, 557, 880
  * Lee et al. (2017) Lee, K.-S., Imada, S., Watanabe, K., Bamba, Y., & Brooks, D. H. 2017, ApJ, 836, 150
  * Lee et al. (1995) Lee, T. T., Petrosian, V., & McTiernan, J. M. 1995, ApJ, 448, 915
  * Li et al. (1993) Li, P., Emslie, A. G., & Mariska, J. T. 1993, ApJ, 417, 313
  * Liu et al. (2013) Liu, W.-J., Qiu, J., Longcope, D. W., & Caspi, A. 2013, ApJ, 770, 111
  * López et al. (2022) López, F. M., Giménez de Castro, C. G., Mandrini, C. H., et al. 2022, A&A, 657, A51
  * Lörinčík et al. (2022) Lörinčík, J., Polito, V., De Pontieu, B., Yu, S., & Freij, N. 2022, Frontiers in Astronomy and Space Sciences, 9, 1040945
  * Lysenko et al. (2018) Lysenko, A. L., Altyntsev, A. T., Meshalkina, N. S., Zhdanov, D., & Fleishman, G. D. 2018, ApJ, 856, 111
  * Maehara et al. (2015) Maehara, H., Shibayama, T., Notsu, Y., et al. 2015, Earth, Planets and Space, 67, 59
  * Maehara et al. (2012) Maehara, H., Shibayama, T., Notsu, S., et al. 2012, Nature, 485, 478
  * Martínez Oliveros et al. (2012) Martínez Oliveros, J.-C., Hudson, H. S., Hurford, G. J., et al. 2012, ApJ, 753, L26
  * Mondal et al. (2021) Mondal, B., Sarkar, A., Vadawale, S. V., et al. 2021, ApJ, 920, 4
  * Nagai & Emslie (1984) Nagai, F., & Emslie, A. G. 1984, ApJ, 279, 896
  * Nama et al. (2023) Nama, L., Mondal, B., Narendranath, S., & Paul, K. T. 2023, Sol. Phys., 298, 55
  * Namekata et al. (2017) Namekata, K., Sakaue, T., Watanabe, K., et al. 2017, ApJ, 851, 91
  * Neupert (1968) Neupert, W. M. 1968, ApJ, 153, L59
  * Osten et al. (2004) Osten, R. A., Brown, A., Ayres, T. R., et al. 2004, ApJS, 153, 317
  * Pereira & Uitenbroek (2015) Pereira, T. M. D., & Uitenbroek, H. 2015, A&A, 574, A3
  * Pietras et al. (2022) Pietras, M., Falewicz, R., Siarkowski, M., Bicz, K., & Preś, P. 2022, ApJ, 935, 143
  * Poland et al. (1984) Poland, A. I., Orwig, L. E., Mariska, J. T., Auer, L. H., & Nakatsuka, R. 1984, ApJ, 280, 457
  * Qiu (2021) Qiu, J. 2021, ApJ, 909, 99
  * Qiu et al. (2013) Qiu, J., Sturrock, Z., Longcope, D. W., Klimchuk, J. A., & Liu, W.-J. 2013, ApJ, 774, 14
  * Reep et al. (2015) Reep, J. W., Bradshaw, S. J., & Alexander, D. 2015, ApJ, 808, 177
  * Reep et al. (2019) Reep, J. W., Bradshaw, S. J., Crump, N. A., & Warren, H. P. 2019, ApJ, 871, 18
  * Reep & Brown (2016) Reep, J. W., & Brown, J. C. 2016, ApJ, 824, 90
  * Reep & Knizhnik (2019) Reep, J. W., & Knizhnik, K. J. 2019, ApJ, 874, 157
  * Reep & Russell (2016) Reep, J. W., & Russell, A. J. B. 2016, ApJ, 818, L20
  * Reep & Toriumi (2017) Reep, J. W., & Toriumi, S. 2017, ApJ, 851, 4
  * Reep et al. (2022) Reep, J. W., Ugarte-Urra, I., Warren, H. P., & Barnes, W. T. 2022, ApJ, 933, 106
  * Ricker et al. (2014) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9143, Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave, ed. J. Oschmann, Jacobus M., M. Clampin, G. G. Fazio, & H. A. MacEwen, 914320
  * Rubio da Costa et al. (2016) Rubio da Costa, F., Kleint, L., Petrosian, V., Liu, W., & Allred, J. C. 2016, ApJ, 827, 38
  * Santos et al. (2004) Santos, N. C., Israelian, G., & Mayor, M. 2004, A&A, 415, 1153
  * Scott et al. (2022) Scott, R. B., Bradshaw, S. J., & Linton, M. G. 2022, ApJ, 933, 72
  * Serio et al. (1991) Serio, S., Reale, F., Jakimiec, J., Sylwester, B., & Sylwester, J. 1991, A&A, 241, 197
  * Simões et al. (2015a) Simões, P. J. A., Graham, D. R., & Fletcher, L. 2015a, A&A, 577, A68
  * Simões et al. (2015b) Simões, P. J. A., Hudson, H. S., & Fletcher, L. 2015b, Sol. Phys., 290, 3625
  * Siskind et al. (2022) Siskind, D. E., Jones, M., Reep, J. W., et al. 2022, Space Weather, 20, e2021SW003012
  * Spitzer (1962) Spitzer, L. 1962, Physics of Fully Ionized Gases
  * Spitzer & Härm (1953) Spitzer, L., & Härm, R. 1953, Physical Review, 89, 977
  * Sutherland & Dopita (1993) Sutherland, R. S., & Dopita, M. A. 1993, ApJS, 88, 253
  * Sylwester et al. (2023) Sylwester, B., Sylwester, J., Phillips, K. J. H., & Kepa, A. 2023, ApJ, 946, 49
  * Sylwester et al. (1998) Sylwester, J., Lemen, J. R., Bentley, R. D., Fludra, A., & Zolcinski, M. C. 1998, ApJ, 501, 397
  * Testa (2010) Testa, P. 2010, Space Sci. Rev., 157, 37
  * Toriumi et al. (2017) Toriumi, S., Schrijver, C. J., Harra, L. K., Hudson, H., & Nagashima, K. 2017, ApJ, 834, 56
  * Veronig et al. (2002) Veronig, A., Vršnak, B., Dennis, B. R., et al. 2002, A&A, 392, 699
  * Warmuth & Mann (2016) Warmuth, A., & Mann, G. 2016, A&A, 588, A115
  * Warren (2014) Warren, H. P. 2014, ApJ, 786, L2
  * Warren et al. (2016) Warren, H. P., Reep, J. W., Crump, N. A., & Simões, P. J. A. 2016, ApJ, 829, 35
  * Washinoue & Suzuki (2023) Washinoue, H., & Suzuki, T. K. 2023, The Astrophysical Journal, 953, 74.  https://dx.doi.org/10.3847/1538-4357/ace106
  * Watanabe et al. (2017) Watanabe, K., Kitagawa, J., & Masuda, S. 2017, ApJ, 850, 204
  * Woodgate et al. (1983) Woodgate, B. E., Shine, R. A., Poland, A. I., & Orwig, L. E. 1983, ApJ, 265, 530
  * Xia et al. (2021) Xia, F., Su, Y., Wang, W., et al. 2021, ApJ, 908, 111
  * Yao et al. (2022) Yao, X., Muñoz, P. A., Büchner, J., et al. 2022, ApJ, 933, 219
  * Zeng et al. (2014) Zeng, Z., Qiu, J., Cao, W., & Judge, P. G. 2014, ApJ, 793, 87
  * Zharkova & Siversky (2011) Zharkova, V. V., & Siversky, T. V. 2011, ApJ, 733, 33
  * Zhu et al. (2018) Zhu, C., Qiu, J., & Longcope, D. W. 2018, ApJ, 856, 27
  * Zimovets et al. (2021) Zimovets, I. V., McLaughlin, J. A., Srivastava, A. K., et al. 2021, Space Sci. Rev., 217, 66