# Observational Analysis of Lyman-alpha Emission in Equivalent Magnitude Solar
Flares

Harry J. Greatorex Astrophysics Research Centre, School of Mathematics and
Physics, Queen’s University Belfast, University Road, Belfast, BT7 1NN, UK
Ryan O. Milligan Astrophysics Research Centre, School of Mathematics and
Physics, Queen’s University Belfast, University Road, Belfast, BT7 1NN, UK
Phillip C. Chamberlin Laboratory for Atmospheric and Space Physics, University
of Colorado Boulder, Boulder, CO, 80309, USA

###### Abstract

The chromospheric Lyman-alpha line of neutral hydrogen (Ly$\rm\alpha$; 1216Å)
is the most intense emission line in the solar spectrum, yet until recently
observations of flare-related Ly$\rm\alpha$ emission have been scarce. Here,
we examine the relationship between nonthermal electrons accelerated during
the impulsive phase of three M3 flares that were co-observed by RHESSI, GOES,
and SDO, and the corresponding response of the chromosphere in Ly$\rm\alpha$.
Despite having identical X-ray magnitudes, these flares show significantly
different Ly$\rm\alpha$ responses. The peak Ly$\rm\alpha$ enhancements above
quiescent background for these flares were 1.5%, 3.3%, and 6.4%. However, the
predicted Ly$\rm\alpha$ enhancements from FISM2 were consistently $<$2.5%. By
comparing the properties of the nonthermal electrons derived from spectral
analysis of hard X-ray observations, flares with a ‘harder’ spectral index
were found to produce a greater Ly$\rm\alpha$ enhancement. The percentage of
nonthermal energy radiated by the Ly$\rm\alpha$ line during the impulsive
phase was found to range from 2.0–7.9%. Comparatively, the radiative losses in
He II (304Å) were found to range from 0.6–1.4% of the nonthermal energy while
displaying enhancements above the background of 7.3–10.8%. FISM2 was also
found to underestimate the level of He II emission in two out of the three
flares. These results may have implications for space weather studies and
modelling the response of the terrestrial atmosphere to changes in the solar
irradiance, and will guide the interpretation of flare-related Ly$\rm\alpha$
observations that will become available during Solar Cycle 25.

##  1 Introduction

During solar flares, a significant amount of magnetic free energy is released
following the reconnection of opposing polarity magnetic fields. In the
generally accepted standard model (CSHKP; Carmichael 1964; Sturrock 1966;
Hirayama 1974; Kopp & Pneuman 1976), liberated energy drives the acceleration
of nonthermal electrons along newly restructured field lines towards the
chromospheric footpoints of the flare loops. The majority of the energy
deposited in the chromosphere during a solar flare is believed to be radiated
by optical and extreme ultraviolet (EUV) emission (Neidig, 1989; Hudson et
al., 2006, 1992; Kleint et al., 2016). Measurements of the Total Solar
Irradiance during solar flares by Woods et al. (2006) suggest that
$\rm\sim$70% of the total flare energy resides in wavelengths $>$270Å, in good
agreement with results from a superposed epoch analysis of solar flares
conducted by Kretzschmar et al. (2010) and Kretzschmar (2011). Determining the
radiated energy budget of solar flares, and how this energy is distributed
across the spectrum, is an important step in constraining solar flare heating
models (e.g. Allred et al. 2005).

In a multi-wavelength study of an X2.2 flare, Milligan et al. (2014) were able
to account for $\sim 15\%$ of the total nonthermal energy deposited in the
chromosphere through observations of various EUV/UV and optical lines and
continua, leaving 85% of the remaining energy unaccounted for. Of the
available observations for this event, the Ly$\rm\alpha$ line of neutral
hydrogen (Ly$\rm\alpha$; 1216Å) as measured by the Extreme Ultraviolet
Variability Experiment (EVE; Woods et al. 2012) onboard the Solar Dynamics
Observatory (SDO; Pesnell et al. 2012) was found to dominate the radiative
losses in the chromosphere, amounting to 5–8% of the total nonthermal energy
as determined from hard X-ray (HXR) observations.

Ly$\rm\alpha$ is the brightest emission line in the solar spectrum (Curdt et
al., 2001) and during solar flares Ly$\rm\alpha$ emission is assumed to
originate from the upper chromosphere/flare footpoints due to the relatively
high abundance of hydrogen in these regions (Chamberlin et al., 2018).
Enhancements in Ly$\rm\alpha$ irradiance during solar flares are generally
marginal due to the intense background at this wavelength. Kretzschmar et al.
(2012) reported a 0.6% enhancement in Ly$\rm\alpha$ above the background
during an M2 flare measured with the Large-Yield Radiometer onboard the
Projects for Onboard Autonomy mission (PROBA2/LYRA; Dominique et al. 2013;
Santandrea et al. 2013), and this was found to be comparable to He II (304Å)
emission. This is consistent with the $<$1% enhancements reported by Raulin et
al. (2013) also using PROBA2/LYRA. By comparison, Woods et al. (2004) found a
peak enhancement of 20% in the Ly$\rm\alpha$ line core and a factor of two
increase in the wings for an X17 flare observed by Solar-Stellar Irradiance
Comparison Experiment onboard the Solar Radiation and Climate Experiment
(SORCE/SOLSTICE; McClintock et al. 2005), whereas Brekke et al. (1996) found a
6% enhancement in Ly$\rm\alpha$ for an X3 flare measured by the SOLSTICE
instrument onboard the Upper Atmosphere Research Satellite (UARS/SOLSTICE;
Rottman et al. 1993; Woods et al. 1993).

In a statistical study of 477 M- and X-class flares observed in the E-channel
of the Extreme Ultraviolet Sensor onboard the Geostationary Operational
Environmental Satellite (GOES/EUVS-E; Viereck et al. 2007; Evans et al. 2010),
Milligan et al. (2020) found that typical increases in Ly$\rm\alpha$ were
$<10\%$, with a maximum of approximately $30\%$. A follow-up study of several
thousand flares using a superposed epoch analysis, revealed average increases
of 0.18–0.35% for B- and C- class flares, and 1–4% for M- and X-class
(Milligan, 2021).

The study of flare-induced changes in Ly$\rm\alpha$ irradiance is also
important for space weather research. Ly$\rm\alpha$ is known to form and
maintain the dayside D-region of Earth’s ionosphere ($\rm\sim$60–90 km)
through the photoionization of Nitric Oxide (NO; Chubb et al. 1957), thus any
fluctuations in solar Ly$\rm\alpha$ emission may have an impact on the
dynamics of the terrestrial atmosphere. Despite this, observational studies of
flare-induced Ly$\rm\alpha$ emission have been relatively scarce due to
instrumental limitations, such as reduced duty cycles and insufficient
sensitivity or cadence to capture changes in Ly$\rm\alpha$ irradiance on flare
timescales (Milligan, 2015). Of the studies into the terrestrial effects of
flare radiation, a number have cited soft X-rays (SXRs) as the dominant driver
of compositional changes in the ionosphere (McRae & Thomson, 2004; Kumar &
Kumar, 2018; Nina et al., 2018; Hayes et al., 2021). Raulin et al. (2013)
investigated the response of the D-region of the ionosphere to seven medium-
sized flares, three of which had detectable Ly$\rm\alpha$ irradiance increases
as observed by PROBA2/LYRA. Their analysis found that the measured Very Low
Frequency (VLF) phase shift resulting from each flare in the sample showed no
dependence on Ly$\rm\alpha$ radiation and concluded that the response of the
ionosphere to Ly$\rm\alpha$ was negligible compared to that produced by the
SXRs.

Contrary to this, Milligan et al. (2020) investigated the ionospheric response
to an X-class flare through measurement of both VLF signal amplitude and
atmospheric conductivity taken from magnetometer data. Comparing these to the
respective Ly$\rm\alpha$ and SXR profiles from EUVS-E and the X-Ray Sensor
(XRS; Hanser & Sellers 1996) they noted an impulsive geomagnetic disturbance
in the Y-component of the magnetic field in the magnetometer data that was
temporally correlated with the peak of the Ly$\rm\alpha$ irradiance and that
peaked a few minutes before the SXR irradiance. This implied that the incident
Ly$\rm\alpha$ produced enhanced current systems in the ionosphere, localised
in the E-region through increased ionization of NO.

Unlike H I Ly$\rm\alpha$ measurements, the Ly$\rm\alpha$ line of Helium II (He
II; 304Å) has been comparatively well observed, with routine images taken by
the Atmospheric Imaging Assembly onboard SDO (SDO/AIA; Boerner et al. 2012),
the Extreme Ultraviolet Imager onboard the Solar Terrestrial Relations
Observatory - Ahead (STEREO-A/EUVI; Wuelser et al. 2004), the Extreme
Ultraviolet Imaging Telescope onboard the Solar and Heliospheric Observatory
(SOHO/EIT; Domingo et al. 1995; Delaboudinière et al. 1995), and others. H I
Ly$\rm\alpha$ and He II Ly$\rm\alpha$ are both formed in similar regions in
the lower solar atmosphere during quiescent conditions and both are therefore
sensitive to changing dynamics in the solar atmosphere due to solar flares
(Simões et al., 2016; Brown et al., 2018). Consequently, given their greater
availability, He II observations have been used as a proxy for H I
Ly$\rm\alpha$ (Auchère 2005). Gordino et al. (2022) examined the relationship
between He II and Ly$\rm\alpha$ for a range of solar phenomena such as
filament eruptions, prominences, and active regions, but did not include
flare-related emission in their study.

In this paper we present an multi-wavelength analysis of three solar flares of
identical GOES class, but with significantly different Ly$\rm\alpha$ and He II
responses. HXR spectroscopy was used to determine whether differing nonthermal
electron distributions are responsible for driving the different responses in
EUV irradiance, and what fraction of the nonthermal energy is radiated by
these two fundamental emission lines. A comparison was also made between the
irradiance observations and semi-empirical model predictions from the Flare
Irradiance Spectral Model (FISM2; Chamberlin et al. 2020). Section 2 outlines
the multi-instrument observations taken for each flare in the sample and
details the applied analysis. Section 3 details the results of this analysis,
and Section 4 summarises the findings from this study and discusses their
broader implications.

Figure 1: Images taken from SDO/AIA for each flare listed in Table 1 in the 94Å (top row) and 304Å (bottom row) filters approximately centred around their Ly$\rm\alpha$ peaks. Each column represents a single flare and the timestamp of each image is shown in the title above each frame. Cyan boxes in each image indicate the flaring region. Table 1: Summary of the observational characteristics of the flare sample. Solar Object Locator | GOES Class | Heliographic Position | GOES start time | GOES peak time | GOES end time
---|---|---|---|---|---
|  |  | (UT) | (UT) | (UT)
SOL2010-10-16T19:07:00 | M2.9 | S19W29 | 19:07:00 | 19:12:30 | 19:15:00
SOL2011-09-24T19:09:00 | M3.0 | N13E45 | 19:09:00 | 19:21:20 | 19:41:00
SOL2014-02-01T07:14:00 | M3.0 | S14E17 | 07:14:00 | 07:23:47 | 07:36:00
Figure 2: Top Row: HXR lightcurves from RHESSI in discrete energy bands. The
A0 and A1 flags (horizontal magenta lines) denote the RHESSI attenuator
states. The A0 state corresponds to both thick and thin aluminum attenuators
out of the detector field of view, the A1 state corresponds to thin in and
thick out. The N flag (horizontal cyan line) denotes RHESSI night, where the
spacecraft field of view is occulted by Earth. The SAA flag (horizontal orange
line) denotes instrument switch off due to crossing of the South Atlantic
Anomoly. SXR lightcurves from GOES/XRS in the 1–8Å wavelength range (black
solid). Bottom Row: Observed Ly$\rm\alpha$ (red) and He II (blue) irradiance
from GOES/EUVS-E and SDO/EVE, respectively. The vertical dashed line in each
panel represents the peak of the 25–50 keV band for each flare.

##  2 Observation & Analysis

This study focuses on the analysis of data from three $\rm\sim$M3 flares that
occurred during Solar Cycle 24. Images of each flare in 94Å and 304Å emission
taken by SDO/AIA are shown in the top and bottom rows of Figure 1,
respectively. The times and locations of each flare are summarized in Table 1.
These flares were selected based on the following criteria:

  * •

The flares are of equivalent GOES class to remove any potential dependence of
EUV irradiance variability on flare magnitude.

  * •

Each flare occurred on-disk, reducing any opacity or foreshortening effects
that may impact the measurement of optically-thick emission (center-to-limb
variation, CLV; Milligan et al. 2020).

  * •

The impulsive phase of each flare must have been jointly observed by Reuven
Ramaty High-Energy Solar Spectroscopic Imager (RHESSI; Lin et al. 2002),
GOES-15/EUVS-E, and SDO/EVE MEGS-A to determine the nonthermal electron
distribution assumed to be responsible for driving the associated increases in
Ly$\rm\alpha$ and He II irradiance.

  * •

The flares must have occurred at times when the observations from
GOES-15/EUVS-E were unaffected by geocoronal absorption, which may attenuate
Ly$\rm\alpha$ emission (see Meier & Prinz 1970; Baliukin et al. 2019).

The top row of Figure 2 shows the SXR (black) and HXR (colored) lightcurves
for each flare from GOES-15/XRS and RHESSI, respectively, while the bottom row
shows the full-disk irradiance lightcurves in Ly$\rm\alpha$ (red curve) and He
II (blue curve), from GOES-15/EUVS-E and SDO/EVE, respectively across the
full-flare duration. The vertical dashed line in each panel represents the
peak of the 25–50 keV emission observed by RHESSI.

###  2.1 GOES/XRS & EUVS

The GOES series of spacecraft operated by the National Oceanic and Atmospheric
Administration (NOAA) have offered continuous observations of solar emissions
since the launch of GOES-1 in 1975. For this study, GOES-15 provided
observations of SXR and Ly$\rm\alpha$ irradiance for each flare from the XRS
and EUVS instruments, respectively. The XRS instrument consists of two
channels observing in the 0.5–4Å and 1–8Å wavelength ranges, respectively, at
2s cadence. The peak flux of the 1-minute averaged SXR observations in the
1–8Å wavelength range are the standard classification system for solar flare
magnitudes (e.g. a peak flux of $3.0\times 10^{-5}$ Wm-2 corresponds to an
M3.0 flare). As of 2020 October 28, GOES-8–15 SXR data accessed using the GOES
object in the SolarSoftWare system (SSWIDL; Freeland & Handy 1998) are
returned as “true flux” values, meaning a scaling factor must be applied to
these data to return the values used for determining flare class. A scaling
factor of 0.7 has been applied to the SXR lightcurves presented in the top row
of Figure 2 to preserve the GOES class of each flare.

The EUVS instrument is comprised of five channels (A–E) spanning the 50–170Å,
240–340Å, 200–620Å, 200–800Å, and 1180–1250Å wavelength ranges. The E-channel
(EUVS-E) is a dedicated channel centred around the Ly$\rm\alpha$ line at
1216Å, taking broadband, full-disk irradiance measurements of the Sun at a
cadence of 10.24s. The daily average values in Ly$\rm\alpha$ are scaled to
those taken by SORCE/SOLSTICE to account for degradation in the EUVS-E
instrument over time. The observed Ly$\rm\alpha$ emission for each flare is
given by the red line in the bottom row of Figure 2. The background
Ly$\rm\alpha$ flux for each flare was determined by calculating the mean flux
over a period of approximately 20 minutes prior to the flare onset. The data
were then normalized to the background flux in order to quantify the
enhancement in Ly$\rm\alpha$ irradiance attributed to the flare.

In order to calculate the energy radiated in Ly$\rm\alpha$ and SXRs, the
irradiance data measured at Earth ($\rm I_{Earth}$) in units of W m-2 was
converted to power radiated at the Sun (PSun) in units of erg s-1 using the
expression:

$P_{Sun}\ =\ 2\pi R^{2}\times 10^{7}~{}I_{Earth}\ \rm erg~{}s^{-1}$ (1)

where $R$ is the Sun-Earth distance corrected for deviations in Earth’s orbit
around the Sun at the time of each flare using the SSWIDL routine get_sun.pro.
The value $10^{7}$ is a conversion factor from J to erg. The total energy
radiated during the impulsive phase was found by integrating the power from
GOES start $\rightarrow$ GOES peak, allowing for a like-for-like comparison
between the energy radiated in Ly$\rm\alpha$ and the energy deposited into the
chrompshere by nonthermal electrons (see Section 2.4). This integration window
was also extended to the full-flare period (GOES start $\rightarrow$ GOES end)
to quantify the additional energy radiated after the initial injection of
energy by the electrons, which may reveal evidence of further contributions to
the overall emission in the latter stages of the flare. It should be noted
that the GOES start time is defined as the first of four consecutive 1-minute
intervals in which the 1–8Å flux is monotonically increasing, with the flux in
the fourth minute being 1.4 times the initial flux. The GOES end time is
defined as the time at which the SXR flux reaches half of the peak
value.111https://www.ngdc.noaa.gov/stp/solar/solarflares.html The uncertainty
in the total energy of the radiated components was calculated as the standard
deviation in the flux of the background period.

###  2.2 SDO/EVE

The EVE instrument onboard SDO acquires full-disk EUV spectra at a cadence of
10s through the A and B components comprising the Multiple EUV Grazing
Spectrographs (MEGS). The MEGS-A spectral range is 60–370Å covering the 304Å
He II line and MEGS-B spans 370–1060Å. From the launch of SDO until 2014 May
26 MEGS-A had a near 100% duty cycle, after which a power anomaly led to an
instrumentation failure. The He II 304Å line was the strongest emission line
measured by MEGS-A and is the brightest emission line in the solar irradiance
spectrum shortward of 1000Å (Simões et al., 2016). EVE data for each flare was
acquired from the Laboratory for Atmospheric and Space Physics
(LASP222https://lasp.colorado.edu/eve/data_access/) database. These data are
available in hour-long Level 2 FITS files with emission line irradiances at 10
s cadence (EVL data). The 304Å line irradiances are derived from Level 2
spectral data integrated over the line centre ($\rm\Delta$Å=2.5). These data
were background subtracted, normalized, and converted using Equation 1 using
the same process as the GOES/EUVS-E data. The observed He II emission for each
flare is given by the blue line in the bottom row of Figure 2.

Figure 3: RHESSI spectra from the HXR peak of each flare with individual fit
components overlaid. Each plot shows the model fits to the count data at the
peak of the impulsive phase, taken from detector 6. The normalized residuals
are shown in the lower panels. The dotted lines indicate the energy range over
which the fits were performed.

###  2.3 FISM2

Prior to the launch of SDO/EVE, synthetic EUV spectra could be generated using
FISM (Chamberlin et al., 2007, 2008), which is a semi-empirical model of solar
UV emission in the 0–1900Å wavelength range, designed to fill temporal and
spectral gaps in observations of solar flare irradiance. These proxy spectra
could serve as inputs into planetary atmospheric models such as Whole
Atmosphere Community Climate Model with thermosphere and ionosphere eXtension
(WACCM-X; Garcia et al. 2007; Marsh et al. 2013; Neale et al. 2013; Qian et
al. 2018; Solomon et al. 2018), which models fluctuations in electron density,
vertical ion drift, zonal electric field, and conductance in the ionosphere
due to solar activity. Following the launch of SDO, irradiance profiles
generated by FISM2 are based on data acquired by SDO/EVE and by theX-ray
Photometer System (XPS) and SOLSTICE instruments onboard SORCE. The data from
FISM2 are provided in 1Å spectral bins and flare products are produced with a
60s cadence. For this study, FISM2 was employed to provide predicted
irradiance profiles in both Ly$\rm\alpha$ and He II for each flare, which
could then be directly compared to the measured profiles from GOES/EUVS-E and
SDO/EVE. Irradiance profiles were acquired for each of the three flares from
the LASP Interactive Solar Irradiance Datacentre
(LISRD333https://lasp.colorado.edu/lisird/). FISM2 data from 1180–1250Å was
scaled to the instrument response data for EUVS-E and integrated to mimic the
broadband Ly$\rm\alpha$ observations from GOES-15. For the He II profiles,
FISM2 data was summed over the bins surrounding the line centre to resemble
the resolution of the irradiance data from SDO/EVE. These were then background
subtracted and normalized using identical processes and periods to their
respective observations.

###  2.4 RHESSI

Nonthermal electrons incident on the chromosphere during solar flares drive
the emission of HXR bremsstrahlung through interaction with the dense plasma
at the flare footpoints (Dennis & Schwartz, 1989). RHESSI provides high-
resolution imaging and spectroscopic measurements of HXR emission in flares
that can be used to discern the properties of the nonthermal electrons assumed
to be responsible for driving impulsive emission at longer wavelengths. HXR
lightcurves for the three flares in this study are given by the colored lines
in the top row of Figure 2, where each line represents each of the discrete
energy bands ranging from 6–300 keV. Assuming the HXR emissions are driven by
thick-target interactions, the electron energy spectrum can be derived from
the count spectra under the assumptions of the Collisional Thick-Target Model
(CTTM; Brown 1971, 1972, 1973).

The total power in the electron distribution is given by:

$P_{nth}(E\geq E_{c})\ =\ \int\limits_{E_{c}}^{\infty}EF(E)\ dE\ \rm erg\
s^{-1}$ (2)

where $E_{c}$ is the low energy cutoff and the electron energy flux
distribution, F(E), can be written in power-law form $AE^{-\delta}$. The
coefficient, $\rm A$, corresponds to the electron number flux propagating
toward the chromosphere and $\rm\delta$ represents the spectral index of the
distribution. The measured value of $\rm E_{c}$ is taken as an upper limit to
the nonthermal cutoff due to the dominance of thermal emission at low
energies. Subsequently, nonthermal electron energy calculated through the HXR
spectral fitting is considered to be a lower limit to the total energy (Holman
et al. 2003; Ireland et al. 2013). With these, Equation 2 then becomes:

$P_{nth}(E\geq E_{c})\ =\ \frac{\kappa_{E}A}{(\delta-2)}E_{c}^{(2-\delta)}\
\rm erg\ s^{-1}$ (3)

where $\rm\kappa{{}_{E}}$ is a conversion factor from keV to erg. For each
flare, RHESSI count spectra were compiled for seven detectors (omitting
detectors 2 and 7 due to their comparatively reduced sensitivity) using native
energy binning from 3–300 keV.

Figure 4: Normalized irradiance profiles for each of the three flares. Red and blue curves correspond to Ly$\rm\alpha$ and He II data, respectively. Solid lines correspond to observational data from GOES/EUVS-E and SDO/EVE while dashed lines correspond to the predicted irradiance profiles from FISM2. Overplotted are vertical lines corresponding to the peaks of the 25–50 keV emission from RHESSI (dashed black) and the 1–8Å emission from GOES/XRS (dotted black). Table 2: Peak irradiance enhancements in Ly$\rm\alpha$ and He II for each flare, both as a percentage normalized to the background flux and as an absolute background-subtracted value.  | Peak Ly$\rm\alpha$ | Peak He II | Peak Ly$\rm\alpha$ | Peak He II
---|---|---|---|---
Flare | Enhancement | Enhancement | Irradiance ($\rm W$ $\rm m^{-2})$ | Irradiance ($\rm W$ $\rm m^{-2})$
| GOES/EUVS-E | FISM2 | SDO/EVE | FISM2 | GOES/EUVS-E | SDO/EVE
SOL2010–10–16 | $6.4\%$ | $2.2\%$ | $10.8\%$ | $10.7\%$ | $4.4\times 10^{-4}$ | $5.1\times 10^{-5}$
SOL2011–09–24 | $3.3\%$ | $0.7\%$ | $7.4\%$ | $0.4\%$ | $2.4\times 10^{-4}$ | $4.3\times 10^{-5}$
SOL2014–02–01 | $1.5\%$ | $0.8\%$ | $7.3\%$ | $3.3\%$ | $1.3\times 10^{-4}$ | $4.3\times 10^{-5}$

The count spectra for each flare were binned into 12s intervals to match the
EUVS-E cadence to the nearest integer number of RHESSI rotations. Each
spectrum was then fit with a combination of an isothermal component, a
Gaussian, and a nonthermal CTTM electron spectrum. An additional component was
also included to account for pulse pile-up using the most up-to-date version
available in OSPEX. The spectra were also corrected for albedo attributed to
Compton backscattered photons within the OSPEX software (Kontar et al. 2006).
Spectra from the HXR peak of each flare are shown in Figure 3 along with their
fit components. For each interval, the power in the nonthermal electrons was
calculated from Equation 3, where the low energy cutoff, spectral index, and
electron rate at each time interval are defined by the nonthermal component
fitted to the spectra. These were then integrated in time to give the total
nonthermal energy in each flare. The nonthermal power over the flare duration
was calculated for each detector from the parameters at each interval. This
was then integrated to return the total nonthermal energy for each detector.
The total nonthermal energy in each flare was taken as the detector-averaged
energy $\rm\pm$ the standard deviation across the detectors.

##  3 Results

###  3.1 Irradiance Variability

The full-flare irradiance profiles for Ly$\rm\alpha$ (red) and He II (blue),
normalized to their respective backgrounds, are shown in Figure 4. Solid
curves denote the observed profiles while dashed lines denote the profiles
produced by FISM2. From the GOES-15/EUVS-E data, Ly$\rm\alpha$ enhancements of
6.4%, 3.3%, and 1.5% were found for SOL2010–10–16, SOL2011–09–24, and
SOL2014–02–01, respectively. The corresponding He II enhancements from SDO/EVE
appear significantly larger than the Ly$\rm\alpha$ across all three flares
with values of 10.8%, 7.4%, and 7.3%, respectively. This is likely due to the
much higher background flux in Ly$\rm\alpha$. SOL2010–10–16 was found to have
both the largest Ly$\rm\alpha$ and He II enhancements. SOL2011–09–24 and
SOL2014–02–01 had significantly different Ly$\rm\alpha$ enhancements yet their
He II enhancements were remarkably similar, differing by only 0.1%. For all
three flares, the peaks in their Ly$\rm\alpha$ emission coincided with the
peaks of the HXR emission as measured by RHESSI (dashed vertical lines in
Figure 4), suggesting that these impulsive enhancements were driven by
nonthermal electron incidence. This is in agreement with a number of previous
studies that report a temporally correlated relationship between Ly$\rm\alpha$
and nonthermal emission (Nusinov et al. 2006; da Costa et al. 2009; Milligan
et al. 2017; Dominique et al. 2018; Lu et al. 2021; Li et al. 2022). However,
each flare also exhibits a prolonged decay time before returning to background
levels. The peak Ly$\rm\alpha$ and He II enhancements for each flare are
summarised in Table 2.

Figure 4 also shows that the peak flare enhancements predicted by FISM2
(dashed red and blue lines) are consistently unrepresentative of the observed
values. While FISM2 also predicts some variation in peak Ly$\rm\alpha$
enhancement between flares, these enhancements are $<$2.5% in all cases. The
range of He II peak values from FISM2 is much broader than the observations. A
good estimate was made for SOL2010–10–16, however the irradiances of
SOL2011–09–24 and SOL2014–02–01 were significantly under-predicted. The values
derived from FISM2 are also presented in Table 2.

Several factors may contribute to the disagreement between FISM2 and
observational data. For one, FISM2 flare models rely on the time derivative of
GOES/XRS observations as a proxy for the impulsive phase of each flare. This
means that the impulsivity of each flare will influence the FISM2 estimations
of the associated emission. SOL2010–10–16 clearly demonstrates a more
impulsive profile compared to SOL2011–09–24 and SOL2014–02–01, therefore while
their SXR magnitudes are equivalent, the magnitude of their SXR derivatives is
varied. Differing associated background levels further drive variation in the
SXR derivative magnitude. Given the flares presented here are of an
intermediate class, their associated background level will have a more
significant impact on the calculated SXR derivative compared to larger
(X-class) flares. Moreover, the observing statistics used in the FISM2 data
sets are biased towards smaller flares due to their higher occurrence rates.
Consequently, the lack of data for larger flares with good signal-to-noise
measurements leads to lower estimations in the associated flux.

When using SXRs as a proxy for optically-thick emission, a correction must be
applied to account for CLV effects dependent on flare location. The method for
correcting for CLV in FISM2 is consistent with that of the first version of
FISM with an increased data set. The full set of algorithms and function
parameters used for computing a CLV correction, as well as a detailed
presentation of the method, are presented in Chamberlin et al. (2008). As
FISM2 is an empirical model, its accuracy is limited by the observations on
which it depends. The high resolution spectral measurements
($\rm\lambda<0.1~{}nm$) used for FISM2 are taken from the SORCE/SOLSTICE
scanning spectrograph, which has missed a large number of flares.
Consequently, while there are a statistically sufficient number of
observations of the gradual phase from SORCE/SOLSTICE to compute a CLV
correction in the 115–190 nm range, a lack of observations means the same
could not be done for the impulsive phase in this range. Ultimately, this will
impact the Ly$\rm\alpha$ irradiance profiles from FISM2. However, given the
constraints on flare position discussed in Section 2, the flares in this study
should experience minimal CLV effects due to their positions on-disk (see
Milligan et al. 2020). In spite of this, the He II irradiance profiles from
FISM2, which have been treated with a CLV correction, still demonstrate
underestimation of the observed emission for two flares in the sample. This
suggests further that the fundamental processes involved in producing the
FISM2 irradiance profiles must be improved upon for future iterations of FISM.

Figure 5: Temporal evolution of the nonthermal parameters for each flare as derived from RHESSI HXR spectroscopy. First row: electron rate ($\rm A$). Second row: spectral index ($\rm\delta$). Third row: low energy cutoff ($\rm E_{c}$). Fourth row: nonthermal electron power ($\rm P_{nth}$). Fifth row: reduced $\rm\chi^{2}$ value for the combined fit model at each interval. The vertical dashed grey line indicates the peak time of the RHESSI 25–50 keV energy band. Table 3: Values of nonthermal fit parameters at the peak of the 25–50 keV HXR emission and the associated peak nonthermal electron power. Flare | $\rm A$ | $\rm\delta$ | $\rm E_{c}$ | $\rm P_{nth}$
---|---|---|---|---
| ($\rm 10^{35}$ electrons$\rm\ s^{-1}$) |  | ($\rm keV$) |  $(\rm 10^{28}$ $\rm erg$ $\rm s^{-1})$
SOL2010–10–16 | $4.2\pm 1.3$ | $4.6\pm 0.1$ | $16.8\pm 1.8$ | $1.5\pm 0.4$
SOL2011–09–24 | $1.1\pm 0.3$ | $5.7\pm 0.2$ | $21.8\pm 1.3$ | $0.5\pm 0.1$
SOL2014–02–01 | $2.7\pm 1.2$ | $7.4\pm 0.4$ | $20.2\pm 0.5$ | $1.0\pm 0.4$

###  3.2 Nonthermal Electron Parameters

From the fits to the HXR spectra from RHESSI, the parameters that describe the
distribution of nonthermal electrons believed to be responsible for driving
the impulsive increases in EUV irradiance were derived. Figure 5 shows the
progression of $\rm A$, $\rm\delta$, $\rm E_{c}$, and nonthermal electron
power ($\rm P_{nth}$) across the impulsive phase for each flare. The reduced
$\chi^{2}$ values are shown in the bottom row. The value of each parameter was
examined at the peak of the 25–50 keV energy band (and therefore the peak of
the Ly$\rm\alpha$ emission), indicated by the vertical grey dashed lines in
Figure 5. These are summarised in Table 3 along with the corresponding $\rm
P_{nth}$ value.

An apparent soft-hard-soft (SHS) behavior is observable from the temporal
evolution of $\rm\delta$ across SOL2010–10–16 (left panel in row 2 of Figure
5), where the value of $\rm\delta$ decreases (the spectrum hardens) as the
flare reaches its peak HXR flux, before increasing (the spectrum softens) once
more after the HXR peak time. SOL2011–09–24 shows similar but less pronounced
SHS behavior compared to SOL2010–10–16 (middle panel in row 2 of Figure 5) .
In SOL2014–02–01, this behavior is particularly weak (right panel in row 2 of
Figure 5). This temporal behavior may give indication to the nature of
chromospheric heating in each flare driven by nonthermal electrons. The SHS
behavior is an intrinsic property of solar flares characterized by the
tendency of flare-related HXR emission to vary with spectral hardness (Grigis
& Benz 2004). A prominent SHS component with a short temporal duration implies
a more impulsive acceleration event, such as in the case of SOL2010–10–16,
whereas a longer duration and less pronounced SHS behavior implies a more
gradual nature, such as in the case of SOL2011–09–24 and SOL2014–02–01.

Figure 6: Nonthermal and radiated power as a function of time for each flare in the sample. The dashed lines indicate the impulsive phase integration limits. The shaded region represents $\rm P_{nth}\pm\sigma$, where $\rm\sigma$ is taken as the as the standard deviation of $\rm P_{nth}$ over the full set of RHESSI detectors used. Table 4: Total energy in nonthermal electrons and radiated components during each flare.  Flare | Time Range (UT) | Total Energy (erg) |  $\rm\nicefrac{{E_{rad}}}{{E_{nth}}}$ (%)
---|---|---|---
|  | $\rm E_{nth}$ | $\rm E_{Ly\alpha}$ | $\rm E_{He~{}II}$ | Ly$\rm\alpha$ | He II
SOL2010–10–16 | 19:07:00 - 19:12:30 | $9.7\pm 1.6\times 10^{29}$ | $4.3\times 10^{28}$ | $7.1\times 10^{27}$ | $4.4\pm 0.7$ | $0.7\pm 0.1$
| 19:07:00 - 19:15:00 |  | $6.3\times 10^{28}$ | $1.3\times 10^{28}$ |  |
SOL2011–09–24 | 19:09:00 - 19:21:20 | $2.0\pm 0.3\times 10^{30}$ | $1.6\times 10^{29}$ | $2.9\times 10^{28}$ | $7.9\pm 1.1$ | $1.4\pm 0.2$
| 19:09:00 - 19:41:00 |  | $3.3\times 10^{29}$ | $6.6\times 10^{28}$ |  |
SOL2014–02–01 | 07:14:00 - 07:23:47 | $2.3\pm 0.7\times 10^{30}$ | $4.7\times 10^{28}$ | $1.5\times 10^{28}$ | $2.0\pm 0.6$ | $0.6\pm 0.2$
| 07:14:00 - 07:36:00 |  | $9.7\times 10^{28}$ | $3.6\times 10^{28}$ |  |

Considering the findings from Section 3.1, there is an apparent tendency for
the flux enhancement in Ly$\rm\alpha$ to scale with the value of $\rm\delta$,
such that a smaller $\rm\delta$, and therefore harder electron energy
spectrum, correlates to a larger enhancement in Ly$\rm\alpha$ irradiance.
SOL2010–10–16 had a $\rm\delta$ value of 4.6 at the peak of the HXR emission
and showed a 6.4% enhancement in Ly$\rm\alpha$. Whereas SOL2014–02–01 had a
$\rm\delta$ value of 7.4 at the peak of the respective 25–50 keV HXRs and
showed a 1.5% Ly$\rm\alpha$ enhancement.

The value of A at the peak of the 25–50 keV HXRs does not seem to have any
relationship to the enhancements in EUV emission. Similarly, the value of $\rm
E_{c}$ at the 25–50 keV peak does not appear to show a link to the enhancement
in Ly$\rm\alpha$ emission. An $\rm E_{c}$ of 16.8, 21.8, and 20.2 keV was
found found for SOL2010–10–16, SOL2011–09–24, and SOL2014–02–01, respectively.
Here we see that the lowest value of $\rm E_{c}$ is attributed to the flare
with the greatest enhancement in Ly$\rm\alpha$, however the largest $\rm
E_{c}$ value is found in the flare with the second largest enhancement in
Ly$\rm\alpha$. It is noted that there is a particular ambiguity in determining
the value of $\rm E_{c}$, especially for weaker flares, due to the dominance
of thermal emission at low energies. Thus, the apparent relationship between
$\rm\delta$ and the Ly$\rm\alpha$ flux is the most compelling result here.

Ly$\rm\alpha$ emission is optically-thick and may form over a range of
chromospheric altitudes. Using RADYN (Allred et al. 2005, 2015) simulations of
the chromospheric response to flare heating, Brown et al. (2018) found that
variation in electron beam parameters may impact plasma upflows in the
chromosphere and subsequently may effect the emission and absorption of
Ly$\rm\alpha$ and other lines in the Lyman series. Thus for the flares studied
here, the variability in nonthermal parameters found from HXR spectral
analysis may directly influence the chromospheric dynamics in each flare, in
turn impacting the levels of associated Ly$\rm\alpha$ emission.

Comparing Table 3 to Table 2, it is apparent that the value of $\rm P_{nth}$
has no clear relationship with the peak enhancement in either Ly$\rm\alpha$ or
He II. While SOL2010–10–16 shows both the largest peak $\rm P_{nth}$ and the
largest peak in Ly$\rm\alpha$ and He II, these values do not appear to be
related in SOL2011–09–24 and SOL2014–02–01. This implies that the total energy
deposited into the chromosphere does not necessarily dictate the degree to
which Ly$\rm\alpha$ or He II emission is enhanced. Instead, it is more likely
that the depth at which the energy is deposited in the chromosphere will play
a greater role in dictating the enhancement in the EUV emission, which will
depend on the nonthermal properties themselves.

###  3.3 Energetics

The power in nonthermal electrons derived from RHESSI fits for each flare is
given by the black line in Figure 6, the red and blue lines show the power
radiated in Ly$\rm\alpha$ and He II calculated from GOES/EUVS-E and SDO/EVE
lightcurves, respectively. Here, the plot limits encapsulate the full-flare
period; the vertical dotted line denotes the GOES peak and therefore the upper
limit of the impulsive phase.

For SOL2010–10–16, the total energy contained in nonthermal electrons was
found to be $9.7\times 10^{29}$ erg. The corresponding energy radiated in
Ly$\rm\alpha$ during the impulsive phase (330s) was found to be $4.3\times
10^{28}$ erg, accounting for 4.4% of the total nonthermal energy budget.
Extending the integration window to the entire GOES flare period, the total
energy radiated in Ly$\rm\alpha$ was found to be $6.3\times 10^{28}$ erg. The
radiated energy in He II was found to be $7.1\times 10^{27}$ erg over the
impulsive phase, amounting to 0.7% of the nonthermal energy budget, and was
found to increase to $1.3\times 10^{28}$ erg over the full-flare period.
Taking the ratio of the impulsive phase and full-flare period energies
($\rm\nicefrac{{E_{FP}}}{{E_{IP}}}$), the radiated energies in Ly$\rm\alpha$
and He II are found to increase by a factor of 1.5 and 1.8 between the
impulsive phase and the entire flare period, respectively.

For SOL2011–09–24, the impulsive phase lasted around 740s and the total
nonthermal energy was found to be $2.0\times 10^{30}$ erg. The energy radiated
in Ly$\rm\alpha$ over the impulsive phase was found to be $1.6\times 10^{29}$
erg accounting for 7.9% of the total nonthermal energy. Integrating over the
entire flare duration, the energy in Ly$\rm\alpha$ increased by a factor of
2.1 to $3.3\times 10^{29}$ erg. For He II, the radiated impulsive phase energy
was $2.9\times 10^{28}$ erg, amounting to 1.4% of the nonthermal energy. Over
the entire flare duration, the He II energy was found to increase to
$6.6\times 10^{28}$ erg, which is 2.3 times the impulsive phase energy.

Finally, for SOL2014–02–01 the total nonthermal energy was found to be
$2.3\times 10^{30}$ erg over the 587s impulsive phase. The corresponding
Ly$\rm\alpha$ energy radiated in this period was $4.7\times 10^{28}$ erg
accounting for 2.0% of the nonthermal electron energy. The energy radiated
over the entire flare period in Ly$\rm\alpha$ was found to increase to
$9.7\times 10^{28}$ erg. Thus, the value of $\rm\nicefrac{{E_{FP}}}{{E_{IP}}}$
for Ly$\rm\alpha$ in SOL2014–02–01 was 2.1. Once again, examining the He II
emission in the impulsive phase, the total energy radiated at this wavelength
was $1.5\times 10^{28}$ erg, 0.6% of the nonthermal energy, increasing by a
factor of 2.4 to $3.6\times 10^{28}$ erg over the full-flare period.

A full summary of the flare energetics is presented in Table 4. The typical
nonthermal energy was found to be approximately on the order of $10^{30}$ erg
in reasonable agreement with previous studies of flare energetics in M-class
flares (Saint-Hilaire & Benz, 2005). It is clear from Table 4 that
Ly$\rm\alpha$ may carry a reasonable portion of the energy deposited in the
chromosphere by nonthermal electrons during the impulsive phase. Ly$\rm\alpha$
was found to radiate approximately 2–8% of the energy deposited in the
chromosphere by nonthermal electrons. The lower limit to this range is lower
than value found by Milligan et al. (2014) and the overall range is broader,
this may be attributed to their use of the MEGS-P instrument within SDO/EVE to
measure the Ly$\rm\alpha$ flux, which has been found to produce distinctly
different Ly$\rm\alpha$ profiles compared to GOES/EUVS-E (Milligan &
Chamberlin 2016), as well as the differing magnitude of the flare examined.

Each flare showed elevated Ly$\rm\alpha$ and He II flux during their decay
phase. A value of $\rm\nicefrac{{E_{FP}}}{{E_{IP}}}$ $\geq 2$ implies at least
half of the total energy radiated during the flare is radiated after the
impulsive phase, such was the case for SOL2011–09–24 and SOL2014–02–01.
Possible contributions to the elevated flux may be attributed to evaporated
material in the corona cooling to chromospheric temperatures (Jing et al.,
2020), failed filament eruptions (Wauters et al., 2022), or evaporated
material driven by thermal conduction (Zarro & Lemen, 1988), for example.
While we cannot necessarily determine the source of the emission using disk-
integrated measurements alone, non-chromospheric contributions should not be
neglected from the interpretation of the results presented here. Ultimately
these may contribute to total incident flux in the Earth’s atmosphere
attributed to each flare, as well as reveal information on the distribution of
energy in the solar atmosphere during flares.

##  4 Summary & Conclusions

In this study, a multi-wavelength analysis was conducted for three solar
flares with equivalent GOES magnitude but distinctly different Ly$\rm\alpha$
profiles. The observed Ly$\rm\alpha$ and He II emission in each flare was
compared to synthetic line profiles generated by FISM2, which despite some
agreement for one flare, generally under-predicted the Ly$\rm\alpha$ and He II
flux. It was hypothesised that the variability in the associated Ly$\rm\alpha$
emission for each flare may be attributed to differing properties of the
nonthermal electrons incident on the chromosphere. This was examined by
carrying out spectral analysis of flare-related HXRs to discern a set of
parameters that describe the energy distribution of the nonthermal electrons
in each flare. Following this, the radiated energy in Ly$\rm\alpha$ and He II
was calculated for each flare and compared to the total nonthermal energy
deposited into the chromosphere over the impulsive phase, determining the
contribution of both wavelengths to the radiated energy budget.

The peak enhancement in Ly$\rm\alpha$ emission between the flare sample was
found to range from 1.5–6.4%. Comparatively, Milligan (2021) reported an
average enhancement in Ly$\rm\alpha$ emission of 1.5% for M-class flares,
placing two of the three flares in this study above that average.
Interestingly, FISM2 estimated Ly$\rm\alpha$ enhancements of $<$2.5% for each
flare.

The enhancement in Ly$\rm\alpha$ emission demonstrated a clear tendency to
scale with the spectral index, with the largest Ly$\rm\alpha$ enhancement
corresponding to the smallest spectral index value at the peak of the 25–50
keV HXR emission. No such behavior was found for the electron rate or the low
energy cutoff.

In some unique cases, RHESSI HXR observations show contributions from super-
hot ($\rm T_{e}>30~{}MK$) coronal plasma, which are spatially and spectrally
distinct from the hot plasma associated with chromospheric evaporation (Caspi
& Lin, 2010). Emission from these plasma may contribute to the thermal
component of flare HXR spectra. However, the flares presented in this study
have GOES temperatures of $\rm T_{GOES}\lesssim 15~{}MK$ and RHESSI
temperatures of $\rm T_{RHESSI}\lesssim 20~{}MK$, therefore lacking a clear
super-hot component. The temporal relationship between the Ly$\rm\alpha$ and
HXR flux in Figure 2 further supports the assumption that the emission for
these flares is primarily chromospheric in origin.

Finally, the energy content of the nonthermal electrons in each flare was
found to be approximately on the order of $10^{30}$ erg. Comparing this to the
radiated energy in Ly$\rm\alpha$, we were able to account for 2.0–7.9% of the
chromospheric energy budget in this line alone. These energies are found to be
consistent with studies by Milligan et al. (2012, 2014) and da Costa et al.
(2009), the former of which found the Ly$\rm\alpha$ energy radiated during
solar flares to be comparable to the total energy radiated across the entire
EVE spectral range (50–1050Å), further demonstrating the energetic
significance of Ly$\rm\alpha$ as a single emission line.

For He II, the peak enhancements were found to be larger than in
Ly$\rm\alpha$, ranging from 7.3–10.8%. However, He II did not demonstrate the
same behavior as Ly$\rm\alpha$ in relation to the spectral index. The total
energy radiated in He II during the impulsive phase was found to equate to
0.6–1.4% of the nonthermal electron energy, which was notably less than
Ly$\rm\alpha$ for all three of the flares examined here.

It has been shown that Ly$\rm\alpha$ emission can show dependence on flare
magnitude (Milligan 2021), however this study has demonstrated that the
variability in Ly$\rm\alpha$ emission in solar flares may also depend on a
more fundamental driver, such as the varying energetic properties of the
nonthermal electrons. Recent studies have examined the effects of flare-
induced radiation on the ionosphere across differing GOES classes (Hayes et
al. 2021; Barta et al. 2022). Milligan et al. (2020) found that increased
conductivity in the ionosphere may be attributed to increased Ly$\rm\alpha$
absorption in the E-region. Thus, irregularity in Ly$\rm\alpha$ emission
between equivalent magnitude flares may impact our ability to predict
ionospheric responses to solar flares based on their class, as well as our
ability to extrapolate our understanding of the terrestrial impact of solar
flares to our planetary neighbors (see Yan et al. 2022).

On the relationship between nonthermal electrons and Ly$\rm\alpha$ emission,
it is understood that the nonthermal electron parameters dictate the transport
of energy into the chromosphere, and therefore the levels of chromospheric
emission (Kennedy et al., 2015; Procházka et al., 2018). Our findings imply
that the level of flare-related Ly$\rm\alpha$ emission may be sensitive to the
energetic properties of the nonthermal electrons incident on the chromosphere
during solar flares. Care must be taken when drawing conclusions regarding the
impact of individual parameters on the observed flare emission, as ultimately
the energy distribution of the nonthermal electrons is dependent on the
combination of the electron rate, spectral index, and low energy cutoff.
Therefore, it may be useful to test a combination of parameters to discern the
magnitude of their impact on chromospheric emission using radiative
hydrodynamic simulation codes such as RADYN and RH (Uitenbroek 2001).
Synthetic emission profiles for both Ly$\rm\alpha$ and He II may then be
examined in an effort to validate the observational findings presented in this
research.

For this study, calculations of the percentage contribution of Ly$\rm\alpha$
and He II to the radiation of energy deposited in the chromosphere were
founded on the assumption that their origin is entirely chromospheric. In the
first instance, this would place the energy calculated in Ly$\rm\alpha$ at a
lower limit, due to the significant optical-thickness of Ly$\rm\alpha$ in the
chromosphere and the potential for self-absorption. With no spatially resolved
observations of Ly$\rm\alpha$ for these flares, one must take great care when
drawing conclusions regarding the energy attributed to chromospheric emission
alone. We also note that our observations of Ly$\rm\alpha$ emission are taken
across a relatively large spectral range given the broadband nature of the
GOES/EUVS-E instrument. This means we cannot determine with absolute certainty
that all of the emission measured for each flare is entirely attributable to
the Ly$\rm\alpha$ line alone, as the E-channel spectral resolution also
encapsulates the Si III line at 1206Å. We assume here that the irradiance data
is dominated by the Ly$\rm\alpha$ emission, but it is impossible to confirm
this exclusively from the observations presented in this study. For future
study of Ly$\rm\alpha$ emission in solar flares, it would be of great value to
have spatially and spectrally resolved measurements.

FISM is the most widely used model for providing estimates of solar emissions
for periods of activity lacking observation. The first version of FISM saw
application in numerous studies into the effect of solar radiation in both the
terrestrial and martian atmospheres (Qian et al. 2010, 2011, 2012; Lollo et
al. 2012), as well as the development of the FISM-M model for the Extreme
Ultraviolet Monitor (EUVM; Eparvier et al. 2015; Thiemann et al. 2017) onboard
Mars Atmosphere and Volatile Experiment (MAVEN; Jakosky et al. 2015). FISM2
offers significant improvements on the initial version of FISM, implementing
more accurate data-sets, a greater number of flares, and longer time series.
As of the release of FISM2, plans for improvements were already identified
(Chamberlin et al., 2020). Further upgrades to address the deficiencies of
FISM2 model have been proposed for FISM3, including incorporating measurements
and proxies from GOES/EUVS in order to improve the impulsive phase
estimations, especially for optically thick emissions such as Ly$\rm\alpha$
and He II. By including observations from GOES/EUVS-E and UARS/SOLSTICE, it
may become possible to compute a CLV correction in the FUV wavelengths for the
impulsive phase. Proposed improvements also plan to incorporate new
measurements from Miniature X-ray Solar Spectrometer and Dual-zone Aperture
X-ray Solar Spectrometer (MinXSS; DAXSS Mason et al. 2016; Woods et al. 2017;
Moore et al. 2018; Mason et al. 2020; Schwab et al. 2020) to improve the soft
X-ray emissions from 0.1-2.5 nm.

Finally, it is apparent that interest in studying flare-related Ly$\rm\alpha$
emission is growing, with newly launched and proposed missions featuring
dedicated apparatus for observing at this wavelength. The new generation of
GOES-R satellites feature a suite of co-observing SXR and Ly$\rm\alpha$
instruments as part of the EUV and X-ray Irradiance Sensors (EXIS) which will
provide pseudo Ly$\rm\alpha$ line profiles as part of its continuous
observations for the next two decades (Eparvier et al. 2009; Chamberlin et al.
2009). The recently launched Solar Orbiter mission includes a dedicated
Ly$\rm\alpha$ channel as part of the High Resolution Imager (HRILyα) onboard
the Extreme Ultraviolet Imager (EUI), which is able to image the Sun in
Ly$\rm\alpha$ at 1″ resolution and 1s cadence (Rochus et al. 2020; Müller et
al. 2020). Future Ly$\rm\alpha$ missions also include Solar Spectral
Irradiance Monitor (SoSpIM) as part of the EUV High-Throughput Spectroscopic
Telescope (EUVST; Shimizu et al. 2019) onboard the Solar–C mission (Watanabe
2014) and the Ly$\rm\alpha$ Solar Telescope (LST) as part of the Advanced
Space-based Solar Observatory (ASO–S; Li et al. 2019; Gan et al. 2023). The
findings presented here suggest that the spectral index, derived from HXR
observations, may be an important tool for future development of models of
Ly$\rm\alpha$ emission in solar flares. Instruments such as the Spectrometer
Telescope for Imaging X-rays (STIX; Krucker et al. 2020) onboard Solar Orbiter
and the Hard X-Ray Imager (HXI; Zhang et al. 2019) onboard ASO-S will provide
further HXR observations of flares within Solar Cycle 25. The results
presented in this study may guide interpretation of the observations taken by
this new generation of instruments as part of future studies.

H.J.G would like to thank the UK’s Science & Technology Facilities Council
(ST/W507751/1) for supporting this research and for the award of Long Term
Attachment funding to facilitate overseas fieldwork as part of this research.
R.O.M would like to thank the UK’s Science & Technologies Facilities Council
for the award of an Ernest Rutherford Fellowship (ST/N004981/2). P.C.C would
like to acknowledge support for this work by the SDO/EVE project at CU/LASP
(PI Dr. Thomas Woods). H.J.G and R.O.M. would both like to thank Prof. Mihalis
Mathioudakis for several stimulating discussions and initial feedback on this
work. H.J.G would like to further extend his gratitude to Dr. Brian Dennis for
his invaluable advice and guidance surrounding the analysis techniques
presented in this paper. The authors would also like to thank the anonymous
reviewer for their useful comments that contributed to improving this work.

##  5 Data Access Statement

No new data was generated as part of this research. The pre-existing
observational data used in this study are openly accessible through
SolarSoftware (SSWIDL; https://www.lmsal.com/solarsoft/). The FISM2 data are
publicly available on the LISIRD data site on the FISM page, or directly here:
http://lasp.colorado.edu/lisird/.

## References

  * Allred et al. (2005) Allred, J. C., Hawley, S. L., Abbett, W. P., & Carlsson, M. 2005, ApJ, 630, 573, doi: 10.1086/431751
  * Allred et al. (2015) Allred, J. C., Kowalski, A. F., & Carlsson, M. 2015, ApJ, 809, 104, doi: 10.1088/0004-637x/809/1/104
  * Auchère (2005) Auchère, F. 2005, ApJ, 622, 737, doi: 10.1086/427903
  * Baliukin et al. (2019) Baliukin, I. I., Bertaux, J. L., Quémerais, E., Izmodenov, V. V., & Schmidt, W. 2019, Journal of Geophysical Research (Space Physics), 124, 861, doi: 10.1029/2018JA026136
  * Barta et al. (2022) Barta, V., Natras, R., Srećković, V., et al. 2022, Frontiers in Environmental Science, 10, doi: 10.3389/fenvs.2022.904335
  * Boerner et al. (2012) Boerner, P., Edwards, C., Lemen, J., et al. 2012, Solar Physics, 275, 41
  * Brekke et al. (1996) Brekke, P., Rottman, G. J., Fontenla, J., & Judge, P. G. 1996, ApJ, 468, 418, doi: 10.1086/177701
  * Brown (1971) Brown, J. C. 1971, Solar Physics, 18, 489, doi: 10.1007/BF00149070
  * Brown (1972) —. 1972, Solar Physics, 26, 441, doi: 10.1007/BF00165286
  * Brown (1973) —. 1973, Solar Physics, 28, 151, doi: 10.1007/BF00152919
  * Brown et al. (2018) Brown, S. A., Fletcher, L., Kerr, G. S., et al. 2018, ApJ, 862, 59, doi: 10.3847/1538-4357/aacc29
  * Brown et al. (2018) Brown, S. A., Fletcher, L., Kerr, G. S., et al. 2018, The Astrophysical Journal, 862, 59, doi: 10.3847/1538-4357/aacc29
  * Carmichael (1964) Carmichael, H. 1964, in NASA Special Publication, Vol. 50, 451
  * Caspi & Lin (2010) Caspi, A., & Lin, R. P. 2010, The Astrophysical Journal Letters, 725, L161, doi: 10.1088/2041-8205/725/2/L161
  * Chamberlin et al. (2007) Chamberlin, P. C., Woods, T. N., & Eparvier, F. G. 2007, Space Weather, 5, doi: https://doi.org/10.1029/2007SW000316
  * Chamberlin et al. (2008) Chamberlin, P. C., Woods, T. N., & Eparvier, F. G. 2008, Space Weather, 6, S05001, doi: 10.1029/2007SW000372
  * Chamberlin et al. (2009) Chamberlin, P. C., Woods, T. N., Eparvier, F. G., & Jones, A. R. 2009, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7438, Solar Physics and Space Weather Instrumentation III, ed. S. Fineschi & J. A. Fennelly, 743802, doi: 10.1117/12.826807
  * Chamberlin et al. (2018) Chamberlin, P. C., Woods, T. N., Didkovsky, L., et al. 2018, Space Weather, 16, 1470, doi: https://doi.org/10.1029/2018SW001866
  * Chamberlin et al. (2020) Chamberlin, P. C., Eparvier, F. G., Knoer, V., et al. 2020, Space Weather, 18, e2020SW002588, doi: https://doi.org/10.1029/2020SW002588
  * Chubb et al. (1957) Chubb, T. A., Friedman, H., Kreplin, R. W., & Kupperian Jr., J. E. 1957, Journal of Geophysical Research (1896-1977), 62, 389, doi: https://doi.org/10.1029/JZ062i003p00389
  * Curdt et al. (2001) Curdt, W., Brekke, P., Feldman, U., et al. 2001, A&A, 375, 591, doi: 10.1051/0004-6361:20010364
  * da Costa et al. (2009) da Costa, F. R., Fletcher, L., Labrosse, N., & Zuccarello, F. 2009, A&A, 507, 1005, doi: 10.1051/0004-6361/200912651
  * Delaboudinière et al. (1995) Delaboudinière, J. P., Artzner, G. E., Brunaud, J., et al. 1995, Sol. Phys., 162, 291, doi: 10.1007/BF00733432
  * Dennis & Schwartz (1989) Dennis, B. R., & Schwartz, R. A. 1989, Sol. Phys., 121, 75, doi: 10.1007/BF00161688
  * Domingo et al. (1995) Domingo, V., Fleck, B., & Poland, A. I. 1995, Sol. Phys., 162, 1, doi: 10.1007/BF00733425
  * Dominique et al. (2013) Dominique, M., Hochedez, J. F., Schmutz, W., et al. 2013, Sol. Phys., 286, 21, doi: 10.1007/s11207-013-0252-5
  * Dominique et al. (2018) Dominique, M., Zhukov, A. N., Heinzel, P., et al. 2018, ApJ, 867, L24, doi: 10.3847/2041-8213/aaeace
  * Eparvier et al. (2015) Eparvier, F., Chamberlin, P., Woods, T., & Thiemann, E. 2015, Space Science Reviews, 195, doi: 10.1007/s11214-015-0195-2
  * Eparvier et al. (2009) Eparvier, F. G., Crotser, D., Jones, A. R., et al. 2009, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7438, Solar Physics and Space Weather Instrumentation III, ed. S. Fineschi & J. A. Fennelly, 743804, doi: 10.1117/12.826445
  * Evans et al. (2010) Evans, J. S., Strickland, D. J., Woo, W. K., et al. 2010, Sol. Phys., 262, 71, doi: 10.1007/s11207-009-9491-x
  * Freeland & Handy (1998) Freeland, S. L., & Handy, B. N. 1998, Sol. Phys., 182, 497, doi: 10.1023/A:1005038224881
  * Gan et al. (2023) Gan, W., Zhu, C., Deng, Y., et al. 2023, Sol. Phys., 298, 68, doi: 10.1007/s11207-023-02166-x
  * Garcia et al. (2007) Garcia, R. R., Marsh, D. R., Kinnison, D. E., Boville, B. A., & Sassi, F. 2007, Journal of Geophysical Research: Atmospheres, 112, doi: https://doi.org/10.1029/2006JD007485
  * Gordino et al. (2022) Gordino, M., Auchère, F., Vial, J. C., et al. 2022, A&A, 657, A86, doi: 10.1051/0004-6361/202141960
  * Grigis & Benz (2004) Grigis, P. C., & Benz, A. O. 2004, A&A, 426, 1093, doi: 10.1051/0004-6361:20041367
  * Hanser & Sellers (1996) Hanser, F. A., & Sellers, F. B. 1996, in GOES-8 and Beyond, ed. E. R. Washwell, Vol. 2812, International Society for Optics and Photonics (SPIE), 344 – 352, doi: 10.1117/12.254082
  * Hayes et al. (2021) Hayes, L. A., O’Hara, O. S. D., Murray, S. A., & Gallagher, P. T. 2021, Solar Physics, 296, doi: 10.1007/s11207-021-01898-y
  * Hirayama (1974) Hirayama, T. 1974, Sol. Phys., 34, 323, doi: 10.1007/BF00153671
  * Holman et al. (2003) Holman, G. D., Sui, L., Schwartz, R. A., & Emslie, A. G. 2003, ApJ, 595, L97, doi: 10.1086/378488
  * Hudson et al. (1992) Hudson, H. S., Acton, L. W., Hirayama, T., & Uchida, Y. 1992, PASJ, 44, L77
  * Hudson et al. (2006) Hudson, H. S., Wolfson, C. J., & Metcalf, T. R. 2006, Sol. Phys., 234, 79, doi: 10.1007/s11207-006-0056-y
  * Ireland et al. (2013) Ireland, J., Tolbert, A. K., Schwartz, R. A., Holman, G. D., & Dennis, B. R. 2013, ApJ, 769, 89, doi: 10.1088/0004-637X/769/2/89
  * Jakosky et al. (2015) Jakosky, B. M., Lin, R. P., Grebowsky, J. M., et al. 2015, Space Sci. Rev., 195, 3, doi: 10.1007/s11214-015-0139-x
  * Jing et al. (2020) Jing, Z., Pan, W., Yang, Y., et al. 2020, ApJ, 904, 41, doi: 10.3847/1538-4357/abbacc
  * Kennedy et al. (2015) Kennedy, M. B., Milligan, R. O., Allred, J. C., Mathioudakis, M., & Keenan, F. P. 2015, A&A, 578, A72, doi: 10.1051/0004-6361/201425144
  * Kleint et al. (2016) Kleint, L., Heinzel, P., Judge, P., & Krucker, S. 2016, The Astrophysical Journal, 816, 88, doi: 10.3847/0004-637x/816/2/88
  * Kontar et al. (2006) Kontar, E. P., MacKinnon, A. L., Schwartz, R. A., & Brown, J. C. 2006, A&A, 446, 1157, doi: 10.1051/0004-6361:20053672
  * Kopp & Pneuman (1976) Kopp, R. A., & Pneuman, G. W. 1976, Sol. Phys., 50, 85, doi: 10.1007/BF00206193
  * Kretzschmar (2011) Kretzschmar, M. 2011, A&A, 530, A84, doi: 10.1051/0004-6361/201015930
  * Kretzschmar et al. (2010) Kretzschmar, M., de Wit, T. D., Schmutz, W., et al. 2010, Nature Physics, 6, 690, doi: 10.1038/nphys1741
  * Kretzschmar et al. (2012) Kretzschmar, M., Dominique, M., & Dammasch, I. E. 2012, Solar Physics, 286, 221, doi: 10.1007/s11207-012-0175-6
  * Krucker et al. (2020) Krucker, S., Hurford, G. J., Grimm, O., et al. 2020, A&A, 642, A15, doi: 10.1051/0004-6361/201937362
  * Kumar & Kumar (2018) Kumar, A., & Kumar, S. 2018, Earth, Planets and Space, 70, 29, doi: 10.1186/s40623-018-0794-8
  * Li et al. (2019) Li, H., Chen, B., Feng, L., et al. 2019, Research in Astronomy and Astrophysics, 19, 158, doi: 10.1088/1674-4527/19/11/158
  * Li et al. (2022) Li, Y., Li, Q., Song, D.-C., et al. 2022, The Astrophysical Journal, 936, 142, doi: 10.3847/1538-4357/ac897c
  * Lin et al. (2002) Lin, R. P., Dennis, B. R., Hurford, G. J., et al. 2002, Sol. Phys., 210, 3, doi: 10.1023/A:1022428818870
  * Lollo et al. (2012) Lollo, A., Withers, P., Fallows, K., et al. 2012, Journal of Geophysical Research: Space Physics, 117, doi: https://doi.org/10.1029/2011JA017399
  * Lu et al. (2021) Lu, L., Feng, L., Li, D., et al. 2021, ApJS, 253, 29, doi: 10.3847/1538-4365/abd79b
  * Marsh et al. (2013) Marsh, D., Mills, M., Kinnison, D., et al. 2013, Journal Of Climate, 26, 7372, doi: 10.1175/JCLI-D-12-00558.1
  * Mason et al. (2020) Mason, J., Woods, T., Chamberlin, P., et al. 2020, Advances in Space Research, 66, 3, doi: 10.1016/j.asr.2019.02.011
  * Mason et al. (2016) Mason, J. P., Woods, T. N., Caspi, A., et al. 2016, Journal of Spacecraft and Rockets, 53, 328, doi: 10.2514/1.A33351
  * McClintock et al. (2005) McClintock, W. E., Rottman, G. J., & Woods, T. N. 2005, Sol. Phys., 230, 225, doi: 10.1007/s11207-005-7432-x
  * McRae & Thomson (2004) McRae, W. M., & Thomson, N. R. 2004, Journal of Atmospheric and Solar-Terrestrial Physics, 66, 77, doi: 10.1016/j.jastp.2003.09.009
  * Meier & Prinz (1970) Meier, R. R., & Prinz, D. K. 1970, Journal of Geophysical Research (1896-1977), 75, 6969, doi: https://doi.org/10.1029/JA075i034p06969
  * Milligan (2015) Milligan, R. O. 2015, Solar Physics, 290, 3399, doi: 10.1007/s11207-015-0748-2
  * Milligan (2021) Milligan, R. O. 2021, Solar Physics, 296, doi: 10.1007/s11207-021-01796-3
  * Milligan & Chamberlin (2016) Milligan, R. O., & Chamberlin, P. C. 2016, A&A, 587, A123, doi: 10.1051/0004-6361/201526682
  * Milligan et al. (2012) Milligan, R. O., Chamberlin, P. C., Hudson, H. S., et al. 2012, ApJ, 748, L14, doi: 10.1088/2041-8205/748/1/l14
  * Milligan et al. (2017) Milligan, R. O., Fleck, B., Ireland, J., Fletcher, L., & Dennis, B. R. 2017, The Astrophysical Journal, 848, L8, doi: 10.3847/2041-8213/aa8f3a
  * Milligan et al. (2020) Milligan, R. O., Hudson, H. S., Chamberlin, P. C., Hannah, I. G., & Hayes, L. A. 2020, Space Weather, 18, doi: 10.1029/2019sw002331
  * Milligan et al. (2014) Milligan, R. O., Kerr, G. S., Dennis, B. R., et al. 2014, ApJ, 793, 70, doi: 10.1088/0004-637x/793/2/70
  * Moore et al. (2018) Moore, C. S., Caspi, A., Woods, T. N., et al. 2018, Solar Physics, 293, 21, doi: 10.1007/s11207-018-1243-3
  * 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
  * Neale et al. (2013) Neale, R. B., Richter, J., Park, S., et al. 2013, Journal of Climate, 26, 5150 , doi: 10.1175/JCLI-D-12-00236.1
  * Neidig (1989) Neidig, D. F. 1989, Sol. Phys., 121, 261, doi: 10.1007/BF00161699
  * Nina et al. (2018) Nina, A., Čadež, V. M., Bajčetić, J., Mitrović, S. T., & Popović, L. Č. 2018, Sol. Phys., 293, 64, doi: 10.1007/s11207-018-1279-4
  * Nusinov et al. (2006) Nusinov, A. A., Kazachevskaya, T. V., Kuznetsov, S. N., Myagkova, I. N., & Yushkov, B. Y. 2006, Solar System Research, 40, 282, doi: 10.1134/S0038094606040034
  * 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
  * Procházka et al. (2018) Procházka, O., Reid, A., Milligan, R. O., et al. 2018, ApJ, 862, 76, doi: 10.3847/1538-4357/aaca37
  * Qian et al. (2010) Qian, L., Burns, A. G., Chamberlin, P. C., & Solomon, S. C. 2010, Journal of Geophysical Research: Space Physics, 115, doi: https://doi.org/10.1029/2009JA015225
  * Qian et al. (2011) —. 2011, Journal of Geophysical Research: Space Physics, 116, doi: https://doi.org/10.1029/2011JA016777
  * Qian et al. (2012) Qian, L., Burns, A. G., Solomon, S. C., & Chamberlin, P. C. 2012, Geophysical Research Letters, 39, doi: https://doi.org/10.1029/2012GL051102
  * Qian et al. (2018) Qian, L., Burns, A. G., Solomon, S. S., et al. 2018, Journal of Geophysical Research: Space Physics, 123, 1006, doi: https://doi.org/10.1002/2017JA024998
  * Raulin et al. (2013) Raulin, J.-P., Trottet, G., Kretzschmar, M., et al. 2013, Journal of Geophysical Research: Space Physics, 118, 570, doi: https://doi.org/10.1029/2012JA017916
  * Rochus et al. (2020) Rochus, P., Auchère, F., Berghmans, D., et al. 2020, A&A, 642, A8, doi: 10.1051/0004-6361/201936663
  * Rottman et al. (1993) Rottman, G. J., Woods, T. N., & Sparn, T. P. 1993, J. Geophys. Res., 98, 10,667, doi: 10.1029/93JD00462
  * Saint-Hilaire & Benz (2005) Saint-Hilaire, P., & Benz, A. 2005, A&A, 435, doi: 10.1051/0004-6361:20041918
  * Santandrea et al. (2013) Santandrea, S., Gantois, K., Strauch, K., et al. 2013, Sol. Phys., 286, 5, doi: 10.1007/s11207-013-0289-5
  * Schwab et al. (2020) Schwab, B. D., Sewell, R. H. A., Woods, T. N., et al. 2020, ApJ, 904, 20, doi: 10.3847/1538-4357/abba2a
  * Shimizu et al. (2019) Shimizu, T., Imada, S., Kawate, T., et al. 2019, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 11118, UV, X-Ray, and Gamma-Ray Space Instrumentation for Astronomy XXI, ed. O. H. Siegmund, 1111807, doi: 10.1117/12.2528240
  * Simões et al. (2016) Simões, P. J. A., Fletcher, L., Labrosse, N., & Kerr, G. S. 2016, in Astronomical Society of the Pacific Conference Series, Vol. 504, Coimbra Solar Physics Meeting: Ground-based Solar Observations in the Space Instrumentation Era, ed. I. Dorotovic, C. E. Fischer, & M. Temmer, 197\.  https://arxiv.org/abs/1512.03477
  * Solomon et al. (2018) Solomon, S. C., Liu, H.-L., Marsh, D. R., et al. 2018, Geophysical Research Letters, 45, 1567, doi: https://doi.org/10.1002/2017GL076950
  * Sturrock (1966) Sturrock, P. A. 1966, Nature, 211, 695, doi: 10.1038/211695a0
  * Thiemann et al. (2017) Thiemann, E. M. B., Chamberlin, P. C., Eparvier, F. G., et al. 2017, Journal of Geophysical Research: Space Physics, 122, 2748, doi: https://doi.org/10.1002/2016JA023512
  * Uitenbroek (2001) Uitenbroek, H. 2001, ApJ, 557, 389, doi: 10.1086/321659
  * Viereck et al. (2007) Viereck, R., Hanser, F., Wise, J., et al. 2007, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6689, Solar Physics and Space Weather Instrumentation II, ed. S. Fineschi & R. A. Viereck, 66890K, doi: 10.1117/12.734886
  * Watanabe (2014) Watanabe, T. 2014, in Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave, ed. J. M. O. Jr., M. Clampin, G. G. Fazio, & H. A. MacEwen, Vol. 9143, International Society for Optics and Photonics (SPIE), 91431O, doi: 10.1117/12.2055366
  * Wauters et al. (2022) Wauters, L., Dominique, M., Milligan, R., et al. 2022, Sol. Phys., 297, 36, doi: 10.1007/s11207-022-01963-0
  * Woods et al. (2006) Woods, T. N., Kopp, G., & Chamberlin, P. C. 2006, Journal of Geophysical Research: Space Physics, 111, doi: https://doi.org/10.1029/2005JA011507
  * Woods et al. (1993) Woods, T. N., Rottman, G. J., & Ucker, G. J. 1993, J. Geophys. Res., 98, 10,679, doi: 10.1029/93JD00463
  * Woods et al. (2004) Woods, T. N., Eparvier, F. G., Fontenla, J., et al. 2004, Geophysical Research Letters, 31, doi: https://doi.org/10.1029/2004GL019571
  * Woods et al. (2012) Woods, T. N., Eparvier, F. G., Hock, R., et al. 2012, Sol. Phys., 275, 115, doi: 10.1007/s11207-009-9487-6
  * Woods et al. (2017) Woods, T. N., Caspi, A., Chamberlin, P. C., et al. 2017, ApJ, 835, 122, doi: 10.3847/1538-4357/835/2/122
  * Wuelser et al. (2004) Wuelser, J.-P., Lemen, J. R., Tarbell, T. D., et al. 2004, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5171, Telescopes and Instrumentation for Solar Astrophysics, ed. S. Fineschi & M. A. Gummin, 111–122, doi: 10.1117/12.506877
  * Yan et al. (2022) Yan, M., Dang, T., Cao, Y.-T., et al. 2022, The Astrophysical Journal, 939, 23, doi: 10.3847/1538-4357/ac92ff
  * Zarro & Lemen (1988) Zarro, D. M., & Lemen, J. R. 1988, ApJ, 329, 456, doi: 10.1086/166391
  * Zhang et al. (2019) Zhang, Z., Chen, D.-Y., Wu, J., et al. 2019, Research in Astronomy and Astrophysics, 19, 160, doi: 10.1088/1674-4527/19/11/160