# Near-Ultraviolet Continuum Modeling of the 1985 April 12 Great Flare of AD
Leo

Adam F. Kowalski 1,2,3,∗

###### Abstract

White-light stellar flares are now reported by the thousands in long-baseline,
high-precision, broad-band photometry from missions like _Kepler_ , K2, and
TESS. These observations are crucial inputs for assessments of biosignatures
in exoplanetary atmospheres and surface ultraviolet radiation dosages for
habitable-zone planets around low-mass stars. A limitation of these
assessments, however, is the lack of near-ultraviolet spectral observations of
stellar flares. To motivate further empirical investigation, we use a grid of
radiative-hydrodynamic simulations with an updated treatment of the pressure
broadening of hydrogen lines to predict the $\lambda\approx 1800-3300$ Å
continuum flux during the rise and peak phases of a well-studied superflare
from the dM3e star AD Leo. These predictions are based on semi-empirical
superpositions of radiative flux spectra consisting of a high-flux electron
beam simulation with a large, low-energy cutoff ($\gtrsim 85$ keV) and a
lower-flux electron beam simulation with a smaller, low-energy cutoff
($\lesssim 40$ keV). The two-component models comprehensively explain the
hydrogen Balmer line broadening, the optical continuum color temperature, the
Balmer jump strength, and the far-ultraviolet continuum strength and shape in
the rise/peak phase of this flare. We use spatially resolved analyses of solar
flare data from the Interface Region Imaging Spectrograph, combined with the
results of previous radiative-hydrodynamic modeling of the 2014 March 29 X1
solar ﬂare (SOL20140329T17:48), to interpret the two-component electron beam
model as representing the spatial superposition of bright kernels and fainter
ribbons over a larger area.

##  1 Keywords:

flares-stars, flares-sun, habitability and astrobiology, spectroscopy, near-
ultraviolet continuum

##  2 Introduction

Rapidly rotating, magnetically active M dwarf (dMe) stars occasionally flare
with energies that are factors of 100–10,000 greater than the most energetic
solar flares that have been observed in the modern era. These so-called
“superflares” provide insight into the physics of extreme plasma conditions
attained in stars (Osten et al., 2007; Testa et al., 2008; Osten et al., 2010,
2016; Karmakar et al., 2017) and possibly also the young Sun (Ayres, 2015;
Maehara et al., 2015; Namekata et al., 2021). Further afield, observations of
these superflares are widely used in the characterization of the high-energy
radiation environments in the habitable zones of low-mass stars (Smith et al.,
2004; Segura et al., 2010), which are a primary target for exoplanet transit
spectroscopy (e.g., Scalo et al., 2007; Belu et al., 2011; de Wit et al.,
2018; Fauchez et al., 2019; Barstow and Irwin, 2016; Barstow et al., 2016)
with the _James Webb Space Telescope_ (JWST). The JWST and future extremely
large telescope facilities provide a means to determine whether exoplanets in
or around the habitable zone of low-mass stars retain atmospheres in the
presence of high fluxes of stellar energetic particles and X-ray and extreme
ultraviolet (XEUV) flare radiation. The evolution of the atmosphere of Mars is
thought to have undergone significant mass loss due to coronal mass ejections
and XEUV heating from the Sun (Jakosky et al., 2018), which motivates
investigations into the evolution of exoplanetary atmospheres that are much
closer to stars that are highly magnetically active for billions of years
(West et al., 2008).

Segura et al. (2010) simulated the impact of a superflare event on a non-
magnetic, but otherwise Earth-like, exoplanet atmosphere in the habitable zone
of the dM3e flare star AD Leo. They found that planetary ozone is largely
depleted due to chemical reactions (e.g., Scalo et al., 2007) that follow from
incident scaled-up fluxes (Belov et al., 2005) of solar energetic protons.
More recent simulations have considered the effects of repeated flaring and
particle events after such a superflare has occurred (Tilley et al., 2019;
Howard et al., 2018). Howard et al. (2018) and Tilley et al. (2019) discuss
the role of UV-C111According to the World Health Organization, the ultraviolet
is comprised of three bands: UV-C ($\lambda=1000-2800$ Å), UV-B
($\lambda=2800-3150$ Å), and UV-A ($\lambda=3150-4000$ Å). We follow Abrevaya
et al. (2020) and other recent studies of the biological impact of UV flares
and denote the continuum radiation at $\lambda=2000-2800$ Å as UV-C; the
ultraviolet radiation at $\lambda=1000-2000$ Å is also known as the very
ultraviolet (VUV). ($\lambda=2000-2800$Å), and in particular the wavelengths
$\lambda=2400-2800$ Å, from repeated flaring in germicidal radiation surface
fluxes following an ozone-depletion event. Recently, Abrevaya et al. (2020)
conducted laboratory measurements of survival curves of microorganisms through
exposure to a UV-C radiation flux inferred from optical observations of a
superflare from the dM5.5e star Proxima Centauri (Howard et al., 2018).

The transient (Loyd et al., 2018a) and secular (Venot et al., 2016) effects on
ozone biosignature photochemistry caused by ultraviolet flares is an ongoing
subject of research, especially in light of lack of direct observation of
stellar energetic proton fluences and exoplanetary magnetic field properties
(see discussion in Tilley et al., 2019). Loyd et al. (2018a) demonstrate that
assumptions of the ultraviolet continuum shape during flares can affect ozone
photolysis rates (see also Howard et al., 2020), while effects spanning
several orders of magnitude on other important atmospheric constituents (CH4,
H2O, O2) are expected. Significant effort has been invested into the empirical
characterization of the quiescent and flaring spectra of low-mass stars in the
far-ultraviolet wavelength region of $\lambda=1100-1800$ Å through the
MUSCLES, Mega-MUSCLES, and HAZMAT treasury programs with the _Hubble Space
Telescope_ (Shkolnik and Barman, 2014; Loyd and France, 2014; Froning et al.,
2019; Loyd et al., 2018b; France et al., 2020; Wilson et al., 2021, and see
also Feinstein et al. 2022). However, much still remains unknown about the
spectral characteristics of transient, impulsive-phase, near-ultraviolet (NUV)
enhancements in flare radiation from $\lambda\approx 2000-3300$ Å (e.g.,
Robinson et al., 2005; Hawley et al., 2007; Brasseur et al., 2019; Fleming et
al., 2022), which is thought to account for a large percentage ($\approx$25%)
of the $\lambda=1200-8000$ Å radiated energy (Hawley and Pettersen, 1991). The
largest X-ray solar flare of Sunspot Cycle 24 was recently studied in
spatially integrated light at continuum wavelengths through a
$\Delta\lambda\approx 300$ Å bandpass around $\lambda\approx 2000$ Å
(Dominique et al., 2018).

Many studies of stellar flares have utilized photometry from high-precision
missions of _Kepler_ , K2, and TESS, which observe through broad white-light
bandpasses in the optical and near-infrared. Photochemistry and habitability
calculations often use extrapolations to shorter ultraviolet wavelengths by
assuming a $T\approx 9000-10,000$ K blackbody spectrum (e.g., Günther et al.,
2020); this assumption has also been widely employed for calculations of
bolometric energies in statistical analyses (e.g., Shibayama et al., 2013;
Yang et al., 2017). We refer the reader to Howard et al. (2020) and Brasseur
et al. (2022, submitted to _ApJ_) for discussions about several unprecedented
problems raised by recent studies of multi-wavelength broadband photometry of
stellar superflares.

Very few stellar flare spectral observations exist with near-ultraviolet
coverage and contemporaneous optical spectra (Hawley and Pettersen, 1991;
Robinson et al., 1993; Wargelin et al., 2017; Kowalski et al., 2019b) that
would facilitate detailed tests of and improvements upon blackbody modeling of
flares. However, ground-based spectra suggest that the NUV flare continuum has
non-negligible contributions from Balmer continuum radiation (Kowalski et al.,
2010, 2013, 2016). Most recently, Kowalski et al. (2019b) analyze
$\lambda\approx 2500-7400$ Å flare spectra and photometry over two events from
the dM4e star GJ 1243. Detailed modeling demonstrates that a $T=9000$ K
blackbody fit to the blue-optical continuum at $\lambda\approx 4000-4800$ Å
underpredicts the NUV flare flux by factors of $2-3$ during these two events;
the discrepency was tied to a moderately-sized jump in the continuum flux
around the Balmer limit, a confluence of Fe II emission lines through the NUV,
and the bright Mg II $h$ and $k$ resonance lines (see also Hawley et al.,
2007). These “hybrid flare” (HF) or “gradual flare” (GF) (see Kowalski et al.,
2013) events exhibit the largest Balmer jumps that have been detected
spectroscopically in dMe flares, and Kowalski et al. (2019b) argue that other
dMe events that are categorized as “impulsive flare” (IF) events according to
their broadband time evolution, smaller Balmer jumps, and hotter blackbody
fits to the optical (Kowalski et al., 2013) and $U$-band ($\lambda=3260-3940$
Å) continua (Fuhrmeister et al., 2008) yet require spectroscopic investigation
at shorter wavelengths. The optical spectral properties of impulsive-type M
dwarf flares are crucial in our understanding of fundamental flare physics
because they are not reproduced in simulations with typical, solar-type
electron beam (Allred et al., 2006) or intense XEUV radiation fields (Hawley
and Fisher, 1992); large continuum optical depths are required in the flare
chromosphere (e.g., Livshits et al., 1981; Kowalski et al., 2015) or
photosphere.

In this paper, we present radiative-hydrodynamic model predictions of the NUV
flare continuum during the Great Flare of AD Leo (Hawley and Pettersen, 1991),
a particularly well-studied, impulsive-type superflare. The optical emission
line data have previously been modeled in detail in Hawley and Fisher (1992)
using NLTE, X-ray backwarming calculations and in Allred et al. (2006) with
electron beam heating with the RADYN code. However, comprehensive models of
the powerful optical continuum radiation and the broadening of the hydrogen
Balmer line series have not yet been addressed. The multi-wavelength spectra
of this event have been widely utilized for empirically-driven models of
exoplanet photochemistry and surface UV dosages (Segura et al., 2010; Venot et
al., 2016; Ranjan et al., 2017; Tilley et al., 2019; Estrela et al., 2020).

This paper is organized as follows. In Section 3, the AD Leo Great Flare
spectrum and photometry data that are used for model fitting are briefly
reviewed. Section 4 describes the radiative-hydrodynamic flare models, which
are fit to the H$\gamma$ emission line in the spectrum corresponding to the
rise and peak phases of the Great Flare (Section 5). We calculate the Balmer
line merging in the spectral region around the Balmer limit (Section 5.3) to
further justify our two-component model fitting approach. We consider the FUV
spectrum during the early impulsive phase and independently fit to the
observed continuum distribution of the Great Flare in order to make a new
continuum model prediction for the NUV wavelength range that was not observed
during this time (Section 5.4). The interpretation of the results is discussed
in Section 6 in terms of the spatial distribution of intensity in an image of
a well-studied solar flare; we present new calculations for habitable zone
UV-C fluxes in Section 6.2. We conclude in Section 7.

##  3 Observations

###  3.1 The 1985 April 12 Great Flare of AD Leo

The Great Flare of AD Leo was a large-amplitude, superflare event with an
energy of nearly $10^{34}$ erg emitted in the $U$ band. The available data are
the broadband $UBVR$ photometry and the optical spectra covering
$\lambda=3560-4400$ Å at a resolving power of $R\approx 1240$. The exposure
times of the spectra varied between one and three minutes (see Hawley and
Pettersen, 1991, for details). Here, we model the spectrum that integrated
over most of the rise and first peak, labeled as the “542s” spectrum in Hawley
and Pettersen (1991) and “S# 36” in the analysis of Kowalski et al.
(2013)222Segura et al. (2010) refers to the spectrum at 915 s as the peak
spectrum; this is S# 39 in the labeling scheme of Kowalski et al. (2013) and
corresponds to the second, lower-amplitude peak in the impulsive phase.. The
light curve of the $U$ band is shown in Figure 1 with the integration time of
the rise/peak spectrum indicated.

The Great Flare exhibits all of the spectral and light curve characteristics
of a highly “impulsive-type” stellar flare, according to the “IF”
classification in Kowalski et al. (2013). Specifically, the blue-optical
spectra were fit with a color temperature of $T\approx 11,600$ K, and small
Balmer jump in the $U$ band suggests that Balmer recombination radiation is
important at shorter wavelengths (Fig. 9 of Kowalski et al., 2013). During the
early impulsive phase, a FUV spectrum (described below) constrains the peak of
the continuum to the $U$ band with a turnover toward shorter wavelengths that
was found to be most consistent with a $T=8500-9500$ K blackbody among the
models that were available at the time (Hawley and Fisher, 1992). Hawley and
Pettersen (1991) analyzed the highly broadened, symmetric wings of the Balmer
series, which were attributed to the Stark effect. The full-width evolution of
the hydrogen Balmer H$\gamma$ emission line from Hawley and Pettersen (1991)
is reproduced in Figure 1. The rise/peak spectrum corresponds to the first
observation that exhibits very broad H$\gamma$ wings.

The AD Leo Great Flare was observed with ultraviolet spectroscopy with the
International Ultraviolet Explorer (IUE). A FUV spectrum was observed in the
short-wavelength channel (SWP; $\lambda=1150-2000$ Å), which integrated over
the first 900 s of the flare (up until 4:55 UT) and included 41 minutes of
quiescence. The fluxes in the major emission lines and in the continuum
longward of 1780 Å saturated the detector. NUV spectral observations in the
long wavelength band (LWP; $\lambda=1900-3100$ Å) of IUE started at 5:00 UT,
which is about midway during the second, fast decay phase in the $U$-band
light curve in Figure 1. The LWP observation from 5:00–5:20 UT was split into
five sub exposures, each $3-8$ minutes in duration (see Fig. 1 of Hawley and
Pettersen (1991) for the $\lambda=2000$ Å and $2800$ Å continuum flux
evolution over the first set of five sub-exposures). In the flare, some
emission lines in the NUV were saturated as well (see Segura et al., 2010, for
details about how these data have been utilized by interpolation and binning).
Since there were no NUV spectra covering the rise and peak phases, we do not
consider the IUE/LWP spectra further in this study. For detailed descriptions
of the reduction and analyses of the IUE spectra, we refer the reader to
Hawley and Pettersen (1991).

###  3.2 2014 Mar 29 X1 Solar Flare

The 2014 March 29 GOES class X1 ﬂare (SOL20140329T17:48) is one of the best-
observed and most-widely studied solar flares from Sunspot Cycle 24 (e.g.,
Heinzel and Kleint, 2014; Battaglia et al., 2015; Aschwanden, 2015; Young et
al., 2015; Rubio da Costa et al., 2016; Woods et al., 2017; Rubio da Costa and
Kleint, 2017; Polito et al., 2018; Kleint et al., 2018; Zhu et al., 2019). The
flare was observed by the Interface Region Imaging Spectrograph (IRIS; De
Pontieu et al., 2014) with FUV and and NUV longslit spectroscopy (see, e.g.,
Kleint et al., 2016; Kowalski et al., 2017a, for detailed descriptions of
these spectra). Narrow-band ($\Delta\lambda\sim 4$ Å), slit jaw images (SJI)
in the NUV at Mg II 2796 (SJI 2796) and at 2830 Å (SJI 2832) are available for
contextual information about the flare brightenings that cross the IRIS slit.
For this study, a level-2, SJI 2832 image at 17:46 UTC is retrieved from the
IRIS data archive hosted at the Lockheed Martin Solar and Astrophysical
Laboratory333https://iris.lmsal.com/search/. The SJI 2832 image corresponds to
the early impulsive phase of the hard X-rays at $E\geq 25$ keV and has been
analyzed in Kowalski et al. (2017a). The IRIS slit location stepped through
the ribbons in this flare, resulting in a cadence of 75 s for the raster and
SJI 2832 images. Following previous analyses (e.g., Kowalski et al., 2017a),
we convert the level-2 data in units of DN s-1 pix-1 to an equivalent,
constant intensity value, $\langle I_{\lambda}\rangle_{\rm{SJI}}$, over the
SJI bandpass using the time-dependent effective area curves (Wülser et al.,
2018) provided by the IRIS mission.

X-ray imaging data from the Reuven Ramaty High-Energy Solar Spectroscopic
Imager (RHESSI; Lin et al., 2002) were retrieved from the new RHESSI data
archive444https://hesperia.gsfc.nasa.gov/rhessi_extras/flare_images/image_archive_guide.html.
We use the 6–12 keV and 50–100 keV imaging over the time-interval of
17:45:58.7 to 17:46:32.0. We refer the reader to Battaglia et al. (2015) and
Kleint et al. (2016) for higher temporal and spatial resolution analyses of
the RHESSI data for the 2014 March 29 flare.

Figure 1: The $U$-band light curve (right axis) of the Great Flare of AD Leo
and the full-width (left axis) of the H$\gamma$ emission line from Hawley and
Pettersen (1991). The $U$-band flux is normalized to quiescence. The start and
stop times of the exposure corresponding to the rise/peak phase spectrum is
indicated with two vertical dotted lines.

##  4 Radiative-Hydrodynamic Flare Models

To model the rise/peak phase spectrum of the AD Leo flare, we use results from
a grid of radiative-hydrodynamic (RHD) models calculated with the RADYN code
(Carlsson and Stein, 1992, 1995, 1997, 2002; Allred et al., 2015). All of the
details about the simulation setup will be described in a separate paper
(Kowalski et al. in preparation), but a brief summary is presented here. To
simulate flare heating, we model the energy deposition from a power-law
distribution (hereafter, “beam”) of electrons, which is calculated in a 1D
magnetic loop of half-length $10^{9}$ cm, a constant surface gravity of log10
$g/[$cm s${}^{-2}]=4.75$, and a uniform cross-sectional area. The effective
temperature of the starting atmosphere is $T_{\rm{eff}}\approx 3600$ K (see
the Appendix of Kowalski et al., 2017b, for details regarding the starting
atmosphere). The equations of mass, momentum, internal energy, and charge are
solved on an adaptive grid with the equations of radiative transfer and level
populations for hydrogen, helium, and Ca II. The electron beam is injected at
the loop apex with a ramping flux to a maximum value at $t=1$ s, followed by a
decrease until $t=10$ s according to the pulsed injection profile prescription
in Aschwanden (2004). The heating rate as a function of depth is determined by
the steady-state solution to the Fokker-Planck equation for energy loss and
pitch angle scattering due to Coulomb collisions using the module that was
further developed into the FP code (Allred et al., 2020). The heating rate is
recalculated at every time-step in the radiative-hydrodynamic simulation (see
Allred et al., 2015). In this first generation of models, return current
electric fields and magnetic mirroring forces (Allred et al., 2020) are not
considered. However, hydrogen Balmer line spectra (H$\alpha$, H$\beta$,
H$\gamma$) are properly modeled using the Doppler-convolved, TB09+HM88 line
profile functions (Smith et al., 1969; Vidal et al., 1970, 1971, 1973; Hummer
and Mihalas, 1988; Tremblay and Bergeron, 2009), which accurately capture the
pressure broadening from ambient, thermal electrons and ions in the density
regimes of flare chromospheres (Kowalski et al., 2017b, 2022).

Stellar flare hard X-ray emission is below current detection limits, except in
the most energetic events (e.g., Osten et al., 2007, 2016), and millimeter /
radio observations at optically thin frequencies have been reported only
recently (MacGregor et al., 2020). The paucity of direct constraints on
accelerated electrons in stellar flares thus necessitates a grid of models
covering a large parameter space of electron beam heating. Our grid of M dwarf
flare models includes a large range of low-energy cutoff ($E_{\rm{c}}$)
values: $17,25,37,85,150,200,350,$ and $500$ keV. All of the selected models
from the grid are calculated for injected electron beam number fluxes with
hard power-law indices of $\delta=2.5-4$, which are consistent with available
stellar flare constraints (Osten et al., 2007; MacGregor et al., 2018, 2020,
2021). The peak injected beam energy flux densities (hereafter, “flux”) span
four orders of magnitude: $10^{10}$ (F10), $10^{11}$ (F11), $10^{12}$ (F12),
and $10^{13}$ (F13) erg cm-2 s-1. For this study, we select five models with
maximum (“m”) injected beam fluxes of $10^{13}$ erg cm-2 s-1 (“mF13”), low-
energy cutoffs of $E_{c}=37,85,150,200,$ and $500$ keV, and a power-law index
of $\delta=3$; these are referred to as the mF13-37-3, mF13-85-3, mF13-150-3,
mF13-200-3, and mF13-500-3 models, respectively555A corresponding grid of
models is calculated using a constant beam flux injection; these models are
indicated with a “c”-prefix, such as cF13-85-3.  These models are especially
notable because they reproduce $T\approx 10,000$ K color temperatures in the
blue-optical wavelength range and small Balmer jump ratios, as reported in
many M dwarf flare spectral observations (e.g., Mochnacki and Zirin, 1980;
Fuhrmeister et al., 2008; Kowalski et al., 2013, 2016). The justification for
selecting these high-flux models will be discussed further in Section (5.2).

The mF13-37-3 model is a recalculation of the RADYN simulation from Kowalski
et al. (2016) (see also Kowalski et al. (2015)) with a pulsed beam flux
injection. The atmosphere in the new model follows a similar evolution with
the development a dense ($n_{e}=5\times 10^{15}$ cm-3), cool chromospheric
condensation at $t\approx 2.2$ s. The mF13-85-3 and mF13-150-3 models produce
relatively small amounts of coronal heating and relatively fast upflows
($\approx 5-20$ km s-1) in the flare chromosphere because most of the beam
energy is deposited into the deep chromosphere. Without magnetic mirroring and
return current electric field forces, long-lasting chromospheric condensations
do not develop as in the mF13-37-3 model. However, large ambient electron
densities ($n_{e}\approx 1-7\times 10^{15}$ cm-3) are attained due to thermal
ionization of hydrogen by the beam heating in low-lying chromospheric layers
(see the Appendices of Kowalski et al., 2017b, for a description of several,
similar large, low-energy-cutoff models). These charge densities refer to the
atmospheric (ambient/thermal) proton and electron densities that pressure
broaden the hydrogen lines that we model in Sections 5.2 – 5.3. The nonthermal
electron densities are many orders of magnitude smaller in the chromosphere.
The large, low-energy-cutoff models represent a semi-empirical approach in the
spirit of the static flare atmospheres of Cram and Woods (1982), but the RADYN
models include time-dependent atmospheric thermodynamics that are calculated
self-consistently with beam heating.

In addition to the F13 models, a lower beam flux that has been used to model
IRIS NUV spectra of a solar flare (Kowalski et al., 2017a) and the broadening
of the hydrogen Balmer series (Kowalski et al., 2022) has been injected into
our M dwarf atmosphere for a duration of $15$ s. The electron beam parameters
($E_{c}=25$ keV, $\delta=4$, flux of $5\times 10^{11}$ erg cm-2 s-1) for this
model (“c15s-5F11-25-4”) were selected to be consistent with those that were
inferred through the collisional thick target modeling of RHESSI hard X-ray
data of the 2014 Mar 29 solar flare (Kleint et al., 2016). Similar to the
analogous simulation in the solar atmosphere, a dense chromospheric
condensation develops by $t\approx 4$ s with densities of $n_{e}\approx
5\times 10^{14}$ cm-3. We also calculate a model (m5F11-25-4) with a shorter,
pulsed injection profile in the same form as for the pulsed F13 beams. Several
other models that are considered in this work are two intermediate flux models
(mF12-37-3 and m2F12-37-2.5) with hard power-law distributions ($\delta=2.5$
and $3$) and intermediate low-energy cutoff values ($E_{c}=37$ keV). A similar
model to the mF12-37-3 beam was analyzed in Namekata et al. (2020), who found
satisfactory agreement between the broadening of the hydrogen Balmer $\alpha$
emission line in the model and in the observation of a superflare event from
AD Leo.

The parameters of the RHD models that are used in the remainder of this work
are summarized in Table 1.

Table 1 – RADYN Electron Beam Heating Models
---
Model |  Beam Flux |  $t_{1/2}$ |  $t_{\rm{end}}$ |  $E_{c}$ |  $\delta$ |  C4170′ |  $F^{\prime}_{\rm{H}\gamma}$ |  $F^{\prime}_{\rm{H}\gamma}\div$
C4170′ |  H$\gamma$ Eff.
Width
|  |  (s) |  (s) |  (keV) |  |  (erg cm-2 s-1 Å-1) |  (erg cm-2 s-1) |  (Å) |  (Å)
mF13-85-3 |  max F13 |  2.3 |  10 |  85 |  3 |  $1.14\times 10^{8}$ |  $1.62\times 10^{9}$ |  $14.2$ |  $9.1$
mF13-150-3 |  max F13 |  2.3 |  10 |  150 |  3 |  $1.54\times 10^{8}$ |  $1.16\times 10^{9}$ |  $7.5$ |  $10.3$
mF13-200-3 |  max F13 |  2.3 |  10 |  200 |  3 |  $1.73\times 10^{8}$ |  $6.21\times 10^{8}$ |  $3.6$ |  $10.4$
mF13-500-3 |  max F13 |  2.3 |  10 |  500 |  3 |  $1.87\times 10^{7}$ |  $-1.37\times 10^{9}$ |  $-107$ |  $-47.7$
mF12-37-3 |  max F12 |  2.3 |  10 |  37 |  3 |  $3.51\times 10^{6}$ |  $5.45\times 10^{8}$ |  $155.2$ |  $3.8$
m2F12-37-2.5 |  max 2F12 |  2.3 |  10 |  37 |  2.5 |  $1.72\times 10^{7}$ |  $8.65\times 10^{8}$ |  $50.3$ |  $5.8$
mF13-37-3 |  max F13 |  2.3 |  10 |  37 |  3 |  $7.94\times 10^{7}$ |  $2.22\times 10^{9}$ |  $27.9$ |  $12.1$
m5F11-25-4 |  max 5F11 |  2.3 |  10 |  25 |  4 |  $6.35\times 10^{5}$ |  $2.95\times 10^{8}$ |  $464.6$ |  $2.2$
c15s-5F11-25-4 |  const 5F11 |  15 |  15 |  25 |  4 |  $3.30\times 10^{6}$ |  $1.33\times 10^{9}$ |  $402.4$ |  $5.3$
Table 1:  $t_{1/2}$ is the full-width-at-half-maximum of the injected beam
heating pulse; $t_{\rm{end}}$ indicates the end of the simulation and the
duration over which the temporal averages of the model spectra are calculated.
The effective width of H$\gamma$ is defined as the integral of the continuum-
subtracted, peak-normalized emission line profile (Kowalski et al., 2022);
note that the H$\gamma$ profile is an absorption profile in the mF13-500-3
model.

##  5 Model Spectrum Analysis

We leverage the new hydrogen pressure broadening profiles that have been
incorporated into RADYN to examine the models that reproduce the Balmer jump
strength and blue-optical continuum color temperature. The Balmer H$\gamma$
emission line broadening and nearby blue-optical continuum fluxes of the Great
Flare are the focus of our modeling analyses (Sections 5.1-5.2). In Section
5.3, we extend the detailed calculations to spectra of the entire Balmer line
series.

###  5.1 Average Model Line-to-Continuum Ratios

We first describe a simple method that allows comparisons of 1D loop models to
the Great Flare spectra, which are not spatially resolved. Over an exposure
time of 180 s, we reasonably expect that many sequentially ignited, spatially
distinct, $\Delta t=10$ s pulses accumulate flare radiation in the spatially
unresolved, observed flare spectrum. For each RHD model, we thus calculate a
coadded spectrum from the radiative surface flux spectra at every $\Delta
t=0.2$ s by temporal averaging over the duration given by $t_{\rm{end}}$ in
Table 1. These coadded spectra are used in all analyses, unless otherwise
indicated.

Several coadded F13 model spectra around the H$\gamma$ emission line are
normalized to the observed continuum flare-only flux averaged over
$\lambda=4155-4185$ Å in the Great Flare (Figure 2). With an older, less
accurate prescription of hydrogen line pressure broadening, Kowalski et al.
(2015) found that a coadded F13 model with a double power-law beam
distribution and $E_{c}=37$ keV was an adequate model of the early/mid rise
phase of a giant flare from the dM4.5e star YZ CMi. As Figure 2 clearly
demonstrates, the mF13-37-3 model profile with the updated hydrogen broadening
is far too broad even though times when the chromospheric condensation is not
extremely dense and the emission lines are relatively narrow are included in
the coadd. The coadded mF13 spectra from the models with large, low-energy-
cutoffs ($E_{c}=85-150$ keV) adequately account for some or all of the flux in
the H$\gamma$ wings, but these models of deep flare heating vastly under-
predict the relative H$\gamma$ line-peak flux.

Figure 2:  Comparisons of several F13 model spectra of H$\gamma$ directly from
RADYN (and thus have relatively coarse wavelength sampling in the far wings)
to the Great Flare rise/peak phase spectrum. Each model has been scaled to the
observed continuum flux, C4170′. The mF13-37-3 model prediction is far too
broad, while the mF13-85-3 and mF13-150-3 models do not exhibit an amount of
broadening that exceeds the observation in the line wings. The dashed line
shows a detailed continuum spectrum that is interpolated to the wavelengths
over the H$\gamma$ line.

To quantitatively assess the models, we calculate several quantities from the
detailed H$\gamma$ line profiles and the continuum spectra. Specifically, we
calculate the continuum-subtracted, preflare-subtracted, line-integrated flux
over the H$\gamma$ emission line (hereafter, $F^{\prime}_{\rm{H}\gamma}$), the
preflare-subtracted flux666Following traditional use, we denote flare-only
quantities with a prime-symbol, and we refer to an observed
spectral/monochromatic/specific flux density at Earth as the “flux”; we use
“spectral luminosity” to refer to the luminosity per unit wavelength. at
$\lambda=4170$ Å (hereafter, C4170′), and the ratio of these quantities (Table
1). The 5F11 and F12 models produce ratios that are far too large compared to
$F^{\prime}_{\rm{H}\gamma}$/C4170${}^{\prime}\approx 20$ that is calculated
from the Great Flare observation (Kowalski et al., 2013), while the ratios
from the F13 models with large low-energy cutoffs are too small. This
motivates linear superpositions of two RHD model components – one high-flux
(F13) component and one lower-flux (5F11 or F12) component – to
comprehensively reproduce the observed line-to-continuum ratios, the shape of
the H$\gamma$ wing broadening, and the H$\gamma$ emission line peak flux.

###  5.2 Two-component Model Fits to H$\gamma$ Line in the AD Leo Great Flare

A linear superposition of a high flux (F13), large low-energy cutoff beam
component and a lower flux (5F11, F12, or 2F12), smaller low-energy cutoff
beam component is represented by the equation (Eq. 1),

$f^{\prime}_{\lambda,\rm{Model}}=\left(X_{\rm{F13}}\,S^{\prime}_{\lambda,\rm{F13}}+X_{\rm{5F11}}\,S^{\prime}_{\lambda,\rm{5F11}}\right)\frac{R^{2}_{\rm{star}}}{d^{2}}$
(1)

where $f^{\prime}_{\lambda,\rm{Model}}$ is the model flux at Earth. The
filling factor, $X$, is the exposure-time-averaged fraction of the visible
hemisphere of the star that is flaring with the temporal coadd of the
radiative surface flux spectrum, $S_{\lambda}$, calculated from each RHD model
component. $R_{\rm{star}}=3\times 10^{10}$ cm is the radius of AD Leo, and
$d=1.5\times 10^{19}$ cm is the distance to the star. The preflare model
surface flux spectrum is subtracted to give the flare-only, model surface flux
spectra, $S^{\prime}_{\lambda}=S_{\lambda}-S_{\lambda,o}$, in Equation 1. To
mitigate systematic errors in the far wings of the H$\gamma$ line (Fig. 2),
which are coarsely sampled at 31 wavelength points in the RADYN calculation,
we recalculate777All analyses have been performed on both the original
31-wavelength array and the 327-wavelength array. the emergent surface flux
spectra using a Feautrier solver on a 327 point wavelength grid with the
frequency-independent, non-LTE source function from RADYN and a four-point,
third-order interpolation of the line profile opacity from the Appendix of
Vidal et al. (1973). The emergent radiative flux spectra of H$\gamma$ are
time-averaged over the duration of each simluation, convolved with a Gaussian
with a full-width-at-half maximum that corresponds to the instrumental
resolution of 3.5 Å, and binned to the wavelengths of the Great Flare spectra.

We perform an inverse-variance-weighted, linear least-squares fit of the two
parameters $X_{\rm{F13}}$ and $X_{\rm{5F11}}$ to the observed spectrum around
the H$\gamma$ line. The model surface flux spectra are the basis functions in
the $n\lambda\times 2$ design matrix, $\Lambda$. The maximum likelihood (ML)
estimates of the parameters are given by the standard matrix equation (Eq. 2),

$\hat{\vec{X}}_{\rm{ML}}=\begin{pmatrix}\hat{X}_{\rm{F13},\rm{ML}}\\\
\hat{X}_{\rm{5F11},\rm{ML}}\end{pmatrix}=(\Lambda^{\rm{T}}C_{\vec{f}^{\prime}_{\lambda}}^{-1}\Lambda)^{-1}(\Lambda^{\rm{T}}C_{\vec{f}^{\prime}_{\lambda}}^{-1}\vec{f}^{\prime}_{\lambda})$
(2)

where $\vec{f}^{\prime}_{\lambda}$ is the observed $n\lambda\times 1$ flare-
only flux at Earth (hereafter dropping the vector notation) as a function of
wavelength and the $n\lambda\times n\lambda$ covariance matrix $C$ is
populated with independent, Gaussian uncertainties, which are estimated from
the data, $f^{\prime}_{\lambda}$. The wavelength range from
$\lambda=4320-4361$ Å is used in the fits, which are performed for all
combinations of two models from the grid. The models in Table 1 are among the
combinations with the lowest values of $\chi^{2}$ and were thus chosen as the
focus of this study. Figure 3 shows the result of one of the best fits with
$\chi_{\rm{dof}}^{2}=1.2$ for $21$ degrees of freedom (dof). The quality of
this fit is representative of many such results with two RHD component spectra
consisting of a high-flux, large low-energy cutoff beam and a lower-flux,
smaller low-energy cutoff beam. In the right panel of Figure 3, we show
likelihood contours for this fit to visualize typical uncertainties on the
maximum-likelihood estimates of the parameters. The comparisons of the
inferred filling factors of the high-flux (e.g., F13) and lower-flux (e.g.,
5F11) models in each fit will be more useful in comparison to solar flare data
(Section 6.3), and therefore we report values of
$X_{\rm{rel}}=\frac{X_{\rm{5F11}}}{X_{\rm{F13}}}$. For the model combination
in Figure 3, the best-fit parameters and standard error propagation give
$X_{\rm{rel}}=2.28\pm 0.08$. The value of
$F^{\prime}_{\rm{H}\gamma}/$C4170${}^{\prime}=19.9$ is remarkably consistent
with this measured quantity from the observed flare spectrum.

Figure 3:  (Left) A representative example of a satisfactory, two-component
(mF13-85-3, m5F11-25-4) RHD model fit to the observed H$\gamma$ line profile
and nearby continuum flux in the Great Flare of AD Leo. (Right) Constant
joint-likelihood contours for the model fit in the left panel. The maximum-
likelihood estimates of the parameters and $1\sigma$, Gaussian marginal
uncertainties are $\hat{X}_{\rm{F13,ML}}=2.78\times 10^{-3}\pm 2\times
10^{-5}$ and $\hat{X}_{\rm{5F11,ML}}=6.34\times 10^{-3}\pm 2.3\times 10^{-4}$
with a correlation coefficient of $\rho=-0.53$.

The results for several representative combinations of models in Table 1 with
small values of $\chi^{2}$ are presented in Table 2, indicating a range of
value of $X_{\rm{rel}}\approx 1.5-10$. Since these fits include only the
H$\gamma$ line data for this flare, the relatively small differences in the
various $\chi^{2}$ values in Table 2 are not strictly indicative of a global
minimum. The vast majority of all model combinations from the entire grid,
however, result in $\chi^{2}$ values far in excess of those shown in Table 2.
We think that an exploratory approach to the model grid predictions is more
productive than an effort to find one model that best satisfies all
constraints from the data, given the many assumptions in the RHD modeling
(e.g., specific choice of low-energy cutoff values in the grid, assumptions of
constant-area loop geometry and a constant power-law index value over each
pulse). The small values of $\chi^{2}$ are thus most informative for limiting
the vast parameter space for further comparisons of our general modeling
paradigm to the multi-wavelength data of the Great Flare.

Table 2 – Least-Squares Fitting Results for H$\gamma$
---
F13 Model |  $X_{\rm{F13}}$ |  Lower-Flux Model |  $X_{\rm{rel}}$ |  $\chi^{2}_{\rm{dof}}$
mF13-85-3 |  0.0028 |  m5F11-25-4 |  2.3 |  1.2
mF13-150-3 |  0.0022 |  m5F11-25-4 |  4.5 |  4.1
mF13-150-3 |  0.0019 |  c15s5F11-25-4 |  1.6 |  7.9
mF13-200-3 |  0.0017 |  mF12-37-3 |  4.9 |  1.9
mF13-500-3 |  0.0009 |  m2F12-37-2.5 |  9.6 |  1.7
Table 2:  Least-squares fitting results for the H$\gamma$ emission line in the
Great Flare of AD Leo.

###  5.3 Models of the Balmer Limit

In order to robustly extrapolate the models of this flare to the NUV
wavelength range that was not observed during most of the impulsive phase, the
Balmer jump strength in the observation should be satisfactorily reproduced.
However, many linear combinations of two models in the RADYN flare grid
produce small Balmer jump ratios that are consistent with the measured range
($\chi_{\rm{flare}}\approx 1.7-1.9$; Kowalski et al., 2013) from the spectrum
of the Great Flare. For supplementary constraints, we compare the details of
the merging of the Balmer series at $\lambda=3646-4000$ Å. The last visible
Balmer emission line is often used as an indication of the electron density,
and Hawley and Pettersen (1991) discusses that the Balmer lines up to and
including H15 or H16 are resolved in the Great Flare spectra. Thus, our RHD
model combinations should reproduce this salient property.

We use the RH code (Uitenbroek, 2001) with a 20-level hydrogen atom and the
occupational probability modifications to the bound-bound and bound-free
opacities that account for level dissolution at the Balmer limit (Dappen et
al., 1987; Hummer and Mihalas, 1988; Nayfonov et al., 1999; Tremblay and
Bergeron, 2009). The RH calculation setup is the same as described in Kowalski
et al. (2017b). These calculations are intensive because they involve a large
numerical convolution at each atmospheric depth, and not every time-step in
all models readily converges to a solution. To demonstrate a representative
solution with the two-component modeling approach from the previous section,
we use atmospheric snapshots from the mF13-150-3 simulation at
$t=0.0,0.4,1.0,2.0,4.0,6.0,8.0,9.8$ s and a snapshot from the m5F11-25-4
simulation at $t=0.8$ s. The F13 model spectra are coadded, and the preflare
spectrum is subtracted from the two model components. We then use the equation
(Eq. 3),

$\left(F^{\prime}_{\rm{H}\gamma}/\rm{C4170}^{\prime}\right)_{\rm{obs}}=\frac{X_{\rm{rel}}\times
F^{\prime}_{\rm{H}\gamma\rm{,\ 5F11}}+F^{\prime}_{\rm{H}\gamma\rm{,\
F13}}}{X_{\rm{rel}}\times\rm{C4170}^{\prime}_{\rm{5F11}}+\rm{C4170}^{\prime}_{\rm{F13}}}$
(3)

to solve for $X_{\rm{rel}}$ given
$(F^{\prime}_{\rm{H}\gamma}/\rm{C4170}^{\prime})_{\rm{obs}}=20$. The total
two-component model flux spectrum is scaled to the continuum flare flux,
C4170′, in the observed spectrum. For the two-component model combination
above, a value of $X_{\rm{rel}}\approx 3.9$ is obtained, which is close to
$4.5$ that is obtained from fitting the H$\gamma$ line profile (Table 2). We
convolve the flare model with the spectral resolution of the data and show the
result against the observations in Figure 4, which demonstrates consistency
with the observed Balmer jump flux ratio, the detailed merging of the line
series wings, and in the bluest visible Balmer line. Without the additional
narrow-line flux from the 5F11, the highest balmer line in emission is H13,
which is inconsistent with the observations. Without the continuum and Balmer
wing flux from the F13 model, the Balmer jump ratio and the
$F^{\prime}_{\rm{H}\gamma}/\rm{C4170}^{\prime}$ value from the 5F11 model are
far larger (Table 2) than measured from the observed spectrum.

Figure 4: (Top) RH calculations with non-ideal opacity effects at the Balmer
limit. A linear superposition of the mF13-150-3 model, averaged over its
duration, and the m5F11-25-4 model at $t=0.8$ s is shown for the combination
that is constrained by a value of
$F^{\prime}_{\rm{H}\gamma}/\rm{C4170}^{\prime}$ that is measured from the
observed spectrum. The individual component model calculations are shown: the
m5F11 contributes to the narrow hydrogen line flux, the highest order Balmer
lines, and optically thin Balmer continuum flux in addition with the optically
thick Balmer continuum flux from the F13. The mF13 model accounts for nearly
all of the optical continuum flux and far wing radiation. (Bottom) The pseudo-
continuum from the merging of the Balmer H8 – H15 line wings, the dissolved
level continuum between the lines, the fading of the emission line fluxes into
the dissolved-level continuum at $\lambda<3700$ Å, and the bluest Balmer line
(H15) are adequately reproduced in the RHD model superposition. The
wavelengths of the hydrogen series and a helium I line noted by Hawley and
Pettersen (1991) are indicated.

###  5.4 Broadband Continuum Fitting

In this section, we fit the observed continuum fluxes from the FUV to the red-
optical during the early impulsive phase of the AD Leo Great Flare to compare
to the results from fitting to the H$\gamma$ spectrum (Table 2) and high-order
series merging (Section 5.3).

Figure 5 shows a comparison of the detailed continuum fluxes for several
combinations of the models that satisfactorily explain the early-impulsive
phase, blue-optical spectrum of the Great Flare of AD Leo. A representative
RHD model combination (mF13-150-3, m5F11-25-4 ($t=0.8$s)) from Section 5.3
exhibits a peak at $\lambda\approx 2350-2400$ Å followed by a turnover toward
shorter wavelengths. Qualitatively, these properties are consistent with FUV
constraints of this superflare and other, smaller flares from AD Leo (Hawley
et al., 2003), but the IUE/SWP observation allows a more detailed comparison.
To adjust the flux calibration of the data for the different exposure times
between the early-impulsive phase, IUE/SWP spectrum ($t<900$ s in Fig. 1) and
the $\lambda>3560$ Å ground-based spectrum ($t=542\pm 90$ s in Fig. 1; see
Section 3), we apply the relative scaling between the $U$-band and the FUV
continuum flux at $\lambda\approx 1600$ Å within the first 900 s of the flare
that is presented in the upper left panel of Fig. 11 of Hawley and Fisher
(1992). The relative scale factor ($1.3$) is used to adjust the lower envelope
of the SWP continuum flux relative to a synthetic $U$-band flux that we
calculate from the blue-optical spectrum. The scaled and original IUE/SWP
spectra are shown in Figure 5. We also include the $V$\- and $R$-band
photometry from the same figure in Hawley and Fisher (1992) and apply the
scaling in the same way as for the FUV continuum. We calculate two-parameter,
linear least-squares fits to the seven flare-only flux measurements of the
continuum in the Great Flare888Instead of $U$ and $B$-band photometry used for
model fitting in Hawley and Fisher (1992), we use C3615′, C4170′, and C4400′
calculated from averages of the continuum fluxes at $\Delta\lambda\approx 30$
Å around $\lambda=3615,4170,4400$Å, respectively.. Minimizing $\chi^{2}$
(Section 5.2) gives several combinations of models with very large low-energy
cutoffs ($E_{c}=350-500$) as the $X_{\rm{F13}}$ model component superimposed
with the m2F12-37-2.5 model spectrum. Note, the m2F12-37-2.5 model is the only
simulation in our RADYN grid with such a hard, $\delta<3$, electron beam
power-law index.

The best-fit superposition of the mF13-500-3 and m2F12-37-2.5 radiative flux
spectra is shown in Figure 5 (top panel) with $X_{\rm{rel}}\approx 11.9$.
Notably, this fit comprehensively accounts for the slope of the lower-envelope
of the FUV flare spectrum, the Balmer jump strength, and the optical continuum
constraints. The middle, left panel of Figure 5 shows the contributions of the
individual model components to the spectral luminosity of the flare continuum
from the top panel. The mF13-500-3 accounts for most of the FUV continuum
luminosity, whereas the Balmer jump in the m2F12-37-2.5 contributes a larger
fraction in the NUV and in the $U$ band. At optical wavelengths, relative
contributions to the blue continuum spectral luminosity are about equal, but
the lower beam-flux model is larger toward near-infrared wavelengths. The
comparisons of the surface flux spectra without adjustments by the best-fit
filling factors emphasize that the F13 model is the much brighter source at
all wavelengths. A fit using these two model component to the observed
H$\gamma$ line profile (Section 5.2) is shown in Figure 5 (bottom, right
panel); the fit is excellent and, moreover, returns a similar, independent
estimate for the parameter $X_{\rm{rel}}=9.6$ (Table 2).

The fully-relativistic electron beam parameters of the mF13-500-3 beam are
rather extreme, but they are not without precedent and sufficient semi-
empirical necessity. Kowalski et al. (2017b) used a superposition of three
RADYN simulations to model the decay phase spectra of a superflare from the
dM4.5e star YZ CMi. A RADYN model with a constant electron beam energy flux
injection of $2\times 10^{12}$ erg cm-2 s-1, a low-energy cutoff of
$E_{c}=500$ keV, and a power-law index of $\delta=7$ was used to explain the
spectra of a secondary flare event, which exhibit features that are similar to
an A-type star photospheric spectrum: namely, broad Balmer lines and a Balmer
jump “in absorption” (see also Kowalski et al., 2012, 2013). Secondary flare
events in the decay phase of a large flare from the young G-dwarf, EK Dra,
were reported in Ayres (2015) to exhibit a response in only the FUV continuum.
Finally, we note that increasing the value of $X_{\rm{rel}}$ after the peak
flare phase may be able to explain the relatively rapid nature of the FUV
continuum radiation that has been reported in other M dwarf flares (Hawley et
al., 2003; MacGregor et al., 2020).

Figure 5:  (Top) Models of the early-impulsive phase broadband continuum flux
distribution of the Great Flare of AD Leo compared to the IUE/SWP spectrum,
optical ground-based spectrum, and broadband $V$ and $R$ photometry. All
observed fluxes have been adjusted to the synthetic $U$-band flux according to
the broadband distribution at $t=0-900$ s in Fig. 11 of Hawley and Fisher
(1992). The wavelength-binned, flare-only fluxes that are used to fit the
models are indicated by square symbols with a best-fit, two-component RHD
model continuum spectrum shown as the solid red line. Other model predictions
are scaled to the observations as follows: the blackbody functions are scaled
to the $R$ band flux observation, and the other two RHD models are scaled to
the average continuum flux at $\lambda=4155-4185$Å. (Bottom, left) Individual
model components in the best-fit mF13-500-3 $+11.9\times$ m2F12-37-2.5
combination, which compares the relative contributions to the spectral
luminosity of the continuum radiation in the Great Flare. The m2F12-37-2.5
model component is also shown without scaling by the best-fit filling factor
to facilitate direct comparison to the radiative surface flux of the
mF13-500-3. (Bottom, right) Best-fit H$\gamma$ line profile model using the
mF13-500-3 and m2F12-37-2.5 flux spectra gives a similar value of
$X_{\rm{rel}}$ as for the fits to the broadband continuum fluxes.

##  6 Discussion

###  6.1 Summary of Fitting Results

We fit the Great Flare impulsive phase (rise/peak) spectrum using simulations
of electron beam heating from a new grid of RADYN flare models. The data
require two, independent RHD model components, resulting in relative filling
factors ($X_{\rm{rel}}$) of the components between $\approx 1-10$, with the
lower beam flux model component exhibiting the larger filling factor. The
time-evolution of the simulated atmosphere and emergent radiative flux spectra
over each heating pulse is included in these comparisons to the data. Several
examples of fits were presented and discussed. The shape of the H$\gamma$ line
profile far into the wings and nearby continuum flux constrains combinations
of a high-flux model (F13) with large low-energy cutoff values ($E_{c}=85-500$
keV) and a lower-flux model (5F11–2F12) with smaller low-energy cutoff values
($E_{c}=25-37$ keV). In the (mF13-85-3, m5F11-25-4) spectral luminosity, most
of the H$\gamma$ wing broadening and blue-optical continuum radiation is
attributed to the F13 model component. In the (mF13-200-3, mF12) model fits,
most of the blue-optical continuum luminosity is due to the F13 spectrum, but
much of the wing broadening can be attributed to the lower beam flux, F12,
component. This is qualitatively consistent with the modeling results from
Namekata et al. (2020), who found that similar F12 electron beam models
produce satisfactory agreement in the broadening of the H$\alpha$ line in a
superflare. In their work, however, detailed comparisons to the spectra of the
blue-optical continuum radiation were not possible.

We examined the prediction of one of the fits to the Great Flare H$\gamma$
line against the spectrum of the hydrogen series at the Balmer limit; there is
remarkable agreement with the highest-order Balmer line in emission and with
other features in the rise/peak spectrum (Figure 4). Models are also
independently fit to the broadband photometry and spectral distribution during
the first 900s of the Great Flare, and the superposition of spectra from the
mF13-500-3 (or mF13-350-3) and the m2F12-37-2.5 RHD model components gives an
excellent fit; moreover, this fit results in about the same relative filling
factor as inferred from the H$\gamma$ line profile fitting. For this
combination of models, the relative contributions to the optical continuum
luminosity are comparable, but the $E_{c}=500$ keV model dominates the FUV
flare luminosity. In all model combinations, the F13 model component produces
the brightest continuum surface flux.

In this section, we use the results from the fits to discuss the implications
for models of the NUV radiation environment of the habitable zones of low-mass
flare stars (Section 6.2). Then, we examine a high-spatial resolution image of
a widely-studied X-class solar flare to speculate on the origin of these two
spectral components in terms of solar flare phenomenology (Section 6.3). We
show how the relative filling factors of the two model components are
consistent with the relative areas of solar flare kernels and ribbon
intensities, respectively, in the impulsive phase of this solar flare. In
Section 6.4, we discuss further empirical investigation to anchor the two-
component continuum and H$\gamma$ broadening models of stellar flares in
reality.

###  6.2 The NUV Radiation Field in Habitable Zones of Low-Mass, Flare Stars

The detailed RHD models provide insight into the magnitude of the possible
systematic errors for the inferred NUV radiation field in the habitable zones
of low-mass flare stars. The RHD spectra in the NUV reveal that simple
extrapolations from flare photometry in the red-optical and near-infrared
(e.g., from the _Kepler_ or TESS bands) that do not account for the Balmer
jump strength, may result in rather large systematic modeling errors. We scale
a $T=9000$ K and $T=10,000$ K blackbody to the observed $R$-band flux of the
Great Flare in Figure 5 (top panel). Compared to the RHD models, the blackbody
models under-predict the $\lambda=1800-3646$ Å flare-only flux by factors
ranging from $1.2$ to $2.0$. The peaks and slopes of the UV and $U$-band
continuum spectra are largely in disagreement as well. Scaling all models to a
common RHD continuum flux at a redder continuum wavelength, $\lambda=7810$ Å,
that is closer to the central wavelength of the TESS white-light band (Figure
5) generally results in larger underestimates of the NUV continuum flare-only
flux by factors up to $2.6$. The inadequacies of single-temperature, blackbody
models are even more evident at $\lambda=1100-1800$ Å and in the expected
amount of Lyman continuum fluxes at $\lambda\lesssim 911$ Å (not shown) that
are present in the RHD model spectra.

The recent laboratory experiments of Abrevaya et al. (2020) measured survival
curves of microorganisms that were irradiated by sustained fluxes of
monochromatic NUV light at $\lambda=2540$ Å. In the worst-case scenario of
direct irradiation, they found that a large UV-C flux from a superflare in the
habitable zone ($d=0.0485$ au; Anglada-Escudé et al., 2016) of the dM5.5e star
Proxima Centauri fails to terminate biological function in a small but non-
negligible fraction of the initial sample. The UV-C flux999For continuity with
these studies, we momentarily express quantities in S.I. units. of $92$ W m-2
was calculated by scaling a $T=9000$ K blackbody curve to the peak magnitude
change in the Evryscope $g^{\prime}$ bandpass as described in Howard et al.
(2018); we refer the reader to Law et al. (2015) and Howard et al. (2019) for
details about the Evryscope survey. We estimate that the peak $B$-band
($\lambda=3910-4890$ Å) luminosity of the Great Flare of AD Leo was at least a
factor of three larger than the $g^{\prime}$-band ($\lambda\approx 4050-5500$
Å) peak luminosity of the Proxima Centauri superflare. If a flare as luminous
as the Great Flare (and the same in all other regards) were to occur on
Proxima Centauri, the RHD models in Figure 5 predict UV-C, impulsive-phase,
habitable-zone fluxes of $800-1000$ W m-2. This range is rather similar to the
habitable-zone, UV-C fluxes inferred in Howard et al. (2020) using
extrapolations from much higher temperature blackbody fits to broadband
optical and near-IR photometry. As Howard et al. (2020) discuss, it would be
interesting for laboratory experiments to determine whether there is an upper
limit to the UV-C flux at which a microbial population achieves a steady-state
survival fraction.

The pioneering study of Segura et al. (2010) combined the multi-wavelength AD
Leo flare spectra for empirically-driven photochemistry and surface UV dosage
modeling of an Earth-like planet in the habitable zone at $d=0.16$ au from a
dM3 star. In their approach, Segura et al. (2010) used the first IUE/LWP (NUV)
spectrum available (starting at 5:00 UT in Figure 1) to bridge the blue-
optical and IUE/SWP spectra during the early impulsive phase in the first 900
s of Figure 1. This approach assumes that the peak impulsive-phase NUV flare
spectrum is the same as that at the end of the fast decay and start of the
gradual decay phase in this event. At longer wavelengths, this assumption is
not consistent with analyses of more recent, time-resolved spectra (Kowalski
et al., 2013). However, we think that this approach is reasonably justified
given the vagaries inherent in such spectral observations with relatively long
exposure times that are not contemporaneous within the Great Flare.

Our scaling of the IUE/SWP spectrum (Figure 5) follows a different approach
and is consistent with the relative surface fluxes at $\lambda\approx 1600$ Å
and the $U$ band that are shown in the upper left panel of Fig. 11 of Hawley
and Fisher (1992) and in Table 5 of Hawley and Pettersen (1991). In the
IUE/LWP decay phase spectra of the Great Flare, the ratio of the
$\lambda=2800$ Å to $\lambda=2000$ Å continuum fluxes is $\approx 5$ (Hawley
and Pettersen, 1991, see also Segura et al. (2010)), which effectively force
the FUV continuum flux to a lower value relative to the fluxes at longer
wavelengths in the NUV and $U$ band. In our best-fit (mF13-500-3,
m2F12-37-2.5) continuum flux model of the early impulsive phase (Figure 5),
the $\lambda=2800$ Å to $2000$ Å continuum flux ratio is only $1.3$. Thus, one
expects the wavelength-integrated, UV-C flux of this model combination to be a
factor of $\approx 1.5$ larger than the empirical model of Segura et al.
(2010), assuming equal top-of-the-atmosphere fluxes at $\lambda\approx 2800$Å.
However, further comparison reveals that the Segura et al. (2010) composite
flare spectrum is similarly flat at $\lambda\gtrsim 2400$Å. This effect is
apparently due to the large number of blended (and saturated) Fe II and Mg II
emission lines, which are generally much more prominent relative to the flare
continuum radiation in the decay phase (Kowalski et al., 2019b). This
coincidence is fortuitous for many follow-up studies (e.g., Venot et al.,
2016; Tilley et al., 2019) that have adopted the composite NUV flare spectra
from Segura et al. (2010) for photochemistry modeling: the pseudo-continuum of
blended, saturated lines in the decay phase of the Great Flare mimics the
shape of our RHD model continuum distribution that best reproduces the
available observations in the rise and peak phases.

###  6.3 A Solar Flare “Kernel$+$Ribbon” Interpretation of the Great Flare
Rise/Peak Phase

Figure 6:  IRIS SJI 2832 image during the hard X-ray impulsive phase of the
2014 Mar 29 solar flare. The spatial resolution of the IRIS image is
$0.\mbox{${}^{\prime\prime}$}4$ ($0.\mbox{${}^{\prime\prime}$}167$ pix-1;
$724$ km arcsec-1). The projected, exclusive areas of $2.3\times 10^{17}$,
$3.9\times 10^{16}$, and $8.6\times 10^{15}$ cm2 correspond to the intensity
ranges indicated in the figure below the equivalent circular areas. Note, an
excess image formed by subtracting the image from 150 s earlier reveals much
fainter emission; in this case, an excess threshold value of $6\times 10^{5}$
erg cm-2 s-1 sr-1 Å-1 (Kowalski et al., 2017a) show that the faintest parts of
the ribbons extend over an area of $\approx 5\times 10^{17}$ cm2. The RHESSI
X-ray contours are plotted at 25, 50, 75, and 90% of the maxima.

In this section, we argue that the results from the two-component model RHD
fits are ostensibly consistent with the relative areas of high-intensity and
medium-intensity sources in the impulsive phase of well-studied solar flare.
The IRIS SJI 2832 image during the impulsive phase of the 2014 March 29 X1
solar flare is shown in Figure 6. We calculate the areas corresponding to
several intensity ranges in the SJI 2832 image: $\langle
I_{\lambda}\rangle_{\rm{SJI}}=2-4\times 10^{6}$ erg cm-2 s-1 sr-1 Å-1 (faint
threshold), $4-8\times 10^{6}$ erg cm-2 s-1 sr-1 Å-1 (medium threshold), and
$\geq 8\times 10^{6}$ erg cm-2 s-1 sr-1 Å-1 (bright threshold). The bright
threshold selects the pixels corresponding to the bright kernel (which is
referred to as BK#1 in Kowalski et al., 2017a), and the medium threshold
corresponds to the elongated ribbons on both sides of the brightest
kernel101010The medium threshold approximately corresponds to the “high
thresh” area calculated from the excess intensity images in Kowalski et al.
(2017a).. In Figure 6, the dashed contour outlines the faint threshold area,
the green-colored pixels correspond to the medium threshold pixels, and the
purple and white pixels isolate the bright threshold kernel. We sum the
exclusive areas within these three intensity ranges, and the equivalent
circular areas that correspond to the assumed stellar footpoint geometry (Eq.
1) are illustrated in the bottom left of the figure. The ratios of these areas
are $\approx$ 25:5:1 for the faint:medium:bright areas, respectively.
Coincidentally, the ratio of medium:bright areas is 5:1, and the ratio of
faint:medium brightness areas is also 5:1. These ratios are very similar to
the areal ratios ($X_{\rm{rel}}$; Table 2) that we inferred between the higher
flux and lower flux models through our spectral fitting to the AD Leo Great
Flare. Thus, we attribute the two model components as representing a bright
kernel – or several bright kernels – and fainter ribbons over a larger area.

The two RHD model components could also represent the faint-intensity and
medium-intensity areas, respectively, which exhibit an areal ratio of 5:1. To
justify this interpretation as the less plausible analogy for stellar flares,
we bring in analyses of a solar, RADYN flare model (“c15s-5F11-25-4.2”) from
Kowalski et al. (2017a) and Kowalski et al. (2022). Kowalski et al. (2017a)
synthesized the SJI 2832 intensity from this model, accounting for the
emission lines and continuum response in this bandpass. At the brightest times
of the 5F11 model ($t\approx 4$ s), they calculate a synthetic SJI 2832
intensity of $\langle I_{\lambda}\rangle_{\rm{SJI}}\approx 10^{7}$ erg cm-2
s-1 sr-1 Å-1, which is consistent with the brightest pixels in Figure 6. Since
this study, a detailed identification of the emission lines that contribute to
the SJI 2832 data in solar flares has been presented (Kowalski et al., 2019a,
see also Kleint et al. 2017), and several updates to the atomic physics of Fe
II have been implemented (which are to be described elsewhere in detail). The
new calculations result in fainter emission line flux but similar redshift
evolution of the Fe II lines. Averaging the solar c15s-5F11-25-4.2 model from
Kowalski et al. (2022) over a simulated SJI 2832 exposure time of 8 s, so as
to be directly comparable to the brightest pixels in the data in Figure 6,
results in a synthetic model intensity of only $\langle
I_{\lambda}\rangle_{\rm{SJI}}\approx 5\times 10^{6}$ erg cm-2 s-1 sr-1 Å-1.
This intensity is above the medium-intensity threshold area corresponding to
the green-colored ribbon pixels in Figure 6, but it is not nearly as bright as
the most intense pixels. The IRIS raster clearly “steps over” the brightest
kernel111111We further confirm this by inspecting the Mg II slit jaw images:
though large regions of the ribbon are saturated, most of the saturation
occurs away from the slit. in Figure 6 (see Fig. 1 in Kowalski et al. (2017a)
and the discussion in Kleint et al., 2016). Thus, the 5F11 electron-beam,
solar flare modeling with a small, low-energy cutoff is apparently most
appropriate for the medium-intensity ribbons instead of the fainter ribbons
that extend beyond the hard X-ray contours or the brightest pixels at the
centroid of the hard X-ray contours. The faintest ribbon intensity in the
impulsive phase121212The next SJI 2832 image corresponds to the beginning of
the fast decay phase of the hard X-rays, and the faint intensity threshold
clearly selects a large area in the “wakes” of the bright ribbons. In these
wakes, the emission lines may exhibit broad, nearly symmetric profiles as the
red-wing asymmetries have coalesced with the line component near the rest
wavelength (Graham et al., 2020) while the flare continuum intensity is still
at a detectable level in the IRIS NUV spectra (Kowalski et al., 2017a; Panos
et al., 2018; Zhu et al., 2019). may correspond to locations of impulsive
energy deposition by thermal conduction (Battaglia et al., 2015; Ashfield et
al., 2022) and/or XEUV backheating over a large area (Fisher et al., 2012).
The latter has been investigated in detail with 1D non-LTE models for the
Great Flare data (Hawley and Fisher, 1992); we speculate that radiative
backheating from an arcade of hot loops (e.g. Kerr et al., 2020) may account
for additional Ca II K line flux in the model spectra in Figure 4 (top panel).

Is there evidence that a much stronger source of heating than a 5F11 beam
contributes to the brightest SJI 2832 kernel pixels in this solar flare? Using
an even brighter intensity threshold of $\langle
I_{\lambda}\rangle_{\rm{SJI}}=10^{7}$ erg cm-2 s-1 sr-1 Å-1 to mask the solar
flare kernel in Figure 6 gives an area of $3.3\times 10^{15}$ cm-2, or four
IRIS pixels. Dividing this area into the nonthermal electron power ($8\times
10^{27}$ erg s-1) above $20$ keV that is inferred through standard collisional
thick target modeling in Kleint et al. (2016) gives an injected electron beam
flux of $2\times 10^{12}$ erg cm-2 s-1 (2F12). This is not as high as the
maximum injected beam fluxes in the F13 models, but it is much larger than
typically considered in solar flare RHD modeling. This line of reasoning
implies that a much higher beam flux model is a more appropriate collisional
thick target inference for the brightest SJI 2832 kernel in this solar flare.
For these large beam fluxes, however, the standard assumptions in collisional
thick target modeling of the hard X-ray footpoints are not applicable when the
ambient coronal densities are small (Krucker et al., 2011). Although the
RHESSI sources are largely unresolved (with a spatial resolution of
$3.\mbox{${}^{\prime\prime}$}6$, or $5.4\times 10^{16}$ cm2 at the Sun;
Battaglia et al., 2015), the spatially integrated hard X-ray and IRIS SJI 2832
powers provide upper limits on necessitated modifications to the thick-target
physics (e.g., Kontar et al., 2008; Brown et al., 2009; Kontar et al., 2012;
Hannah et al., 2013; Alaoui and Holman, 2017; Allred et al., 2020) that are
implemented in future modeling of the heterogeneous atmospheric response
within the hard X-ray source contours. Graham et al. (2020) investigated the
deficiencies in chromospheric condensation model predictions of the red-wing
asymmetry evolution of Fe II flare lines in IRIS spectra. Resolving the
disagreements, and drawing on implications for the standard collisional thick
target inferences of beam parameters, would greatly enhance the realism of the
analogous stellar flare RHD component with small low-energy cutoff values.

###  6.4 Future Observational Constraints

Graham et al. (2020) used two intensity thresholds in faster-cadence SJI 2796
imaging of a different X1 solar flare to quantify newly brightened areas as a
function of time. The ratios of these areas are $\approx 10:1$, and the areal
evolution is rather similar to the timing of the spatially integrated, hard
X-ray emission peaks from Fermi/GBM. Further verification of the heterogeneity
between kernel and the medium-brightness ribbon pixels are clearly needed from
solar flare spectral observations. One such unexplored constraint is the NUV
and FUV continuum evolution from IRIS flare spectra. In Figure 5 (top), we
show the locations of two continuum windows in the IRIS spectra around
$\lambda\approx 2826$ Å and $\lambda\approx 1349$ Å. The two spectral
components obtained from our fits exhibit distinct, time-dependent
C2826′/C1349′ emergent intensity ratios. A detailed investigation of the
relative continuum intensities for a large sample of solar flares would help
to determine the heterogeneity of the atmospheric response between the
brightest kernels and nearby bright ribbons.

Reality checks could also be attained through spatially resolved
characterization of the hydrogen Balmer line broadening along the slit length
of observations of solar flares with the Visible Spectropolarimeter (ViSP; de
Wijn et al., 2022) on the Daniel K. Inouye Solar Telescope (DKIST; Rimmele et
al., 2020). Our stellar flare phenomenological modeling paradigm (Section 6.3)
implies that the continuum-subtracted effective widths of the H$\gamma$
emission line (Kowalski et al., 2022) from the emergent intensity spectra of
the brightest kernels should exhibit much larger effective widths (Table 1,
rightmost column) than the medium-brightness ribbon component. In solar
observations, the pixels with the largest H$\gamma$ effective widths should
also show the brightest blue-optical continuum intensity. A statistical
classification of hydrogen line spectra should reveal distinct components that
correlate with timing and position along the solar flare ribbons, similar to
the groupings that were reported for a large sample of Mg II flare lines in
IRIS spectra (Panos et al., 2018).

On the stellar side, high-cadence spectral observations of low-mass stellar
flares at $\lambda=1800-3200$ Å during the impulsive phase would clarify how
the NUV continuum flux peaks and turns over into the FUV in events like the
Great Flare, which exhibits a small Balmer jump and a highly-impulsive,
broadband temporal evolution. The Cosmic Origins Spectrograph on the _Hubble
Space Telescope_ provides such an opportunity: the G230L grating with a
central wavelength at $\lambda=3000$ Å gives simultaneous spectral coverage at
$\lambda=1700-2100$ Å and $\lambda=2800-3200$ Å, which would provide the
necessary observations to test the RHD models. Recently, Kowalski et al.
(2019b) reported on flare spectra from the Cosmic Origins Spectrograph (using
a different central wavelength) and constrained the peak continuum flux to the
$U$ band. It was argued that these events are gradual-flare (GF) events with
large Balmer jumps and large line-to-continuum ratios, which are in stark
contrast to the measured quantities from the Great Flare optical spectra. We
briefly comment that our two-component modeling can readily reproduce the
properties of these gradual-type flare events as well. For example, a two-
component model consisting of the mF13-150-3 and the m5F11-25-4 spectra with
$X_{\rm{rel}}\approx 90$ results in a Balmer jump ratio of $3.9$ and a value
of $F^{\prime}_{\rm{H}\gamma}/\rm{C4170}^{\prime}\approx 150$, which are
consistent with the quantities from HST-1 in Kowalski et al. (2019b). A
parameter study of the detailed hydrogen line broadening and NUV spectra are
planned as the subject of Paper II in that series.

##  7 Summary & Conclusions

We have comprehensively modeled the multi-wavelength spectra during the rise
and peak phase of the Great Flare of AD Leo (Hawley and Pettersen, 1991). We
have shown that fitting two RHD spectral components to the detailed properties
of the hydrogen series using an updated treatment of the pressure broadening,
combined with a mechanism that heats deep chromospheric heights to $T\gtrsim
10^{4}$ K, is readily feasible with satisfactory statistical significance.
This semi-empirical modeling approach accounts for the evolution of height-
and wavelength-dependent emission line and continuum opacities in the flare
atmosphere, which is self-consistently calculated in response to high-flux
electron beam heating. A simulation (Kowalski et al., 2015, 2016) with a large
electron beam flux and the largest low-energy cutoff value range ($\lesssim
40$ keV) that is inferred from solar flare data (Holman et al., 2003; Ireland
et al., 2013) produces a dense chromospheric condensation and hydrogen Balmer
wings that are far too broad compared to the observation. Models that exhibit
a large ($\gtrsim 85$ keV), low-energy cutoff and high electron beam flux
($\approx 10^{13}$ erg cm-2 s-1) are able to explain the observed continuum
distribution and highly broadened Balmer line wings that are within the
constraints of the Balmer H$\gamma$–H16 emission line series. Large, low-
energy cutoffs are sometimes inferred in the so-called “late impulsive peaks”
in solar flares (Holman et al., 2003; Warmuth et al., 2009), and significant
progress has been made to improve the hard X-ray modeling of these events
beyond the physics in the standard collisional thick target model (Alaoui and
Holman, 2017).

A second, lower electron beam flux model exhibiting more similarities to
nonthermal electron parameters that are typically inferred in collisional
thick target modeling of hard X-ray data of solar flares (e.g., fluxes of
$5\times 10^{11}$ erg cm-2 s-1, low-energy cutoffs of $E_{c}\approx 25$ keV)
is necessary to fit the narrower hydrogen Balmer emission peak fluxes and
account for the bluest Balmer line in the AD Leo Great Flare spectrum. We
suggest that this second component represents heterogeneity of nonthermal beam
injection and the differences between bright, larger area ribbons and
brightest kernel morphologies that are readily seen in the impulsive phase of
solar flare imagery. However, further verification is needed from solar
observations: specifically, comparisons of hydrogen spectra at different
locations in early flare development are critical. The implementation of this
two-component, semi-empirical RHD modeling approach to Balmer line profiles of
other M dwarf flares with higher resolving-power, echelle observations is
underway (Kowalski et al. 2022, in preparation; Notsu et al. 2022, in
preparation) and will further constrain plausible linear combinations of RHD
model spectra.

The effects of transient UV radiation during flares is a relatively new topic
in the study of exoplanet habitability (e.g., surface dosages) and atmospheric
photochemistry (e.g., ozone photodissociation). These studies would benefit
from new NUV spectral observations of stellar flares. The semi-empirical
combination of RHD model spectra that are fit to the Great Flare observations
predict unexpected properties of the NUV continuum spectra of impulsive-type M
dwarf flares with small Balmer jumps, highly broadened Balmer lines, and small
line-to-continuum ratios. We conclude that small Balmer jumps, which appear as
relatively small deviations from a $T=9000$ K or $T=10,000$ K blackbody fit to
optical data in some flares, actually are consistent with much more energetic
sources of ultraviolet radiation than previously thought were possible from
solar and stellar chromospheres.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

AFK contributed by analyzing the AD Leo flare data, running the RHD models,
and performing the least squares fitting analysis. AFK lead the analysis of
the solar flare imaging, and he wrote the manuscript and produced the figures.

## Funding

AFK acknowledges funding support from NSF Award 1916511, NASA ADAP
80NSSC21K0632, and NASA grant 20-ECIP20_2-0033.

## Acknowledgments

We thank two anonymous referees for their detailed comments that helped
improve this work. AFK thanks Suzanne L. Hawley for many stimulating
conversations about flare spectra, flare modeling, and the AD Leo Great Flare
data. AFK thanks Joel C. Allred and Mats Carlsson for access to and assistance
with the RADYN code, and Han Uitenbroek for access to and assistance with the
RH code. AFK thanks Pier-Emmanuel Tremblay for access to the TB09$+$HM88
hydrogen broadening profiles. Many discussions with Rachel A. Osten, Clara
Brasseur, Isaiah Tristan, Ward Howard, and John P. Wisniewski about NUV and
optical flare continuum radiation were important for the motivation and
development of the ideas in this work. AFK thanks Eduard Kontar for helpful
discussions about the collisional thick target model of solar flare hard X-ray
emission. AFK thanks Eric Agol for suggesting ten years ago to explore large,
low-energy cutoffs in flare heating models. Many discussions about Balmer jump
solar flare spectra and instrumentation with Gianna Cauzzi, Hoasheng Lin, and
Tetsu Anan led to refining the ideas about the solar-stellar connection
presented in this work. IRIS is a NASA small explorer mission developed and
operated by LMSAL with mission operations executed at NASA Ames Research
Center and major contributions to downlink communications funded by ESA and
the Norwegian Space Centre.

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