presented for the large flare on CN Leo in [241]. Broadened Paschen lines have
been reported in the decay phase of a large flare from the very low mass star
VB10 [371]. Similar comparisons are, in principle, possible with the
broadening of the high-order Paschen series and the Ca II infrared triplets
[536], for which opacity broadening is perhaps less of a concern in
comparisons to Ca II H and K.

Figure 26:  Dramatic broadening differences between hydrogen and metallic
emission lines, such as the Ca II resonance lines, in flare spectra are
considered evidence for the important role of pressure broadening of hydrogen.
The pressure broadening of hydrogen constrains the charge densities (and
optical depths) in the flare chromosphere. This conclusion is supported by
detailed spectral synthesis with, e.g., the RADYN, RH, MULTI codes. The top
figure shows these effects during a flare on the dM4 star Gl 699, reproduced
from Paulson et al [579], and the bottom panel shows the coarse temporal
evolution of the broadening differences during a flare [743] on the dM4.5e
star YZ CMi.

####  10.2.1 Recombination Edge Modeling

Accurate interpretation of stellar spectra from EUV through NIR wavelengths
depends on modeling how the wings of the hydrogen lines overlap and merge near
bound-free ionization limits [338]. Accurate modeling, which should lack any
sharp edge feature, opens up diagnostic power from the rich spectral region at
wavelengths just longward of a recombination limit. For example, the Inglis-
Teller relation is [355, 644]

$\log_{10}n_{e}=23.2-7.5\log_{10}n_{\rm{max}}^{\rm{Balmer}}$ (23)

which has commonly been used to determine electron density from the last
discernible Balmer emission line near the Balmer recombination limit [430,
297]. Besides stellar flares, many types of astrophysical and laboratory
phenomena also lack discontinuities in spectra around the expected location of
hydrogen recombination edges [312, 209, 497, 677]. In solar flares, features
such as a “blue continuum bump”, a shift of the edge to redder wavelengths, or
a completely flat continuum have been reported in observations [533, 173]. The
complete lack of any type of edge feature, and a lack of a jump in flux across
the expected edge location, is ostensibly an indication of blackbody
radiation.

However, the lack of an edge is also expected from recombination theory that
includes non-ideal, level dissolution. The fundamental question pertaining to
stellar flares is thus if there is enough instrumental blending of the wings
and/or overlapping, linearly-superposed wing opacities to account for the lack
of the observed discontinuities. A remarkable example of stellar flare spectra
with no edge feature is shown in Fig. 11(a). Instead, these spectra (and
others) show a series of faint, broad hydrogen lines that fade into a
continuum. Other spectra, albeit at much lower resolving power, show some
noisy features here, however [297]. Ostensibly, the conservation of oscillator
strength density should result in a smooth transition of observed flux from a
pseudo-continuum of a confluence of high-order, highly broadened and blended
emission lines to the onset of recombination radiation. A phenomenological
description of the transfer of bound-bound opacity to bound-free opacity was
developed by [154] using the occupational probability ($w_{n}$) formalism in
the non-ideal equation-of-state of [348]. The “dissolved-level” bound-free
Balmer continuum opacity longward of the Balmer edge has been recently
incorporated into spectral models of stellar flares [418, 408]. However, self-
consistent modeling of the pressure broadening of the bound-bound Balmer lines
[721] was not also included in the first models [415]. This incorrectly
supported the extremely dense chromospheric condensations in 1D RADYN models
(Sect. 8.2) as realistic models of flare density enhancements at cool
temperatures [416]. Otherwise, the dissolved-level continuum opacities have
largely reconciled observations with models by explaining the lack of an edge
around $\lambda=3646-3700$ Å and the range of observed pseudo-continuum
properties in between the longer-wavelength, high-order Balmer lines at
$\lambda=3700-4000$ Å. The non-discontinuous, transition of dissolved-level
opacities at $T\approx 10^{4}$ K have apparently reconciled the hydrogen
recombination (Balmer jump) theory of the NUV with the blackbody-like theory
of the optical continuum radiation.

Figure 27:  Direct comparisons of two methods [173, 418] for calculating the
Balmer jump spectral region in solar and stellar flares. The emergent LTE
intensity is calculated from a slab with $n_{e}=3.9\times 10^{13}$ cm-3. The
method from [173] predicts a “blue continuum bump”, whereas the current
treatment with occupational probabilities predicts an extension of the Balmer
continuum intensity to wavelengths longward of the “ideal” recombination limit
(indicated by an $\infty$ symbol) at $\lambda=3646$ Å.

The physical cause of the continuous, dissolved-level opacity that was
parameterized in [154] for hydrogen is still not well understood. Experiments
with Rydberg atoms support the idea that Landau-Zener transitions [e.g., 802]
at avoided level crossings [800, 640, 593] facilitate a cascading ionization
process. In this picture, an electron is photo-excited to a high-lying level
$n$ that is pressure-broadened to overlap with the broader, next higher-energy
level $n+1$, thus effectively “dissolving” the level $n$. This description
apparently explains the Inglis-Teller relationship through dynamic microfield
ionization [348]. There are actually many types of calculations [455, 153, 64,
667, 59, 218, 219, 130, and see the review in Sections 86.2.2 - 86.2.3 in
Drake 186, Weisheit and Murillo 766]484848The dynamic microfield ionization
hydrogen is assumed to be analogous to microwave field ionization of Rydberg
atoms [348]. The rather ambiguous term “pressure ionization” is usually
reserved for the situation in which ground-state hydrogen atoms are so close
together that electron degeneracy pressure causes the ionization (e.g., in the
cores of stars and in white dwarfs).  that may contribute to “lowering the
ionization potential” or “continuum lowering” in a hydrogen plasma with
microfield fluctuations and collisional perturbations494949Nonetheless, one
should note that Landau-Zener transitions have been invoked in microwave
ionization experiments with Rydberg atoms in order to account for more rapid
ionization than is predicted by the classical, hydrogenic saddle-point
estimate of the lowering of the ionization potential; see Fig. 40 in Appendix
D. .

Figure 28: A variety of approximations to opacity line profile functions for
the Balmer series have been used in stellar flare modeling codes over many
decades. This figure summarizes the differences in models of the line profile
function, $\phi_{\lambda}$, for the H11 Balmer line [see also 367, for a
similar figure for the H10 line]. The calculations use $n_{e}=n_{p}=3.16\times
10^{14}$ cm-3, or about 10x larger than in Fig. 27, and thermal Doppler
broadening at $T=10^{4}$ K is included. The “ideal” profiles, $\phi_{VCS}$,
are the VCS73 calculations from [443], and do not account for line narrowing
due to occupational probability / level dissolution. Several Voigt profiles
are also plotted for a range of Lorentzian FWHM values, $\Gamma$. Currently,
the most accurate theory corresponds to the $\phi_{\rm{TB09+HM88}}$ (solid
black) curve.

Due to level-dissolution in dense regions of flare atmospheres that are not
very optically thick in the Balmer continuum, applying a simple relationship,
such as the Inglis-Teller, that connects the last-visible hydrogen line to a
single electron density may be misleading. [421] discuss a scenario wherein
level-dissolution of hydrogen in a chromospheric condensation can generate an
approximately grey opacity window at $\lambda\approx 3700-3750$ Å. Thus, the
Balmer lines that are in emission in the emergent spectra actually originate
from deeper layers with smaller ambient (thermal) electron densities and no
apparent gas velocities. Due to the heterogeneous vertical stratification of
the flare atmosphere, the high-order hydrogen Balmer series in emergent
intensity spectra can be as broad as or less broad than the low-order Balmer
lines (H$\gamma$, H$\beta$, H$\alpha$), which are very optically thick in some
of the more recent RHD flare models. Some models of stellar flare spectra at
the Balmer limit also incorporate spatial and temporal heterogeneity within
the flare ribbons [408]. These seek to explain the inconsistencies in large
$n_{\rm{max}}^{\rm{Balmer}}\gtrsim 15$ (Eq. 23) being observed (indicative of
low electron densities, $n_{e}=10^{13}-10^{14}$ cm-3) with the necessity to
also produce the observed continuum properties from a location on the star
with much larger electron density ($n_{e}\gtrsim 10^{15}$ cm-3;
$n_{\rm{max}}^{\rm{Balmer}}\approx 11$). Note that in occupational probability
theory, however, there is not one single maximum upper level for all atoms in
a plasma.

Fig. 27 directly compares two different modeling techniques within the Balmer
jump spectral region. The emergent intensity is calculated in LTE from a
static slab with $dl=400$ km, $T=10^{4}$ K, and $\rho=10^{-10}$ g cm-3
($n_{e}=3.9\times 10^{13}$ cm-3). [173] modeled several solar flare spectra
(with high resolving power) using Voigt profiles up to $n_{\rm{max}}=2\times
10^{4}T^{0.25}n_{e}^{-0.25}\approx 65-85$; this is a saddle-point estimate
with Debye-screening of the proton field [499]. This method convolves the
Balmer edge with the Voigt function corresponding to the line with
$n_{\rm{max}}$. The “current (2022)” method uses the TB09$+$HM88 profiles
[721] and the dissolved level bound-free continuum opacity. Many other various
approximations for the pressure broadening of hydrogen have been used in
models of solar and stellar flares. The ambiguities were discussed extensively
in [367] and were resolved only in recent years. In Fig. 28, we summarize the
wide variety of broadening prescriptions for the Balmer H11 line, whose
integrated flux is sensitive to the amount of level dissolution in the
densities of flare atmosphere models [415]. For a description of each method
presented in the figure, we refer the reader to [421] where similar
comparisons are shown for the Balmer H$\gamma$ line.

##  11 A Comprehensive Multi-Wavelength Stellar Flare Dataset

What fundamental physical parameters can one infer using a multi-wavelength
dataset of a stellar (super)flare? A comprehensive analysis of a series of two
consecutive superflares from the binary dM4e$+$dM4e system DG CVn was
presented in [571], who summarize the plasma and geometric properties that can
be constrained in a flaring atmosphere. This culminates an extensive history
of stellar flare research, and we briefly review the salient points to the
algorithm.

###  11.1 Inferring Nonthermal Electron Properties from Stellar Data

The nonthermal radiation from power-law electrons in stellar flares occurs as
hard X-rays ($E\gtrsim 10$ keV) and as incoherent gyrosynchrotron radiation in
the radio/microwaves at cm wavelengths. The radiative processes are thought to
be similar to solar flares, in which the hard X-rays are emitted through
collisional bremsstrahlung in the chromosphere, whereas the radio flux
originates from predominantly higher energy electrons ($E\approx 100-300$ keV)
trapped in magnetic fields; see the review in [772]. The distribution of
electrons is parameterized in the hard X-rays and radio using two power-law
indices, $\delta_{x}$ and $\delta_{r}$, respectively. From hard X-rays, the
particle flux distribution is

$F(E)=F_{o}\left(\frac{E}{E_{c}}\right)^{-\delta_{x}}$ (24)

where $E$ is the electron kinetic energy in keV, $E_{c}$ is the low-energy
cutoff, and $F_{o}$ is the differential (specific) electron flux density (el
s-1 cm-2 keV-1) around the cutoff energy. Similarly, a nonthermal electron
beam density distribution determined from the radio is written as

$n(E)=n_{o}\left(\frac{E}{E_{c}}\right)^{-\delta_{r}}$ (25)

where $n_{o}$ is the differential number density (el cm-3 keV-1) of electrons
around energy $E_{c}$. If the same population of electrons is responsible for
both X-rays and radio, then $\delta_{x}\approx\delta_{r}$ is expected for
electrons with velocity $v$ very close to the speed of light $c$; otherwise,
$\delta_{x}\approx\delta_{r}-0.5$. The different forms (_i.e.,_ , flux density
_vs._ number density) of these equations is because the injected power-law
flux distribution is inferred through the collisional thick target model of
the hard X-ray spectrum, whereas a power-law density distribution is
determined through observations at optically thin microwave frequencies (see
below).

Note that the integral of Eq. 24 multiplied by $E$ gives the total beam energy
flux density (erg cm-2 s-1), which is an important input for flare modeling
(e.g., F10, F13, etc…). Integrating Eq. 25 over energy gives the total beam
density, $N$, where $N$ is the quantity that is used in the gyrosynchrotron
expressions in [195] if $E_{c}$ is set to $10$ keV. In the studies of [687,
571, 196, 195], Eq. 25 is integrated from $E_{c}$ to $\infty$ to replace our
$n_{o}$ with the prefactor $N(t)\frac{\delta_{r}-1}{E_{c}}$ where $N(t)$ is
the total electron density (el cm-3) above the cutoff (and the same is usually
done for Eq. 24). Eq. 24 can be multiplied by a probability density function
for the angular distribution, $M(\theta)$, to specify the directivity of
particles with respect to the magnetic field [11]. A highly-collimated
Gaussian distribution is usually chosen for $M(\theta)$. Then, a “1.5D”
solution to the Fokker-Planck equation (Sect. 9.2) accounts for the changes in
the pitch angles of the particles.

The best diagnostic of accelerated particles in stellar flares is optically
thin frequencies in the mm or radio. If the observations are in the radio at
GHz frequencies, one must ensure that these frequencies are above the peak
frequency ($\approx 10$ GHz) so as to be in the optically thin part of the
gyrosynchrotron spectrum. An expression from [195] converts from a power-law
index in the optically thin flux spectrum, $S_{\nu}\propto\nu^{\alpha_{r}}$,
to the power-law index in the electron distribution, $\delta_{r}$:

$\alpha_{r}=1.22-0.90\delta_{r}$ (26)

The power-law index in the optically thick part of the spectrum is much less
sensitive to the power-law index of nonthermal electrons.

Following [687], the kinetic energy of nonthermal particles can be constrained
from the radio spectrum as follows. The power per volume in nonthermal
electrons above the assumed low-energy cutoff value, $E_{c}$, is given by Eq.
9 of [571]. In this equation, the total number of nonthermal electrons,
$N(t)V(t)$, over the flaring volume $V(t)$ is an unknown, but it is directly
proportional to the optically thin spectral radio luminosity, $L_{\nu}$. By
constraining the value of $\delta_{r}$ in this part of the spectrum and
invoking some prior knowledge or assumption about the peak frequency, an
integral over the entire radio spectrum can be calculated – including the
optically thick part – to give the time-integrated energy of nonthermal
electrons in the flare. This leaves one unknown, the magnetic field, $B$, in
the gyrosynchrotron emissivity formula for the optically thin part of the
spectrum (see below for further constraints on the magnetic field, but it must
be assumed that the $B$ derived from the radio originates from the same $B$
that is derived from the soft X-rays). If the radio or mm data are available
at only one frequency, and thus a spectral index cannot be determined from the
data, then contours of nonthermal energies and plausible ranges of magnetic
field strengths and spectral indices are reported, as in Fig. 29, which is
reproduced from [571] [see also 471, where a spectral index was derived from
mm observations].

Figure 29: A range of possible nonthermal electron beam power-law indices and
magnetic field strengths that can be constrained from gyrosynchrotron data of
stellar flares; reproduced from Figure 6 in [571] with permission. See also
[687] and [471]. Contours of constant electron beam kinetic energies add a
third constraint. Cross sectional footpoint areas from optical data can
constrain the beam flux density for forward modeling inputs [569, 571].

X-ray and optical data can constrain the physical parameters of flares
further. From time-resolved X-ray spectra, one obtains the volume emission
measure, $\mathcal{VEM}$. The $\mathcal{VEM}$ is $\int n_{e}^{2}dV$, which is
the amount of optically thin emitting plasma at a fixed X-ray temperature,
$T_{X}$. Typical values of the $\mathcal{VEM}$ (in M-dwarf flares) are
$10^{51}-10^{52}$ cm-3. In a $\log_{10}$ $T_{X}$ vs. $\log_{10}\
\sqrt{\mathcal{VEM}}$ diagram, the peak $T_{X}$ may clearly occur before the
peak $\mathcal{VEM}$ [213, 34, 448]. Over the decay phase evolution of a
flare, the slope in this diagram provides one of three inputs for the widely
employed flare loop modeling of [607]. Combined with an exponential decay
constant from the X-ray light curve and the maximum X-ray temperature, a semi-
circular loop’s half-length, $l$, can be inferred. The loop sizes in giant
X-ray flares are typically reported to be on the order of the stellar radius
[e.g., 213, 569, 434, 571]. Note, large $50-200$ Mm loop lengths are also
inferred under some interpretations of QPPs (Section 6.2) in optical broadband
light curves [16, 184].

Optical and/or NUV data provide data to relax the geometrical assumption of a
single, giant monolithic loop (Sect. 12), while also providing constraints on
the electron density and the magnetic field strength in the flare corona. The
optical and NUV probe the footpoint radiation, from which a dimensionless
ratio of $R_{\rm{foot}}/(2l)$ can be calculated, where $R_{\rm{foot}}$ is an
equivalent radius of the area of an optical flare footpoint:
$A_{\rm{flare,opt}}=2\pi R_{\rm{foot}}^{2}$. By combining with an alternative
expression for the $\mathcal{VEM}$ in terms of these quantities and using the
value of the $\mathcal{VEM}$ from X-ray spectral fitting, a representative
value of the coronal $n_{e}$ is readily obtained that is independent of the
number of identical loops in a hypothetical stellar arcade [Eq. 25 of 571].
One also can assume that the magnetic pressure must be at least equal to the
gas pressure of the evaporated MK plasma for it to be confined to the loops
instead of cooling through expansion, as in a scenario analogous to a filament
eruption or coronal mass ejection [145]. The magnetic field confinement
condition places a lower-limit on the coronal magnetic field of the flaring
loops.

[523] homogeneously analyzed a large sample of many sizes of X-ray stellar
flares from various spectral types and evolutionary stages using “Haisch’s
simplified analysis” [HSA; 292], which involve scaling relations [see also
731] that infer coronal temperature, electron density, loop length, and
confining magnetic field strength from an emission measure and light curve
decay timescale. Typical X-ray temperatures derived from multi-component fits
to $E=0.5-10$ keV spectra are similar to those in solar flares, $T_{X}\approx
15-30$ MK [e.g., 567, 448, 527]. At the low coronal temperature end, $T\approx
2-15$ MK, electron density variations, or lack thereof, are constrained with
X-ray spectra of the $f$ (forbidden) and $i$ (intercombination) components of
helium-like triplets [e.g., bottom right spectra in Fig. 18; 283, 565, 708,
567, 448]. When significantly detected, the flare-enhanced electron densities
are reported in the range of $10^{11}-10^{12}$ cm-3 at these cooler coronal
temperatures.

Osten et al [571] summarize the physical parameters in three of the largest
dMe flares that have been observed with X-ray spectra (_cf._ their Table 5).
The inferred temperatures are $T\approx 50$, $140$, $290$ MK; the electron
densities are 3 $\times 10^{11}$, 3 $\times 10^{12}$ 3 $\times 10^{11}$ cm-3;
the confining magnetic field flux densities are 230, 1100, 580 G,
respectively; and $\mathcal{VEM}$ ranges between $10^{54}-10^{55}$ cm-3.
Generally similar numbers are found in the largest RS CVn flares, which
produce integrated X-ray energies and luminosities one to two orders of
magnitude larger [568, 376]. YSO’s also exhibit such extreme volume emission
measures and temperature [$40-80$ MK; 748]. These superhot temperatures
correspond with X-ray luminosities that are close to or in excess of the
bolometric luminosity of the star505050Before the term “superflare” became
widely associated with white-light $E\gtrsim 10^{33.5}$ erg superflares on G
dwarfs in _Kepler_ data, [568] proposed the term for flares that are super-
bolometric in X-ray luminosity. It is remarkable that so many types of stars –
from RS CVn’s to tiny, red-dwarf stars – can release more energy per second
momentarily in their outer atmospheres than they do through nuclear fusion in
their cores; perspective provided by R. Osten, priv. communication, 2014. .
Note that in an X-ray superflare from the dM3.5e star EV Lac with comparable
temperatures ($T_{X,\rm{peak}}\approx 70$ MK) that were derived from the
hydrodynamic modeling analysis of [607], [213] argue that accounting for the
$E>10^{34}$ erg of energy release during the flare places further constraints
on coronal magnetic fields in the $\approx 4$ kG range before the flare.

The peak stellar flare values of $\mathcal{VEM}$ and $T_{X}$ generally compare
well with scaling relations over many orders of magnitude, from superflares to
solar flares, in the famous “Hertzsprung-Russell (H-R) diagram of solar and
stellar flares” [790]. The H-R diagram of solar and stellar flares was first
theoretically explained using MHD scaling relations in [672]. This diagram was
investigated further in [34], and the tightest correlation was reported in the
empirical relationship

$\mathcal{VEM}(T_{p})=10^{50.8}\big{(}\frac{T_{p}}{\rm{10\ MK}}\big{)}^{4.5\pm
0.4}[cm^{-3}]$ (27)

where $T_{p}$ is the peak flare X-ray temperature. Their relation for decay
time and peak temperature is, in principle, useful for inferring physical
parameters if only the decay phase is caught in the observations [43];
however, the empirical scatter about the trends is an order of magnitude in
both quantities. [257] and [571] suggest that the hottest ($T_{X}>100$ MK)
stellar flares exhibit a “saturated”-like volume emission measure of $\approx
10^{55}$ cm-3 and do not follow a solar extrapolation in the flare H-R
diagram. [672] include two additional parameters (either loop length or
magnetic field and ambient thermal electron density) in their scaling relation
analogue [see also further alternatives summarized in 309] to Equation 27 to
explain the apparent offset in volume emission measure from the solar
extrapolation. [335] include a much smaller flare from an active M-dwarf flare
in the context of stellar superflares and solar flares. [571] report on a
stellar superflare (their “F2” event) with $T_{X}\approx 50$ MK that is within
a factor of 2.5 in the predicted $\mathcal{VEM}$ given by Equation 27.

Large stellar flares occasionally trigger _Swift_ ’s Burst Alert Telescope
(BAT trigger numbers 172969, 310125, 331592, 596958, 625898), which is
intended to automatically slew to gamma-ray bursts and other extra-galactic
transients515151The _Swift_ project maintains a webpage of flare stars whose
count rates are monitored;
https://swift.gsfc.nasa.gov/results/transients/Flare_stars.html.. The
combination of X-ray data from the BAT and at softer X-ray energies from the
_Swift_ ’s XRT facilitates classical model hypothesis testing among thermal
and nonthermal models out to $E\approx 100$ keV. In superflares from II Peg
and DG CVn, nonthermal bremsstrahlung models were found to be equally
statistically robust fits as a superhot thermal model fit. However, in the
superflare event from DG CVn [571] that the BAT spectra triggered on, the
$T_{X}=290$ MK thermal model was favored over a nonthermal model because the
time-evolution of the X-rays was delayed with respect to the optical $V$-band
data [in line with the thermal, empirical Neupert effect; 108], and a
footpoint area from the optical gives (even by stellar flare standards)
unrealistic collisional thick target electron beam energy and flux into the
lower atmosphere. The collisional relaxation times were estimated to be rather
short compared to the (classical) thermal conduction times, thus further
supporting the thermal interpretation. Additional evidence for thermal
interpretation of superflare X-ray emission from YSO’s is discussed in
Vievering et al [748].

In soft X-ray flare spectra, features around the Fe 6.4 keV K$\alpha$ emission
line and the 6.7 keV Fe XXV thermal emission line complex are also reported.
These features have been interpreted as either electron beam excitation [202]
or non-LTE fluorescence of a larger-area photospheric ‘halo’ [568, 192, 569].
However, there are some calibration issues in _Swift_ data around these
features that have been reported only recently [572]. The recalibrations
greatly diminish the 6.4 keV feature while enhancing the 6.7 keV complex. Fe
K$\alpha$ fluorescence was studied in a flare from an active giant star using
a different observatory (Chandra) without instrumental problems and higher
resolving power [710]; see Sect. 12.2.

Table 8 highlights a few important multi-wavelength diagnostics in stellar
flares that have been discussed in this review. Of course, the interpretations
of these diagnostics are complicated by issues pertaining to heterogeneities
of the unresolved nature of stellar flare sources (in time, and in all three
spatial directions), and thus are generally model-dependent. In the final
section of this review, we discuss assumptions for and inferences of stellar
flare geometries and the associated implications for heterogeneity.

Table 8: A summary of flare diagnostics

Measurement/Data | Physical parameter of flare
---|---
X-ray exponential decay constant, $T_{X,\rm{max}}$, slope in $T_{X}$ vs. $\mathcal{VEM}$ | semi-loop length $l$
Slope at optically thin radio/mm frequencies | power-law index of NT electrons, $B$, $E_{\rm{NT,kinetic}}$
Optical ($\lambda>4000$ Å) color temperature | Footpoint area
Optical or $U$-band light curve timing _vs._ X-rays, impulsiveness index | non-thermal vs. thermal interpretation of hard X-rays
Balmer jump ratio (or H$\gamma$/C4170′) | Optical depth at $T\approx 10,000$ K
H$\epsilon$, H$\delta$, or H$\gamma$ vs. Ca II H or K | electric pressure (Stark) broadening vs. turbulent Doppler broadening
Fe K$\alpha$ spectral region | fluorescence vs. NT/beam excitation, viewing angle, halo geometry
$f/i$ ratios from X-ray spectra of helium-like triplets (e.g., O VII $\lambda\approx 22$Å) | thermal electron densities at $T\lesssim 15$ MK

5151footnotetext: “NT” is an abbreviation for nonthermal.

##  12 Stellar Flare Geometries

Stellar flares are spatially unresolved point sources. Yet, from spectral
data, model fitting, and sufficient time-resolution, there are many properties
of the spatial dimensions of flaring sources that can be inferred. These
reveal similarities to solar flare phenomenology at similar wavelengths and
strongly reinforce the solar-stellar connectivity of flare physics. In this
section, we review the methods and results of stellar flare geometry
calculations.

Optical and NUV observations constrain the chromospheric/photospheric flare
areas. A blackbody temperature, $T_{\rm{flare}}$, and the fraction (“filling
factor”) of the visible stellar hemisphere that is flaring,
$X_{\rm{flare}}=R_{\rm{flare}}^{2}/R_{\rm{star}}^{2}$, can be fit to spectral
data or multi-band photometry through the following equation [298, 302]:

$f_{\lambda,\rm{Model}}^{\prime}=\Big{(}\pi
B_{\lambda}(T_{\rm{flare}})-S_{\lambda,q}\Big{)}X_{\rm{flare}}\frac{R^{2}_{\rm{star}}}{d^{2}}$
(28)

where $B_{\lambda}(T_{\rm{flare}})$ is the Planck function intensity model of
the flare, $S_{\lambda,q}$ is the surface flux of the quiescent star, and
$f_{\lambda,\rm{Model}}^{\prime}$ is the ‘flare-only’ model flux at Earth
above the atmosphere that is to be fit to the observed, calibrated [erg cm-2
s-1 Å-1] flare-only flux, $f_{\lambda}^{\prime}$. The ratio of flare-only
fluxes at two wavelengths is a flare color and is the minimum information that
is needed to solve for $T_{\rm{flare}}$ in Eq. 28 (through linearization and
iteration or a lookup table). An $m$-component, linear superposition of RHD
surface flux flare spectra, $S_{\lambda,\mathrm{RHD},m}$, or a multi-thermal
blackbody model are typically required to reproduce the optical and NUV
continuum observations over a wavelength range spanning more than
$\Delta\lambda\approx 1000$ Å. Additional terms are thus included in Equation
28. For an $m$-component RHD model superposition, the solution is a linear
least squares fit to an observed flare-only spectrum, $f_{\lambda}^{\prime}$,
resulting in the estimated parameters, $\hat{X}_{\mathrm{flare},m}$ [see 571,
418, 408]. For further discussion, see Appendix C.

Constraints on $X_{\rm{flare}}$ from blackbody fitting to optical flare-only
spectra result in inferences that a small fraction of the visible stellar
hemisphere flares for even the largest, $E\approx 10^{34}$ erg, dMe events.
For example, the broadband photometry in the Great Flare of AD Leo [297]
constrain $X_{\rm{flare}}$ to be 0.5% [295] in the first impulsive peak. Fits
to the flare-only, optical continuum in the rise/peak spectrum of this event
are consistently small (0.2%) but with a correspondingly larger blue-optical
blackbody color temperature of $T_{\rm{flare}}\approx T_{BB}=11,600$ K [414].
An $m=2$ component RHD model of this spectrum gives a similar filling factor
for a model with large electron-beam heating fluxes, which are combined with
the second RHD model component with filling factors that are $\approx 2-10$x
larger in order to account for most of the narrow emission line flux [408, see
also Cram and Woods 140, Hawley and Fisher 295, Kowalski et al 412]. The
corresponding projected area at the star is $A_{\rm{flare}}=X_{\rm{flare}}\pi
R_{\rm{star}}^{2}$, which may instead consist of a series of $K$ circular
flare kernels. Inferences in smaller dMe flares from broadband photometry
[302] and optical spectroscopy [416]525252See Table 5 of [416] for different
inferences of optical color temperature and filling factors using different
wavelength ranges over the optical. are consistent with very small areas
$\approx 5-10\times 10^{16}$ cm2, thus reinforcing the analogy to compact
white-light kernels that are observed at the footpoints of impulsively heated
loops in solar flares [e.g., 231]. During the most energetic dMe events,
however, peak-phase flare areas are inferred to be several orders of magnitude
larger than solar flare white-light and NUV areas [425, 417].

In the gradual decay phase, and in the impulsive phase of gradual-type dMe
flare events with large Balmer jump ratios, blackbody fits to broadband
photometry may be misleading (Sect. 7.2.4). Blackbody fits to the gradual
phase continuum give a rather inconsistent picture of the post-peak footpoint
areal evolution from event to event; perhaps, heterogeneous RHD model
superpositions in the gradual decay phase are much more important for accurate
areal inferences than in the $10^{4}$ K blackbody-like rise and peak phase. A
superposition of RHD model components in the gradual decay phase of a
megaflare on the dM4.5e star YZ CMi is shown in Fig. 30 from [418]. This model
was previously developed to explain the NUV to $V$-band ratio in the DG CVn
superflare event reported in [571], and it also adequately modeled the NUV
continua and Balmer jumps in much smaller, gradual-type events [420]. This “DG
CVn superflare multi-thread (F13) model” consists of several RHD model spectra
over the evolution of the atmosphere that is shown in Fig.21, thus superposing
snapshots from a range of optical depths, temperatures, and densities (see
Sect. 12.1) in order to explain the gradual decay phase.

Figure 30:  Gradual-decay phase, flare-only spectrum of a megaflare on the
MV4.5e star YZ CMi [412]. Multi-thread RHD modeling of another superflare on
DG CVn [571] adequately reproduces the Balmer jump, optical continuum shape,
and some of the hydrogen Balmer emission line properties (but some important
shortcomings remain). Each RHD model component is calculated self-
consistently, but the linear superposition of the model components is semi-
empirical; see discussion in [418] and [419].

Various size scales in solar and stellar flares are shown in Fig. 31.

Figure 31:  Sizes to scale. An image of the Sun at $\lambda=6173$ Å from the
Helioseismic and Magnetic Imager [HMI; 652] on SDO, a representative size of
an active mid-type (MV3e-MV4.5e) M star ($R_{\rm{star}}=0.3R_{\rm{Sun}}$), and
a circular flare area ($\approx 5\times 10^{18}$ cm2 ) that corresponds to
about 0.5% of the visible hemisphere of this hypothetical M star. For
comparison, Earth and Jupiter are shown to scale. The insets each have a
field-of-view of $\approx 10^{19}$ cm2, and they show an image of the solar
active region in the NW quadrant during the hard X-ray impulsive phase of the
SOL2014-03-29T17:48 X1.0 flare (total intensity left, difference image right).
For a detailed analysis of the HMI data of this solar flare, we refer the
reader to [392]. Jupiter image credit: NASA, ESA, and J. Nichols; Earth image
credit: GOES/NOAA/NESDIS.

###  12.1 A Dominant Loop or an Arcade?

There are two general approaches to modeling spatially unresolved flare light
curves and spectra [448]. The first approach is summarized in [376], and
references therein, where it is most recently applied to an RS CVn X-ray
superflare. This modeling follows the analysis of [607] and describes the rise
and decay of a flare as result of a dominant monolithic loop (or a
superposition of a few dominant loops that are heated and cooled
simultaneously). The heating and cooling timescales in the loop are on the
order of tens of minutes to hours [see also 608, 709, 20]. In [608], the
initial X-ray event in the same flare in Fig. 18 was modeled with this method,
and the subsequent X-ray flare in the event was modeled as an arcade of
$\approx$ five loops that were sequentially heated in a nearby region.
Coincidentally, a solar analogy inspired a similar hypothetical explanation
for the NUV and optical footpoint ignition during the decay phase of a
different M dwarf flare in Kowalski et al 418. Longitudinal MHD oscillations
in a monolithic, giant (200 Mm) loop have been proposed as an explanation for
the triggering of quasi-periodic, $U$-band, light-curve oscillations in the
decay phase of this megaflare event [16]. The comparisons of particularly
dense snapshots [e.g., $t=2.2$ s; 415] from RHD models to observations also
inherently make the assumption that a single dominant kernel at any one time
contributes to the bulk of the flux received at Earth [but see 528, for recent
tests of this assumption].

An alternative hypothesis is known as “multi-thread modeling”, which was
pioneered by [324], [615], and [762] to explain Sun-as-a-star soft X-ray solar
flare light curves. These works addressed the large discrepancy between the
observations of long decay times in GOES soft X-rays and the very short decay
times in models of individual loops; multithread techniques have further been
used to model a variety of solar flare phenomena [e.g., 616, 642, 611].
Elements of this approach and of observations of sequential footpoint
brightenings in the hard X-ray and optical wavelengths in solar flare ribbons
[e.g., 405, 601] have inspired stellar flare RHD model superpositions [571,
418] for more direct comparisons to spatially unresolved observations. [258]
argue that the rise phase of a giant X-ray flare from the PMS star DQ Tau
consists of a multi-threaded source; see also [661] in reference to the
modeling approach showcased in Fig. 23. Doyle et al [184] suggest that a
series of tens of loops contribute to the geometrical dimensions, as inferred
from their QPP (Section 6.2) analysis, in two high-energy flares. [487]
discuss the loop triggering scenario of [201] as an alternative hypothesis for
short-duration QPPs in white-light. [567] hypothesize that an arcade of
heterogeneous gyrosynchrotron-emitting loops may explain the microwave
spectral indices in a large dMe flare.

Multi-thread modeling approaches seek to be more representative of a solar
flare arcade geometry, following arguments previously made in stellar [295]
and solar modeling [296]. However, ad hoc geometrical assumptions and
simplifications must be imposed due to the absence of appropriate (i.e., broad
spectral coverage, high-time resolution, unsaturated data at comparable
wavelengths) constraints from solar data. Recent analyses of IRIS solar data
at high-time resolution, such as the “new area” calculations of [274], and
spatially resolved spectral line analyses [528], may be able to provide
clearer guidance to stellar models of sequential footpoint ignition.
Nonetheless, there may be important differences between solar and stellar
flares that complicate a direct application of the thermal multi-thread
modeling of [762]: shorter-during footpoint pulses [e.g., 487], greater
importance of the gradual-phase white-light continuum radiation [297, 309] and
particle acceleration [567], and larger apparent loop height growth over the
flare evolution [$\tau_{c}\propto l^{2}$; e.g., 567, 571] all are examples of
properties that distinguish stellar flares.

Novel stellar flare applications of the two-ribbon solar flare reconnection
model of [404] are described in [595] and Güdel et al 282 [see also 292].
[282] model the $\Delta t\approx 8000$ s rising phase of a flare on the RS CVn
UX Ari in this framework. This flare exhibited a remarkable peak X-ray
luminosity of $>10^{32}$ erg s-1, and they infer the evolution of the
reconnection heights and electron densities (which evolve through $\approx
10^{11}$ cm-3 at peak). A long-duration decay phase is predicted by the model,
and the paper discusses how the lack of nonthermal particles in the early rise
phase is a potential limitation of the original assumptions.

###  12.2 Stellar Core-Halo

Multi-dimensional geometrical analysis of flare sources is possible by
combining hydrodynamic modeling of loop lengths and high resolution spectral
observations at Fe K$\alpha$. The technique is described in [710], who infer
an X-ray source height that irradiates a large area of the photosphere, from
which the Fe K$\alpha$ is observed at a viewing angle $\phi$. The geometry is
shown in Fig. 32. The combined constraints from the Palermo-Harvard code and
the MOCASSIN 3D Monte Carlo radiation code [207, 208, 191] are reproduced from
their paper on the right of Fig. 32. Recently, Werner band H2 flare emission
has been reported in [235] and interpreted as fluorescence due to O VI
$\lambda=1032$ Å irradiation of the temperature minimum region ($T\leq 1500$
K).

Figure 32:  Geometrical inference from spectral analysis of the 6.4 keV (1.94
Å) Fe K$\alpha$ emission line during a flare on the giant star HR 9024,
reproduced from [710] with permission. The X-rays from hot flare loops
fluoresce the neutral iron in the surrounding photosphere. The modeling
leverages hydrodynamic modeling from [709].

In 1D radiative-hydrodynamic modeling, XEUV backwarming of the photosphere is
approximated using the treatments of [248], [295], [296] and [2] through
exponential integrals and detailed radiative transfer [10, 753]. A multi-
dimensional extension of UV and optical radiative backwarming is illustrated
in a cartoon of Figure 4 of [225], and the geometrical dilution of
chromospheric radiative flux impinging on the photosphere is given by their
Eq. 25.

The remarkable, state-of-the-art 3D MHD codes that have been recently
developed with the primary goal to study quiet Sun magnetism are not ready to
address some of the most pressing current problems in stellar flare physics.
Flare models require accurate treatments of non-equilibrium physics and
detailed radiative transfer for comparisons to chromospheric observations. An
approximation to nonthermal particle heating can in principle be included as
$Q_{\rm{flare}}$ in the energy equation in 3D simulations of small flares
[236]. The geometrical grid spacing in 3D MHD models, however, is immobile and
too coarse (e.g., see Sect. 8.2) to resolve the gradients and contribution
function features for the optically thick spectral lines and continua. Section
4.6.1 of [76] discusses the issues facing a 3D MHD solar flare simulation that
predict chromospheric emission lines in response to thermally conducted
fluxes. The RADYN code (for example) was originally a “quiet” Sun code, but it
had implemented the 1D adaptive grid scheme of [175] and was flexible enough
to simulate particle beam heating in a non-solar gravity. These features
proved fortuitous in applications to stellar flare models. The stellar
community is in urgent need of collaborative development of multi-dimensional
extensions of Carlsson’s and Dorfi’s works, such as [699], in order to
understand the comprehensive, radiative-hydrodynamic response of stellar
atmospheres to the impulsive release of magnetic energy.

##  13 Conclusions and Future Outlook

In conclusion, we summarize six, non-exhaustive big-picture questions that are
(in our view) at the forefront of stellar flare research.

  * •

Stellar flares routinely attain energies and peak luminosities that are
factors of $10^{2}-10^{4}$ greater than the largest solar flares. How are the
flare energies connected to photospheric magnetic flux densities/geometries,
particle acceleration properties, flaring footpoint areas, and energy
transport processes in the atmospheres of other stars

  * •

Stellar superflares exceed the quiescent bolometric stellar luminosity. The
rate of superflares, however, is not characterized with high statistical
significance. What are the rates of events in the superflare regime for solar
analogs and other low-mass cool stars as a function of stellar age?

  * •

On the Sun, the brightness of chromospheric flaring sources is transient at
any single location for (arguably) $\lesssim 10-20$ s. A comprehensive, self-
consistent model (e.g., within frameworks similar to Reeves and Forbes 614 and
Rempel et al 623 pertaining to solar flare soft X-rays) has not yet been
developed for optical or radio stellar flare light curves that last for tens
of minutes to hours. What are the principal physical parameters that drive the
observed temporal scales of stellar flare light curves of optically thick
radiation in the rise and decay phases?

  * •

Blue asymmetries in Balmer lines are often interpreted as evidence of mass
eruptions from a star. Are there direct analogues [e.g., as in Mg II; 707] on
the Sun, and does chromospheric evaporation in stellar flares contribute to
blueshifted spectral features in cool lines?

  * •

Turbulent mass motions and other non-equilibrium macroscale processes (e.g.,
UV and X-ray radiative backwarming) in stellar flares remain poorly understood
in three spatial dimensions. If gas-dynamic turbulence is important for
explaining the observed spectral line shapes, what processes generate large
Reynold’s numbers in flare atmospheres and, more specifically, in flare
chromospheres?

  * •

Optical and NUV stellar flare spectra are inconsistent with optically thin
hydrogen recombination theory. Electron-beam and thermally-conducted heating
fluxes that are within our expectations of the standard solar flare model
paradigm do not explain the observed strength and spectral shape of the
continuum radiation at NUV and optical wavelengths in M-dwarf flares. What is
the source of stellar flare white-light continuum radiation, which is
prominent in almost all of the _Kepler_ (and TESS) broadband flare data?

Supplementary information Additional data are provided in supplementary online
material hosted on Zenodo at https://doi.org/10.5281/zenodo.10641273.

Acknowledgments I thank Dr. Yuta Notsu for providing helpful comments and
corrections. I gratefully acknowledge an anonymous referee for valuable
comments and suggestions. I am grateful to Dr. Suzanne Hawley for much
guidance and many stimulating discussions about stellar flares over the years.
I thank Dr. Mihalas Mathioudakis and the solar physics group at Queen’s
University Belfast for many productive discussions. I thank Dr. Rachel Osten
for many discussions about stellar flares, radio and X-ray observations, and
about energy partitions. I thank Dr. John Wisniewski for many conversations
about near-ultraviolet stellar flares. I thank Dr. Thomas Gomez for helpful
discussions about line broadening theory, Dr. Eugene Oks for discussions about
plasma line broadening, Dr. Jim Drake and Dr. Joel Dahlin for discussions
about PIC simulations and related plasma physics topics, Dr. Joel Allred for
extremely helpful guidance in radiative-hydrodynamics and assistance with
flare modeling, Dr. Mats Carlsson with invaluable assistance with flare
modeling, Dr. Han Uitenbroek for helpful conversations about stellar
atmosphere codes, Dr. Lyndsay Fletcher, Dr. Gianna Cauzzi, Dr. Dana Longcope,
Dr. Eduard Kontar, and Dr. Markus Aschwanden for stimulating discussions on
solar flare topics, Dr. Gregory Fleishman and Dr. Stephen White for
discussions about solar flare radio observations, Dr. Andy Inglis for helpful
discussion about quasi-periodic pulsations in solar and stellar flares, Dr.
Matthias Rempel and Cole Tamburri for conversations about 3D flare modeling,
and Dr. Manuel Güdel for discussions about the Neupert effect and stellar
flares on Proxima Centauri. I acknowledge the International Space Science
Institute (ISSI) for my participation in several solar-stellar workshops that
were organized and lead by Dr. Lyndsay Fletcher, Dr. Sven Wedemeyer, Dr.
Louise Harra, Dr. Graham Kerr, Dr. Vanessa Polito, and Dr. Paola Testa.
Lastly, I thank all of the researchers in this field for developing such an
engrossing area of study within astrophysics.

##  Appendix A Optical Filter Reference

We include several appendices in this Living Review. This appendix, Appendix
A, serves as a quick-reference for optical broadband filter curves. Appendix B
shows supplementary spectral constraints from optical dMe flare data, Appendix
C reviews two continuum modeling approaches with slab geometries (blackbody
fitting and optically thin hydrogen emissivity spectra) and the continuum
modeling approach with parameterized chromospheric condensations [410], and
Appendix D reviews the salient physics (through a mostly qualitative
exposition) of the pressure broadening of the spectral lines of hydrogen.

Figure 33:  Quantum efficiency curves of common optical filters, normalized to
peak transmission. An example of a terrestrial atmospheric transmission curve
from the Sloan Digital Sky Survey is shown for an airmass of 1.0 at the Apache
Point Observatory. One should note that the quantum efficiency of the CCD,
telescope primary mirror, and other optical elements are not included. For
reference, the impulsive phase flare spectrum from the IF4 event in [416] is
shown. The wavelengths that are affected by the dichroic are in a lighter
shade of red. The ULTRACAM photometry (NBF3500, NBF4170, NBF6010) at the peak
of this flare were calibrated independently, and they are scaled to account
for the factor of ten shorter integration time. The flare data were obtained
from the Zenodo repository at: https://doi.org/10.5281/zenodo.45878.

In Fig. 33, we show the optical broadband and narrowband filters that are
commonly used in stellar flare studies. There are numerous databases that
provide these, such as the SVO Filter Profile Service at http://svo2.cab.inta-
csic.es/theory/fps/, and some are published alongside photometry calibration
surveys [67]. For comparison, we show the low-resolving power, flare-only
spectrum [from Figure 3 of 416] over the impulsive phase of the IF event from
the bottom panel of Fig. 8. The spectral shape from the calibrated ULTRACAM
narrowband filters at the flare peak time is in agreement with the spectrum.

##  Appendix B A Summary of Supplemental Model Constraints in the Optical

In this appendix, we supplement the impulsive phase flare colors from spectra,
shown in Fig. 12, with the peak colors calculated from narrow-band continuum
filters. Fig. 34(top) is a summary of the peak-phase ULTRACAM filter (Appendix
A) ratios from [416]. These flare colors were calculated at higher time
resolution than from the spectra that were discussed in the main text (Sect.
7.2). Figure 34(middle) shows the evolution of the ULTRACAM filter ratios in
the large IF1 event described in row (1a) of Table 7; see also Kowalski [409].
Average fluxes in the ULTRACAM narrowband filters are calculated from the
spectra of the energetic IF3 event (also on YZ CMi) from [414] and are shown
for comparison in Fig. 34(middle). Note, the peak flare spectrum for this
event is showcased in Fig. 10. In the bottom panel, we show Balmer jump ratios
against the ratios of the H$\gamma$ line flux to the flare-only continuum flux
around $\lambda=4170$ Å at the time corresponding to the optical continuum
peak flux. (The H$\gamma$ line flux is continuum-subtracted, integrated over
the wavelengths of the line, and pre-flare subtracted; C4170′ is the average
the flare-only flux over $\lambda=4155-4185$ Å). Fig. 34(bottom) combines the
impulsive-phase spectral measurements from [414] with the same quantities
obtained in the four flares (indicated on the top axis) with low-resolving-
power, optical spectra that were analyzed in [416] and [420]. It is our hope
that as more flares are observed, these figures will be updated to provide
compelling and comprehensive constraints for future multi-dimensional flare
models.

Figure 34:  (_Top_) Flare-peak colors from ULTRACAM narrowband filter ratios;
similar to Fig. 12 and see text for details. (_Middle_) Evolution of flare
colors in two large flares on YZ CMi. Time increases over the tracks in a
clockwise direction. These data have small statistical uncertainties (which
are shown) but non-negligible systematic errors. The uncertainties are
correlated, which we show for representative 5% systematic absolute color
calibration uncertainties in both flare colors located at $(x,y)\approx(2,2)$.
The error-ellipses indicate representative joint probabilities: the outer is
the joint 68% confidence interval, while the inner translates to the marginal
68% uncertainties. (_Bottom_) Flare-only, line-to-continuum flux ratios from
the same peak spectra in Fig. 12 in the main text.

##  Appendix C Slab Model Reference

###  C.1 Blackbody Fitting to Stellar Flare Optical Data

In this appendix, we summarize the assumptions and techniques employed by
fitting isothermal blackbody functions to spectra, broadband photometry, and
narrowband photometry of flares. A blackbody “color” temperature refers to the
temperature of a blackbody that is fit to a ratio of two fluxes (e.g., a flare
color calculated from a ratio of two flare-only continuum fluxes; Sect.
7.2.1), where color is equivalently a difference of magnitudes in two
bandpasses. The term also refers to the temperature of a blackbody that is fit
to the absolute value of more than two continuum flux values (outside of
prominent emission lines) over a limited spectral range ($\Delta\lambda\approx
1000-2000$ Å). Generally, a color temperature contains rather limited
information about the actual nature of the radiation, such as whether it is
optically thick, thin, an intermediate optical thickness, or which atomic
processes contribute to the emergent radiative flux. A color temperature does
not unambiguously constrain how the spectrum of continuum radiation extends to
wavelengths outside of the range over which it is calculated. Within the total
wavelength span that a flare color is calculated (e.g., the ratio of flare
fluxes in narrowband filters around 4170 Å and 6010 Å), there may be emission
lines that affect the calculated value, and the actual spectrum between two
wavelengths may be best represented by a more complex continuum model.
Therefore, flare color temperatures are _equivalent_ color temperatures of
blackbody functions.

Other measures of temperature include gas temperature, radiation (or
brightness) temperature, and effective temperature (in addition to the color
temperatures of optically thin recombination and bremsstrahlung spectra). None
of these are possible to robustly measure directly from optical stellar flare
observations and must be inferred from detailed modeling. In principle, flare
effective temperatures could be measured, but there has not yet been a more
complete broadband spectral observation of a flare than analyzed in Hawley and
Pettersen 297. To demonstrate the ambiguities in these measures of
temperature, it is interesting to note that for an optically thin hydrogen
recombination and free-free spectrum calculated from LTE level populations for
a slab with a gas temperature of $T_{\rm{gas}}=10,000$ K (Appendix C.2, Fig.
37), the radiation temperature at $\lambda=4170$ Å is $T_{\rm{rad}}=3600$ K.
The emergent intensities are calculated for a slab thickness of $dl=0.1$ km to
ensure that $\tau(\lambda)<0.005$ across the optical regime. The color
temperature measured from the ratio of intensities at $\lambda=4170$ Å to 6010
Å is $T_{\rm{FcolorR}}=5300$ K. Thus,
$T_{\rm{gas}}>T_{\rm{FcolorR}}>T_{\rm{rad}}$ in this example.

Another interesting ambiguity in inferred gas temperatures occurs when the
emergent continuum flux is formed over significant ($\tau\approx 1$) optical
depth in dynamic, non-LTE, non-isothermal flare atmospheres. In Fig. 35, we
summarize the argument in [409], which is also closely related to the analyses
in the appendices of [418]. Two Planck functions with $T=13,000$ K and
$T=14,000$ K are shown. If wavelength-dependent, continuum optical depths are
large over an inhomogeneous temperature structure, then one could possibly,
for example, measure radiation temperatures of $T_{\rm{rad}}=13,000$ K at
$\lambda=4170$ Å and $T_{\rm{rad}}=14,000$ K at $\lambda=6010$ Å. The ratio of
the corresponding emergent intensities gives a color temperature that is only
$T_{\rm{FcolorR}}=10,500$ K. Thus, $T_{\rm{rad}}>T_{\rm{FcolorR}}$ in this
example. For semi-infinite atmospheres that exhibit a linear dependence
between the optical depth and the (LTE or non-LTE) source function, the
Eddington-Barbier approximation relates the radiation temperature to the gas
temperature at the depth corresponding to $\tau(\lambda)=1$; however, dynamic
flare atmosphere calculations are usually much more complicated over the large
physical depth ranges of continuum formation (e.g., in chromospheric
condensations and the stationary layers below; _cf._ Fig. 2(right) and Fig.
21), which results in differences between the radiation temperature and the
gas temperatures over which the emergent continuum intensity forms [see 409,
for details].

Figure 35: Demonstration showing the relationship between a color temperature
and wavelength-dependent radiation (brightness) temperatures at $\lambda=4170$
Å and $\lambda=6010$ Å.

The temperatures and filling factors calculated from optical and NUV flare
data are in general obtained through two methods:

  1. 1.

From a ratio of flare-only fluxes in two bandpasses, one can solve for the
temperature and filling factor ($T_{\rm{flare}}$, $X_{\rm{flare}}$) of a
Planck function following Hawley et al 298 (i.e., by equating ratios of the
RHS of Eq. 28 in Sect. 12 to the observed flare-only ratio). The ratio
equation is a transcendental equation for $T_{\rm{flare}}$ and can be solved
numerically using linearization and iteration (i.e., Newton-Raphson) or
through a simple lookup table. The result for $T_{\rm{flare}}$ is used to
infer $X_{\rm{flare}}$.

As discussed in [392] and [121], by not accounting for the fraction of stellar
flux ($X_{\rm{flare}}$) that no longer shines through with the quiescent
brightness, such as in a blackbody flare [see 298] or a flare with a transient
overlying absorbing layer, it is possible to calculate a color temperature of
$\approx 8500$ K from flare-only irradiance observations of solar flares, as
reported in Kretzschmar [422]. In Fig. 36, we demonstrate what happens to the
calculated color temperature of an optically thick ($\tau\gg 1$) flare source
if one does not use the proper equation in [298]. For example, a 5800 K Planck
function subtracted from a 7800 K Planck function at $\lambda_{1}=4170$ Å and
$\lambda_{2}=6010$ Å gives a blackbody color temperature of 9000 K (_c.f._ ,
where the 9000 K solid line and the 7500 K dashed line approximately
intersect). However, for certain conditions, $S_{\lambda,\rm{flare}}\gg
S_{\lambda,q}$ where $S_{\lambda,\rm{flare}}$ is the model flare surface flux
spectrum, one may ignore the background subtraction errors (in which case, the
problem must still be solved numerically or through a simple lookup table).
Using the actual quiet Sun photospheric intensity, which is larger than a
Planck function at these wavelengths, a 7500 K Planck function is sufficient
to produce a 9000 K blackbody shape in the subtraction residuals. The surface
fluxes of M dwarfs are much fainter than on G dwarfs. M-dwarf surface fluxes
are lower than blackbody functions given by their effective temperatures, and
the errors indicated in Fig. 36 are consequently smaller. For optically thick
emission lines, similar issues arise if the model surface fluxes are
comparable to the pre-flare surface fluxes (thus, it is fortuitous that large
electron beam flux models produce spectra that are extremely bright compared
to the pre-flare surface flux spectrum!).

Figure 36: Lines of constant blackbody flare temperatures ($T_{\rm{flare}}$)
and lines of constant blackbody color temperatures that are calculated after
subtracting a quiescent blackbody surface flux spectrum,
$S_{\lambda,q}=B_{\lambda}(T_{q})$, from the flare model spectrum,
$S_{\lambda,\mathrm{flare}}=B_{\lambda}(T_{\rm{flare}})$. The color
temperature calculations, here shown for $\lambda_{1}=4170$ Å and
$\lambda_{2}=6010$ Å, from stellar flare data usually account for or consider
these issues in M dwarf flares [298, 416], but hotter flare color temperatures
in higher-mass stars could be a result of “oversubtraction” effects.

  2. 2.

Second, a color temperature can be fit to a spectrum of observed fluxes. A
Planck function is typically fit to the excess (flare-only) spectrum, or to
flare-only multi-band photometry data [302], with $n\lambda$ points ($n>2$)
and Gaussian uncertainties, thus deriving the maximum likelihood parameters
$\hat{T}_{\rm{flare}},\hat{X}_{\rm{flare}}$. This can be accomplished with
non-linear least squares through linearization and iteration for unconstrained
parameter fits and with the Levenberg-Marquardt algorithm (e.g., IDL’s
_mpcurvefit_) for constrained parameter fits. The color temperatures (e.g.,
$T_{\rm{BB}}$) that were quoted from blackbody fitting in [414] were obtained
by fitting Planck functions directly to the observed flare-only spectral
fluxes (i.e., assuming $B_{\lambda}(T_{\rm{flare}})\gg S_{\lambda,q}$). Also
in that paper, the excess RHD model flux spectra were similarly compared to a
Planck function temperature without subtracting a pre-flare from the Planck
function. In this approach, the color temperature is just a convenient
parametrization for comparing models to flare-only spectral observations,
since temperatures are easier to compare than color indices in different
filters or at different wavelengths (as noted in the main text). Whatever one
does to the data, one should also do analogously to the model, whether the
model is a blackbody or an RHD spectrum, irrespective of spectral type of the
star (from RHD models of M dwarf flares that are optically thin in the
chromospheric continuum radiation, it is actually desirable to isolate the
chromospheric flare radiation by subtracting the pre-flare surface flux
because the model photospheric spectrum, which does not have proper molecular
opacities, otherwise shines through and obfuscates comparisons to
observations).

The main purpose of subtracting a pre-flare spectrum from the flare
observations is to remove the relatively large, wavelength-dependent flux from
the area on the star that is not flaring so that the flare emission is more
accurately characterized. A method for determining an accurate subtraction of
the preflare spectrum based on molecular bandpass veiling by the flare
continuum is described in [414]. Alternatively, a very wide slit may be
employed, but spectral purity and wavelength resolution are sacrificed; see
[416].

In summary, the first method uses two filters and assumes that the flare
spectrum is inherently blackbody radiation emitting at a single temperature
$T_{\rm{flare}}$, and the second method is used for calculating color
temperatures ($T_{\rm{BB}},T_{\rm{FcolorR}}$) directly from a spectrum of
flare-only fluxes assuming that the effects in Figure 35 are not important
(although this assumption can readily be relaxed in the second method).

###  C.2 Hydrogen Continuum Emissivity Models

[428] calculated a grid of optically thin, LTE hydrogen recombination models
with free-free contributions. These are often used as a starting place for
comparisons to flare observations (e.g., Fig. 12). The equations are not
included in [428] but were restated in [406]. Here, we update the calculations
with free-free Gaunt factors from CHIANTI and include the dissolved-level
Balmer continuum components (longward of the Balmer edge at $\lambda=3646$ Å)
in the spectra. These are shown in Fig. 37 for a range of temperatures and a
gas density of $\rho=10^{-8}$ g cm-3. The Balmer jump ratios are calculated
from these spectra and are shown in Fig. 38. All spectra are available as FITS
tables for a range of gas densities in the supplemental online material hosted
on Zenodo.

Figure 37: Hydrogen LTE emissivity calculations with bound-free
(recombination) emissivity from H I and H- and free-free (thermal
bremsstrahlung) emissivity from H II. The spectral shapes here are identical
to the shapes of intensity spectra, $I_{\lambda}$, that emerge from optically
thin, homogeneous slabs. A blackbody function and a mono-energetic synchrotron
spectrum is shown for comparison. The smooth transitions from the Balmer jump
to the optical wavelengths are caused by the “dissolved level Balmer
continuum” [154], which extends to redder wavelengths for larger electron
densities.  Figure 38: Balmer jump ratios calculated from the LTE emissivity
models in Fig. 37 and from the same calculations of the LTE emissivity in a
hydrogen gas with a density of $\rho=10^{-9}$ g cm-3.

At constant slab temperature, a large range of flare colors result from
adjusting the continuum optical depths [512]. In Fig. 39, we show the LTE
continuum spectra from RHD model parameterizations [410]; the colors of these
spectra are calculated and are shown in Fig. 12 as the “KA18 CC Models”. A
range of reference column masses ($m_{\rm{ref}}$) were chosen to increase the
continuum optical depths within the model chromospheric condensations.
Occupational probabilities ($w_{n}$) for the dissolved level Balmer continuum
opacities are included as in Figure 37.

Figure 39: (Left) LTE continuum flux spectra calculated from the parameterized
evolution of chromospheric condensations from RHD models. (Right) Examples of
two of the parameterized atmospheres.

##  Appendix D Pressure Broadening of Hydrogen Lines in Flares

Electric fields split the degenerate energy levels of hydrogen atoms, Rydberg
atoms, and hydrogen-like ions by an amount directly proportional to the field
magnitude [$\propto E^{1}$; e.g., 137, 68, 247, 267], which is commonly known
as the linear Stark, or Stark-Lo Surdo535353This effect was discovered nearly
simultaneously in experiments conducted by Johannes Stark and Antonino Lo
Surdo in 1913 and 1914. Additional ground-breaking experiments were conducted
by Toshio Takamine and Noboru Kokubu. The classical splitting of energy levels
by electric fields was predicted by Woldemar Voigt to be very small, and the
large splitting in hydrogen that was subsequently observed by Stark and Lo
Surdo was one of the first major achievements of quantum theory, thanks to
pioneering theoretical efforts of Bohr, Sommerfield, Epstein, K.
Schwarzschild, Schrodinger, and Kramers. The relevant history is thoroughly
summarized in [444] and [460]. See Kleppner et al [393] for an excellent high-
level overview. effect. The splitting increases for larger $n$, and $n=1$
experiences only second-order energy shifts, which are small. Within a
partially ionized gas, the electric fields from ambient, thermal charges
(electrons and protons) within a Debye radius broaden spectral lines. This
explains the broad hydrogen absorption profiles in main sequence A stars and
white dwarfs (combined with, of course, large optical depths in their
photospheric layers).

The symmetric, atomic energy-level shifts that are linearly proportional to an
external, uniform electric field $\mathcal{E}$ are described by the quantum
numbers $nqm$, according to the following absolute value (in _cgs_ units):

$|\Delta E^{(1)}|=|\frac{3a_{0}e}{2Z}nq\mathcal{E}|$ (29)

In Eq. 29, $n$ is the principal quantum number, $q$ is the difference of two
parabolic quantum numbers, $k_{1}-k_{2}$, that satisfy $n=k_{1}+k_{2}+\mid
m\mid+1$, $m=m_{l}$ is the magnetic quantum number, $a_{0}$ is the Bohr
radius, $e$ is the electron charge, and $Z$ is the nuclear charge (where $Z=1$
for hydrogen).

Fig. 40 shows the splitting of the energy levels of hydrogen in a static
electric field. The energy shifts were calculated using Equation 29 and the
quadratic (second-order) and cubic (third-order) perturbation terms for
hydrogen [137, 68, 186, 267]. The second-order and third-order terms create
asymmetric splittings but are only important for hydrogen at very high
electric field strengths. The maximum possible wavelength shift for an allowed
Balmer transition ($n_{\rm{lower}}=2$) and $\Delta m\pm 1,0$ follows from Eq.
29 and is proportional to [733]:

$\big{(}\frac{n_{j}^{2}}{n_{j}^{2}-4}\big{)}^{2}(n_{j}(n_{j}-1)+2)n_{e}^{2/3}$
(30)

where $\big{(}\frac{n_{j}^{2}}{n_{j}^{2}-4}\big{)}^{2}$ is the appropriate
conversion from $dE$ to $d\lambda$, and $n_{j}$ is the upper level of the
Balmer transition. This formula, which goes as $n_{j}^{2}$ in the limit of
large $n_{j}$, is sometimes used as an equivalent Doppler width that is added
in quadrature with the true Doppler width in a Voigt profile. However, this
approximation, among others [703], predicts inaccurate scalings for the line
shapes and widths within a series. Thus, we recommend that these
approximations not be used in place of proper line broadening calculations,
which we briefly summarize next.

Figure 40: Energy level splitting for hydrogen in a static external electric
field as a function of the electric field (in SI units). A principal level $n$
is split into roughly $2n-1$ (including the unshifted) sub-states. For even
$n$, there is an unshifted component. Every split sub-state ($k_{1},k_{2},m$)
is a mix of all the $l$-states for that value of $n$. The maximum splitting of
the $n=5$ level shown here corresponds to 0.016 eV, which would correspond to
a maximum wavelength displacement of $\Delta\lambda\approx 24$ Å for the
H$\gamma$ transition. Only levels that can result in a dipole transition with
$n=2$ have been plotted. The transitions with $\Delta m=0$ are known as the
$\pi$ components, and $\Delta m\pm 1$ are the $\sigma$ components. The field
strength that forms a “saddle point” (potential energy maximum) and destroys
all levels $>n$ in hydrogen is shown as a function of $n$ [if there is a
saddle-point in the potential, then tunneling is allowed as well; 68]. The
field ionization of hydrogen has been observed (see the photograph from [606]
that is reproduced in Montgomery et al 509). The Inglis-Teller relation
corresponds to the critical field that causes the “bluest” sub-state of $n$ to
intersect with the “reddest” sub-state of $n+1$, which occurs at a much
smaller electric field for a given $n$ than the saddle point estimate (the
ratio of the critical field to the saddle-point field is given by the factor
known as $K_{n}$ in the literature). Holtsmark probability distributions of
electric field magnitudes are shown in the top panel to illustrate an
occupational probability ($w_{n}$) calculation for $n=11$ and $n_{e}=10^{15}$
cm-3. A “normal field”, $F_{0}$, is the electric field magnitude at an average
interparticle separation; the critical fields for $n_{e}=10^{14}$ and
$10^{15}$ cm-3 are indicated by vertical dotted lines in the top panel.

In a partially ionized gas, the electric microfield amplitude and direction
vary from atom to atom and are not constant in time even at one atom because
the charge distribution changes. The microfield amplitude probability is
described by a probability distribution function (e.g., in the top panel of
Fig. 40). A brief phenomenological summary of the microphysics of the
broadening of hydrogen spectral lines follows. See [51], [262], [271], [337]
for more comprehensive and rigorous reviews545454My description herein
combines elements from classical and quantum theory, while generally following
the classical path “unified theory” framework of [744, 745, 746] (VCS73) that
was extended to incorporate the [348] model of non-ideal effects in the line
profile shapes following [665] and [721]. Note that some of the important
approximations made by VCS73, and other groups at the time, are no longer
necessary in newer, numerical simulations of the low-order Balmer lines [270,
269, 130]. . The densities of perturbing, ambient (thermal) protons ($n_{p}$)
and electrons ($n_{e}$) affect the total electric pressure broadening of
radiative (dipole) transitions in hydrogen differently. The slow-moving
protons and ions produce quasi-static [meaning that the microfield changes
slowly compared to radiative/orbital timescales; see 275, 348] electric
microfields that split the energy levels of hydrogen into sub-states (Fig.
40), which are superpositions of the various orbital angular momentum states,
$l$, within each $n$ (“manifold”). The microfield magnitude is given by a
Hooper probability distribution [which is a modified Holtsmark distribution
that accounts for Debye screening and plasma correlations; 532]. The
collisional perturbations from ambient electrons are more complicated.
Classically, the individual elastic electron impacts on each atom are randomly
occurring and well-separated in time, thus causing a damping/Lorentzian
profile function with broadening FWHM $\Gamma$ that is proportional to
$n_{e}$. However, on very short timescales, which correspond to large
frequency displacements from line center (in the far line wings) in Fourier-
conjugate space, the surrounding electrons are nearly still and thus produce a
quasi-static (Holtsmark) broadening proportional to $n_{e}^{2/3}$. The
transition in electron broadening from its impact to its quasistatic regimes
occurs at wavelength shifts, of order $|\lambda-\lambda_{\rm{rest}}|\approx
20$ Å (for the H$\gamma$ line). Thus, large optical depths can make the
effects of the quasi-static limit of electrons important in the wings.
Additionally, lifetime broadening occurs if there are inelastic effects, such
as collisional transitions to states of different principle quantum number $n$
[e.g., 348, 86], but these are generally small for well-separated levels (and
are ignored in the “no-quenching” assumption in the VCS73 theory).

The energy-level perturbations from protons and from electron collisions are
not independent random variables, and the line profiles thus cannot be modeled
as convolutions of two probability distribution functions, such as a
Lorentzian convolved with a Holtsmark (or Hooper) distribution. This is
because the electron collisions with hydrogen are transient (short duration)
and non-adiabatic555555 _cf._ the Adiabatic Theorem which is described in most
textbooks on quantum mechanics. [686, 689, 262, 337]: they can cause
transitions and redistribution among the microfield-split sub-states of the
same $n$. Quantum calculations [Eq 10 of 721, which is reproduced from Vidal
et al 744] that account for the perturbing electrons in both of their limiting
regimes are crucial. If the dynamical aspects [692] of the ions are ignored,
the normalized line profile functions are then the ionic microfield averages
of the electronic broadening profiles calculated at each value of the quasi-
static (QS), ionic microfield $\mathcal{E_{\rm{QS}}}$. When the spacing
between the upper levels $n_{j}$ becomes small and the broadening causes the
states to have a significant probability of overlapping, level dissolution
occurs and the bound-bound transitions become narrower [665, 721] and less
Holtsmark-like in the wings. The extensions of the VCS73 profiles that account
for the effects of level dissolution through occupational probabilities (Sect.
10.2.1) are known as the TB09$+$HM88 profiles. [130] discuss comparisons of
high-density analytic profiles to new numerical calculations without level
dissolution effects [see also 694, 269]. To our knowledge, the TB09$+$HM88
profiles are yet the only quantum calculations extended to large $n_{j}$ that
also include a framework, albeit phenomenological, for ionization-limit
lowering consistently with the bound-bound opacities and line shapes. This
method is adequate for model spectra of the A0 V star Vega [e.g., 418], and
thus by extension for flare models, given that there is an unfortunate paucity
of echelle observations in this spectral region in stellar [241, 243] and
solar [173] flares.

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