# The efficiency of electron acceleration during the impulsive phase of a
solar flare

Eduard P. Kontar School of Physics & Astronomy, University of Glasgow,
Scotland, UK  A. Gordon Emslie Department of Physics & Astronomy, Western
Kentucky University, Bowling Green, KY 42101, USA  Galina G. Motorina
Astronomical Institute of the Czech Academy of Sciences, 251 65 Ondřejov,
Czech Republic
Central Astronomical Observatory at Pulkovo of Russian Academy of Sciences,
St. Petersburg, 196140, Russia
Ioffe Institute, Polytekhnicheskaya, 26, St. Petersburg, 194021, Russia  Brian
R. Dennis Solar Physics Laboratory, Code 671, Heliophysics Science Division,
NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

###### Abstract

Solar flares are known to be prolific electron accelerators, yet identifying
the mechanism(s) for such efficient electron acceleration in solar flare (and
similar astrophysical settings) presents a major challenge. This is due in
part to a lack of observational constraints related to conditions in the
primary acceleration region itself. Accelerated electrons with energies above
$\sim$20 keV are revealed by hard X-ray (HXR) bremsstrahlung emission, while
accelerated electrons with even higher energies manifest themselves through
radio gyrosynchrotron emission. Here we show, for a well-observed flare on
2017 September 10, that a combination of _RHESSI_ hard X-ray and and SDO/AIA
EUV observations provides a robust estimate of the fraction of the ambient
electron population that is accelerated at a given time, with an upper limit
of $\,\lower 3.0pt\hbox{$\sim$}\hbox to0.0pt{\hss\raise
2.0pt\hbox{$<$}}\,10^{-2}$ on the number density of nonthermal ($\geq 20$ keV)
electrons, expressed as a fraction of the number density of ambient protons in
the same volume. This upper limit is about two orders of magnitude lower than
previously inferred from microwave observations of the same event. Our results
strongly indicate that the fraction of accelerated electrons in the coronal
region at any given time is relatively small, but also that the overall
duration of the HXR emission requires a steady resupply of electrons to the
acceleration site. Simultaneous measurements of the instantaneous accelerated
electron number density and the associated specific electron acceleration rate
provide key constraints for a quantitative study of the mechanisms leading to
electron acceleration in magnetic reconnection events.

##  1 Introduction

Spatially resolved hard X-ray (HXR) observations of solar flares reveal that
electron acceleration likely occurs in magnetic reconnection outflow regions
(see, e.g., Holman et al., 2011; Benz, 2017, for reviews). Indeed,
observations from the Ramaty High Energy Solar Spectroscopic Imager (_RHESSI_
; Lin et al., 2002) observations have demonstrated (Sui & Holman, 2003; Asai
et al., 2007; Liu et al., 2008; Battaglia & Kontar, 2013; Krucker & Battaglia,
2014; Battaglia et al., 2019) the presence of faint sources of HXR emission
above the tops of coronal loop structures as evidence for magnetic
reconnection, as first noted by Masuda et al. (1994). These sources are
consistent with regions where magnetic turbulence could provide sufficient
power to accelerate electrons to the required energies (Kontar et al., 2017;
Ruan et al., 2019; Chitta & Lazarian, 2020; Stores et al., 2021). It is well
known (e.g., Miller et al., 1997; Zharkova et al., 2011) that the total number
of electrons accelerated during a large solar flare can exceed the total
number of electrons in the coronal source region, so that continuous resupply
of the acceleration region (e.g., by a cospatial return current; Knight &
Sturrock, 1977; Emslie, 1980; Zharkova & Syniavskii, 1997; Alaoui & Holman,
2017) is an essential ingredient of a viable electron acceleration model.

Spatially-integrated bremsstrahlung hard X-ray emission, produced during
collisions of non-thermal electrons on ambient ions, provides direct
diagnostics of the total electron acceleration rate $\dot{N}$ (s-1) at
energies above a specified energy (typically taken to be 20 keV). In such a
“thick-target” calculation, the deduced electron acceleration rate is
generally independent of the density structure of the flaring region, but the
relationship between the accelerated electron distribution and the emitted HXR
spectrum does depend on whether a cold (Brown, 1971; Syrovatskii & Shmeleva,
1972) or warm (Kontar et al., 2015) thick target model is used. (The
dependency of total acceleration rate arises through the difference in the
energy loss term and transport through plasma (Kontar et al., 2019) – where
the average energy of the ambient electrons is much smaller than the energy of
the accelerated electrons – versus warm targets, where the energies of the
accelerated and target particles can be comparable.) The inferred total
electron acceleration rate depends strongly on the value of the low-energy
spectral cutoff, which is difficult to determine from observations (e.g.
Aschwanden et al., 2019, as a review).

By contrast with the spatially-integrated case, the HXR flux spectrum from a
given spatial sub-region in the flare provides information on the density-
weighted mean electron flux spectrum in that region. For this analysis, “thin-
target” modeling (Brown et al., 2003) is used; this does not require any
assumptions about the nature of the prevailing electron transport mechanism.
This density weighting implies that bright hard X-ray footpoints are
conspicuously evident (e.g., Emslie et al., 2003; Kontar et al., 2008) because
of the high density in the chromosphere, while strong coronal HXR sources are
relatively rare (Holman et al., 2011) because the density is so low. Given the
limited dynamic range of _RHESSI_ and similar instruments that employ indirect
(Fourier-transform based) imaging techniques, the problem of characterizing
the HXR emission from low-density coronal acceleration regions is exacerbated
by the presence of much brighter footpoint emission in the field of view.

Of considerable interest in understanding the nature of the process that
accelerates particles to high energies are the ratios of the number densities
(cm-3) of nonthermal and thermal electrons ($n_{nth}$ and $n_{th}$,
respectively) to the total number density of background electrons, which, from
considerations of charge neutrality, is equal to the number density of protons
in the same region: $n_{p}=n_{th}+n_{nth}$ in hydrogen plasma. The
instantaneous fraction of accelerated electrons in the flaring corona has been
previously discussed in the literature, but the results, methods, and the
region that defines the “coronal source” and “above-the-looptop” differ
somewhat and hence necessitates addition investigations. Krucker et al. (2010)
reported $n_{nth}(>\text{16~{}keV})/n_{p}\simeq 0.1$, while Oka et al. (2013)
re-analysed the flare and deduced $n_{nth}(>$$20$ keV)$/n_{p}\simeq 0.01$ (see
their Equation (4)). Krucker & Battaglia (2014) analysed a limb flare in the
presence of bright footpoint emission and found
$n_{nth}(>(10-15)~{}\text{keV})/n_{p}\simeq 1$ in the region above the coronal
source, assuming a power-law spectrum with a sharp low-energy cut-off in the
range from $10-15$ keV. However, this high value was obtained using a
Maxwellian+power-law fit to the hard X-ray spectrum. Mean source electron
spectra (Brown et al., 2003) in some coronal hard X-ray sources have been
found to be more consistent with a kappa distribution (Kašparová & Karlický,
2009), which rolls over smoothly from a power-law at high energies to a
Maxwellian at low energies. Compared to fitting the thermal and nonthermal
(power-law) parts of the spectrum separately, fitting the data with a kappa
distribution generally results (as in Oka et al., 2013) in a significant
reduction in the required electron flux near the rollover energy and hence a
lower inferred value of $n_{nth}$. Indeed, Battaglia et al. (2015) demonstrate
that using a kappa-distribution fit to _RHESSI_ data, while also incorporating
EUV emission line data to constrain the fit, results in a reduction in
$n_{nth}$ by a factor of up to $\sim$30 compared with the commonly-used power-
law-only fit. Early in a flare, electron distributions in magnetic
reconnection outflow regions could form a bulk thermal distribution with a
relatively steep (e.g., power-law) non-thermal tail, so that the electrons
above, say, $20$ keV represent only a small fraction of the total ambient
population (Battaglia et al., 2019). However, the spectrum could change during
the peak of the flare. Radio emission produced by gyrosynchrotron radiation of
accelerated electrons with relatively high energies $\,\lower
3.0pt\hbox{$\sim$}\hbox to0.0pt{\hss\raise 2.0pt\hbox{$>$}}\,100$ keV is also
commonly observed in the flaring atmosphere (see Dulk & Marsh, 1982; Nindos,
2020, for reviews) and provides insight into the properties of electrons with
energies higher than those that generate most of the HXR emission (e.g. White
et al., 2011; Musset et al., 2018).

In this paper, we combine thin-target and thick-target modeling of _RHESSI_
X-ray observations of a well-observed solar flare with contemporaneous EUV
observations from the Solar Dynamics Observatory Atmospheric Imaging Assembly
(SDO/AIA; Lemen et al., 2012) in order to better constrain both the total
number of accelerated electrons and the all-important ratio $n_{nth}/n_{p}$.
The flare occurred on 2017 September 10 and revealed clear evidence (Chen et
al., 2020) for a reconnection current sheet located above the flare loop-top.
Fleishman et al. (2022) have shown from Expanded Owens Valley Solar Array
(EOVSA; Gary et al., 2018) radio observations of this event that there was a
relative dearth of thermal emission from an extended coronal volume but with
considerable non-thermal emission from the same volume. They suggested that an
inductive electric field of order 20 V cm-1 was generated by a magnetic field
decay rate of order 5 G s-1 in a reconnection region of extent $\ell\simeq
4\times 10^{8}$ cm, and concluded that almost all of the electrons within that
extended volume were accelerated to energies in excess of 20 keV,
corresponding to $n_{nth}/n_{p}\simeq 1$ (and $n_{th}/n_{p}\simeq 0$). Our HXR
and EUV observations conclusively demonstrate, however, that the instantaneous
accelerated fraction $n_{nth}/n_{p}$ is much less than unity, specifically
$\,\lower 3.0pt\hbox{$\sim$}\hbox to0.0pt{\hss\raise 2.0pt\hbox{$<$}}\,0.01$.
Moreover, we argue that the accelerating electric fields are likely
considerably smaller in magnitude than suggested by Fleishman et al. (2020)
and Fleishman et al. (2022), indicative of a much smaller scale length
$\ell\simeq 10^{3}$ cm associated with the reconnecting magnetic fields in the
acceleration region.

##  2 X-Ray and EUV Observations

Figure 1: GOES and _RHESSI_ X-ray lightcurves of the X8.2 flare on 2017
September 10, in the indicated energy ranges. Note that _RHESSI_ data are only
available from 15:52:30 UT at the start of the daytime part of the orbit until
16:17:30 UT on entry into the South Atlantic Anomaly. The horizontal purple
lines indicate the times in the different attenuator states - A0, A1, and A3.
The plotted count rates are shown for the front segment of Detector #3 (3f),
corrected to first order to the rates that would have been recorded in the A1
attenuator state, i.e., with only the thin attenuators in place. The dashed
vertical line at 15:58 UT indicates the time interval of interest. Note that
the periodic ripples in the lightcurves after about 16:06 UT are the result of
uncorrected changes in the orientation of the instrument axis with respect to
the spacecraft spin axis, caused by the attenuator motions (see Inglis et al.,
2011).

We consider the well-observed solar flare GOES X-class flare
SOL2017-09-10T15:58, the GOES lightcurve of which is shown in Figure 1. This
flare appeared at the western limb of the solar disk, so that its vertical
geometry can be well ascertained. The hard X-ray emission evolved over tens of
minutes, but we here focus on a narrow time interval near the peak of the
impulsive phase, around 15:58 UT. The spatially-integrated hard X-ray spectrum
at this time (Figure 2) shows both thermal and non-thermal components, with
the spectrum dominated by non-thermal emission above about $30$ keV.

The spatially-integrated hard X-ray spectrum can be used to infer the total
number of accelerated electrons above a chosen reference energy, and dividing
this by the volume of the hard X-ray source (obtained from imaging
spectroscopy data), this leads to a measure of the accelerated electron number
density $n_{nth}$ (cm-3). To quantify these relationships precisely, consider
a region of volume $V$ (cm3), populated by accelerated electrons with number
density spectrum $n(E)$ (electrons cm-3 keV-1). The associated electron flux
$F(E)$ (electrons cm-2 s-1 keV-1) is $F(E)=n(E)\,v(E)$, where the (non-
relativistic) electron speed $v(E)=\sqrt{2E/m_{e}}$, $m_{e}$ being the
electron mass. This population of electrons produces a bremsstrahlung hard
X-ray spectrum $I(\epsilon)$ (photons cm-2 s-1 keV-1) at a distance $R$ from
the emitting region given by (Brown et al., 2003)

$I(\epsilon)=\frac{1}{4\pi R^{2}}\,\int_{\epsilon}^{\infty}\langle
n_{p}VF(E)\rangle\,\sigma(\epsilon,E)\,dE\,\,\,.$ (1)

Here $\sigma(\epsilon,E)$ is the bremsstrahlung cross-section111see OSPEX
software
https://hesperia.gsfc.nasa.gov/rhessi3/software/spectroscopy/spectral-
analysis-software/index.html . The quantity $\langle n_{p}VF(E)\rangle$ is the
mean source electron spectrum, the volume integral of the local electron flux
(differential in energy), weighted by the volume-averaged number density of
target protons $n_{p}$, and $\langle\cdots\rangle$ emphasizes that the
quantity is a volume average. It is important to note that $\langle
n_{p}VF(E)\rangle$ is directly determined from the observed hard X-ray
spectrum, requiring only knowledge of the bremsstrahlung cross-section
involved; no knowledge or assumptions regarding the source geometry or the
physics governing the propagation of the accelerated electrons is required.

Figure 2: Spatially integrated hard X-ray (HXR) spectrum measured by Detector
#3, one of the four working _RHESSI_ detectors near the peak of the flare, in
a 4 s interval starting at 15:58 UT. The HXR spectrum is well fitted by a
combination of a thermal (“vth,” green) and nonthermal (“thin,” yellow)
components. The “thin” function is used here for the nonthermal component
assuming thin-target interactions of electrons with a power-law spectrum. The
listed fit parameters are as follows: A(0) (here $19.8$) is the value of
$\langle n_{p}V{\cal F}_{o}\rangle$ (in units of $10^{55}$ electrons cm-2 s-1)
at energies above reference energy $E_{o}$, A(1) is the electron power-law
index $\delta$ (here 2.92), and A(4) is the value of $E_{o}$ (here 20 keV).
The parameters A(2), A(3) and A(5) define the extended electron spectrum at
energies above the range of interest in the present analysis, i.e., the upper
limit (“break energy”; here $1000$ keV) to this power law, the spectral index
above this break energy (here $3.3$), and the maximum energy for which the
spectrum is computed (here $32$ MeV). The thermal and nonthermal contributions
are comparable at an X-ray energy $\epsilon\simeq 27$ keV. The total spectrum
(red) also includes the pileup component (lower red curve) that peaks near 30
keV; the background spectrum is shown in magenta. The bottom panel shows
residuals from the fit in units of the the $1\sigma$ statistical uncertainties
on the measured count rates at each energy.

Because the mean source electron spectrum corresponds to the instantaneous
distribution of high-energy electrons in the coronal target, its form is
obtained by a “thin-target” fit to the bremsstrahlung hard X-ray spectrum.
Figure 2 shows that the instantaneous spectrum of nonthermal electrons that
best fits the hard X-ray spectrum222thin-target fit using OSPEX
https://sohoftp.nascom.nasa.gov/solarsoft/packages/spex/doc/ospex_explanation.htm
for this event is well-represented by a power-law of the form

$F(E)=(\delta-1)\,\frac{{\cal
F}_{o}}{E_{o}}\,\left(\frac{E}{E_{o}}\right)^{-\delta}\,\,\,,$ (2)

where $E_{o}$ is a reference energy (e.g., a low-energy cutoff), ${\cal
F}_{o}$ is the total electron flux (electrons cm-2 s-1) above that energy, and
$\delta$ is the power-law spectral index.

From this follows the accelerated number density spectrum (electrons cm-3
keV-1)

$n(E)=\frac{F(E)}{v(E)}=\sqrt{\frac{m_{e}}{2}}\,\,(\delta-1)\,{\cal
F}_{o}\,E_{o}^{\delta-1}\,E^{-\delta-1/2}\,\,\,,$ (3)

so that the number density of nonthermal electrons $n_{nth}$ (defined as the
number of electrons per cm3 that are accelerated to energies above $E_{o}$) is

$n_{nth}=\int_{E_{o}}^{\infty}n(E)\,dE=\frac{(\delta-1)}{\left(\delta-\frac{1}{2}\right)}\,\frac{{\cal
F}_{o}}{v_{o}}\,,$ (4)

where $v_{0}=\sqrt{2E_{0}/m_{e}}\simeq 8\times 10^{9}$ cm s-1 for $E_{0}=20$
keV. The total mean source electron flux (electrons cm-2 s-1 above energy
$E_{o}$) may be written as the integral of the electron flux spectrum
(Equation (2)) multiplied by the proton density and emitting volume, so that

$\langle n_{p}V{\cal
F}_{o}\rangle=n_{p}\,V\,\int_{E_{0}}^{\infty}F(E)\,dE\,\,\,.$ (5)

Combining Equations (4) and (5), we find that

$n_{nth}\,n_{p}=\frac{(\delta-1)}{\left(\delta-\frac{1}{2}\right)}\,\frac{\langle
n_{p}V{\cal
F}_{o}\rangle\,}{v_{o}\,V}=\frac{(\delta-1)}{\left(\delta-\frac{1}{2}\right)}\,\sqrt{\frac{m_{e}}{2E_{o}}}\,\,\frac{\langle
n_{p}V{\cal F}_{o}\rangle}{V}\,\,\,.$ (6)

The observationally-inferred value of $\langle n_{p}V{\cal F}_{o}\rangle$,
coupled with estimates of the source volume $V$, thus straightforwardly
constrains the product of the number of accelerated electrons $n_{nth}$ and
the mean number density of target protons $n_{p}$ in the region under study.

Figure 3: Left: Clean _RHESSI_ X-ray images using Detectors #3 and #6 for the
20 s interval centered on 15:58 UT, with three regions of interest identified.
The color-scale image represents thermal (10-12 keV) emission. The blue dashed
lines are the contours at 20, 70, and 90% of the peak 50-100 keV non-thermal
emission. The region of interest labeled ROI-0 is defined by the solid blue
contour at the 50% level of the 50-100 keV image. Regions of interest ROI-1
and ROI-2 from Fleishman et al. (2022) are depicted by solid black and magenta
boundary lines, respectively. Right: Thermal plasma density map, also showing
ROI-1 and ROI-2. This was constructed by applying a regularized differential
emission measure algorithm (Hannah & Kontar, 2012) to SDO/AIA data for the
event in question, assuming the same line-of-sight distance of
$8^{\prime\prime}$ used by Fleishman et al. (2022).

Figure 4: Left: Hard X-ray flux spectrum in the selected regions: the
integrated emission is shown in blue, the emission from ROI-1 is shown in
magenta, and the emission from ROI-2 is shown in black; Right: Electron flux
spectrum in ROI-1 (magenta) and ROI-2 (black), normalized by the image-
integrated spectrum.

Fleishman et al. (2022) have identified two “regions of interest” shown in
Figure 3, ROI-1 (magenta line) and ROI-2 (black line). ROI-1 is the extended
curved elliptical region that outlines the volume in which they claim nearly
100% electron acceleration efficiency. This claim is based on EOVSA imaging
spectroscopy analysis showing that ROI-1 has (1) a large nonthermal electron
population, (2) a relative dearth of thermal electrons over an extended (4
minute) period of time, and (3) a rapid magnetic field decay (reported earlier
by Fleishman et al., 2020) that leads to a large induced electric field,
capable of accelerating the entire ambient population in a very short time.
For comparison, ROI-2 is identified as a region “of more typical flare plasma,
outside the acceleration region.”

X-ray imaging spectroscopy of this event can be made by analyzing _RHESSI_
observations of the same flare. Various image reconstruction algorithms,
described by Piana et al. (2022), can be used such as Clean, MEM_NJIT,
uv_smooth, and Vis_FwdFit. The X-ray image in Figure 3 was produced with the
Clean method using observations over a 20 s interval centered at 15:58:00 UT.
It reveals a source which we term ROI-0, with a FWHM area of $256$ arcsec2 and
a centroid at a slightly lower altitude (some 15 arcsec) than ROI-1 and ROI-2.
This image shows that the vast majority of the hard X-ray emission at this
time comes not from ROI-1, but rather from the compact ROI-0 source, the
structure of which is rather poorly resolved by _RHESSI_.

X-ray spectra for each region of interest, produced using the _RHESSI_ imaging
spectroscopy technique described by Emslie et al. (2003), Battaglia & Benz
(2006), Piana et al. (2007), Saint-Hilaire et al. (2008), and Jeffrey et al.
(2014), are presented in Figure 4. These imaging spectroscopy results are
somewhat limited for two reasons: (1) only four of the nine _RHESSI_ germanium
detectors (#s 1, 3, 6, and 8, with FWHM angular resolutions of 2.3, 6.8, 35.3,
and 106 arcsec, respectively) were operational at this time, and (2) the count
rates were sufficiently high to cause significant pulse pileup, where two or
more low-energy photons arriving within a short time ($\sim$$1~{}\mu$s) of
each other are recorded as a single high-energy count. (One can see in Figure
2 a clear signature of pulse pileup at or below the $10-20\%$ level in the
spectrum recorded by Detector #3 near 30 keV; the effects of pile-up are
weaker for other energies.) In general, at energies above the peak in the
count-rate spectrum (here in the A3 attenuator state at $\sim$$18$ keV) pulse
pileup tends to increase visibility amplitudes. This is especially true for
the detectors with the coarser grids (#s 6 and 8), since the signals from
these detectors have the greatest modulation amplitudes and the pileup effect
is greater at the peaks of the modulation cycles than in the valleys. Detector
#6 has the highest sensitivity of all the operating detectors, so that the
estimated source extent at the characteristic 35-arcsec scale sampled by this
detector is subject to an artificial increase, so the source could appear
larger than the actual source extent and hence the X-ray intensity in both
ROI-1 and ROI-2 could be overestimated.

Performing spatial integrations over the three regions of interest (ROI-0
through ROI-2) results in the three HXR source spectra shown in Figure 4.
Similar to the methodology of previous studies (e.g., Emslie et al., 2003),
the hard X-ray flux $I_{o}(\epsilon)$ from ROI-0 in each energy range is
calculated by integrating over the FWHM area, while the spectra
$I_{1}(\epsilon)$ and $I_{2}(\epsilon)$ from the weaker regions ROI-1 and
ROI-2 are obtained by integrating the images over the entire areal extent of
each respective region. We find that ROI-2 accounts for $\sim$$30$% of the
total emission, while ROI-1 is rather faint, accounting for less than 10% of
the total. Indeed, the HXR flux that appears to originate from ROI-1 in these
reconstructed images is likely to be due mostly to a number of artifacts
associated with pulse pileup, incomplete removal of side lobes from the
intense compact source, and the limited number of spatial Fourier-transform
components (“visibilities”) measured with _RHESSI_ (Hurford & Curtis, 2002).
Thus, the already small inferred ROI-1 HXR fluxes $I_{1}(\epsilon)$ shown in
Figure 4 represent upper limits to the actual HXR intensity from that region,
so that

$\frac{I_{ROI-1}}{I_{\rm total}}\leq 0.1\,\,\,.$

Using the thin-target spectral fit to the spatially-integrated HXR spectrum in
Figure 2, the mean electron flux spectrum $\langle n_{p}V{\cal
F}_{o}\rangle\,(E_{o})$ for the entire flare is $\simeq 2\times 10^{56}$
electrons cm-2 s-1 above reference energy $E_{o}=20$ keV. Given the
considerations of the previous paragraph, an upper limit for the mean source
electron flux above 20 keV from ROI-1 is 10% of this value, i.e., $\langle
n_{p}V{\cal F}_{o}\rangle_{ROI-1}\,\lower 3.0pt\hbox{$\sim$}\hbox
to0.0pt{\hss\raise 2.0pt\hbox{$<$}}\,2\times 10^{55}$ electrons cm-2 s-1.
Using the cube of the source FWHM to estimate the volume of ROI-1 gives
$V\simeq 1.6\times 10^{27}$ cm3, which agrees very well with the value
$V=1.7\times 10^{27}$ cm3 obtained by Fleishman et al. (2022) from radio
observations. Substituting these values with $\delta=2.92$ and
$v_{o}=8.4\times 10^{9}$ cm s-1 in Equation (6) gives

$n_{nth}\,n_{p}\,\,\lower 3.0pt\hbox{$\sim$}\hbox to0.0pt{\hss\raise
2.0pt\hbox{$<$}}\,\,0.79\times\frac{2\times 10^{55}}{(8.4\times
10^{9})\,(1.6\times 10^{27})}\simeq 1.2\times 10^{18}~{}{\rm cm}^{-6}\,.$ (7)

EUV images from SDO/AIA allow us to make an independent estimate the density
in ROI-1. Dividing the total emission measure $n_{p}^{~{}2}V$ of the EUV-
emitting plasma in this region by the previously-determined source volume, we
find that the ambient proton density in this region (right panel of Figure 3)
is $n_{p}\simeq 10^{10}$ cm-3, i.e., consistent with densities commonly found
in the preflare corona.

With this constraint in mind, we now consider the implications of Equation
(7).

  1. 1.

Fleishman et al. (2022) claim that all of the electrons in ROI-1 are
accelerated, i.e., $n_{nth}\simeq 10^{10}$ cm-3, and moreover (their Figure
2c) that the number density of thermal electrons is depleted in ROI-1.
However, due to quasi-neutrality of a plasma, the proton density must be
$n_{p}\simeq n_{nth}$; Equation (7) then shows that the number of non-thermal
electrons cannot exceed $10^{9}$ cm-3. Extended Data Figure 3 of Fleishman et
al. (2022) shows a best-fit thermal plasma density of $10^{8.3}$ cm-3 (with an
uncertainty of an order of magnitude or so) in their sample ROI-1 pixel.
Extended Data Figure 5b from the same Fleishman et al. (2022) work uses
SDO/AIA data to conclude that the thermal number density $n_{th}$ in ROI-2 has
a value of order $10^{10}$ cm-3. This value is consistent with our own
estimates of the thermal density in that region (right panel of our Figure 3),
and we would also note that our Figure 3 shows the SDO/AIA-inferred thermal
densities in ROI-1 and ROI-2 to be comparable. The thermal density in ROI-1
inferred from SDO/AIA data (our Figure 3) is some two orders of magnitude
greater than the radio-observation-based estimates of the same quantity shown
in Figure 2c of Fleishman et al. (2022).

  2. 2.

If the average ambient (thermal) plasma density in ROI-1 is $n_{p}=10^{10}$
cm-3, as evidenced by the SDO/AIA observations (right panel of Figure 3), then
$n_{nth}/n_{p}\simeq 0.012$, a value that is consistent with other estimates
(e.g., Simões & Kontar, 2013) and kappa-like electron distribution in hard
X-ray coronal sources (Kašparová & Karlický, 2009; Oka et al., 2013) and
magnetic reconnection outflow regions (Battaglia et al., 2019).

We believe that $n_{nth}/n_{p}\simeq 0.01$ in region ROI-1 is consistent with
all the available observations of this flare and with reasonable expectations
for the physical nature of ROI-1. The value of $n_{nth}(>20\text{keV})$
obtained from this reasoning can also be corroborated by appealing to thick-
target (Brown, 1971) modeling of the event. A thick-target fit to the
spatially-integrated hard X-ray spectrum in Figure 2 gives an injected
electron rate $\dot{N}\simeq 9\times 10^{35}$ electrons s-1 above $20$ keV. In
a comparison of the electron rate at the loop top and at the footpoints,
Simões & Kontar (2013) showed that the coronal $\dot{N}$ is on average $\sim
1.7-1.8$ times larger than the rate of electron precipitation toward the loop
footpoints, with this difference resulting from some form of coronal trapping
(e.g., turbulent scattering or magnetic mirroring). Applying this correction
factor to the thick target $\dot{N}$ gives333It should be noted that the event
in question was a West limb event, with no obvious footpoint sources visible.
It is possible that the coronal density is high enough that the corona acts as
a thick target (Veronig & Brown, 2004), effectively stopping the electrons
before they reach the chromosphere. However, if the footpoint sources were
indeed occulted by the solar photosphere, the observed HXR intensity does not
represent the entire HXR emission, and thus the precipitation rate
$\dot{N}_{LT}$, and hence the inferred value of $n_{nth}$, is higher than that
obtained from the observed HXR flux. a “loop-top acceleration rate”
$\dot{N}_{LT}=n_{nth}\,v_{o}\,A\simeq 5\times 10^{35}$ electrons s-1,
corresponding to the rate at which electrons are injected from the loop-top
acceleration region into the surrounding target. Assuming that the cross-
sectional area of the acceleration region is at least
$A=8^{\prime\prime}\times 8^{\prime\prime}\simeq 3.3\times 10^{17}$ cm2, and
using $v_{o}\simeq 8.4\times 10^{9}$ cm s-1 (corresponding to the low energy
cut-off electron energy $E_{o}=20$ keV), gives $n_{nth}<2\times 10^{8}$ cm-3,
which is much lower than the ambient plasma density from SDO/AIA and gives
$n_{nth}/n_{p}<0.02$, consistent with the results from thin-target modeling of
the ROI-1 region alone.

Figure 5: Thermal plasma (left panel) and non-thermal (above 20 keV, center
panel) electron number densities as inferred by Fleishman et al. (2022). The
right panel shows the corresponding simulated HXR emission, while the blue
contour shows the 50% level ($0.6$ photons cm-2 sec-1 arcsec-2) obtained from
_RHESSI_ imaging spectroscopy observations in the $30-40$ keV range over the
same time interval used for Figure 2.

As a final self-consistency check, the nonthermal and thermal electron
distribution maps reported by Fleishman et al. (2022) can be used in a thin-
target model, without any additional assumptions, to produce simulated HXR
maps. The simulated hard X-ray emission map in the range $30-40$ keV is shown
in the right panel of Figure 5, together with the 50% contour of the HXR
emission observed by _RHESSI_ in the same energy range. This panel shows that
the spatial distribution of simulated HXR emission based on the on the
electron distribution maps of Fleishman et al. (2022) is not consistent with
that observed by _RHESSI_. Specifically, the observed HXR flux between $30-40$
keV within the 50% contour level (Figure 5) is about $0.6$ photons sec-1
arcsec-2 cm-2, but the nonthermal and thermal electron densities inferred by
Fleishman et al. (2022) would predict a significantly higher HXR count rate in
many pixels of ROI-1. In fact, the simulated flux for some
$2^{\prime\prime}\times 2^{\prime\prime}$ pixels exceeds the observed emission
from the entire flare volume.

We further note that the radio spectrum analysis by Fleishman et al. (2022) is
performed over $2^{\prime\prime}$ pixels. This scale is well below the EOVSA
beam resolution of $(45-5)^{\prime\prime}$ for the $(2-18)$ GHz range, so that
their non-thermal electron maps must be interpreted with a considerable degree
of caution. The required angular resolution ($1^{\prime\prime}$ at 15 GHz)
with better sensitivity, and unprecedented u-v coverage could, however, be
achieved by the Square Kilometre Array (SKA) that is under construction
(Nindos et al., 2019), and we encourage such observations444The Frequency
Agile Solar Radiotelescope (FASR), which is currently under development (Gary
et al., 2022), could also be very useful in this regard., ideally in concert
with simultaneous HXR imaging spectroscopy measurements (e.g., from the STIX
instrument on Solar Orbiter).

##  3 Summary and discussion

The results of the previous section, based on _RHESSI_ hard X-ray and SDO/AIA
observations of the 2017 September 10 flare, show a discrepancy with the
results presented by Fleishman et al. (2020). Our analysis indicates that the
ratio of nonthermal electrons to ambient electrons in ROI-1 at a time near the
peak of the X-ray emission is $n_{nth}/\,n_{p}\simeq 0.01-0.02$, while the
analysis by Fleishman et al. (2022) using radio observations alone suggests
that $n_{nth}/\,n_{p}\simeq 1$. In light of these widely discrepant values of
$n_{nth}/n_{p}$ (or, equivalently, the significant excess of HXR emission
predicted by the model of Fleishman et al. (2022), compared with the actual
_RHESSI_ observations of the same event — Figure 5), it is worthwhile to
revisit the arguments of Fleishman et al. (2020) regarding the values of the
physical parameters that characterize the primary reconnection process.

First, we remark that the mean kinetic energy per electron $\langle
E\rangle=(1/n_{nth})\,\int_{E_{o}}^{\infty}E\,n(E)\,dE$ inferred by Fleishman
et al. (2022) for ROI-1 corresponds to $\sim$$25$ keV or $\langle
E\rangle/k_{B}\simeq 250$ MK, in a region (Figure 5) where the thermal
electron density is claimed to be negligible. This vastly exceeds commonly-
accepted values of $20-30$ MK for most flare bulk plasma components (see,
e.g., Doschek et al., 1979; Dennis, 1988; Holman et al., 2011; Benz, 2017;
Aschwanden et al., 2017, for reviews), and even exceeds that of the so-called
“super-hot” component at a temperature of $30-50$ MK that has been inferred in
some events (Lin et al., 1985; Phillips, 1996; Caspi et al., 2014). The bulk
energization of a volume of solar plasma to such high equivalent temperatures
therefore represents an unprecedented situation.

Second, Fleishman et al. (2022) use Faraday’s law $\nabla\times{\cal
E}=(1/c)\,\partial B/\partial t$ and write $\nabla\times{\cal E}\simeq{\cal
E}/\ell$, where $\ell$ is a characteristic scale length associated with the
gradient in the reconnecting magnetic field region. This gives555the factor
300 converts the cgs statvolt units into Volts $\cal{E}\simeq$ (300 $\ell/c$)
$\partial B/\partial t=(\ell/10^{8})\,\partial B/\partial t$ V cm-1. Using a
value $\partial B/\partial t\simeq 5$ G s-1, based on observations reported by
Fleishman et al. (2020), and further estimating that $\ell=3.65\times 10^{8}$
cm (corresponding to $5^{\prime\prime}$ on the solar disk), Fleishman et al.
(2022) obtain ${\cal E}\simeq 20$ V cm-1, more than five orders of magnitude
greater than the Dreicer (1959) field ${\cal E}_{D}\simeq 4\times
10^{-4}\,(n/10^{10}~{}{\rm cm}^{-3})\,(10^{6}\,{\rm K}/T)\simeq 1.3\times
10^{-4}$ V cm-1 (for a density $n=10^{10}$ cm-3 and a temperature $T=3\times
10^{6}$ K). Under the influence of such a large electric field, all of the
electrons in the ambient Maxwellian distribution would be accelerated, and
they would reach an energy of 20 keV over a very short distance $L\simeq
10^{3}$ cm, some five orders of magnitude less than the assumed reconnection
scale $\ell$. This is not consistent with the claim of Fleishman et al. (2022)
that “this strong super-Dreicer field must be present over a substantial
portion of ROI-1.” Such a value of $L$ also corresponds to an acceleration
time $L/v\simeq 10^{-7}$ s and thus, with an accelerated fraction of unity, to
a specific acceleration rate (fraction of electrons accelerated to 20 keV per
unit time; Emslie et al., 2008; Guo et al., 2012, 2013) $\eta$ (20 keV)
$\simeq 1/10^{-7}=10^{7}$ s-1, nine orders of magnitude greater than
previously inferred for other flares (Guo et al., 2013).

Significant runaway acceleration of the electrons in the ambient Maxwellian
will, however, occur for electric field strengths ${\cal E}$ that are merely
of order the Dreicer field, much lower than the field strength claimed by
Fleishman et al. (2022). Furthermore, once this runaway acceleration
commences, the fundamental electrodynamic properties in the acceleration
region (including the replacement of accelerated particles by a co-spatial
return current) will change sufficiently that further buildup of the electric
field is not necessary and, indeed, is unlikely. Using an illustrative
reconnection scale length value $\ell=10^{3}$ cm and the same 5 G s-1 rate of
change of magnetic field inferred by Fleishman et al. (2020), the resulting
induced electric field $\cal{E}$ $\simeq 5\times 10^{-8}\,\ell\simeq 5\times
10^{-5}$ V cm-1, approximately one-third of the Dreicer field. Such a field
will cause runaway acceleration of electrons with $v\,\lower
3.0pt\hbox{$\sim$}\hbox to0.0pt{\hss\raise
2.0pt\hbox{$>$}}\,v_{crit}=\sqrt{{\cal E}_{D}/{\cal
E}}\,v_{th}\simeq\sqrt{3}\,v_{th}$, representing a fraction $0.5\times{\rm
erfc}(\sqrt{3}/\sqrt{2})\simeq$ 5% of the ambient Maxwellian distribution.
However, it is very likely that the acceleration region is highly
inhomogeneous, so that the required electron energies are reached through a
more stochastic process involving a succession of smaller impulses acting in
different directions and with different efficiencies, so that electrons are
accelerated through a Fermi acceleration process (e.g., Miller et al., 1997;
Bian et al., 2012; Zank et al., 2015; Gordovskyy et al., 2020; Arnold et al.,
2021), rather than as the result of a single large-scale unidirectional
acceleration event.

With 5% of the electron population undergoing runaway acceleration to deka-keV
energies at any given time, and a collisional repopulation of the tail on a
timescale of $0.5$ s, the instantaneous number density of accelerated
electrons, $n_{nth}$, corresponding to the number density of electrons in the
high-energy runaway tail of the distribution, is a relatively small fraction
of the ambient number density $n_{p}$. Further, the associated value of the
specific acceleration rate is $\eta$ (20 keV) $\simeq 0.05/0.5=0.1$ s-1,
consistent with the _RHESSI_ and SDO/AIA observations of Section 2 and
comparable to the specific acceleration rates inferred previously using
different methods (e.g., Guo et al., 2013). In contrast, a much larger overall
population of electrons is successively accelerated over significantly longer
timescales. For an ambient density $n_{p}\simeq 10^{10}$ cm-3 and a source
volume $V\simeq 10^{27}$ cm3, the total number of available electrons is
$\simeq 10^{37}$, so that the electron acceleration rate of $9\times 10^{35}$
s-1 inferred from thick-target modeling of the event (Section 2) corresponds
to an acceleration of all the electrons in the corona in $\sim 10$ s, shorter
than the duration of the HXR burst. This constitutes the well-known “number
problem” (e.g., Brown, 1971) that requires a continual replenishment of the
acceleration region, e.g., by a cospatial return current carried by ambient
thermal electrons (Knight & Sturrock, 1977; Emslie, 1980; Alaoui & Holman,
2017).

The dramatically different values for $(n_{nth}/\,n_{p})$, from $\simeq$$0.01$
to $\simeq$$1$, clearly presents us with a dilemma. How are such disparate
results to be reconciled? One possibility lies in the fact that the emitted
microwave flux is rather insensitive to the value of the low-energy cutoff
energy $E_{o}$ (see Figure 1 in Holman, 2003). Further, Extended Data Figure 3
of Fleishman et al. (2022) claims a relatively high spectral index
$\delta\simeq 5-6$ for a “typical” pixel in ROI-1, much higher than the value
$\delta\simeq 3$ obtained from the total spatially integrated HXR spectrum
(Figure 2). This indicates (probably similarly to Chen et al., 2021) that such
a high value of $\delta$ is not applicable to the deka-keV regime from which
most of the contribution to $n_{nth}$ arises. Such a flattening of the
electron spectrum at lower energies would reduce the value of $n_{nth}$ from
that claimed by Fleishman et al. (2022). Indeed, since the total number of
accelerated electrons ($\int_{E_{o}}^{\infty}E^{-\delta-1/2}\,dE\propto
E_{o}^{1/2-\delta}$; Equation (3)) depends rather sensitively on the the
values of both $E_{o}$ and $\delta$, it is possible that a much lower value of
$n_{nth}$, comparable with that deduced from HXR and EUV observations, is also
consistent with the observed microwave emission. Interestingly, the self-
consistent simulations of electron acceleration during magnetic reconnection
in a macroscale system (Arnold et al., 2021), as well as spatially-extended
turbulent electron acceleration (Stores et al., 2023), also suggests that the
instantaneous number density of nonthermal electrons remains small.

The authors are thankful to G. Fleishman for sharing data and useful
discussions. We also thank the referee for a thorough review of the manuscript
and for some excellent suggestions for improving it. This work was supported
by STFC consolidated grant ST/T000422/1. AGE was supported by NASA Kentucky
under NASA award number 80NSSC21M0362. GGM was supported by grant 21-16508J of
the Grant Agency of the Czech Republic, the project RVO:67985815, the project
LM2018106 of the Ministry of Education, Youth and Sports of the Czech
Republic, and the State Assignment 0040-2019-0025. _RHESSI_ data are available
freely from https://hesperia.gsfc.nasa.gov/rhessi3/

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