The Efficiency of Electron Acceleration during the Impulsive Phase of a Solar Flare

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 ∼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 HXR and and the Solar Dynamics Observatory/Atmospheric Imaging Assembly (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 ≲10−2 on the number density of nonthermal (≥20 keV) electrons, expressed as a fraction of the number density of ambient protons in the same volume. This upper limit is about 2 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 ares reveal that electron acceleration likely occurs in magnetic reconnection out ow 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 rst noted by Masuda et al. (1994).These sources are consistent with regions where magnetic turbulence could provide sucient 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 are 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 ̇ (s −1 ) at energies above a speci ed energy (typically taken to be 20 keV).In such a "thicktarget" calculation, the deduced electron acceleration rate is generally independent of the density structure of the aring 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 di erence 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 lowenergy spectral cuto , which is di cult to determine from observations (e.g.Aschwanden et al. 2019, as a review).
By contrast with the spatially-integrated case, the HXR ux spectrum from a given spatial sub-region in the are provides information on the density-weighted mean electron ux spectrum in that region.For this analysis, "thintarget" 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 eld 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 ( ℎ and ℎ , 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: = ℎ + ℎ in hydrogen plasma.The instantaneous fraction of accelerated electrons in the aring corona has been previously discussed in the literature, but the results, methods, and the region that de nes the "coronal source" and "above-the-looptop" di er somewhat and hence necessitates addition investigations.Krucker et al. (2010) reported ℎ (> 16 keV)∕ ≃ 0.1, while Oka et al. (2013) re-analysed the are and deduced ℎ (>20 keV)∕ ≃ 0.01 (see their Equation ( 4)).Krucker & Battaglia (2014) analysed a limb are in the presence of bright footpoint emission and found ℎ (> (10 − 15) keV)∕ ≃ 1 in the region above the coronal source, assuming a powerlaw spectrum with a sharp low-energy cut-o in the range from 10 − 15 keV.However, this high value was obtained using a Maxwellian+power-law t 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 powerlaw at high energies to a Maxwellian at low energies.Compared to tting the thermal and nonthermal (power-law) parts of the spectrum separately, tting the data with a kappa distribution generally results (as in Oka et al. 2013) in a signi cant reduction in the required electron ux near the rollover energy and hence a lower inferred value of ℎ .Indeed, Battaglia et al. (2015) demonstrate that using a kappa-distribution t to RHESSI data, while also incorporating EUV emission line data to constrain the t, results in a reduction in ℎ by a factor of up to ∼30 compared with the commonly-used power-law-only t.Early in a are, electron distributions in magnetic reconnection out ow 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 are.Radio emission produced by gyrosynchrotron radiation of accelerated electrons with relatively high energies ∼ > 100 keV is also commonly observed in the aring 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 are 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 ℎ ∕ .The are occurred on 2017 September 10 and revealed clear evidence (Chen et al. 2020) for a reconnection current sheet located above the are 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 eld of order 20 V cm −1 was generated by a magnetic eld decay rate of order 5 G s −1 in a reconnection region of extent ≃ 4 × 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 ℎ ∕ ≃ 1 (and ℎ ∕ ≃ 0).Our HXR and EUV observations conclusively demonstrate, however, that the instantaneous accelerated fraction ℎ ∕ is much less than unity, speci cally ∼ < 0.01.Moreover, we argue that the accelerating electric elds 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 ≃ 10 3 cm associated with the reconnecting magnetic elds in the acceleration region.
2. X-RAY AND EUV OBSERVATIONS We consider the well-observed solar are GOES X-class are SOL2017-09-10T15:58, the GOES lightcurve of which is shown in Figure 1.This are 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 accel- erated electron number density ℎ (cm −3 ).To quantify these relationships precisely, consider a region of volume (cm 3 ), populated by accelerated electrons with number density spectrum ( ) (electrons cm −3 keV −1 ).The associated electron ux ( ) where the (non-relativistic) electron speed ( ) = √ 2 ∕ , being the electron mass.This population of electrons produces a bremsstrahlung hard X-ray spectrum ( ) (photons cm −2 s −1 keV −1 ) at a distance from the emitting region given by (Brown et al. 2003) Here ( , ) is the bremsstrahlung cross-section 1 .The quantity ⟨ ( )⟩ is the mean source electron spectrum, the volume integral of the local electron ux (di erential in energy), weighted by the volume-averaged number density of target protons , and ⟨⋯⟩ emphasizes that the quantity is a volume average.It is important to note that ⟨ ( )⟩ 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.The HXR spectrum is well tted 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 t parameters are as follows: A(0) (here 19.8) is the value of ⟨ ℱ ⟩ (in units of 10 55 electrons cm −2 s −1 ) at energies above reference energy , A(1) is the electron powerlaw index (here 2.92), and A(4) is the value of (here 20 keV).
The parameters A(2), A(3) and A(5) de ne 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 ≃ 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 t in units of the the 1 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" t to the bremsstrahlung hard X-ray spectrum.Figure 2 shows that the instantaneous spectrum of nonthermal electrons that best ts the hard X-ray spectrum 2 for this event is well-represented by a power-law of the form where is a reference energy (e.g., a low-energy cuto ), ℱ is the total electron ux (electrons cm −2 s −1 ) above that energy, and is the power-law spectral index.
From this follows the accelerated number density spectrum (electrons cm −3 keV −1 ) so that the number density of nonthermal electrons ℎ (de ned as the number of electrons per cm 3 that are accelerated to energies above ) is where 0 = √ 2 0 ∕ ≃ 8 × 10 9 cm s −1 for 0 = 20 keV.
The total mean source electron ux (electrons cm −2 s −1 above energy ) may be written as the integral of the electron ux spectrum (Equation ( 2)) multiplied by the proton density and emitting volume, so that Combining Equations ( 4) and ( 5), we nd that (6) The observationally-inferred value of ⟨ ℱ ⟩, coupled with estimates of the source volume , thus straightforwardly constrains the product of the number of accelerated electrons ℎ and the mean number density of target protons in the region under study.Fleishman et al. (2022) have identi ed 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 e ciency.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 eld decay (reported earlier by Fleishman et al. 2020) that leads to a large induced electric eld, capable of accelerating the entire ambient population in a very short time.For comparison, ROI-2 is identi ed as a region "of more typical are plasma, outside the acceleration region." X-ray imaging spectroscopy of this event can be made by analyzing RHESSI observations of the same are.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 arcsec 2 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 Je rey 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 su ciently high to cause significant pulse pileup, where two or more low-energy photons arriving within a short time (∼1 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 e ects of pile-up are weaker for other energies.)In general, at energies above the peak in the countrate spectrum (here in the A3 attenuator state at ∼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 e ect 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 arti cial increase, so the source could appear larger than the actual source extent and hence the Xray 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 Xray ux ( ) from ROI-0 in each energy range is calculated by integrating over the FWHM area, while the spectra 1 ( ) and 2 ( ) 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 nd that ROI-2 accounts for ∼30% of the total emission, while ROI-1 is rather faint, accounting for less than 10% of the total.Indeed, the HXR ux 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 uxes 1 ( ) shown in Figure 4 represent upper limits to the actual HXR intensity from that region, so that Using the thin-target spectral t to the spatially-integrated HXR spectrum in Figure 2, the mean electron ux spec-trum ⟨ ℱ ⟩ ( ) for the entire are is ≃ 2 × 10 56 electrons cm −2 s −1 above reference energy = 20 keV.
(7) EUV images from SDO/AIA allow us to make an independent estimate the density in ROI-1.Dividing the total emission measure 2 of the EUV-emitting plasma in this region by the previously-determined source volume, we nd that the ambient proton density in this region (right panel of Figure 3) is ≃ 10 10 cm −3 , i.e., consistent with densities commonly found in the pre are corona.With this constraint in mind, we now consider the implications of Equation ( 7).
1. Fleishman et al. (2022) claim that all of the electrons in ROI-1 are accelerated, i.e., ℎ ≃ 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 ≃ ℎ ; 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-t 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 ℎ 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).
We believe that ℎ ∕ ≃ 0.01 in region ROI-1 is consistent with all the available observations of this are and with reasonable expectations for the physical nature of ROI-1.The value of ℎ (> 20keV) obtained from this reasoning can also be corroborated by appealing to thicktarget (Brown 1971) modeling of the event.A thick-target t to the spatially-integrated hard X-ray spectrum in Figure 2 gives an injected electron rate ̇ ≃ 9 × 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 ̇ is on average ∼ 1.7 − 1.8 times larger than the rate of electron precipitation toward the loop footpoints, with this di erence resulting from some form of coronal trapping (e.g., turbulent scattering or magnetic mirroring).Applying this correction factor to the thick target ̇ gives 3 a "loop-top acceleration rate" ̇ = ℎ ≃ 5×10 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 = 8 × 8 ≃ 3.3 × 10 17 cm 2 , and using ≃ 8.4 × 10 9 cm s −1 (corresponding to the low energy cut-o electron energy = 20 keV), gives ℎ < 2 × 10 8 cm −3 , which is much lower than the ambient plasma density from SDO/AIA and gives ℎ ∕ < 0.02, consistent with the results from thin-target modeling of the ROI-1 region alone.
As a nal 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.Speci cally, the observed HXR ux 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 signi cantly higher HXR count rate in many pixels of ROI-1.In fact, the simulated ux for some 2 × 2 pixels exceeds the observed emission from the entire are volume.
We further note that the radio spectrum analysis by Fleishman et al. ( 2022) is performed over 2 pixels.This scale is well below the EOVSA beam resolution of (45−5) 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 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 observations4 , ideally in concert with simultaneous HXR imaging spectroscopy measurements (e.g., from the STIX instrument on Solar Orbiter).

SUMMARY AND DISCUSSION
The results of the previous section, based on RHESSI hard X-ray and SDO/AIA observations of the 2017 September 10 are, 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 ℎ ∕ ≃ 0.01 − 0.02, while the analysis by Fleishman et al. (2022) using radio observations alone suggests that ℎ ∕ ≃ 1.In light of these widely discrepant values of ℎ ∕ (or, equivalently, the signi cant 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 ⟨ ⟩ = (1∕ ℎ ) ∫ ∞ ( ) inferred by Fleishman et al. (2022) for ROI-1 corresponds to ∼25 keV or ⟨ ⟩∕ ≃ 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 are 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.
Signi cant runaway acceleration of the electrons in the ambient Maxwellian will, however, occur for electric eld strengths ℰ that are merely of order the Dreicer eld, much lower than the eld 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 su ciently that further buildup of the electric eld is not necessary and, indeed, is unlikely.Using an illustrative reconnection scale length value = 10 3 cm and the same 5 G s −1 rate of change of magnetic eld inferred by Fleishman et al. (2020), the resulting induced electric eld ℰ ≃ 5 × 10 −8 ≃ 5 × 10 −5 V cm −1 , approximately one-third of the Dreicer eld.Such a eld will cause runaway acceleration of electrons with ∼ > = √ ℰ ∕ℰ ℎ ≃ √ 3 ℎ , representing a fraction 0.5 × erf c( √ 3∕ √ 2) ≃ 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 di erent directions and with di erent e ciencies, 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, ℎ , 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 .Further, the associated value of the speci c acceleration rate is (20 keV) ≃ 0.05∕0.5 = 0.1 s −1 , consistent with the RHESSI and SDO/AIA observations of Section 2 and comparable to the speci c acceleration rates inferred previously using di erent methods (e.g., Guo et al. 2013).In contrast, a much larger overall population of electrons is successively accelerated over signi cantly longer timescales.For an ambient density ≃ 10 10 cm −3 and a source volume ≃ 10 27 cm 3 , the total number of available electrons is ≃ 10 37 , so that the electron acceleration rate of 9 × 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 ∼ 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 di erent values for ( ℎ ∕ ), from ≃0.01 to ≃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 ux is rather insensitive to the value of the low-energy cuto energy (see Figure 1 in Holman 2003). Further, Extended Data Figure 3 of Fleishman et al. (2022) claims a relatively high spectral index ≃ 5−6 for a "typical" pixel in ROI-1, much higher than the value ≃ 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 is not applicable to the deka-keV regime from which most of the contribution to ℎ arises.Such a attening of the electron spectrum at lower energies would reduce the value of ℎ from that claimed by Fleishman et al. (2022).Indeed, since the total number of accelerated electrons (∫ depends rather sensitively on the the values of both and , it is possible that a much lower value of ℎ , 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.

Figure 1 .
Figure 1.GOES and RHESSI X-ray lightcurves of the X8.2 are 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 di erent attenuator states -A0, A1, and A3.The plotted count rates are shown for the front segment of Detector #3 (3f), corrected to rst 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).

Figure 2 .
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 are, in a 4 s interval starting at 15:58 UT.The HXR spectrum is well tted 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 t parameters are as follows: A(0) (here 19.8) is the value of ⟨ ℱ ⟩ (in units of 10 55 electrons cm −2 s −1 )

Figure 3 .
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 identi ed.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 de ned 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 di erential emission measure algorithm(Hannah & Kontar 2012) to SDO/AIA data for the event in question, assuming the same line-of-sight distance of 8 used byFleishman et al. (2022).

Figure 4 .
Figure 4. Left: Hard X-ray ux 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 ux spectrum in ROI-1 (magenta) and ROI-2 (black), normalized by the image-integrated spectrum.

Figure 5 .
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.