Energy-Containing Electrons in Solar Flares: Improving Hard X-Ray and EUV Diagnostics

TL;DR

This work addresses the challenge of constraining the low-energy, energy-containing nonthermal electron population in solar flares by integrating the warm-target model (WTM) with a kappa-distributed injected electron spectrum and EUV-derived differential emission measures (DEM). By applying this approach to two GOES M-class flares, it demonstrates that the WTM can robustly recover nonthermal electron properties across energies from a few keV to tens of keV, and that EUV DEMs provide independent thermal constraints that reconcile the low-energy regime with HXR data. The results indicate that flare-accelerated electrons comprise a small fraction of the ambient coronal population, with transport in the corona dominated by diffusion and acceleration timescales closely linked to the observed escape times, supporting coronal thermalization. Overall, the combined WTM+EUV framework improves diagnostic leverage on energy-containing electrons and clarifies their acceleration, transport, and energy-partition roles in flare energetics.

Abstract

Solar flares effectively accelerate particles to non-thermal energies. These accelerated electrons are responsible for energy transport and subsequent emissions in HXR, radio, and UV/EUV radiation. Due to the steeply decreasing electron spectrum, the electron population and consequently the overall flare energetics, are predominantly influenced by low-energy non-thermal electrons. However, deducing the electron distribution in this energy-containing range remains a significant challenge. In this study, we apply the warm-target HXR emission model with kappa-form injected electrons to two well-observed GOES M-class flares. Moreover, we utilize EUV observations to constrain the flaring plasma properties, which enables us to determine the characteristics of accelerated electrons across a range from a few keV to tens of keV. We demonstrate that the warm-target model reliably constrains the properties of flare-associated electrons, even accounting for the uncertainties that had previously been unaddressed. The application of a kappa distribution for the accelerated electrons allows for meaningful comparisons with electron distributions inferred from EUV observations, specifically for energy ranges below the detection threshold of RHESSI. Our results indicate that the accelerated electrons constitute only a small fraction of the total electron population within the flaring region. Moreover, the physical parameters, such as electron escape time and acceleration time scale, inferred from both the warm-target model and the EUV observations further support the scenario in which electrons undergo thermalization within the corona. This study highlights the effectiveness of integrating the warm-target model with EUV observations to accurately characterize energy-containing electrons and their associated acceleration and transport processes.

Figures (14)

• Figure 1: Schematic representation of flare coronal geometry, illustrating the key parameters used throughout this study. The diagram shows the loop-top source with diameter $d_{\rm LT}$ and length $L_{\rm LT}$, footpoint diameter $d_{\rm FP}$, and half-loop length $L_{\rm loop}$. The corresponding injection areas ($A_{\rm LT}$ and $A_{\rm FP}$) and volume $V_{\rm LT}$ are used in estimating electron densities and fluxes from both HXR and EUV observations. This schematic serves as a general reference for the geometric configuration assumed in both flare events analyzed.
• Figure 2: Left panel: RHESSI light curves showing the flare evolution. The red shaded region marks the HXR burst interval used for spectral analysis (07:30:00--07:30:44 UT), while the black shaded region indicates the time used for DEM reconstruction (07:30:10 UT). Right panel: AIA 131 Å image at 07:30:10 UT, de-saturated for clarity, overlaid with RHESSI HXR contours at the 50%, 70%, and 90% levels in four energy bands (6--12, 12--25, 25--50, and 50--100 keV). The contours highlight the loop-top and footpoint HXR sources used in geometric analysis. The geometric parameters $d_{\rm{LT}}$, $L_{\rm{LT}}$, and $d_{\rm{FP}}$ are marked as red, blue, and magenta lines, respectively, and are used for volume and area estimates.
• Figure 3: DEM analysis of the 50% loop-top source region for the 2011 February 24 flare using AIA EUV observations. Left panel: Reconstructed DEM distribution $\xi(T)$ obtained from regularized inversion applied to de-saturated AIA images. The DEM curve shows two peaks: a cooler background component and a hotter flare-heated component. Right panel: Comparison between observed AIA data counts and simulated counts generated from the DEM solution, showing good agreement across the six EUV channels.
• Figure 4: Electron number density and temperature maps derived from per-pixel DEM inversion of AIA EUV data for the 2011 February 24 flare. Left panel: Electron number density map, calculated from the total emission measure using a column depth of $d = d_{\rm LT} = 10.2\arcsec = 7.3$ Mm (magenta line). Right panel: DEM-weighted average temperature map.
• Figure 5: HXR spectral fitting results for the 2011 February 24 flare using the warm-target model with a kappa-distributed electron population. Left panel: Fitted photon spectra using thermal input parameters from the isothermal pre-burst fit (blue curves) and from DEM analysis (green curves). Both fits use the same $f_{\rm vth}$ + $f_{\rm thick\_warm\_kappa}$ framework. Normalized residuals for each fit are shown in the bottom panel. Right panel: Corresponding injected electron spectra for the two fits, demonstrating close agreement above 10 keV.