Flare Accelerated Electrons in Kappa-Distribution from X-Ray Spectra with Warm-Target Model

X-ray observations provide essential and valuable insights into the acceleration and propagation of non-thermal electrons during solar flares. Improved X-ray spectral analysis requires a deeper understanding of the dynamics of energetic electrons. Previous studies have demonstrated that the dynamics of accelerated electrons of a few thermal speeds are more complex. To better describe the energetic electrons after injection, a model considering energy diffusion and thermalization effects in flare conditions (warm-target model) has recently been developed for Hard X-ray spectral analysis. This model has demonstrated how the low-energy cut-off, which can hardly be constrained in cold-target modeling, can be determined. However, the power-law form may not be the most suitable representation of injected electrons. The kappa distribution, which is proposed as a physical consequence of electron acceleration, has shown successful application in RHESSI spectral analysis. In this study, we employ the kappa-form injected electrons in the warm-target model to analyze two M-class flares, observed by RHESSI and STIX, respectively. The best-fit results show that the kappa-form energetic electron spectrum generates lower non-thermal energy when producing a similar photon spectrum in the fit range compared to the power-law form. We also demonstrated that the fit parameters associated with kappa-form electron spectrum can be well determined with small fit uncertainty. Further, the kappa distribution, which covers the entire electron energy range, enables the determination of key electron properties such as total electron number density and average energy in the flare site, providing valuable information on electron acceleration processes.