The Formation of Kappa-Distribution Accelerated Electron Populations in Solar Flares

TL;DR

This paper models how stochastic acceleration and Coulomb collisions produce kappa-distributed electron populations in solar flares, explaining observed X-ray sources and energy-dependent acceleration times.

Contribution

It introduces a theoretical framework showing the formation of kappa distributions through stochastic acceleration in solar flare environments, including the effects of electron escape.

Findings

Electron distributions reach a stationary kappa form under certain conditions.

Higher energy electrons are accelerated later, with acceleration time scaling as E^{3/2}.

Escape at high energies influences the distribution shape in finite regions.

Abstract

Driven by recent RHESSI observations of confined loop-top hard X-ray sources in solar flares, we consider stochastic acceleration of electrons in the presence of Coulomb collisions. If electron escape from the acceleration region can be neglected, the electron distribution function is determined by a balance between diffusive acceleration and collisions. Such a scenario admits a stationary solution for the electron distribution function that takes the form of a kappa distribution. We show that the evolution toward this kappa distribution involves a "wave front" propagating forwards in velocity space, so that electrons of higher energy are accelerated later; the acceleration time scales with energy according to $\tau_{\rm acc} \sim E^{3/2}$. At sufficiently high energies escape from the finite-length acceleration region will eventually dominate. For such energies, the electron velocity distribution function is obtained by solving a time-dependent Fokker-Planck equation in the "leaky-box" approximation. Solutions are obtained in the limit of a small escape rate from an acceleration region that can effectively be considered a thick target.