Weightiness of the Dispersive Rate in Stochastic Acceleration Processes

Abstract

AbstractWe present analytical solutions to the transport equation of the accelerated particles that are valid over the entire energy range (nonrelativistic, transrelativistic, and ultrarelativistic) for the time-dependent and equilibrium cases. On the basis of these overall spectra, we have studied the relative importance of the term of average (systematic) energy increase and the term of diffusion (spread) in energy that define the evolution of accelerated particles within the frame of a simplified diffusion-convection transport equation, when the particle distribution function N(E, t) is assumed to be independent of spatial pitch-angle diffusion (isotropic). Since in some astrophysical works, only the term of average energy increase in the simplified transport equation in energy space is considered to be leading directly to a power-law-type spectrum, we have established the conditions under which such an approximation may be justified. The analysis is illustrated for acceleration by two different kinds of turbulence: MHD and Langmuir waves. The omission of the term of energy spread leads, in general, to an important depletion or overproduction of the accelerated particle flux that must be seriously considered in any calculation of the flux of secondary radiation produced by the accelerated particles. The presented analytical spectra may be of particular relevance to gamma-ray, neutron, and pion production in solar flares as well as in other astrophysical sites.Subject headings: acceleration of particles — MHD — Sun: general — turbulence — wave motions