DERIVATION OF ANALYTICAL PARTICLE SPECTRA FROM THE SOLUTION OF THE TRANSPORT-EQUATION BY THE WKBJ METHOD

We propose the use of the WKBJ method to simplify the solution of tran sport equations in collisionless plasmas. This technique is illustrate d with the solution of a Fokker-Planck type equation, for the specific case of particle acceleration in solar flare sources. We derive analy tical expressions for the time-dependent and steady state spectra that are valid over the entire energy range of the accelerated particles a nd fit correctly the numerical results of other authors. The derived s pectra correspond to a momentum diffusion coefficient D(p) similar to p(n)/B (n = 2, corresponding to Fermi-type acceleration). The analytic al spectrum in the transrelativistic domain is of particular importanc e, for instance, in the study of solar radiation emissions produced by the interaction of the energetic particles with matter and electromag netic fields. The study is developed for two kinds of accelerating tur bulence, MHD and Langmuir waves, and for monoenergetic injection, but we also present semianalytical spectra for thermal-type injection and for injection from magnetic reconnection in a neutral current sheet. W e calculate the time for the time-dependent spectra to reach equilibri um at different energies, and we show that the equilibrium time is sen sitive to the kind of accelerating turbulence and slightly sensitive t o the kind of injection process. We analyze the effect of an energy-de pendent particle escape, tau(E) similar to B--mu for mu = 0, 1, and 2 on the particle spectrum.