Derivation of Analytical Particle Spectra from the Solution of the Transport Equation by the WKBJ Method

Details Derivation of Analytical Particle Spectra from the Solution of the Transport Equation by the WKBJ Method Derivation of Analytical Particle Spectra from the Solution of the Transport Equation by the WKBJ Method

Jun 1995

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995apj...446..400g&link_type=abstract

Astrophysical Journal v.446, p.400

Astronomy and Astrophysics

Astronomy

20

Acceleration Of Particles, Plasmas, Sun: Particle Emission

Scientific paper

We propose the use of the WKBJ method to simplify the solution of transport equations in collisionless plasmas. This technique is illustrated with the solution of a Fokker-Planck type equation, for the specific case of particle acceleration in solar flare sources. We derive analytical expressions for the time-dependent and steady state spectra that are valid over the entire energy range of the accelerated particles and fit correctly the numerical results of other authors. The derived spectra correspond to a momentum diffusion coefficient D(p) ˜ pn/β (n = 2, corresponding to Fermi-type acceleration). The analytical spectrum in the transrelativistic domain is of particular importance, for instance, in the study of solar radiation emissions produced by the interaction of the energetic particles with matter and electromagnetic fields. The study is developed for two kinds of accelerating turbulence, 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. We calculate the time for the time-dependent spectra to reach equilibrium at different energies, and we show that the equilibrium time is sensitive to the kind of accelerating turbulence and slightly sensitive to the kind of injection process. We analyze the effect of an energy-dependent particle escape, τ(E) ˜ β for μ = 0, 1, and 2 on the particle spectrum.

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