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Astron. Astrophys. 331, 1103-1107 (1998)

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Analytic description of collisionally evolving fast electrons, and solar loop-top hard X-ray sources

A.J. Conway , A.L. MacKinnon , J.C. Brown and G. McArthur

Departmet of Physics & Astronomy, University of Glasgow, UK (e-mail: conway@astro.gla.ac.uk)

Received 31 July 1997 / Accepted 20 November 1997

Abstract

We present a new approach to the problem of particle transport described by the linearised Fokker-Planck equation. Instead of attempting to solve for the distribution function directly, exact and analytic expressions for the moments of the distribution are derived from the equivalent stochastic differential equation. Although the moments themselves will be of greatest use, we also show how these moments can be used to construct an exact, analytic solution to the Fokker-Planck equation. In addition, we explain how mean scattering theory naturally emerges from the first order moments. The derivation of the second (and higher) order moments means that the spatial spreading of electrons due to the changing pitch angle distribution can be described analytically for any injected pitch angle - previously, such a description was not possible with mean scattering and, in general, numerical simulation was the only method available. The treatment also explicitly reveals a simple scaling relationship between the distribution of particles along the magnetic field and the square of the particle's injection energy. We check our results against numerical simulations and point out how the results here can be extended to more general cases. Uses of these results are illustrated in relation to the spatial distribution of Hard X-Ray (HXR) emission and its relevance to solar HXR "above the loop top" sources.

Key words: Sun: flares – scattering – X-rays: bursts

Send offprint requests to: A.J. Conway

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© European Southern Observatory (ESO) 1998

Online publication: March 3, 1998
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