## Analytic description of collisionally evolving fast electrons, and solar loop-top hard X-ray sources
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.
## Contents- 1. Introduction
- 2. Fokker-Planck equation
- 3. Solving the Îto equations
- 4. Comparison with numerical results
- 5. Application to HXR loop top sources
- 6. Conclusions and discussion
- Acknowledgements
- References
© European Southern Observatory (ESO) 1998 Online publication: March 3, 1998 |