Solution of the electron continuity equation in an inhomogeneous medium via the method of characteristics

Many astrophysical problems require a knowledge of the spatial and temporal evolution of fast electron distributions as they interact (e.g., collisionally) with an ambient medium; an example is provided by hard X-ray burst modelling in solar flares. In the mean scattering-rate approximation (which neglects dispersion in the distribution function), this evolution is governed by the continuity equation in phase space. Here we present a rigorous solution of the continuity equation, derived via the method of characteristics, which confirms and generalises the results of previous authors, and gives greater physical and mathematical insight into these solutions. We illustrate the calculation of this solution for mildly relativistic electrons undergoing Coulomb collisions, and briefly indicate the extension of the method to other physical situations.On leave from Department of Mathematics, University of Waikato, Hamilton, New Zealand.