6. Conclusions and discussion
We have shown how to find an exact analytic solution to the Fokker-Planck equation using moments of the distribution function. However, the moments themselves, which are relatively simple in form, are almost certainly of much greater use. In particular, as illustrated in the previous section, the 2nd order moment allows a mathematical description of the spreading of electrons, reducing the need for lengthy numerical simulations.
Also, we note that (4), (11) and (13) can be combined to give the standard mean scattering results (1). The current treatment therefore provides a formal link between the Fokker-Planck equation and standard mean scattering theory. Note that, like mean scattering theory, since we deal with column depth, the density distribution is arbitrary. The method developed here may be extended in various ways. Non-relativistic forms of drift and diffusion coefficients were employed here for simplicity, but the relativistic form is well known (e.g. Hamiltron and Petrosian 1990) and their employment presents no extra difficulty of principle. Inclusion of spatially varying magnetic field strength would be more involved and we note that even in the mean-scattering case (Chandrasekhar and Emslie 1987), no analytical solution can be found.
Here we have shown how the first and second order statistical properties of the electron distribution in pitch angle cosine µ and field line normalised column depth y can be obtained. The method can, in principle, be extended to the higher moments, which should only involve a greater degree of algebraic complexity. We note that since the distribution in y is not symmetric about the mean (see numerical results in MC), the skewness of the distribution will be non-zero.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998