Pitch-Angle Diffusion in Canonical Coordinates: A Theoretical Formulation.

The equation for pitch-angle diffusion at constrant particle energy E and shell parameter L in a dipole field can be transformed into canonical form, by the introduction of a new coordinate called z. The new coordinate is obtained by integrating the bounce period of a particle at fixed E and L with respect to y sq, where y is the sine of the equatorial pitch angle alpha sub 0. A potpourri of applications serves to illustrate the usefulness of such a transformation. For example, if D sub zz is a suitably simple function of z, one can specify the eigenfunctions g sub nz of the diffusion operator is closed form. If D sub zz differs only slightly from such a simple function of z, then the corresponding eigenfunctions can be generated from the above set by procedures analogous to the Rayleigh-Schrodinger perturbation theory used in quantum mechanics. The availability of such eigenfunctions enables one to evaluate quantitatively the manner in which geomagnetically trapped particles are redistributed in alpha sub 0 and lost from the magnetosphere as the phase-space density f evolves in time.