On the bounce-averaging of scattering rates and the calculation of bounce period

For many applications to planetary magnetospheres and elsewhere in the Universe, it is desirable to average physical quantities such as particle and plasma transport coefficients over a charged particle’s bounce motion between magnetic mirror points along field lines. In this paper, we perform such bounce-averaging in a way that avoids singularities in the integrands of expressions that arise in calculations of this sort. Our method applies in principle to an almost arbitrary magnetic field model. We illustrate the advantage of using our method for removing the integrand’s singularity through a change of variables (rather than by truncating the integral over latitude at points progressively nearer to the mirror point) by computing the 〈Dpp〉 component of the bounce-averaged momentum-space diffusion tensor in a dipolar magnetic field both ways for resonance interaction of geomagnetically trapped relativistic (1 MeV) electrons with field-aligned whistler-mode plasma waves at L = 6 (a field line that passes near synchronous altitude). Moreover, we develop improved analytical approximations for particle bounce periods in a dipolar magnetic field by minimizing mean square errors with respect to expansion coefficients in algebraic representations of the bounce period.