The radial distribution of radiation belt protons: Approximate solution of the steady state transport equation at arbitrary pitch angle

Abstract

We present quasi‐analytic solutions to the proton diffusion equation, with energy degradation and charge exchange losses included. The transport mode is modeled as D ∝ M−l Lg(K,l), where M, K, and L are the adiabatic invariants, and l refers to the spectral index of the fluctuating electric field. At the outer boundary, say L = 7, a distribution function of the form f ∝ exp −[M/M0(K)]v is maintained; at the inner boundary (L ≈ 1), f is required to vanish. We find that (1) the diffusion flux is directed outward, when energy loss and charge exchange are in approximate balance, (2) the shape of the proton flux is strongly affected by the spectral index of the fluctuating fields, (3) the proton flux exhibits both positive and negative gradients with respect to energy, and (4) the anisotropy of the proton flux increases with decreasing L value, and may pass through a minimum with respect to energy.