Precipitation from an asymmetric magnetic flux tube

Abstract

A simple model for precipitation of trapped particles from an asymmetric magnetic flux tube is formulated and analytic solutions are found in the strong diffusion and weak diffusion limits. The loss cones α <α1 and α >π ‐ α2 (α = pitch angle) are assumed unequal (α1 <α2), the pitch angle diffusion coefficient D(α) is assumed asymmetric (D(α)≠D(π‐α)), and the particles are assumed to be injected at an angle αs≠π/2. The steady state solution in the strong diffusion limit implies that the ratio of the precipitation rates at the two feet is equal to the ratio of the solid angles filled by the loss cones, R1/R2=α1²/α2². An explicit result for R1 is obtained in the weak diffusion limit. For α2 ‐ α1 ≫αD1, where αD1 is roughly the typical angle a particle is deflected in one transit across the flux tube from foot 2 to foot 1, R1 is smaller than R2 by a factor of order exp [−(α2‐α1) /αD1]. The effect of asymmetric diffusion, which is to favor precipitation at foot 1 for D(α1) >D(α2) and at foot 2 for D(α2) >D(α1), is described in terms of a factor F(d) of an asymmetry parameter d given by equations (35) and (24), respectively. Applications to the solar corona, to Jupiter's radio emission and to the terrestrial magnetosphere are discussed briefly.