Limit on stably trapped particle fluxes in planetary magnetospheres

We reexamine the Kennel-Petschek concept of self-limitation of stably trapped particle fluxes in a planetary magnetosphere. In contrast to the original Kennel-Petschek formulation, we carry out a fully relativistic analysis. In addition, we replace the wave reflection criterion in the Kennel-Petschek theory by the condition that the limit on the stably trapped particle flux is attained in the steady state condition of marginal stability when electromagnetic waves generated at the magnetic equator acquire a specified gain over a given convective growth length. We derive relativistic formulae for the limiting electron integral and differential fluxes for a general planetary radiation belt at a given L shell. The formulae depend explicitly on the spectral index and pitch angle index of the assumed particle distribution and on the ratio of the electron gyrofrequency to the electron plasma frequency. We compare the theoretical limits on the trapped flux with observed energetic electron fluxes at Earth, Jupiter, and Uranus.