Limiting energy spectrum of an electron radiation belt

To determine the Kennel-Petschek limiting particle flux in a planetary radiation belt in a fully relativistic regime, without assuming a predetermined form for the particle energy distribution, has been a long-standing challenge in space physics. In this paper, for the case of whistler mode wave-electron interaction, we meet this challenge. The limiting flux is determined by a steady state marginal stability criterion in which a convective wave gain condition is applied over all frequencies for which wave growth occurs. This condition produces an integral equation for the trapped flux. We find that in the relativistic regime the limiting electron energy spectrum varies asymptotically as 1/E, for large energy E, just as in the nonrelativistic case. However, the scaling coefficient in the relativistic case is twice that in the nonrelativistic result. We compare numerical solutions for the limiting spectra with measured energetic electron spectra at Jupiter.