Proton trajectories in the radiation belts

Trajectories of charged particles in a static dipole magnetic field are considered for the case of small equatorial pitch angles under the assumption of the conservation of the adiabatic invariants. By physical reasoning, higher order correction terms to the conventional first invariant sin2α/B, are deduced. These terms, being functions of particle mass, velocity, and gyration angle around the field line, produce the following effects in the radiation belts: (1) Near the equatorial plane at large L, the oscillation of the proton pitch angle at the cyclotron frequency is significant. For example, at L = 10, the 150-kev proton pitch angle oscillates by as much as 10°. (2) At low altitudes the pitch angular distribution of protons accelerated near the equatorial plane is limited to certain general forms. Results of detailed calculations are presented.