FEM code simulation of the magnetospheric proton fluxes

We report a study of the numeric solution to the diffusive transport equation for energetic protons magnetically trapped in the Earth's equatorial magnetosphere. The analysis takes into account the pertinent physical processes in this region, including deceleration of protons by Coulomb collisional interactions with free and bound electrons, the charge exchange process, cosmic ray albedo neutron decay source, and electric and magnetic radial diffusion. These results were obtained using the Finite Element Method with magnetic moment and geomagnetic L-shell as free variables. Steady state boundary conditions were imposed at L=1 as zero distribution function and at L=7 with proton distribution function extracted from ATS 6 satellite observations. The FEM-code yields unidirectional proton flux in the energy range of 0.1–1000 MeV at the equatorial top of the geomagnetic lines, and the results are found to be in satisfactorily agreement with the empirical NASA AP-8 model proton flux within the energy range of 0.5–100 MeV. Below 500 keV, the empirical AP-8 model proton fluxes are several orders of magnitude greater than those computed with the FEM-code at L<3. This discrepancy is difficult to explain by uncertainties of boundary spectrum parameters or transport coefficients.