The Keplerian regime of charged particles in planetary magnetospheres

The dynamics of a charged particle orbiting around a rotating magnetic planet is studied. The system is modelled by the two-body Hamiltonian perturbed by an axially-symmetric function which goes to infinity as soon as the particle approaches the planet. The perturbation consists in a magnetic dipole field and a corotational electric field. When it is weak compared to the Keplerian part of the Hamiltonian, we average the system with respect to the mean anomaly up to first order in terms of a small parameter defined by the ratio between the magnetic and the Keplerian interactions. After dropping higher-order terms, we use invariant theory to reduce the averaged system by virtue of its continuous and discrete symmetries, determining also the successive reduced phase spaces. Then, we study the flow of the resulting system in the most reduced phase space, describing all equilibria and their stability, as well as the different classes of bifurcations. Finally, we connect the analysis of the flow on these reduced phase spaces with the one of the original system.