Orbits of charged particles trapped in a dipole magnetic field

Motion of a test charged particle in a dipole magnetic field can be reduced to a two degree-of-freedom Hamiltonian system due to the axisymmetry of the dipole field. We carried out a systematic study of orbits of low-energy trapped charged particles in the dipole field via calculation of their Lyapunov characteristic exponents (LCEs) with random initial conditions in the four-dimensional phase space. Since there is at most one positive LCE, these orbits are classified as chaotic ones with one positive LCE and quasi-periodic ones with vanishing LCEs. The dependence of the fraction of quasi-periodic orbits in the phase space on the particle energy is given, which reveals a discrete spectrum, confirming the results of earlier studies. It is also found that most quasi-periodic orbits are confined near the equatorial plane and away from the dipole except for some at very low energies, while chaotic ones are ergodic. The distribution of the maximum LCE (mLCE) appears to vary gradually in the phase space and chaotic orbits with very low values of the mLCE wander near quasi-periodic orbits for a significant amount of time before merging into the sea of chaos.