Discrete Periodic St\"{o}rmer Orbits

Motion of a test charged particle in a dipole magnetic field can be described as a Hamiltonian system with 2 degree of freedom (2 d.o.f.) due to axisymmetry of the dipole field. It has been suggested that at a given energy and angular momentum, there exists a unique orbit entering the dipole, the so-called St\"{o}rmer orbits \citep{1970braun}. Via extensive numerical calculations, here we report the discovery of discrete symmetric open periodic orbits entering the dipole along discrete curves, connecting two families of symmetric open periodic orbits terminating at the equatorial plane. Since each open periodic orbit has two static points in the meridian plane, these St\"{o}rmer orbits are singular with only one static point at the dipole and ambiguous potential energies, connecting all families of open periodic orbits terminating at the equatorial plane. Numerical results also indicate that the particle rotates around a guiding magnetic field line for infinite circles at these St\"{o}rmer orbits.Asymmetric periodic orbits have similar properties except that they have two distinct static points in the meridian plane: one at the dipole and the other at the center of spiral structures in the phase plane of their locations.We also find that two families of symmetric open periodic orbits with the highest energies terminate at the equatorial plane at two distinct orbits, while lower energy family pairs appear to terminate at the same orbits.