Planar asymmetric periodic orbits in four dipole problem

Abstract

In the rings of Saturn the Lorentz field, co-acting with the gravity field of the planet, limits the motion of all charged particles and small size grains with surface charges inside a layer of about 200 meters thickness. In these regions there is enormous three dimensional instability. So, the motion of the charged particles cannot be studied by the theory of Stormer’s problem of the magnetic field of a single dipole, where generally there is not instability in the third dimension. For this reason we follow the assumption that the magnetic field of Saturn is composed by many dipoles in rotation and we have extended Sormer’s problem considering four magnetic dipoles in motion, where in this environment there is three dimensional instability. In this way we study the motions of the charged particles and we reveal families of planar asymmetric periodic orbits, we calculate their horizontal and vertical stability and we give their bifurcations with families of three dimensional asymmetric periodic orbits.