Lorentz resonances and the vertical structure of dusty rings: Analytical and numerical results

Abstract

Dust grains orbiting the giant outer planets carry a weak and nearly constant electrical charge. Through interaction with the planet's irregular magnetic field, the inclinations i and eccentricities e of these charged grains can be pumped up significantly. This occurs especially when a period of the Lorentz force matches the radial or out-of-plane epicyclic periods. Since orbiting particles sample the spatial structure of the planetary magnetic field at differing rates depending on their mean orbital radius a, these Lorentz resonances (LR) occur at specific orbital radii. Several of the strongest LR have been related to structural features of the Jovian dust ring system. In this paper we use a combination of analytical and numerical techniques in order to understand the nature of these resonances. A simple extension of the perturbation theory for the LRs yields the charge-to-mass ratio ( q/ m) corrections to the periods of the in-plane and out-of-plane motion, thus allowing for an accurate determination of the resonance locations. The analysis also indicates that the two modes of oscillation are weakly coupled. We then show that the mean orbital radius a, assumed constant in previous analytical developments, undergoes a small but significant resonant forcing as well, resulting in shifts in a which break the exact linear resonance condition. An analysis of the energy of the grain's orbit - for small values of eccentricity e and inclination i - shows that it can be decomposed into three parts: energies of vertical and horizontal oscillators, and the energy of the circular equatorial orbit from which these grains are perturbed by the Lorentz force. In a reference frame corotating with the planet where no electric field is present, the Lorentz force cannot do work but it can exert torque; hence it can transfer energy between the circular orbit and the horizontal and vertical oscillators. Numerical integrations in the resonance zones support this view of the dynamical interaction, and also confirm the existence of a small range of charge-to-mass ratios where the vertical and horizontal resonances overlap, thereby producing what appears to be chaotic motion. Resonance zones can be traversed as orbits evolve in semimajor axis, e.g., due to plasma drag. Using numerical simulations to study this passage through resonance, we find that the orbital elements a, e and i undergo large jumps, and we determine the size of these jumps as a function of ( q/ m) and drag evolution rates. We also ascertain the distribution of final orbital parameters, after passage, for a distribution of initial launch longitudes. In particular we show that the longitude of nodes, Ω, is uniformly distributed over the interval 0 < Ω < 2π. In this case, a cross-section of the Jovian halo would be predicted to have symmetry about the equatorial plane, as confirmed through analysis of the Voyager data by Showalter et al. (1987, Icarus, 69, 458–498). Using adiabatic invariants of the motion, we show that the large inclinations induced in passing through one resonance survive during the excursion to the next resonance encounter. Thus many features of the Jovian ring halo are explained by the Lorentz resonance mechanism.