It is well known that equatorial dust grains are deeply trapped in their potential wells. However, it is not obvious that the much smaller halo grains are sufficiently deeply trapped to survive the perturbations discussed in the following sections. To this end we define the well depth
which varies with q/m, becoming vanishingly small at a bifurcation point. While care must be taken in defining EmaX} this detail need not concern us here; usually it suffices to take Emax = min(Es, E^), where Es is the saddle point energy. Figure 2 compares potential profiles for the equatorial and halo orbits shown in Figure 1, with that for the halo well taken along a line joining the elliptic and hyperbolic critical points.
First consider nonzero J? and g2, for which the Hamiltonian remains axisymmetric. For equatorial orbits the dynamics is primarily Keplerian for aß > 0.2 /im. In such cases it is well known that the dipole field and oblateness both cause Keplerian ellipses to precess and can in fact cancel, with important consequences for E-Ring particles . However, this precession is of minor importance for particle trapping, since the locations of the critical points of Ue are only slightly perturbed. Furthermore, Ue is structurally stable, i.e. its critical points do not change type under small perturbations . In fact the orbits in Figures 1 and 2 all include J2 and g? and are seen to be firmly confined to their ideal two dimensional potential wells. Extensive orbital calculations show that the effects of g2 are miniscule compared to J2 and we therefore omit it from further discussion.
The effects of radiation pressure are more subtle and can have large long-term effects, non-magnetic Mars and Venus. Dynamically the presence of radiation pressure breaks the axisymmetry of (1), so that the motion becomes truly three dimensional, and the canonical momentum p^ is no longer conserved. Nevertheless, for magnetic planets with k << 1, is still conserved on the average, as guaranteed by the KAM theorem, with the result that orbits can be trapped in 3D potential wells. For the orbit of Figure la varies by only 0.24%. In general only positively charged conducting grains at large distance from the planet are significantly perturbed from their two dimensional potential wells by radiation pressure.
We have seen that stable nonequatorial ('halo') orbits may exist about Saturn. These orbits are composed of very small (js 100 nm) grains and are insensitive to the influence of J2 and g2. Radiation pressure is relatively small for dielectric grains but can be large for distant conducting grains. Work is currently underway to determine the effects of time-dependent charging on these intriguing orbits.
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