# Hamilton, D.P. 1994. Orbital dynamics and the structure of faint dusty rings. Doctoral Dissertation, Cornell University.

The orbital perturbations that act on objects circling a planet vary in strength depending on the sizes of both the particle and its orbit. We examine three cases that are difficult to treat with the standard tools of celestial mechanics: {\it i}) large distant satellites, {\it ii}) small objects on distant orbits, and {\it iii}) tiny particles orbiting near a planet.

The dominant perturbation in the first case is the tidal component of solar gravity. Taking as our example an asteroid on a circular orbit about the Sun, we numerically determine the size and three-dimensional shape of the surface beyond which circum-asteroidal debris is unlikely to be present. We present scaling laws that allow this result to be applied to objects with different masses, semimajor axes, and eccentricities. Small objects on distant orbits are highly perturbed by radiation pressure, which rapidly causes many of them to escape or to impact the asteroidal surface. We determine that, for the asteroid Gaspra (radius $\approx10\km$), debris smaller than centimeter-sized will disappear from distant orbits in just a few years. We generalize our results for application to arbitrary asteroids.

Micron-sized grains, the principal constituents of the many diffuse rings circling within a few planetary radii of the giant planets, are dominantly perturbed by electromagnetic and radiation forces. We derive orbit-averaged equations that govern the evolution of such grains subject to these perturbations; our expressions are valid at all non-resonant locations. Resonant locations are treated by expanding the electromagnetic perturbation analogously to the derivation of the disturbing function of celestial mechanics. We compare our electromagnetic expansion to previous gravitational expansions; similarities lead to the discovery of simple orbital symmetries that constrain the possible consequences of any perturbation.

We use the above expressions to explore the dynamics of the micron-sized grains that make up Saturn's E ring and find that a coupling between planetary oblateness, electromagnetism, and radiation pressure generates highly-eccentric orbits. The distribution of material along an ensemble of these elliptical orbits agrees well with the E ring's observed radial and vertical structure. As a consequence of their highly-elliptical orbits, dust grains strike embedded satellites and nearby rings at large velocities. We argue that these energetic collisions sustain the E ring at its current optical depth against the erosive effects of grain-grain collisions.