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.
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