DYNAMICS OF DISTANT MOONS OF ASTEROIDS

We introduce a new analytic method for treating the orbital motions of objects about asteroids and planets. For an asteroid following a circ ular path around the Sun, we rewrite Jacobi's integral of the motion i n terms of the orbital elements relative to the asteroid. This procedu re is similar to the derivation of Tisserand's Constant, but here we m ake the approximation that the satellite is bound to the asteroid rath er than far from it. In addition, we retain high order terms that Tiss erand ignored and make no assumptions about the relative masses of the asteroid and its satellite. We then average our expression over one c ircuit of the binary asteroid about its center of mass and obtain the ''Generalized Tisserand Constant.'' We use the Generalized Tisserand C onstant to elucidate properties of distant orbits and test our predict ions against numerical integrations. In particular, we show analytical ly that planar prograde orbits are elongated along the Sun-asteroid li ne, that planar retrograde orbits extend furthest perpendicular to the Sunasteroid line, and that retrograde orbits are more stable than pro grade ones. Our formalism can be extended (i) to three dimensions and (ii) to apply to faint dusty rings around planets by including the eff ects of planetary oblateness, radiation pressure, and the electromagne tic force from a rotating dipolar magnetic field. (C) 1997 Academic Pr ess.