Dynamics of Distant Moons of Asteroids

solar perturbations dominate. Since the detailed mass distribution within the asteroid is relatively unimportant in We introduce a new analytic method for treating the orbital motions of objects about asteroids and planets. For an asteroid this regime, orbits about one asteroid are similar to those following a circular path around the Sun, we rewrite Jacobi’s about another as long as the eccentricities of the asteroidal integral of the motion in terms of the orbital elements relative orbits do not differ greatly. to the asteroid. This procedure is similar to the derivation of In this paper, we investigate two distinct types of orTisserand’s Constant, but here we make the approximation bits—prograde and retrograde—in which the Sun, asterthat the satellite is bound to the asteroid rather than far from oid, and asteroidal satellite all move within a fixed plane. it. In addition, we retain high order terms that Tisserand igPrograde orbits are those for which the angular momentum nored and make no assumptions about the relative masses of vectors of the satellite’s motion about the asteroid and the the asteroid and its satellite. We then average our expression asteroid’s motion around the Sun are parallel; for retroover one circuit of the binary asteroid about its center of mass and obtain the ‘‘Generalized Tisserand Constant.’’ We use the grade orbits, these vectors are antiparallel. Several numeriGeneralized Tisserand Constant to elucidate properties of discal studies have focused on the differences between protant orbits and test our predictions against numerical integragrade and retrograde orbits and found the latter are more tions. In particular, we show analytically that planar prograde stable than the former (for historical references, consult orbits are elongated along the Sun–asteroid line, that planar Szebehely 1978; for more recent work see, e.g. Innanen retrograde orbits extend furthest perpendicular to the Sun– 1979, Huang and Innanen 1983, Zhang and Innanen 1988, asteroid line, and that retrograde orbits are more stable than Chauvineau and Mignard 1990a, 1990b, and Hamilton and prograde ones. Our formalism can be extended (i) to three Burns 1991). Approximate analytic escape criteria have dimensions and (ii) to apply to faint dusty rings around planets by including the effects of planetary oblateness, radiation presbeen derived, most of which follow from considerations sure, and the electromagnetic force from a rotating dipolar of the Jacobi constant (e.g. Szebehely 1978, and Markellos magnetic field.  1997 Academic Press and Roy 1981) or equating forces in a rotating frame (e.g. King 1962, Innanen 1979). Interestingly, all previous estimates which made use of the Jacobi constant show that,