We introduce a new analytic method for treating the orbital motions of objects about asteroids and planets. For an asteroid following a circular path around the Sun, we rewrite Jacobi's integral of the motion in terms of the orbital elements relative to the asteroid. This procedure is similar to the derivation of Tisserand's Constant, but here we make the approximation that the satellite is bound to the asteroid rather than far from it. In addition, we retain high order terms that Tisserand ignored and make no assumptions about the relative masses of the asteroid and its satellite. We then average our expression over one circuit of the binary asteroid about its center of mass and obtain the ``Generalized Tisserand Constant.''

We use the Generalized Tisserand Constant to elucidate properties of distant orbits and test our predictions against numerical integrations. In particular, we show analytically that planar prograde orbits are elongated along the Sun-asteroid line, that planar retrograde orbits extend furthest perpendicular to the Sun-asteroid-line, and that retrograde orbits are more stable than prograde ones. Our formalism can be extended i) to three dimensions and ii) to apply to faint dusty rings around planets by including the effects of planetary oblateness, radiation pressure, and the electromagnetic force from a rotating dipolar magnetic field.

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