Hamilton, D.P and A.V. Krivov 1997. Dynamics of Distant Moons of
Asteroids. Icarus 128, 141-149.
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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