Spacecraft motion about slowly rotating asteroids

Spacecraft motion about an arbitrary second-degree and second-order gravity field is investigated. We assume that the gravity field is in uniform rotation about an axis of its maximum moment of inertia and that it rotates slowly as compared to the spacecraft orbit period. We derive the averaged Lagrange planetary equations for this system and use them, in conjunction with the Jacobi integral, to give a complete description of orbital motion. We show that, under the averaging assumptions, the problem is completely integrable and can be reduced to quadratures. For the case of no rotation, these quadratures can be expressed in terms of elliptic functions and integrals. For this problem, the orbit plane will experience nutation in addition to the precession that is found for orbital motion about an oblate body. It is possible for the orbit plane to be trapped in a 1:1 resonance with the rotating body, the plane essentially being dragged by the rotating asteroid. As the asteroid rotation rate is increased, this resonant motion disappears for rotation rates greater than a specific value. Finally, we validate our analysis with numerical integrations of cases of interest, showing that the averaging assumptions apply and give a correct prediction of motion in this system. These results are applicable to understanding spacecraft and particle motion about slowly rotating asteroids.