Spacecraft Motion About Slowly Rotating Asteroids

Abstract

Spacecraft motion about an arbitrary second-degree and second-order gravity eeld is investigated. We assume thatthegravityeeldisinuniformrotation aboutanaxisofitsmaximummomentofinertiaandthatitrotatesslowly 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 speciec 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. I. Introduction T HIS paper studies the orbital motion of a spacecraft about slowly rotating asteroids. The asteroid gravity e eld is represented by the second-degree and second-order gravity coefecients C20 and C22. The asteroid itself is assumed to be uniformly rotating about the axis of its major moment of inertia. When the asteroid is slowly rotating, we can apply averaging techniques to the Lagrange planetary equations to derive a simplie edset of equations that describe a spacecraft’s orbit. Also, because this is a uniformly rotatinggravitye eld,the Jacobiintegralthatistraditionally used for restricted three-body problems is deened for this problem. 1 Using the Jacobi integral in conjunction with the averaged equations, we can show that the problem can be reduced to quadratures, that is, the problem is integrable. This is an extension of the results that show that the averaged problem is integrable in terms of elliptic, hyperbolic, and trigonometric functions ifthe central body does not rotate. 2 ByproperinterpretationoftheJacobiintegral,wecanderive