Relative Orbital Motion Dynamical Models for Orbits about Nonspherical Bodies

Abstract

Relative orbital motion dynamical models are presented and discussed. Two types of models are primarily discussed in this work: a linear relative motion model accounting for J2 (the gravitational parameter associated with the oblateness of the Earth), and a new linear relative motion model accounting for both nonzero second degree and order gravity terms C20 = - J2 and C22. The latter model, referred to as the ``second-order model,'' is useful for simulating and studying spacecraft relative motion in orbits about uniformly rotating asteroids. This model is derived in two alternate forms. The first makes use of averaging in the kinematics and the second avoids any use of averaging. Additional work is devoted to analyzing the stability of relative orbital motion in rotating second degree and order gravity fields. To facilitate this, a parameter called the ""relative orbit angular momentum"" is introduced. For commensurate angular rates of primary body rotation and orbital mean motion, a special case linear time-invariant (LTI) model is obtained from the unaveraged second-order model. This is connected to the topic of libration points in the body frame of rotating gravitating triaxial ellipsoids, and shown to successfully predict the instability of libration points collinear with the long axis and the stability of libration points collinear with the short axis. Analytical and numerical results confirm the accuracy of all models discussed.