Out-of-plane equilibrium points and invariant manifolds about an asteroid with gravitational orbit-attitude coupling perturbation

By considering the spacecraft as an extended, rigid body with a prior known attitude instead of a point mass, the attitude-restricted orbital dynamics can improve the precision of the classical point-mass orbital dynamics in close proximity to an asteroid, because it includes the perturbation caused by the gravitational orbit-attitude coupling of the spacecraft (GOACP). The GOACP is defined as the difference between the gravity acting on a non-spherical, extended body (the real case of a spacecraft) and the gravity acting on a point mass (the approximation of a spacecraft in classical orbital dynamics). In-plane equilibrium points that are within the principal planes of the asteroid have been investigated for the attitude-restricted orbital dynamics in previous studies, including equatorial and in-plane non-equatorial equilibrium points. In this study, out-of-plane equilibrium points outside the principal planes of the asteroid were examined. Out-of-plane equilibrium points cannot exist in the classical point-mass orbital dynamics but do exist in the attitude-restricted orbital dynamics owing to the effects of the GOACP. The previously investigated in-plane equilibrium points and the out-of-plane ones examined in this study provide a complete map of the equilibrium points in close proximity to an asteroid with the GOACP. Equatorial and in-plane non-equatorial equilibrium points have extended the longitude and latitude ranges of the classical equilibrium points without the GOACP, respectively, while the out-of-plane ones examined in the present study extend both the longitude and latitude ranges. Additionally, the invariant manifolds of out-of-plane equilibrium points were calculated, and the results indicated that the attitude of spacecraft significantly affects the invariant manifolds. In practice, these equilibrium points can provide natural hovering positions for operations in proximity to asteroids, and their invariant manifolds can be used for transfers to or from the equilibrium points.