Bounded trajectories of a spacecraft near an equilibrium point of a binary asteroid system

With a growing interest in asteroid exploration, combined with the fact that numerous asteroids in nature occur in pairs, it is likely that future missions will include the exploration of binary asteroid systems. Thus, it is useful to study the motion of a spacecraft in the vicinity of such systems, modeled as the three-body problem. In this paper, the circular restricted full three-body problem is considered. The zeroth-order equations of motion near an equilibrium point are similar in form to those in the classical case with point-masses or spherical primaries. For most asteroid pairs found in practice, all five equilibrium points are unstable. However, with selection of appropriate initial conditions, it is possible to obtain bounded solutions to the zeroth-order equations, corresponding to the Lissajous trajectories near collinear points, and bounded trajectories near noncollinear points. Numerical simulations confirm that when including the additional perturbations due to the asphericity of the asteroid pair, the motion of the spacecraft is unbounded. Thus, control laws are developed by utilizing an appropriate Lyapunov function, with the solutions to the zeroth-order equations as reference trajectories. These were found to be sufficient to maintain the spacecraft in bounded trajectories.