Searching for Time Optimal Periodic Orbits Near Irregularly Shaped Asteroids by Using an Indirect Method

Periodic orbits over irregularly shaped asteroids and comets are fundamental for understanding vicinal dynamical behaviors and space explorations. In this paper, a new method is proposed to obtain natural periodic orbits, which is based on the optimal control framework with respect to a general form of the irregular gravitational field. Natural elongated asteroids are taken as representatives, whose potentials are approximated by the rotating mass dipole with appropriate parameters. An indirect method is used to transform the optimal control model into a two-point boundary value problem, which can be solved by using a shooting method. Numerical simulations are performed to validate the effectiveness of the proposed method regarding the asteroid 951 Gaspra. Three types of periodic orbits are identified, including the Lyapunov orbit around the collinear equilibrium point, the equatorial retrograde orbit, and the inclined orbit. The connection between the latter two types of orbits is also briefly discussed via numerical continuation.