Ballistic and powered capture of asteroids in the Sun-Earth-Moon system

Space missions aimed at in-situ exploration of near-Earth asteroids have long attracted the interest of both space agencies and private research groups. Several solutions have been proposed in the last decades, all associated by the critical but pivotal phase of asteroid capture. In fact, the dynamics of asteroid trajectories is that of a negligible mass in a multibody environment, resulting from the combined effect of the gravitational potentials of the surrounding celestial bodies, called the primaries. In the Sun-Earth-Moon system here examined, asteroids typically orbit the center of mass of the system, nevertheless it can happen that, under some conditions, the asteroid gets trapped in the gravitational well of the Earth (or the Moon). Such a scenario is called a ballistic capture and corresponds to an opportune energy condition for an exploration mission. Because of the strong gravitational perturbations characterizing multibody environments, ballistic captures have typically limited duration and all the mission operations, including landing, experiments and take-off, shall be performed within this interval of time. The occurrence and duration of an asteroid capture can be in principle controlled by means of a spacecraft capable of docking the asteroid and providing the thrust acceleration necessary to adequately modify its dynamical state, performing a powered capture. In this work, the conditions for both ballistic and powered capture of a near-Earth asteroid are investigated in the dynamical framework of the elliptic restricted 4-body problem. The problem is modeled using the Hamiltonian formalism and canonical transformations. The dynamic equations of motion are first linearized about quasi-equilibrium points, which represent an extension of libration points existing in the circular restricted 3-body problem, and then set to a normal form of the type saddle-center-center, by means of canonical transformations. The resulting mathematical system allows defining the topological location of capture trajectories, and estimating the capture time, thus representing a powerful tool in the design of ballistic asteroid capture. Furthermore, once identified the parameters characterizing capture condition, expressions for the magnitude and direction of low thrust acceleration to be provided for powered capture are inferred. The accuracy of the proposed method is verified by means of numerical simulations on a number of scenarios for possible missions to be performed in the next future.