Periodic solutions in the Sun–Jupiter–Trojan Asteroid–Spacecraft system

In this paper we investigate families of non-symmetric periodic orbits of the restricted four-body problem where the three primary bodies are set in the stable Lagrangian equilateral triangle configuration. More precisely, we consider the primary bodies Sun, Jupiter and an hypothetical Trojan Asteroid lie at the apices of an equilateral triangle and the fourth massless body (Spacecraft) is moving under the Newtonian gravitational attraction of the primaries. The problem admits eight non-collinear equilibrium points. Four of them are close to the Asteroid, two are stable and two are unstable. We focus our investigation on families of periodic orbits around Jupiter and/or the Asteroid. We also study analytically the solutions in the neighborhood of the stable equilibrium points. New families, namely short and long-period ones of non-symmetric periodic orbits, for each stable equilibrium point exist. The linear stability of each periodic solution is also studied. Special generating horizontal and vertical critical periodic orbits of each family are calculated.