Practical Applications of the Invariant Structures of the Spatial Restricted Three-Body Problem in the Dynamics of the Solar System

Abstract

This research project comprises the study of dynamical invariant structures of the Spatial Restricted Three-Body Problem (SR3BP) and their practical applications in Astronautics and Astrodynamics, particularly in the design of trajectories for space mission projects and in the dynamics of subsystems of the Solar System. The mathematical model is described by a autonomous non-integrable Hamiltonian with three degrees of freedom. We will employ an analytical-numerical approach and Dynamical Systems tools in the investigation proposed. We are particularly interested in the open question of stability of solutions in the six-dimensional phase space. It is known that domains of effective stability around the triangular equilibrium points of the SR3BP are related, for example, to the motion of asteroids and comets known as NEO (Near Earth Objects), to the dynamics of the rings of Uranus and to the existence of the Trojan asteroids that follow the orbit of Jupiter. Particles remain confined within those domains due to the presence of codimension-1 hyperbolic invariant manifolds that act as effective barriers between captured and escaping trajectories. Also, the invariant objects near the collinear equilibrium points of the SR3BP have practical applications in the design of modern space missions. This is because periodic and quasi-periodic orbits and the hyperbolic invariant manifolds of periodic orbits and two-dimensional tori can guide trajectories, taking advantage of the non-linear character of the model to decrease the amount of fuel needed. Thus, we propose a systematic investigation of relevant invariant objects, initially, in the Sun-Jupiter and the Earth-Moon systems aiming to characterize the transport processes provided by these structures with possible applications, respectively, to the explanation of the dynamics of the Trojans and to the construction of possible parking orbits for space platforms and low-energy transfers.