Invariant manifolds associated to periodic orbits around Mars

Abstract

[en] The main purpose of this work is to explore if it is possible to find a transfer trajectory from the vicinity of the Earth to Mars that uses the stable invariant manifolds associated to some families of periodic orbits around Mars. This means that the study in a theoretic level of both periodic orbits and invariant manifolds is in the roadmap. The study of the Solar System is complex because of the interaction between all the planets and the Sun, however, there are some simplifications that can be done. Due to our interest in the study of spacecraft periodic orbits around Mars it is adopted the approach of a three-body problem, considering only the Sun, Mars and an artificial satellite. There has been extensively research in the field throughout the years and although the two-body problem is a well-known problem with well-known solutions, the three-body problem is neither solved nor is the behaviour of the dynamical system completely understood. It has to be said that whereas the two-body problem is completely integrable, the three body problem is not. The restricted three-body problem is of special interest because of its application in celestial mechanics, stellar dynamics or space mechanics. Due to the fact that the objective is to apply these theoretic aspects to the restricted three-body problem with Sun and Mars as primaries, the necessary software, using the C++ language, has to be developed in order to compute the desired objects and run the simulations. This work is organised in the following way: In chapter 1 it is introduced the planar circular restricted three-body problem, which describes the framework of our problem. The problem is formulated in dimensionless coordinates and its Hamiltonian is constructed. Chapter 2 is mainly divided in two parts, one related to periodic orbits and another to invariant manifolds. In the first part, devoted to periodic orbits, it is seen the CR3BP as a perturbation of the Hill’s problem, that allows us to use circular Keplerian orbits of the two body problem as a good approximation of periodic orbits of the CR3BP around Mars. Then, it is exposed the necessary theory in order to show that the orbits are grouped in families, that concludes with the Cylinder Theorem. Finally, this first part ends explaining the numerical methods used to continue the orbits along the family. The second part briefly exposes the theory of invariant manifolds and its application to periodic orbits. In chapter 3, there is a more detailed implementation of the methods and there are exposed the results of the computations. The work ends with some conclusions and future work. Although most of the proofs of theorems or propositions are broadly found in bibliography, in this work it has been opted for redo most of this demonstrations in order to gain a better insight of the theory.