Two-manoeuvres transfers between LEOs and Lissajous orbits in the Earth–Moon system

Abstract

The purpose of this work is to compute transfer trajectories from a given Low Earth Orbit (LEO) to a nominal Lissajous quasi-periodic orbit either around the point L 1 or the point L 2 in the Earth–Moon system. This is achieved by adopting the Circular Restricted Three-Body Problem (CR3BP) as force model and applying the tools of Dynamical Systems Theory. It is known that the CR3BP admits five equilibrium points, also called Lagrangian points, and a first integral of motion, the Jacobi integral. In the neighbourhood of the equilibrium points L 1 and L 2, there exist periodic and quasi-periodic orbits and hyperbolic invariant manifolds which emanate from them. In this work, we focus on quasi-periodic Lissajous orbits and on the corresponding stable invariant manifolds. The transfers under study are established on two manoeuvres: the first one is required to leave the LEO, the second one to get either into the Lissajous orbit or into its associated stable manifold. We exploit order 25 Lindstedt–Poincaré series expansions to compute invariant objects, classical manoeuvres and differential correction procedures to build the whole transfer. If part of the trajectory lays on the stable manifold, it turns out that the transfer’s total cost, Δ v tot , and time, t tot , depend mainly on: 1. the altitude of the LEO; 2. the geometry of the arrival orbit; 3. the point of insertion into the stable manifold; 4. the angle between the velocity of insertion on the manifold and the velocity on it. As example, for LEOs 360 km high and Lissajous orbits of about 6000 km wide, we obtain Δ v tot ∈ [ 3.68 , 4.42 ] km / s and t tot ∈ [ 5 , 40 ] days . As further finding, when the amplitude of the target orbit is large enough, there exist points for which it is more convenient to transfer from the LEO directly to the Lissajous orbit, that is, without inserting into its stable invariant manifold.