Eight-shaped Lissajous orbits in the Earth-Moon system
MathematicS In Action, Volume 4 (2011) no. 1, pp. 1-23.

Euler and Lagrange proved the existence of five equilibrium points in the circular restricted three-body problem. These equilibrium points are known as the Lagrange points (Euler points or libration points) L 1 ,...,L 5 . The existence of families of periodic and quasi-periodic orbits around these points is well known (see [20, 21, 22, 23, 37]). Among them, halo orbits are 3-dimensional periodic orbits diffeomorphic to circles. They are the first kind of the so-called Lissajous orbits. To be selfcontained, we first provide a survey on the circular restricted three-body problem, recall the concepts of Lagrange point and of periodic or quasi-periodic orbits, and recall the mathematical tools in order to show their existence. We then focus more precisely on Lissajous orbits of the second kind, which are almost vertical and have the shape of an eight – we call them eight-shaped Lissajous orbits. Their existence is also well known, and in the Earth-Moon system, we first show how to compute numerically a family of such orbits, based on Linsdtedt Poincaré’s method combined with a continuation method on the excursion parameter. Our original contribution is in the investigation of their specific stability properties. In particular, using local Lyapunov exponents we produce numerical evidences that their invariant manifolds share nice global stability properties, which make them of interest in space mission design. More precisely, we show numerically that invariant manifolds of eight-shaped Lissajous orbits keep in large time a structure of eight-shaped tubes. This property is compared with halo orbits, the invariant manifolds of which do not share such global stability properties. Finally, we show that the invariant manifolds of eight-shaped Lissajous orbits (viewed in the Earth-Moon system) can be used to visit almost all the surface of the Moon.

Published online:
DOI: 10.5802/msia.5
Classification: 70F07, 37N05
Keywords: Lagrange points, Lissajous orbits, stability, mission design
Grégory Archambeau 1; Philippe Augros 1; Emmanuel Trélat 2

1 EADS SPACE Transportation SAS, Flight Control Group, 66 route de Verneuil, BP 3002, 78133 Les Mureaux Cedex, France
2 Université d’Orléans, Laboratoire MAPMO, CNRS, UMR 6628, Fédération Denis Poisson, FR 2964, Bat. Math., BP 6759, 45067 Orléans cedex 2, France
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Grégory Archambeau; Philippe Augros; Emmanuel Trélat. Eight-shaped Lissajous orbits in the Earth-Moon system. MathematicS In Action, Volume 4 (2011) no. 1, pp. 1-23. doi : 10.5802/msia.5. https://msia.centre-mersenne.org/articles/10.5802/msia.5/

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