97-54 Gomez G., Masdemont J.J., Simo C.

LISSAJOUS ORBITS AROUND HALO ORBITS (738K, Postscript, gzipped and uuencoded) Feb 6, 97

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Abstract. The analysis of the phase space around the collinear equilibrium points of the Restricted Three Body Problem shows different kinds of motions in a (large) vicinity of these points, depending on the values of the energy. Beyond some critical value of the energy, there appear: an almost vertical periodic orbit that crosses the $z=0$ plane through a point close to the equilibrium point position, a family of Lissajous orbits around the almost vertical periodic orbits, two halo orbits which are symmetric with respect to the $z$=0 plane and two families of Lissajous orbits around these halo orbits. Finally, the planar Lyapunov orbit ``closes'' the level of energy. Halo orbits, which are three dimensional periodic orbits around the equilibrium points are useful, as nominal orbits, for space missions and this is one of the main reasons to study this kind of motions or other ones close to them. This paper is devoted to the study of the two dimensional tori around halo orbits on which the motion is quasiperiodic with two basic frequencies. The orbits look like Lissajous curves. The first part of the paper is devoted to the computation of these orbits using the Lindstedt-Poincar\'e method. In the second part we study the validity of the formal expansions obtained.

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