A note on the dynamics around the L1,2 Lagrange points of the Earth-Moon system in a complete solar system model

The purpose of this paper is the study of the phase space around the collinear libration points\nL1 and L2 of the Earth–Moon system when the gravitational effects of the remaining bodies of the\nSolar System are taken into account. In the simplified model defined by the Circular Restricted\nThree Body Problem (CR3BP), the description of the phase space around these two points has\nalready been done in the past, either using semi-analytical techniques1 or numerical ones.2 When\nusing more realistic models of motion, the refinement of the different kinds of libration point orbits\naround both points (see G´omez et al.3) has some problems when the time interval used is large.\nThese problems are more evident for the Earth–Moon L2 point due to a 1:2 resonance between the\nnatural frequency of some halo orbits (!h) and the external frequency due to the perturbation of\nthe Sun (!s); in fact !h ' 2!s for some orbits of the halo family of periodic orbits around this\npoint. To analyse more closely this fact, Andreu4 introduced an intermediate model between the\nCR3BP and the restricted n-body problem, which is the so called Quasi-Bicircular Problem. It is a\nrestricted four body problem in which the three primaries move following a “true” solution of the\nthree body problem along orbits close to circular (the Earth and the Moon around their barycenter,\nand the Earth-Moon barycenter around the Sun). The description of the different types of orbits in\nthe neighbourhood of L2 is also done in Reference 4 by means of the reduction of the Hamiltonian of\nthe problem to the central manifold. The reduced Hamiltonian is then studied by means of Poincar´e\nmaps at different energy levels.