Short-period effects of the planetary perturbations on the Sun-Earth Lagrangian point L3

Context. The Lagrangian point L3 of the Sun–Earth system, and its Lyapunov orbits, have been proposed to perform station-keeping, although L3 is only rigorously defined for the extremely simplified model represented by the reduced Sun–Earth–spacecraft system. As in L3 the planetary perturbations (mainly from Jupiter and Venus) are stronger than Earth’s attraction, it is necessary to understand whether or not the dynamics close to L3 persist under such a strong perturbation, allowing for a definition of dynamical substitutes for models that are more realistic than the circular restricted three-body problem. Aims. In this paper we address the problem of the existence of motions that remain close to L3 for a time-span which is relevant for space missions in a model of the Solar System compatible with the precision of JPL digital ephemerides. Methods. First, we computed analytically the main short-period effects of planetary perturbations in a simplified model of the Solar System with the orbits of all the planets co-planar and circular. We then applied the Fast Lyapunov Indicator method in order to find dynamical substitutes that exist for time-spans of hundreds of years in the model of the Solar System that is used to produce the modern ephemerides. Results. We find that the original system is conjugate by a canonical transformation to an averaged system that has an equilibrium close to L3: even if Venus and Jupiter each move the position of this equilibrium by about 218 and 176 km, respectively, in opposite directions, in the model where both the planets are included, their effects almost perfectly compensate for one another, leaving a displacement of about 40 km only. This equilibrium is then mapped in the original system to a quasi-periodic dynamical substitute; the contributions of each planet to the amplitude of this quasi-periodic libration around L3 do not compensate for one another, and sum to about 10000 km. The Fast Lyapunov Indicator method allowed us to find orbits of any amplitude bigger than this one (up to 0.03 AU) for time-spans of hundreds of years in the model of the Solar System that is used to produce the modern ephemerides. Conclusions. Using a combination of the Hamiltonian averaging method with a new implementation of the Fast Lyapunov Indicator method we find orbits useful for astrodynamics originating at the Sun–Earth Lagrangian point L3 for a realistic model of the Solar System. In particular, this usage of the chaos indicator provides an innovative application of dynamical systems theory to astrodynamics, here the short-period perturbations represent a relevant part of the model.