Space Trajectories. Three-Body Problem

The three-body problem concerns the motion of three gravitationally interacting material points. Unlike the two-body problem, it has no analytical solution. The problem is said to be restricted when one of the bodies has negligible mass, as in the case of a spacecraft subject to the attraction of two stars. The two stars then follow a Keplerian motion, almost circular in the case of Sun-Earth or Earth-Moon systems.

The restricted circular problem, although not integrable, lends itself more to study. Because of its practical importance, it has been the subject of much work since the announcement of collinear equilibrium points by Euler in 1767, then triangular by Lagrange in 1772. Poincaré carried out mathematical work on the existence of periodic orbits (1890), and the discovery of hundreds of Trojan asteroids in the Sun-Jupiter system from 1906 onwards confirmed the relevance of mathematical models.

Lagrange points L ₁ and L ₂ are located on the axis of the two stars on either side of the less massive star. They are particularly interesting for space exploration due to their stable position relative to the attracting stars. In the Sun-Earth system, these points are 1.5 million kilometers from the Earth. They offer a constant thermal environment, conducive to observing the Sun in L ₁ (ISEE, SOHO, LISA missions) or space in L ₂ (MAP, Gaia, Herschell-Planck, JWST missions). In the Earth-Moon system, these points are 60,000 km from the Moon. They are favorable for the installation of permanent space stations, as envisaged by Arthur Clarke as early as 1950. These stations would make it easier to serve the lunar surface or to explore the solar system.

These mission projects are made possible by the existence of periodic orbits in the vicinity of points L ₁ and L ₂ . The halo orbits discovered by Farquhar in 1966 have sufficient amplitude to avoid occultation by the Moon and maintain a constant link with the Earth. These orbits are associated with a set of trajectories naturally arriving at or departing from them. These trajectories, called invariant varieties, result from gravitational currents intrinsic to the dynamics of the three-body problem. The connections between varieties associated with different orbits form a complex network opening up the prospect of low-cost space transfers to the Moon or further afield.

This article presents the basic elements of the restricted circular problem, in particular the determination of equilibrium points, the...