The leading long distance quantum corrections to the Newtonian potential imply tiny but observable effects in the restricted three-body problem of celestial mechanics, i.e., at the Lagrangian libration points of stable equilibrium the planetoid is not exactly at equal distance from the two bodies of large mass, but the Newtonian values of its coordinates are changed by a few millimeters in the Earth-Moon system. We show that such a theoretical calculation is governed by a couple of quintic algebraic equation, whereas coordinates of collinear Lagrangian points are given in terms of the solution of an algebraic equation of ninth degree.
We also discuss the prospects to measure, with the help of laser ranging, the above departure from the equilateral triangle picture, which is a challenging task. On the other hand, a modern version of the planetoid is the solar sail, and much progress has been made, in recent years, on the displaced periodic orbits of solar sails at all libration points, both stable and unstable. By taking into account the quantum corrections to the Newtonian potential, displaced periodic orbits of the solar sail at libration points are again found to exist.