  The one-loop long distance quantum corrections to the Newtonian potential
imply tiny but observable effects in the restricted three-body problem of
celestial mechanics, i.e., both at the Lagrangian points of stable equilibrium
and at those of unstable equilibrium the Newtonian values of planetoid's
coordinates are changed by a few millimetres in the Earth-Moon system. First,
we find that the equations governing the position of both noncollinear and
collinear quantum libration points are algebraic fifth degree and ninth degree
equations, respectively. Second, we 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.
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.
