  Recent work in the literature has shown that 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., 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. First, we assess such a theoretical calculation by exploiting the full
theory of the quintic equation, i.e., its reduction to Bring-Jerrard form and
the resulting expression of roots in terms of generalized hypergeometric
functions. By performing the numerical analysis of the exact formulas for the
roots, we confirm and slightly improve the theoretical evaluation of quantum
corrected coordinates of Lagrangian libration points of stable equilibrium.
Second, we prove in detail that also for collinear Lagrangian points the
quantum corrections are of the same order of magnitude in the Earth-Moon
system. Third, 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, both stable and
unstable. The present paper investigates therefore, eventually, a restricted
three-body problem involving Earth, Moon and a solar sail. By taking into
account the one-loop quantum corrections to the Newtonian potential, displaced
periodic orbits of the solar sail at libration points are again found to exist.
