Quantum effects on Lagrangian points and displaced periodic orbits in the Earth-Moon system

Abstract

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.

Earth-moon Lagrangian points as a test bed for general relativity and effective field theories of gravity

“…This topic is investigated in Sec. V, where we also show that, among all quantum coefficients κ 1 and κ 2 in the long-distance corrections to the Newtonian potential available in literature [5,[12][13][14][15], the most suitable ones to describe the gravitational interactions involving (at least) three bodies are those connected to the bound-states potential. Finally, conclusions and open problems are discussed in Sec.…”

“…Moreover, following Ref. [15], we have the classical values ξ p = 1.87528148802 × 10 8 m and η p = 3.32900165215 × 10 8 m. If we initially set S = 0 in (2.1), we obtain that the perturbative effect of the Sun makes the spacecraft ultimately escape from the stable equilibrium point after about 700 days [52], as is shown in Figs. 1 and 2.…”

“…These predictions become more realistic if we include the perturbations due to the gravitational presence of the Sun, in other words we have to face up the restricted problem of four bodies, consisting of the Sun, the Earth and the Moon as the three primaries and the fourth body (e.g. a laser-ranged test mass, a spacecraft or by exploiting the solar sail technology [15,[44][45][46][47][48][49][50][51]) which has an infinitesimal mass, to avoid affecting the motion of the primaries. As we know, in the restricted three-body problem the motion of the two primaries is exactly described by the equations of motion governing the two-body problem.…”

Quantum time delay in the gravitational field of a rotating mass

“…(3.12), (3.13), and Unfortunately this deviation from the underlying classical theory has no chance to be tested, unlike, for example, the effects some of us predicted in Refs. [10,11,14,15], where it is shown that Earth-Moon Lagrangian points are expected to undergo a displacement of the order of few millimetres from their classical position. Furthermore, we have proved that the result reported in Eq.…”

Triangular Libration Points in the CR3BP with Radiation, Triaxiality and Potential from a Belt

“…The collinear libration points L 1 , L 2 , L 3 are unstable, while the triangular libration points L 4 , L 5 are stable for the mass ratio of the primaries less than 0.03852 · · · , [38]. The 3BP, under different forms and aspects, has been used for studies in fundamental physics, general relativity and alternative theories of gravity [1,2,5,10,20,24,31,41].…”