On the anomalous increase of the lunar eccentricity

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based on an analysis of a long LLR data record spanning 38.7 yr performed by Williams and Boggs 2 with the dynamical force models of the DE421 ephemerides 3,4 including all the known relevant Newtonian and Einsteinian effects. Notice that Eq. 1 is statistically significant at a 3σ-level. The first account 5 of this effect appeared in 2001 by Williams et al., who gave an extensive discussion of the state-of-the-art in modeling the tidal dissipation in both the Earth and the Moon. Later, Williams and Dickey 6 , relying upon the 2001 study 5 , released an anomalous eccentricity rate as large as ėmeas = (1.6 ± 0.5) × 10 -11 yr -1 . Anderson and Nieto 1 commented that Eq. 1 is not compatible with present, standard knowledge of the dissipative processes in the interiors of both the Earth and Moon, which were, actually, modeled by Williams and Boggs 2 .

Naive, dimensional evaluations of the effect caused on e by an additional anomalous acceleration A can be made by noticing that

for the geocentric orbit of the Moon, whose mass is denoted as m. In it, a is the orbital semimajor axis, while n . = µ/a 3 is the Keplerian mean motion in which µ .

= GM (1 + m/M ) is the gravitational parameter of the Earth-Moon system: G is the Newtonian constant of gravitation and M is the mass of the Earth. It turns out that an extra-acceleration as large as

would satisfy Eq. 1. In fact, a mere order-of-magnitude analysis based on Eq. 2 would be inadequate to infer meaningful conclusions: finding simply that this or that dynamical effect induces an extra-acceleration of the right order of magnitude may be highly misleading. Indeed, exact calculations of the secular variation of e caused by such putative promising candidate extra-accelerations A must be performed with standard perturbative techniques in order to check if they, actually, cause an averaged non-zero change of the eccentricity. Moreover, it may well happen, in principle, that the resulting analytical expression for ė retains multiplicative factors 1/e j , j = 1, 2, 3, … or e j , j = 1, 2, 3… which would notably alter the size of the found non-zero secular change of the eccentricity with respect to the expected values according to Eq. 2.

It is well known that a variety of theoretical paradigms 7,8 allow for Yukawa-like deviations 9 from the usual Newtonian inverse-square law of gravitation. The Yukawa-type correction to the Newtonian gravitational potential U N = -µ/r, where µ . = GM is the gravitational parameter of the central body which acts as source of the supposedly modified gravitational field, is

in which µ ∞ is the gravitational parameter evaluated at distances r much larger than the scale length λ. In order to compute the long-term effects of Eq. 5 on the eccentricity of a test particle it is convenient to adopt the Lagrange perturbative scheme 10 . In such a framework, the equation for the long-term variation of e is 10

where ω is the argument of pericenter, M is the mean anomaly of the test particle, and R denotes the average of the perturbing potential over one orbital revolution. In the case of a Yukawa-type perturbation, Eq. 5 yields

where I 0 (x) is the modified Bessel function of the first kind I q (x) for q = 0. An inspection of Eq. 6 and Eq. 7 immediately tells us that there is no secular variation of e caused by an anomalous Yukawa-type perturbation.

The size of the general relativistic Lense-Thirring 11 acceleration experienced by the Moon because of the Earth’s angular momentum 12 S = 5.86 × 10 33 kg m 2 s -1 is just

i.e. close to Eq. 4. On the other hand, it is well known that the Lense-Thirring effect does not cause long-term variations of the eccentricity. Indeed, the integrated shift of e from an initial epoch corresponding to f 0 to a generic time corresponding to f is 13

in which I ′ is the inclination of the Moon’s orbit with respect to the Earth’s equator and f is the true anomaly. From Eq. 9 it straightforwardly follows that after one orbital revolution, i.e. for f → f 0 + 2π, the long-term gravitomagnetic shift of e vanishes.

A promising candidate for explaining the anomalous increase of the lunar eccentricity is, at least in principle, a trans-Plutonian massive body X of planetary size located in the remote peripheries of the solar system. Indeed, the perturbation induced by it would, actually, cause a non-vanishing long-term variation of e. Moreover, since it depends on the spatial position of X in the sky and on its tidal parameter

where m X and d X are the mass and the distance of X, respectively, it may happen that a suitable combination of them is able to reproduce Eq. 1. Let us recall that, in general, the perturbing potential felt by a test particle orbiting a centr

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