An Empirical Explanation of the Anomalous Increases in the Astronomical Unit and the Lunar Eccentricity

Such phenomena attracted the attention of various scientists dealing with them in different contexts [9,10,11,12,13,14,15,16,17,18,19,20,21]. Thus, several more or less sound attempts to find, or to rule out, possible explanations [2,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,1,41,42,43] for both the anomalies were proposed so far, both in terms of standard known gravitational physical phenomena and of long-range modified models of gravity.

Here we propose an empirical formula which is able to accommodate both the anomalies, at least as far as their orders of magnitude are concerned.

Let us assume that, in addition to the usual Newtonian inverse-square law for the gravitational acceleration imparted to a test particle by a central body orbited by it, there is also a small radial extra-acceleration of the form

In it k is a positive numerical parameter of the order of unity to be determined from the observations, H 0 = (73.8 ± 2.4) km s -1 Mpc -1 = (7.47 ± 0.24)×10 -11 yr -1 [44] is the Hubble parameter at the present epoch, defined in terms of the time-varying cosmological scaling factor S(t) as H 0 . = Ṡ/S 0 , and v r is the component of the velocity vector v of the test particle’s proper motion about the central body along the common radial direction. The radial velocity for a Keplerian ellipse is [45]

where n is the Keplerian mean motion, a is the semi-major axis, and f is the true anomaly reckoning the instantaneous position of the test particle along its orbit: v r vanishes for circular orbits.

The consequences of eq. ( 1) on the trajectory of the particle can be straightforwardly worked out with the standard Gauss equations for the variation of the Keplerian orbital elements [45] which are valid for any kind of perturbing acceleration, whatever its physical origin may be. For the semi-major axis and the eccentricity they are

In eq. ( 3) p . = a(1e 2 ) is the semi-latus rectum, and A R and A T are the radial and transverse components of the disturbing acceleration, respectively: in our case, eq. ( 1) is entirely radial. In a typical first-order perturbative 1 calculation like in the present case, the right-hand-sides of eq. ( 3) have to be computed onto the unperturbed Keplerian ellipse, characterized by

and integrated over one orbital period by means of

It turns out that both the semi-major axis a and the eccentricity e of the test particle’s orbit secularly increase according to

The formulas in eq. ( 6), which were obtained by taking an average over a full orbital revolution, are exact to all order in e.

Since e Moon = 0.0647, it turns out that eq. ( 6) is able to reproduce the measured anomalous increase of the lunar orbit for 2.5 k 5. Moreover, for such values of k eq. ( 6) yields an increase of the lunar semi-major axis of just 0.3 -0.6 mm yr -1 . It is, at present, undetectable, in agreement with 1 Indeed, it can be easily inferred that eq. ( 1) is of the order of 10 -15 m s -2 for the Earth’s motion around the Sun, while its Newtonian solar monopole term is as large as 10 -3 m s -2 . The same holds for the Earth-Moon system as well. Indeed, eq. ( 1) yields about 10 -16 m s -2 for the lunar geocentric orbit, while the Newtonian monopole acceleration due to the Earth is of the order of 10 -3 m s -2 .

the fact that, actually, no anomalous secular variations pertaining such an orbital element of the lunar orbit have been detected so far. If we assume the terrestrial semi-major axis2 a ⊕ = 1.5 × 10 13 cm as an approximate measure of the astronomical unit and consider that e ⊕ = 0.0167, eq. ( 6) and the previous values of k yield a secular increase of just a few cm yr -1 . Also in this case, it can be concluded that eq. ( 6), if applied to other situations for which accurate data exist, does not yield results in contrast with empirical determinations for a and e. Indeed, for the eccentricity of the Earth eq. ( 6), with 2.5 k 5, yields ė = (1.7 -3.4) × 10 -12 yr -1 . Actually, such an anomalous effect cannot be detectable since, according to Table 3 of Ref. [51], the present-day formal, statistical accuracy in determining e from the observations amounts just to 3.6 × 10 -12 ; it is well known that the realistic uncertainty can be up to one order of magnitude larger. Similar considerations hold for the other planets.

Here we do not intend to speculate too much about possible viable physical mechanisms yielding the extra-acceleration of3 eq. ( 1).

It might be argued that, reasoning within a cosmological framework, the Hubble law may give eq. ( 1) for k = 1 if the proper motion of the particle about

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