On the Orbit of the Moon

Abstract

The study of the role of high-integer near commensurabilities among lunar months has led to the discovery that the lunar orbit is very close to a set of 8 long-period periodic orbits of the restricted circular 3-dimensional Sun-Earth-Moon problem in which also the secular motion of the argument of perigee is involved (Valsecchi et al., 1993). In each of these periodic orbits 223 synodic months are equal to 239 anomalistic and 242 nodical ones, a relationship that approximately holds in the case of the observed Saros cycle, and the various orbits differ from each other for the initial phases. These integer ratios imply that, in one cycle of the periodic orbit, the argument of perigee makes exactly 3 revolutions, i.e., the difference between the 242 nodical and the 239 anomalistic months (in fact, these two months differ from each other just for the prograde rotation of the argument of perigee).

The periodic orbits associated with the Saros cycle, even if of long duration when compared to those usually found in literature, are by no means the longest ones that can be found close to that of the Moon: actually, according to a conjecture of Poincaré there should be infinitely many, of longer and longer period. It is possible to show, with the help of Delaunay’s expressions for the motion of the lunar perigee and node, that the longer periodic orbits are arranged in the eccentricity-inclination plane in a rather characteristic pattern, that is simply a deformation of the arrangement, in frequency space, of the set of points corresponding to the frequencies of the periodic orbits themselves.

This finding allows to set up a numerical scheme to find the periodic orbits automatically; this tool can then be used to make a systematic exploration of the Main Lunar Problem.