On the Occurrence of Commensurable Mean Motions in the Solar System: The Mirror Theorem

Abstract

In Part I it is shown that the preference for near-commensurability of mean motions demonstrated in a previous paper extends to the small satellites of Jupiter, including the retrograde ones, and to the retrograde satellite of Saturn, implying that stability of near-commensurable configurations is the reason for such a preference. In Part II it is proved that if, at a certain epoch, a system of n gravitating point-masses has each radius vector from the (assumed stationary) centre of mass of the system perpendicular to every velocity vector (hereinafter called a “mirror configuration”), then the behaviour of each of the point-masses under the internal gravitational forces of the system after the epoch will be a mirror image of its behaviour prior to the epoch. It is further shown that, if a mirror configuration of the system exists at two separate epochs, then the orbit of each point-mass is periodic. The authors argue that such periodic orbits are the more stable (under the action of external forces) the shorter the interval of time between mirror configurations, and they show that the frequent occurrence of mirror configurations between any two point-masses requires that the mean motions of the two point-masses be nearly commensurable. In Part III, various near-commensurable pairs of orbits in the solar system are shown to behave according to the above arguments. The relationship of the present work to the more restricted “symmetry theorem” of Griffin is discussed.