The Laplace relation in the asteroidal belt

Abstract

The motion of asteroids which, together with Jupiter and Saturn, satisfy the commensurability relation n-3n'+2n''=0 (where the mean motions correspond to the asteroid, Jupiter and Saturn, respectively) has been studied. Through a generalization of Wilkens's method possible librators were sought, and five asteroids were found in such a condition. These results were then checked by direct numerical integration of the equations of motion for one hundred thousand years, finding temporary libration in three asteroids. We conclude that this particular triple commensurability affects very significantly the dynamics of these bodies (in some cases, chaotic motion was observed), even though no sign of orbital evolution was found. This paper will be published elsewhere.