Three-body Mean Motion Resonances and the Chaotic Structure of the Asteroid Belt

We discuss the existence and the properties of a new type of mean motion resonance populated by a large number of asteroids, the resonant angle being deÐned as a linear combination (with integer coefficients) of the mean longitudes of asteroid, Jupiter, and Saturn. We call these resonances the threebody mean motion resonances. In the present paper, we show that the anomalous large oscillation of the mean semimajor axis of about 250 numbered asteroids is associated with some of the most prominent three-body mean motion resonances, and we conjecture that the actual number of the resonant asteroids is much larger. The positive Lyapunov exponent detected for the majority of resonant bodies in our numerical integrations suggests the chaotic nature of their orbits. Moreover, we show, using frequency analysis, that orbits in the three-body mean motion resonances may slowly and chaotically di†use in eccentricity. The existence of such di†usion may have major consequences for our present understanding of the long-term evolution of the asteroidal belt and the delivery of objects to near-Earth orbits.