Chaos in high order mean motion resonances
This paper argues, via numerical examples, that the short Lyapunov times, shown by certain minor planets in the outer asteroid belt, can probably be traced to the overlap of a high order mean motion resonance with a type of secondary resonance. The latter occurs when a 1-to-1 commensurability exists, generally temporarily, between the rate at which a conjunction with Jupiter circulates and the apsidal line of the minor planet precesses. [Such a behavior finds a more well-known parallel within first order mean motion resonances: there secondary resonance arises when the libration and apsidal frequencies are commensurate.] The resulting chaotic zones for orbits with low eccentricity have narrow widths, usually about 0.003 AU for four of the minor planets considered here, even one of the ninth order. But objects lying within them have typical Lyapunov times even somewhat less than 500 Jovian orbital periods.