ASYMMETRIC LIBRATIONS IN EXTERIOR RESONANCES

The purpose of this paper is to present a general analysis of the planar circular restricted problem of three bodies in the case of exterior mean-motion resonances. Particularly, our aim is to map the phase space of various commensurabilities and determine the singular solutions of the averaged system, comparing them to the well-known case of interior resonances. In some commensurabilities (e.g. 1/2, 1/3) we show the existence of asymmetric librations; that is, librations in which the stationary value of the critical angle theta = (p + q)lambda1 - plambda - qomegaBAR is not equal to either zero or pi. The origin, stability and morphogenesis of these solutions are discussed and compared to symmetric librations. However, in some other resonances (e.g. 2/3, 3/4), these fixed points of the mean system seem to be absent. Librations in such cases are restricted to theta = 0 mod(pi). Asymmetric singular solutions of the planar circular problem are unknown in the case of interior resonances and cannot be reproduced by the reduced Andoyer Hamiltonian known as the Second Fundamental Model for Resonance. However, we show that the extended version of this Hamiltonian function, in which harmonics up to order two are considered, can reproduce fairly well the principal topological characteristics of the phase space and thereby constitutes a simple and useful analytical approximation for these resonances.