Asymmetric Librations in Exterior Resonances
Abstract
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 meanmotion 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 wellknown 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 ϑ=(p+q)λ_{1}pλqω is not equal to either zero or π. 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 ϑ=0 mod(π). Asymmetric singular solutions of the planar circular problem are unkown 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.
 Publication:

Celestial Mechanics and Dynamical Astronomy
 Pub Date:
 October 1994
 DOI:
 10.1007/BF00693323
 Bibcode:
 1994CeMDA..60..225B
 Keywords:

 Celestial Mechanics;
 Libration;
 Orbit Perturbation;
 Orbital Mechanics;
 Orbital Resonances (Celestial Mechanics);
 Three Body Problem;
 Astronomical Models;
 Distribution Functions;
 Hamiltonian Functions;
 Perturbation Theory;
 PoyntingRobertson Effect;
 Astrophysics;
 Orbital resonance;
 librations;
 Andoyer Hamiltonians