Asymmetric librations in exterior resonances
type: Item
instance of: scholarly article
title: Asymmetric librations in exterior resonances
zbMATH Open document ID: 0852.70008
DOI: 10.1007/BF00693323
published in: Celestial Mechanics and Dynamical Astronomy
publication date: 1996-12-12T00:00:00Z
review text: (The author's abstract.) The purpose is to present a general analysis of the planar circular restricted three-body problem 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 \(\theta= (p+ q) \lambda_1- p\lambda- q\overline {\omega}\) 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 \(\theta =0\text{\;mod} (\pi)\). 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.
Mathematics Subject Classification ID: 70F07
Mathematics Subject Classification ID: 70F15
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 70M20
zbMATH DE Number: 756390
zbMATH Keywords: planar circular restricted three-body problem
zbMATH Keywords: phase space
zbMATH Keywords: commensurabilities
zbMATH Keywords: singular solutions
zbMATH Keywords: stability
zbMATH Keywords: reduced Andoyer Hamiltonian
zbMATH Keywords: second fundamental model for resonance
zbMATH Keywords: topological characteristics
author: Cristian Beaugé
reviewed by: Helmut Rüssmann
MaRDI profile type: Publication
cites work: Periodic orbits of the planetary type and their stability
cites work: Capture into resonance: An extension of the use of adiabatic invariants
cites work: A second fundamental model for resonance
cites work: High-order resonances in the restricted three-body problem
cites work: The main mean motion commensurabilities in the planar circular and elliptic problem
full work available at URL: https://doi.org/10.1007/bf00693323
OpenAlex ID: W2157896290
description: scientific article; zbMATH DE number 756390