Mathematical Problems in Engineering
Volume 2007 (2007), Article ID 40475, 14 pages
Research Article

Third-Body Perturbation Using a Single Averaged Model: Application in Nonsingular Variables

Carlos Renato Huaura Solórzano and Antonio Fernando Bertachini de Almeida Prado

National Institute for Space Research (INPE), São José Dos Campos 12227-010, SP, Brazil

Received 22 October 2005; Revised 13 December 2006; Accepted 13 May 2007

Academic Editor: José Manoel Balthazar

Copyright © 2007 Carlos Renato Huaura Solórzano and Antonio Fernando Bertachini de Almeida Prado. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Lagrange's planetary equations written in terms of the classical orbital elements have the disadvantage of singularities in eccentricity and inclination. These singularities are due to the mathematical model used and do not have physical reasons. In this paper, we studied the third-body perturbation using a single averaged model in nonsingular variables. The goal is to develop a semianalytical study of the perturbation caused in a spacecraft by a third body using a single averaged model to eliminate short-period terms caused by the motion of the spacecraft. This is valid if no resonance occurs with the moon or the sun. Several plots show the time histories of the Keplerian elements of equatorial and circular orbits, which are the situations with singularities. In this paper, the expansions are limited only to second order in eccentricity and for the ratio of the semimajor axis of the perturbing and perturbed bodies and to the fourth order for the inclination.