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Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 510759, 22 pages
Research Article

Some Initial Conditions for Disposed Satellites of the Systems GPS and Galileo Constellations

1Departamento de Estatística, Matemática Aplicada e Computação, IGCE, UNESP campus Rio Claro, Caixa Postal 178, CEP: 13506-900 Rio Claro, SP, Brazil
2Departamento de Física, IGCE, UNESP campus Rio Claro, CEP: 13506-900 Rio Claro, SP, Brazil
3Departamento de Física, Centro de Ciências Exatas e Tecnológicas, Universidade Federal de Viçosa, UFV, CEP: 36571-000 Viçosa, MG, Brazil

Received 31 July 2009; Accepted 20 October 2009

Academic Editor: Silvia Maria Giuliatti Winter

Copyright © 2009 Diogo Merguizo Sanchez et al. 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.


Through the averaged equations we revisit theoretical and numerical aspects of the strong resonance that increases the eccentricity of the disposed objects of GPS and Galileo Systems. A simple view of the phase space of the problem shows that the resonance does not depend on the semi-major axis. Therefore, usual strategies of changing altitude (raising perigee) do not work. In this problem we search for a set of initial conditions such that the deactivated satellites or upper-stages remain at least for 250 years without penetrating in the orbits of the operational satellites. In the case that Moon's perturbation is not significant, we can identify, in the phase space, the regions where eccentricity reaches maximum and minimum values so that possible risks of collision can be avoided. This is done semi-analytically through the averaged system of the problem. Guided by this idea, we numerically found the ( 𝜔 , Ω ) values of the real unaveraged problem. In particular, for the Galileo case, the theoretical results predicted in the averaged system are in good agreement with numerical results. We also show that initial inclination of the Moon plays an important role in the search of these conditions.