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Mathematical Problems in Engineering
Volume 2012, Article ID 168632, 24 pages
Research Article

A Numerical Study of Low-Thrust Limited Power Trajectories between Coplanar Circular Orbits in an Inverse-Square Force Field

1Departamento de Matemática, Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos, SP, Brazil
2EMBRAER S. A., Divisão de Ensaio em Voo, 12227-901 São José dos Campos, SP, Brazil
3Subdepartamento Técnico do Departamento de Ciência e Tecnologia Aeroespacial, 12228-900 São José dos Campos, SP, Brazil

Received 15 November 2011; Accepted 19 January 2012

Academic Editor: Silvia Maria Giuliatti Winter

Copyright © 2012 Sandro da Silva Fernandes 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.


A numerical study of optimal low-thrust limited power trajectories for simple transfer (no rendezvous) between circular coplanar orbits in an inverse-square force field is performed by two different classes of algorithms in optimization of trajectories. This study is carried out by means of a direct method based on gradient techniques and by an indirect method based on the second variation theory. The direct approach of the trajectory optimization problem combines the main positive characteristics of two well-known direct methods in optimization of trajectories: the steepest-descent (first-order gradient) method and a direct second variation (second-order gradient) method. On the other hand, the indirect approach of the trajectory optimization problem involves two different algorithms of the well-known neighboring extremals method. Several radius ratios and transfer durations are considered, and the fuel consumption is taken as the performance criterion. For small-amplitude transfers, the results are compared to the ones provided by a linear analytical theory.