Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 753653, 18 pages
Research Article

Hill Problem Analytical Theory to the Order Four: Application to the Computation of Frozen Orbits around Planetary Satellites

1Celestial Mechanics Division, Real Observatorio de la Armada, 11110 San Fernando, Spain
2Departamento de Ingeniería Matemática e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain

Received 7 June 2009; Accepted 24 August 2009

Academic Editor: Silvia Maria Giuliatti Winter

Copyright © 2009 Martin Lara and Jesús F. Palacián. 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.


Frozen orbits of the Hill problem are determined in the double-averaged problem, where short and long-period terms are removed by means of Lie transforms. Due to the perturbation method we use, the initial conditions of corresponding quasi-periodic solutions in the nonaveraged problem are computed straightforwardly. Moreover, the method provides the explicit equations of the transformation that connects the averaged and nonaveraged models. A fourth-order analytical theory is necessary for the accurate computation of quasi-periodic frozen orbits.