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

Solórzano, Carlos Renato Huaura; Prado, Antonio Fernando Bertachini de Almeida

doi:https://doi.org/10.1155/2007/40475

Mathematical Problems in Engineering

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Volume 2007 | Article ID 040475 | https://doi.org/10.1155/2007/40475

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

Carlos Renato Huaura Solórzano¹and Antonio Fernando Bertachini de Almeida Prado¹

Academic Editor: José Manoel Balthazar

Received22 Oct 2005

Revised13 Dec 2006

Accepted13 May 2007

Published17 Oct 2007

Abstract

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.

References

A. M. Garofalo, “New set of variables for astronomical problems,” The Astronomical Journal, vol. 65, pp. 117–121, 1960.
View at: Google Scholar | MathSciNet
S. Pines, “Variation of parameters for elliptic and near circular orbits,” The Astronomical Journal, vol. 66, pp. 5–7, 1961.
View at: Publisher Site | Google Scholar | MathSciNet
C. J. Cohen and E. C. Hubbard, “A nonsingular set of orbit elements,” The Astronomical Journal, vol. 67, pp. 10–15, 1962.
View at: Publisher Site | Google Scholar | MathSciNet
R. A. Broucke and P. J. Cefola, “On the equinoctial orbit elements,” Celestial Mechanics and Dynamical Astronomy, vol. 5, no. 3, pp. 303–310, 1972.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
G. E. O. Giacaglia, “The equations of motion of an artificial satellite in nonsingular variables,” Celestial Mechanics and Dynamical Astronomy, vol. 15, no. 2, pp. 191–215, 1977.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
P. E. Nacozy and S. S. Dallas, “The geopotential in nonsingular orbital elements,” Celestial Mechanics and Dynamical Astronomy, vol. 15, no. 4, pp. 453–466, 1977.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. J. H. Walker, B. Ireland, and J. Owens, “A set modified equinoctial orbit elements,” Celestial Mechanics and Dynamical Astronomy, vol. 36, no. 4, pp. 409–419, 1985.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
I. Wytrzyszczak, “Nonsingular elements in description of the motion of small eccentricity and inclination satellites,” Celestial Mechanics and Dynamical Astronomy, vol. 38, no. 2, pp. 101–109, 1986.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
R. A. Broucke, “Long-term third-body effects via double averaging,” Journal of Guidance, Control, and Dynamics, vol. 26, no. 1, pp. 27–32, 2003.
View at: Google Scholar
A. F. B. A. Prado, “Third-body perturbation in orbits around natural satellites,” Journal of Guidance, Control, and Dynamics, vol. 26, no. 1, pp. 33–40, 2003.
View at: Google Scholar
C. R. H. Solórzano and A. F. B. A. Prado, “Third-body perturbation using a single averaged: lunisolar perturbation,” Nonlinear Dynamic and System Theory, vol. 7, no. 4, pp. 1–9, 2007.
View at: Google Scholar
S. Aoki, “Contribution to the theory of critical inclination of close earth satellites,” The Astronomical Journal, vol. 68, no. 6, pp. 355–365, 1963.
View at: Publisher Site | Google Scholar
S. Aoki, “Contribution to the theory of critical inclination of close earth satellites. 2. Case of asymmetrical potential,” The Astronomical Journal, vol. 68, no. 6, pp. 365–381, 1963.
View at: Publisher Site | Google Scholar
A. H. Jupp, “The problem of the critical inclination revisited,” Celestial Mechanics and Dynamical Astronomy, vol. 11, no. 3, pp. 361–378, 1975.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
A. H. Jupp, “The critical inclination problem with small eccentricity—I: general theory,” Celestial Mechanics and Dynamical Astronomy, vol. 21, no. 4, pp. 361–393, 1980.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

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.

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