Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 260830, 11 pages http://dx.doi.org/10.1155/2013/260830 Research Article A Study of Single- and Double-Averaged Second-Order Models to Evaluate Third-Body Perturbation Considering Elliptic Orbits for the Perturbing Body R. C. Domingos, 1 A. F. Bertachini de Almeida Prado, 1 and R. Vilhena de Moraes 2 1 Instituto Nacional de Pesquisas Espaciais (INPE), 12227-010 S˜ ao Jos´ e dos Campos, SP, Brazil 2 Universidade Federal de S˜ ao Paulo (UNIFESP), 12231-280 S˜ ao Jos´ e dos Campos, SP, Brazil Correspondence should be addressed to A. F. Bertachini de Almeida Prado; bertachiniprado@bol.com.br Received 6 November 2012; Accepted 29 April 2013 Academic Editor: Maria Zanardi Copyright © 2013 R. C. Domingos et al. Tis 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. Te equations for the variations of the Keplerian elements of the orbit of a spacecraf perturbed by a third body are developed using a single average over the motion of the spacecraf, considering an elliptic orbit for the disturbing body. A comparison is made between this approach and the more used double averaged technique, as well as with the full elliptic restricted three-body problem. Te disturbing function is expanded in Legendre polynomials up to the second order in both cases. Te equations of motion are obtained from the planetary equations, and several numerical simulations are made to show the evolution of the orbit of the spacecraf. Some characteristics known from the circular perturbing body are studied: circular, elliptic equatorial, and frozen orbits. Diferent initial eccentricities for the perturbed body are considered, since the efect of this variable is one of the goals of the present study. Te results show the impact of this parameter as well as the diferences between both models compared to the full elliptic restricted three-body problem. Regions below, near, and above the critical angle of the third-body perturbation are considered, as well as diferent altitudes for the orbit of the spacecraf. 1. Introduction Most of the papers on this topic consider the third-body per- turbation due to the Sun and due to the Moon in a satellite around the Earth. Tis is the most immediate application of the third-body perturbation. One of the frst studies was made by Kozai [1] that developed the most important long- period and secular terms of the perturbing potential of the lunisolar perturbations, written as a function of the orbital elements of the Sun, the Moon, and the satellite. Moe [2], Musen [3], and Cook [4] studied long-period efects of the Sun and the Moon on artifcial satellites of the Earth. Tey applied Lagrange’s planetary equations to study the variation of the orbital elements of the satellite and its rate of variation. Tis idea was expanded by Musen et al. [5] that included the parallactic term in the perturbing potential. Kozai [6] studied the secular perturbations in asteroids that are in orbits with high inclination and eccentricity, assuming that the bodies are perturbed by Jupiter. Blitzer [7] made estimates for the lunisolar perturbation for the secular terms. Around the same time, Kaula [8] obtained the disturbing function considering the perturbations of the Sun and the Moon. Later, Giacaglia [9] calculated the disturbing function due to the Moon using equatorial elements for the satellite and ecliptic elements for the Moon. All terms were expressed in closed forms. Kozai [10] worked again on that problem and expressed the perturbing function as a function of the polar geocentric coordinates of the Moon, the Sun, and the orbital elements of the satellite. Te short-period terms are shown in an analytical form, and the secular and long-period terms are obtained by numerical integration. Hough [11] considered the efects of the perturbation of the Sun and the Moon in orbits near the inclinations of 63.4 ∘ and 116.6 ∘ (critical inclinations when considering the geopo- tential of the Earth) and showed that those efects are important only in high altitudes. Ash [12] also studied this problem using the double-averaged technique for a high-alti- tude satellite around the Earth. Collins and Cefola [13] used