Semianalytic Integration of High-Altitude Orbits under Lunisolar Effects

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2012, Article ID 659396, 17 pages doi:10.1155/2012/659396 Research Article Semianalytic Integration of High-Altitude Orbits under Lunisolar Effects Martin Lara,1 Juan F. San Juan,2 and Luis M. López3 1 C/Columnas de Hércules 1, ES-11100 San Fernando, Spain Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, Spain 3 Departamento de Ingenierı́a Mecánica, Universidad de La Rioja, 26004 Logroño, Spain 2 Correspondence should be addressed to Martin Lara, Received 6 November 2011; Accepted 20 January 2012 Academic Editor: Josep Masdemont Copyright q 2012 Martin Lara 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. The long-term effect of lunisolar perturbations on high-altitude orbits is studied after a double averaging procedure that removes both the mean anomaly of the satellite and that of the moon. Lunisolar effects acting on high-altitude orbits are comparable in magnitude to the Earth’s oblateness perturbation. Hence, their accurate modeling does not allow for the usual truncation of the expansion of the third-body disturbing function up to the second degree. Using canonical perturbation theory, the averaging is carried out up to the order where second-order terms in the Earth oblateness coefficient are apparent. This truncation order forces to take into account up to the fifth degree in the expansion of the lunar disturbing function. The small values of the moon’s orbital eccentricity and inclination with respect to the ecliptic allow for some simplification. Nevertheless, as far as the averaging is carried out in closed form of the satellite’s orbit eccentricity, it is not restricted to low-eccentricity orbits. 1. Introduction In an increasingly saturated space about the Earth, aerospace engineers confront the mathematical problem of accurately predicting the position of Earth’s artificial satellites. This is required not only for the correct operation of satellites but also for preserving the integrity of space assets. Thus, operational satellites are threatened by the not so remote possibility of a collision with a defunct satellite 1 but most probably by the impact with other uncontrolled man-made space objects as spent rocket stages or collision fragments—all of them commonly called space debris. Precise predictions require the integration of complete force models including both gravitational and nongravitational effects, like a high-degree and order geopotential, 2 Mathematical Problems in Engineering ephemerides-based lunisolar perturbations, drag, solar radiation pressure including eclipses, and so forth see 2, 3, for instance. The most accurate integration is expected from numerical methods, although precision ephemeris can also be obtained by means of semianalytical integration 4. In fact, both approaches, numerical and semi-analytical, do not need to enter a competition. Thus, for instance, while semi-analytical methods may be efficient in keeping a running catalog of hundreds of thousands of space objects within a reasonable accuracy, if the probability of impact with an operational satellite is detected to surpass a certain threshold, then a more accurate numerical integration will help control engineers in deciding whether a collision avoidance maneuver is required or, on the contrary, the integrity of the satellite is not in risk 5. In a semi-analytical approach the highest frequencies of the motion, which normally have small amplitudes, are filtered analytically via averaging procedures. This filtering allows the numerical integration of the averaged system to proceed with very long step sizes. Then, the short-period terms are recovered analytically, if desired, at any step of the numerical integration 6–8. Averaging techniques are also used for exploring questions affecting stability, such as those derived from tesseral resonances or third-body perturbations, in a reduced-phase space 9. In this respect, much attention has been recently paid to the long-term evolution of classical GNSS constellations, either for operational or disposal orbits 10. While the noncentralities of the Earth gravitational potential play a key role in the motion of low altitude satellites, third-body perturbations have also a decisive influence in the long-term evolution of medium- and high-altitude Earth orbits. The third-body disturbing function is commonly given by a series expansion in Legendre polynomials. Often, the series is truncated to the first term in the expansion 11, but this early truncation is not always accurate enough to represent the real dynamics 4, 12, 13. Nevertheless, recursions in the literature allow to extend the Legendre polynomial expansion to any desired order either in classical or nonsingular elements 14–16. Because of the physical characteristics of the Earth gravitational potential, where the second-order zonal coefficient J2  clearly dominates over all other harmonics, second-order effects of J2 may be important and must be taken into account when the effect of higherorder harmonics is studied. Correspondingly, the truncation in the expansion of third-body perturbations must include terms of magnitude comparable to J2 -squared. Because the thirdbody disturbing function is expanded in the ratio semi-major axis of the satellite’s orbit to semi-major axis of the third-body’s orbit, the degree at which the expansion must be truncated depends on the altitude of the satellites to be studied. In this paper we study the effect of lunisolar perturbations on high-altitude orbits about a noncentral Earth, which is assumed to be oblate although without equatorial symmetry. More specifically, we are interested in the semi-analytical integration of satellites on altitudes of classical global navigation satellite system GNSS constellations such as GPS, Glonass, or Galileo. Note, however, that the assumption of an axisymmetric geopotential prevents to tackle the tesseral resonance problem that commonly suffer this kind of orbits. With respect to previous research 17, where we approached the general case of third-body perturbations via double averaging, we release here the common simplification of assuming the third-body in circular orbit. Also, we focus on the case of Earth’s artificial satellites dealing with more real lunisolar perturbations. We do that because recent results 18 seem to contradict the claim that taking the third-body in circular orbit does not produce any noticeable degradation in the long-term propagation of real Earth orbits 19. Besides, for the actual Mathematical Problems in Engineering 3 values of the orbits of the sun and moon, neglecting terms in the eccentricity is not consistent with a higher-order expansion of the lunis (...truncated)