9/5/2023 0 Comments Elliptical orbit![]() For orbits above Low Earth Orbits, considering only the second order is not enough to accurately estimate the effect of luni-solar perturbations. However, it does consider the obliquity of the Sun and the Moon over the equator and the precession of the Moon plane due to the Earth's oblateness (in a period of 18.6 years with respect to the ecliptic). It assumes circular orbit for the disturbing bodies and considers only the second term of a/ a′, where a and a′ are, respectively, the spacecraft and the disturbing body semi-major axis about the Earth ( Blitzer, 1970). For example, Cook's formulation gives the secular and long-periodic perturbation due to luni-solar perturbation obtained through averaging over one orbit revolution of the satellite ( Cook, 1962). In averaged development the potential is usually truncated to the second order. The effect of third body is usually modeled as a series expansion of the potential with respect to the ratio between the orbit semi-major axis and the distance to the third body. Moreover, the use of the average dynamics reduces the computational time for numerical integration as the stiffness of the problem is reduced, while maintaining sufficient accuracy compatible with problem requirements also for long-term integrations. The resulting system allows a deeper understanding of the dynamics ( Shapiro, 1995 Krivov and Getino, 1997). Indeed, averaging corresponds to filtering the higher frequencies of the motion (periodic over one orbit revolution), which typically have small amplitudes ( Ely, 2014). ![]() The short-term effect of perturbations is eliminated by averaging the variational equations, or the corresponding potential, over one orbit revolution of the small body. It separates the constant, short periodic and long-periodic terms of the disturbing function. The semi-analytical technique based on averaging is an elegant approach to analyze the effect of orbit perturbations. The dynamics of HEOs with high apogee altitude is mainly influenced by the effect of third body perturbations due to the Moon and the Sun, which induces long-term variations in the eccentricity and the inclination, corresponding to large fluctuations of the orbit perigee and the effect of the Earth's oblateness. This paper, whose preliminary version was presented at the 25th AAS/AIAA Space Flight Mechanics Meeting, in Williamsburg (VA) in January 2015 ( Colombo, 2015), investigates the long-term evolution of spacecraft in HEOs through the exploitation and development of semi-analytical techniques. If the inclination is properly selected, HEO can minimize the duration of the motion inside the eclipses. In addition, geo-synchronicity is opted to meet coverage requirements, enhanced at the apogee, and optimize the ground station down-link. HEOs guarantees spending most of the time at an altitude outside the Earth's radiation belt therefore, long periods of uninterrupted scientific observation are possible. ![]() Moreover, elliptical Geostationary Transfer Orbits are commonly selected to inject spacecraft in geostationary orbit. Highly Elliptical Orbits (HEOs) about the Earth are often selected for astrophysics and astronomy missions, as well as for Earth missions, such as Molniya or Tundra orbits, as they offer vantage point for the observation of the Earth and the Universe ( Draim et al., 2002). On the opposite side, unstable conditions can be exploited to target Earth re-entry at the end-of-mission. In addition, to allow meeting specific mission constraints, quasi-frozen orbits can be selected as graveyard orbits for the end-of-life of HEO missions, in the case re-entry option cannot be achieved due to propellant constraints. The behavior of these long-term orbit maps is studied for increasing values of the initial orbit inclination and argument of the perigee with respect to the Moon's orbital plane. Maps of long-term orbit evolution are constructed by measuring the maximum variation of the orbit eccentricity to identify conditions for quasi-frozen, long-lived libration orbits, or initial orbit conditions that naturally evolve toward re-entry in the Earth's atmosphere. The double averaged potential is also derived in the Earth-centered equatorial system. The single averaged disturbing potential due to luni-solar perturbations, zonal harmonics of the Earth gravity field is written in mean Keplerian elements. ![]() This paper investigates the long-term evolution of spacecraft in Highly Elliptical Orbits (HEOs). Department of Aerospace Science and Technology, Politecnico di Milano, Milan, Italy.
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