## Controlling the eccentricity of polar lunar orbits with low-thrust propulsion.(English)Zbl 1184.70012

Summary: It is well known that lunar satellites in polar orbits suffer a high increase on the eccentricity due to the gravitational perturbation of the Earth. That effect is a natural consequence of the Lidov-Kozai resonance. The final fate of such satellites is the collision with the Moon. Therefore, the control of the orbital eccentricity leads to the control of the satellite’s lifetime. In the present work we study this problem and introduce an approach in order to keep the orbital eccentricity of the satellite at low values. The whole work was made considering two systems: the 3-body problem, Moon-Earth-satellite, and the 4-body problem, Moon-Earth-Sun-satellite. First, we simulated the systems considering a satellite with initial eccentricity equals to $$0.0001$$ and a range of initial altitudes between 100 km and 5000 km. In such simulations we followed the evolution of the satellite’s eccentricity. We also obtained an empirical expression for the length of time needed to occur the collision with the Moon as a function of the initial altitude. The results found for the 3-body model were not significantly different from those found for the 4-body model. Secondly, using low-thrust propulsion, we introduced a correction of the eccentricity every time it reached the value 0.05. These simulations were made considering a set of different thrust values, from $$0.1 N$$ up to $$0.4 N$$ which can be obtained by using Hall Plasma Thrusters. In each run we measured the length of time, needed to correct the eccentricity value (from $$e=0.04$$ to $$e=0.05$$). From these results we obtained empirical expressions of this time as a function of the initial altitude and as a function of the thrust value.

### MSC:

 70F15 Celestial mechanics
Full Text:

### References:

 [1] B. H. Foing and P. Ehrenfreund, “Journey to the Moon: recent results, science, future robotic and human exploration,” Advances in Space Research, vol. 42, no. 2, pp. 235-237, 2008. [2] T. A. Ely, “Stable constellations of frozen elliptical inclined lunar orbits,” Journal of the Astronautical Sciences, vol. 53, no. 3, pp. 301-316, 2005. [3] T. A. Ely and E. Lieb, “Constellations of elliptical inclined lunar orbits providing polar and global coverage,” Journal of the Astronautical Sciences, vol. 54, no. 1, pp. 53-67, 2006. [4] I. Wytrzyszczak, S. Breiter, and W. Borczyk, “Regular and chaotic motion of high altitude satellites,” Advances in Space Research, vol. 40, no. 1, pp. 134-142, 2007. [5] J. H. Souza, Estudo da dinâmica de partículas em um propulsor a plasma do tipo Hall com ímãs permanentes, M.S. thesis, Universidade de Brasília, Brasília, Brazil, 2006. [6] M. L. Lidov, “Evolution of artificial planetary satellites under the action of gravitational perturbations due to external bodies,” Iskusstviennye Sputniki Zemli, vol. 8, pp. 5-45, 1961 (Russian). [7] Y. Kozai, “Secular perturbations of asteroids with high inclination and eccentricity,” Astrophysical Journal, vol. 67, no. 9, pp. 591-598, 1962. [8] 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. [9] M. A. Vashkov’yak and N. M. Teslenko, “Peculiarities of the orbital evolution of the Jovian satellite J34 (Euporie),” Astronomy Letters, vol. 33, no. 11, pp. 780-787, 2007. [10] J. Kovalevsky, Introduction to Celestial Mechanics, Bureau dês Longitudes, Paris, France, 1967. · Zbl 0189.24502 [11] A. A. Sukhanov, Lectures on Astrodynamics, INPE, São José dos Campos, Brazil, 5th edition, 2007.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.