Invariant Relative Orbits Taking into Account Third-Body Perturbation

Affiliation(s)

Department of Astronomy, Faculty of Science, Cairo University, Cairo, Egypt.

Observatoire de la C?te d’Azur, Grasse, France.

Department of Astronomy, Faculty of Science, Cairo University, Cairo, Egypt.

Observatoire de la C?te d’Azur, Grasse, France.

ABSTRACT

For a satellite in an orbit of more than 1600 km in altitude, the effects of Sun and Moon on the orbit can’t be negligible. Working with mean orbital elements, the secular drift of the longitude of the ascending node and the sum of the argu-ment of perigee and mean anomaly are set equal between two neighboring orbits to negate the separation over time due to the potential of the Earth and the third body effect. The expressions for the second order conditions that guaran-tee that the drift rates of two neighboring orbits are equal on the average are derived. To this end, the Hamiltonian was developed. The expressions for the non-vanishing time rate of change of canonical elements are obtained.

For a satellite in an orbit of more than 1600 km in altitude, the effects of Sun and Moon on the orbit can’t be negligible. Working with mean orbital elements, the secular drift of the longitude of the ascending node and the sum of the argu-ment of perigee and mean anomaly are set equal between two neighboring orbits to negate the separation over time due to the potential of the Earth and the third body effect. The expressions for the second order conditions that guaran-tee that the drift rates of two neighboring orbits are equal on the average are derived. To this end, the Hamiltonian was developed. The expressions for the non-vanishing time rate of change of canonical elements are obtained.

Cite this paper

W. Rahoma and G. Metris, "Invariant Relative Orbits Taking into Account Third-Body Perturbation,"*Applied Mathematics*, Vol. 3 No. 2, 2012, pp. 113-120. doi: 10.4236/am.2012.32018.

W. Rahoma and G. Metris, "Invariant Relative Orbits Taking into Account Third-Body Perturbation,"

References

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[8] R. C. Domingos, R. V. deMoraes and A. F. Prado, “Third-Body Perturbation in the Case of Elliptic Orbits for the Disturbing Body,” Mathematical Problems in Engineering, Vol. 2008, 2008, p. 14. doi:10.1155/2008/763654

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[1] H. Schaub and K. Alfriend, “J2 Invariant Relative Orbits for Spacecraft Formations,” Celestial Mechanics and Dynamical Astronomy, Vol. 79, No. 2, 2001, pp. 77-95. doi:10.1023/A:1011161811472

[2] Y. Zhang and J. Dai, “Satellite Formation Flying with J2 Perturbation,” Journal of National University of Defense Technology, Vol. 24, No. 2, 2002, pp. 6-10.

[3] W. S. Koon and J. E. Marsden, “J2 Dynamics and Formation Flight,” Proceedings of AIAA Guidance, Navigation, and Control Conference, Montreal, August 2001, p. 4090.

[4] X. Li and J. Li, “Study on Relative Orbital Configuration in Satellite Formation Flying,” Acta Mechanica Sinica, Vol. 21, No. 1, 2005, pp. 87-94. doi:10.1007/s10409-004-0009-3

[5] X. Meng, J. Li and Y. Gao, “J2 Perturbation Analysis of Relative Orbits in Satellite Formation Flying,” Acta Mechanica Sinica, Vol. 38, No. 1, 2006, pp. 89-96.

[6] J. D. Biggs and V. M. Becerra, “A Search for Invariant Relative Satellite Motion,” 4th Workshop on Satellite Constellations and Formation Flying, Sao Jose dos Campos, 2005, pp. 203-213.

[7] F. A. Abd El-Salam, I. A. El-Tohamy, M. K. Ahmed, W. A. Rahoma and M. A. Rassem, “Invariant Relative Orbits for Satellite Constellations: A Second Order Theory,” Applied Mathematics and Computation, Vol. 181, No. 1, 2006, pp. 6-20. doi:10.1016/j.amc.2006.01.004

[8] R. C. Domingos, R. V. deMoraes and A. F. Prado, “Third-Body Perturbation in the Case of Elliptic Orbits for the Disturbing Body,” Mathematical Problems in Engineering, Vol. 2008, 2008, p. 14. doi:10.1155/2008/763654

[9] A. A. Kamel, “Expansion Formulae in Canonical Transformations Depending on a Small Parameter,” Celestial Mechanics and Dynamical Astronomy, Vol. 1, No. 2, 1969, pp. 190-199. doi:10.1007/BF01228838