×

Analytical and semi-analytical treatment of the satellite motion in a resisting medium. (English) Zbl 1235.70013

Summary: The orbital dynamics of an artificial satellite in the Earth’s atmosphere is considered. An analytic first-order atmospheric drag theory is developed using Lagrange’s planetary equations. The short periodic perturbations due to the geopotential of all orbital elements are evaluated. And to construct a second-order analytical theory, the equations of motion become very complicated to be integrated analytically; thus we are forced to integrate them numerically using the method of Runge-Kutta of fourth order. The validity of the theory is checked on the already decayed Indian satellite ROHINI where its data are available.

MSC:

70B15 Kinematics of mechanisms and robots
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] I. Newton, Philosophiae Naturalis Principia Mathematica, Book 2, Section 4, University of California Press, Berkeley, Calif, USA, 1934, English translation by F. Cajori, Newton.s Principia. · Zbl 0732.01044
[2] P. S. Laplace, Traité de Mécanique Céleste, Tom 4, Par 2 Courcier, 1805.
[3] D. Brouwer, “Review of celestial mechanics,” Annual Review of Astronomy and Astrophysics, vol. 1, pp. 219-234, 1963.
[4] D. G. King-Hele, Theory of Satellite Orbits in an Atmosphere, Butterworth, London, UK, 1964. · Zbl 0125.26502
[5] D. G. King-Hele, A Tapestry of Orbits, Cambridge University Press, Cambridge, UK, 1992.
[6] D. Brouwer and G. M. Clemence, Methods of Celestial Mechanics, Academic Press, New York, NY, USA, 1961. · Zbl 0132.23506
[7] V. A. Chobotov, Ed., Orbital Mechanics, AIAA, Washington, DC, USA, 1991. · Zbl 0925.70317
[8] D. Boccaletti and G. Pucacco, Theory of Orbits. Vol. 1, Springer, Berlin, Germany, 1996. · Zbl 1194.93101
[9] P. D. Van Kamp, Principles of Astronomy, Freemann & Co., San Francisco, Calif, USA, 1967.
[10] R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, AIAA, Reston, Va, usa, 1999. · Zbl 0972.70001
[11] S. Herrick, Astrodynamics. Vol. 2, Van Nostrand Reinhold, London, uk, 1972. · Zbl 0276.70026
[12] D. A. Vallado, Fundamentals of Astrodynamics and Applications, Springer, Berlin, Germany, 3rd edition, 2007. · Zbl 1191.70002
[13] J. Meeus, Astronomical Algorithms, Willmann-Bell, Richmond, Va, USA, 1991.
[14] O. Montenbruck and E. Gill, Satellite Orbits: Models, Methods and Applications, Springer, Heidelberg, Germany, 2000. · Zbl 0949.70001
[15] G. Seeber, Satellite Geodesy: Foundations, Methods, and Applications, Walter de Gruyter, Berlin, Germany, 2003.
[16] G. Xu, GPS-Theory, Algorithms and Applications, Springer, Heidelberg, Germany, 2nd edition, 2007.
[17] G. Xu, Orbits, Springer, Heidelberg, Germany, 2008. · Zbl 1183.70001
[18] G. Xu, T. H. Xu, T. K. Yeh, and W. Chen, “Analytic solution of satellite orbit disturbed by atmospheric drag,” Monthly Notices of the Royal Astronomical Society, vol. 410, no. 1, pp. 654-662, 2010.
[19] C. Cui, Deutsche Geodätische Kommission bei der Bayerischen Akademie der Wissenschaften, Reihe C, 357, 1990.
[20] M. Schneider and C. F. Cui, Deutsche Geodätische Kommission bei der Bayerischen Akademie der Wissenschaften, Reihe A: Theoretische Geodäsie, 121, 2005.
[21] S. M. Kudryavtsev, “Long-term harmonic development of lunar ephemeris,” Astronomy & Astrophysics, vol. 471, pp. 1069-1075, 2007.
[22] J. Licandro, A. Alvarez-Candal, J. De León, N. Pinilla-Alonso, D. Lazzaro, and H. Campins, “Spectral properties of asteroids in cometary orbits,” Astronomy and Astrophysics, vol. 481, no. 3, pp. 861-877, 2008.
[23] A. Pál, “An analytical solution for Kepler’s problem,” Monthly Notices of the Royal Astronomical Society, vol. 396, no. 3, pp. 1737-1742, 2009.
[24] D. Lynden-Bell, “Analytic orbits in any central potential,” Monthly Notices of the Royal Astronomical Society, vol. 402, no. 3, pp. 1937-1941, 2010.
[25] W. Torge, Geodesy, Walter de Gruyter, Berlin, Germany, 1991. · Zbl 1244.86003
[26] J. Desmars, S. Arlot, J.-E. Arlot, V. Lainey, and A. Vienne, “Estimating the accuracy of satellite ephemerides using the bootstrap method,” Astronomy and Astrophysics, vol. 499, no. 1, pp. 321-330, 2009. · Zbl 1177.85008
[27] J. R. Touma, S. Tremaine, and M. V. Kazandjian, “Gauss’s method for secular dynamics, softened,” Monthly Notices of the Royal Astronomical Society, vol. 394, no. 2, pp. 1085-1108, 2009.
[28] A. Bezdeka and D. Vokrouhlicky, “Semianalytic theory of motion for close-Earth spherical satellites includingdragand gravitational perturbations,” Planetary and Space Science, vol. 52, pp. 1233-1249, 2004.
[29] G. Xu, X. Tianhe, W. Chen, and T.-K. Yeh, “Analytical solution of a satellite orbit disturbed by atmospheric drag,” Monthly Notices of the Royal Astronomical Society, vol. 410, no. 1, pp. 654-662, 2011.
[30] L. Sehnal, “Dynamics and Astrometry of natural and artificial celestial bodies,” in Proceedings of the Conference on Astrometry and Celestial Mechanics, pp. 325-332, Poznan, Poland, 1994.
[31] P. M. Fitzpatrick, Principles Of Celestial Mechancics, Academic Press, London, UK, 1970. · Zbl 0214.23804
[32] F.A. Abd El- Salam and S. Abd El-Bar, “Computation of the different errors in the ballistic missiles range,” Journal of the Applied Mathematics, vol. 2011, Article ID 349737, 16 pages, 2011. · Zbl 1238.70008
[33] R. K. Sharma and L. Mani, “Study of RS-1 orbital decay with KS differential equations,” Indian Journal of Pure and Applied Mathematics, vol. 6, pp. 833-842, 1985.
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.