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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 94
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero
Paper 37

Numerical Methods for Orbit Determination of Artificial Satellites

V. Arroyo, A. Cordero and J.R. Torregrosa

Institute for Multidisciplinary Mathematics, Universidad Politécnica de Valencia, Spain

Full Bibliographic Reference for this paper
V. Arroyo, A. Cordero, J.R. Torregrosa, "Numerical Methods for Orbit Determination of Artificial Satellites", in B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru, M.L. Romero, (Editors), "Proceedings of the Seventh International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 37, 2010. doi:10.4203/ccp.94.37
Keywords: orbit determination, Gauss method, nonlinear systems, Newton method, iterative function, convergence, efficiency.

Summary
The classical Gauss method can be used to determine the orbits of artificial satellites, from two position vectors and times, by solving a nonlinear equation using the fixed point method [1]. We aim to introduce more efficient techniques to solve this specific problem, which make it possible to increase the order of convergence of the process, by means of solving the nonlinear system of Gauss equations with different higher order iterative methods.

Orbit determination consists in obtaining, as far as possible, the most accurate description of a celestial body orbit by means of several observations [1,2,3]. This orbit will be expressed by its orbital elements, which will let object's future positions be determined. The first preliminary orbit will be determined using the premise of the two body problem, not taking into account any force other than mutual gravitational attraction between both bodies.

After introducing the reference coordinate systems where the orbit will be determined, and the Gauss method equations and processes, different schemes to solve the problem are analyzed, as the classical method and the higher order Newton's method variants. As the results show, it is possible to reduce the number of iterations to find a solution with the Newton method variants, obtaining even greater accuracy than the classical scheme, allowing the use of more restrictive tolerances without increasing the number of iterations. Some limitations of the classical scheme are still present in these variants, such as the inability to distinguish between direct and retrograde orbits, or the spread limit in observations, although using higher order schemes does not increase the number of iterations as the spread grows, as occurs with the classical scheme.

References
1
P.R. Escobal, "Methods of Orbit Determination", Robert E. Krieger Publishing Company, 1975.
2
J.M.A. Danby, "Fundamentals of Celestial Mechanics", The MacMillan Company, 1962.
3
M.J. Sevilla, "Mecánica Celeste Clásica", Instituto de Astronomía y Geodesia, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 1989.
4
J.F. Traub, "Iterative methods for the solution of equations", Chelsea Publishing Company, New York, 1982.

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