Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite
American Journal of Applied Mathematics
Volume 2, Issue 3, June 2014, Pages: 85-91
Received: May 19, 2014;
Accepted: Jun. 7, 2014;
Published: Jun. 20, 2014
Views 2919 Downloads 125
Hany R. Dwidar, Astronomy, Meteorology and Space Science Dept., Faculty of Science - Cairo University, Giza, Egypt
In this paper, literal analytical solution in power series forms which is one of the semi-analytical solution, are developed for the regularized Burdet equations to estimate the motion of an artificial satellite under the influence of J2-Earth’s gravitational field. Also a numerical solution of the regularized Burdet equations is applied using eighth order Dormand-Prince Rung-Kutta method. Comparison between the power series solution and the numerical solution applied to high eccentric frozen satellite orbit is also given and showed excellent agreement.
Hany R. Dwidar,
Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite, American Journal of Applied Mathematics.
Vol. 2, No. 3,
2014, pp. 85-91.
I. Aparicio and L. Floria, "A BF-regularizationof a nonstationary two-body problem under the Maneff perturbing potential," Extracta Mathematicae, vol. 12, no. 3, 1997.
R. Broucke, "Solution of the N-Body Problem with Recurrent Power Series," Celestial Mechanics, vol. 4, pp. 110-115, Sep. 1971.
C. A. Burdet, "Regularization of the two body problem," Zeitschrift für angewandte Mathematik und Physik, vol. 18, no. 3, pp. 434-438, 1967.
J. R. Dormand and P. J. Prince, "A Family of Embedded Runge-Kutta Formulae," Journal of Computational and Applied Mathematics, vol. 6, p. 19–26, 1980.
W. Flury and G. Janin, "Accurate integration of geostationary orbits with Burdet's focal elements," Astrophysics and Space Science, vol. 36, pp. 495-503, Sep. 1975.
T. Fukushima, "Numerical Comparison of Two-Body Regularizations," The Astronomical Journal, vol. 133, no. 6, pp. 2815-2824, Jun. 2007.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Seventh Edition ed., A. Jeffrey and D. Zwillinger, Eds., New York: Academic Press Inc., 2007.
E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I (Nonstiff Problems), Third ed., Springer-Verlag, 2008.
D. J. Jezewski, "A Comparative Study of Newtonian, Kustaanheimo/Stiefel, and Sperling/Burdet Optimal Trajectories," Celestial Mechanics, vol. 12, no. 3, pp. 297-315, Nov. 1975.
W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, NUMERICAL RECIPES The Art of Scientific Computing, Third Edition ed., Cambridge: CAMBRIDGE UNIVERSITY PRESS, 2007.
M. A. Sharaf and M. E. Awad, "Prediction of trajectories in Earth's gravitational field with axial symmetry.," Proc. Math. Phys. Soc. Egypt, vol. 60, 1985.
M. A. Sharaf, M. R. Arafah and M. E. Awad, "Prediction of satellites in earth's gravitational field with axial symmetry using Burdet's regularized theory," Earth Moon and Planets, vol. 38, pp. 21-36, May 1987.
M. Silver, "A short derivation of the Sperling-Burdet equations," Celestial Mechanics, vol. 11, no. 1, pp. 39-41, Feb. 1975.
G. Sitarski, "Recurrent power series integration of the equations of comet's motion," Acta Astrono., vol. 29, no. 3, pp. 401-411, 1979.
G. Sitarski, "Recurrent-power-series integration of equations of comet's motion including the nongravitational terms in Marsden's form," Acta Astrono., vol. 34, no. 1, pp. 53-63, 1984.
G. Sitarski, "Solution of the ADONIS problem," Acta Astrono., vol. 29, no. 3, pp. 413-424, 1979.
H. Sperling, "Computation of Keplerian Conic Sections," American Rocket Society Journal, vol. 31, no. 5, pp. 660-661, 1961.
D. A. Vallado, Fundamentals of Astrodynamics and Applications, 3rd ed., D. A. Vallado, Ed., Microcosm Press/Springer, 2007.