Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite

Hany R. Dwidar

doi:doi:10.11648/j.ajam.20140203.13

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Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite

Hany R. Dwidar

Published in American Journal of Applied Mathematics (Volume 2, Issue 3)

Received: 19 May 2014 Accepted: 7 June 2014 Published: 20 June 2014

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Abstract

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.

Published in	American Journal of Applied Mathematics (Volume 2, Issue 3)
DOI	10.11648/j.ajam.20140203.13
Page(s)	85-91
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Astrodynamics, Satellite Orbit Determination, Power Series, Numerical integration

References

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[2]	R. Broucke, "Solution of the N-Body Problem with Recurrent Power Series," Celestial Mechanics, vol. 4, pp. 110-115, Sep. 1971.
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[4]	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.
[5]	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.
[6]	T. Fukushima, "Numerical Comparison of Two-Body Regularizations," The Astronomical Journal, vol. 133, no. 6, pp. 2815-2824, Jun. 2007.
[7]	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.
[8]	E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I (Nonstiff Problems), Third ed., Springer-Verlag, 2008.
[9]	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.
[10]	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.
[11]	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.
[12]	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.
[13]	M. Silver, "A short derivation of the Sperling-Burdet equations," Celestial Mechanics, vol. 11, no. 1, pp. 39-41, Feb. 1975.
[14]	G. Sitarski, "Recurrent power series integration of the equations of comet's motion," Acta Astrono., vol. 29, no. 3, pp. 401-411, 1979.
[15]	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.
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[18]	D. A. Vallado, Fundamentals of Astrodynamics and Applications, 3rd ed., D. A. Vallado, Ed., Microcosm Press/Springer, 2007.

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APA Style

Hany R. Dwidar. (2014). Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite. American Journal of Applied Mathematics, 2(3), 85-91. https://doi.org/10.11648/j.ajam.20140203.13

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ACS Style

Hany R. Dwidar. Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite. Am. J. Appl. Math. 2014, 2(3), 85-91. doi: 10.11648/j.ajam.20140203.13

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AMA Style

Hany R. Dwidar. Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite. Am J Appl Math. 2014;2(3):85-91. doi: 10.11648/j.ajam.20140203.13

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@article{10.11648/j.ajam.20140203.13,
  author = {Hany R. Dwidar},
  title = {Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite},
  journal = {American Journal of Applied Mathematics},
  volume = {2},
  number = {3},
  pages = {85-91},
  doi = {10.11648/j.ajam.20140203.13},
  url = {https://doi.org/10.11648/j.ajam.20140203.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20140203.13},
  abstract = {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.},
 year = {2014}
}

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TY  - JOUR
T1  - Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite
AU  - Hany R. Dwidar
Y1  - 2014/06/20
PY  - 2014
N1  - https://doi.org/10.11648/j.ajam.20140203.13
DO  - 10.11648/j.ajam.20140203.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 85
EP  - 91
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20140203.13
AB  - 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.
VL  - 2
IS  - 3
ER  -

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Author Information

Hany R. Dwidar

Astronomy, Meteorology and Space Science Dept., Faculty of Science - Cairo University, Giza, Egypt

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APA Style

Hany R. Dwidar. (2014). Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite. American Journal of Applied Mathematics, 2(3), 85-91. https://doi.org/10.11648/j.ajam.20140203.13

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ACS Style

Hany R. Dwidar. Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite. Am. J. Appl. Math. 2014, 2(3), 85-91. doi: 10.11648/j.ajam.20140203.13

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AMA Style

Hany R. Dwidar. Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite. Am J Appl Math. 2014;2(3):85-91. doi: 10.11648/j.ajam.20140203.13

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@article{10.11648/j.ajam.20140203.13,
  author = {Hany R. Dwidar},
  title = {Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite},
  journal = {American Journal of Applied Mathematics},
  volume = {2},
  number = {3},
  pages = {85-91},
  doi = {10.11648/j.ajam.20140203.13},
  url = {https://doi.org/10.11648/j.ajam.20140203.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20140203.13},
  abstract = {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.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Semi-Analytical and Numerical Solution of Regularized Burdet Equations to Predict the Motion of an Artificial Satellite
AU  - Hany R. Dwidar
Y1  - 2014/06/20
PY  - 2014
N1  - https://doi.org/10.11648/j.ajam.20140203.13
DO  - 10.11648/j.ajam.20140203.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 85
EP  - 91
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20140203.13
AB  - 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.
VL  - 2
IS  - 3
ER  -

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