Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2013, Article ID 570127, 11 pages
http://dx.doi.org/10.1155/2013/570127
Research Article

Averaging Tesseral Effects: Closed Form Relegation versus Expansions of Elliptic Motion

1C/Columnas de Hercules 1, San Fernando, Spain
2Universidad de La Rioja, Logroño, Spain

Received 7 February 2013; Accepted 26 March 2013

Academic Editor: Antonio F. Bertachini A. Prado

Copyright © 2013 Martin Lara et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Deprit, “The main problem in the theory of artificial satellites to order four,” Journal of Guidance, Control and Dynamics, vol. 4, no. 2, pp. 201–206, 1981. View at Google Scholar
  2. S. Coffey and A. Deprit, “Third-order solution to the main problem in satellite theory,” Journal of Guidance and Control, vol. 5, no. 4, pp. 366–371, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. L. M. Healy, “The main problem in satellite theory revisited,” Celestial Mechanics & Dynamical Astronomy. An International Journal of Space Dynamics, vol. 76, no. 2, pp. 79–120, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Deprit, “The elimination of the parallax in satellite theory,” Celestial Mechanics, vol. 24, no. 2, pp. 111–153, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. Deprit, J. Palacián, and E. Deprit, “The relegation algorithm,” Celestial Mechanics & Dynamical Astronomy, vol. 79, no. 3, pp. 157–182, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. F. Palacián, “An analytical solution for artificial satellites at low altitudes,” in Dynamics and Astrometry of Natural and Artificial Celestial Bodies, K. Kurzyńska, F. Barlier, P. K. Seidelmann, and I. Wytrzyszczak, Eds., pp. 365–370, 1994. View at Google Scholar
  7. J. F. Palacián, J. F. San-Juan, and P. Yanguas, “Analytical theory for the spot satellite,” Advances in the Astronautical Sciences, vol. 95, pp. 375–382, 1996. View at Google Scholar
  8. M. Lara, J. F. San-Juan, and L. López-Ochoa, “Precise analytical computation of frozen-eccentricity, low Earth orbits in a tesseral potential,” Mathematical Problems in Engineering, vol. 2013, Article ID 191384, 13 pages, 2013. View at Publisher · View at Google Scholar
  9. A. M. Segerman and S. L. Coffey, “An analytical theory for tesseral gravitational harmonics,” Celestial Mechanics and Dynamical Astronomy, vol. 76, no. 3, pp. 139–156, 2000. View at Google Scholar · View at Scopus
  10. J. F. San, A. Abad, M. Lara, and D. J. Scheeres, “First-order analytical solution for spacecraft motion about (433) Eros,” Journal of Guidance, Control, and Dynamics, vol. 27, no. 2, pp. 290–293, 2004. View at Google Scholar · View at Scopus
  11. D. Brouwer and G. M. Clemence, Methods of Celestial Mechanics, Academic Press, New York, NY, USA, 1961. View at MathSciNet
  12. W. M. Kaula, Theory of Satellite Geodesy. Applications of Satellites to Geodesy, Dover, Mineola, NY, USA, 1966.
  13. N. X. Vinh, “Recurrence formulae for the Hansen’s developments,” Celestial Mecshanics, vol. 2, no. 1, pp. 64–76, 1970. View at Google Scholar
  14. E. Wnuk, “Tesseral harmonic perturbations for high order and degree harmonics,” Celestial Mechanics, vol. 44, no. 1-2, pp. 179–191, 1988. View at Google Scholar
  15. F. Delhaise and J. Henrard, “The problem of critical inclination combined with a resonance in mean motion in artificial satellite theory,” Celestial Mechanics & Dynamical Astronomy, vol. 55, no. 3, pp. 261–280, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  16. L. D. D. Ferreira and R. Vilhena de Moraes, “GPS satellites orbits: resonance,” Mathematical Problems in Engineering, vol. 2009, Article ID 347835, 12 pages, 2009. View at Publisher · View at Google Scholar
  17. J. C. Sampaio, A. G. S. Neto, S. S. Fernandes, R. Vilhena de Moraes, and M. O. Terra, “Artificial satellites orbits in 2:1 resonance: GPS constellation,” Acta Astronautica, vol. 81, pp. 623–634, 2012. View at Google Scholar
  18. J. C. Sampaio, R. Vilhena de Moraes, and S. D. S. Fernandes, “The orbital dynamics of synchronous satellites: irregular motions in the 2:1 resonance,” Mathematical Problems in Engineering, Article ID 405870, 22 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. M. Lara, J. F. San-Juan, Z. J. Folcik, and P. J. Cefola, “Deep resonant GPS-dynamics due to the geopotential,” Journal of the Astronautical Sciences, vol. 58, no. 4, pp. 661–676, 2012. View at Google Scholar
  20. S. L. Coffey, H. L. Neal, A. M. Segerman, and J. J. Travisano, “An analytic orbit propagation program for satellite catalog maintenance,” Advances in the Astronautical Sciences, vol. 90, no. 2, pp. 1869–1892, 1995. View at Google Scholar
  21. W. D. McClain, “A recursively formulated first-order semianalytic artificial satellite theory based on the generalized method of averaging, volume 1: the generalized method of averaging applied to the artificial satellite problem,” Computer Sciences Corporation CSC/TR-77/6010, 1977. View at Google Scholar
  22. A. R. Golikov, “Numeric-analytical theory of the motion of artificial satellites of celestial bodies,” Preprint No. 70, Keldysh Institute of Applied Mathematics; USSR Academy of Sciences, 1990. View at Google Scholar
  23. R. J. Proulx and W. D. McClain, “Series representations and rational approximations for Hansen coefficients,” Journal of Guidance, Control, and Dynamics, vol. 11, no. 4, pp. 313–319, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  24. B. Garfinkel, “The disturbing function for an artificial satellite,” The Astronomical Journal, vol. 70, no. 9, pp. 699–704, 1965. View at Google Scholar
  25. B. Garfinkel, “Tesseral harmonic perturbations of an artificial satellite,” The Astronomical Journal, vol. 70, no. 10, pp. 784–786, 1965. View at Publisher · View at Google Scholar · View at MathSciNet
  26. S. Coffey and K. T. Alfriend, “Short period elimination for the tesseral harmonics,” Advances in the Astronautical Sciences, vol. 46, no. 1, pp. 87–101, 1982. View at Google Scholar
  27. A. Deprit and S. Ferrer, “Simplifications in the theory of artificial satellites,” American Astronautical Society, vol. 37, no. 4, pp. 451–463, 1989. View at Google Scholar · View at MathSciNet
  28. M. Lara, J. F. San-Juan, and L. M. Lopez-Ochoa, “Efficient semi-analytic integration of GNSS orbits under tesseral effects,” in Proceedings of the 7th International Workshop on Satellite Constellations and Formation Flying, Paper IWSCFF-2013-04-02, Lisbon, Portugal, March 2013.
  29. M. Irigoyen and C. Simó, “Non integrability of the J2 problem,” Celestial Mechanics and Dynamical Astronomy, vol. 55, no. 3, pp. 281–287, 1993. View at Google Scholar
  30. A. Celletti and P. Negrini, “Non-integrability of the problem of motion around an oblate planet,” Celestial Mechanics & Dynamical Astronomy, vol. 61, no. 3, pp. 253–260, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. D. J. Jezewski, “An analytic solution for the J2 perturbed equatorial orbit,” Celestial Mechanics, vol. 30, no. 4, pp. 363–371, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. A. Deprit, “Canonical transformations depending on a small parameter,” Celestial Mechanics, vol. 1, pp. 12–30, 1969. View at Publisher · View at Google Scholar · View at MathSciNet
  33. A. Deprit and A. Rom, “Lindstedt’s series on a computer,” The Astronomical Journal, vol. 73, no. 3, pp. 210–213, 1968. View at Google Scholar
  34. G. R. Hintz, “Survey of orbit element sets,” Journal of Guidance, Control and Dynamics, vol. 31, no. 3, pp. 785–790, 2008. View at Google Scholar
  35. E. Brumberg and T. Fukushima, “Expansions of elliptic motion based on elliptic function theory,” Celestial Mechanics & Dynamical Astronomy, vol. 60, no. 1, pp. 69–89, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. G. Metris, P. Exertier, Y. Boudon, and F. Barlier, “Long period variations of the motion of a satellite due to non-resonant tesseral harmonics of a gravity potential,” Celestial Mechanics & Dynamical Astronomy, vol. 57, no. 1-2, pp. 175–188, 1993. View at Publisher · View at Google Scholar · View at Scopus
  37. A. Deprit, “Delaunay normalisations,” Celestial Mechanics, vol. 26, no. 1, pp. 9–21, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. K. Aksnes, “A note on ‘The main problem of satellite theory for small eccentricities, by A. Deprit and A. Rom, 1970’,” Celestial Mechanics, vol. 4, no. 1, pp. 119–121, 1971. View at Publisher · View at Google Scholar · View at Scopus
  39. A. Deprit and A. Rom, “The main problem of artificial satellite theory for small and moderate eccentricities,” Celestial Mechanics, vol. 2, no. 2, pp. 166–206, 1970. View at Publisher · View at Google Scholar · View at Scopus