Advertisement

Semi-analytical model for third-body perturbations including the inclination and eccentricity of the perturbing body

  • Tao NieEmail author
  • Pini Gurfil
  • Shijie Zhang
Original Article

Abstract

A general third-body perturbation problem, considering the perturbing body in an elliptic and inclined orbit, is investigated by using a semi-analytical theory. Previous works have contributed to deriving the averaged third-body-perturbed dynamics, but did not provide a transformation between osculating and mean elements in the general case. In this paper, an analytical transformation between osculating and mean elements is developed explicitly using von Zeipel’s method, in addition to developing the long-term dynamical equations. The resulting dynamical model is improved, because the disturbing function is averaged as a whole, instead of separating the disturbing function into many terms and averaging them independently. The simulation results indicate that the new singly averaged dynamical model behaves much better than the doubly averaged dynamics in propagating the long-term evolution of the orbital elements. Moreover, it is shown that the perturbing body’s inclination and eccentricity have a vital influence on the evolution of the satellite’s inclination and eccentricity.

Keywords

Third-body perturbation von Zeipel’s method Periodic corrections 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest associated with the work reported in this article.

Supplementary material

10569_2019_9905_MOESM1_ESM.pdf (183 kb)
Supplementary material 1 (pdf 182 KB)

References

  1. Bertachini de Almeida Prado, A.F.: Third-body perturbation in orbits around natural satellites. J. Guidance Control Dyn. 26(1), 33–40 (2003).  https://doi.org/10.2514/2.5042 ADSCrossRefGoogle Scholar
  2. Battin, R.H.: An introduction to the mathematics and methods of astrodynamics, revised edition. American Institute of Aeronautics and Astronautics (1999).  https://doi.org/10.2514/4.861543
  3. Blitzer, L.: Lunar-solar perturbations of an Earth satellite. Am. J. Phys. 27(9), 634–645 (1959).  https://doi.org/10.1119/1.1934947 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. Broucke, R.A.: Long-term third-body effects via double averaging. J. Guidance Control Dyn. 26(1), 27–32 (2003).  https://doi.org/10.2514/2.5041 ADSCrossRefGoogle Scholar
  5. Brouwer, D.: Solution of the problem of artificial satellite theory without drag. Astron. J. 64, 378–397 (1959).  https://doi.org/10.1086/107958 ADSMathSciNetCrossRefGoogle Scholar
  6. Cook, G.: Luni-solar perturbations of the orbit of an Earth satellite. Geophys. J. R. Astron. Soc. 6(3), 271–291 (1962).  https://doi.org/10.1111/j.1365-246X.1962.tb00351.x ADSCrossRefzbMATHGoogle Scholar
  7. De Saedeleer, B.: Analytical theory of a lunar artificial satellite with third body perturbations. Celest. Mech. Dyn. Astron. 95(1–4), 407–423 (2006).  https://doi.org/10.1007/s10569-006-9029-6 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. Deprit, A.: Canonical transformations depending on a small parameter. Celest. Mech. 1(1), 12–30 (1969)ADSMathSciNetCrossRefGoogle Scholar
  9. Domingos, RdC, de Moraes, R.V., de Almeida Prado, A.: Third-body perturbation in the case of elliptic orbits for the disturbing body. Math. Probl. Eng. 2008, 1–14 (2008).  https://doi.org/10.1155/2008/763654 MathSciNetCrossRefzbMATHGoogle Scholar
  10. Folkner, W.M., Williams, J.G., Boggs, D.H.: The planetary and lunar ephemeris DE421. In: Technical Report IOM 343R-08-003, Jet Propulsion Laboratory (2008)Google Scholar
  11. Giacaglia, G.E., Murphy, J.P., Felsentreger, T.L.: A semi-analytic theory for the motion of a lunar satellite. Celest. Mech. Dyn. Astron. 3(1), 3–66 (1970).  https://doi.org/10.1007/BF01230432 CrossRefzbMATHGoogle Scholar
  12. Hori, Gi: Theory of general perturbation with unspecified canonical variable. Publ. Astron. Soc. Jpn. 18, 287 (1966)ADSGoogle Scholar
  13. Jupp, A.: A comparison of the Bohlin-Von Zeipel and Bohlin-Lie series methods in resonant systems. Celest. Mech. 26(4), 413–422 (1982)ADSMathSciNetCrossRefGoogle Scholar
  14. Kozai, Y.: The motion of a close Earth satellite. Astron. J. 64, 367–377 (1959a).  https://doi.org/10.1086/107957 ADSMathSciNetCrossRefGoogle Scholar
  15. Kozai, Y.: On the effects of the Sun and the Moon upon the motion of a close Earth satellite. Smithsonian Astrophysical Observatory, Special Report No. 22 pp. 7–10 (1959b)Google Scholar
  16. Kozai, Y.: Second-order solution of artificial satellite theory without air drag. Astron. J. 67, 446 (1962a)ADSMathSciNetCrossRefGoogle Scholar
  17. Kozai, Y.: Secular perturbations of asteroids with high inclination and eccentricity. Astron. J. 67, 591 (1962b).  https://doi.org/10.1086/108790 ADSMathSciNetCrossRefGoogle Scholar
  18. Kozai, Y.: Motion of a lunar orbiter. Publ. Astron. Soc. Jpn. 15(3), 301–312 (1963)ADSGoogle Scholar
  19. Lara, M., Palacián, J.F.: Hill problem analytical theory to the order four: application to the computation of frozen orbits around planetary satellites. Math. Probl. Eng. 2009, 1–18 (2009).  https://doi.org/10.1155/2009/753653 CrossRefzbMATHGoogle Scholar
  20. Liu, X., Baoyin, H., Ma, X.: Long-term perturbations due to a disturbing body in elliptic inclined orbit. Astrophys. Space Sci. 339(2), 295–304 (2012).  https://doi.org/10.1007/s10509-012-1015-8 ADSCrossRefGoogle Scholar
  21. Lyddane, R.: Small eccentricities or inclinations in the Brouwer theory of the artificial satellite. Astron. J. 68, 555–558 (1963).  https://doi.org/10.1086/109179 ADSMathSciNetCrossRefGoogle Scholar
  22. Musen, P., Bailie, A., Upton, E.: Development of the lunar and solar perturbations in the motion of an artificial satellite. In: Technical Report, NASA-TN-D-494 (1961)Google Scholar
  23. Nie, T., Gurfil, P.: Lunar frozen orbits revisited. Celest. Mech. Dyn. Astron. 130(10), 61 (2018).  https://doi.org/10.1007/s10569-018-9858-0 ADSMathSciNetCrossRefGoogle Scholar
  24. Nie, T., Gurfil, P., Zhang, S.: Bounded lunar relative orbits. Acta Astronaut. 157(4), 500–516 (2019).  https://doi.org/10.1016/j.actaastro.2019.01.018 ADSCrossRefGoogle Scholar
  25. Roscoe, T.W.C., Vadali, S.R., Alfriend, K.T.: Third-body perturbation effects on satellite formations. J. Astron. Sci. 60(3–4), 408–433 (2015).  https://doi.org/10.1007/s40295-015-0057-x CrossRefGoogle Scholar
  26. Schaub, H., Alfriend, K.T.: \(J_2\) invariant relative orbits for spacecraft formations. Celest. Mech. Dyn. Astron. 79(2), 77–95 (2001).  https://doi.org/10.1023/A:1011161811472 ADSCrossRefzbMATHGoogle Scholar
  27. Tresaco, E., Carvalho, J.P.S., Prado, A.F.B.A., Elipe, A., de Moraes, R.V.: Averaged model to study long-term dynamics of a probe about Mercury. Celest. Mech. Dyn. Astron. 130(2), 9 (2018).  https://doi.org/10.1007/s10569-017-9801-9 ADSMathSciNetCrossRefGoogle Scholar
  28. Vallado, D.A.: Fundamentals of Astrodynamics and Applications, vol. 12. Springer, New York (2001)zbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Research Center of Satellite TechnologyHarbin Institute of TechnologyHarbinChina
  2. 2.Faculty of Aerospace EngineeringTechnion - Israel Institute of TechnologyHaifaIsrael

Personalised recommendations