Cosmic Research

, Volume 51, Issue 4, pp 289–303 | Cite as

Closed form perturbation solution of a fast rotating triaxial satellite under gravity-gradient torque

  • M. LaraEmail author
  • S. Ferrer


The attitude dynamics of a fast rotating triaxial satellite under gravity-gradient is revisited. The essentially unique reduction of the Euler-Poinsot Hamiltonian, which can be performed in different sets of variables, provides a suitable set of canonical variables that expedites the perturbation approach. Two canonical transformations reduce the perturbed problem to its secular terms. The secular Hamiltonian and the transformation equations of the averaging are computed in closed form of the triaxiality coefficient, thus being valid for any triaxial body. The solution depends on Jacobi elliptic functions and integrals, and applies to non-resonant rotations under the assumption that the rotation rate is much higher than the orbital or precessional motion.


Canonical Variable Cosmic Research Canonical Transformation Gravity Gradient Hamilton Jacobi Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andoyer, M.H., Cours de Mécanique Céleste, Paris: Gauthier-Villars et cie, vol. 1, p. 57.Google Scholar
  2. 2.
    Arnold, V.I., Les Méthodes Mathématiques de la Mécanique Classique, Mir, 1976.zbMATHGoogle Scholar
  3. 3.
    Barkin, Yu.V., Unperturbed chandler motion and perturbation theory of the rotation motion of deformable celestial bodies, Astron. Astrophys. Trans., 1998, vol. 17, no. 3, pp. 179–219.ADSCrossRefGoogle Scholar
  4. 4.
    Beletskii, V.V., Motion of an Artificial Satellite About Its Center of Mass, S. Monson, Jerusalem: Israel Program for Scientific Translations, 1966.Google Scholar
  5. 5.
    Byrd, P.F. and Friedman, M.D., Handbook of Elliptic Integrals for Engineers and Scientists, Berlin Heidelberg, New York: Springer-Verlag, 1971.zbMATHCrossRefGoogle Scholar
  6. 6.
    Campbell, J.A. and Jefferys, W.H., Equivalence of the Perturbation Theories of Hori and Deprit, Celestial Mechan., 1970, vol. 2, no. 4, pp. 467–473.ADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Cochran, J.E., Effects of gravity-gradient torque on the rotational motion of a triaxial satellite in a precessing elliptic orbit, Celestial Mechan., 1972, vol. 6, no. 2, pp. 127–150; Celestial Mechan., 1974, vol. 8(E), no. 4, pp. 534–534.ADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Crenshaw, J.W. and Fitzpatrick, P.M., Gravity effects on the rotational motion of a uniaxial artificial satellite, AIAA J., 1968, vol. 6, no. 11, pp. 2140–2145.ADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Deprit, A., Free rotation of a rigid body studied in the phase space, Amer. J. Phys., 1967, vol. 35, pp. 424–428.ADSCrossRefGoogle Scholar
  10. 10.
    Deprit, A., Canonical transformations depending on a small parameter, Celestial Mechan., 1969, vol. 1, no. 1, pp. 12–30.MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Efroimsky, M. and Escapa, A., The theory of canonical perturbations applied to attitude dynamics and to the earth rotation. Osculating and nonosculating Andoyer variables, Celestial Mechan. Dynam. Astron., 2007, vol. 98, no. 3, pp. 251–283.MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Ferrer, S. and Lara, M., On the free rigid body integration as a perturbed spherical rotor, Astron. J., 2010, vol. 139, no. 5, pp. 1899–1908.ADSCrossRefGoogle Scholar
  13. 13.
    Ferrer, S. and Lara, M., Families of canonical transformations by Hamilton-Jacobi-Poincare Equation. Application to rotational and orbital motion, J. Geomet. Mechan., 2010, vol. 2, no. 3, pp. 223–241.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Fukushima, T., New canonical variables for orbital and rotational motions, Celestial Mechan. Dynam. Astron., 1994, vol. 60, no. 1, pp. 57–68.MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Getino, J. and Ferrändiz, J.M., On the effect of the mantle elasticity on the Earth’s rotation, Celestial Mechan. Dynam. Aatron., 1995, vol. 61, no. 2, pp. 117–180.ADSCrossRefGoogle Scholar
  16. 16.
    Goldstein, H., Poole, C.P., and Safko, J.L., Classical Mechanics, 3rd Ed., Addison-Wesley, 2001.Google Scholar
  17. 17.
    Holland, R.L. and Sperling, H.J., A first-order theory for the rotational motion of a triaxial rigid body orbiting an oblate primary, Astron. J., 1969, vol. 74, no. 3, pp. 490–496.ADSCrossRefGoogle Scholar
  18. 18.
    Hitzl, D.L. and Breakwell, J.V., Resonant and nonresonant gravity-gradient perturbations of a tumbling triaxial satellite, Celestial Mechan., 1971, vol. 3, no. 3, pp. 346–383.ADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Hori, G.-i., Theory of general perturbation with unspecified canonical variables, Publ. Astron. Soc. Jpn., 1966, vol. 18, pp. 287–296.ADSGoogle Scholar
  20. 20.
    Kinoshita, H., First-order perturbations of the two finite body problem, Publ. Astron. Soc. Jpn., 1972, vol. 24, pp. 423–457.MathSciNetADSGoogle Scholar
  21. 21.
    Kozlov, V.V., Qualitative Analysis Methods in Rigid Body Dynamics, Regular & Chaotic Dynamics, 2000, p. 256 (in Russian).zbMATHGoogle Scholar
  22. 22.
    Kraige, L.G. and Junkins, J.L., Perturbation formulations for satellite attitude dynamics, Celestial Mechan., 1976, vol. 13, no. 1, pp. 39–64.MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. 23.
    Lara, M., Fukushima, T., and Ferrer, S., Ceres’ rotation solution under the gravitational torque of the Sun, Monthly Notices of the Royal Astron. Soc., 2011, vol. 415, pp. 461–469.ADSCrossRefGoogle Scholar
  24. 24.
    Lara, M. and Ferrer, S., Complete closed form solution of a tumbling triaxial satellite under gravity-gradient torque, American Astronautical Society, Paper AAS 12-119.Google Scholar
  25. 25.
    MacCullagh, J., On the rotation of a solid body, Proc. Royal Irish Acad., 1840, vol. 2, pp. 520–545.Google Scholar
  26. 26.
    Sadov, Yu.A., The action-angles variables in the Euler-Poinsot problem, J. Appl. Math. Mechan., 1970, vol. 34, no. 5, pp. 922–925.ADSzbMATHCrossRefGoogle Scholar
  27. 27.
    Sadov, Yu.A., The action-angle variables in the Euler-Poinsot problem, Preprint Russian Academy of Sciences Moscow, KIAM, 1970, no. 22.Google Scholar
  28. 28.
    Sadov, Yu.A., Using action-angle variables in problems of disturbed Motion of a solid body about its center of mass, Preprint Russian Academy of Sciences Moscow, KIAM, 1984, no. 33.Google Scholar
  29. 29.
    Shuster, M.D., A survey of attitude representations, J. Astron. Sci., 1993, vol. 41, pp. 439–517.MathSciNetGoogle Scholar
  30. 30.
    Tchernousko, F.L. Study of satellite motion about center of mass using averaging method. Proc. XIVth International Astronautical Congress. 1963, vol. IV, pp. 143–154.Google Scholar
  31. 31.
    Touma, J., Wisdom, J. Lie-Poisson integrators for rigid body dynamics in the solar system. Astronomical J. 1994, vol. 107, no. 3, pp. 1189–1202.ADSCrossRefGoogle Scholar
  32. 32.
    Zanardi, M.C. Study of the terms of coupling between rotational and translational motion. Celestial Mechanics. 1986, vol. 39, no. 2, pp. 147–164.ADSzbMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.San FernandoSpain
  2. 2.Grupo de Dinámica EspacialUniversidad de MurciaMurciaSpain

Personalised recommendations