Advertisement

Celestial Mechanics and Dynamical Astronomy

, Volume 118, Issue 3, pp 221–234 | Cite as

Short-axis-mode rotation of a free rigid body by perturbation series

  • Martin Lara
Original Article

Abstract

A simple rearrangement of the torque free motion Hamiltonian shapes it as a perturbation problem for bodies rotating close to the principal axis of maximum inertia, independently of their triaxiality. The complete reduction of the main part of this Hamiltonian via the Hamilton–Jacobi equation provides the action-angle variables that ease the construction of a perturbation solution by Lie transforms. The lowest orders of the transformation equations of the perturbation solution are checked to agree with Kinoshita’s corresponding expansions for the exact solution of the free rigid body problem. For approximately axisymmetric bodies rotating close to the principal axis of maximum inertia, the common case of major solar system bodies, the new approach is advantageous over classical expansions based on a small triaxiality parameter.

Keywords

Free rigid body Short-axis-mode Action-angle variables Lie transforms Hamiltonian formulation Andoyer variables 

Notes

Acknowledgments

Support is acknowledged from projects AYA 2009-11896 and AYA 2010-18796 of the Government of Spain.

References

  1. Andoyer, M.H.: Cours de Mécanique Céleste. Gauthier-Villars et cie, Paris (1923)zbMATHGoogle Scholar
  2. Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, New York (1989)CrossRefGoogle Scholar
  3. Barkin, Y.V.: Unperturbed Chandler motion and perturbation theory of the rotation motion of deformable celestial bodies. Astron. Astrophys. Trans. 17(3), 179–219 (1998). doi: 10.1080/10556799808232092 ADSCrossRefGoogle Scholar
  4. Chernous’ko, F.L.: On the motion of a satellite about its center of mass under the action of gravitational moments. PMM J. Appl. Math. Mech. 27(3), 708–722 (1963)CrossRefzbMATHMathSciNetGoogle Scholar
  5. Cicalò, S., Scheeres, D.J.: Averaged rotational dynamics of an asteroid in tumbling rotation under the YORP torque. Celest. Mech. Dyn. Astron. 106(4), 301–337 (2010). doi: 10.1007/s10569-009-9249-7 ADSCrossRefzbMATHGoogle Scholar
  6. Cottereau, L., Souchay, J., Aljbaae, S.: Accurate free and forced rotational motions of rigid Venus. Astron. Astrophys. 515, A9 (2010). doi: 10.1051/0004-6361/200913785 ADSCrossRefGoogle Scholar
  7. Dehant, V., de Viron, O., Karatekin, O., van Hoolst, T.: Excitation of Mars polar motion. Astron. Astrophys. 446(1), 345–355 (2006). doi: 10.1051/0004-6361:20053825 ADSCrossRefGoogle Scholar
  8. Deprit, A.: Free rotation of a rigid body studied in the phase space. Am. J. Phys. 35, 424–428 (1967)ADSCrossRefGoogle Scholar
  9. Deprit, A.: Canonical transformations depending on a small parameter. Celest. Mech. 1(1), 12–30 (1969). doi: 10.1007/BF01230629 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  10. Escapa, A.: Corrections stemming from the non-osculating character of the Andoyer variables used in the description of rotation of the elastic Earth. Celest. Mech. Dyn. Astron. 110(2), 99–142 (2011). doi: 10.1007/BF00051485 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. Ferrandiz, J.M., Sansaturio, M.E.: Elimination of the nodes when the satellite is a non spherical rigid body. Celest. Mech. Dyn. Astron. 46(4), 307–320 (1989). doi: 10.1007/BF00051485 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. Ferrer, S., Lara, M.: Families of canonical transformations by Hamilton–Jacobi–Poincaré equation. Application to rotational and orbital motion. J. Geom. Mech. 2(3), 223–241 (2010a). doi: 10.3934/jgm.2010.2.223
  13. Ferrer, S., Lara, M.: Integration of the rotation of an Earth-like body as a perturbed spherical rotor. Astron. J. 139(5), 1899–1908 (2010b). doi: 10.1088/0004-6256/139/5/1899 ADSCrossRefGoogle Scholar
  14. Fukushima, T.: Efficient integration of torque-free rotation by energy scaling method. In: Brzezinski, A., Capitaine, N., Kolaczek, B. (eds.) Proceedings of the Journées Systèmes de Référence Spatio-Temporels 2005. Space Research Centre PAS, Warsaw, Poland (2006)Google Scholar
  15. Fukushima, T.: Simple, regular, and efficient numerical integration of rotational motion. Astron. J. 135(6), 2298–2322 (2008). doi: 10.1088/0004-6256/135/6/2298 ADSCrossRefGoogle Scholar
  16. Getino, J., Ferrándiz, J.M.: A Hamiltonian theory for an elastic Earth—first order analytical integration. Celest. Mech. Dyn. Astron. 51(1), 35–65 (1991). doi: 10.1007/BF02426669 ADSCrossRefGoogle Scholar
  17. Getino, J., Escapa, A., Miguel, D.: General theory of the rotation of the non-rigid Earth at the second order. I. The rigid model in Andoyer variables. Astron. J. 139(5), 1916–1934 (2010). doi: 10.1088/0004-6256/139/5/1916 ADSCrossRefGoogle Scholar
  18. Golubev, V: Lectures on Integration of the Equations of Motion of a Rigid Body About a Fixed Point. Israel Program for Scientific Translations, S. Monson, Jerusalem (1960)Google Scholar
  19. Henrard, J.: Virtual singularities in the artificial satellite theory. Celest. Mech. 10(4), 437–449 (1974). doi: 10.1007/BF01229120 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. Henrard, J., Moons, M.: Hamiltonian theory of the libration of the Moon. In: Szebehely, V.G. (ed.) Dynamics of Planets and Satellites and Theories of Their Motion, Proceedings of the International Astronomical Union colloquium no. 41, vol. 72, pp. 125–135. D. Reidel Publishing Company, Dordrecht; USA, Astrophysics and Space Science Library, Holland/Boston (1978)Google Scholar
  21. Hitzl, D.L., Breakwell, J.V.: Resonant and non-resonant gravity-gradient perturbations of a tumbling tri-axial satellite. Celest. Mech. 3(5), 346–383 (1971). doi: 10.1007/BF01231806 ADSCrossRefzbMATHGoogle Scholar
  22. Hori, G.: Theory of general perturbation with unspecified canonical variables. Publ. Astron. Soc. Jpn. 18(4), 287–296 (1966)Google Scholar
  23. Kinoshita, H.: First-order perturbations of the two finite body problem. Publ. Astron. Soc. Jpn. 24, 423–457 (1972)ADSMathSciNetGoogle Scholar
  24. Kinoshita, H.: Theory of the rotation of the rigid earth. Celest. Mech. 15(3), 277–326 (1977). doi: 10.1007/BF01228425 ADSCrossRefMathSciNetGoogle Scholar
  25. Kinoshita, H.: Analytical expansions of torque-free motions for short and long axis modes. Celest. Mech. Dyn. Astron. 53(4), 365–375 (1992). doi: 10.1007/BF00051817 ADSCrossRefzbMATHGoogle Scholar
  26. Kozlov, V.V.: La Géométrie des variables action-angle dans le problème d’Euler-Poinsot. Vestnik Moskovskogo Universiteta Seriya I Matematika, Mekhanika 5, 74–79 (1974); (in Russian)Google Scholar
  27. Kubo, Y.: The kinematical mechanism for the perturbation of the rotational axis in the rotation of the elastic Earth. Celest. Mech. Dyn. Astron. 112(1), 99–106 (2012). doi: 10.1007/s10569-011-9385-8 ADSCrossRefMathSciNetGoogle Scholar
  28. Lara, M., Ferrer, S.: Closed form perturbation solution of a fast rotating triaxial satellite under gravity-gradient torque. Cosm. Res. 51(4), 289–303 (2013)ADSCrossRefGoogle Scholar
  29. Lara, M., Fukushima, T., Ferrer, S.: First-order rotation solution of an oblate rigid body under the torque of a perturber in circular orbit. Astron. Astrophys. 519, A1 (2010). doi: 10.1051/0004-6361/200913880 ADSCrossRefGoogle Scholar
  30. Lara, M., Fukushima, T., Ferrer, S.: Ceres’ rotation solution under the gravitational torque of the Sun. Mon. Notices R. Astron. Soc. 415(1), 461–469 (2011). doi: 10.1111/j.1365-2966.2011.18717.x ADSCrossRefGoogle Scholar
  31. Newhall, X.X., Williams, J.G.: Estimation of the lunar physical librations. Celest. Mech. Dyn. Astron. 66(1), 21–30 (1996). doi: 10.1007/BF00048820 ADSCrossRefGoogle Scholar
  32. Sadov, Y.A.: The action-angles variables in the Euler-Poinsot problem. PMM J. Appl. Math. Mech. 34(5), 922–925 (1970)Google Scholar
  33. Sidorenko, V.V.: Capture and escape from resonance in the dynamics of the rigid body in viscous medium. J. Nonlinear Sci. 4(1), 35–57 (1994). doi: 10.1007/BF02430626 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  34. Sidorenko, V.V., Scheeres, D.J., Byram, S.M.: On the rotation of comet Borrelly’s nucleus. Celest. Mech. Dyn. Astron. 102(1–3), 133–147 (2008). doi: 10.1007/s10569-008-9160-7 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  35. Souchay, J., Bouquillon, S.: The high frequency variations in the rotation of Eros. Astron. Astrophys. 433(1), 375–383 (2005). doi: 10.1051/0004-6361:20035780 ADSCrossRefGoogle Scholar
  36. Souchay, J., Folgueira, M., Bouquillon, S.: Effects of the triaxiality on the rotation of celestial bodies: application to the Earth. Mars and Eros. Earth Moon Planets 93(2), 107–144 (2003a). doi: 10.1023/B:MOON.0000034505.79534.01
  37. Souchay, J., Kinoshita, H., Nakai, H., Roux, S.: A precise modeling of Eros 433 rotation. Icarus 166(2), 285–296 (2003b). doi: 10.1016/j.icarus.2003.08.018
  38. Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 2nd edn. Cambridge University Press, Cambridge (1917)Google Scholar
  39. Zanardi, M.C.: Study of the terms of coupling between rotational and translational motions. Celest. Mech. 39(1), 147–158 (1986). doi: 10.1007/BF01230847 ADSCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Columnas de Hercules 1San FernandoSpain

Personalised recommendations