Skip to main content
SpringerLink
Log in

Analytical and numerical results on the stability of a planetary precessional model

  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript

Abstract

A model for planetary precession is investigated using analytical and numerical techniques. A Hamiltonian function governing the model is derived in terms of action-angle Andoyer-Déprit variables under the assumption of equatorial symmetry. As a first approximation a simplified Hamiltonian with zero-eccentricity is considered and stability estimates are derived using KAM theory. A validation of the analytical results is performed computing Poincaré surfaces of section for the circular and elliptical model. We also investigate the role of the eccentricity and its connection with the appearance of resonances. Special attention is devoted to the particular case of the Earth–Moon system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andoyer, H.: Cours de Mécanique Classique, Gauthier-Villars, 1923.

  2. Arnold, V. I.: 'Proof of a theorem by A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian', Russ. Math. Surv. 18(9) (1963).

  3. Benettin, G. and Fassó, F.: 'Fast rotations of the rigid body: a study by Hamiltonian perturbation theory', Part I, Nonlinearity 9 (1996), 137-186.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Benettin, G., Fassó, F. and Guzzo M.: 'Fast rotations of the rigid body: a study by Hamiltonian perturbation theory, Part II, Nonlinearity 10 (1997), 1695-1717.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Burns, T. J.: Celest. Mech. & Dyn. Astr. 19 (1979), 297.

    MATH  MathSciNet  ADS  Google Scholar 

  6. Celletti, A.: 'Analysis of resonances in the spin-orbit problem in Celestial Mechanics: the synchronous resonance', Part I, J. Appl. Math. Phys. (ZAMP) 41 (1990), 174-204.

    Article  MATH  MathSciNet  Google Scholar 

  7. Celletti, A.: 'Analysis of resonances in the spin-orbit problem in Celestial Mechanics: the synchronous resonance', Part II, J. Appl. Math. Phys. (ZAMP) 41 (1990), 473.

    Google Scholar 

  8. Celletti, A.: 'Construction of librational invariant tori in the spin-orbit problem', J. Appl. Math. Phys. (ZAMP) 45 (1993), 61.

    Article  MathSciNet  Google Scholar 

  9. Celletti, A. and Chierchia, L.: 'Rigorous estimates for a computer-assisted kam theory', J. Math. Phys. 28 (1981), 2078.

    Article  MathSciNet  ADS  Google Scholar 

  10. Celletti, A. and Chierchia, L.: 'A constructive theory of Lagrangian tori and computer-assisted applications', Dynamics Reported (new series) 4 (1995), 60-129.

    MATH  MathSciNet  Google Scholar 

  11. Celletti, A. and Chierchia, L.: 'On the stability of realistic three body problems', Comm. Math. Phys. 186 (1977), 413-449.

    Article  MathSciNet  ADS  Google Scholar 

  12. Chierchia, L. and Gallavotti, G.: 'Drift and diffusion in phase space', Ann. Inst. Henri Poincaré 60 (1994), 79-133.

    MathSciNet  Google Scholar 

  13. Cochran, J. E.: 'Effects of gravity-gradient torque on the rotational motion of a triaxial in a precessing elliptic orbit', Celest. Mech. & Dyn. Astr. 6 (1972), 127-150.

    MATH  ADS  Google Scholar 

  14. Colombo, G.: Nature 208 (1965), 575.

    Article  ADS  Google Scholar 

  15. Colombo, G. and Shapiro, I. I.: 'The rotation of the planet mercury', Astrophys. J. 145 (1966), 296.

    Article  ADS  Google Scholar 

  16. Counselman, C. C. and Shapiro, I. I.: Symposia Mathematica 3 (1970), 121.

    ADS  Google Scholar 

  17. Déprit, A.: 'Free rotation of a rigid body studied in phase plane', Am. J. Phys. 55 (1967), 424.

    Article  Google Scholar 

  18. Duboshin, G. N.: 'On the generalized restricted problem of three bodies', Celest. Mech. & Dyn. Astr. 4 (1971), 423-441.

    MATH  MathSciNet  ADS  Google Scholar 

  19. Duboshin., G. N.: 'About the first integrals of the generalized problem of translatory-rotary motion of rigid bodies', Celest. Mech. & Dyn. Astr. 6 (1972), 27-39.

    MATH  MathSciNet  ADS  Google Scholar 

  20. Froeschlé, C.: 'Numerical study of a four-dimensional mapping', Astron. Astrophys. 16 (1972), 172.

    MATH  ADS  Google Scholar 

  21. Goldreich, P. and Peale, S.: 'Spin-orbit coupling in the solar system', Astron. J. 71 (1966), 425.

    Article  ADS  Google Scholar 

  22. Goldreich, P. and Peale, S.: 'The dynamics of planetary rotations', Ann. Rev. Astron. Astroph. 6, 287.

  23. Henrard, J.: In: S. Ferraz-Mello and W. Sessin (eds), Spin-Orbit Resonance and the Adiabatic Invariant, Sao Paulo, 1985.

  24. Hitzl, D. L. and Breakwell, J. V.: 'Resonant and non-resonant gravity-gradient perturbations of a tumbling tri-axial satellite', Celest. Mech. & Dyn. Astr. 3 (1971), 346-383.

    MATH  ADS  Google Scholar 

  25. Kolmogorov, A. N.: 'On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian', Dokl. Akad. Nauk. SSR 98 (1954), 469.

    MathSciNet  Google Scholar 

  26. Laskar, J.: 'Secular evolution of the solar system over 10 million years', Astron. Astrophys. 198 (1988), 341-362.

    ADS  Google Scholar 

  27. Laskar, J.: 'The chaotic motion of the solar system. A numerical estimate of the size of the chaotic zones', Icarus 88 (1990), 266-291.

    Article  ADS  Google Scholar 

  28. Laskar, J.: 'Frequency analysis for multi-dimensional systems. Global dynamics and diffusion', Physica D 67 (1993), 257-281.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. Laskar, J.: 'Large scale chaos in the solar system', Astron. Astrophys. 287 (1994), 9-12.

    ADS  Google Scholar 

  30. Laskar, J.: 'Large scale chaos and marginal stability in the solar system', Celest. Mech. & Dyn. Astr. 64 (1996), 115-162.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  31. Laskar, J. and Robutel, P.: 'The chaotic obliquity of the planets', Nature 361 (1993a), 608-612.

    Article  ADS  Google Scholar 

  32. Laskar, J., Joutel, F. and Robutel, P.: 'Stabilization of the Earth's obliquity by the Moon', Nature 361 (1993b), 615-617.

    Article  ADS  Google Scholar 

  33. Moser, J.: 'On invariant curves of area-preserving mappings of an annulus', Nach. Akad. Wiss. Gottingen Math. Phys. Kl. II 1 (1962), 1.

    MATH  Google Scholar 

  34. Murdock, J. A.: 'Some mathematical aspects of spin-orbit resonance I', Celest. Mech. & Dyn. Astr. 18 (1978), 237.

    MATH  MathSciNet  ADS  Google Scholar 

  35. Murdock, J. A.: 'Some mathematical aspects of spin-orbit resonance II', Celest. Mech. & Dyn. Astr. 24 (1981), 83.

    MATH  MathSciNet  ADS  Google Scholar 

  36. Nekhoroshev, N. N.: 'An exponential estimate of the time of stability of nearly integrable Hamiltonian systems', Usp. Mat. Nauk 32 (1977), 5.

    MATH  Google Scholar 

  37. Poincaré, H.: Les Méthodes Nouvelles de la Méchanique Céleste, Gautier-Villars, 1982.

  38. Wisdom, J.: 'Chaotic behaviour in the solar system', Proc. R. Soc. Lond. A 413 (1987), 109.

    Article  MathSciNet  ADS  Google Scholar 

  39. Wisdom, J. and Peale, S. J.: 'The chaotic rotation of Hyperion', Icarus 58 (1984), 137.

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Observatoire de Nice, BP 4229, F–06304, Nice Cedex 4, France, e-mail

    Gabriella Della Penna

Authors
  1. Gabriella Della Penna
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Penna, G.D. Analytical and numerical results on the stability of a planetary precessional model. Celestial Mechanics and Dynamical Astronomy 75, 103–124 (1999). https://doi.org/10.1023/A:1008369122327

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008369122327

Search

Navigation