Skip to main content
SpringerLink
Log in

Analysis of resonances in the spin-orbit problem in Celestial Mechanics: The synchronous resonance (Part I)

  • Original Papers
  • Published:
Zeitschrift für angewandte Mathematik und Physik ZAMP Aims and scope Submit manuscript

Abstract

We study the stability of spin-orbit resonances in Celestial Mechanics, namely the exact commensurabilities between the periods of rotation and revolution of satellites or planets. We introduce a mathematical model describing an approximation of the physical situation and we select a set of satellites for which such simplified model provides a good approximation.

Applying the Kolmogorov-Arnold-Moser theory we are able to construct invariant surfaces trapping the synchronous resonance from above and below. The existence of such surfaces, established for the natural values of the physical and orbital parameters, allows to prove the stability of the 1∶1 resonance. Furthermore we try to construct KAM tori with frequencies as close as possible to one so to trap the synchronous resonance in a finer region.

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. V. I. Arnold,Proof of a theorem by A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surveys18, 9 (1963).

    Google Scholar 

  2. D. Braess and E. Zehnder,On the numerical treatment of a small divisor problem. Numer. Math.39, 269 (1982).

    Google Scholar 

  3. A. Cayley,Tables of the developments of functions in the theory of elliptic motion. Mem. Roy. Astron. Soc.29, 191 (1859).

    Google Scholar 

  4. A. Celletti,Analysis of Resonances in the Spin-Orbit Problem in Celestial Mechanics. Ph.D. Thesis, ETH-Zürich (1989).

  5. A. Celletti,Analysis of resonances in the spin-orbit problem in Celestial Mechanics: Higher order resonances and some numerical experiments (Part II). To appear in ZAMP.

  6. A. Celletti and L. Chierchia,Rigorous estimates for a computer-assisted KAM theory. J. Math. Phys.28, 2078 (1987).

    Google Scholar 

  7. A. Celletti and L. Chierchia,Construction of analytic KAM surfaces and effective stability bounds. Commun. Math. Phys.118, 119 (1988).

    Google Scholar 

  8. A. Celletti and L. Chierchia,A computer-assisted approach to small-divisors problems arising in Hamiltonian mechanics. Proc. conf. on Computer aided proofs in analysis, Cincinnati (Ohio), March 1989.

  9. A. Celletti and A. Giorgilli,On the numerical optimization of KAM estimates by classical perturbation theory. J. Appl. Math. and Phys. (ZAMP)39, 743 (1988).

    Google Scholar 

  10. G. Colombo and I. I. Shapiro,The rotation of the planet Mercury. Astrophys. J.145, 296 (1966).

    Google Scholar 

  11. J. M. A. Danby,Fundamentals of Celestial Mechanics. Macmillan, New York 1962.

    Google Scholar 

  12. R. De La Llave and D. Rana,Proof of accurate bounds in small denominator problems. Preprint (1986).

  13. J. Denzler,Mather sets for plane Hamiltonian systems. J. of Applied Math. and Phys. (ZAMP)38, 791 (1987).

    Google Scholar 

  14. J.-P. Eckmann and P. Wittwer,Computer methods and Borel summability applied to Feigenbaum's equation. Springer Lect. Notes in Phys.227, Berlin 1985.

  15. H. Eliasson,Absolutely convergent series expansions for quasiperiodic motions. Preprint, Univ. of Stockholm (1987).

  16. D. F. Escande and F. Doveil,Renormalization method for computing the threshold of the large-scale stochastic instability in two degrees of freedom Hamiltonian systems. J. Stat. Phys.26, 257 (1981).

    Google Scholar 

  17. G. Gallavotti,The Elements of Mechanics. Springer-Verlag, New York 1983.

    Google Scholar 

  18. G. Gallavotti,Perturbation theory for classical Hamiltonian systems. InScaling and Self-Similarity in Physics (Ed. J. Fröhlich), Birkhäuser, Boston 1984.

    Google Scholar 

  19. G. Gallavotti,Quasi-integrable mechanical systems. InCritical Phenomena, Random Systems, Gauge Theories (Eds. K. Osterwalder and R. Stora), Les HouchesXLIII, 539 (1986).

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

    Google Scholar 

  21. P. Goldreich and S. Peale,The dynamics of planetary rotations. Ann. Rev. Astron. Astroph.6, 287 (1970).

    Google Scholar 

  22. P. Goldreich and S. Soter,Q in the solar system. Icarus5, 375 (1966).

    Google Scholar 

  23. J. M. Greene,A method for determining a stochastic transition. J. of Math. Phys.20, 1183 (1979).

    Google Scholar 

  24. O. Hald,On a Newton-Moser type method. Numer. Math.23, 411 (1975).

    Google Scholar 

  25. G. H. Hardy and E. M. Wright,The Theory of Numbers, Oxford 1979.

  26. M. Herman,Sur les courbes invariantes pour les difféomorphismes de l'anneau. Astérisque2, 144 (1986).

    Google Scholar 

  27. M. Herman,Recent results and some open questions on Sigel's linearization theorems of germs of complex analytic diffemorphisms of C n near a fixed point. Preprint (1987).

  28. A. N. Kolmogorov,On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. Dokl. Akad. Nauk. SSR98, 469 (1954).

    Google Scholar 

  29. O. E. Lanford III,Computer assisted proofs in analysis. Physics A124, 465 (1984).

    Google Scholar 

  30. G. J. F. MacDonald,Tidal friction. Rev. Geophys.2, 467 (1964).

    Google Scholar 

  31. R. S. MacKay,A renormalization approach to invariant circles in area-preserving maps. Physica7D, 283 (1983).

    Google Scholar 

  32. R. S. MacKay,Transition to chaos for area-preserving maps. Lect. Notes in Phys.247, 390 Springer, Berlin 1985.

    Google Scholar 

  33. J. Mather,Nonexistence of invariant circles. Erg. Theory and Dynam. Syst.4, 301 (1984).

    Google Scholar 

  34. J. Moser,On invariant curves of area-preserving mappings of an annulus. Nach. Akad. Wiss. Göttingen, Math. Phys. Kl. II1, 1 (1962).

    Google Scholar 

  35. J. Moser,A rapidly convergent iteration method and non-linear partial differential equations. Ann. Scuola Norm. Sup. Pisa20, 265 (1966).

    Google Scholar 

  36. J. Moser,Minimal solutions of variational problems on a torus. Ann. Inst. Henri Poincaré3, 229 (1986).

    Google Scholar 

  37. J. Moser,Recent developments in the theory of Hamiltonian systems, SIAM Rev.28, 459 (1986).

    Google Scholar 

  38. J. Moser,Minimal foliations on a torus, Four lectures at C.I.M.E.; Conf. on “Topics in Calculus of Variations” (1987).

  39. J. Moser,Ausgewählte Kapitel der Variationsrechnung. Lect. held at the ETH in the summer semester 1988.

  40. J. A. Murdock,Some mathematical aspects of spin-orbit resonance I. Celestial Mech.18, 237 (1978).

    Google Scholar 

  41. J. A. Murdock,Some mathematical aspects of spin-orbit resonance II. Celestial Mech.24, 83 (1981).

    Google Scholar 

  42. D. P. Parent,Exercises in Number Theory. Springer-Verlag, Berlin 1984.

    Google Scholar 

  43. S. J. Peale,Rotation histories of the natural satellites. InPlanetary Satellites (Ed. J. A. Burns), Univ. of Arizona Press 1977.

  44. I. C. Percival,Chaotic boundary of a Hamiltonian map. Physica6D, 67 (1982).

    Google Scholar 

  45. D. Rana,Proof of accurate upper and lower bounds to stability domains in small denominator problems. Ph.D. thesis, Princeton 1987.

  46. D. Salamon and E. Zehnder,KAM theory in configuration space. Comment. Math. Helvetici64, 84 (1989).

    Google Scholar 

  47. J. Wisdom,Chaotic dynamics in the solar system, Urey lecture (1986).

  48. J. Wisdom,Chaotic behaviour in the solar system. Proc. R. Soc. Lond.A413, 109 (1987).

    Google Scholar 

  49. J. Wisdom and S. J. Peale,The chaotic rotation of Hyperion. Icarus58, 137 (1984).

    Google Scholar 

  50. (no author listed)Vax architecture handbook. Digital Equipment Corporation (1981).

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, II Universita' di Roma, Via Fontanile di Carcaricola, 00133, Roma, Italia

    Alessandra Celletti

Authors
  1. Alessandra Celletti
    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

Celletti, A. Analysis of resonances in the spin-orbit problem in Celestial Mechanics: The synchronous resonance (Part I). Z. angew. Math. Phys. 41, 174–204 (1990). https://doi.org/10.1007/BF00945107

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00945107

Keywords

Search

Navigation