Abstract
We investigate the stability of the synchronous spin-orbit resonance. In particular we construct invariant librational tori trapping periodic orbits in finite regions of phase space. We first introduce a mathematical model describing a simplification of the physical situation. The corresponding Hamiltonian function has the formH(γ,x,t)=(γ2/2) + εV(x,t), whereV is a trigonometric polynomial inx, t and ε is the “perturbing parameter” representing the equatorial oblateness of the satellite.
We perform some symplectic changes of variables in order to reduce the initial Hamiltonian to a form which suitably describes librational tori. We then apply Birkhoff normalization procedure in order to reduce the size of the perturbation. Finally the application of KAM theory allows to prove the existence of librational tori around the synchronous periodic orbit. Two concrete applications are considered: the Moon-Earth and the Rhea-Saturn systems. In the first case one gets the existence of trapping orbits for values of the perturbing oblateness parameter far from the real physical value by a factor ∼ 5. In the Rhea-Saturn case we construct the trapping tori for values of the parameters consistent with the astronomical measurements.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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. Surveys18, 9 (1963).
Arnold, V. I. (ed.),Encyclopedia of Mathematical Sciences, Dynamical System III, Springer-Verlag 3 (1988).
Aubry, S. and Le Daeron, P. Y.,The discrete Frenkel-Kontorova model and its extensions I, Physica8D, 381 (1983).
Cayley, A.,Tables of the developments of functions in the theory of elliptic motion, Mem. Roy. Astron. Soc.29, 191 (1859).
Celletti, A.,Analysis of resonances in the spin-orbit problem in Celestial Mechanics:The synchronous resonance (Part I), J. Appl. Math. and Phys. (ZAMP)41, 174 (1990).
Celletti, A.,Analysis of resonances in the spin-orbit problem in Celestial Mechanics: Higher order resonances and some numerical experiments (Part II), J. Appl. Math. and Phys. (ZAMP)41, 453 (1990).
Celletti, A. and Chierchia, L.,Rigorous estimates for a computer-assisted KAM theory, J. Math. Phys.28, 2078 (1987).
Celletti, A. and Chierchia, L.,Construction of analytic KAM surfaces and effective stability bounds, Commun. Math. Phys.118, 119 (1988).
Celletti, A. and Chierchia, L.,A constructive theory of Lagrangian tori and computer-assisted applications, to appear in Dynamics Reported.
Celletti, A., Falcolini, C. and Porzio, A.,Rigorous numerical stability estimates for the existence of KAM tori in a forced pendulum, Ann. Inst. Henri Poincaré47, 85 (1987).
Celletti, A. and Giorgilli, A.,On the numerical optimization of KAM estimates by classical perturbation theory, J. Appl. Math, and Phys. (ZAMP)39, 743 (1988).
Danby, J. M. A.,Fundamentals of Celestial Mechanics, Macmillan, New York 1962.
Eckmann, J.-P. and Wittwer, P.,Computer methods and Borel summability applied to Feigenbaum's equation, Springer Lect. Notes in Phys. 227 (1985).
Gallavotti, G.,The Elements of Mechanics, Springer-Verlag, New York 1983.
Goldreich, P. and Peale, S.,Spin-orbit coupling in the solar system, Astron. J.71, 425 (1966).
Greene, J. M.,A method for determining a stochastic transition, J. of Math. Phys.20, 1183 (1979).
Kolmogorov, A. N.,On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian, Dokl. Akad. Nauk. SSR98, 469 (1954).
Lanford, III, O. E.,Computer assisted proofs in analysis, Physics A124, 465 (1984).
Mather, J.,Nonexistence of invariant circles, Erg. Theory and Dynam. Systems4, 301 (1984).
Moser, J.,On invariant curves of area-preserving mappings of an annulus, Nach. Akad. Wiss. Göttingen, Math. Phys. Kl. II1, 1 (1962).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Celletti, A. Construction of librational invariant tori in the spin-orbit problem. Z. angew. Math. Phys. 45, 61–80 (1994). https://doi.org/10.1007/BF00942847
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00942847