Abstract
We studied the stability of the restricted circular three-body problem. We introduced a model Hamiltonian in action-angle Delaunay variables. which is nearly-integrable with the perturbing parameter representing the mass ratio of the primaries. We performed a normal form reduction to remove the perturbation in the initial Hamiltonian to higher orders in the perturbing parameter. Next we applied a result on the Nekhoroshev theorem proved by Pöschel [13] to obtain the confinement in phase space of the action variables (related to the elliptic elements of the minor body) for an exponentially long time. As a concrete application. we selected the Sun-Ceres-Jupiter case, obtaining (after the proper normal form reduction) a stability result for a time comparable to the age of the solar system (i.e., 4.9 · 109 years) and for a mass ratio of the primaries less or equal than 10−6.
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References
Benettin, G. and Gallavotti, G.: 1986, ‘Stability of Motions Near Resonances in Quasi-Integrable Hamiltonian Systems’,J. of Stat. Phys. 44, 293.
Benettin, G., Galgani, L. and Giorgilli, A.: 1985, ‘A Proof of Nekhoroshev's Theorem for the Stability Times in Nearly Integrable Hamiltonian Sytems’,Celest. Mach. 37, 1.
Celletti, A. and Chierchia, L.: 1995, ‘A Construction Theory of Lagrangian Tori and ComputerAssisted Applications’,Dynamics Reported, N.Ser. Vol. 4, Springer, Berlin, p. 60.
Celletti, A. and Chierchia, L.: 1995, ‘Quasi-Periodic Motions for Realistic Three-body Problems’, Preprint.
Celletti, A. and Giorgilli, A.: 1991, ‘On the Stability of the Lagrangian Points in the Spatial Restricted Problem of Three Bodies’,Celest. Mech. Dynam. Astron. 50, 31.
Delaunay, C.: 1860, Théorie du Mouvement de la Lune,Mém. Acad. Sci. 1, Tome XXVIII, Paris.
Gallavotti, G.: 1983,The Elements of Mechanics, Springer, New York.
Giorgilli, A., Delshams, A., Fontich, E., Galgani, L. and Simó, C.: 1989, Effective Stability for a Hamiltonian System Near an Elliptic Equilibrium Point,J. Diff. Eq. 77, 167.
Giorgilli, A. and Zehnder, E.: 1992, Exponential Stability for Time Dependent Potentials,Z. Angew. Math. Phys. (ZAMP) 43, 827.
Ferrara, L.: 1994, ‘Stime di Stabilitá per Tempi Esponenziali con Applicazione al Problema dei Tre Corpi Ristretto’,Tesi di Laurea, Universitá di L'Aquila.
Nekhoroshev, N.N.: 1977, An Exponential Estimate of the Time of Stability of Nearly Integrable Hamiltonian Systems,Russian Math. Surv. 32, 1.
Niederman, L.: 1993,Résonances et Stabilité dans le Probléme Planétaire, Thése de Doctorat de l'Université de Paris 6.
Pöschel, J.: 1993, Nekhoroshev Estimates for Quasi-Convex Hamiltonian Systems,Math. Z. 213, 187.
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Cellett, A., Ferrara, L. An application of the Nekhoroshev theorem to the restricted three-body problem. Celestial Mech Dyn Astr 64, 261–272 (1996). https://doi.org/10.1007/BF00728351
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DOI: https://doi.org/10.1007/BF00728351