Long-time stability of the secular part of a planetary problem with more than three bodies
speaker: Ugo Locatelli (Dip. di Matematica dell'Universita' di Roma "Tor Vergata")abstract: This work is part of a research project I'm carrying out in collaboration with A. Giorgilli and M. Sansottera.
We study the stability of the planetary problem including Sun, Jupiter, Saturn and Uranus (SJSU, respectively) in the framework of the secular model of order~$2$ in the masses. In a previous work of us, we applied a rather standard scheme of estimates to the partial construction of the Birkhoff normal form in the vicinity of the origin of the phase space (that is an elliptic equilibrium point). This allowed us to show that our secular Hamiltonian model is ``effectively stable'' for times larger than the age of the solar system ``just'' for all the initial conditions corresponding to eccentricities smaller than (about) $12$ of the real values.
In the present talk, we reconsider the same problem in order to extend the result to the ``real'' initial conditions. Therefore, we changed our approach by firstly looking for a KAM torus in the vicinity of the ``real'' orbit. Such a KAM torus is explicitly constructed up to a high order of approximation, by using algebraic manipulations on a computer. As a final step of our new approach, we evaluate the stability time related to the Birkhoff normal form centered about that KAM torus. This strongly improves our previous result.
timetable:
Sat 19 Feb, 10:00 - 10:30, Aula Bianchi Scienze
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