Construction of librational invariant tori in the spin-orbit problem

Details Construction of librational invariant tori in the spin-orbit problem Construction of librational invariant tori in the spin-orbit problem

Jan 1994

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994zamp...45...61c&link_type=abstract

ZAMP Zeitschrift für angewandte Mathematik und Physik, Volume 45, Issue 1, pp.61-80

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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 form H(γ, x, t)=(γ2/2) + ɛ V( x, t), where V is a trigonometric polynomial in x, t and ɛ 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.

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