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We give a constructive proof of the existence of elliptic lower dimensional tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov normalization algorithm to the case of planetary systems, for which elliptic tori may be used as replacements of elliptic Keplerian orbits in Lagrange-Laplace theory. With this paper we support with rigorous convergence estimates the semi-analytic work in our previous article (Sansottera et al., Celest Mech Dyn Astron 111:337–361, 2011), where an explicit calculation of an invariant torus for a planar model of the Sun-Jupiter-Saturn-Uranus system has been made. With respect to previous works on the same subject we exploit the characteristic of Lie series giving a precise control of all terms generated by our algorithm. This allows us to slightly relax the non-resonance conditions on the frequencies.

We investigate the long time stability in Nekhoroshev's sense for the Sun-Jupiter-Saturn problem in the framework of the problem of three bodies. Using computer algebra in order to perform huge perturbation expansions we show that the stability for a time comparable with the age of the universe is actually reached, but with some strong truncations on the perturbation expansion of the Hamiltonian at some stage. An improvement of such results is currently under investigation., Comment: 17 pages, 2 figures

We consider a dissipative spin-orbit model where it is assumed that the orbit of the satellite is Keplerian, the obliquity is zero, and the dissipative effects depend linearly on the relative angular velocity. The measure of the basins of attraction associated to periodic and quasi-periodic attractors is numerically investigated. The results depend on the interaction among the physically relevant parameters, namely, the orbital eccentricity, the equatorial oblateness and the dissipative constant. In particular, it appears that, for astronomically relevant parameter values, for low eccentricities (as in the Moon’s case) about 96% of the initial data belong to the basin of attraction of the 1/1 spin-orbit resonance; for larger values of the eccentricities higher order spin-orbit resonances and quasi-periodic attractors become dominant providing a mechanism for explaining the observed state of Mercury into the 3/2 resonance.

The dumbbell satellite is a simple structure consisting of two point masses connected by a massless rod. We assume that it moves around the planet whose gravity field is approximated by the field of the attracting center. The distance between the point masses is assumed to be much smaller than the distance between the satellite’s center of mass and the attracting center, so that we can neglect the influence of the attitude dynamics on the motion of the center of mass and treat it as an unperturbed Keplerian one. Our aim is to study the satellite’s attitude dynamics. When the center of mass moves on a circular orbit, one can find a stable relative equilibrium in which the satellite is permanently elongated along the line joining the center of mass with the attracting center (the so called local vertical). In case of elliptic orbits, there are no stable equilibrium positions even for small values of the eccentricity. However, planar periodic motions are determined, where the satellite oscillates around the local vertical in such a way that the point masses do not leave the orbital plane. We prove analytically that these planar periodic motions are unstable with respect to out-of-plane perturbations (a result known from numerical investigations cf. Beletsky and Levin Adv Astronaut Sci 83, 1993). We provide also both analytical and numerical evidences of the existence of stable spatial periodic motions.

We review analytical (rigorous) results about the existence of invariant tori for planetary many-body problems.