Celletti, A., Universita' di Roma "Tor Vergata", Froeschle', C., Observatoire de Nice, Lega, E., Observatoire de Nice
Frequency analysis of the stability of asteroids in the framework of the
restricted, three-body problem
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ABSTRACT.  The stability of some asteroids, in the framework of the restricted three-
body problem, has been recently proved in \cite{CC03} by developing an
isoenergetic KAM theorem. More precisely, having fixed a level of
energy related to the motion of the asteroid, the stability can be
obtained by showing the existence of nearby trapping invariant tori
living on the same energy level. The analytical results are compatible
with the astronomical observations, since the theorem is valid for the
realistic mass-ratio of the primaries.
The model adopted in \cite{CC03} is the planar, circular, restricted
three-body model, in which only the most significant contributions of
the Fourier development of the perturbation are retained. In this paper
we investigate numerically the stability of the same asteroids
considered in \cite{CC03} (namely, Iris, Victoria and Renzia). In
particular, we implement the nowadays standard method of frequency-
map analysis and we compare our investigation with the analytical
results on the planar, circular model with the truncated perturbing
function. By means of frequency analysis, we study the behaviour of the
bounding tori and henceforth we infer stability properties on the
dynamics of the asteroids. In order to test the validity of the truncated
Hamiltonian, we consider also the complete expression of the perturbing
function on which we perform again frequency analysis. We investigate
also more realistic models, taking into account the eccentricity of the
trajectory of Jupiter (planar-elliptic problem) or the relative inclination
of the orbits (circular-inclined model). We did not find a relevant
discrepancy among the different models, except for the case
of Renzia, that we explain in terms of its proximity to a resonance.