SEMINAIRE MERCREDI 7 JANVIER 2004
14 heures
Salle Séminaire 5
Centre de Physique Théorique
Marseille-Luminy

Ugo Locatelli
Université Rome 2

Title: A very challenging problem for the Hamiltonian perturbation
theory: the study of the stability of the Trojan asteroids"

Abstract: The Trojans constitute a family of asteroids orbiting close
the vertexes of the two equilateral triangles having the segment-line
connecting Sun and Jupiter as basis. The problem of the apparent
stability of their motion is debated since the discovery of the first
Trojan asteroids (i.e. almost one century ago).
We will study the problem of the stability of the Trojan asteroids in
the very simple framework of the planar restricted (circular)
three-body problem.
We first numerically investigate the size of the stability region by
using the frequency analisys. We will see how this region is very much
larger than those where the previously known (semi)analytical results
apply.
Most of the time will be dedicated to explain how the KAM theory can
be adapted to the general problem of constructing invariant tori close
to an equilibrium point (i.e., in our system, a vertex of the
equilateral triangles in the non-inertial frame corotating with Sun
and Jupiter). We will see that the particular problem of the Trojan
asteroids needs a few further refinements of the general approach.
Finally, we will compare the orbits obtained by the tori constructed
using our approach with those starting from some initial conditions
given by the "projections on the model" of the positions and
velocities of some real asteroids. This will give us the possibility
of appreciating the improvements given by our approaches.