  We construct a Nekhoroshev-like result of stability with sharp constants for
the planar three body problem, both in the planetary and in the restricted
circular case, by using the periodic averaging technique. Our constructions can
be generalized to any near-integrable hamiltonian system whose unperturbed
hamiltonian is quasi-convex. The dependence of the constants on the analyticity
widths of the complex hamiltonian is carefully taken into account. This allows
for a deep analytical understanding of the limits of such techniques in
insuring Nekhoroshev stability for high magnitudes of the perturbation and
suggests hints on how to overcome such obstructions in some cases. Finally, two
examples with concrete values are considered, one for the planetary case and
one for the restricted one.
