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We construct a Nekhoroshev-like result of stability with sharp constants forthe planar three body problem, both in the planetary and in the restrictedcircular case, by using the periodic averaging technique. Our constructions canbe generalized to any near-integrable hamiltonian system whose unperturbedhamiltonian is quasi-convex. The dependence of the constants on the analyticitywidths of the complex hamiltonian is carefully taken into account. This allowsfor a deep analytical understanding of the limits of such techniques ininsuring Nekhoroshev stability for high magnitudes of the perturbation andsuggests hints on how to overcome such obstructions in some cases. Finally, twoexamples with concrete values are considered, one for the planetary case andone for the restricted one.

Sharp Nekhoroshev estimates for the three body problem around periodic orbits