  The discovery in [G. Pinzari. PhD thesis. Univ. Roma Tre. 2009], [L.
Chierchia and G. Pinzari, Invent. Math. 2011] of the Birkhoff normal form for
the planetary many--body problem opened new insights and hopes for the
comprehension of the dynamics of this problem. Remarkably, it allowed to give a
{\sl direct} proof of the celebrated Arnold's Theorem [V. I. Arnold. Uspehi
Math. Nauk. 1963] on the stability of planetary motions. In this paper, using a
"ad hoc" set of symplectic variables, we develop an asymptotic formula for this
normal form that may turn to be useful in applications. As an example, we
provide two very simple applications to the three-body problem: we prove a
conjecture by [V. I. Arnold. cit] on the "Kolmogorov set"of this problem and,
using Nehoro{\v{s}}ev Theory [Nehoro{\v{s}}ev. Uspehi Math. Nauk. 1977], we
prove, in the planar case, stability of all planetary actions over
exponentially-long times, provided mean--motion resonances are excluded. We
also briefly discuss perspectives and problems for full generalization of the
results in the paper.
