  In the potential theory, isochrony was introduced by Michel H\'enon in 1959
to characterize astrophysical observations of some globular clusters. Today,
Michel Henon's isochrone potential is mainly used for his integrable property
in numerical simulations, but is generally not really known. In a recent paper
[29], we have presented new fundamental and theoretical results about isochrony
that have particular importance in self-gravitating dynamics and which are
detailed in this paper. In particular, new characterization of the isochrone
state has been proposed which are investigated in order to analyze the product
of the fast relaxation of a self-gravitating system. The general paradigm
consists in considering that this product is a lowered isothermal sphere (King
Model). By a detailed numerical study we show that this paradigm fails when the
isochrone model succeeds in reproducing the quasi-equilibrium state obtained
just after fast relaxation.
