  The asymptotic method of post-Newtonian (PN) expansion for weak gravitational
fields, recently developed, is compared with the standard method of PN
expansion, in the particular case of a massive test particle moving along a
geodesic line of a weak Schwarzschild field. First, the expression of the
active mass in Schwarzschild's solution is given for a barotropic perfect
fluid, both for general relativity (GR) and for an alternative, scalar theory.
The principle of the asymptotic method is then recalled and the PN expansion of
the active mass is derived. The PN correction to the active mass is made of the
Newtonian elastic energy, augmented, for the scalar theory, by a term due to
the self-reinforcement of the gravitational field. Third, two equations, both
correct to first order, are derived for the geodesic motion of a mass particle:
a "standard" one and an "asymptotic" one. Finally, the difference between the
solutions of these two equations is numerically investigated in the case of
Mercury. The asymptotic solution deviates from the standard one like the square
of the time elapsed since the initial time. This is due to a practical
shortcoming of the asymptotic method, which is shown to disappear if one
reinitializes the asymptotic problem often enough. Thus, both methods are
equivalent in the case investigated. In a general case, the asymptotic method
seems more natural.

General Relativity and Quantum Cosmology;Astrophysics