Comparison between two methods of post-Newtonian expansion for the motion in a weak Schwarzschild field

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.