Description of orbits with polar Gaussian coordinates in the post-Newtonian approximation of a spherically symmetric field

Abstract

The orbital motion about a central symmetric massive body appears very well described in terms of polar Gaussian coordinates because these are the only coordinates that can be materialized by a proper reference frame with origin at the centre of the massive body. In fact, by means of the relativistic world function in the post-Newtonian approximation of the exterior field written in these coordinates, it also results that the relativistic Gaussian post-Newtonian distance between two orbiting particles reduces to their Euclidean distance when these particles become in conjunction with the central massive body and the distance is calculated by an observer at rest with respect to this body. Besides, the differential equation in the reciprocal of this polar Gaussian radial coordinate for any bounded orbit is governed by a function which is asymptotically a cubic, so that these orbits can be given in terms of elliptic functions. Since, on the other hand, the most relevant (non-linear) relativistic term appearing in other post-Newtonian approximations of the exterior field disappears when these coordinates are used, to see how the relativistic corrections are now incorporated, an approximate solution for this kind of trajectories, as well as for their respective advances of perihelion is, finally, derived.