Post-Newtonian effects of planetary gravity on the perihelion shift

TL;DR

This paper analyzes the tiny post-Newtonian effects of a secondary body's gravity on the perihelion shift of a test particle in a planetary system, concluding these effects are negligible for near-future observations.

Contribution

It provides a detailed calculation of the post-Newtonian contributions from a secondary body to perihelion precession, showing they are too small to be detected soon.

Findings

Post-Newtonian effects are split into two parts related to angular momentum and orbital radius.

Numerical factors increase the expressions but effects remain small.

Effects are negligible for upcoming measurements.

Abstract

We consider a coplanar system comprised of a massive central body (a star), a less massive secondary (a planet) on a circular orbit, and a test particle on a bound orbit exterior to that of the secondary. The gravitational pull exerted on the test particle by the secondary acts as a small perturbation, wherefore the trajectory of the particle can be described as an ellipse of a precessing perihelion. While the apsidal motion is defined overwhelmingly by the Newtonian portion of the secondary's gravity, the post-Newtonian portion, too, brings its tiny input. We explore whether this input may be of any astrophysical relevance in the next few decades. We demonstrate that the overall post-Newtonian input of the secondary's gravity can be split into two parts. One can be expressed via the orbital angular momentum of the secondary, another via its orbital radius. Despite some moderately large numerical factors showing up in the expressions for these two parts, the resulting post-Newtonian contributions from the secondary's gravity into the apsidal motion of the test particle turn out to be small enough to be neglected in the near-future measurements.