Revisiting the two-body problem in Yukawa gravity and in a gravitational extension of the Buckingham potential

TL;DR

This paper revisits the two-body problem in Yukawa gravity, resolves previous inconsistencies, extends the analysis to a gravitational extension of the Buckingham potential, and tests the models against Solar System and star data, finding no definitive deviation from general relativity.

Contribution

It provides a corrected and extended Keplerian-like parametrization for two-body problems in Yukawa and Buckingham-inspired gravity models, with empirical testing against astronomical data.

Findings

Pericenter advance predictions are consistent with observations.

Current data cannot exclude deviations from general relativity.

The models offer a viable alternative framework for gravitational interactions.

Abstract

We revisit the Keplerian-like parametrization of the two-body problem in Yukawa gravity studied in the literature. Some inconsistencies, which spoil Bertrand's theorem, observed in the $\eta$ parametrization of the true anomaly $\theta$ and in the formulae for the pericenter's advance are resolved. Moreover, inspired in this kind of study, we couple the Buckingham potential, a variation of the Lennard-Jones intermolecular potential, with the gravitational Newtonian potential and find a Keplerian-like parametrization for the solution of the two-body problem in this sort of gravity. The outcomes for the advance of the pericenter in both types of gravity are corroborated by using the Landau and Lifshitz's method. We also tested the expressions thus obtained against Solar System and S2 star data. The result for both models is that while some deviation from general relativity (GR) is allowed, GR cannot be discarded by the current analysis.