New series expansion method for the periapsis shift

TL;DR

This paper introduces a new series expansion method for calculating the periapsis shift in various spacetimes, offering simple, accurate formulas that outperform traditional post-Newtonian expansions especially for small eccentricities.

Contribution

The paper develops a novel series expansion approach that provides analytical, highly accurate periapsis shift formulas without special functions, applicable to Kerr and Chazy-Curzon spacetimes.

Findings

New series representations for periapsis shift in Kerr and Chazy-Curzon spacetimes.

The new formulas outperform post-Newtonian expansions for small eccentricities.

Exact periapsis shift formula for quasi-circular orbits in Chazy-Curzon spacetime.

Abstract

We propose a new series expansion method for the periapsis shift. The method formulates the periapsis shift in various spacetimes analytically without using special functions and provides simple and highly accurate approximate formulae. We derive new series representations for the periapsis shift in the Kerr and the Chazy-Curzon spacetimes by using the method, where the expansion parameter is defined as the eccentricity divided by the non-dimensional quantity that vanishes in the limit of the innermost stable circular orbit. That is to say, the expansion parameter denotes how eccentric the orbit is and how close it is to the innermost stable circular orbit. The smaller the eccentricity, the higher the accuracy of the formulae that are obtained by truncating the new series representations up to a finite number of terms. If the eccentricity is sufficiently small, the truncated new representations have higher accuracy than the post-Newtonian expansion formulae even in strong gravitational fields where the convergence of the post-Newtonian expansion formula is not guaranteed. On the other hand, even if the orbit is highly eccentric, the truncated new representations have comparable or higher accuracy than the post-Newtonian expansion formulae if the semi-major axis is sufficiently large. An exact formula for the periapsis shift of the quasi-circular orbit in the Chazy-Curzon spacetime is also obtained as a special case of the new series representation.