On the 2PN pericentre precession in the general theory of relativity and the recently discovered fast orbiting S-stars in Sgr A$^\ast$

TL;DR

This paper revisits the 2PN pericentre precession calculation in general relativity, correcting previous errors, and discusses its potential measurability in the orbits of recently discovered S-stars near Sgr A*.

Contribution

The paper identifies and corrects errors in previous 2PN precession calculations, providing a consistent analytical expression and exploring its observational implications for S-stars.

Findings

Corrected the 2PN pericentre precession formula.

Demonstrated agreement among different calculation approaches.

Suggested potential measurability of effects in S-star orbits.

Abstract

Recently, the secular pericentre precession was analytically computed to the second post-Newtonian (2PN) order by the present author with the Gauss equations in terms of the osculating Keplerian orbital elements in order to obtain closer contact with the observations in astronomical and astrophysical scenarios of potential interest. A discrepancy with previous results by other authors was found. Moreover, some of such findings by the same authors were deemed as mutually inconsistent. In this paper, it is demonstrated that, in fact, two calculational errors plagued the most recent calculation by the present author. They are explicitly disclosed and corrected. As a result, all the examined approaches mutually agree yielding the same analytical expression for the total 2PN pericentre precession once the appropriate conversions from the adopted parameterizations are made. It is also shown that, in future, it may become measurable, at least in principle, for some of the recently discovered short-period S-stars in Sgr A$^\ast$ like S62 and S4714.