New general relativistic contributions to Mercury's orbital elements and their measurability

TL;DR

This paper calculates the first-order post-Newtonian effects of a distant third body on Mercury's orbit, showing these effects are comparable to current measurement errors and could be detected with upcoming missions.

Contribution

It provides analytical and numerical estimates of relativistic orbital effects due to a third body, highlighting their potential measurability and methods to distinguish them from other influences.

Findings

Relativistic effects on Mercury's orbital elements are at the same level as current measurement errors.

The perihelion precession due to a third body is smaller than some recent estimates.

Linear combinations of orbital element precessions can isolate relativistic effects from other perturbations.

Abstract

We numerically and analytically work out the first-order post-Newtonian (1pN) orbital effects induced on the semimajor axis $a$, the eccentricity $e$, the inclination $I$, the longitude of the ascending node $\Omega$, the longitude of perihelion $\varpi$, and the mean longitude at epoch $\epsilon$ of a test particle orbiting its primary, assumed static and spherically symmetric, by a distant massive third body X. For Mercury, the rates of change of the linear trends found are $\dot I_\mathrm{1pN}^\mathrm{X} = -4.3\,\mathrm{microarcseconds\,per\,century}\,\left(\mu\mathrm{as\,cty}^{-1}\right)$, $\dot\Omega_\mathrm{1pN}^\mathrm{X} = 18.2\,\mu\mathrm{as\,cty}^{-1}$, $\dot\varpi_\mathrm{1pN}^\mathrm{X} = 30.4\,\mu\mathrm{as\,cty}^{-1}$, $\dot\epsilon_\mathrm{1pN}^\mathrm{X} = 271.4\,\mu\mathrm{as\,cty}^{-1}$, respectively. Such values, which are due to the added actions of the other planets from Venus to Saturn, are essentially at the same level of, or larger by one order of magnitude than, the latest formal errors in the Hermean orbital precessions calculated with the EPM2017 ephemerides. The perihelion precession $\dot\varpi_\mathrm{1pN}^\mathrm{X}$ turns out to be smaller than some values recently appeared in the literature in view of a possible measurement with the ongoing BepiColombo mission. Linear combinations of the supplementary advances of the Keplerian orbital elements for several planets, if determined experimentally by the astronomers, could be set up in order to disentangle the 1pN $N$-body effects of interest from the competing larger precessions like those due to the Sun's quadrupole moment $J_2$ and angular momentum $\boldsymbol{S}$.