A post-Newtonian gravitomagnetic effect on the orbital motion of a test particle around its primary induced by the spin of a distant third body

TL;DR

This paper investigates a relativistic gravitomagnetic 3-body effect caused by a distant body's spin, analyzing its impact on the orbital elements of a test particle and assessing potential observability in the solar system.

Contribution

It provides an analytical and numerical study of the post-Newtonian gravitomagnetic effects induced by a third body's spin on a test particle's orbit, including potential measurable signals.

Findings

Relativistic precessions can reach 10-50 mas/yr around moons of giant planets.

Range-rate signals for a Europa orbiter can exceed current Doppler measurement accuracy.

Sun's angular momentum effects are at the microarcsecond per year level.

Abstract

We study a general relativistic gravitomagnetic 3-body effect induced by the spin angular momentum ${\boldsymbol S}_\textrm{X}$ of a rotating mass $M_\textrm{X}$ orbited at distance $r_\textrm{X}$ by a local gravitationally bound restricted two-body system $\mathcal{S}$ of size $r\ll r_\textrm{X}$ consisting of a test particle revolving around a massive body $M$. At the lowest post-Newtonian order, we analytically work out the doubly averaged rates of change of the Keplerian orbital elements of the test particle by finding non-vanishing long-term effects for the inclination $I$, the node $\Omega$ and the pericenter $\omega$. Such theoretical results are confirmed by a numerical integration of the equations of motion for a fictitious 3-body system. We numerically calculate the magnitudes of the post-Newtonian gravitomagnetic 3-body precessions for some astronomical scenarios in our solar system. For putative man-made orbiters of the natural moons Enceladus and Europa in the external fields of Saturn and Jupiter, the relativistic precessions due to the angular momenta of the gaseous giant planets can be as large as $\simeq 10-50~\textrm{milliarcseconds~per~year}~\left(\textrm{mas~yr}^{-1}\right)$. A preliminary numerical simulation shows that, for certain orbital configurations of a hypothetical Europa orbiter, its range-rate signal $\Delta\dot\rho$ can become larger than the current Doppler accuracy of the existing spacecraft Juno at Jupiter, i.e. $\sigma_{\dot\rho}=0.015~\textrm{mm~s}^{-1}$, after 1 d. The effects induced by the Sun's angular momentum on artificial probes of Mercury and the Earth are at the level of $\simeq 1-0.1~\textrm{microarcseconds~per~year}~\left(\mu\textrm{as~yr}^{-1}\right)$.