Post-Newtonian accelerations of a Mercury orbiter

TL;DR

This paper develops a first-post-Newtonian Mercury-centric framework to model spacecraft motion using the DSX/Brumberg–Kopeikin approach, enabling accurate treatment of Mercury's multipole field and external gravito-electric/magnetic perturbations. It constructs Mercury-centered coordinates with carefully defined time scales and scalings, and implements a quadruple-precision PN integrator to evaluate accelerations along BepiColombo's MPO and Mio trajectories, validating the local-to-barycentric transformations against the EIH-based barycentric model. The results show Schwarzschild and solar geodetic precession are the dominant PN effects, with the gravito-electric term reaching ~1e-12 m/s^2 for Mio, while other PN terms are negligible for the MORE experiment; nevertheless, the approach provides a practical, high-accuracy model for Mercury orbiters and can inform future planetary missions. The work thus delivers a robust methodology for relativistic orbit modeling in planet-centered frames, with clear guidance on which PN terms must be included for cm- to cm-level trajectory accuracy and how to extend the framework to other planets and observables.

Abstract

We investigate the relativistic modeling of spacecraft motion in Mercury's post-Newtonian local coordinates. This investigation is motivated by the fact that Mercury's post-Newtonian gravitational field (as well as that of any other planet) admits an expansion in terms of multipole moments, which are most appropriately defined in the local reference system. The equations of motion in the Mercury-centric local frame include relativistic local perturbations, given by the Schwarzschild term, Lense-Thirring precession, and the acceleration due to the quadrupole moment, and relativistic third-body perturbations, which are the gravito-electric and gravito-magnetic accelerations, along with a coupling term between Mercury and other solar system bodies. The relativistic third-body perturbations are usually neglected in all practical applications. In this study, we analyze the magnitude of the post-Newtonian terms of the equations of motion formulated in the Mercury-centric frame, evaluating them along the trajectories of the two BepiColombo spacecrafts. Based on this analysis, we provide a practical approach for constructing a high-accuracy relativistic orbital model suitable for a Mercury orbiter.

Figures (10)

• Figure 1: Quasi-periodic part of the TM-TDB difference evaluated at the center of mass of Mercury from the 1st January 1997 TDB. This term has an amplitude of approximately 0.0127 s. The linear part of the transformation amounts to $-3.825\time10^{-8}$.
• Figure 2: Norm of post-Newtonian terms in $\Phi_\text{M}^a$, evaluated on the MPO's trajectory.
• Figure 3: Norm of post-Newtonian $\Phi_\text{el}^a$, $\Phi_\text{mg}^a/c^2$, $\Phi_\text{coup}^a/c^2$, and of the difference between $\Phi_\text{mg}^a/c^2$ and the geodetic precession, evaluated on the MPO's trajectory.
• Figure 4: Norm of post-Newtonian terms in $\Phi_\text{M}^a$, evaluated on Mio's trajectory.
• Figure 5: Norm of post-Newtonian $\Phi_\text{el}^a$, $\Phi_\text{mg}^a/c^2$, $\Phi_\text{coup}^a/c^2$, and of the difference between $\Phi_\text{mg}^a/c^2$ and the geodetic precession, evaluated on Mio's trajectory.