Testing General Relativity with the BepiColombo Mission: Post-Newtonian Modeling and Applications

This thesis presents two scientific investigations, both conducted in the context of testing general relativity with the BepiColombo mission to Mercury. BepiColombo is a joint space mission between the European Space Agency (ESA) and the Japan Aerospace Exploration Agency (JAXA), launched in October 2018 and planned for orbit insertion around Mercury in November 2026. It is composed of two spacecraft, the Mercury Planetary Orbiter (MPO) and the Mercury Magnetospheric Orbiter (Mio), which will be placed in different orbits around Mercury and will study the surface, the interior structure, and the magnetosphere of the planet. The Mercury Orbiter Radio Science Experiment (MORE) is one of the experiments on board the MPO spacecraft, whose scientific goals focus on the measurement of Mercury’s gravity field, the determination of Mercury’s rotational state, along with performing a test of Einstein's theory of general relativity and of alternative theories of gravity. Indeed, Mercury's unique position in the solar system makes it a natural setting for fundamental physics experiments. Being the closest planet to the Sun and moving at high orbital speeds, it experiences a gravitational environment where relativistic effects are particularly significant. During the hermean phase of the mission, MORE will be performed by accurately determining the orbit of the MPO around Mercury and the orbits of Mercury and the Earth around the solar system barycenter, using state-of-the-art on-board and on-ground instrumentation. This will allow the estimation of several parameters of interest: Mercury's gravity field coefficients, its gravitational tidal response, the orientation and spin rate of its pole, and the amplitude and phase of its physical librations in longitude, which are relevant to the gravimetry and rotation experiments, as well as the parameters of the parameterized post-Newtonian (PPN) formalism, a framework used to constrain multiple theories of gravitation, including general relativity, in the context of the relativity experiment. In addition to tests during the orbital phase of the mission, general relativity can also be probed during BepiColombo’s cruise phase via a solar conjunction experiment (SCE). When the Earth, spacecraft, and Sun align, radio signals passing near the Sun experience a delay in their propagation due to spacetime curvature. Tracking the spacecraft near a solar conjunction and measuring the time delay in the propagation of radio signals and their Doppler shift allows us to obtain an estimate of the PPN parameter $\gamma$ (whose predicted value by general relativity is 1). The current state-of-the-art estimate of $\gamma$ was provided by the 2002 NASA Cassini-Huygens mission SCE, yielding $\gamma - 1 = (2.1 \pm 2.3) \times 10^{-5}$, consistent with general relativity. The first scientific objective of this thesis is related to the orbital phase of the BepiColombo mission, and concerns the construction of a high-accuracy relativistic orbital model of the motion of the MPO around Mercury. According to the International Astronomical Union (IAU) 2000 Resolutions, which represent practical guidelines for applications in astrometry, celestial mechanics and metrology, the motion of a Mercury orbiter (as well as of an orbiter of any other planet) should be described in the planet's corresponding local relativistic reference system, to be defined similarly to the Geocentric Celestial Reference System (GCRS) for the Earth. This is because 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. We investigate the full set of post-Newtonian equations of motion in the Mercury-centric local frame, which 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. We present analytical and numerical estimates of the post-Newtonian terms of the equations of motion in the Mercury-centric frame, by evaluating them along the trajectories of the two BepiColombo spacecrafts. Based on this analysis, and starting from the basic principles of the theory of astronomical reference systems, we provide a practical approach for building a high-accuracy relativistic model suitable for a Mercury orbiter. The second investigation of this thesis consists of the analysis of the radio-tracking data of the first BepiColombo SCE, conducted with the orbit determination software Orbit14. Orbit14 was developed by the Celestial Mechanics Group of the Department of Mathematics at the University of Pisa, under contract with the Italian Space Agency (ASI). It has been used for the analysis of the radio science data of the Juno mission to Jupiter and to conduct simulations of the BepiColombo MORE experiment. For this thesis, it has also been employed for the analysis of the BepiColombo SCEs, after appropriate modifications and additions to the software. Out of the total six conjunction experiments where general relativity was testable, we chose to focus on the first one, carried out from March 10, 2021 to March 24, 2021, as it was performed in the quietest dynamical environment. Indeed, because of its proximity to the Sun, BepiColombo is subject to strong non-gravitational perturbations affecting the accuracy of the radio science experiments, some of which cannot be described by available models or measurements, and therefore a different strategy must be adopted in order to compensate for these disturbances.