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Orbital motions as gradiometers for post-Newtonian tidal effects

The direct long-term changes occurring in the orbital dynamics of a local gravitationally bound binary system $S$ due to the post-Newtonian tidal acceleration caused by an external massive source are investigated. A class of systems made of a test particle $m$ rapidly orbiting with orbital frequency $n_{\rm b}$ an astronomical body of mass $M$ which, in turn, slowly revolves around a distant object of mass $M^{'}$ with orbital frequency $n_{\rm b}^{'}\ll n_{\rm b}$ is considered. The characteristic frequencies of the non-Keplerian orbital variations of $m$ and of $M$ itself are assumed to be negligible with respect to both $n_{\rm b}$ and $n_{\rm b}^{'}$. General expressions for the resulting Newtonian and post-Newtonian tidal orbital shifts of $m$ are obtained. The future missions BepiColombo and JUICE to Mercury and Ganymede, respectively, are considered in view of a possible detection. The largest effects, of the order of $\approx 0.1-0.5$ milliarcseconds per year (mas yr$^{-1}$), occur for the Ganymede orbiter of the JUICE mission. Although future improvements in spacecraft tracking and orbit determination might, perhaps, reach the required sensitivity, the systematic bias represented by the other known orbital perturbations of both Newtonian and post-Newtonian origin would be overwhelming. The realization of a dedicated artificial mini-planetary system to be carried onboard and Earth-orbiting spacecraft is considered as well. Post-Newtonian tidal precessions as large as $\approx 1-10^2$ mas yr$^{-1}$ could be obtained, but the quite larger Newtonian tidal effects would be a major source of systematic bias because of the present-day percent uncertainty in the product of the Earth's mass times the Newtonian gravitational parameter.