Dynamical measurements of deviations from Newton's $1/r^2$ law

In a previous work (arXiv:1609.05654v2), an experimental setup aiming at the measurement of deviations from the Newtonian $1/r^2$ distance dependence of gravitational interactions was proposed. The theoretical idea behind this setup was to study the trajectories of a "Satellite" with a mass $m_{\rm S} \sim {\cal O}(10^{-9})$ $\mathrm{g}$ around a "Planet" with mass $m_{\rm P} \in [10^{-7},10^{-5} ]$ $\mathrm{g}$, looking for precession of the orbit. The observation of such feature induced by gravitational interactions would be an unambiguous indication of a gravitational potential with terms different from $1/r$ and, thus, a powerful tool to detect deviations from Newton's $1/r^2$ law. In this paper we optimize the proposed setup in order to achieve maximal sensitivity to look for {\em Beyond-Newtonian} corrections. We study in detail possible background sources that could induce precession and quantify their impact on the achievable sensitivity. We conclude that a dynamical measurement of deviations from newtonianity can test Yukawa-like corrections to the $1/r$ potential with strength as low as $\alpha \sim 10^{-2}$ for distances as small as $\lambda \sim 10 \, \mu\mathrm{m}$.