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A novel experimental setup to measure deviations from the $1/r^2$ distancedependence of Newtonian gravity was proposed in arXiv:1609.05654. Theunderlying theoretical idea was to study the orbits of a microscopically-sizedplanetary system composed of a ``Satellite'', with mass $m_{\rm S} \sim {\calO}(10^{-9})$ g, and a ``Planet'', with mass $M_{\rm P} \sim {\cal O} (10^{-5})$ g at an initial distance of hundreds of microns. The detection of precessionof the orbit in this system would be an unambiguous indication of a centralpotential with terms that scale with the distance differently from $1/r$. Thisis a huge advantage with respect to the measurement of the absolute strength ofthe attraction between two bodies, as most electrically-induced backgroundpotentials do indeed scale as $1/r$. Detection of orbit precession isunaffected by these effects, allowing for better sensitivities. InarXiv:2106.08611, the impact of other subleading backgrounds that may induceorbit precession, such as, {\em e.g.}, the electrical Casimir force or generalrelativity, was studied in detail. It was found that the proposed setup couldtest Yukawa-like corrections, $\alpha \times \exp(-r/\lambda)$, to the $1/r$potential with couplings as low as $\alpha \sim 10^{-2}$ for distances as smallas $\lambda \sim 10$ $\mu$m, improving by roughly an order of magnitude presentbounds. In this paper, we start to move from a theoretical study of theproposal to a more realistic implementation of the experimental setup. As afirst step, we study the impact of air viscosity on the proposed setup and seehow the setup should be modified in order to preserve the theoreticalsensitivity achieved in our previous works.

Towards a realistic setup for a dynamical measurement of deviations from Newton's $1/r^2$ law: the impact of air viscosity