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The wide binary orbit of Proxima Centauri around $\alpha$ Centauri A and Bdiffers significantly between Newtonian and Milgromian dynamics (MOND). Bycombining previous calculations of this effect with mock observations generatedusing a Monte Carlo procedure, we show that this prediction can be tested usinghigh precision astrometry of Proxima Centauri. This requires ${\approx 10}$years of observations at an individual epoch precision of $0.5 \, \mu$as,within the design specifications of the proposed Theia mission. In general, therequired duration should scale as the 2/5 power of the astrometric precision. Along-period planet could produce a MOND-like astrometric signal, but only if ithas a particular ratio of mass to separation squared and a sky position closeto the line segment connecting Proxima Centauri with $\alpha$ Centauri.Uncertainties in perspective effects should be small enough for this test ifthe absolute radial velocity of Proxima Centauri can be measured to within${\approx 10}$ m/s, better than the present accuracy of 32 m/s. We expect therequired improvement to become feasible using radial velocity zero pointsestimated from larger samples of close binaries, with the Sun providing ananchor. We demonstrate that possible astrometric microlensing of ProximaCentauri is unlikely to affect the results. We also discuss why it should bepossible to find sufficiently astrometrically stable reference stars.Adequately addressing these and other issues would enable a decisive test ofgravity in the currently little explored low acceleration regime relevant tothe dynamical discrepancies in galactic outskirts.

Directly testing gravity with Proxima Centauri