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Astrometric and spectroscopic monitoring of individual stars orbiting thesupermassive black hole in the Galactic Center offer a promising way to detectgeneral relativistic effects. While low-order effects are expected to bedetected following the periastron passage of S2 in Spring 2018, detectinghigher-order effects due to black hole spin will require the discovery ofcloser stars. In this paper, we set out to determine the requirements such astar would have to satisfy to allow the detection of black hole spin. We focuson the instrument GRAVITY, which saw first light in 2016 and which is expectedto achieve astrometric accuracies $10-100 \mu$as. For an observing campaignwith duration $T$ years, $N_{obs}$ total observations, astrometric precision$\sigma_x$ and normalized black hole spin $\chi$, we find that$a_{orb}(1-e^2)^{3/4} \lesssim 300 R_S \sqrt{\frac{T}{4 \text{years}}}\left(\frac{N_{obs}}{120}\right)^{0.25} \sqrt{\frac{10 \mu as}{\sigma_x}}\sqrt{\frac{\chi}{0.9}}$ is needed. For $\chi=0.9$ and a potential observingcampaign with $\sigma_x = 10 \mu$as, 30 observations/year and duration 4-10years, we expect $\sim 0.1$ star with $K<19$ satisfying this constraint basedon the current knowledge about the stellar population in the central 1". Wealso propose a method through which GRAVITY could potentially measure radialvelocities with precision $\sim 50$ km/s. If the astrometric precision can bemaintained, adding radial velocity information increases the expected number ofstars by roughly a factor of two. While we focus on GRAVITY, the results canalso be scaled to parameters relevant for future extremely large telescopes.

What stellar orbit is needed to measure the spin of the Galactic center black hole from astrometric data?