What stellar orbit is needed to measure the spin of the Galactic center
  black hole from astrometric data?
  Astrometric and spectroscopic monitoring of individual stars orbiting the
supermassive black hole in the Galactic Center offer a promising way to detect
general relativistic effects. While low-order effects are expected to be
detected following the periastron passage of S2 in Spring 2018, detecting
higher-order effects due to black hole spin will require the discovery of
closer stars. In this paper, we set out to determine the requirements such a
star would have to satisfy to allow the detection of black hole spin. We focus
on the instrument GRAVITY, which saw first light in 2016 and which is expected
to achieve astrometric accuracies $10-100 \mu$as. For an observing campaign
with 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 observing
campaign with $\sigma_x = 10 \mu$as, 30 observations/year and duration 4-10
years, we expect $\sim 0.1$ star with $K<19$ satisfying this constraint based
on the current knowledge about the stellar population in the central 1". We
also propose a method through which GRAVITY could potentially measure radial
velocities with precision $\sim 50$ km/s. If the astrometric precision can be
maintained, adding radial velocity information increases the expected number of
stars by roughly a factor of two. While we focus on GRAVITY, the results can
also be scaled to parameters relevant for future extremely large telescopes.
