What stellar orbit is needed to measure the spin of the Galactic centre 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, total observations Nobs, astrometric precision sigma(x), and normalized black hole spin., we find that a(orb)(1 - e(2))(3/4) less than or similar to 300RS root T/4yr (N-obs/120)(0.25) root 10 mu as/sigma(x) root x/0.9 is needed. For x = 0.9 and a potential observing campaign with sigma(x) = 10 mu as, 30 observations yr(-1) and duration 4-10 yr, we expect similar to 0.1 star with K < 19 satisfying this constraint based on the current knowledge about the stellar population in the central 1 arcsec. We also propose a method through which GRAVITY could potentially measure radial velocities with precision similar to 50 km s(-1). If the astrometric precision can be maintained, adding radial velocity information increases the expected number of stars by roughly a factor of 2. While we focus on GRAVITY, the results can also be scaled to parameters relevant for future extremely large telescopes.