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 μas. For an observing campaign with duration T years, total observations N_obs, astrometric precision σ_x, and normalized black hole spin χ, we find that a_orb(1-e^2)^{3/4} ≲ 300 R_S √{T/4 {yr}} (N_obs/120)^{0.25} √{10 μ as/σ _x} √{χ /0.9} is needed. For χ = 0.9 and a potential observing campaign with σ _x = 10 μas, 30 observations yr^-1 and duration 4-10 yr, we expect ∼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 ∼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.