Catching Up with the Fastest Star in the Galaxy: A Two-Passage GRAVITY+ Campaign to Detect S301's Schwarzschild Precession

Abstract

S301 is the fastest known star in the Galaxy. It orbits Sgr A* every 8.68 yr on an e = 0.9832 orbit that carries it within 136 Schwarzschild radii at 25,600 km/s (GRAVITY Collaboration 2026). An orbit like that must precess, by the same Schwarzschild effect that turns Mercury's perihelion, whatever the spin of the black hole. We ask whether an astrometry-only GRAVITY+ campaign covering the 2031.8 and 2040.5 periapsis passages can catch it. We validate a relativistic light-curve model, design a 131-epoch Fisher-optimal cadence with a periapsis floor, fit a seven-parameter orbit including the light-travel-time delay, and build a systematic error budget. With white noise at the 207 microarcseconds achieved on S301, the fit recovers the injected rate, 0.2307 deg/yr, as 0.2324 +/- 0.0023 deg/yr (+/-0.0012 at the forecast 100 microarcseconds). Real astrometric noise is not white. We add per-run calibration offsets and a wandering Sgr A* photocentre at GRAVITY's published amplitudes, marginalize over the reference zero point, mass and distance, add the bounded systematics in quadrature, and inflate by the excess scatter of real GRAVITY orbit fits. What survives is 0.0074 deg/yr, a 31 sigma detection. The Sun hides the first periapsis for 153 d, at a cost to the period and periapsis epoch but not to the precession rate. Radial velocities are out of reach at S301's magnitude; photometry only validates the light-curve model. The spin of Sgr A* could shift the rate by 5.17-5.74 per cent and cannot be bounded, so the campaign is a spin-agnostic test of general relativity, and a forecast until the star comes round.