Schwarzschild Precession of the Star S2: A Relativistic Budget at the Galactic Centre

The orbit of the star S2 around the compact radio source Sagittarius A* advances in its own plane, and the size of that advance is a direct reading of relativistic gravity in a regime no solar-system test reaches. Using the published orbital elements and the mass and distance of GRAVITY Collaboration (2022), this paper recomputes the Schwarzschild precession from first principles: with a semi-major axis of 1035.1 AU, an eccentricity of 0.885 and a period of 16.07 years, the predicted advance is 12.21 arcminutes per orbit, or 0.760 arcminutes per year. The published interferometric measurement corresponds to a scaling factor of 1.10 plus or minus 0.19 relative to the relativistic prediction, a deviation from unity of only 0.53 standard deviations, while a purely Newtonian point-mass orbit predicts no advance at all. The pericentre of 119.4 AU lies 1408 Schwarzschild radii from the centre, which explains both why the effect is measurable and why it remained hidden until milliarcsecond interferometry matured. Every number is computed from stated inputs; nothing here is a new observation.