The dark mass signature in the orbit of S2
Aims. We explore a strategy for how the Schwarzschild and mass precessions can be separated from each other despite their secular interference, by pinpointing their signatures within a single orbit. From these insights, we then seek to assess the prospects for improving the dark mass constraints in the coming years. Methods. We analysed the dependence of the osculating orbital elements and of the observables on true anomaly, and we compared these functions for models with and without extended mass. We then translated the maximum astrometric impacts within one orbit to detection thresholds given hypothetical data of different accuracies. These theoretical investigations were then supported and complemented by an extensive mock-data fitting analysis. Results. We have four main results. 1. While the mass precession almost exclusively impacts the orbit in the apocentre half, the Schwarzschild precession almost exclusively impacts it in the pericentre half, allowing for a clear separation of the effects. 2. Data that are limited to the pericentre half are not sensitive to a dark mass, while data limited to the apocentre half are, but only to a limited extent. 3. A full orbit of data is required to substantially constrain a dark mass. 4. For a full orbit of astrometric and spectroscopic data, the astrometric component in the pericentre half plays the stronger role in constraining the dark mass than the astrometric data in the apocentre half. Furthermore, we determine the 1σ dark mass detection thresholds given different datasets on one full orbit. In particular, with a full orbit of data of 50 microarcseconds (VLTI/GRAVITY) and 10 km/s (VLT/SINFONI) precision, the 1σ bound would improve to about 1000 solar masses, for example.