The dark mass signature in the orbit of S2

Context. The Schwarzschild precession of star S2, which orbits the massive black hole at the centre of the Milky Way, has recently been detected with the result of similar to 12 arcmin per orbit. The same study also improved the 1 sigma upper bound on a possibly present dark continuous extended mass distribution (e.g. faint stars, stellar remnants, stellar mass black holes, or dark matter) within the orbit of S2 to similar to 4000 M-circle dot. The secular (i.e. net) effect of an extended mass onto a stellar orbit is known as mass precession, and it runs counter to the Schwarzschild precession. 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 1 sigma dark mass detection thresholds given different datasets on one full orbit. In particular, with a full orbit of data of 50 microarcsec (VLTI/GRAVITY) and 10 km s(-1) (VLT/SINFONI) precision, the 1 sigma bound would improve to similar to 1000 M-circle dot, for example. Conclusions. The current upper dark mass bound of similar to 4000 M-circle dot has mainly been obtained from a combination of GRAVITY and VLT/NACO astrometric data, as well as from SINFONI spectroscopic data, where the GRAVITY data were limited to the pericentre half. From our results 3 and 4, we know that all components were thereby crucial, but also that the GRAVITY data were dominant in the astrometric components in constraining the dark mass. From results 1 and 2, we deduce that a future population of the apocentre half with GRAVITY data points will substantially further improve the dark mass sensitivity of the dataset, and we note that at the time of publication, we already entered this regime. In the context of the larger picture, our analysis demonstrates how precession effects that interfere on secular timescales can clearly be distinguished from each other based on their distinct astrometric signatures within a single orbit. The extension of our analysis to the Lense-Thirring precession should thus be of value in order to assess future spin detection prospects for the galactic centre massive black hole.