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 1{\sigma} 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 1{\sigma} bound would improve to about 1000 solar
masses, for example.
