  Precise measurements of the S-stars orbiting SgrA* have set strong
constraints on the nature of the compact object at the centre of the Milky Way.
The presence of a black hole in that region is well established, but its
neighboring environment is still an open debate. In that respect, the existence
of dark matter in that central region may be detectable due to its strong
signatures on the orbits of stars: the main effect is a Newtonian precession
which will affect the overall pericentre shift of S2, the latter being a target
measurement of the GRAVITY instrument. The exact nature of this dark matter
(e.g., stellar dark remnants or diffuse dark matter) is unknown. This article
assumes it to be an scalar field of toroidal distribution, associated with
ultra-light dark matter particles, surrounding the Kerr black hole. Such a
field is a form of "hair" expected in the context of superradiance, a mechanism
that extracts rotational energy from the black hole. Orbital signatures for the
S2 star are computed and shown to be detectable by GRAVITY. The scalar field
can be constrained because the variation of orbital elements depends both on
the relative mass of the scalar field to the black hole and on the field mass
coupling parameter.
