Constraining the accretion flow density profile near Sgr A* using the
  $L'$-band emission of the S2 star
  The density profile of the ambient medium around a supermassive black hole
plays an important role in understanding the inflow-outflow mechanisms in the
Galactic Centre. We constrain the spherical density profile using the stellar
bow shock of the star S2 which orbits the supermassive black hole at the
Galactic Centre with the pericentre distance of 14.4 mas ($\sim$ 1500
R$_\text{s}$). Assuming an elliptical orbit, we apply celestial mechanics and
the theory of bow shocks that are at ram pressure equilibrium. We analyse the
measured infrared flux density and magnitudes of S2 in the L'-band (3.8 micron)
obtained over seven epochs in the years between 2004-2018. We detect no
significant change in S2 flux density until the recent periapse in May 2018.
The intrinsic flux variability of S2 is at the level of 2 - 3%. Based on the
dust-extinction model, the upper limit on the number density at the S2 periapse
is $\sim$1.87$\times$10$^9$ cm$^{-3}$, which yields a density slope of at most
3.20. Using the synchrotron bow-shock emission, we obtain an ambient density of
$\leq$ 1.01$\times$10$^5$ cm$^{-3}$ and a slope of $\leq$ 1.47. These values
are consistent with a wide variety of media from hot accretion flows to
potentially colder and denser media comparable in properties to broad-line
region clouds. A standard thin disc can be, however, excluded at the distance
of the S2 pericentre. Based on our sensitivity of 0.01 mag, we can distinguish
between hot accretion flows and thin, cold discs, where the latter can be
excluded at the scale of the S2 periapse. Future observations of stars with
smaller pericentre distances in the S cluster using instruments such as
METIS@ELT with a photometric sensitivity of as much as 10$^{-3}$ mag will allow
to probe the Galactic Centre medium at intermediate scales at densities as low
as $\sim$ 700 cm$^{-3}$ in case of non-thermal bow-shock emission.
