Gaia-DR2 extended kinematical maps. Part III: Rotation curves analysis,
  dark matter, and Modified Newtonian dynamics tests
  Recent statistical deconvolution methods have produced extended kinematical
maps in a range of heliocentric distances that are a factor of two to three
larger than those analysed in the Gaia Collaboration based on the same data. In
this paper, we use such maps to derive the rotation curve both in the Galactic
plane and in off-plane regions and to analyse the density distribution. By
assuming stationary equilibrium and axisymmetry, we used the Jeans equation to
derive the rotation curve. Then we fit it with density models that include both
dark matter and predictions of the MOND (Modified Newtonian dynamics) theory.
Since the Milky Way exhibits deviations from axisymmetry and equilibrium, we
also considered corrections to the Jeans equation. To compute such corrections,
we ran N-body experiments of mock disk galaxies where the departure from
equilibrium becomes larger as a function of the distance from the centre. The
rotation curve in the outer disk of the Milky Way that is constructed with the
Jeans equation exhibits very low dependence on $R$ and $z$ and it is
well-fitted both by dark matter halo and MOND models. The application of the
Jeans equation for deriving the rotation curve, in the case of the systems that
deviate from equilibrium and axisymmetry, introduces systematic errors that
grow as a function of the amplitude of the average radial velocity. In the case
of the Milky Way, we can observe that the amplitude of the radial velocity
reaches $\sim 10\%$ that of the azimuthal one at $R\approx 20$ kpc. Based on
this condition, using the rotation curve obtained from the Jeans equation to
calculate the mass may overestimate its measurement.
