A method to calculate the local density distribution of the Galaxy from the Tycho-Gaia Astrometric Solution data

New and more reliable distances and proper motions of a large number of stars in the Tycho-Gaia Astrometric Solution (TGAS) catalogue allow to calculate the local matter density distribution more precisely than earlier. We devised a method to calculate the stationary gravitational potential distribution perpendicular to the Galactic plane by comparing the vertical probability density distribution of a sample of observed stars with the theoretical probability density distribution computed from their vertical coordinates and velocities. We applied the model to idealised test stars and to the real observational samples. Tests with two mock datasets proved that the method is viable and provides reasonable results. Applying the method to TGAS data we derived that the total matter density in the Solar neighbourhood is $0.09\pm 0.02 \text{M}_\odot\text{pc}^{-3}$ being consistent with the results from literature. The matter surface density within $|z|\le 0.75 \text{kpc}$ is $42\pm 4 \text{M}_\odot\text{pc}^{-2}$. This is slightly less than the results derived by other authors but within errors is consistent with previous estimates. Our results show no firm evidence for significant amount of dark matter in the Solar neighbourhood. However, we caution that our calculations at $|z| \leq 0.75$ kpc rely on an extrapolation from the velocity distribution function calculated at $|z| \leq 25$ pc. This extrapolation can be very sensitive to our assumption that the stellar motions are perfectly decoupled in R and z, and to our assumption of equilibrium. Indeed, we find that $\rho (z)$ within $|z|\le 0.75$ kpc is asymmetric with respect to the Galactic plane at distances $|z| = 0.1-0.4$ kpc indicating that the density distribution may be influenced by density perturbations.