Kinematic modelling of the Milky Way using the RAVE and GCS stellar surveys

We investigate the kinematic parameters of the Milky Way disc using the RAVE and GCS stellar surveys. We do this by fitting a kinematic model to the data taking the selection function of the data into account. For stars in the GCS we use all phase-space coordinates, but for RAVE stars we use only $(l,b,v_{\rm los})$. Using MCMC technique, we investigate the full posterior distributions of the parameters given the data. We investigate the `age-velocity dispersion' relation for the three kinematic components ($\sigma_R,\sigma_{\phi},\sigma_z$), the radial dependence of the velocity dispersions, the Solar peculiar motion ($U_{\odot},V_{\odot}, W_{\odot} $), the circular speed $\Theta_0$ at the Sun and the fall of mean azimuthal motion with height above the mid-plane. We confirm that the Besan\c{c}on-style Gaussian model accurately fits the GCS data, but fails to match the details of the more spatially extended RAVE survey. In particular, the Shu distribution function (DF) handles non-circular orbits more accurately and provides a better fit to the kinematic data. The Gaussian distribution function not only fits the data poorly but systematically underestimates the fall of velocity dispersion with radius. We find that correlations exist between a number of parameters, which highlights the importance of doing joint fits. The large size of the RAVE survey, allows us to get precise values for most parameters. However, large systematic uncertainties remain, especially in $V_{\odot}$ and $\Theta_0$. We find that, for an extended sample of stars, $\Theta_0$ is underestimated by as much as $10\%$ if the vertical dependence of the mean azimuthal motion is neglected. Using a simple model for vertical dependence of kinematics, we find that it is possible to match the Sgr A* proper motion without any need for $V_{\odot}$ being larger than that estimated locally by surveys like GCS.