A new formula for disc kinematics
  In a disc galaxy the distribution of azimuthal components of velocity is very
skew. In the past this skewness has been modelled by superposed Gaussians. We
use dynamical arguments to derive an analytic formula that can be fitted to
observed velocity distributions, and validate it by fits to the velocities
derived from a dynamically rigorous model, and to a sample of local stars with
accurate space velocities. Our formula is much easier to use than a full
distribution function. It has fewer parameters than a multi-Gaussian fit, and
the best-fitting model parameters give insight into the underlying disc
dynamics. In particular, once the azimuthal velocities of a sample have been
successfully fitted, the apparatus provides a prediction for the corresponding
distribution of radial velocities vR . An effective formula like ours is
invaluable when fitting to data for stars at some distance from the Sun because
it enables one to make proper allowance for the errors in distance and proper
motion when determining the underlying disc kinematics. The derivation of our
formula elucidates the way the horizontal and vertical motions are closely
intertwined, and makes it evident that no stellar population can have a scale
height and vertical velocity dispersions that are simultaneously independent of
radius. We show that the oscillation of a star perpendicular to the Galactic
plane modifies the effective potential in which the star moves radially in such
a way that the more vertical energy a star has, the larger is the mean radius
of its orbit.
