A new formula for disc kinematics

TL;DR

This paper introduces a new analytic formula for modeling the skewed azimuthal velocity distribution in disc galaxies, simplifying analysis and providing insights into disc dynamics and vertical-radial motion coupling.

Contribution

The paper derives and validates a new, simpler formula for disc kinematics that captures velocity distribution skewness and links vertical and radial motions, improving data fitting and interpretation.

Findings

The formula fits observed and model velocities accurately.

It predicts radial velocity distributions from azimuthal fits.

Vertical motions influence radial orbit characteristics.

Abstract

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.