Stellar kinematics using a third integral of motion: method and application on the Andromeda galaxy

TL;DR

This paper introduces a method to model stellar kinematics in axisymmetric galaxies using a third integral of motion, demonstrating its application on the Andromeda galaxy to reproduce observed kinematics.

Contribution

The paper develops a new approach to incorporate a third integral of motion into Jeans equation-based models for galaxy dynamics, improving the description of stellar kinematics.

Findings

Successfully applied the method to the Andromeda galaxy.

Reproduced observed stellar kinematics with good accuracy.

Found that velocity dispersion ratios vary with radius and height.

Abstract

We probe the feasibility of describing the structure of a multi-component axisymmetric galaxy with a dynamical model based on the Jeans equations while taking into account a third integral of motion. We demonstrate that using the third integral in the form derived by G. Kuzmin, it is possible to calculate the stellar kinematics of a galaxy from the Jeans equations by integrating the equations along certain characteristic curves. In cases where the third integral of motion does not describe the system exactly, the derived kinematics would describe the galaxy only approximately. We apply our method to the Andromeda galaxy, for which the mass distribution is relatively firmly known. We are able to reproduce the observed stellar kinematics of the galaxy rather well. The calculated model suggests that the velocity dispersion ratios ${\sigma}_z^2/{\sigma}_R^2$ of M31 decrease with increasing R. Moving away from the galactic plane, ${\sigma}_z^2/{\sigma}_R^2$ remains the same. The velocity dispersions ${\sigma}_{\theta}^2$ and ${\sigma}_R^2$ are roughly equal in the galactic plane.