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

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.