Quasi integral of motion for axisymmetric potentials

TL;DR

This paper introduces a new method to estimate a third integral of motion in axisymmetric galactic potentials using a Staeckel approximation, which accurately conserves the integral for disc stars across various models.

Contribution

It provides an explicit formula for the third integral based on the potential, tested on galactic models, and demonstrates its utility in modeling stellar distributions.

Findings

The third integral is conserved with 0.1% to 1% accuracy.

The method works well for heights up to 6 kpc and radii from 5 to 15 kpc.

The distribution function remains approximately stationary using this integral.

Abstract

We present an estimate of the third integral of motion for axisymmetric three-dimensional potentials. This estimate is based on a Staeckel approximation and is explicitly written as a function of the potential. We tested this scheme for the Besancon Galactic model and two other disc-halo models and find that orbits of disc stars have an accurately conserved third quasi integral. The accuracy ranges from of 0.1% to 1% for heights varying from z = 0~kpc to z= 6 kpc and Galactocentric radii R from 5 to 15kpc. We also tested the usefulness of this quasi integral in analytic distribution functions of disc stellar populations: we show that the distribution function remains approximately stationary and that it allows to recover the potential and forces by applying Jeans equations to its moments.