Quasi integral of motion for axisymmetric potentials

¹ Observatoire astronomique de Strasbourg, Université de Strasbourg, CNRS, UMR 7550, 11 rue de l’Université, 67000 Strasbourg, France
e-mail: olivier.bienayme@unistra.fr
² Institut Utinam, CNRS UMR 6213, Université de Franche-Comté, OSU THETA Franche-Comté-Bourgogne, Observatoire de Besançon, BP 1615, 25010 Besançon Cedex, France

Received: 12 May 2015
Accepted: 20 July 2015

Abstract

We present an estimate of the third integral of motion for axisymmetric three-dimensional potentials. This estimate is based on a Stäckel approximation and is explicitly written as a function of the potential. We tested this scheme for the Besançon 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 15 kpc. 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.