On the Stäckel potential approximation in the extended solar neighbourhood

Abstract

The problem of the third integral of motion has been studied for decades. Its existence was determined at the beginning of numerical experiments with orbits in an axisymmetric potential and by observations of triaxiality in the velocity dispersion of nearby stars. In general it is not possible to derive an exact expression for a general axisymmetric potential analytically. One way to find an exact third integral of motion is to approximate the gravitational potential with a potential of the Stackel form. In addition to the third integral of motion, another important property of the Stackel potential is that it separates the Hamilton-Jacobi equation in confocal spheroidal coordinates. This results in an alignment of the velocity ellipsoid with the spheroidal coordinate system in the meridional plane. In this thesis we construct a Stackel model, fixing the focal point of these coordinates based on recent observations of the velocity ellipsoid (Binney et al. 2014a). The aim of this PhD thesis is to show how a universal Stackel potential works. This Stackel model is focused on the region of the extended solar neighbourhood, up to 2 kpc radius around the Sun. The free parameters are constrained with the observational values of the local volume and surface density. Studying the vertical structure we can predict the behaviour of the local vertical force and radial dependencies at 1.1 kpc far from the plane. The value of the focal point is very important, because for different values we obtain different profiles for the radial density, different values for the scale length and different behaviour for the vertical force at large heights. We derive again the orientation of the velocity ellipsoid in the meridional plane based on RAVE data. The result for the tilt angle is in agreement with Binney et al. (2014a), both for the complete sample of red clump giants and also for different metallicities. This result is a confirmation of the derived Stackel model.