Models of our Galaxy - II

Stars near the Sun oscillate both horizontally and vertically. In a previous paper by Binney it was assumed that the coupling between these motions can be modelled by determining the horizontal motion without reference to the vertical motion, and recovering the coupling between the motions by assuming that the vertical action is adiabatically conserved as the star oscillates horizontally. Here, we show that, although the assumption of adiabatic invariance works well, more accurate results can be obtained by taking the vertical action into account when calculating the horizontal motion. We use orbital tori to present a simple but fairly realistic model of the Galaxy's discs in which the motion of stars is handled rigorously, without decomposing it into horizontal and vertical components. We examine the ability of the adiabatic approximation to calculate the model's observables, and find that it performs perfectly in the plane, but errs slightly away from the plane. When the new correction to the adiabatic approximation is used, the density, mean-streaming velocity and velocity dispersions are in error by less than 10 per cent for distances up to 2.5 kpc from the Sun. The torus-based model reveals that at locations above the plane, the long axis of the velocity ellipsoid points almost to the Galactic centre, even though the model potential is significantly flattened. This result contradicts the widespread belief that the shape of the Galaxy's potential can be strongly constrained by the orientation of velocity ellipsoid near the Sun. An analysis of individual orbits reveals that in a general potential the orientation of the velocity ellipsoid depends on the structure of the model's distribution function as much as on its gravitational potential, contrary to what is the case for Stäckel potentials. We argue that the adiabatic approximation will provide a valuable complement to torus-based models in the interpretation of current surveys of the Galaxy.