Trapped orbits and solar-neighbourhood kinematics

Abstract

Torus mapping yields constants of motion for stars trapped at a resonance. Each such constant of motion yields a system of contours in velocity space at the Sun and neighbouring points. If Jeans' theorem applied to resonantly trapped orbits, the density of stars in velocity space would be equal at all intersections of any two contours. A quantitative measure of the violation of this principle is defined and used to assess various pattern speeds for a model of the bar recently fitted to observations of interstellar gas. Trapping at corotation of a bar with pattern speed near $36\, \mathrm{Gyr}^{-1}$ is favoured and trapping at the outer Lindblad resonance is disfavoured. As one moves around the Sun the structure of velocity space varies quite rapidly, both as regards the observed star density and the zones of trapped orbits. The data seem consistent with trapping at corotation.