Kinematic properties of stellar dynamical discs and spheroids

Abstract

The stellar hydrodynamical equations are solved for discs and oblate and prolate spheroids generated by Stäckel potentials to give the components of stress required to prevent gravitational collapse. The velocity ellipsoids are aligned on confocal quadric surfaces, rather than less realistic spheroidal or cylindrical polar coordinate surfaces. The solutions for the velocity second moments are projected to give plots of line-of-sight dispersions and rotation curves, which are compared with observational data. The spheroidal models behave as if the velocity ellipsoids are aligned in cylindrical polars close to the centre but in spherical polars asymptotically. For oblate spheroids, the velocity ellipsoids are largest at the position of the focus of the spheroidal coordinates, but diminish and become isotropic in the core. When viewed at inclinations of 40° or greater, radially anisotropic oblate spheroids show a peak in the line-of-sight velocity dispersion along the isophotal minor axis at or just beyond the projected position of the focus (typically about 0.25–1.0 kpc from the centre). For prolate spheroids, the velocity ellipsoids are usually largest at the centre. The line-of-sight velocity dispersions along the isophotal major and minor axes are much flatter than in the oblate case. The maxima of these curves, if present, tend to be shallow. It is concluded that the existence of a maximum in the velocity dispersion, offset from the centre along the minor axis, is a signature of oblate models dominated by radial anisotropy in the outer parts and generated by distribution functions depending on three integrals of motion.