On the axis ratio of the stellar velocity ellipsoid in disks of spiral galaxies

The spatial distribution of stars in a disk of a galaxy can be described by a radial scale length and a vertical scale height. The ratio of these two scale parameters contains information on the axis ratio of the velocity ellipsoid, i.e. the ratio of the vertical to radial stellar velocity dispersions of the stars, at least at some fiducial distance from the center. The radial velocity dispersion correlates well with the amplitude of the rotation curve and with the disk integrated magnitude, as was found by Bottema (1993). These relations can be understood as the result of the stellar disk being (marginally) stable against local instabilities at all length scales. This is expressed by Toomre's well-known criterion, which relates the sheer in the rotation to a minimum value that the radial stellar velocity dispersion should have for stability for a given surface density. Via the Tully-Fisher (1977) relation, the velocity dispersion then becomes related to the integrated magnitude and hence to the scale length. The vertical velocity dispersion relates directly to the scale height through hydrostatic equilibrium. It can be shown that the ratio of the two length scales relates to the axis ratio of the velocity ellipsoid only through the Toomre parameter Q and in particular does not require a choice of the mass-to-light ratio or a distance scale. We have applied this to the statistically complete sample of edge-on galaxies, for which de Grijs (1997) has performed surface photometry and has determined the length scales in the stellar light distribution.