Streamlines of the Mean Stellar Motions in Elliptical Galaxies

The stellar velocity fields of elliptical galaxies hold clues to their dynamical structure and origin. The construction of velocity-field models is greatly simplified by assuming an approximate geometrical form for the streamlines of the mean stellar motions, for example, representing the streamlines of short-axis and long-axis tube orbits by coordinate lines in a confocal coordinate system. A single confocal system precisely fits the mean motions of all tube orbits in a Stackel potential, but these potentials are not sufficiently general. Here we test the conjecture that confocal streamlines may still be a valid approximation for more realistic triaxial systems. We numerically integrate orbits in Schwarzschild's logarithmic potential. Six sets of axis ratios are used; in each set, ~50 orbits, comprising short-axis and long-axis tubes as well as some resonant families, are integrated for ~20,000 dynamical times, and the average velocity is found in each of ~4000 spatial cells. Confocal streamlines are compared with the velocity field by finding the rms magnitude of the cross product between the velocity vectors and the streamlines. Minimizing this quantity yields a best-fit confocal system for each orbit. We find that the great majority of orbits at a given energy in each potential can be fitted by nearly identical confocal systems. There are statistically significant differences between the average streamline parameters obtained for different orbit families, but the differences are small. We show that the fitted parameters reproduce, to high accuracy, the location of the boundary between short-axis and outer long-axis tubes, which is a direct measure of the triaxiality of the potential. These results strongly support efforts to obtain accurate statistical measurements of triaxiality from kinematic observations and reasonably simple velocity-field models.