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The stellar velocity fields of elliptical galaxies are important diagnostics of their intrinsic shapes, which in turn hold clues to their origin and evolution. The construction of dynamical velocity field models is a complex task, but is greatly simplified if one has a geometrical approximation for the streamlines of the mean stellar motions. In the special class of St\"ackel potentials, a single confocal ellipsoidal coordinate system gives a set of streamlines that precisely fit the mean motions of all orbits in a given potential. Unfortunately, the St\"ackel potentials are not realistic models for ellipticals; but confocal streamlines may nonetheless be a valid approximation for the mean motions in realistic triaxial systems. Here we test this conjecture, by fitting confocal streamlines to the mean velocities of individual orbits in realistic potentials.
We numerically integrate orbits in Schwarzschild's (1993) logarithmic potential and find the average velocity vector in each of $\sim 2000$ spatial cells. Integrations are performed over typically $\sim 50000$ orbital periods in each case. Six sets of axis ratios are used; in each, $\sim 50$ orbits, comprising short-axis and long-axis tubes, as well as the circulating resonant families, are integrated. Confocal streamlines are compared to the orbital velocity vectors by finding the RMS magnitude of the cross product between the true velocity and streamline unit vectors. Minimizing this quantity yields a best fit confocal system for each orbit. Initial results for non-rotating potentials indicate that the parameters of the fitted coordinate system do not vary much among different orbits in the same potential; thus confocal streamlines will be a good approximation to the total mean velocity field. Results for rotating potentials will also be presented.
\noindent Schwarzschild, M. 1993, ApJ, 409, 563.
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