2.3. How Many PNe Velocities Are Needed for Orbital and Potential Analysis?
With a few tens of PN velocities it is possible to detect a clear trend of rotation in the outer halo; see, e.g., the work of Arnaboldi et al. for NGC 4406 and NGC 1316 [13, 14]. To determine the detailed shape of the rotation velocity field requires substantially larger samples. Fig. 1, due to N. Sambhus, shows the results of fitting the velocities of approximately 200 PNe from PN.S observations  in the E0/1 galaxy NGC 3379 with a non-parametric spline method. The velocity field shows a peak-to-peak rotation of ~ 100 km s-1 at ~ 100", with a subsequent decline out to ~ 300", confirmed by the rotation along the line of maximum gradient. However, both the detailed shape of the rotation curve and the reality of the apparent asymmetry in the velocity field remain uncertain. For comparison, SAURON data show a clear rotation gradient in the central 20", which is unresolved in the PN velocities . The conclusion from this is that PN velocities should be used as an outward extension of the ALS velocity measurements whenever possible .
Figure 1. Velocity field and streaming velocities from ~ 200 PNe in NGC 3379. The right-hand figure shows a smoothed spline model for the projected mean velocity, derived from the PN velocities; contour spacing is 10 km s-1 and the galaxy center is at (0,0). The two plots on the left show fits with different smoothing to the distribution of PN velocities versus distance along the direction of maximum gradient.
The second important application of PN velocities is to help determine the masses of ellipticals in their outer regions, where the dark matter halos should begin to dominate. Because of the large freedom in the orbital structure and velocity anisotropy in elliptical galaxies, this is not an easy task. If no other kinematic constraints are available, Merritt has estimated that at least many hundreds of PN velocities would be required . Thus, it is again advantageous to use the PN data in conjunction with ALS velocities and other data. From ALS data out to ~ 2Re, including line profile shape parameters h3 and h4, a range of dynamical models and gravitational potentials can be determined that are consistent with these data. Within these models, the implied orbital anisotropy and halo mass are correlated. Then only a few tens of PN velocities beyond 2Re are needed to further narrow down the permitted dynamics and mass distribution .