The SAURON project -- X. The orbital anisotropy of elliptical and lenticular galaxies: revisiting the (V/sigma,epsilon) diagram with integral-field stellar kinematics

We analyse the orbital distribution of E and S0 galaxies using SAURON integral-field stellar kinematics. We construct the anisotropy (V/sigma,epsilon) diagram, for the 48 E/S0 galaxies from the SAURON survey. For a subsample of 24 galaxies consistent with axisymmetry, we use three-integral axisymmetric Schwarzschild dynamical models to recover the detailed orbital distribution and we find good agreement with the anisotropy derived from the (V/sigma,epsilon) diagram. In the companion Paper IX we show that the early-type galaxies can be subdivided into two classes of systems with or without a significant amount of specific stellar angular momentum. Here we show that the two classes have different distributions on the (V/sigma,epsilon) diagram. The slow rotators are more common among the most massive systems and are generally classified as E from photometry alone. Those in our sample tend to be fairly round (epsilon<0.3), but can have significant kinematical misalignments, indicating that as a class they are moderately triaxial, and span a range of anisotropies (delta<0.3). The fast rotators are generally fainter and are classified either E or S0. They can appear quite flattened (epsilon<0.7), do not show significant kinematical misalignments (unless barred or interacting), indicating they are nearly axisymmetric, and span an even larger range of anisotropies (delta<0.5). These results are confirmed when we extend our analysis to 18 additional E/S0 galaxies observed with SAURON. All these results support the idea that fast rotators are nearly oblate and contain disk-like components. The role of gas must have been important for their formation. The slow rotators are weakly triaxial. Current collisionless merger models seem unable to explain their detailed observed properties. (Abridged)