On the Alignment of the Stress Tensor in Galaxies

We show that, provided the principal axes of the second velocity moment tensor of a stellar population are generally unequal and are oriented perpendicular to a set of orthogonal surfaces at each point, then those surfaces must be confocal quadric surfaces and the potential must be separable or Stackel. This is true under the mild assumption that the even part of the distribution function is invariant under time reversal $v_i \rightarrow -v_i$ of each velocity component. In particular, if the second velocity moment tensor is everywhere exactly aligned in spherical polar coordinates, then the potential must be of Stackel form (excepting degenerate cases where two or more of the semiaxes of ellipsoid are everywhere the same). The theorem also has consequences for alignment in cylindrical polar coordinates, which is used in the popular Jeans Anisotropic Models (JAM). We analyse data on the radial velocities and proper motions of a sample of $\sim 7400$ stars in the stellar halo of the Milky Way. We provide the distributions of the tilt angles or misalignments from the spherical polar coordinate systems. We show that in this sample the misalignment is always small (usually within $3^\circ$) for radii between 7 and 12 kpc. The velocity anisotropy is very radially biased ($β\approx 0.7$), and almost invariant across the volume in our study. Finally, we construct a triaxial stellar halo in a triaxial NFW dark matter halo using a made-to-measure method. Despite the triaxiality of the potential, the velocity ellipsoid of the stellar halo is nearly spherically aligned within $\sim6^\circ$ for large regions of space, particularly outside the scale radius of the stellar halo. We conclude that the second velocity moment ellipsoid can be close to spherically aligned for a much wider class of potentials than the strong constraints that arise from exact alignment might suggest.