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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 ($beta approx 0.7$), 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.
We explore the distribution of position angles (PA) of galaxies in clusters. We selected for study the isolated clusters, since the distribution of the galaxy orientation in clusters with close neighbors could be altered by gravitational influence of
We study correlation functions involving generalized ANEC operators of the form $int dx^- left(x^-right)^{n+2} T_{--}(vec{x})$ in four dimensions. We compute two, three, and four-point functions involving external scalar states in both free and holog
It has been shown, both in simulations and observationally, that the tidal field of a large galaxy can torque its satellites such that the major axis of satellite galaxies points towards their hosts. This so-called `shape alignment has been observed
The alignment between satellites and central galaxies has been studied in detail both in observational and theoretical works. The widely accepted fact is that the satellites preferentially reside along the major axis of their central galaxy. However,
The correlation between the spins of dark matter halos and the large-scale structure (LSS) has been studied in great detail over a large redshift range, while investigations of galaxies are still incomplete. Motivated by this point, we use the state-