Efficient solution of the anisotropic spherically aligned axisymmetric Jeans equations of stellar hydrodynamics for galactic dynamics

I present a flexible solution for the axisymmetric Jeans equations of stellar hydrodynamics under the assumption of an anisotropic (three-integral) velocity ellipsoid aligned with the spherical polar coordinate system. I describe and test a robust and efficient algorithm for its numerical computation. I outline the evaluation of the intrinsic velocity moments and the projection of all first and second velocity moments, including both the line-of-sight velocities and the proper motions. This spherically-aligned Jeans Anisotropic Modelling (JAM(sph)) method can describe in detail the photometry and kinematics of real galaxies. It allows for a spatially-varying anisotropy, or stellar mass-to-light ratios gradients, as well as for the inclusion of general dark matter distributions and supermassive black holes. The JAM(sph) method complements my previously derived cylindrically-aligned JAM(cyl) and spherical Jeans solutions, which I also summarize in this paper. Comparisons between results obtained with either JAM(sph) or JAM(cyl) can be used to asses the robustness of inferred dynamical quantities. As an illustration, I modelled the Atlas(3D) sample of 260 early-type galaxies with high-quality integral-field spectroscopy, using both methods. I found that they provide statistically indistinguishable total-density logarithmic slopes. This may explain the previously-reported success of the JAM method in recovering density profiles of real or simulated galaxies. A reference software implementation of JAM(sph) is included in the publicly-available JAM software package.