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I present a flexible solution for the axisymmetric Jeans equations of stellarhydrodynamics under the assumption of an anisotropic (three-integral) velocityellipsoid aligned with the spherical polar coordinate system. I describe andtest a robust and efficient algorithm for its numerical computation. I outlinethe evaluation of the intrinsic velocity moments and the projection of allfirst and second velocity moments, including both the line-of-sight velocitiesand the proper motions. This spherically-aligned Jeans Anisotropic Modelling(JAM_sph) method can describe in detail the photometry and kinematics of realgalaxies. It allows for a spatially-varying anisotropy, or stellarmass-to-light ratios gradients, as well as for the inclusion of general darkmatter distributions and supermassive black holes. The JAM_sph methodcomplements my previously derived cylindrically-aligned JAM_cyl and sphericalJeans solutions, which I also summarize in this paper. Comparisons betweenresults obtained with either JAM_sph or JAM_cyl can be used to asses therobustness of inferred dynamical quantities. As an illustration, I modelled theAtlas3D sample of 260 early-type galaxies with high-quality integral-fieldspectroscopy, using both methods. I found that they provide statisticallyindistinguishable total-density logarithmic slopes. This may explain thepreviously-reported success of the JAM method in recovering density profiles ofreal or simulated galaxies. A reference software implementation of JAM_sph isincluded in the publicly-available JAM software package.

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