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
Atlas3D 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.
