A general spectral solver for the axisymmetric Jeans equations: fast dynamical modelling of galaxies with arbitrary anisotropy
Axisymmetric Jeans modelling is widely used to infer galaxy mass profiles from integral-field kinematics, but existing implementations maintain tractability by adopting highly restricted anisotropy prescriptions. I present a new spectral method that solves the axisymmetric Jeans equations as a two-dimensional boundary-value problem. Remarkably, this breaks the traditional trade-off between model flexibility and computational cost, accommodating completely general anisotropy distributions $β(r,θ)$ while executing significantly faster than standard restrictive techniques. The method relies on three key choices: (i) solving for the intrinsic dispersion $\overline{v_r^2}$ rather than the rapidly varying pressure $ν\overline{v_r^2}$ to improve numerical conditioning; (ii) working in logarithmic radius to efficiently resolve the large dynamic range of galaxies, uniquely matching scale-free (power-law) regimes; and (iii) imposing a Robin outer boundary condition that enforces the correct asymptotic decay on a finite computational domain. Orbit integrations in realistic galaxy potentials motivate spherical alignment of the velocity ellipsoid as a physically plausible default, though the framework easily adapts to other alignments. Validated against exact analytic benchmarks -- including new analytic Jeans solutions derived herein -- the solver recovers intrinsic second moments with high accuracy, showing radially uniform residuals for power-law tests. In practice, it delivers orders-of-magnitude speed-ups over high-accuracy quadrature schemes and is naturally suited to massive GPU parallelization. Released in the public JamPy package, this enables the routine application of highly general Jeans models to large surveys and the extensive parameter-space exploration required for rigorous uncertainty quantification.