Flattening–anisotropy virial relation (${\Pi }_{\perp }/(2{\Pi }_{zz})=1+\gamma (q^{-2}-1)/5+O((q^{-2}-1{)}^2)$)

For a pressure-supported scale-free tracer $\rho \propto (R^2+z^2/q^2{)}^{-\gamma /2}$ in a spherical Newtonian gravitational potential, the tensor virial theorem constrains integrated directional velocity dispersions. Modest oblate flattening and a decreasing radial mass density require greater in-plane random support per direction. The displayed expansion assumes finite global virial integrals; the corresponding ratio of force moments also has a cutoff-independent angular interpretation.

For $h=({\sin }^2\theta +q^{-2}{\cos }^2\theta {)}^{-\gamma /2}$, separation in spherical coordinates makes both stellar potential-energy tensor integrals contain $\int r^{3-\gamma }{\Phi }^{\prime }(r)dr$. Their ratio is the displayed positive finite angular expression. If the radial integrals diverge, identical spherical cutoffs give this same ratio of force moments, but there is no finite global pressure-energy ratio without further hypotheses. A finite-aperture tensor virial theorem retains its surface tensor.