Planar virial trace (${\Pi }_{\perp }={\Pi }_{xx}+{\Pi }_{yy}$)

The planar trace of the stellar pressure-energy tensor is the sum of its two Cartesian in-plane components. For an axisymmetric nonstreaming population it equals $\int \rho (⟨v_R^2⟩+⟨v_{\varphi }^2⟩)d^{3}x$, not merely the cylindrical radial second moment. A spherical isotropic population has ${\Pi }_{\perp }=2{\Pi }_{zz}$.