 Oblate tensor-virial shape factor

For a finite stationary axisymmetric self-gravitating system with similar oblate density surfaces, let $q=b/a$, $s=\sqrt{1-q^2}$, $A_1=q\arccos (q)/s^3-q^2/s^2$, and $A_3=2/s^2-2q\arccos (q)/s^3$. Its stellar potential-energy tensor has $Q=A_1/(q^2{A}_3)$; the common radial integral cancels. The spherical limit has $A_1=A_3=2/3$ and $Q=1$. With mass-weighted in-plane velocity dispersion ${\sigma }_x$ and constant azimuthal streaming $v$, the tensor virial theorem gives $v^2/{\sigma }_x^2=2[(1-\beta )Q-1]$, where ${\sigma }_z^2=(1-\beta ){\sigma }_x^2$. Neither $Q=1/q$ nor the omission of the Cartesian streaming factor is exact.