Rotation-anisotropy diagram ($v/\sigma \text{ versus }e$)

A rotation-anisotropy diagram compares ordered rotation divided by velocity dispersion with a galaxy's flattening. An oblate isotropic rotator provides a reference curve. Falling below it can indicate anisotropic random support, but inclination, measurement apertures, and the definition of rotation change the interpretation. Intrinsic axis ratios and projected ellipticities are different observables.

The simplified moment prescription $q={\sigma }_z^2/({\sigma }_x^2+v_p^2)$, with $e=1-q$ and $\beta =1-{\sigma }_z^2/{\sigma }_x^2$, gives the displayed relation by solving for $v_p^2$. The prescription is an approximation, not a consequence of the tensor virial theorem for arbitrary oblate density profiles. For constant azimuthal streaming $v_{\mathrm{az}}$, the one-component streaming second moment is $v_p^2=v_{\mathrm{az}}^2/2$.