Past exam of the astrophysics course of the University of Oxford / 2015 / 7 / a / i / Solution

For an ensemble of nearly circular stellar orbits with randomly oriented planes, radial motion is small while tangential motion is substantial: $\boxed{{\sigma }_r^2\ll {\sigma }_t^2}$. The two tangential variances remain equal by spherical symmetry. There need not be bulk rotation even though individual stars move on nearly circular paths. The anisotropy parameter $\beta =1-{\sigma }_t^2/{\sigma }_r^2$ is strongly negative; its exactly circular limit is singular as ${\sigma }_r\to 0$.