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

For a steady spherical, nonrotating, bounded collisionless galaxy, the mean velocity vanishes, the mixed velocity moments vanish, and ${\sigma }_{\theta \theta }^2={\sigma }_{\varphi \varphi }^2\equiv {\sigma }_t^2$. A steady radial stellar flux would satisfy $r^2\nu {\overline{v}}_r=\mathrm{const}$; regularity and no central source/sink set that constant to zero in the equilibrium model. The Spherical Jeans equation is thereforeHere $\nu$ is the tracer number density, not necessarily the gravitating mass density. Random motions supply the velocity stress balancing the gravitational field.

Two conventions in the printed general equation need correction before interpreting it outside this special limit. Its first standalone derivative of mean radial velocity should be a time derivative, $\nu {\partial }_t{\overline{v}}_r$, not $\nu {\partial }_r{\overline{v}}_r$: the latter has incorrect acceleration dimensions. Also a velocity dispersion is the centered moment $\overline{v_i{v}_j}-{\overline{v}}_i{\overline{v}}_j$. The printed raw-moment definition equals it only when the mean velocities vanish, as they do in the equilibrium used here. These issues do not change the simplified result above.

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 ${\sigma }_r\to 0$.

Nearly radial orbits have small transverse motion: $\boxed{{\sigma }_t^2\ll {\sigma }_r^2,\beta \to 1}$. A bounded steady population contains both inward and outward phases, so large radial dispersion is compatible with zero mean radial velocity. Positivity of the distribution function and stability can impose further restrictions on particular radial models.

An isotropic velocity distribution has equal variance in every direction and zero mixed moments:This is a statement about random velocity directions at each radius, not a requirement that every star follows a circular orbit.