 Galactic dynamical time

A galaxy of characteristic radius $R$ and velocity dispersion $\sigma$ has crossing time $R/\sigma$. Gravitational balance gives $GM/R\sim {\sigma }^2$, hence $t_{\mathrm{dyn}}\sim GM/{\sigma }^3$. A numerical coefficient depends on the radius definition, density profile, and whether one uses a crossing, orbital, or free-fall time; these are not identical clocks.