Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 60 / 3 / i / Solution

Let $R$ be a characteristic radius enclosing mass $M$, and let $\sigma$ denote a one-dimensional velocity dispersion. Gravity and random-motion support give a relation $GM/R=\eta {\sigma }^2$, where the dimensionless structure coefficient $\eta$ depends on the chosen radius and density profile. A crossing-based galactic dynamical time is thereforeThe scaling follows by equating acceleration scales ${\sigma }^2/R$ and $GM/R^2$, not by identifying a complete orbital period with a crossing time. An explicit scale-free example checks the coefficient: constant circular speed and spherical symmetry give $M(<r)=v_c^{2}r/G$ and $\rho =v_c^2/(4\pi G{r}^2)$. With isotropic constant velocity dispersion, the Spherical Jeans equation becomesso $v_c^2=2{\sigma }^2$, $M(<R)=2{\sigma }^{2}R/G$, and $R/\sigma =GM/(2{\sigma }^3)$. This is the singular isothermal sphere scaling, valid over an interior radial range; a literal infinite model has no finite total mass, and a hard finite boundary needs a pressure term or an outer transition. A free-fall time of a uniform sphere or orbital period changes the numerical factor. The subsequent estimates use the nominal scale $GM/{\sigma }^3$, keeping these order-unity structural uncertainties explicit.