Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 65 / 2 / i / Solution

Use a steady, isotropic, self-gravitating dark matter halo, with $V$ its constant one-dimensional velocity dispersion. The Spherical Jeans equation and spherical mass conservation readFor the scale-free outer region, put $\rho =A{r}^{-p}$. Its enclosed mass scales as $r^{3-p}$, so $GM(r)/r$ is constant only if $p=2$; the Spherical Jeans equation then gives $2V^2=GM(r)/r=4\pi GA$. Thus the singular isothermal sphere hasIts circular speed is $v_c=\sqrt{2}V$. If a speed denoted $V$ were instead the circular speed, the coefficient would be $V^2/(4\pi G)$, but the printed variable is a velocity dispersion.

The assumptions matter. A constant velocity dispersion and spherical shape alone do not specify a unique central mass density or orbital distribution. Even with isotropy, the Spherical Jeans equation integrates to $\rho ={\rho }_0\exp [-(\Phi -{\Phi }_0)/V^2]$, and the self-gravitating Poisson equation for Newtonian gravity givesThere are finite-central-density isothermal solutions as well as the singular one; the $r^{-2}$ expression is the intended outer scale-free halo approximation. A finite halo must also end or steepen at large radius, since the untruncated singular isothermal sphere has infinite total mass.