Singular isothermal sphere

Example

Let $\sigma > 0$ be a velocity dispersion and $G$ the gravitational constant. The singular isothermal sphere is the spherical mass distribution with density $$\rho(r) = \frac{\sigma^2}{2 \pi G r^2},$$ whose mass inside radius $r$ is $M(r) = \int_0^r 4\pi s^2 \rho(s) \, ds = 2 \sigma^2 r / G$.

Check of the spherical isotropic Jeans equation $\frac{d(\rho \sigma^2)}{dr} = -\rho \frac{G M(r)}{r^2}$ with constant $\sigma$: the left side is $\sigma^2 \rho'(r) = -2 \sigma^2 \rho / r$, and the right side is $-\rho \cdot 2 \sigma^2 r / r^2 = -2 \sigma^2 \rho / r$. So stars with the same isotropic dispersion $\sigma$ everywhere are in equilibrium in this mass distribution.

The circular speed is $v_c(r) = \sqrt{G M(r)/r} = \sqrt{2} \, \sigma$, independent of $r$, so the model has a flat rotation curve.