Free fall in a fixed singular isothermal sphere ($t_{\mathrm{ff}}=(8G{\rho }_0{)}^{-1/2}$)

A test particle released from rest at $r_0$ in a fixed spherical mass density $\rho (r)={\rho }_0(r_0/r{)}^2$ obeys $\ddot{r}=-4\pi G{\rho }_0{r}_0^2/r$. Integrating once gives ${\dot{r}}^2=8\pi G{\rho }_0{r}_0^2\ln (r_0/r)$. The substitution $u=\sqrt{\ln (r_0/r)}$ then gives $t(r)=(8G{\rho }_0{)}^{-1/2}\mathrm{erf}(u)$, and the arrival time at the centre is $(8G{\rho }_0{)}^{-1/2}$. This is free fall in a frozen singular isothermal sphere, not the self-consistent pressureless collapse of the background. An initially resting material shell with the same initial enclosed mass instead arrives after $\sqrt{\pi /(32G{\rho }_0)}$, because its enclosed mass stays fixed while the cloud contracts.