 Pressure-truncated singular isothermal gas sphere

For $p=a_T^2\rho$, spherical hydrostatic equilibrium admits $\rho =a_T^2/(2\pi G{r}^2)$ with $M(r)=2a_T^{2}r/G$. Matching an external pressure $p_0$ fixes $r_s=a_T^2/\sqrt{2\pi G{p}_0}$. The central density is singular although the enclosed mass tends to zero; this is one branch, not every finite-centred isothermal cloud.