Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2015 / Paper 1 / 2 / a

With constant isothermal sound speed $a_T$, the equation of state is $p=a_T^2\rho$. Spherical hydrostatic equilibrium and mass continuity requireTry $\rho =A/r^2$. With no central point mass, $M(r)=4\pi Ar$, and hydrostatic balance becomes $2a_T^{2}A=4\pi G{A}^2$. The positive-density branch is thereforeThe constant external pressure fixes the truncation radius $r_s$ through $p(r_s)=p_0$, givingThis is the pressure-truncated singular isothermal gas sphere, one hydrostatic solution, not a claim that every isothermal cloud has a singular centre. Its central density diverges, but its enclosed mass tends to zero as $r\to 0$.