Pressure-confined singular isothermal gas sphere ($\rho =c_s^2/(2\pi G{r}^2),2U+\Omega =3p_0\mathcal{V}$)

Truncating the singular isothermal gas profile at radius $r_0$ gives $M=2c_s^2{r}_0/G$ and boundary pressure $p_0=c_s^4/(2\pi G{r}_0^2)$. The internal energy of a monatomic gas is $U=3M{c}_s^2/2$, while shell integration gives gravitational self-energy $\Omega =-G{M}^2/r_0=-2M{c}_s^2$. Thus $2U+\Omega =3p_0\mathcal{V}$, with $\mathcal{V}=4\pi r_0^3/3$. The confining pressure supplies the virial surface term. This singular-central-density model differs from a regular pressure-confined isothermal sphere.