= Pressure-confined singular isothermal gas sphere
{title2=$\rho=c_s^2/(2\pi Gr^2),\quad 2U+\Omega=3p_0\mathcal V$}
Truncating the singular isothermal gas profile at radius $r_0$ gives $M=2c_s^2r_0/G$ and boundary <pressure> $p_0=c_s^4/(2\pi Gr_0^2)$. The <internal energy> of a <monatomic gas> is $U=3Mc_s^2/2$, while shell integration gives gravitational self-energy $\Omega=-GM^2/r_0=-2Mc_s^2$. 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.
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