= Gravitational energy of a stellar polytrope
{title2=$\Omega=-3GM^2/[(5-n)R]$}
For a spherical <stellar polytrope> of finite radius and zero surface pressure, the <specific enthalpy> is $(n+1)P/\rho$. <Hydrostatic equilibrium> gives $h+\Phi=-GM/R$, hence $(n+1)\int P\,dV=-GM^2/R-2\Omega$. Eliminating the pressure integral with the <stellar virial theorem> proves the displayed formula. For a gas <adiabatic stellar polytrope>, $n=1/(\gamma-1)$ and $U=GM^2/[(5\gamma-6)R]$; a general <polytropic index> need not have this relation to the <specific-heat ratio>.
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