Polytropic vertical structure in stellar gravity ($T=T_0(1-z^2/H^2),\rho \propto T^n$)

In harmonic stellar gravity, an ideal gas with a polytropic equation of state of index $n>0$ has a quadratic temperature profile and finite semi-thickness $H^2=2(n+1)\mathcal{R}{T}_0/{\Omega }^2$. This differs from the polytropic vertical structure of a self-gravitating disk.