 Polytropic vertical structure of a self-gravitating disk

For $P=K{\rho }^{1+1/n}$, dimensionless pseudo-enthalpy $w=n(\rho /{\rho }_0{)}^{1/n}$, and $\zeta =\Omega z/c_0$, vertical hydrostatic balance and the Poisson equation reduce toThe midplane conditions are $w(0)=n$ and $w^{\prime }(0)=0$, and the first zero gives the disk surface.