Polytropic vertical structure of a self-gravitating disk
= 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 to
$$
w''+\frac4{n^nQ_0}w^n=-1.
$$
The midplane conditions are $w(0)=n$ and $w'(0)=0$, and the first zero gives the disk surface.