= Polytropic vertical structure of a self-gravitating disk
For a locally plane-parallel <thin disk>, assume a <polytropic equation of state> $P=K\rho^{1+1/n}$ with $K>0$ and <polytropic index> $n>0$. Let $\rho_0$ be the midplane <mass density> and $c_0^2=(1+1/n)K\rho_0^{1/n}$ the midplane squared <sound speed>. Take the external vertical <gravitational acceleration> to be $-\Omega^2z$, as for a local <Keplerian disk>.
The specific enthalpy is $\mathcal H=(n+1)K\rho^{1/n}$. Define $w=\mathcal H/c_0^2=n(\rho/\rho_0)^{1/n}$ and $\zeta=\Omega z/c_0$. <Hydrostatic equilibrium> gives $d\mathcal H/dz=-\Omega^2z-d\Phi_d/dz$, while the plane-parallel <Poisson equation> gives $d^2\Phi_d/dz^2=4\pi G\rho$. Combining them yields
$$
\frac{d^2w}{d\zeta^2}+\frac4{n^nQ_0}w^n=-1,\qquad Q_0=\frac{\Omega^2}{\pi G\rho_0}.
$$
This $Q_0$ is a midplane-density parameter; it is not the surface-density <Toomre parameter>. Midplane symmetry imposes $w(0)=n$ and $w'(0)=0$. The first positive zero of $w$ defines the model's surface, where the <mass density> vanishes.
In the negligible-self-gravity limit $Q_0\to\infty$, $w=n-\zeta^2/2$ and the half-thickness is $\sqrt{2n}\,c_0/\Omega$. This finite polytropic surface differs from an isothermal Gaussian atmosphere with no sharp edge.<Dynamics of Astrophysical Discs by Gordon I. Ogilvie>[Lecture 7, section 7.1, external vertical gravity and hydrostatic/polytropic structure; the equation above also retains disk self-gravity explicitly]{ref}
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