{title2=$T=T_0(1-z^2/H^2),\quad\rho\propto T^n$}

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>.


Polytropic vertical structure in stellar gravity

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.


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 Incoming links: Polytropic vertical structure of a self-gravitating disk