 Solution

For a plane-parallel slab, all equilibrium quantities depend only on height $z$. The vertical hydrostatic equilibrium equation and the Poisson equation for Newtonian gravity areWith the polytropic equation of state $p=K{\rho }^2$ and constant $K$,In the gas interior, where $\rho >0$, divide by $\rho$ and differentiate once:ConsequentlyThe coefficient $2\pi G/K$ has units inverse square metres. The index-one plane-parallel polytrope illustrates the resulting structure: symmetry about $z=0$ gives $\rho (z)={\rho }_c\cos (kz)$, with $k=\sqrt{2\pi G/K}$. A slab bounded by vacuum ends at its first zero, $|z|=\pi /(2k)$; the negative continuation of the cosine is not physical mass density.