Index-one plane-parallel polytrope ($\rho (z)={\rho }_c\cos (kz),k^2=2\pi G/K$)

For a symmetric self-gravitating slab with polytropic equation of state $p=K{\rho }^2$ and constant $K>0$, hydrostatic equilibrium gives $2K{\rho }^{\prime }=-{\psi }^{\prime }$. The plane-parallel Poisson equation for Newtonian gravity then yields ${\rho }^{\prime \prime }+(2\pi G/K)\rho =0$. Reflection symmetry at the midplane fixes the title profile. For a vacuum surface, retain the nonnegative profile only on $|z|\le z_s=\pi /(2k)$ and put $\rho =0$ outside. The half-thickness is independent of central density in this ideal index-one model, while the surface mass density is $\Sigma =2{\rho }_c/k$.