 Orbits in a self-gravitating isothermal slab

For a self-gravitating isothermal slab with $\rho ={\rho }_0{\mathrm{sech}}^2(z/H)$ and $H^2={\sigma }^2/(2\pi G{\rho }_0)$, hydrostatic equilibrium gives the displayed Newtonian gravitational potential. A test particle has constant in-plane velocity and conserved vertical energy $E_z={\dot{z}}^2/2+\Phi (z)-\Phi (0)$. Its vertical motion is periodic for every finite $E_z>0$, with turning height $H\mathrm{arcosh}\exp [E_z/(2{\sigma }^2)]$. Near the plane it is a harmonic oscillator of squared frequency $4\pi G{\rho }_0$; at large height the force tends to the constant $2\pi G\Sigma$, where $\Sigma =2{\rho }_{0}H$. The ideal infinite slab has no finite vertical escape energy.