= Orbits in a self-gravitating isothermal slab
{title2=$\Phi(z)-\Phi(0)=2\sigma^2\log\cosh(z/H)$}
For a <self-gravitating isothermal slab> with $\rho=\rho_0\operatorname{sech}^2(z/H)$ 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 $H\operatorname{arcosh}\exp[E_z/(2\sigma^2)]$. 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 $\Sigma=2\rho_0H$. The ideal infinite slab has no finite vertical escape energy.
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