Self-gravitating incompressible slab (${\rho }_0=\Sigma /(2H)$)

A uniform mass density slab with free surfaces at $z=\pm H$ has ${\rho }_0=\Sigma /(2H)$, internal gravity $4\pi G{\rho }_{0}z$ and hydrostatic pressure $2\pi G{\rho }_0^2(H^2-z^2)$ above ambient pressure. Its surface modes of a self-gravitating incompressible slab separate into thickness and bending parities.

A test star inside an infinite planar slab of uniform mass density $\rho$ experiences acceleration $\ddot{z}=-4\pi G\rho z$. Reflection symmetry gives zero vertical gravitational field at the mid-plane, and integrating the gravitational Poisson equation gives the field linear in height. The interior orbit is therefore simple harmonic motion; release from rest at any height within the slab gives its first mid-plane crossing after $\pi /(2\sqrt{4\pi G\rho })$. Finite radial extent, a varying vertical mass density or leaving the slab changes this result.

The even surface branch is unstable for $0<kH<q_c$, where $q_c=(1+e^{-2q_c})/2\simeq 0.6392322714$, and stable above this threshold. All positive-wavenumber odd modes are stable because $1-e^{-2q}<2q$. The equilibrium nevertheless has a long-wavelength column-density instability.

For $q=kH$, the even and odd scalar-potential branches have ${\omega }_e^2=(2\pi G\Sigma /H)[q\tanh q-1/(1+\coth q)]$ and ${\omega }_o^2=(2\pi G\Sigma /H)[q\coth q-1/(1+\tanh q)]$. Even potential symmetry moves the surfaces oppositely; odd potential symmetry bends them together. Hydrostatic free-surface pressure gives the positive term and perturbed self-gravity the negative term.