Isothermal displacement energy of a self-gravitating slab (${\omega }^{2}I=c_s^2\int |\delta \rho {|}^2/\rho dz-(4\pi G{)}^{-1}\int (|{\varphi }_z{|}^2+k^2|\varphi {|}^2)dz$)

For an isothermal gas slab, a horizontal Fourier mode has $\delta \rho =-ik\rho {\xi }_x-(\rho {\xi }_z{)}_z$, $W=c_s^2\delta \rho /\rho +\varphi$, ${\omega }^2{\xi }_x=ikW$, ${\omega }^2{\xi }_z=W_z$ and $({\partial }_z^2-k^2)\varphi =4\pi G\delta \rho$. Multiply momentum by $\rho {\mathbf{\xi }}^{\ast }$ and integrate by parts, then use the Poisson equation for Newtonian gravity. Vanishing pressure-work and gravitational boundary terms give the displayed real quadratic identity, with $I=\int \rho |\mathbf{\xi }{|}^{2}dz>0$. Thus every admissible squared eigenfrequency is real; a negative value gives gravitational growth rather than an oscillatory instability.

For mass density $\rho ={\rho }_0{\mathrm{sech}}^2(z/H)$ with $H^2=c_s^2/(2\pi G{\rho }_0)$, a zero-frequency gravitational mode with $k\ne 0$ obeys $W=0$. Its potential satisfies ${\varphi }_{zz}+[2{\mathrm{sech}}^2(z/H)/H^2-k^2]\varphi =0$. The substitution $\tau =\tanh (z/H)$ gives the associated Legendre differential equation of degree one and order $|k|H$. Solutions decaying at both infinities exist only at $|k|H=1$, when $\varphi \propto \mathrm{sech}(z/H)$. Longer horizontal wavelengths are unstable. Neutral mass-preserving relabeling displacements and the uniform vertical translation at $k=0$ are distinct from this fragmentation threshold.