Vertically integrated pressure of an isothermal disk
= Vertically integrated pressure of an isothermal disk
{title2=$P=c_s^2\Sigma=\Omega_z^2\Sigma H^2$}
A locally isothermal disk in vertically harmonic external gravity has <hydrostatic equilibrium> $c_s^2\rho_z=-\Omega_z^2z\rho$, hence $\rho=\rho_0e^{-z^2/(2H^2)}$ with $H=c_s/\Omega_z$. Because $c_s$ is independent of height, integrating $p=c_s^2\rho$ gives the displayed pressure. The harmonic-gravity approximation has the accuracy of the <thin disk> expansion.