= Isothermal vertical structure in stellar gravity
{title2=$\rho(z)=\rho_0e^{-z^2/(2H^2)},\quad H=c_s/\Omega_K$}
For a <thin disk> in the harmonic vertical <gravitational potential> $\Omega_K^2z^2/2$, <hydrostatic equilibrium> and $p=c_s^2\rho$ with height-independent <isothermal sound speed> give the displayed Gaussian density. For perturbations with <adiabatic index> $\gamma$, the <Brunt–Väisälä frequency> is $N^2=(1-1/\gamma)\Omega_K^2z^2/H^2$. An isothermal background can therefore support buoyancy for adiabatic perturbations; isothermal perturbations instead have $\gamma=1$ and zero buoyancy frequency.
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