= Isothermal stellar and gas density equivalence
{title2=$\sigma^2\rho'=-\rho\Phi',\quad \sigma^2=k_BT/m_0$}
Without an escape-energy cutoff, a <galactic distribution function> proportional to $\exp[(\Psi-v^2/2)/\sigma^2]$ gives $\rho\propto e^{\Psi/\sigma^2}$ and a constant one-dimensional <velocity dispersion>. Since $\Psi=\Phi_0-\Phi$, differentiation gives the displayed hydrostatic relation. An <ideal gas> at constant <temperature> has pressure $P=\rho k_BT/m_0$, so its <hydrostatic equilibrium> equation is identical at the displayed dispersion-temperature identification. Coupling either relation to the same <Poisson equation for Newtonian gravity>, with matched central density and boundary data, therefore gives the same density structure. Truncating the stellar <velocities> at escape changes this equivalence.
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