Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2015 / Paper 1 / 2 / e

First specify the perturbation: take fixed gas density, fixed ionic abundance and the same temperature-independent heating, so thermal stability requires $d\mathcal{C}/dT>0$ at an equilibrium. Let $\theta =\Delta E/k_B$, $n_e$ be the collider density and $n_A=n_1+n_2$ the atomic density. A two-level optically thin balance isThe critical density of an atomic transition is approximately $A_{21}/q_{21}$. At low density, each rare collisional excitation is followed by an escaping photon, soThe isochoric thermal stability of a collisionally excited line therefore givesAt $T=2\theta$ the linear restoring coefficient vanishes; the cooling maximum is marginal to first order and is not robust two-sided stability.

At high density, collisional de-excitation competes successfully with radiative decay, and collisions establish local thermodynamic equilibrium level populations. Detailed balance gives $n_2/n_1=g{e}^{-\theta /T}$, where $g=g_2/g_1$. The cooling is thenThus the high-density fixed-density branch is stable for finite $T$, becoming weakly restoring as the upper fraction saturates at high temperature. It is incorrect to retain the low-density factor $T^{-1/2}$ in the radiative cooling law once populations are thermalized.^[ref]

The isobaric thermal stability of a two-level coolant is a distinct question. If the intended heating is a fixed rate per particle, then $n\propto T^{-1}$ and stability tests the cooling per particle. The low-density boundary becomes $T<2\theta /3$; the high-density cooling per particle remains proportional to $f_u$, so remains stable. If instead heating is constant per volume, the isobaric low-density boundary is $T<2\theta /5$, and the high-density stability condition is $\theta (1-f_u)/T>1$. The source gives no pressure constraint or explicit heating normalization: the boxed $2\theta$ result is the fixed-density answer, with these conditional alternatives stated to avoid asserting a universal boundary.