Gas Cooling

We use two cooling models in this work, denoted C1 and C2, which differ in that C2 has significant cooling in the neutral gas. Our first model, C1, contains four elements:

1. Radiative losses due to recombining Hydrogen, calculated from the non-equilibrium ion fraction and temperature in each cell according to rates tabulated in Hummer (1994).
2. Collisional ionisation of Hydrogen: this is relatively unimportant because the rates are typically very low, but we subtract the ionisation energy from the gas for each collisional ionisation.
3. Cooling due to heavy elements at high temperatures, using the collisional ionisation equilibrium (CIE) cooling curve tabulated in Sutherland (1993) and shown in their fig. 18. This provides strong cooling in ionised gas with temperatures significantly larger than $10\,000\,$K. Note that in CIE at $10\,000\,$K, Hydrogen, Nitrogen and Oxygen are neutral so this fitting function does not double count the other terms in our cooling function, at least for the gas temperatures encountered in our simulations.
4. A linear fit to collisionally excited emission from photo-ionised Oxygen and Nitrogen (Osterbrock, 1989).

The last term is the most important for this work, since these ionic species are the dominant coolants in H II regions and set the equilibrium temperature in ionised gas of $T_{eq}\simeq8000\,$K. In experiments with different cooling functions for ionised gas, we found that the most important factor for the dynamical evolution of our models was the equilibrium temperature. If the normalisation of the cooling function is kept fixed at $8000\,$K, its slope has little effect on the resulting dynamics so long as the slope is positive. If we had strong shocks in the ionised gas this aspect of the cooling function would have more influence, but the photo-ionised gas in our simulations has a very narrow temperature range.

In this prescription the neutral atomic gas has no efficient cooling avenue, and shocked neutral gas is typically at $100-10\,000\,$K. This is undoubtedly a limitation in our modelling, but we do not yet model the formation of molecules, or the formation/destruction of dust, which are the primary neutral gas coolants in star forming regions. To assess the effects of significant neutral gas cooling we also use an alternate cooling function, C2, consisting of the previous components in C1 plus additional exponential cooling (Newton's Law) in neutral gas with a rate given by

$$\dot{T} = \left[\frac{(1-x)^2}{10^N\mathrm{yrs}}\right] \left(T_{\infty}-T\right) \;,$$ (1.2)

where $T$ is gas temperature, $T_{\infty}$ is the temperature to which this cooling law relaxes at late times, $x$ is the ionisation fraction of the gas, and $N$ is a parameter specifying the chosen cooling time-scale, $t_c=10^N\mathrm{yrs}/(1-x)^2$. The scaling with $(1-x)^2$ ensures only mostly neutral gas is affected. We set $T_{\infty}=100\,$K and $N=4$ for the alternate models run in this paper. This is not an extreme model either in terms of the equilibrium temperature or the cooling time, having less cooling in dense gas than e.g. Henney (2009). It is a very simple prescription, with an effect which is intermediate between C1 and a two-temperature isothermal model (e.g. Lora, 2009; Gritschneder, 2009; Williams, 2001).