8 The effect of heating and cooling

Let us next examine the effects of altering our treatments of heating and cooling. Figure 11 shows the effect of heating and cooling on the wind structure. The solid line is the reference model. The dashed line is a model where the cooling function $\Lambda$ has been artificially reduced by a factor 1000. The dotted line is a model where the floor temperature is $T_{\rm eff}/2$, as opposed to $T_{\rm eff}$ for the reference model. This can be seen as reducing the effectiveness of radiative heating.

Figure 11: Effect of heating and cooling on the wind structure. The solid line is the reference model, the dashed line a model where the cooling function $\Lambda$ has been reduced by a factor 1000, and the dotted line a model with floor temperature equal to $T_{\rm eff}/2$.

8.1 Reduced cooling

In a smooth CAK wind, the maintenance of the wind temperature at values near the stellar effective temperature ensures that there are lots of ions with lines at energies near the peak of the star's radiative flux spectrum. But as gas is heated in shocks or other compressions, the associated ionisation runup means there are generally fewer line transitions with energies that can scatter the star's UV radiative flux. As such, the line-driving force in such hot gas should be substantially reduced.

In our calculations this ionisation runup effect is mimicked by reducing the line-driving opacity of the hot gas by a factor $\exp(1-T/T_{\rm floor})$. When the cooling function is reduced, a much greater fraction of the wind mass can remain above the floor temperature (i.e., ${>}40{\%}$, vs. ${<}5{\%}$in the reference model). The reduction in line-force associated with the reduced cooling can have a strong effect on the structure; but since the wind outflow is already close to its terminal speed at the radius where such hot structure forms, the effect on the overall bulk outflow is not too great. The influence of reduced cooling on the terminal velocity is barely $10{\%}$. The presence of hot gas significantly reduces the amount of structure, however: when a parcel of gas subjected to a positive velocity perturbation is further accelerated due to the instability, it is hindered by the neighbouring hot gas that is not subject to any such instability. This prevents the formation of very strong rarefactions and shocks. Reducing the cooling function therefore produces a wind with almost the same mean flow, but significantly less structure (Fig. 11, dashed line).

8.2 Reduced heating

The effect of reducing the floor temperature is shown by the dotted line in Fig. 11. The model with a reduced floor temperature is more strongly clumped. This can be understood in terms of the lower expansion speed of the dense clumps. Around $5\ R_*$both models are equally structured and the cooler floor temperature results in subsequent collisions of narrower clumps than for the reference model, where $T_{\rm floor}=T_{\rm eff}$. This also results in a reduced filling-in of the interclump gas, as radiative cooling is less efficient. Eventually though, cooling wins, so the outer wind is at the floor temperature for both models.

Effectively all of the dense clumps (and therefore nearly all of the wind mass, as the clumps comprise most of the wind) are at the floor temperature. It should be emphasised that the assumption of a constant floor temperature is a crude approximation of the effect of radiative heating. If, for some reason, the clumps could shield each other from ionising stellar radiation, the dense gas might become substantially cooler than the $T_{\rm eff}/2$ used in this section.