We follow CAK’s use of a power-law line-number distribution with exponent \(\alpha\), but parametrize power law normalization with \(\bar{Q}\) (Gayley 1995) instead of the traditional CAK \(\kappa\). Following Gayley 1995, we use a \(2 \times 10^3\) value of \(\bar{Q}\) for our reference simulation with galactic metal content.
We are forced to apply an exponentially truncated line-strength cutoff \(Q_{max}\) (see Sundqvist 2013) \[\frac{\text{d}N}{\text{d}q} = \frac{1}{\Gamma(\alpha) \bar{Q}} \left( \frac{q}{\bar{Q}} \right)^{\alpha-2} e^{-q/Q_{max}},\] where \(\Gamma\) is the complete gamma function. Preliminary work to determine the effects of this line strength cutoff suggests an increase in structure and clumping when not applying the cutoff, but this causes perturbations at the grid scale and numerical artifacts. (FUTURE WORK OR SOMETHING).
The line-strength parameter \(\bar{Q}\) should scale directly with wind metal content to first order, since it is sensitive to the strongest lines (which come from metals). (I don’t understand Gayley’s argument about a shifting towards weaker lines). Following equation 60 of Gayley 1995, we have \[\bar{Q} \cong 10^5 Z.\]
(Gayley 1995 also shows a table from Shimada et al 1993 which suggests a strong dependence of \(\alpha\) on \(Z\), which is real questionable...)