Appendix C: On the difference of $\bar Q$ and $Q_{\rm o}$

In this appendix, we want to discuss the relation between $\bar Q$and Gayley's cut-off parameter $Q_{\rm o} = {v_{\rm th}}/c \,k_{\rm max}$. In particular and related to the discussion in Sect. 3.3, the inequality $Q_{\rm o} > \bar Q$ which has been found for cooler atmospheres shall be inspected.

Using the definition of $\bar Q$ and the power-law line-strength distribution with exponential cutoff, Eq. (B1), $\bar Q$ can be expressed as

$$\bar Q \frac{{v_{\rm th}}}{c}\, \sum \frac{L_{\nu_{\rm i}} \nu_{\rm i}}{L} k_{\rm L}^{\rm i}$$ =

$$\rightarrow N_{\rm o}\left\langle \frac{L_{\nu}\nu}{L}\right\rangle\frac{{v_{... ...{\infty}} k_{\rm L}^{\alpha-1} \exp(-k_{\rm L}/k_{\rm max}) {\rm d}k_{\rm L}\,=$$

$$N_{\rm o}\left\langle \frac{L_{\nu}\nu}{L}\right\rangle\frac{{v_{\rm th}}}{c} \Gamma(\alpha) k_{\rm max}^{\alpha},$$ = (C1)

where $\langle\frac{L_{\nu}\nu}{L}\rangle$ is an appropriate average of the weight factor, e.g.,

$$\left\langle \frac{L_{\nu}\nu}{L}\right\rangle= \frac {\sum ... ...}} \nu_{\rm i}}{L} k_{\rm L}^{\rm i}}{\sum k_{\rm L}^{\rm i}}.$$ (C2)

Note, that both $\bar Q$ and the average weight factor depend essentially on the corresponding values for the strongest lines, due to the linear dependence on $k_{\rm L}$, a behaviour which was already addressed in Gayley's ([1995]) paper. In the previous expression, we have implicitely assumed that curvature effects concerning $\alpha$ are unimportant, i.e., that $\hat \alpha\approx \alpha$ everywhere except at highest line-strengths.

On the other hand, the flux (times frequency) weighted cumulative line number, evaluted at maximum line-strength, gives

$${\left\langle N(k_{\rm max})\right\rangle\approx \left\langle \frac{L_{\nu}\nu}{L}\right\rangle\vert _{k_{\rm max}}\, \approx}$$

$$\approx N_{\rm o}\left\langle \frac{L_{\nu}\nu}{L}\right\rangle\vert _{k_... ...{\infty}} k_{\rm L}^{\alpha-2} \exp(-k_{\rm L}/k_{\rm max}) {\rm d}k_{\rm L}\,=$$

$$N_{\rm o}\left\langle \frac{L_{\nu}\nu}{L}\right\rangle\vert _{k_{\rm max}} \Gamma(\alpha-1,1) k_{\rm max}^{\alpha-1}\,=$$ =

$$N_{\rm o}\left\langle \frac{L_{\nu}\nu}{L}\right\rangle\vert _{k_... ...lpha-1,1) k_{\rm max}^{\alpha}\left(\frac{c}{{v_{\rm th}}}Q_{\rm o}\right)^{-1}$$ = (C3)

where in the rhs of the first line this number (without weight!) has been set to unity (i.e., $Q_{\rm o}$ shall be the line-strength of the single strongest line) and the subscript " $k_{\rm max}$'' accounts for the weighting process at the according frequency. By comparing the above two expressions, we obtain for the flux-weighted number of this single strongest line

$$\left\langle N(k_{\rm max})\right\rangle \frac{\left\langle \frac{L_{\nu}\nu}{L}\right\rangle\vert _{k_{\r... ...)} {\left\langle \frac{L_{\nu}\nu}{L}\right\rangle Q_{\rm o} \Gamma(\alpha)}\,=$$ =

$$\frac{\left\langle \frac{L_{\nu}\nu}{L}\right\rangle\vert _{k_{\r... ...{\rm o}}\, \frac{({\rm e}^{-1} - \Gamma(\alpha,1))}{(1-\alpha) \Gamma(\alpha)},$$ = (C4)

which can be inverted (again by requiring $N(k_{\rm max}) = 1$) to yield the corresponding line-strength

$$Q_{\rm o} \approx \left\langle \frac{L_{\nu}\nu}{L}\right\rangle^{-1} \bar Qf(\alpha).$$ (C5)

The last factor in this equation ( $=\Gamma(\alpha-1,1)/\Gamma(\alpha)$) has a value of 0.14, 0.10 and 0.077 for typical $\alpha$'s = 0.66, 0.5 and 0.4.

Thus, in cases where the frequential line-distribution is essentially independent on line-strength and is distributed according to CAK's assumption, ${\rm d}N \propto {\rm d}\nu/\nu$ and hence $\left\langle \frac{L_{\nu}\nu}{L}\right\rangle= 1$, $Q_{\rm o}$ should be (slightly) smaller than $\bar Q$, in contrast to what is "observed'' especially for cooler atmospheres. Vice versa, by accounting for the actual similarity of $\bar Q$ and $Q_{\rm o}$ in hotter atmospheres (Table 2), one might argue that the "effective'' number of lines with strength $Q_{\rm o}$shall be of order $f(\alpha)$. Independently from these more "philosophical'' questions (involving uncertainties of order one dex), the plain fact that the ratio $Q_{\rm o}/\bar Q$ is larger than unity and increasing for decreasing $T_{\rm eff}$ inevitably leads to the conclusion that the average factor has to be significantly below unity and is decreasing in parallel with $T_{\rm eff}$. Both by exploring the frequential line-distribution of the strongest lines (cf. Fig. 4 and the accompanying text) as well as by simply calculating the average weight factor (Eq. C2), it turns out that this is actually the case: for the same model atmospheres as in Sect. 3.2, we find $\left\langle \frac{L_{\nu}\nu}{L}\right\rangle= 0.4$ for $T_{\rm eff}= 50\,000$ K and $\left\langle \frac{L_{\nu}\nu}{L}\right\rangle= 0.003$ for $T_{\rm eff}= 10\,000$ K. Inserting these values and using the derived $\alpha$-values from Table 2, $Q_{\rm o}/\bar Q$ = 0.35 and 28 are predicted by means of (C5). The differences to the actual values of 1.16 and 16, respectively, remain to be attributed to the deviations from a perfect power-law, especially at the predominantely contributing high $k_{\rm L}$-end of the distribution.