6 Summary and discussion

In this paper, we have tried to analyze various aspects concerning the relation between line-statistics and radiative driving in massive stars with not too thick winds (i.e., we have excluded the problem of WR winds).

In the following, we will summarize our results, point to additional aspects which have not been discussed in the previous sections and give some caveats where necessary.

In Sect. 2, we found an alternative interpretation of the total line-force as the integral of the cumulative flux weighted line-strength distribution function over line-strength, which allowed for an instructive visualization of the line-force and further investigations: For arbitrary distribution functions, the local logarithmic slope $\alpha(k_{\rm L})$ can be identified with the f.m. parameter $\hat \alpha(k_{\rm 1})$, if this slope is not too steep around $k_{\rm L}= k_{\rm 1}$ (essentially, $\alpha$ has to be larger than zero). If the latter condition is violated locally, $\hat \alpha$ should follow at least the basic trend of $\alpha$, however remaining larger/equal than zero with respect to its definition as the ratio of optically thick to total line acceleration. (A steep increase with $\alpha < 0$ over a larger range of line-strength well below $k_{\rm L}= k_{\rm 1}$, finally, would prohibit the parameterization of the f.m. in its usual form $\propto k_{\rm 1}^{\hat \alpha}$ completely.) These statements were checked for various conditions throughout the paper, and turned out to be fulfilled always.

In order to understand the principal behaviour and numerical value of $\hat \alpha$ itself and to allow for predictions concerning its behaviour under different conditions, we have performed a rigorous discussion of the line-strength distribution as function of atmospheric conditions. This discussion relied on our extensive data base and an approximate NLTE description, provided in Sect. 3.

For some typical atmospheric conditions, we checked at first the applicability of Gayley's ([1995]) $\bar Q$-approach. As long as the dependence on the maximum line-strength $Q_{\rm o}$ is correctly accounted for, a perfect consistence with the older CAK approach using the parameter $k_{\rm CAK}$ is found. We concluded, however, that the $\bar Q$ formalism is only advantageous in those cases when the maximum line-strength $Q_{\rm o}$ is of the same order as $\bar Q$, a prerequisite which was assumed by Gayley to be valid always. By means of our detailed calculations, however, this assumption could be validated only for hot winds ( 35000 K). For cooler ones, the difference between $\bar Q$ and $Q_{\rm o}$ is significant and was attributed to the increasing mismatch between the frequential positions of the strongest lines and flux-maximum. In those cases, a "blind'' application of the final scaling relations provided by Gayley (which include the assumed equality) will inevitably lead to erroneous conclusions.

In Sect. 4, we turned to the central question concerning the slope (or shape) of the line-strength distribution function. At first, we considered the (simpler) case of hydrogen (or hydrogenic ions) and derived the important result that the according slope in the essential $k_{\rm L}$ range is almost exactly -1/3, so that $\alpha = 2/3$. This result was shown to be the final consequence of the corresponding oscillator strength distribution, in particular the dependence $gf \propto n^{-3}$ with n the principal quantum number of the upper level of the contributing transitions. The predicted behaviour is actually seen at the hydrogen-Lyman dominated, high line-strength end calculated for distribution functions under A-type conditions. Additionally we showed that, if neutral hydrogen is a trace ion, the corresponding $\delta$ term (resulting from the $n_{\rm e}/W$ dependent part of the normalization constant $N_{\rm o}$) is of order 1/3, and that for arbitrary trace ions one stage below the major one the equality $\alpha + \delta \approx 1$ should hold in general.

In order to derive line-strength distribution functions for arbitrary metallic ions, we followed the approach suggested by Allen ([1966]), modified for the inclusion of non-uniform oscillator strength distributions. This approach and the above one are mutually exclusive, due to the rather specific behaviour of level density as function of energy in hydrogenic ions.

At first, we considered the so-called line-intensity distribution, which is the LTE analogon to (NLTE-)line-strength distribution functions for specific ions. By a number of Monte-Carlo simulations for the oscillator strength distribution we have convinced ourselves that the principal description, resulting in predictions of $\Delta N$ as function of line intensity, is valid both for the frequency dependent as well as the frequency integrated line intensity distribution.

We showed that three different slopes are possible, namely $2t/\sigma, t/\sigma$ and $\gamma-1$, if t is the temperature in units of 625 K, $\sigma$ is the slope of the level-density with respect to energy and $\gamma$ the negative exponent of the differential oscillator strength distribution function. In dependence of $\gamma$ being larger or smaller than a critical value $\gamma_{\rm crit}= 1 + 2t/\sigma$, specific predictions for the line intensity distribution functions can be made, which depend uniquely on the properties of the level- and line-lists. Thus, by comparing these analytical results with actual distribution functions, it is very easy to test for the completeness of the underlying data base of specific ions.

Under "normal'' conditions, the resulting frequency integrated distribution function consists of two regimes, namely a steeper, excitation-dominated one with slope $2t/\sigma$, and a second one with slope $\gamma-1$. The division is given at a line intensity $l=x_{\rm max}/t -{\rm log}\,gf_{\rm max}$, if only those lines with a lower energy level below a certain cutoff energy $x_{\rm max}$ are considered. Thus, the direct influence of $\gamma$regards only the slope of the weaker lines' distribution. Additionally, however, $\gamma$ has an indirect impact which turns out to be of major relevance for the final result: By affecting the normalization constant of the line intensity distribution, it controls the absolute number of lines as function of intensity (or strength).

By translating our findings from line intensity to line-strength distributions, two important points have to be considered: The final distribution function consists of a number of contributing ions, so that the product of ionization-fraction times relative abundance becomes important. Second, under stellar wind conditions, NLTE effects have to be accounted for.

In order to separate NLTE/ionization- from abundance effects, we investigated at first the case of equal abundances and considered two groups of elements, namely "iron group'' and "light'' elements, respectively, which should behave rather similarly amongst each other due to their similar electronic structure. NLTE effects were treated in our approximate way by allowing for only three participating classes of lines, namely those with a lower ground- or meta-stable level and those lines directly connected to the former.

Under these assumptions, it turned out that the major difference between line intensity and -strength distribution is (in the log-log representation) a horizontal shift due to ionization, and the appearance of a low effective value for $x_{\rm max}$, giving rise to a rather narrow excitation dominated range. Within the specific subgroups, the individual ions behave rather similarly, so that the total distribution functions can be described in fairly simple terms ("staircase structure''), since only two or three different ionization stages contribute to the interesting range (six to eight dex) in $k_{\rm L}$. Iron group elements display a significant increase in line number with decreasing temperature due to their increasing complexity in electronic structure, whereas light ions show a comparable number of lines for all considered temperatures. The excitation dominated slope was found to be of order unity for both subgroups, independent of temperature, and relates to the increase of $\sigma$ (lower level density) with increasing ionization stage. Additionally, the maximum value of $k_{\rm L}$ reached by iron group elements is smaller than for light ions (lines to meta-stable levels vs. lines to ground-states).

The latter effect - difference in maximum line-strength between iron-group and light ions - is essentially increased if one accounts for (relative) solar abundances. Thus, at large line-strengths the resulting total distribution is determined by transitions from light ions (plus hydrogen at cooler temperatures), whereas for intermediate and lower $k_{\rm L}$ the iron group elements (most important: Fe itself due to its abundance) dominate the distribution. Consequently, at highest line-strengths the distribution is rather steep (excitation dominated part of light ions' distribution), however smaller than unity due to the variety of abundances present. The intermediate region is primarily controlled by the difference in line number between both groups and by the temperature dependence of the iron group contribution (as mentioned above, the light ions' distribution function remains rather constant): For hotter temperatures, there are fewer iron group lines, thus the slope is smaller and $\alpha$ accordingly higher; for lower temperatures, the iron group line number is much larger, inducing a larger slope and smaller $\alpha$. This explains the decrease of $\alpha$(and $\hat \alpha$) with decreasing temperature. At lowest line-strengths ( $k_{\rm L}\le 0$), finally, the oscillator strength distribution dominates the slope. The staircase-like structure "observed'' for equal abundances is smeared out by the actual abundance pattern.

The deeper wind region is controlled by the line-distribution at lower $k_{\rm L}$values. Thus, the mass-loss rate follows the radiative acceleration by iron group elements. Since increasing $k_{\rm L}$ means also increasing distance from the star, the outer velocity law and especially the terminal velocity is controlled by light ions. Here, details of the distribution (e.g., decreasing $\hat \alpha$) are essential for a quantitative description.

Our results explain easily the contribution of various elements to the depth dependent force-multiplier as shown by Pauldrach ([1987], Fig. 10). They also explain the extreme sensitivity of terminal velocities on subtle effects (small variations in temperature, density and composition, cf. Pauldrach et al. [1990], Fig. 8) having a decent influence on the light ions distribution function. Note that only a small number of lines affect the acceleration in the outer part of the wind! In so far, the large observed variance in $v_{\infty}/v_{\rm esc}$ for O-star Supergiants (Howarth & Prinja [1989]) is not surprising at all.

We have stressed the importance of relative abundances, esp. with respect to the their mean value for the two groups of elements. If there were no differences, the line distribution would be much more curved (and the corresponding slopes more depth- and temperature-dependent) compared to solar conditions. Note that different mean values (e.g., in Pop. III stars) might result in wind properties which are significantly different from "present-day'' objects.

Regarding the total distribution function, the direct influence of $\gamma$concerning the slope (as discussed above for the cases of hydrogenic or individual ions, respecively) is almost completely lost. (Only at weak line-strengths $({\rm log}\, k_{\rm L}< 0)$, which are marginally contributing to the line-force, the $\gamma-1$ slope becomes visible again.) We have seen that a variety of different oscillator strength distributions lead to almost the same shape of the resulting line-strength statistics. What survives, however, is the indirect influence: The lower the value of $\gamma$, the more lines are present in the decisive $k_{\rm L}$ range! Thus, $\gamma$ controls the vertical offset of the distribution, or, in other words, $k_{\rm CAK}$respectively $\bar Q$. By comparing with the actual case, it turns out that the effective value of $\gamma$ is of order 1.2 to 1.3, i.e., is similar to the corresponding value for hydrogenic ions, although the specific value for certain ions can be different (e.g., Fe IV has $\gamma \approx 1$).

Since the line acceleration results from the flux (times frequency) weighted line-strength distribution, we have briefly shown the influence of the corresponding operation (cf. also Abbott [1982]). Although some differences between flux-weighted and "normal'' distribution become visible and turn out to be of importance for quantitative calculations under specific conditions, the principal behaviour and all prior conclusions are not affected:

Unfortunately and contrasted to the case of hydrogenic ions which are controlled almost exclusively by the underlying oscillator strength statistics, there is no single process which dominates the final shape of the total line strength distribution. Summarizing our above results, the most important properties which have to be considered instead are

• the horizontal offset in the distribution of light and heavy ions, as function of relative abundance;
• the different number of lines from both groups and their specific dependence on temperature;
• the fact that the distribution at its high $k_{\rm L}$ end is controlled by excitation effects for the predominant ion species;
• the total line number depending on the (average) oscillator strength distribution.

As a consequence of these effects, we have explained the following properties which should be valid under typical conditions (relative abundances are solar):

$\hat \alpha(k_{\rm 1})$ decreases with decreasing $T_{\rm eff}$, increasing $k_{\rm 1}$ and decreasing global metallicity z,

where the latter effect relies on the argument that a change in z simply shifts the line-strength distribution horizontally. From these considerations, it is obvious that low metallicity and/or low density winds (from dwarfs) should have a smaller average $\hat \alpha$ than high metallicity and denser winds, with the implication of lower terminal velocities and a steeper wind-momentum luminosity relation. Present observations are consistent with any of the above theoretical considerations.

In most cases, the variation of $\hat \alpha$ (and the corresponding $\hat \delta$) throughout the wind is essential and has to be accounted for at least in consistent hydrodynamical solutions aiming at a quantitative description; the usual power-law with constant $\hat \alpha, \hat \delta$, however, may be justified in hotter (O-star) winds which tend to have more constant f.-m. parameters than cooler ones.

In view of this summary, one might question about the central ingredients which are inevitable to allow for a radiation force $\propto k_{\rm 1}^{\hat \alpha}$, with $\hat \alpha$ a rather constant function over the contributing range of $k_{\rm 1}$. Most important to this regard is the requirement that the slope of the line-distribution function is not to steep over several dex in line-strength. Otherwise, the line-force can no longer be described in the presumed way, and all scaling relations derived here or elsewhere would change. (E.g., Gayley [1995] considered a Gaussian line-strength distribution as an "academic'' example.) Thus, a different world might be possible only if the gf-distribution were much steeper as it actually is and the level-density were much higher. If only the first condition were met and $\sigma$ reasonably large, the distribution would be controlled by $2t/\sigma$ globally, since $\gamma > \gamma_{\rm crit}$, and $\alpha = 1 - 2t/\sigma >0$ again. If, on the other hand, $\sigma$ were much smaller, however $\gamma$ as it is, no dramatic effects are to be expected since $x_{\rm max}$ remains small, as a final consequence of the dilution of the radiation field and density in an expanding medium.

Thus, although the shape of the line-distribution is determined by various processes, it would require a significantly different atomic physics (or coupling constants) in order to prevent radiation driven winds to behave as we think they do.

We finish this paper with one important comment. Although our results for line-strength distribution functions are valid under rather general circumstances (at least, if one accounts for a consistent description of the ionizing and illuminating radiation field), the performed transition to radiative accelerations (and scaling laws!) presumes at least two conditions to be valid:

First, the effects of line-overlap should be marginal or at least describable by some (almost) depth-independent correction factor (applied to $k_{\rm CAK}$ or $\bar Q$, e.g. Puls [1987]). In case of stratified ionization structures with increased efficiency of multi-line scattering, which are attributed to be responsible for the large observed performance numbers of WR-winds (Lucy & Abbott [1993]; Springmann & Puls [1998]), additional considerations are required, including the role of the optically thick continuum in the wind. Note, however, that Gayley ([1995]) pointed to the formal possibility of inducing large ${\dot M}$ by decreasing the value of $\hat \alpha$ while, e.g., keeping the "standard assumption'' of $Q_{\rm o} \approx \bar Q\approx 2\,000$. In our perspective, this might be feasible if the number of lines from iron group elements were essentially increased in the sub-critical region, e.g. if additional strong lines from excited levels were present.

Second, the prediction that thin winds should exhibit a smaller average $\hat \alpha$ value with accordingly modified scaling relations may describe only part of the actual situation. Due to the importance of velocity curvature terms in the transonic region (neglected in any CAK-like hydrodynamical approach), additional effects may be present which further reduce the mass-loss rate (cf. Owocki & Puls [1999]).