Radiation driven winds of hot luminous stars. XIV. Line statistics and radiative driving

This paper analyzes the inter-relation between line-statistics and radiative driving in massive stars with winds (excluding Wolf-Rayets) and provides insight into the qualitative behaviour of the well-known force-multiplier parameters k_cak, alpha and delta , with special emphasis on alpha . After recapitulating some basic properties of radiative line driving, the correspondence of the local exponent of (almost) arbitrary line-strength distribution functions and alpha , which is the ratio of optically thick to total line-force, is discussed. Both quantities are found to be roughly equal as long as the local exponent is not too steep. We compare the (conventional) parameterization applied in this paper with the so-called bar Q-formalism introduced by Gayley (\cite{Gayley95}) and conclude that the latter can be applied alternatively in its most general form. Its ``strongest form'', however (requiring the Ansatz bar Q = Q_o to be valid, with Q_o the line-strength of the strongest line), is justified only under specific conditions, typically for Supergiants with T_eff >~ 35 000 K. The central part of this paper considers the question concerning the shape of the line-strength distribution function, with line-strength k_L as approximate depth independent ratio of line and Thomson opacity. Since k_L depends on the product of oscillator strength, excitation- and ionization fraction as well as on elemental abundance, all of these factors have their own, specific influence on the final result. At first, we investigate the case of hydrogenic ions, which can be treated analytically. We find that the exponent of the differential distribution is -4/3 corresponding to alpha = 2/3, as consequence of the underlying oscillator strength distribution. Furthermore, it is shown that for trace ions one stage below the major one (e.g., Hi in hot winds) the equality alpha + delta ~ 1 is valid throughout the wind. For the majority of non-hydrogenic ions, we follow the statistical approach suggested by Allen (\cite{Allen66}), refined in a number of ways which allow, as a useful by-product, the validity of the underlying data bases to be checked. Per ion, it turns out that the typical line-strength distribution consists of two parts, where the first, steeper one is dominated by excitation effects and the second one follows the oscillator strength distribution of the specific ion. By summing up the contributions of all participating ions, this direct influence of the oscillator strength distribution almost vanishes. It turns out, however, that there is a second, indirect influence controlling the absolute line numbers and thus k_cak. From the actual numbers, we find an average exponent of order -1.2... -1.3, similar to the value for hydrogen. Most important for the shape of the total distribution is the difference in line-statistics between iron group and light ions as well as their different (mean) abundance. Since the former group comprises a large number of meta-stable levels, the line number from iron group elements is much higher, especially at intermediate and weak line-strengths. Additionally, this number increases significantly with decreasing temperature (more lines from lower ionization stages). In contrast, the line-strength distribution of light ions remains rather constant as function of temperature. Since the line-strength depends linearly on the elemental abundance, this quantity controls the relative influence of the specific distributions on the total one and the overall shape. For solar composition, a much more constant slope is found, compared to the case if all abundances were equal. In result, we find (for solar abundances) that iron group elements dominate the distribution at low and intermediate values of line-strength (corresponding to the acceleration in the inner wind part), whereas light ions (including hydrogen under A-star conditions) dominate the high k_L end (outer wind). Typically, this part of the distribution is steeper than the rest, due to excitation effects. Finally, the influence of global metallicity z is discussed. We extend already known scaling relations (regarding mass-loss, terminal velocity and wind-momentum rate) with respect to this quantity. In particular, we demonstrate that, besides the well-known direct effect (k_cak ~ z^1-alpha ), the curvature of the line-strength distribution at its upper end induces a decrease of alpha for low metallicity and/or low wind density. Summarizing the different processes investigated, the force-multiplier parameter alpha becomes a decreasing function of decreasing T_eff, increasing k_1 = d v/d r/rho and decreasing global metallicity z, consistent with the findings of earlier and present empirical results and observations.