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

This paper analyzes the inter-relation between line-statistics and radiativ e 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 c orrespondence of the local exponent of (almost) arbitrary line-strength dis tribution functions and a, which is the ratio of optically thick to total l ine-force, is discussed. Both quantities are found to be roughly equal as l ong as the local exponent is not too steep. We compare the (conventional) parameterization applied in this paper with t he so-called Q-formalism introduced by Gayley (1995;) and conclude that the latter can be applied alternatively in its most general form. Its "stronge st form", however (requiring the Ansatz (Q) over bar = Q(o) to be valid, wi th Q, the line-strength of the strongest line), is justified only under spe cific conditions, typically for Supergiants with T-eff greater than or simi lar to 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 appr oximate depth independent ratio of line and Thomson opacity. Since k(L) dep ends on the product of oscillator strength, excitation and ionization fract ion as well as on elemental abundance, all of these factors have their own, specific influence oil 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 oscill ator strength distribution. Furthermore, it is shown that for trace ions on e stage below the major one (e.g., HI in hot winds) the equality alpha+delt a approximate to 1 is valid throughout the wind. For the majority of non-hydrogenic ions, we follow the statistical approach suggested by Alien (1966): refined in a number of ways which allow, as a u seful 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 effec ts and the second one follows the oscillator strength distribution of the s pecific ion. By summing up the contributions of all participating ions, this direct infl uence of the oscillator strength distribution almost vanishes. It turns out , however, that there is a second, indirect influence controlling the absol ute line numbers and thus k(CAK). From the actual numbers, we find an avera ge exponent of order -1.2...-1.3, similar to the value for hydrogen. Most important fur the shape of the total distribution is the difference in line-statistics between iron group and light ions as well as their differe nt (mean) abundance. Since the former group comprises a large number of met a-stable levels, the line number from iron group elements is much higher, e specially at intermediate and weak line-strengths. Additionally, this numbe r increases significantly with decreasing temperature (more lines from lowe r 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 q uantity controls the relative influence of the specific distributions on th e total one and the overall shape. For solar composition, a much more const ant 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 (correspo nding to the acceleration in the inner wind part), whereas light ions (incl uding hydrogen under A-star conditions) dominate the high k(L) end (outer w ind). Typically, this part of the distribution is steeper than the rest, du e to excitation effects. Finally, the influence of global metallicity z is discussed. We extend alre ady known scaling relations (regarding mass-loss; terminal velocity and win d-momentum rate) with respect to this quantity. In particular, we demonstra te that. besides the well-known direct effect (k(CAK) proportional to z(1-a lpha)), the curvature of the line-strength distribution at its upper end in duces a decrease of alpha for low metallicity and/or low wind density. Summarizing the different processes investigated, the force-multiplier para meter a becomes a decreasing function of decreasing T-eff, increasing k(1) = d upsilon/dr/rho and decreasing global metallicity z, consistent with the findings of earlier and present empirical results and observations.