1 Introduction

Over the last three decades the idea of radiative driving by metal-line absorption/scattering has been applied with increasing success. Besides the original motivation of explaining the supersonic outflows of OB-star winds and predicting their strengths (Lucy & Solomon [1970]; Castor et al. [1975] (CAK)), a much wider domain of sophisticated physical processes is considered now. To name only a few recent and exciting developments, there is, e.g., the possibility of compressing wind material into the stellar equatorial zone and even creating a disk due to the combined effects of (radial) line and centrifugal force in case of large rotational rates (Bjorkman & Cassinelli [1993]) and the counteracting role of non-radial line-forces which might inhibit this process (Owocki et al. [1996]). Another example (although originating from the very beginning: again Lucy & Solomon [1970]) would be the theory of line-force instability (Owocki & Rybicki [1984], [1985]) and its ongoing refinements (Owocki & Puls [1996], [1999]), which still awaits direct observational proof.

Going extragalactic, it turned out that the observed wind-momentum rate of supergiant winds allows for a determination of distances (Kudritzki et al. [1995]) and may finally become an independent alternative to using Cepheids as distance indicators on the intermediate distance scale up to the Virgo Cluster. Even further out, the theory of radiative line driving and its possible instability is providing a first step towards explaining the physics of BAL-QSOs (Arav & Li [1994]; Arav et al. [1994]; Feldmeier et al. [1997]).

All of these investigations and theories have one thing in common: Although the specific line transfer for obtaining the radiation force per line is treated differently to meet the required physics, the total line force arising from thousands of contributing lines is calculated - either directly or indirectly - by means of a certain statistical description of the line distribution.

This procedure still follows the ingenious Ansatz by CAK and improvements obtained by Abbott ([1982]), which to its end requires the knowledge of only three "numbers'' $k_{\rm CAK}, \alpha$ and $\delta$, the so-called force-multiplier parameters, in order to extrapolate from the behaviour of one line to the effects of the total line ensemble.

Unfortunately, the physical explanation of their origin and the discussion of their behaviour under various physical conditions is rather unsatisfactory. The only exception (to our knowledge) is the publication by Gayley ([1995]), which provides a profound insight into the efficiency of line driving (compared to electron scattering) and suggests a modified parameterization of the line-force, which however does not give any further clue concerning the involved line-statistics itself.

Nevertheless and especially in view of the wide use of the (improved) CAK- parameterization, the present situation is mostly unclear. This even more so, since it is the actual numerical value of one or more of the force-multiplier parameters which may allow for certain effects to arise or rather to inhibit the process.

We want to mention here only two examples: The theoretical basis of the wind-momentum luminosity relation relies on the parameter $\alpha$ (actually $\alpha-\delta$) to be close to 2/3, and a consistent calculation of a wind-compressed disk (neglecting non-radial forces) requires a (maybe too) large value of $\delta$.

Recently, Kudritzki et al. ([1998]) have presented a method and first results on how to obtain an improved parameterization of the line-force by allowing for a depth dependence of the force-multiplier parameters. However, also this paper is based on a purely descriptive approach, namely by calculating line-forces under various conditions and then by fitting the result to the modified parameterized expression. Although the final outcome of this project will prove to be useful for many applications, the question about the underlying physics still remains.

Since radiation driven wind theory has proven to work under different conditions and the physics of radiation driving is not only a black box process requiring the collection (or guess!) of various values for $k_{\rm CAK}, \alpha$ and $\delta$ in order to solve the hydro-equations, we feel that a more thorough investigation of the underlying statistics, physics and consequences has to be performed. The present paper is intended to answer at least some of the obvious questions and is organized as follows. In Sect. 2, we review the problem of how to calculate the line-force arising from an ensemble of lines by means of the so-called line-strength distribution function, where in a first step the "standard'' representation is used. Additionally, however, we investigate also different distribution functions and interpret the ensemble line-force in a more general way. In Sect. 3, we turn to "reality'' and present our method to derive NLTE line-distribution functions valid for the considered wind plasma conditions. We show typical examples and comment on Gayley's ([1995]) $\bar Q$- formalism. In Sect. 4, we try to understand the fundamental physics which controls the slope of the line-strength distribution function and hence the actual value of $\alpha$. Section 5 focuses on the consequences if the plasma conditions are changed, especially due to a different metallicity and/or mean wind density, and Sect. 6 summarizes our results and gives some caveats regarding the limits of our investigation.