2 The line-force from an ensemble of lines and the line-strength distribution function

In this section, we review the basic physics of radiative driving of an ensemble of lines and interpret the final outcome in terms of a so-called "line-strength distribution function''. To this end, we allow for a number of approximations which will turn out to be of either minor importance or will be relaxed during further proceedings. Specifically,

• we will neglect the finite-disk effect and consider only radially streaming photons. The corresponding modification to the total line-force (cf. Friend & Abbott [1986] and Pauldrach; Puls & Kudritzki [1986]) can be easily incorporated afterwards;

• We will use the Sobolev approximation for calculating the line-force of a single line, neglecting line-overlap, the diffuse radiation field and curvature terms of the velocity law. We will briefly comment on these restrictions and their consequences in Sect. 6. Since these approximations are generally used in the considered framework and may influence the results only under very specific conditions, we accept them until further discussion. Note already here, however, that by neglecting the influence of line-overlap we can and will describe only the situation for "normal'' stars, whereas the case of WRs requires additional considerations (e.g., Lucy & Abbott [1993]; Schmutz [1997]; Springmann & Puls [1998] and Owocki & Gayley [1999]);

• The classification of lines to be either optically thin or thick with interaction probabilities of $\tau$ or unity, respectively, instead of using the "exact'' expression $(1 - \exp(-\tau)$, see below) leads to an only marginal error. We prefer this procedure throughout the major part of our discussion since it yields more insight into the relevant physical processes;

• Finally, we consider only stationary and one-dimensional winds, since our reasoning is directed to understanding the reaction of an ensemble of lines projected from the behaviour of a single line, and the usual procedure to allow for this projection is similar in cases of time-dependent (e.g., Owocki & Rybicki [1984]) or multidimensional (e.g., Owocki et al. [1994]) calculations.

2.1 The radiative acceleration provided by one line

Within these approximations and assumptions, the radiative acceleration provided by scattering of photons in a single line (transition frequency $\nu_{\rm i}$) to the material in a spherically expanding shell of size ${\rm d}r$, mass ${\rm d}m = 4 \pi \rho r^2 {\rm d}r$ and velocity $v(r) \ldots v(r) + {\rm d}v$ is given by the average transferred momentum per unit time and ${\rm d}m$, i.e.,

 

$$g_{\rm rad}^{\rm i} \frac{\left\langle\Delta P\right\rangle}{\Delta t {\rm d}m} =$$ =

$$\frac{1}{4 \pi \rho r^2 {\rm d}r} N_{\nu}{\rm d}\nu \frac{h \nu}{c} \bigl(1 - \exp(-\tau_{\rm S})\bigr) =$$ =

$$\frac{1}{4 \pi r^2 c^2} L_{\nu} \nu_{\rm i} \frac{1}{\rho} \frac{{\rm d}v}{{\rm d}r} \bigl(1 - \exp(-\tau_{\rm S})\bigr)$$ = (1)

where $N_{\nu}{\rm d}\nu$ is the number of photospherically emitted photons per unit time in the frequency range $\nu \ldots \nu + {\rm d}\nu$, $L_{\nu}$the stellar luminosity and

$$\left\langle\Delta P\right\rangle N_{\nu}{\rm d}\nu \frac{h \nu}{c} =$$ =

$$\frac{L_{\nu} {\rm d}\nu}{h \nu} \frac{h \nu}{c}$$ = (2)

the average transferred momentum by line absorption or scattering (radially streaming photons provided and accounting for the cancellation of the foreaft-symmetric reemission processes). Finally,

$${\rm d}\nu = \nu_{\rm i}\frac{{\rm d}v}{c}$$ (3)

is the frequency range which can actually contribute to line scattering inside the shell via the Doppler effect, and $\bigl(1 - \exp(-\tau_{\rm S})\bigr)$ is the interaction probability for a line optical depth (in Sobolev approximation)

 

$$\tau_{\rm S} \frac{\bar{\chi}_{\rm i}\lambda_{\rm i}}{{\rm d}v/{\rm d}r}$$ = (4)

$$\bar{\chi}_{\rm i} \frac{\pi e^2}{m_{\rm e} c} gf_{\rm i} \left({n_{\rm l}\over g_{\rm l}} - {n_{\rm u}\over g_{\rm u}} \right)$$ = (5)

with $\bar{\chi}_{\rm i}$ the frequency integrated line opacity, gf-value $gf_{\rm i}$ and $n_{\rm l}, n_{\rm u}, g_{\rm l}, g_{\rm u}$ the occupation numbers and statistical weights of the participating lower and upper level.

2.2 Line-strength and optical depth

To proceed further, we have to sum up the contribution of all individual lines to obtain the total radiative acceleration. To this end, we define the dimensionless line-strength of a single line as

 

$$k_{\rm L} \frac{\chi_{\nu}(\nu_{\rm i}) \sqrt{\pi}}{\sigma_{\rm e}} = \frac... ... \frac{\bar{\chi}_{\rm i}\lambda_{\rm i}}{\rho} \frac{1}{s_{\rm E}{v_{\rm th}}}$$ = (6)

$$\chi_{\nu} \bar{\chi}_{\rm i}\Phi_{\nu};\; \Phi_{\nu} = \frac{1}{\sqrt{\pi} \Delta \nu_{\rm D}} {\rm e}^{-(\frac{\nu-\nu_{\rm i}}{\Delta \nu_{\rm D}})^2}.$$ =

$\sigma_{\rm e}=: s_{\rm E}\rho$ is the Thomson scattering opacity, $\Delta \nu_{\rm D}$ the Doppler width of the line and ${v_{\rm th}}$ the ionic thermal velocity. From Eq. (6), the line-strength can be interpreted twofold. On the one side, it is, except from a factor $\sqrt{\pi}$, the maximum opacity of the considered line in units of the minimum continuum opacity present in the wind (=Thomson). Alternatively, it can be considered, except a factor of $2 N, N = 2 \ldots 4$, as the ratio of frequency integrated line opacity $\bar{\chi}_{\rm i}$ to minimum frequency integrated continuum opacity $2 N \Delta \nu_{\rm D} \sigma_{\rm e}$.

One comment is necessary here: The incidence of ${v_{\rm th}}$ at this stage of reasoning seems to be "natural'' in terms of understanding the physical meaning of line-strength. Moreover, it is actually needed if we require the latter to be dimensionless, which is important for our future statistical analysis. As it will turn out, however, ${v_{\rm th}}$ will reappear in various combinations with other quantities after summing up the line-strength contributions of different metals, i.e., ${v_{\rm th}}$ would be no longer unique due to its dependence on atomic mass. Thus, from now on we will concentrate on the most important aspect of ${v_{\rm th}}$ in this context, namely that it has a dimension (also noting that its value is smaller than the sound-speed), however use a value independent of atomic mass, in particular, the value for hydrogen.

The relation of the so defined line-strength to Sobolev optical depth is given by

$$\tau_{\rm S}= \frac {\sigma_{\rm e}{v_{\rm th}}}{{\rm d}v/{\rm d}r} k_{\rm L}= t k_{\rm L}$$ (7)

where t is the optical depth parameter defined by CAK. The advantage of using $k_{\rm L}$ instead of opacities or optical depths is that $k_{\rm L}$ is a quantity which remains rather constant throughout the wind (at least in the typical case of frozen-in ionization), and whose distribution can be described in an almost depth independent statistical way. Before doing this, however, we will sum up the contributions of all individual lines, in the spirit outlined above, i.e., by dividing lines into two categories. Optically thick lines are those with $\tau_{\rm S}\ge 1$ and interaction probability "1'', whereas optically thin lines shall have $\tau_{\rm S}< 1$ and interaction probability $\tau_{\rm S}$. Defining

$$k_{\rm 1}= k_{\rm L}(\tau_{\rm S}= 1) = \frac {{\rm d}v/{\rm ... ...s_{\rm E}{v_{\rm th}}}\frac {{\rm d}v/{\rm d}r}{\rho} = t^{-1}$$ (8)

as the line-strength where the division $\tau_{\rm S}= 1$ is reached, we find for the total line acceleration

$$g_{\rm rad}^{\rm tot}= \frac{s_{\rm E}{v_{\rm th}}}{4 \pi r^... ...{\rm 1}} k_{\rm L}^{\rm i}L_{\nu_{\rm i}} \nu_{\rm i}\right\}.$$ (9)

The first term inside the bracket gives the contribution by optically thick lines and depends only on the hydrodynamical structure via $k_{\rm 1}\propto ({\rm d}v/{\rm d}r)/\rho$, whereas the second term gives the optically thin line contribution and is independent of the hydro-structure, however depends on the specific line-strengths, i.e., atomic properties and level population.

2.3 Line-strength distribution function and total acceleration

2.3.1 Perfect power-law distribution

Before going into further detail and in concert with most previous investigations related to this topic, we will assume that the number of lines in a frequency interval $\nu, \nu + {\rm d}\nu$ and line-strength $k_{\rm L}, k_{\rm L}+ {\rm d}k_{\rm L}$ can be represented by a power-law

$${\rm d}N(\nu, k_{\rm L}) = \,-\,N_{\rm o}\,f_{\nu}(\nu)\, k_{\rm L}^{\alpha-2}\, {\rm d}\nu \,{\rm d}k_{\rm L},$$ (10)

with $0 < \alpha < 1$, where the frequential distribution shall be independent from the line-strength distribution. The negative sign accounts for the fact that the number of lines increases for decreasing line-strength. So far, the normalization "constant'' $N_{\rm o}$ is allowed to have some additional depth dependence. With (10) and substituting the sums in (9) by appropriate (double-) integrals with frequencies from $0 \ldots {\infty}$ and line-strengths from $k_{\rm 1}\ldots {\infty}$and $0 \ldots k_{\rm 1}$, respectively, we find

  

$$g_{\rm rad}^{\rm tot} \frac{s_{\rm E}{v_{\rm th}}}{4 \pi r^2 c^2}\, \Biggl\{ k_{\rm 1}\... ...t_{k_{\rm 1}}^{{\infty}} L_{\nu} \nu \, \vert\,{\rm d}N(\nu,k_{\rm L})\vert \,+$$ =

$$\int_0^{{\infty}} \int_{0}^{k_{\rm 1}} k_{\rm L}L_{\nu} \nu \, \vert\,{\rm d}N(\nu,k_{\rm L})\vert \Biggr\} =$$ + (11)

$$\frac{s_{\rm E}{v_{\rm th}}N_{\rm o}\, \int_0^{\infty}L_\nu \nu f... ...{1}{1{-}\alpha} k_{\rm 1}^\alpha {+} \frac{1}{\alpha} k_{\rm 1}^\alpha \Bigr\}.$$ = (12)

Note, that in case of $\alpha < 0$ the contribution from optically thin lines (second term) would diverge at its lower boundary. From Eq. (12), two points are obvious: Both the line-force provided by optically thick and by optically thin lines scales with the same power $\alpha$, and we can interpret this exponent as the ratio of line-force from optically thick lines to total force,

$$\alpha = \frac{g_{\rm rad}^{\rm thick}}{g_{\rm rad}^{\rm tot}},$$ (13)

a result, which we will later on discuss carefully. Collecting terms and using the radiative acceleration provided by Thomson-scattering as a scaling factor,

$$g_{\rm rad}^{\rm TH}= \frac{s_{\rm E}L}{4 \pi r^2 c}$$ (14)

with stellar luminosity L, we can express the total line acceleration in terms of the so-called force-multiplier f.m.(r),

$${\rm f.m.}(r) = \frac{g_{\rm rad}^{\rm tot}}{g_{\rm rad}^{\rm TH}} = k_{\rm CAK}k_{\rm 1}^\alpha = k_{\rm CAK}t^{-\alpha},$$ (15)

where the force-multiplier parameter $k_{\rm CAK}$ is defined by

$$k_{\rm CAK}= \frac{\int_0^{\infty}L_\nu \nu f_{\nu}(\nu)\,{\... ... \frac{{v_{\rm th}}}{c} \frac{N_{\rm o}}{\alpha(1 - \alpha)}.$$ (16)

This result was given firstly by CAK. Note, however, that they additionally assumed $f_{\nu}(\nu)\, {\rm d}\nu = {\rm d}\nu /\nu$, in which case the first factor in (16) is unity.

"Exact'' result.

Let us now consider the error we have made by dividing the lines into optically thick and thin ones exclusively. Accounting for the definition of the line-strength and its relation to the Sobolev optical depth, we can reformulate the radiative acceleration provided by one line, Eq. (1),

 

$$g_{\rm rad}^{\rm i} \frac{s_{\rm E}{v_{\rm th}}L_{\nu} \nu_{\rm i}}{4 \pi r^2 c^2} k_{\rm L}^{\rm i}\frac{1 - \exp(-\tau_{\rm S})}{\tau_{\rm S}} =$$ =

$$\frac{s_{\rm E}{v_{\rm th}}L_{\nu} \nu_{\rm i}}{4 \pi r^2 c^2} k_{\rm 1}\left(1 - \exp\left(-\frac{k_{\rm L}^{\rm i}}{k_{\rm 1}}\right)\right).$$ = (17)

The first equation above can be interpreted as the product of the radiative acceleration if the line were optically thin, multiplied by the correction for self-shadowing due to the actual optical depth $\tau_{\rm S}$ (cf. also Gayley [1995]). Integrating over the line-strength distribution function without performing the division at $\tau_{\rm S}= 1$, we obtain directly the "exact'' version of Eq. (12),

$$g_{\rm rad}^{\rm tot}= \frac{s_{\rm E}{v_{\rm th}}N_{\rm o}\... ... c^2} \; \frac{\Gamma(\alpha)}{1 - \alpha} k_{\rm 1}^\alpha$$ (18)

with Gamma-function $\Gamma(\alpha)$. Thus, since $0 < \alpha < 1$ (by assumption), the error introduced by our approximation is given by $1/(\alpha \Gamma(\alpha)) = 1/(\Gamma(1 + \alpha))$, i.e., an overestimation of at most 13%.

2.3.2 Distribution functions with arbitrary dependence on line-strength

In the previous section, we have shown that the assumption of power-law distributed line-strengths with exponent $\alpha - 2$ directly leads to an ensemble line-force being proportional to $k_{\rm 1}^\alpha$, and that in this case $\alpha$ turns out to represent the ratio of optically thick to total line acceleration, provided that $0 < \alpha < 1$.

In so far, it seems quite natural to consider line strength distribution function and line force as interchangeable quantities and to identify the line-force exponent instantaneously as a manifestation of the underlying line-statistics. Although this perspective is widely spread, a closer inspection of the above procedure immediately necessitates a major caveat:

Due to the different weighting of line-strength in the expression for the line-force (Eq. 11), $g_{\rm rad}^{\rm thin} \propto \int k_{\rm L}{\rm d} N(k_{\rm L})$ and $g_{\rm rad}^{\rm thick} \propto \int {\rm d}N(k_{\rm L})$, a strict correspondence between the exponent of line-strength distribution and force-multiplier parameter can be expected a priori only if the distribution follows a power-law over a large range of line-strengths. If the distribution is curved in the log, this equality remains questionable and has to be considered with caution, even if one accounts for locally defined values $\alpha=\alpha(k_{\rm L}=k_{\rm 1})$.

To investigate this problem, we consider a generalized line-strength distribution function

$${\rm d}N(\nu, k_{\rm L}) = \,-\, f(k_{\rm L},\nu) \, {\rm d}\nu \,{\rm d}k_{\rm L},$$ (19)

with arbitrary (positive) function $f(k_{\rm L}, \nu)$, where we even allow for a dependence of line-strength on frequency. Then, by realizing that

$$\left\langle N(k_{\rm L})\right\rangle= \int_{k_{\rm L}}^{\i... ...ty}\frac{L_\nu \nu}{L} f(k_{\rm L}',\nu) \,{\rm d}\nu \right)$$ (20)

is the flux (times frequency) weighted cumulative number of lines stronger than $k_{\rm L}$, we can integrate the optically thin contribution in (11) by parts and obtain, after adding the optically thick contribution

$$g_{\rm rad}^{\rm tot}\propto \int_0^{k_{\rm 1}} \left\langle N(k_{\rm L})\right\rangle\,{\rm d}k_{\rm L}.$$ (21)

Here we have assumed that the total (flux weighted) number of lines $\left\langle N(0)\right\rangle$ remains finite. (Actually, the much weaker requirement that $k_{\rm L} \left\langle N(k_{\rm L})\right\rangle\rightarrow0$ for $k_{\rm L}\rightarrow0$ is sufficient.)

Equation (21) can be alternatively expressed as

$$g_{\rm rad}^{\rm tot}= \frac{L s_{\rm E}{v_{\rm th}}}{4 \pi ... ...e N_{\rm thin}(k_{\rm L})\right\rangle{\rm d}k_{\rm L}\Bigr\},$$ (22)

with $\left\langle N_{\rm thick}\right\rangle= \left\langle N(k_{\rm 1}) \right\rangle$ the weighted number of lines stronger than $k_{\rm 1}$ (which are then optically thick by definition) and $\left\langle N_{\rm thin}(k_{\rm L})\right\rangle$ the cumulative number of optically thin lines ( $= \left\langle N(k_{\rm L})\right\rangle-\left\langle N(k_{\rm 1})\right\rangle$). Figure 1 displays this result graphically. Note, that in order to derive this result, we have again used our approximation of replacing the interaction probability $1-\exp(-\tau_{\rm S})$ by $(1,\tau_{\rm S})$, respectively.

Before we proceed further, let us mention that Eq. (22) and Fig. 1 allow for an useful visualization of two extreme cases. At first, assume that either the velocity gradient is so small or the density so high that all lines are effectively stronger than/equal to $k_{\rm 1}$. In this optically thick case, $k_{\rm 1}$ is situated in the left part of the previous figure, and $\left\langle N(k_{\rm L}) \right\rangle= \left\langle N(k_{\rm 1}) \right\rangle= {\rm const}$for all $k_{\rm L}< k_{\rm 1}$ by definition (no lines weaker than $k_{\rm 1}$ present). The acceleration is then found from the rectangular area between $k_{\rm L}= 0\ldots k_{\rm 1}$, i.e., $g_{\rm rad}^{\rm tot}\propto k_{\rm 1}$ and thus $\alpha = 1$. From the figure, it is also clear that the resulting acceleration is enormously reduced compared to the case of all lines being optically thin (cf. also Eq. (17)), which is the other extreme. In the latter case (arising for large velocity gradients or low densities), $k_{\rm 1}$ lies at the rightmost point of the abscissa, and all lines have strengths lower than $k_{\rm 1}$. Hence, the integral becomes independent of $k_{\rm 1}$, and the radiative acceleration obtains its maximum value with $\alpha = 0$. In all other cases, the $\alpha$-parameter of the line-force corresponds to the ratio of "dark'' area (optically thick force) and the sum of light and dark area (total force), cf. Eq. (13) for the case of a strict power-law and Sect. 2.4 otherwise.

Figure 1: The line-force as integral of (flux weighted) cumulative line number over line-strength; contribution by optically thin (light) and optically thick lines (dark). Compare Eq. (22) and text

On the basis of our alternative and general expression (21) for the line-force, we can now answer the question raised above, namely under which conditions this line-force can be represented by the CAK law $g_{\rm rad}^{\rm tot} \propto k_{\rm 1}^{\alpha}$, where $\alpha$ corresponds at least to some local exponent of our arbitrary line-strength distribution function.

To this end, we define two line-strengths $k_{-}< k_{\rm 1}< k_{+}$, where k_- and k₊ are chosen in such a way that $N(k_{\rm L})$ follows roughly a power-law in between, i.e., is roughly linear with slope $\alpha -1$ in the log-log representation. Actually, this is almost always possible if the range $k_{-}\ldots k_{+}$ is not too large, say of order two dex. (Here and in the following we assume that the wind is not too thin, so that $k_{\rm 1}$ lies well below the maximum line-strength.)

The number of lines at k₊ is then N₊, N_- is N(k_-) (flux weighting always provided, however brackets suppressed to simplify notation) and the distribution function in between can be approximated by

$$N(k_{\rm L}) N_{+}k_{+}^{1 - \alpha}k_{\rm L}^{\alpha - 1};\, k_{-}< k_{\rm L}< k_{+}$$ =

$$\alpha 1 + \frac{{\rm log}(N_{+}/N_{-})}{{\rm log}(k_{+}/k_{-})}.$$ = (23)

Since by definition $N_{-}\ge N_{+}$ and k₊> k_-, the maximum value of $\alpha$ is constrained to be unity, which occurs in those cases when the maximum line number is reached at a certain $k_{\rm L}$-value. $\alpha$-values below zero are not excluded from now on in our local description.

Under these conditions and using (22), the total line acceleration is given by

$$g_{\rm rad}^{\rm tot}\propto \int_0^{k_{-}} N(k_{\rm L})\,{\r... ...{1 - \alpha}k_{\rm L}^{\alpha - 1}\,{\rm d}k_{\rm L}\nonumber$$

$$= \left\{ \bar N_{\{0,k_{-}\}}- \frac{N_{+}}{\alpha} \left(... ...ac{N_{+}k_{+}^{1 - \alpha}}{\alpha}\right)\,k_{\rm 1}^{\alpha}$$ (24)

( $\alpha \ne 0$), where $\bar N_{\{0,k_{-}\}}$ is the appropriate average of $N(k_{\rm L})$ in the range $0 \ldots k_{-}$

$$\bar N_{\{0,k_{-}\}}\,k_{-}=:\int_0^{k_{-}} N(k_{\rm L})\,{\rm d}k_{\rm L}.$$ (25)

From Eq. (24), it is obvious that the ensemble line-force can be represented by $k_{\rm 1}^\alpha$ with local $\alpha$, if and only if the first term is small compared to the second one, where the former is just the difference between actual and "fitted'' area (i.e., acceleration) in the range $0 \ldots k_{-}$.

In Appendix A, we show that under fairly general assumptions this is actually the case if $\alpha > 0$, i.e., as long as the local slope of the flux-weighted cumulative line-strength distribution (in the log-log representation) is larger than -1. In contrast, line distributions with a steep slope over a large $k_{\rm L}$ range will decouple from the line-force parameterization (cf. also Sect. 4.2.8), leading to effective $\alpha$values in the line-force (see below) different from those defining the line-strength statistics.

2.4 Effective value of $\,\alpha$

Usually, the force multiplier parameters are not derived from the line-strength distribution function, however from the line-force itself, i.e., accouting explicitely for the additional weighting with $k_{\rm L}$mentioned above.

Thus, in the following we postulate the line force to be a function of $k_{\rm 1}^\alpha$ with no a priori knowledge of $\alpha$. Instead, we define an effective value $\hat \alpha$ by

$$\hat \alpha= \frac{{\rm d}\ln g_{\rm rad}^{\rm tot}}{{\rm d}... ...}}{{\rm d}k_{\rm 1}}\,\frac{k_{\rm 1}}{g_{\rm rad}^{\rm tot}}.$$ (26)

From this definition and using ensemble line-forces calculated by summing up their individual components (e.g., Abbott [1982]; Pauldrach et al. [1994]), we obtain by straightforward differentiation (cf. Eq. 17)

$$\frac{{\rm d}g_{\rm rad}^{\rm tot}}{{\rm d}k_{\rm 1}} = \frac... ...m 1}}\right) \right) L_{\nu_{\rm i}} \nu_{\rm i}\nonumber \,=$$

$$= \frac{s_{\rm E}{v_{\rm th}}}{4 \pi r^2 c^2} \sum_i \left(1 ... ...}^{\rm i}}{k_{\rm 1}}\right)\right) L_{\nu_{\rm i}} \nu_{\rm i}$$

$$\approx \frac{g_{\rm rad}^{\rm tot}}{k_{\rm 1}} \,-\,\frac{g_... ...thin}}{k_{\rm 1}} = \frac{g_{\rm rad}^{\rm thick}}{k_{\rm 1}},$$ (27)

where we have performed our approximation $1- \exp(-\tau_{\rm S}) = (1,\tau_{\rm S})$ for $k_{\rm L}^{\rm i}> k_{\rm 1}, k_{\rm L}^{\rm i}< k_{\rm 1}$, respectively, and neglected 2nd order terms. Hence, from Eq. (26)

$$\hat \alpha\approx \frac{g_{\rm rad}^{\rm thick}}{g_{\rm rad}^{\rm tot}},$$ (28)

we find the same result as in Eq. (13), however independent of any underlying line statistics! Especially, this result does not rely on any separability of frequency and line-strength.

In a first interpretation of $\hat \alpha$, Abbott ([1980], his Eq. 10) found a result different from our Eq. (28). This difference, however, bases on Abbott's implicit assumption that the average (flux times frequency weighted) line-strength of optically thin lines,

$$\bar k_{\rm thin} = \frac{\sum_{k_{\rm L}^{\rm i}< k_{\rm 1}}... ...um_{k_{\rm L}^{\rm i}< k_{\rm 1}} L_{\nu_{\rm i}} \nu_{\rm i}}$$ (29)

does not depend on the hydro variable $k_{\rm 1}$. If one relaxes this assumption and accounts also for the variation of $\bar k_{\rm thin}$ as function of $k_{\rm 1}$, the nominator in Abbott's Eq. (10) becomes unity and the same result for $\hat \alpha$ is recovered as given in (28).

From this expression, it is obvious that $\hat \alpha$ must lie in the range 0...1. Combined with our previous notion that the line-force can be parameterized in the form $k_{\rm 1}^\alpha$ if the local $\alpha$ value of the distribution function is larger than zero, this leads to the result that we should have $\hat \alpha(k_{\rm 1}) \approx \alpha(k_{\rm 1})$ for a large range of line-strengths as long as this condition is met. If the distribution has a significant steepness locally, $\hat \alpha(k_{\rm 1}) > \alpha(k_{\rm 1})$ is to be expected.

Furthermore, the value of $\hat \alpha$ is independent of $k_{\rm 1}$ if and only if the flux-weighted line-strength distribution follows an exact power law with exponent $\alpha - 2$. In this case then, $\hat \alpha= \alpha$globally. Otherwise, $\hat \alpha$ becomes a function of $k_{\rm 1}$ and thus a function of depth. An instructive example is given in Appendix B.

2.5 The interpretation of $\,\hat \alpha$: Acceleration vs. line-number ratio

It has often been argued that $\hat \alpha$ is closely related to the number ratio of optically thick to thin lines, contrasted to the above formulated acceleration ratio. In the following, however, we will show that the former gives little (if any) insight into the behaviour of $\hat \alpha$.

At first note that in view of Eq. (11) the optically thin line-force can be expressed by

$$g_{\rm rad}^{\rm thin} \propto \left\langle N_{\rm thin}\right\rangle\bar k_{\rm thin}.$$ (30)

Thus, the product of (flux weighted) number of optically thin lines times average line-strength $\left\langle N_{\rm thin}\right\rangle\bar k_{\rm thin}$ remains bounded, although the number itself may formally diverge, e.g. for a typical power-law index $\alpha-2 < -1$. In this case then, the average line-strength of optically thin lines $\bar k_{\rm thin}$ approaches zero!

To avoid this possible divergence and also to keep the computational effort as small as possible, one usually defines a minimum line-strength $k_{\rm min}< k_{\rm 1}$ as a lower boundary for the contributing lines, resulting in a modified acceleration

$$g_{\rm rad}^{\rm thin}(k_{\rm min}) \propto \frac{k_{\rm 1}^\... ...,-\,\left(\frac{k_{\rm min}}{k_{\rm 1}}\right)^\alpha \right).$$ (31)

If one allows for a relative error $\epsilon$ in the line-force and supposes the power-law distribution function to be valid throughout the range $0 < k < \infty$, one finds as a constraint for $k_{\rm min}$

$$\frac{k_{\rm min}}{k_{\rm 1}} = \Bigl( \frac{\epsilon}{1\,-\,\alpha} \Bigr)^{1 \over \alpha}.$$ (32)

Hence and under "normal'' conditions ( $k_{\rm 1}= {\rm O}(1\,000)$), it is sufficient to account for lines stronger than $k_{\rm min}= 1$, i.e., stronger than Thomson-scattering, if one calculates the line-force (this was done, e.g., in Fig. A1). However, in cases of high wind densities, this limit has to be lowered, since the contribution of weaker lines with $k_{\rm L}< 1$becomes considerable then.

Using this cutoff, the number ratio of optically thick to thin lines is given by

$$\frac{N_{\rm thick}}{N_{\rm thin}} = \frac{1}{(k_{\rm 1}/k_{... ... -1}\,\rightarrow\, 0\,\,{\rm for}\,\,k_{\rm min}\rightarrow0.$$ (33)

Thus, it depends strongly on $k_{\rm 1}$ (usually increasing throughout the wind) as well as on the value chosen for $k_{\rm min}$, contrasted to the accordingly modified acceleration ratio

$$\frac{g_{\rm rad}^{\rm thick}}{g_{\rm rad}^{\rm thin}} = \fr... ...w\frac{\alpha}{1-\alpha} \,{\rm for}\,k_{\rm min}\rightarrow0.$$ (34)

The latter limit, of course, is only valid for $\alpha > 0$. In consequence, even for the simple picture of a perfect power-law, the "knowledge'' or anticipated behaviour of $N_{\rm thick}/N_{\rm thin}$ gives only little (if any) insight into the value of $\hat \alpha$. E.g., the expectation that $N_{\rm thick}/N_{\rm thin} \ll 1$ implies $\hat \alpha\rightarrow0$ is, in view of Eq. (33), by no means justified. As one example of this kind of misinterpretation, we want to mention the argument given by Kudritzki et al. ([1987]) to explain the lower $\hat \alpha$-values resulting from NLTE-calculations for winds with reduced metallicity. It was argued that this effect can be "easily understood in terms of the metallicity'', since the "ratio of strong to weak lines must decrease accordingly'' if the metallicity is lowered. With respect to Eq. (33), this argument is simply wrong, since the plain number ratio becomes inevitably smaller for a reduced wind density ($k_{\rm 1}$ larger) and thus cannot be used to give any predictions concerning $\hat \alpha$. What really matters - if one prefers a discussion in terms of line numbers - is the ratio of optically thick lines to the average number of lines in the range $k_{\rm L}= 0\ldots k_{\rm 1}$,

$$\hat \alpha= \frac{\left\langle N_{\rm thick}\right\rangle}{\left\langle \bar N_{\{0,k_{\rm 1}\}}\right\rangle},$$ (35)

by means of Eq. (21). Further comments on the behaviour of $\hat \alpha$ in a low metallicity environment are given in Sect. 5.

2.6 Relation to Gayley's $\bar Q$

As we have mentioned in the introduction, Gayley ([1995]) considered the problem of radiative line driving in a concept somewhat different from the conventional approach. After discussing the physical origin why radiative line driving is so much more efficient than the radiative continuum acceleration, he introduces the meanwhile well known quantity $\bar Q$ and relates it to the alternative modified CAK approach. The reason that his approach seems to be somewhat favourable compared to the latter is his finding that $\bar Q$ should be much more constant than the line-force parameter $k_{\rm CAK}$, which is somewhat messed up with implicit dependencies on $\alpha$, cf. Eq. (16). By comparing with various published values of force-multipliers, he concludes that $\bar Q$ should be of the order 1000 to 2000. In order to compare to his approach and since in the remaining part of the paper we are mostly concerned with the parameter $\alpha$, we will briefly relate his findings to our concept in the following, and comment on some problems if one considers realistic line-distribution functions in Sect. 3.3.

At first note that Gayley's line-strength parameter q relates to our definition as

$$q = \frac{{v_{\rm th}}}{c} k_{\rm L},$$ (36)

where we have argued previously that in our notation ${v_{\rm th}}$ is evaluated for hydrogen, i.e., is temperature, however not mass dependent.

Similarly, Gayley's cutoff parameter $Q_{\rm o}$ deviates from our quantity $k_{\rm max}$ (Eq. B1) by the same factor. Noting these correspondences and keeping our radial streaming approximation, $\bar Q$ is defined by

 

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

$$\rightarrow \frac{{v_{\rm th}}}{c}\, \int_{0}^{{\infty}}\int_{0}^{{\infty}} k_{\rm L}\frac{L_{\nu} \nu}{L} \, \vert\,{\rm d}N(k_{\rm L},\nu)\vert$$ (37)

which is nothing else than the force-multiplier if all lines were optically thin (cf. Eq. 11). On the other hand, from Eq. (16) we find an intuitive interpretation of $k_{\rm CAK}$, which is roughly the fraction of the total stellar flux which would be blocked already in the photosphere if all lines were optically thick (assuming that each optically thick line blocks a fraction $L_\nu \Delta \nu_{\rm D}$), divided by $\alpha$. In so far, the physical upper limit of $k_{\rm CAK}$ is of order $1/\alpha$. (Much earlier, however, the lines would overlap in the wind, and this effect would have to be accounted for, e.g. Puls [1987].)

With definition (37) and using a line-distribution function with exponential cutoff at $Q_{\rm o}$ (cf. B1) in order to prevent the number of strong lines from becoming smaller than unity, Gayley showed the correspondence (his Eq. (56))

$$k_{\rm CAK}= \frac{1}{1-\alpha} \Bigl(\frac{{v_{\rm th}}}{c}... ...m o}^{-\alpha} =\frac {\bar Q}{1-\alpha} k_{\rm max}^{-\alpha}$$ (38)

in our notation. So far, the $\bar Q$ formalism seems to be of no major conceptual advantage compared to the CAK formalism. The interesting point, however, is the following: For power-law distributed line-strength distribution functions with (roughly) constant $\alpha$ and conditions valid for hot winds, one easily finds (cf. Sect. 3.3 and Appendix C) that

$$\bar Q\approx Q_{\rm o},$$ (39)

a relation which was invoked by Gayley as a generally valid Ansatz, i.e.,

$$k_{\rm CAK}= \frac{1}{1-\alpha} \Bigl(\frac{{v_{\rm th}}}{c}\Bigr)^\alpha \bar Q^{1-\alpha}.$$ (40)

If this Ansatz were correct and since $\bar Q$ is rather constant (cf. also Sect. 3.2), all scaling laws for mass-loss rates would become much easier to interpret and would especially depend only on the quantity $\alpha$ via

$$\frac{{\dot M}}{L} \propto \frac{\alpha}{1-\alpha} \,F_{\rm ... ...igl(\frac{\Gamma \bar Q}{1-\Gamma}\Bigr)^{{1 \over \alpha} -1}$$ (41)

(Gayley's Eq. (43), with stellar continuum flux $F_{\rm c}$ and usual Eddington $\Gamma$). In contrast, the standard formulation (in the same normalization) implies

$$\frac{{\dot M}}{L} \propto \frac{\alpha {v_{\rm th}}}{c}\, ... ...\frac{(1-\alpha) \Gamma}{1-\Gamma}\Bigr)^{{1 \over \alpha} -1}$$ (42)

with a much more varying value of $k_{\rm CAK}$. (Actually, both expressions have to be slightly modified for the so-called $\delta$-term accounting for ionization effects, if present, cf. Abbott [1982] and Kudritzki et al. [1989]). By comparing both equations, the different philosophy of $\bar Q$ and $k_{\rm CAK}$ is evident: the force-multiplier like quantity $\bar Q$ acts on the acceleration ratio $\Gamma$, whereas the flux-ratio like quantity $k_{\rm CAK}$ acts on the stellar flux!

If, on the other hand, $\bar Q$ differs significantly from $Q_{\rm o}$, Eq. (41) has to be modified by an additional factor $\bar Q/Q_{\rm o}$, and the gain of the $\bar Q$ formalism were lost, since then the variation in $k_{\rm CAK}$ would be found again in a variation of $Q_{\rm o}$. In how far this might be a problem will be discussed in the next section, after we have described how we calculate line-distribution functions and derive the appropriate force-multiplier parameters.