4 What determines the slope?

Having calculated the (flux weighted) line-strength distribution and appropriate force-multipliers as function of local variables, one could argue that the problem is solved, since, after performing the required regressions with respect to $\hat \alpha$ and $\hat \delta$ (see Kudritzki et al. [1998]), these values can be tabulated and the hydro-equations solved. In the following, we want to proceed a step further and try to understand the basic physics which determines the slope of the distribution in some detail. Note again that the actual value of this quantity is decisive for all astrophysical problems involving radiatively driven mass-loss (cf. Sect. 1), and that only a thorough understanding of the individual processes which determine this quantity enables us to draw correct conclusions and to give quantitatively correct predictions. Moreover and although it is empirically known that $\alpha$ is of order 0.4 ...0.6 in most situations (leading to the aforementioned similarity of $\alpha$ and $\hat \alpha$), there is no a priori reason that the steepness of the line-strength distribution function lies in this range, and, especially, that $\alpha > 0$ over a large range of line-strengths.

From the definition of the flux-weighted line-strength distribution, there are three essential quantities to be considered in order to predict its behaviour, namely the oscillator strength distribution of contributing lines, the level population of the absorbing/re-emitting levels and the irradiating flux. In a first and more tutorial step, we consider the most simple case being possible, namely the case of pure hydrogen (or, more generally, hydrogenic ions), which states the complete problem (including the influence of NLTE-effects) in an analytically understandable way and leads to a number of interesting results.

4.1 gf- and line-strength distribution for hydrogenic ions

4.1.1 gf-distribution

The gf-value of a given transition between principal quantum number n'and n (summed over all contributing angular momenta, i.e. accounting for selection rules) is given by the well known Kramers formula (neglecting Gaunt-factors of order unity)

$$gf(n',n) = \frac{64}{3 \pi \sqrt 3} \Bigl(\frac{1}{n'^2} - \frac{1}{n^2} \Bigr)^{-3} \left(\frac{1}{n^3 n'^3}\right)\cdot$$ (44)

At first, consider only resonance transitions n' = 1, in which case $gf(1,n) \approx C/n^3$ with C the numerical constant in (44). The number of possible transitions up to a certain principal quantum number n is

$$N(n) = n - 1 \approx \left(\frac{C}{gf(1,n)}\right)^{{1 \over 3}} - 1,$$ (45)

so that the (cumulative) number of transitions with gf-values stronger/equal than a certain value gf reads

$$N(gf) = \Bigl({gf \over C}\Bigr)^{-{1 \over 3}} - 1.$$ (46)

Thus, the resulting distribution function

$$\frac{{\rm d}N}{{\rm d}gf} \propto - gf^{-\frac{4}{3}}$$ (47)

is given by a power-law, where the exponent corresponds to an $\alpha$-value of 2/3, i.e., is just the canonical value which would lead to a mass-independent WLR and is consistent with the observations of O-Supergiants (cf. Puls et al. [1996]).

Thus, from the above arguments one might conclude that the major problem is solved, and that the calculated/observed $\alpha/\hat \alpha$ values are dominated by the oscillator-strength statistics. Note already here, however, that the majority of driving lines (non-hydrogenic!) cannot be described similarily, since the dominant ingredient in the above derivation - the rather simple and specific dependency of atomic quantities (oscillator strength and energy levels) on principal quantum number - is no longer valid in more complex ions.

Two additional points are worth mentioning. First, by using the "exact'' Kramers law accounting for the (1 - n^-2) term, the apparent slope at large gf-values becomes steeper than -4/3, leading to $\alpha < 2/3$, which again is consistent with the behaviour of "realistic'' line-strength distribution function.

Second, by accounting also for transitions between excited levels, one finds (see below) the same statistics, i.e., the exponent -4/3 (with lower values for large gf) is universal for hydrogen-like ions.

Figure 5: gf-distribution for hydrogen, calculated by Kramers-formula (Eq. 44) for $n' \le 10$ and $n \le 200$. Fully drawn: Lines from all series with $n' \le 10$; dotted: Reference line with constant slope of -1/3 (with respect to log-log). Crosses: Different series, starting with Lyman series (leftmost). Size of crosses decreases with increasing n'. Triangles: analytic approximation for small gf-values, Eq. (49)

Figure 5 displays the cumulative gf-distribution function for principal quantum numbers $n' \le 10$ (with sufficiently large $n \le 200$), both for the individual series (n' fixed) denoted by crosses as well as for all combinations of n',n (fully drawn). Following a steep increase from the maximum value, the distribution displays a rather constant slope (in the log) of -1/3 (cf. Eq. 46) over four dex, before it reaches its final, constant value resulting from running out of lines (depending on the maximum value chosen for n). This behaviour compares well to our case "A'' discussed in Appendix A. Concerning the individual series, the higher ones follow exactly the predicted slope from the approximation given below (49), whereas the high gf-tails of the lower series have a somewhat steeper slope, as discussed above. Note also, that the gf-values from the higher series are generally larger than those from the lower ones.

A simple expansion of Eq. (44) clarifies the behaviour of the gfdistribution for not too large gf-values: To first order, the upper level n for given gf-value and lower level n' results to

$$n(gf, n')/n' = \left({gf \over C}\right)^{-{1 \over 3}} + O\left({gf \over C}\right)^{{1 \over 3}}.$$ (48)

Thus, the total number of lines with gf-values larger than a certain one is given by

 

$$\sum_{n'=1}^{n'_{\rm max}} n(gf,n') - n' = \sum_{n'=1}^{n'_{\rm max}} n' \left(\left({gf \over C}\right)^{-{1 \over 3}} -1 \right) =$$ N(gf) =

$$\frac{n'_{\rm max}(n'_{\rm max}+1)}{2}\, \left[\left({gf \over C}\right)^{-{1 \over 3}} -1 \right], \,gf \ll C,$$ = (49)

where C is the numerical constant in Eq. (44) and $n'_{\rm max}$ the maximum lower level of the considered transitions, which controls the vertical offset of the distribution function. Note, that the last equation (all transitions) compares exactly to Eq. (46) (only resonance lines), except from the generalization to $n'_{\rm max}> 1$.

Figure 6: Cumulative line-strength distribution function for hydrogen (see Eq. 50). "Atomic model'' as in Fig. 5, gf-values from Kramers-formula and excitation in LTE. Triangles: analytic approximation for small $k_{\rm L}$; dotted: $T_{\rm e}= 40\,000$ K; short dashes: $T_{\rm e}= 30\,000$ K; dashed-dotted: $T_{\rm e}= 20\,000$ K. Long dashes and asterisks: $T_{\rm e}= 30\,000$ K, only Lyman series. Fully drawn: flux-weighted line distribution ( $T = 30\,000$ K), all lines, flux approximated by Planck-function

4.1.2 Line-strength distribution in LTE

Although it is quite interesting that the gf-distribution follows an exponent of -4/3, our final interest is, of course, the behaviour of the line-strength distribution, i.e., the influence of excitation has to be accounted for. Figure 6 visualizes the corresponding function in the LTE approximation for three different temperatures, $T_{\rm e}= 20\,000, 30\,000$ and $40\,000$ K, as well as the following analytical result, which is derived in Appendix D:

 

$$N(k_{\rm L},T_{\rm e}) k_{\rm L}^{-{1 \over 3}} \bigl(C'{\rm e}^{-\frac{h\nu_1}{kT_{\rm e}}}\bigr)^{1 \over 3}f(T_{\rm e}) -$$ =

$$\frac{n'_{\rm max}(n'_{\rm max}+1)}{2}, \,\,k_{\rm L}< C'{\rm e}^{-\frac{h\nu_1}{kT_{\rm e}}}.$$ - (50)

(For the definition of $C' \propto n_1/\rho$ with n₁ the ground-state occupation number and $f(T_{\rm e})$, see Eq. D5.) Obviously, also the line-strength distribution follows the "canonical'' power-law ${\rm d}N/{\rm d}k_{\rm L}\propto -k_{\rm L}^{-4/3}$, which is, of course, the final consequence of the primary dependence of $gf \propto n^{-3}$ for each series. Excitation plays only a minor role, controlling the function $f(T_{\rm e})$ and thus the vertical offset (or, in other words, the normalization constant): the higher the temperature, the more lines are present if the ionization structure would remain constant, i.e., if $n_1/\rho$ would not change. Again, we have compared the analytical prediction with the numerical simulation (Fig. 6), which gives a perfect agreement in the valid range and even beyond! Note also, that the asymptotic behaviour $N(k_{\rm L}) \propto k_{\rm L}^{-1/3}$ is consistent with our requirement that $k_{\rm L} \left\langle N(k_{\rm L})\right\rangle\rightarrow0$ for $k_{\rm L}\rightarrow0$, which has to be fulfilled in order to validate Eq. (21).

4.1.3 Flux weighting

The next question which arises in our simple example is that concerning the influence of flux-weighting. One might note, also in a more general context, that this "final operation'' in our course of deriving radiative accelerations (enough irradiating photons have to be present!) leads to the actual possibility to perform quantitative simulations: Since we have seen, that the strongest $k_{\rm L}$ values arise from IR-transitions between excited levels (cf. the high line-strength end of Fig. 6), without a final flux-weighting we would have to account for atomic models which were by far too large to be treated numerically!

Fortunately, the flux irradiating these strong IR-transitions is small, and the primary contribution to the line-force is only due to transitions near the flux maximum, a fact which is exploited a priori in our simplified NLTE-calculations (cf. Sect. 3.1.2 and Abbott & Lucy [1985]). With respect to the hydrogen atom under consideration, we expect therefore the Lyman series (including the Balmer series for A-stars) to be the major contributor for radiative momentum. To this end, the long-dashed curve in Fig. 6 displays the line-strength distribution for the Lyman-Series only ( $T_{\rm e}= 30\,000$ K), together with the analytical approximation (asterisks), whereas the fully drawn one corresponds to the final, flux (times frequency) weighted distribution accounting for all considered lines: Obviously, our expectation is met precisely. Note, however, that Fig. 6 serves only as a tutorial and overestimates the real situation by far, since due to line convergence near the ionization edge only a few lines ( ) can be treated as individual ones, thus cutting the distribution function at this value.

Influence of NLTE-effects: The parameter $\delta$

So far, we were concerned only with the slope of the line-strength distribution and its relation to the line-force parameter $\hat \alpha$. Almost nothing has been said on the normalization constant $N_{\rm o}$(Eq. 10), which controls the absolute value of the acceleration due to its relation to the total line number. We have pointed out that $N_{\rm o}$ may have an additional depth dependence, which transforms, via Eq. (16), into a depth dependent line-force parameter $k_{\rm CAK}$. This additional depth dependence originates, of course, from the behaviour of the mass absorption coefficient $\bar{\chi}_{\rm i}/ \rho$ present in the definition of the line-strength $k_{\rm L}$. Although this ratio remains more or less constant if we consider the primary source of radiative driving, namely resonance lines and lines with a lower meta-stable level from main ionization stages (denoted in the following by "j''), there will be always a "contamination'' by lines from excited levels and, most important, by lines from minor ionization stages.

Since H I is such a trace ion in hot star winds, we will investigate the effects of this depth dependence in our tutorial chapter. It is well known (e.g. Mihalas [1978], p. 125) that the NLTE ground state departure coefficient of a trace ion one stage below the major one in a spherical atmosphere is primarily a function of the inverse of the dilution factor

$$W(r) = {1 \over 2}\bigl[1-\bigl(1-(R_{\ast}/r)^2\bigr)^{1 \over 2}\bigr].$$

One finds that

$$n_{1,j-1} \sim \frac{n_{\rm e}}{W} n_j \Phi(T_{\rm e}),$$ (51)

if n_j is the population of the dominant ion (here: H II), $\Phi$ the Saha-Boltzmann factor and the radiation temperature at the ionization edge is not too different from the electron temperature. Equation (51) simply states that in a spherical atmosphere the ground-state of ion (j-1) becomes overpopulated $\propto 1/W$, since the radiation field controlling the ionization is diluted with respect to the LTE-situation, whereas the recombination operates at its usual value. Note, that the above equation, with a couple of refinements, allows also for the approximate solution of the equations of statistical equilibrium in a spherical wind, as was described in Sect. 3.1.2.

Since n_j varies in concert with the local density, the ratio $n_1/\rho$becomes a function of $n_{\rm e}/W$, which finally leads - besides an additional temperature dependence - to a variation of $N_{\rm o}$ as (cf. Eq. 50)

$$N_{\rm o}\propto \left(\frac{n_{\rm e}}{W}\right)^{1 \over 3}.$$ (52)

In a mixture of major and minor ionization stages, as present in a stellar wind, we have, of course, a different dependence. However, the notion that also in this case the ionization structure is primarily controlled by the factor $n_{\rm e}/W$ (with exponents different from unity if one includes trace ions of stages j+1, j-2 etc.) lead Abbott ([1982]) to the introduction of the famous $\delta$-term to the force-multiplier, which in terms of our line-strength distribution function reads

$${\rm d}N(\nu, k_{\rm L}, r) = \,-\,N_{\rm o}\left(\frac{n_{\... ...)^\delta \,f(\nu, k_{\rm L}) \, {\rm d}\nu \,{\rm d}k_{\rm L},$$ (53)

with $n_{\rm e11}$ the electron density in units of $10^{11}~{\rm cm}^{-3}$ and $N_{\rm o}$ now independent of depth. Equation (52) shows that for hydrogen $\delta$ should be of order 1/3. More generally and using the fact that the appropriately scaled variable for all considerations is $\tilde k \propto k_{\rm L}/C'$ (Eq. D2), a plasma dominated by trace-ions of stage j-1 must have a $\delta$-value which is just the negative of the line-strength exponent in the $N(k_{\rm L})$-distribution, i.e, $\delta = - (\alpha - 1)$ or

$$\alpha +\delta = 1, \,\,\,\mbox{trace ions of stage $(j-1)$\space only!}$$ (54)

Figure 7: Iso-contours of force-multiplier parameters $\hat \alpha$ and $\hat \delta$ as function of ${\rm log}(n_{\rm e}/W)$ and $-{\rm log}\, t = {\rm log}\, k_{\rm 1}$, for hydrogen with ten levels. $T_{\rm e}= 40\,000$ K, dilution factor W evaluated at $r/R_{\ast }= 5.,4.,3.,2.5,2.,1.75,1.50,1.25$ and 1, respectively. Fully drawn: $\hat \alpha$, dashed: $\hat \delta$. Thick curves enclose the region of $0.55 < \hat \alpha< 0.75$. A trajectory throughout a typical wind would stretch from the lower right corner to the upper left one

Figure 7 verifies this equation for a number of different conditions. We have calculated the force-multiplier arising from hydrogen in a hot stellar wind plasma under different conditions (see caption), and obtained the effective values $\hat \alpha, \hat \delta$ by first order derivatives. Although $\hat \alpha$ varies from 0 ...1, the equality (54) is almost perfectly reproduced. On a first glance, it is somewhat puzzling that $\hat \alpha$ is so much varying although the according line-strength distribution has an almost constant slope. In the situation described here (trace ion of stage (j-1)), however, we have a strong dependence on local conditions, $k_{\rm L} \propto \rho/W$, contrasted to the usual case of $k_{\rm L}\approx$const throughout the wind for the (resonance) lines of major ion species. Thus, there is only a small strip in the $k_{\rm 1}, (n_{\rm e}/W)$ plane where we do not encounter the case of all lines being either optically thick ( $\hat \alpha\rightarrow1$) or optically thin ( $\hat \alpha\rightarrow0$). Even in those cases, however, Eq. (54) is still valid!

Finally, $\hat \alpha= 1$ and $\hat \delta= 0$ holds generally (i.e., for any type of ionization/excitation) in the optically thick case, since then the dependence of line-force on occupation numbers vanishes completely and only the relation $g_{\rm rad}^{\rm tot}\propto k_{\rm 1}\left\langle N_{\rm thick}\right\rangle$ survives.

We want to finish this section with two summarizing comments: First, we have shown that for hydrogen (generally: hydrogenic ions) the gf- distribution $gf(n',n) \propto n^{-3}$ (for each series) leads quite naturally to a line-strength distribution function with a slope corresponding to $\alpha \approx 2/3$. Second, the dependence of the line-strength on the specific type of ionization (major one or trace ion) gives rise to a depth dependent normalization constant $N_{\rm o}$, finally leading to the $\delta$ term in the force-multiplier and line-strength statistics. Since $\delta$ is of order 1/3 for trace ions of stage j-1 (which are usually the dominating species of minor ions, cf. Sect. 4.2.5) and $\delta = 0$ for major ions, the effective value of $\delta$ resulting from an appropriate mixture should be a small, positive number.

4.2 Arbitrary ions

Since the predominant radiative acceleration in hot star winds is certainly due to non-hydrogenic ions (e.g., from iron-group elements), we turn now to the line-strength statistics of these elements.

To our knowledge, there are only two previous investigations related to this topic. On the one hand, Learner ([1982]) found a line-strength distribution with a typical power-law index corresponding to $\alpha = 0.5$by means of experimental data (mostly ionization stages I and II), a result recently used by Wehrse et al. ([1998]) in their stochastic approach of line transfer in moving atmospheres. The first step, however, was taken by Allen ([1966], [1974]) who performed a thorough analysis of line distributions of neutral elements, aimed at the goal of establishing a "statistical spectroscopy''. His approach provides a significant understanding of the resulting statistical description especially under LTE conditions. The basic philosophy, however, has turned out to be extremely useful also regarding the problem under our consideration, at least if some additional considerations are taken into account. Before we comment on these, we will firstly introduce the basic concept and convince ourselves that it is applicable for our purposes. Note that the following results, which allow to understand the line-strength statistics of individual ions in an almost completely analytical way (as a function of level density, oscillator-strength distribution, ionization potential, temperature etc.), will turn out to be useful also for future investigations related to line-blanketing/blocking calculations. Finally, by a comparison between actual data and analytic simulations based on our results, the degree of completeness of present atomic data bases can be easily checked.

4.2.1 Allen's approach

Following Allen ([1966], [1974]) and neglecting subtleties, the first important assumption concerns the number of energy levels in the energy range $x,x+{\rm d}x$, which will be described by the distribution $p(x)\,{\rm d}x$. Although there are certain irregularities, the basic trend of p(x) is to increase with excitation energy, and Allen adopted - per ion - a power-law

$$p(x,\,{\rm per}\,\, {\rm d}x) = a \, 10^{x/\sigma}$$ (55)

in the range $0 < x < i_{\rm e}$ with (effective) ionization energy $i_{\rm e}$. Note that this parameterization can be validated for rather complex ions (see the tests performed below), is however much less appropriate for hydrogenic ions due to their specific dependence of energy on level number (cf. the examples given by Allen [1966], Fig. 3). Thus, the approach presented here and the one given in the previous section are almost mutual exclusive.

With the above distribution, the number of lines between energies x₁ and x₂ and per excitation ranges ${\rm d}x_{1}$ and ${\rm d}x_{2}$ is

$$\Delta N(x_{1}, x_{2},\,{\rm per}\,\, {\rm d}x_{1},{\rm d}x_{2}) = j\,p(x_{1})\,p(x_{2}),$$ (56)

where $j \le 1$ shall be the "selection'' factor accounting for selection rules (typically of order 0.2...0.3 for iron group elements). Since the transition frequency is given by $\nu = x_{2}- x_{1}$, when we calculate in appropriate units (say, kiloKayser = "kK''), the number of lines with lower levels between $x_{1}, x_{1}+ {\rm d}x_{1}$ and transition frequencies $\nu, \nu + {\rm d}\nu$ is

$$\Delta N(x_{1}, \nu,\,{\rm per}\,\, {\rm d}x_{1},{\rm d}\nu) = j a^2\,10^{\frac{ 2x_{1}+ \nu}{\sigma}}.$$ (57)

In his further derivation, Allen assumed the logarithm of the gf-values ( ${\rm log}\, gf=:r$) to be equally distributed within the range $r_{\rm max}> r >r_{\rm min}$, with $r_{\rm max}-r_{\rm min}$ of order 3 dex. This assumption, being equivalent to a gf-distribution $\vert{\rm d}N/{\rm d}gf\vert \propto gf^{-1}$, will be relaxed in the following. Instead, we consider a distribution with arbitrary, however constant exponent $\gamma$, again independent of frequency (and excitation energy), so that the number of lines with ${\rm log}\, gf\,$-values within $r, r+{\rm d}r$ is given by

$$\Delta N(x_{1}, \nu, r,{\rm per}\, {\rm d}x_{1},{\rm d}\nu,{\... ...a^2}{w}\, 10^{\frac{ 2x_{1}+ \nu}{ \sigma}{+ r(1 - \gamma)}},$$ (58)

where w is the normalization constant with respect to the adopted gf-statistics,

$$\frac{10^{\displaystyler_{\rm max}(1-\gamma)} - 10^{\displaystyler_{\rm min}(1-\gamma)}} {(1 -\gamma)\ln10},\;\;\gamma \ne 1$$ w =

$$r_{\rm max}-r_{\rm min},\;\; \gamma = 1.$$ w = (59)

Now we define the line intensity by

$$l = -{\rm log}\left[gf\,\exp\left(-\frac{1\,000 x_{1}h c}{kT}\right)\right] = \frac{x_{1}}{t} - r$$ (60)

if we measure energies and frequencies in kK and the temperature in units of 625 K, t = T/625 K (not to be confused with the optical depth parameter defined by CAK).

Note that by introducing the line intensity in this way, one implicitly assumes that the plasma is in LTE and that all levels connected by lines play an equally important role, if one uses this quantity as the primary statistical variable. Note also that under LTE conditions l is closely related to the negative logarithmic line-strength if the wavelength dependence of $k_{\rm L}$ is ignored.

If we convert the x₁-dependence of $\Delta N$ into an l-dependence and integrate over ${\rm d}r$, we find the number of lines for given line intensity and transition frequency as

 

$$\Delta N (\,l, \nu,\,{\rm per}\,\, {\rm d}\,l,{\rm d}\nu) =$$

$$\frac{j a^2 t}{w} 10^{\frac{ 2lt+\nu}{ \sigma}} \int^{\tilde r_{\... ...n}} 10^{{\left(\frac{2t}{\sigma}+ 1-\gamma\right)}{\displaystyle r}} {\rm d}r =$$ =

$$\frac{j a^2 t}{A w \ln10} 10^{\frac{ 2lt +\nu}{ \sigma}} \bigl(10^{\displaystyle A \tilde r_{\rm max}}-10^{\displaystyle Ar_{\rm min}}\bigr)$$ = (61)

$${\left(\frac{2t}{\sigma}+ 1-\gamma\right)}.$$ A = (62)

(Here and in the following, we always assume $A \ne 0$. The case A = 0 can be treated by a somewhat different expansion). In case of $\gamma = 1$, the second term in the difference can be neglected with respect to the first one, and we recover the result given by Allen,

$$\Delta N(\,l,\nu,\,{\rm per}\,\, {\rm d}\,l,{\rm d}\nu) \stac... ...displaystyle(l + \tilde r_{\rm max}) + }\frac{ \nu}{ \sigma}.}$$ (63)

The difference between $\tilde r_{\rm max}$ and $r_{\rm max}$ is the following. Whereas $r_{\rm max}$ relates to the maximum ${\rm log}\, gf$-value for the considered ion, $\tilde r_{\rm max}$ is the maximum value which is possible for given $\,l$ and $\nu$ and underlies the following restrictions:

 

$$r_{\rm max}= -l_{\rm min}$$ r <

$$\nu < i_{\rm e},\;\;{\rm i.e.,}$$ x₁ +

$$\frac{i_{\rm e} - \nu}{t} - l$$ r <

$$\rightarrow\tilde r_{\rm max} {\rm Min}\left(\frac{i_{\rm e} - \nu}{t} - l,\, -l_{\rm min}\right),$$ = (64)

where the transition occurs at

$$l_{\rm T}= \frac{i_{\rm e} - \nu}{t} + l_{\rm min}.$$ (65)

Let us first consider the implications of this restriction in the case of $\gamma = 1$. In the following, the line-number $\Delta N$ is always to be understood as function of $(\,l\,,\nu)$ per interval ${\rm d}\,l$ and ${\rm d}\nu$.

  

$${\rm log}\Delta N \stackrel{l < l_{\rm T},\gamma=1}{=} {\rm log}\frac{j a^2 \sigma}{2 w \ln10} +\frac{2t}{\sigma}(l-l_{\rm min}) +\frac{\nu}{\sigma}$$ (66)

$${\rm log}\Delta N \stackrel{l \ge l_{\rm T},\gamma=1}{=} {\rm log}\frac{j a^2 \sigma}{2 w \ln10} +\frac{2i_{\rm e}}{\sigma}\,-\,\frac{\nu}{\sigma}\cdot$$ (67)

The interpretation of these expressions is straightforward. If the r-values are equally distributed between $r_{\rm max}$ and $r_{\rm min}$, the log of $\Delta N$ should increase linearly between $l_{\rm min}\ldots~l_{\rm T}$, where the slope is controlled both by the slope of the level density and the temperature. For all line intensities larger than $l_{\rm T}$, the number of lines should become constant, until $\tilde r_{\rm max}\approx r_{\rm min}$ and the number of lines approaches zero for $l > (i_{\rm e} - \nu)/t - r_{\rm min}$.

In order to check this and the following predictions, we have calculated the line intensity statistics for Fe II (comprising roughly 200000 lines) from our present data base (Sect. 3.1). Instead of using the actual gf-values, however, we firstly simulated different distributions by a Monte-Carlo process, with random variable $x \in (0,1]$,

  

$$\frac{\int_{gf}^{gf_{\rm max}} gf^{-\gamma} {\rm d}gf} {\int_{gf_{\rm min}}^{gf_{\rm max}} gf^{-\gamma} {\rm d}gf}$$ x =

$$\stackrel{\gamma \ne 1}{=} gf_{\rm max}\Bigl[1-x\Bigl(1-\left(\frac{ gf_{\rm min}} {gf_{\rm max}}\right)^{1-\gamma}\Bigr)\Bigr]^ \frac{ 1}{1-\gamma},$$ gf(x) (68)

$$\stackrel{\gamma = 1}{=} 10^{\displaystyler_{\rm max}+x(r_{\rm min}-r_{\rm max})}.$$ gf(x) (69)

Figure 8: $\Delta N$ statistics for Fe II as function of line intensity, for three different temperatures (cf. text). Lines are counted per 5 kK in frequency and per 0.5 dex in intensity. The vertical line displays $l_{\rm T}$(Eq. 65). gf-distribution calculated by Monte-Carlo simulation adopting $\gamma = 1$, cf. Eq. (69)

Figure 9: As Fig. 8, for T= 10000 K and $\gamma = 0.66$(left), $\gamma = 1.15$ (middle) and $\gamma = 2.0$ (right). For the right panel, ${\rm log}\, gf_{\rm min} = -3$ was used, and the results of the corresponding ($\gamma$ = 1)-distribution are overplotted as rectangles

and replaced the actual gf-value by the value drawn from the above distribution, with $gf_{\rm max} = 1$ and $gf_{\rm min} = 10^{-7}$. Thus, by this simulation we primarily investigate in how far the assumptions leading to Eq. (57) are justified and inspect the validity of the overall approach, under the restriction of gf-distributions with constant slope.

In the following plots, we display the result for the frequency with the highest line-density, namely $\nu = 65.5 \pm 2.5$ kK corresponding to $\lambda = 1538\ldots1667~$Å, and the vertical line gives the transition line intensity $l_{\rm T}$, with $i_{\rm e} \approx 130$ kK.

The first series (Fig. 8) has been calculated for the case discussed above, namely $\gamma = 1$, and three different temperatures $T = 5\,000$, 10000, 15000 K. As predicted, the constant, temperature dependent slope and the kink at $l_{\rm T}$ are present, as well as the constant line number for $l > l_{\rm T}$, until the maximum possible l (as function of t, see above) is reached and $\Delta N \rightarrow0$.

Next, we consider the case of $\gamma \ne 1$. Here, we have to perform the following distinction. Let us first assume that ${\left(\frac{2t}{\sigma}+ 1-\gamma\right)}$ is not significantly smaller than $2t/\sigma$, i.e., in the case of Fe II. Then, we can neglect again the 2nd term in the difference in Eq. (61) with respect to the first one, and obtain, accounting for the restrictions concerning $\tilde r_{\rm max}$,

$${\rm log}\Delta N \stackrel{l < l_{\rm T}}{=} {\rm log}\frac{j a^2 \sigma}{\displaystyle\bigl(2 + \frac{\sigma}{t}(1 -\gamma)\bigr) w \ln10} +\frac{2t}{\sigma}(l-l_{\rm min})$$

+ (70)

$${\rm log}\Delta N \stackrel{l \ge l_{\rm T}}{=} {\rm log}\frac{j a^2 \sigma}{\displaystyle\bigl(2 + \frac{\sigma}{t}(1 -\gamma)\bigr) w \ln10} -(1-\gamma)\,l$$

+ (71)

These equations are similar to the case $\gamma = 1$ (Eqs. 66, 67), except from the offset and one decisive difference: For line intensities larger than $l_{\rm T}$, the line number is no longer constant, but becomes directly coupled to the oscillator strength statistics via the term $-(1-\gamma)\,l$. Thus, a declining line number is expected for $\gamma <1$, whereas for $\gamma$ (slightly) larger than unity the distribution function should increase. Note, that the predicted slope for $l > l_{\rm T}$ is independent of temperature!

Figure 9 impressively verifies our predictions. Here, we have simulated an oscillator strength distribution with $\gamma =.66$ (left panel), whereas in the middle one $\gamma = 1.15$ was assumed. Note the abrupt change in the slope at $l \approx l_{\rm T}$.

Finally, we consider the case of a rather steep gf-distribution $\gamma > 1 + 2t/\sigma$, e.g. for Fe II. Then, the gf-distribution is dominated by its lower end, and the values taken at $r_{\rm min}$ are dominating both the normalization constant and the difference term, in contrast to the situation discussed above. Accordingly, $\tilde r_{\rm max}$ and $l_{\rm T}$ do no longer play any role, and we obtain a line number statistics

$${\rm log}\Delta N\stackrel{\gamma\,{\rm large}}{=} {\rm log}... ... \,+\,\frac{2t}{\sigma}(l+r_{\rm min})\,+ \,\frac{\nu}{\sigma}$$ (72)

with uniform, temperature dependent slope $2t/\sigma$. Note, that the only $\gamma$ dependence shows up in the offset. The right panel in Fig. 9 gives the corresponding result, for $\gamma = 2$ and $r_{\rm min}= - 3$, which was used in order to obtain a statistically significant number of lines at the high gf end. Obviously, no kink is present any longer, and the distribution lies parallel to the low-$l\,$ part of a corresponding ($\gamma$ = 1)-distribution, overplotted as rectangles.

So far, our investigations can be summarized as follows. Allan's approach (extended for $\gamma \ne 1$) has been validated for a complex ion in those cases when $\gamma$ is constant, and the slope of the distribution function can be predicted: At low intensities (corresponding to high line-strengths), it is controlled both by the level density as a function of excitation energy (slope $\sigma$) as well as by the population of these levels (excitation temperature t), resulting from our LTE assumption. In this domain, the oscillator strength distribution seems to be of no importance for the slope, since - for - the line number is primarily controlled by the highest gf-value, and for $\gamma > 1 + 2t/\sigma$ by the lowest one, both of which do "only'' control the offset of the distribution. For large line intensities (weak lines!), we encounter a certain inter-relation between energetic neighbourhood to the ionization limit and maximum possible gf-value (cf. Eq. 64), which causes a slope dominated by the gf-distribution, provided $\gamma$ is not too large. In the opposite case, finally, the small gf-values become decisive over the complete range, and the slope of the distribution retains its previous slope for all $\,l\,$values.

Thus, it seems that we have to know only the principal behaviour of the gf-distribution to predict the LTE line intensity/line-strength statistics for a certain ion. Unfortunately, Fig. 10 shows that this is at least not so simple. Here, we have plotted ${\rm log} \Delta N$ using the actual oscillator strengths. As is obvious, this distribution looks rather different from the cases discussed so far, primarily in the domain $\,l > l_{\rm T}$.

Figure 10: As Fig. 9, for T= 10000 K and actual gf-distribution

Figure 11: Cumulative line number as function of logarithmic oscillator strength r. Asterisks: Fe II, in the interval $\lambda$ = 1538 ...1667 Å(as in Fig. 8); triangles: Fe II, all lines of our data base; crosses: all lines of elements H to Zn present in our data bases. For comparison, a reference line with constant slope (in the log) of -1/3 (corresponding to $\gamma = 4/3$) is indicated

The reason for this different functional behaviour becomes evident from Fig. 11, displaying the actual run of the oscillator strength distribution in the considered frequency interval as well as for "all'' lines of Fe II and all lines (for the atoms H to Zn, cf. Sect. 3.1) present in our data base.

At first note that the specific shapes of the particular distributions are extremely similar, where the major differences concern the total line number $N(gf_{\rm min})$ and the highest gf-value being present. In contrast to the case of hydrogenic ions, however, the distribution does no longer show a more or less constant slope, but is curved. At the high gf-end, the distribution is rather steep, with an approximate slope over the first 2.5 dex, and may be described afterwards again by the canonical value $\gamma \approx 4/3$, before our line list becomes incomplete (for ${\rm log}\, gf < -5$). To our understanding, the steep increase results from the very strong lines connecting the (effective) ground states with low lying levels. These lines follow statistics different from the other ones, a feature which we have also found in case of hydrogenic ions (cf. also the related discussion in Allen [1966]).

4.2.2 Frequency integrated distribution functions

Although the gf-distribution is significantly curved leading to certain subtleties in the intensity distribution if considered in a specified frequency range (as it was done, e.g., in Fig. 10), it might be suspected that on a larger average the description should become more uniform again. Note, e.g., that the largest part of the distribution function ( can be described by a more or less constant slope, with a significantly steeper one only over the first two decades, which consequently comprise only few lines.

In order to check this hypothesis and to proceed towards our aim at describing flux-weighted line-strength distribution functions under stellar wind conditions, we will follow our analytical description assuming constant slope $\gamma$. From now on, however, we will concentrate on the distribution of "all'' lines per ion, i.e., we will consider frequency integrated distribution functions. Most important to this regard is the frequency dependence of the threshold value $l_{\rm T}$ (Eq. 65), dividing the two different domains of influence, namely either by excitation and/or by gf-distribution. Since $l_{\rm T}$ decreases with frequency, lines with higher frequencies (transitions from the lowest levels, dominating the line-force) should be much more coupled to the oscillator strength distribution than lines with lower frequencies.

After considering the limitations resulting from the frequency and line intensity dependence of $r_{\rm max}$ (cf. Eq. 64), the distribution function Eq. (61) can be integrated over frequency, and the result is given by

 

$$\Delta N(l) \propto {\rm sign}(A)\,10^{\frac{{2lt}}{{\sigma}}}\times$$

$$\times F(l, t, \sigma, i_{\rm e}, x_{\rm max}, \nu_{\rm max}, \gamma,l_{\rm min},r_{\rm min})$$ (73)

$$\left(\frac{2t}{\sigma}+ 1-\gamma\right)$$ A =

$$l_{\rm min} l < l_{\rm max}= \frac{x_{\rm max}}{t} - r_{\rm min}.$$ <

The functional behaviour of F is given in Appendix E, $x_{\rm max}$ is the highest energetic level considered (acting as lower one for the lines accounted for) and $\nu _{\rm max}$ the maximum integration frequency chosen. In those cases where the level and line list are complete (or the complete ones are used), we have $x_{\rm max}= \nu _{\rm max}= i_{\rm e}$. Note, that the distribution actually depends on certain ratios, e.g., $t/\sigma, i_{\rm e}/\sigma$, $x_{\rm max}/\sigma$ and $\nu_{\rm max}/\sigma$ and not on the absolute numbers.

Summarizing the results concerning F derived in Appendix E, we find that the frequency integrated distribution behaves rather similarly to the frequency dependent one. In total, three different slopes are possible, namely $2t/\sigma, t/\sigma$ and $\gamma-1$, were the occurrence and position of the former two are controlled by the value of $\gamma$ being larger or smaller than a critical value $\gamma_{\rm crit}= 1 + 2t/\sigma$.

Under typical conditions, however, the function consists of only two parts, namely a steeper, excitation-dominated one with slope $2t/\sigma$, and a second one with slope $\gamma-1$, similar to the frequency dependent distribution. The division is given at line intensity $l=x_{\rm max}/t -{\rm log}\,gf_{\rm max}$, when we consider only those lines with a lower energy level below the cutoff energy $x_{\rm max}$ introduced above. Furthermore, in case of a frequency integration between 0 and $\nu_{\rm max}< i_{\rm e}$, the function preserves its shape for all $\nu _{\rm max}< i_{\rm e} - x_{\rm max}$, an effect which we have called the "saturation effect''.

In principle, the constant of proportionality in Eq. (73) depends also on temperature and level distribution for the specified ion as well as on the exponent of the oscillator-strength distribution. However, in the following we derive this constant from the requirement that the total line number is known (and to be found from the cumulative line number at $l_{\rm max}$, which is the maximum possible line intensity).

In order to check the validity and applicability of the above expressions, we have performed a number of test calculations in the same spirit in the previous section, i.e., for our atomic model of Fe II and simulating the gf-distribution by a Monte Carlo process.

Figure 12: Frequency integrated line intensity distribution functions of Fe II at T = 15000 K. Frequency range 0 < x < 65 kK (see text) and resolution 0.5 dex in intensity. Oscillator strengths simulated by Monte Carlo with $\gamma = 1.0, 0.66, 1.15$(asterisks, diamonds and triangles). Curves display the analytical result according to Eq. (73) with $i_{\rm e} = 130$ kK, $i_{\rm e}/\sigma = 1.7$, $x_{\rm max}=63$ kK, $l_{\rm min}= 0$ and $r_{\rm min}= -7$, where the latter two quantities have been also used in the Monte Carlo simulations

Figure 13: As Fig. 12, using actual oscillator strengths (asterisks). Overplotted is the analytical result according to Eq. (73) with $i_{\rm e}$, $i_{\rm e}/\sigma$ and $x_{\rm max}$ as before; $l_{\rm min}= -0.75$ and $r_{\rm min}= -4.25$ (see text). Fully drawn: $\gamma = 1.33$; dotted: $\gamma = 1.03$; dashed: $\gamma = 1.63$

For our specific atomic model, we have $i_{\rm e} \approx 130$ kK (lines with larger energies resulting from levels ionizing to excited levels of Fe III were removed "by hand'' in order to simplify the test), and, from our constraint of considering only those levels which under NLTE-conditions have large enough occupation numbers to form lines of significant strength (see Sect. 3.1 and next section), we have $x_{\rm max}=63$ kK. For reasons of consistency, the integration was performed in the corresponding range $0 < \nu < 65$ kK. From a first comparison, it turned out that $i_{\rm e}/\sigma = 1.7$, which is also consistent with the average slope of the level distribution. The normalization constant was chosen in such a way that a total line number of ${\rm log}N = 4.35$ was reached in the considered frequency range.

Since all other quantities defining the frequency integrated distribution functions are given as parameters of the Monte Carlo simulation ( $\gamma, l_{\rm min}= 0, r_{\rm min}= -7$), a comparison of the analytical result and the simulated line intensity distribution should coincide for all temperatures and all values of $\gamma$, if the above expression were correct. An example is given in Fig. 12, and the agreement is obvious, note in particular the predicted steep slope for strong lines and the dependence of the weak lines' distribution on $\gamma$.

In the next figure, we have investigated the most interesting question in this respect, namely how far the analytical description can deal with the real case. We have considered the same situation as above, however used the actual oscillator strengths. Thus, by this comparison we can check our hypothesis whether the frequency integrated distribution is less influenced by curvature effects than the distribution defined at specific frequencies, and ask for the effective $\gamma$-exponent of the underlying oscillator strength distribution.

Compared to the last simulations, we have changed only the parameters $l_{\rm min}$ and $r_{\rm min}$ in the analytical expression, and tried to fit the actual distribution (asterisks) by varying the value of $\gamma$. The choice of the former quantities relates to the oscillator strength distribution function of Fe II (Fig. 11, triangles), with a maximum ${\rm log}\, gf$-value ( $= -l_{\rm min}$) of 0.75 and the distribution becoming incomplete at $r_{\rm min}= -4.25$. Figure 13 gives the results for three different values of $\gamma$.

Two points are worth noticing. First, the actual distribution can be fitted extremely well by our analytical approach (the same degree of precision was reached at different temperatures), and the effective $\gamma$ is of the order 1.3, i.e., again representing the "canonical'' value. This is not too surprising since a major part of the distribution actually follows this slope (Fig. 11). Second, a comparison with the other simulations with different $\gamma$ illuminates the role of this quantity: Although (and in agreement with our findings for the frequency dependent line statistics) the slope of the distribution for strong lines remains almost unaffected by $\gamma$ and is much more influenced by temperature via excitation, the width of the distribution and its vertical offset depend strongly on this quantity!

This behaviour is readily understood, if one remembers the fact that $\gamma$ primarily controls the distribution of weak lines: for $\gamma = 1$, the number of these lines remains constant (until $l_{\rm max}$), whereas for $\gamma <1$ it decreases with increasing l and vice versa for $\gamma > 1$. This is not only true for frequency dependent distribution functions but even more for integrated ones, since the threshold line intensity $l_{\rm T}$ decreases with increasing frequency. The presence of this effect is particularly demonstrated by our simulations for the high temperature case shown here for this reason.

An additional constraint for the resulting distribution is the conservation of total line number: If $\gamma <1$ and the number of lines is small for large l, this has to be compensated for by a larger number of lines at low l. In contrast, for larger $\gamma$ and a consequently increasing number of weak lines the number of strong lines has to be smaller. Finally, also the width of the distribution at maximum $\Delta N$ is affected, since this region becomes flat for $\gamma = 1$, and the according width displays a maximum.

For reasons of brevity, we have presented here only the case of Fe II, due to its importance for radiative driving and its large number of lines leading to a low degree of statistical noise. Of course, we have checked our approach also for different ions (e.g., belonging to the CNO-group), and found a satisfactory agreement in any case.

Thus, the frequency integrated line intensity distribution of various ions can be described analytically assuming a power-law distribution for the gf-values, and we have illuminated the role of the corresponding exponent above.

The results presented here may turn out to be useful also in another regard. Since the dependency of the frequency integrated line-intensity distribution on the various parameters is understood, the procedure can be inverted to check for the completeness of atomic data bases. This check can be easily performed, since the calculation of the distribution function (per ion) is simple, involving only Boltzmann excitation. From an analysis of the turn-over points and the width of the distribution (as a function of input temperature), it is thus possible to derive, e.g., the effective values of $x_{\rm max}$ and $gf_{\rm min}$, and data gaps will show up immediately. By a variation of $\nu _{\rm max}$, it will be also possible to constrain the completeness as a function of frequency, at least in those cases where the aforementioned "saturation effect'' does not play a role.

4.2.3 From line intensity to line-strength distribution functions

The next important step concerns the transition from line intensities to line-strengths, where we have the following relation (cf. Eq. (60) vs. 6):

 

$$gf \left(\frac{n_l/g_l}{n_1/g_1}\right)^*$$ 10^-l = (74)

$$\frac{k_{\rm L}}{10^{-l}} \approx b_l \,\frac{\pi e^2}{m_{\rm e} c}\, \frac{n_1}{g_1}\, \frac{\lambda}{\sigma_e {v_{\rm th}}} =$$

$$\bigl(b_l X_{jk} \epsilon_k \bigr)\; \frac{\displaystyle\frac{\pi e^2}{m_{\rm e} c}\,\lambda} {(1+I_{\rm He} Y)\,\sigma_{\rm T} g_1 {v_{\rm th}}}.$$ = (75)

The asterisk denotes the LTE-value with respect to the ion's ground-state "1'', b_l is the NLTE departure coefficient of the lower level of the considered transition - again defined with respect to the ground-state (cf. Sect. 3.1.2), X_jk the ionization fraction of ion "j'' with abundance (relative to hydrogen) $\epsilon_k$ of element "k''. $I_{\rm He}$ is the number of free electrons per Helium atom, Y the Helium abundance, g₁the statistical weight of the ion's ground-state and $\sigma_{\rm T}$ the cross-section of electron scattering. Note, that the thermal speed ${v_{\rm th}}$refers to the nominal value for hydrogen (cf. Sect. 2.2). In the above equation, we have neglected the contribution of stimulated emission.

Inserting typical parameters, this relation can be expressed as

  

$${\rm log}\, k_{\rm L}\approx -l + 6 + {\rm log}\, S_{ljk}$$ (76)

$$S_{ljk} = \left(b_l X_{jk} \frac{\epsilon_k}{\epsilon_{\rm Fe_{\o... ...[\frac{g_1}{4}\right]\, \left[\frac{{v_{\rm th}}}{27\,{\rm km~s}^{-1}}\right]},$$ (77)

i.e., ${\rm log}\, k_{\rm L}\approx -l + 6$ for iron ions in hot star winds if the ionization fraction were unity and NLTE-effects were small. From the definition of S_ljk it is obvious, that - per ion - the only quantities which vary as a function of considered transition are b_l and $\lambda$. Thus, it should be possible to relate the line-strength distribution function of a given ion to the corresponding line intensity distribution function, since the (effectively) contributing range in wavelength (factor $\lambda$) is not too large (this even more, if we consider flux-weighed quantities) and the variation of the departure coefficients is moderate as well, if we recall the scaling relations of important NLTE levels (Sect. 3.1.2). Thus, the variation concerns only those excited levels which are connected to a ground or meta-stable level and is proportional to the dilution factor of the chosen reference point (or a steeper function of r, if optical depth effects of lines were to be included). These levels, however, play a role almost only in light ions, whereas all decisive lines of heavy ions are connected directly to meta-stable levels (and the ground state).

Before we can relate the line intensity to the line-strength distribution function, we have to account for a parameter which is essential concerning this objective. Whereas for the (complete) distribution of line intensities all levels up to the ionization edge are significant (excitation by Boltzmann law), the line-strength distribution accounts for a lower number of levels and lines, namely those which are actually occupied under NLTE conditions. These are mainly those lines with a lower level attributed to one of the three categories defined in Sect. 3.1.2.

In other words, NLTE effects introduce an effective cutoff for contributing (lower) levels, $x_{\rm max}< i_{\rm e}$, already introduced in Eq. (73). As discussed in Appendix E, this quantity (instead of the ionization potential in the LTE-case) now controls the transition value between the excitation vs. oscillator strength dominated part of the distribution function, i.e., this point is shifted significantly towards higher line-strengths. In consequence, the apparent distribution resembles that of an ion with low-lying ionisation potential! Of course, only part of this effect becomes visible in the following, since the level lists in our data base have been designed a priori to be complete only up to essentially occupied levels.

An example for the correspondence of line-strengths vs. line intensities is given in Fig. 14. The asterisks display the NLTE (Sect. 3.1) line-strength distribution for Fe IV, and the fully drawn curve shows the corresponding analytical line intensity distribution, however plotted as a function of $-\,l-0.1$, with cutoff energy at $x_{\rm max}=$ 170 kK (for details, see caption). Note, that the highest level in our data base lies at 210 kK, and $i_{\rm e} \approx$ 440 kK.

From the perfect agreement, it is evident that the line-strength distribution can be actually described in analogy to our previous results for line intensities, where - in view of Eq. (76) - the "average'' shift is given by ${\rm log}S = -6.1$, and this shift originates mainly from the rather low ionization fraction of Fe IV at 40000 K, of order 10^-5. In this example, the distribution is not "saturated'', i.e., $\nu_{\rm max}> i_{\rm e} - x_{\rm max}$, which should lead to two distinctive slopes in the first part of the distribution, namely $2t/\sigma$ and $t/\sigma$ (Appendix E). Even this subtle effect is visible in the actual distribution!

Figure 14: Asterisks: (Frequency integrated) Line- strength distribution function for Fe IV, $T_{\rm eff}= 40\,000$ K, dilution factor W = 0.5, $n_{\rm e}/W = 10^{12}$ and resolution 0.5 dex in line-strength. Fully drawn: Analytical line intensity distribution function, plotted over $-\,l-0.1$ (i.e., ${\rm log}S \approx -6.1$). Parameters (except $x_{\rm max}$, which results from the calculated distribution) consistent with our data base: $\gamma =1.0, i_{\rm e}/\sigma = 6.5, i_{\rm e} = 440$ kK, $\nu _{\rm max}= 400$ kK, $x_{\rm max}= 170$ kK, $l_{\rm min}= -0.7, r_{\rm min}= -5.5$ and ${\rm log}N(l_{\rm max}) = 4.1$

4.2.4 Summation over all contributing ions

As we have understood now, the distribution function per ion consists of a steep and a flatter part, were the transition is controlled by the ratio $x_{\rm max}/t$. The major problem left is the summation over all contributing ions, since, of course, each ion has its own specific S_jkvalue. Thus, even if the line intensities were similarly distributed for each ion, the transformation (= horizontal shift) to the line-strength space might produce unpredictable results if all ions are considered in parallel, as required, e.g., for the calculation of the line-force.

To facilitate the investigations, we have performed some test calculations before considering the real case, again by using the actual atomic models and NLTE occupation numbers (ionization and excitation have been calculated according to Sect. 3.1), however simulating the oscillator strength distribution via Monte Carlo.

Moreover, at first we have concentrated on ions with should behave rather similarly due to their electronic structure (here: Ti to Cu, in the following "iron group elements'', and later the "light ions'', C to Ca) and assumed an equal abundance in order to distinguish between ionization and abundance effects. The chosen abundance resembles the maximum solar value for elements of the iron group, namely for Fe itself ( ${\rm log}\, \epsilon = -4.5$).

Figure 15: Asterisks: Line- strength distribution function for the elements Ti to Cu (all ionization stages), with equal abundances ${\rm log}\, \epsilon = -4.5$. Oscillator strength distribution simulated by Monte-Carlo with $\gamma = 1$. Fully drawn: Analytical result of line intensity distribution (summing up the three components of Fig. 16), plotted as function of $-\,l + 6$

Figure 15 (asterisks) gives the resulting frequency integrated line-strength distribution function for those iron group elements and a simulated gf-distribution with $\gamma = 1,\, r_{\rm max}= 0$ and $r_{\rm min}= -7$, for the same atmospheric conditions as in Fig. 14. Obviously, three different groups are present, which can be easily disentangled due to our knowledge of $\gamma$ (input for Monte Carlo simulation).

This is done in Fig. 16, by means of our analytical description (Eq. 73) and at first in line intensity space. Note, that the displayed solution is only one of a number of other possibilities: For low-lying values of $x_{\rm max}$, which have been derived from the onset of the flat $\gamma = 1$ distribution and result from the effective "NLTE cutoff''(here: $\approx$ 140 kK) and as long as the function is "saturated'', which is the case in our example, the actual value of the ionization energy is unimportant (cf. Eq. E8). Decisive is only the parameter $\sigma$, controlling the steeper part of the distribution function via $(2)t/\sigma$.

Figure 16: Analytic line intensity distribution functions for the three components visible in Fig. 15. Common parameters: $\gamma =1.0, i_{\rm e} = 800$ kK, $\nu _{\rm max}= 400$ kK, $x_{\rm max}= 140$ kK, $l_{\rm min}= 0, r_{\rm min}= -7$. Note that the actual value of $i_{\rm e}$ is unimportant as long as $\nu _{\rm max}< i_{\rm e} - x_{\rm max}$ (see text). Individual parameters: strong lines (fully drawn): $i_{\rm e}/\sigma = 8, {\rm log}N(l_{\rm max}) = 4.25$, "ionization shift'' (see Eq. (76) and text) ${\rm log}S = -1$; weak lines (dotted): $i_{\rm e}/\sigma = 2.5, {\rm log}N(l_{\rm max}) = 5.3, {\rm log}S = -6$; very weak lines (dashed): $i_{\rm e}/\sigma = 8, {\rm log}N(l_{\rm max}) = 6.0, {\rm log}S = -12$

After summing up the three different components and plotting them as a function of $-\,l + 6$, the line-strength distribution found in Fig. 15 (asterisks) can be easily simulated and is displayed by the bold line in this figure. Thus, although the actual function consists of a variety of contributing ions (at least 20 important stages in the considered case), only a small number of clearly different groups behaving similarly amongst each other is finally present. The first group consists of dominant ionization stages ( ${\rm log}S \approx -1$), the second one of minor stages comparable to Fe IV in Fig. 14 ( ${\rm log}S \approx -6$), and the third one comprises the weakest lines from ions with negligible populations ( ${\rm log}S \approx -12$). Before commenting on this similarity in behaviour, let us firstly demonstrate that our findings are not only by chance.

Figure 17 shows the same situation as displayed in Fig. 15 (asterisks), with the only change of $\gamma$ from 1 to 0.8. Actually, even without simulating the new function, it is clear that our argumentation still holds. Again, we can distinguish three groups, and, consistent with our earlier findings, the plateaus from Fig. 15 related to $\gamma = 1$ have changed into declining slopes related to the new slope of $\gamma = 0.8$. If we now apply the identical parameters as in Fig. 16 (accounting, of course, for $\gamma = 0.8$) and overplot the summed result, we find again a satisfactory agreement. This clearly indicates that the combination of parameters chosen for the individual components are of the correct order, and that the line-strength distribution function actually consists of three different components, with similar effective cutoff energies.

Figure 17: As Fig. 15, however with exponent $\gamma = 0.8$and corresponding line intensity distributions, plotted as functions of -l + 6. Parameters of individual line intensity distributions as in Fig.  16

Figure 18: Asterisks: Actual line-strength distribution function for the elements Ti to Cu (all ionization stages), with equal abundances ${\rm log}\, \epsilon = -4.5$, compared to the most important individual components. Atmospheric parameters as in Fig. 14. Lower panel: contribution by most important ions of stage IV (dotted), V (dashed) and VI (dashed-dotted), as well as the sum of these three components (bold line). Upper panel: most important individual components of ionization stage IV. Fe (fully drawn), Ni (dotted), Cu(dashed), Co (dashed-dotted), Mn (dashed-dotted-dotted and Cr (long-dashed), as well as the sum of these ions (bold). Second panel: same as above, however for ionization stage V and elements Fe, Ni, Cu, Co, Mn and sum of them. Third panel: ionization stage VI for Fe, Ni, Cu and sum

Figure 18 (asterisks) displays now the actual case, i.e., without any manipulation of the oscillator strengths. A comparison with the $\gamma = 1.0$ simulation (Fig. 15) shows that the differences are only small: Again, three distinct distributions are visible, where the transition from group one to two (roughly at ${\rm log}\, k_{\rm L}\approx 0$) is no longer as pronounced as before. This might indicate that the actual gf-distribution for the lines of this group is steeper than $\gamma = 1.0$The second group, however, is consistent with $\gamma = 1.0$, as is obvious from the plateau at ${\rm log}\,k_{\rm L}= -3 \ldots -8$.

The details of this figure give an the answer to the question raised above, namely, why the sum of rather complex individual distributions (for the dominant iron group ions at $T_{\rm eff}= 40\,000$ K, see caption) can be described in such simple terms as above.

From the last panel, it is obvious that ionization stages V and VI represent the first group, and stage IV is identical with the second one. Additionally, from the sharp decline of the bold line (sum of stages IV, V, VI) at the end of the plateau, where the complete distribution function including all ions is rising again, it is clear that the third group (not analyzed here) consists of "real'' trace ions (mostly stage III).

The upper three panels show the distribution of the individual species amongst the various ionization stages (as well as the appropriate sums). Neglecting certain subtleties, all elements of a given ionization stage behave similarly. Thus, we can speak of line-strength distribution functions of specific ionization stages instead of individual ions (equal abundances provided).

The origin of this similarity bases on the only minor differences (in a statistical sense) in atomic structure of the iron group elements under consideration, especially with respect to ionization rates (giving rise to similar ionization fractions), a rather low lying effective cutoff energy and a level density parameter $\sigma$ which is small enough to induce the steep increase in the first part of the distribution. Note already here that the according slope (the steeper one!) is roughly equal for all kinds of ions and consequently also for the summed distributions, independent of temperature (provided, of course, the abundances were equal). This equality is clearly shown in Figs. 18, 19 and 20 and translates to a similarity in $2t/\sigma$, being of order unity. Since the value of 2t varies from 32 to 160 in the appropriate units (corresponding to 10000 ...50000 K), this indicates that the (effective) level-density parameter $\sigma$ (the smaller, the steeper is the level-distribution as function of energy) has to vary in concert with these numbers. With respect to the cases discussed already as well as from the argument that an increase in ionization stage/potential inevitably leads to an increase of $\sigma$ (fewer levels distributed over a larger energy interval), this behaviour is not surprising at all.

In addition to this similarity in excitation dominated slope, the gf-distributions are similar as well, with $\gamma$ between 1...1.3 ( $< \gamma_{\rm crit}\approx 2$), dominating the individual distributions after the first two or three decades of steep incline.

In consequence, the total line-strength distribution has a "staircase''-like structure, where each staircase corresponds to a specific stage and is controlled by the sequence excitation/oscillator strength distribution. The horizontal width of these staircases depends mostly on the prevalent ionization fraction.

4.2.5 Temperature dependence

Since "only'' the first seven to ten decades of line-strength are important for line-driving, only those trace ions do contribute to the total distribution which have a significant ionization fraction (and abundance, cf. Sect. 4.2.6) not below roughly $10^{-4} \ldots 10^{-5}$ as well as a larger number of lines than the dominant ionization species. In the other case, i.e., if the line number is smaller, these trace ions are barely visible in the summed distribution: Then, the $\gamma-1$ power-law increase of lines from major species (which extends $r_{\rm min}\approx 5\ldots 7$ decades in line-strength from the turnover point to the "left'') dominates the essential part of the distribution.

Especially for iron group elements, the (total) line number per ion increases strongly with decreasing ionization stage due to the increasing complexity of electronic structure. Thus, at higher temperatures a significant contribution from trace ions of lower stages is actually possible, since these have the required larger line number. At the lower temperature end of radiatively driven winds (roughly $T_{\rm eff}\approx$8000 K), however, all (important) trace ions have necessarily a higher degree of ionization than the major ones and consequently do not (or only marginal) contribute to radiative driving.

This effect is clearly visible in Fig. 19, where we have plotted the line-strength distribution for iron group elements (again using equal abundances) as function of temperature. At the lowest temperature displayed ( $T_{\rm eff}=$ 10000 K, bold line), the enormous line number from ionization stages II and III dominates the first twelve decades. Trace ions (below ${\rm log}\,k_{\rm L}= -5$) have too few lines (as well as negligible ionization fractions) to be of any importance. At 20000 K (dotted), the situation is slightly different. Here, stages III and IV are essential, however a 2nd peak shows up indicating the presence of stages II. This trend continues to higher temperatures, e.g., for the case discussed above ( $T_{\rm eff}=$ 40000 K, asterisks), two kinds of trace ions become visible, namely stage IV in the middle part and stage III with even more lines at weakest line-strengths. At the highest temperature ( $T_{\rm eff}=$ 50000 K, dashed-dotted), the dominant species are VI and partly VII. Ions from stage V contribute significantly, whereas stage IV with its typical $\gamma = 1$ distribution is visible only at weakest line-strengths. Thus and in total, we see a clear dominance of one or two major ionization species in the complete temperature regime.

Let us now concentrate on the decisive part of the distribution (down to, say, ${\rm log}\, k_{\rm L}= -2$). With decreasing temperature, the maximum line-strength $k_{\rm max}$ increases, which is primarily related to the presence of low-lying meta-stable levels acting as quasi ground states, which are missing in the higher ionization stages. Most important, however, is the difference in total line-number! Whereas at the hottest temperatures the transition point occurs at a line number ${\rm log}\Delta N \approx 3$, at lower temperatures a factor of 10 more lines are present at this point. This difference, of course, bases on the increasing number of lines with decreasing ionization stage $\Delta N\propto a^2 \sigma \sim 1/\sigma \sim 2t$ (Eq. (61), Allen ([1966]), his Eq. (3.6) and accounting for $2t/\sigma \approx 1$). Note, however, that the position of the transition point itself ( $x_{\rm max}$!) and both slopes (before and after) remain essentially unaffected.

Figure 19: Line-strength distribution function for the elements Ti to Cu, with equal abundances ${\rm log}\, \epsilon = -4.5$, as a function of temperature (remaining atmospheric parameters as in Fig. 14). $T_{\rm eff}=$10000 K (fully drawn), 20000  K (dotted), 30000 K (dashed), 40000 K (asterisks, cf. Fig. 18) and 50000 K (dashed-dotted). For comparison, the straight line shows a power-law distribution with slope corresponding to $\gamma$ = 1.2

Figure 20: As Fig. 19, however for the light ions C to Ca and equal abundances ${\rm log}\, \epsilon = -4.5$

Figure 21: Summing up the line-strength distribution functions for iron group elements and light ions. Atmospheric parameters as in Fig. 14, for a temperature of 40000 K (left) and 10000 K (right panel). Upper row: Distribution functions for iron group elements (asterisks) and light ions, (triangles), assuming equal abundances ${\rm log}\, \epsilon = -4.5$. The dashed line gives the total line-strength distribution function. $\alpha$ (40000 K) = 0.41, $\alpha$ (10000 K) = 0.34, derived fron least square fit to cumulative distribution in decisive $k_{\rm L}$-range. Lower row: As above, for solar abundances and including H, He. $\alpha$ (40000 K) = 0.57, $\alpha$ (10000 K) = 0.48

So far, we have concentrated on iron group elements. As we will see soon, light ions play an equally important role, although the total number of lines from those elements is significantly smaller. Figure 20 shows the corresponding line-strength distribution, again with equal abundances and as function of temperature. In contrast to above, the maximum value of $k_{\rm L}$ remains rather constant, since the strongest lines are formed by resonance transitions, so that excitation effects are unimportant for the definition of $k_{\rm max}$. The largest differences occur at intermediate line-strengths. They are connected to the large number of resonance lines from lower ionization stages in the region around 600 Å(convergence to ionization edges), whereas the corresponding lines of the higher stages are situated well below our frequential cutoff at 250 Å. On the whole, however, the distribution functions are much more similar when the temperature is varied, compared to the iron group case, and the overall line number is smaller everywhere. The reason for this difference is readily understood, if we account for the vanishing number of meta-stable levels in light ions, so that the group of lines with a meta-stable level as lower one (which comprises the majority of lines for iron group elements) is completely missing. Again, the excitation dominated part shows (virtually) no reaction on temperature, i.e., $2t/\sigma \approx 1$ as discussed above.

4.2.6 Total line-strength distribution functions and the influence of relative abundances

In Fig. 21, we have added now the contribution of light and heavy ions, both for the case of a hot wind at 40000 K as well as for a rather "cool'' wind at 10000 K. Additionally, we study the influence of accounting for realistic abundances, e.g., a solar mixture. Most importantly, by giving up the uniform composition, the distribution function of heavy elements is shifted to the left (to lower line-strengths), since all contributing elements behave rather similarly (Fig. 18), however the (solar) abundance of elements different from Fe is smaller than the value ${\rm log}\, \epsilon = -4.5$ adopted so far. Accordingly, the light ions' distribution function is shifted (to a lesser extent) to the right ($\epsilon$ larger than for Fe on the average).

Thus, the difference between maximum line-strengths is increased if a solar composition is accounted for. The effect seems to be especially large for the cooler wind, where $k_{\rm max}$ (light ions) is increased by 2 dex: At these temperatures, the Hydrogen Lyman lines (and, to a lesser extent, the He II Lyman lines), which are insignificant at hotter temperatures, show up at largest line-strengths, both because of the larger ionization fraction of neutral hydrogen (and He II) as well as their much higher abundance, compared to the metals.

Additionally, the resulting distribution functions (solar case) show even less structure than in the case of uniform abundance, simply because there is a larger scatter of the product $(X_{jk} \epsilon_k)$ (Eq. 75), which leads to a larger variation of S_ljk (Eq. 76) and consequently to a smoothing of any inherent ("staircase'') structure which is still visible in the case of uniform abundances (e.g., Fig. 20).

Accounting now for these differences as a function of abundance as well as the intrinsic differences in the line statistics of iron group elements vs. light ions discussed previously, it becomes evident what controls the slope of the total line-strength distribution.

At first and for large line-strengths, the distribution is dominated by the behaviour of light ions, and it is the steeper, excitation influenced part of their distribution which plays the important role. Since a variety of abundances is present, the distribution is smoother and wider (more, however less pronounced staircases!). Thus, the according slope is flatter than the value of unity found in the case of equal abundances. Since the local slope translates to $\alpha = 1 - s$ with $s = \vert{\rm d}{\rm log}\Delta N/\ {\rm d} {\rm log}\, k_{\rm L}\vert$, $\alpha$ becomes larger than zero in this domain (see also Fig. 25). For a cool plasma, the influence of the $\gamma = 4/3$ distribution of H (Sect. 4.1) flattens the curve additionally.

On the other hand, the left part of the distribution (low $k_{\rm L}$) is controlled by iron group elements, due to their much larger line number. To obtain a situation where the light ions were of any influence in this range would require a mixture with a very small abundance of heavy elements, compared to the CNO group (Pop. III stars?).

The specific influence of the solar composition (actually, only the ratio and not the absolute numbers is relevant!) is evident from a comparison of both rows in Fig. 21 and the corresponding dashed lines, giving the total distribution functions. As discussed above, this abundance ratio introduces a larger separation of the two components. If the abundances were equal (upper row), the transition region between strong and weak line-strengths controlled by the light and heavy ions, respectively, is rather small. Thus, below the cutoff (effective $x_{\rm max}$!) of the light ions, the distribution is suddenly dominated by the distribution of the heavy elements with their much larger line number, and the steep slope (order unity) of the first part of the total distribution function is continued, until finally the gf-dominated part becomes visible.

For solar abundance ratios (lower row), the intermediate range is much wider, and, accordingly, the transition to the flatter, gf-dominated part from iron group elements occurs in a rather smooth way. In connection with the fact that for a mixture of abundances the first part is flatter anyway, we find $\alpha > 0$ for almost all $k_{\rm L}$, since $\alpha = 1 - s$ with s < 1 in the first part, $\alpha \approx 2-\gamma$ with $\gamma = 1\ldots 1.3$ for the lowest contributing line-strengths and has values in between at intermediate strengths.

Again: If there were no difference in the abundance of light and heavy ions, the total distribution function would be steeper (significantly smaller $\alpha$) and much more curved compared to the solar case.

From Fig. 21 it becomes also clear why the derived $\hat \alpha$values decrease for decreasing temperature (cf. Table 2). At lower temperatures (right panel), there are simply more iron group lines present (esp. Fe II, III, IV), compared to the rather constant line number of light ions. Thus, by lowering the temperature, the line-distribution becomes progressively steeper, especially at intermediate line-strengths, which reduces the corresponding $\alpha$'s (see also Sect. 5.1).

The actual role of $\gamma$

One might now question in how far the underlying gf-distribution is of any importance for the final result, since it is much more the (relative) difference in abundance and especially in line number between iron group elements and light ions which leads to the "observed'' line statistics. Accordingly, it is much more the mixture of different contributing ions with different ionization fractions and abundances, which plays a role, whereas the gf-dominated part of any specific ionization stage becomes visible only at the lowest end of contributing line-strengths.

To answer the above question, we have simulated the line-strength statistics resulting from different gf-distributions, again by Monte-Carlo, and compare the outcome with the actual situation in Fig. 22, both for the hotter and the cool wind. In contrast to the case of individual ions or to the case of uniform abundances (e.g., Figs. 15 and 17), the slope of the distribution in the decisive $k_{\rm L}> 1$ range seems to be almost unaffected by the various gf-distributions, neglecting certain subtleties (e.g., the expected presence of small staircases for $\gamma \le 1$) which are insignificant for any result derived from the cumulative distribution (Fig. 22, lower panel).

Figure 22: Total line-strength distribution function for solar abundances, with different gf-distributions. Asterisks: actual case, lines: Monte Carlo simulation with constant exponent $\gamma$ = 1.5 (dotted), 1.0 (dashed) and 0.8 (dashed dotted). $r_{\rm max}= 2, r_{\rm min}= -6$. Upper panel $T_{\rm eff}$ = 40000 K, $\gamma =$ 1.2 (fully drawn). Middle panel $T_{\rm eff}$ = 10000 K, $\gamma =$ 1.3 (fully drawn). Lower panel: as upper one, however cumulative line number. The slope remains almost unaffected by any change in $\gamma$!

What really differs, is the vertical offset of the different distributions, where this offset is monotonically increasing for decreasing $\gamma$, and the actual case (asterisks) is met almost precisely for $\gamma$ between $\gamma = 1.3\,(T_{\rm eff}= 10\,000$ K) and $\gamma = 1.2\,(T_{\rm eff}= 40\,000$ K), in agreement with the average situation (Fig. 11). This behaviour is readily understood if we remember the discussion at the end of Sect. 4.2.2: In addition to controlling the slope of the weaker lines for individual ions, $\gamma$ controls the absolute line number $\Delta N$. In mathematical terms, this occurs via the $\gamma$-dependent normalization constant in Eq. (61) (note, that $w = w(\gamma)$). The physical interpretation is given in the discussion referred to: If the number of weak lines decreases for decreasing $\gamma$, the number of strong lines must consequently grow. This is the effect we observe in Fig. 22.

Thus, the final role of $\gamma$ is an important, however implicit one. Due to its relevance for the vertical offset and with respect to derived force-multiplier parameters, it is much more decisive for the value of $k_{\rm CAK}$ (or $\bar Q$) than for the local slope $\alpha$ and consequently $\hat \alpha$.

Flux-weighted line-strength distribution functions and the difference of $\hat \alpha$ vs. $\alpha$

Our final task in order to describe the radiative line acceleration is to weight the line-strength distribution functions obtained so far by the appropriate flux distribution $L_\nu \nu /L$. This is done in Fig. 23, both for the model with equal as well as with solar abundances. With respect to the shape of the distribution, no dramatic effects are encountered, if we compare the non-weighted distributions (lines) with the corresponding flux-weighted ones (symbols). In terms of our discussion in Appendix E concerning the "saturation'' effect, this is by no means surprising. The major impact of flux-weighting is at moderate line-strengths, where the weighted distribution becomes slightly flatter, since a number of high-frequency resonance lines of minor ions are blended out due to missing flux.

The corresponding force-multiplier parameters $\hat \alpha$ and $\hat \delta$are displayed in Fig. 24 as iso-contours in dependence of ${\rm log}\, k_{\rm 1}$ and ${\rm log}(n_{\rm e}/W)$ (see Eq. (26) and the according derivative with respect to $n_{\rm e11}/W$). Compared to our findings from the last section, nothing new has to be added: For solar composition, the resulting $\hat \alpha$ values are much more constant and larger than for the simulation with equal abundances. By inspection of the displayed values for $\hat \delta$, we find that they are rather small (much lower than the value of 1/3 found for hydrogenic trace ions), indicating the dominance of major ionization stages and the frozen in ionization of stellar winds.

In order to account for more realistic fluxes, we have calculated additionally the case of an irradiation by Kurucz fluxes (consistently used also in the ionization equilibrium). Although some quantitative differences become visible (which turn out to be important for a correct description of B-star winds, cf. Petrenz [1999]), the general effects are small and do not change any qualitative conclusion derived so far.

Since we have included now all ingredients required to calculate line-force and force-multiplier parameters, we can come back to one of the problems stated in Sect. 2, namely the difference of $\hat \alpha$ (derived from the line-acceleration itself) and the local slope of the flux-weighted distribution function, $\alpha$. In accordance with our analytical results from Sects. 2.3.2 and 2.4, Fig. 25 displays the following, by means of our $T_{\rm eff}=$ 40000 K model: For not too large $k_{\rm 1}$respectively $k_{\rm L}$, both numbers are fairly similar. At the steep end of the distribution, however, where the local slope (symbols) becomes large (excitation dominated part of light ions) and $\alpha = 1 - s$ converges to small values (solar abundances) or values (equal abundances, $2t/\sigma \approx 1$, cf. Sect. 4.2.6) the effective force-multiplier parameter $\hat \alpha$ remains positive (as it should, by definition). Thus, even at large $k_{\rm 1}, k_{\rm L}$ (i.e., in the outer wind part), the value of $\hat \alpha$ remains much more constant as if it were coupled to the local slope. Nevertheless, especially for equal abundances the decrease of $\hat \alpha$ at large line-strengths is significant! This decrease in $\hat \alpha$ can have severe consequences for low metallicity or thin winds, as we will see in the following section.

Figure 23: Cumulative flux (times frequency) weighted line-strength distribution function for atmospheric parameters as in Fig. 14 and $T = 40\,000$ K. Flux assumed to be Planck. Asterisks: solar, triangles: equal abundances. For comparison, the unweighted cumulative line-strength distribution functions (corresponding to the left panel in Fig. 21) have been overplotted. Dashed: solar, dashed-dotted: equal abundances

Figure 24: Iso-contours of force-multiplier parameters $\hat \alpha$ and $\hat \delta$ as function of ${\rm log}(n_{\rm e}/W)$ and $-{\rm log}\, t = {\rm log}\, k_{\rm 1}$. Radiation field Planck, $T_{\rm e}= 40\,000$ K, dilution factor W = 0.33. Fully drawn: $\hat \alpha$, dashed: $\hat \delta$. Thick curves stress the values $\hat \alpha= 0.55, 0.65$ and 0.75. Upper panel: solar abundances; lower panel: equal abundances

Figure 25: Comparison of $\hat \alpha(k_{\rm 1})$ and local slope of flux-weighted line-strength distribution function. Model as in Fig. 24, ${\rm log}(n_{\rm e}/W)$ = 12. Dashed: $\hat \alpha$ (solar), dashed-dotted: $\hat \alpha$ (equal abundances). Asterisks (solar) and triangles (equal abundances) give local slope as a function of $k_{\rm L}$, corresponding to the distributions displayed by similar symbols in Fig. 23