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Astron. Astrophys. 358, 956-992 (2000)

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5. Properties of the force-multiplier

In a perfect numerical model, one would determine the line force at location [FORMULA] by summing up the contribution of some [FORMULA] lines, which is presently not possible because of computational restrictions, however becomes feasible by utilizing the force multiplier concept (cf. Sect. 2) with parameters usually adopted to be constant. Usually, however, detailed 1-D non-LTE calculations (cf. Pauldrach et al. 1994) yield depth-dependent force-multiplier parameters. Only a posteriori , it is possible to find representative global values [FORMULA], [FORMULA], [FORMULA] which can reproduce the results of the hydrodynamic simulation with depth dependent force-multiplier parameters. Unfortunately, the knowledge of these global parameters requires enormous computational effort, and moreover, these force-multiplier parameters have been calculated only for spherically-symmetric winds . To allow for a realistic parameterization of the line force in winds from rapidly rotating stars, we generalize the depth-dependent approach for 1-D winds proposed by Kudritzki et al. (1998).

For this purpose, we (re)investigate the behaviour of the force-muliplier as function of different physical quantities. Since we will concentrate primarily on B-star winds, as outlined in Sect. 3.3, we will restrict ourselves to the according parameter space.

Dependence on temperature. Even with our significantly extended line list, we recover the earlier result by Abbott (1982) that the force-multiplier M varies only mildly with temperature (at most by a factor of 2... 3) in the B star domain 3 ([FORMULA] kK), as shown in Fig. 2 for an irradiation by Kurucz fluxes. The reason for this remarkable constancy - note that the ionization balance is changing significantly throughout the considered temperature range - is given by the fact that the strongest driving lines remain concentrated close to the maximum of the ionizing radiation field ([FORMULA]). This coincidence arises from the influence of the radiation field on the ionization balance : For wind densities typical for OB-stars the ionization potential of the major ionization stage is approximately [FORMULA], and the wavelength of resonance transitions from the major ionization stage is of the order [FORMULA]. Since the maximum of the function [FORMULA] approximately equals [FORMULA], the strongest lines are situated close to this value, almost independent on the actual ionization structure.

[FIGURE] Fig. 2. Force-multiplier M as a function of temperature, for different values of optical depth parameter [FORMULA]. Irradiation by Kurucz flux distributions for [FORMULA].

For temperatures [FORMULA] K, the flux maximum shifts to the Paschen continuum, whereas the most important transitions (even if neutral elements would play a rôle) are still in the Balmer continuum. As a consequence, the line acceleration rapidly decreases for temperatures below 10000 K.

Dependence on density. The quantity [FORMULA] and thus the force-multiplier depends on density in two different ways: Firstly, [FORMULA] scales inversely with density via [FORMULA] (see Eq. (22)) because the atoms that absorb radiation in the frequency interval [FORMULA] have a mass per unit area of [FORMULA]. This explicit dependence is absorbed in the optical depth parameter [FORMULA] 4 and the [FORMULA] term, i.e., the larger [FORMULA] the smaller is the resulting force-multiplier [FORMULA], as shown in Fig. 4. Secondly, [FORMULA] implicitly depends on density via the ionization structure. Because lower ionization stages have more lines (close to the flux maximum), the line force grows with enhanced density for given [FORMULA]. Note, that the effective wind density controlling the ionization balance is given by [FORMULA] (Eq. (28)). This ratio varies significantly only in the lowest part of the wind and becomes almost constant for larger radii, since the electron density as well as the dilution factor asymptotically scale with [FORMULA]. Fig. 3 illustrates this implicit dependence of the force-multiplier.

[FIGURE] Fig. 3. Dependence of force-multipliers M on effective wind density [FORMULA] for different values of [FORMULA]. Irradiation by Kurucz fluxes, [FORMULA] K and [FORMULA].

[FIGURE] Fig. 4. Force-multiplier [FORMULA]. Illumination with Planck spectrum (left) and Kurucz flux distribution for [FORMULA] (right). Note that [FORMULA] is usually a decreasing function of radial wind velocity.

Dependence on dilution factor. As already shown by Springmann (1997, Fig. 4.4), the force-multiplier varies with W via the contribution of subordinate lines. For these lines, the occupation numbers of the corresponding levels explicitly depend on W. However, photoionizations from subordinate levels affect the ionization equilibrium only close to the star ([FORMULA]), and subordinate lines contribute anyway less than 10% to the line force in the considered temperature range (see Abbott 1982). Accordingly, the explicit dependence of M on W is only weak.

Dependence on ionization structure. The ionization structure is determined by the ionizing radiation field (and the effective density) and is decisive for the force-multiplier. In the following, we discuss some physical consequences for M as the result of different temperatures and/or ionization structures

Fig. 4 displays the quantity M in the relevant [FORMULA] range for [FORMULA], [FORMULA] and Planck (left) and Kurucz (right) flux distributions ([FORMULA]), respectively, as function of [FORMULA] [FORMULA] [FORMULA] K. For fixed [FORMULA], M varies with temperature by a factor less than 2... 3 ([FORMULA] dex) in both cases and is a monotonically decreasing function of [FORMULA], as already pointed out. Note that [FORMULA] is usually a decreasing function of radial wind velocity, i.e., the right parts of the figures correspond to the situation in the inner wind.

For Planck irradiation, the force-multipliers show the strongest variations for different temperatures at low [FORMULA] ([FORMULA], for [FORMULA]), whereas for Kurucz flux distributions this behaviour is not found, and M is markedly larger (roughly 0.8 dex) for large values of [FORMULA].

To understand these differences, let us take a closer look at the underlying physics. The contribution of the various elements to the total line acceleration is displayed in Fig. 5 ([FORMULA] and [FORMULA]), where we have plotted the logarithm of the quantity


over the atomic number Z and [FORMULA] 5. In both cases, the largest contributors are the CNO group (Z = 6 ... 8), the elements silicon, phosphorus, sulfur and argon (Z = 14, 15, 16, 18), and the elements of the iron group (chromium, manganese, iron and nickel (Z = 24, 25, 26, 28). For very small values of [FORMULA] ([FORMULA]), only the CNO group (for [FORMULA] K, additionally phosphorus and sulfur) contributes significantly to [FORMULA]. For large [FORMULA] ([FORMULA]), iron dominates the other elements by far (note the logarithmic scale for [FORMULA]!) (cf. Puls et al. 2000).

[FIGURE] Fig. 5. Contribution of different atomic species to the line acceleration for [FORMULA] [FORMULA] 15000 K (left) and 25000 K (right), plotted over atomic number Z and optical depth parameter [FORMULA]. For the definition of [FORMULA], see text. Kurucz irradiation, W [FORMULA] 0.5, [FORMULA] [FORMULA] 11.

Since the considered parameter space represents winds which are not optically thick yet, the line transitions of the most abundant elements are mostly unsaturated (except for single meta-stable and resonance transitions), and [FORMULA] follows closely the adopted abundance pattern (solar abundances, taken from Anders & Grevesse 1989 and Grevesse et al. 1996).

For Kurucz irradiation, the force-multipliers are markedly larger than for the black body case, owing to systematically lower ionization stages in the former. Fig. 6 shows the major ionization stages for [FORMULA] K, [FORMULA] and [FORMULA] present in both cases. For Kurucz fluxes, in particular the major ionization stages of the most contributing elements are one stage below the ones in the Planck case. Since the lower ionization stages provide more driving lines, also the force-multipliers are larger. This behaviour is a consequence of the reduced ionizing flux below the Lyman edge (911 Å).

[FIGURE] Fig. 6. Major ionization stages for illumination with black-body and Kurucz fluxes for [FORMULA] 20000 K and [FORMULA].

Fig. 7 displays the frequency dependent radiation temperature [FORMULA] of the adopted Kurucz fluxes: the radiation temperatures at the decisive transitions (see figure) are much smaller than in the Planck case, of course due to line-blocking, which is one of the most important physical ingredients which has to be accounted for in calculating realistic stellar model fluxes in the EUV. As a consequence, lower ionization stages dominate for Kurucz irradiation.

[FIGURE] Fig. 7. Radiation temperature [FORMULA] for Kurucz fluxes at [FORMULA] = 20000 K. We have marked the ionization energies of iron, Fev/VI (164.22 Å), Feiv/V (226.26 Å), Feiii/IV (404.53 Å), Feii/III (766.14 Å), and oxygen, Oiv/V (160.16 Å), Oiii/IV (225.64 Å), Oii/III (352.67 Å), Oi/II (910.44 Å).

An inspection of the contribution from the different groups of line transitions mentioned in Sect. 4 and in the temperature range around 20000 K shows that the CNO group contributes most at small [FORMULA] via only a few resonance transitions and transitions to meta-stable levels, whereas the iron group dominates via numerous transitions to meta-stable levels at large [FORMULA] (cf. Abbott 1982and Puls et al. 2000). Note that this behaviour of the different line groups is very similar for irradiation with Planck or line-blocked fluxes, respectively.

The properties of the force-multiplier can be summarized as follows: The force-multiplier mainly depends on the optical depth parameter [FORMULA] and the effective wind density [FORMULA]. For fixed [FORMULA], M varies comparatively weakly with temperature (except in certain cases) and dilution factor W. For large [FORMULA] (i.e., high densities and/or small velocity gradients), numerous unsaturated transitions of the iron group elements (in particular, iron itself) dominate the line acceleration; for small [FORMULA] (i.e., low densities) a few strong resonance or quasi-resonance transitions (especially from the CNO group) are decisive. The contribution from subordinate lines has only minor impact in the temperature range considered here. (Those lines become more important for [FORMULA] K when the corresponding levels are stronger populated).

The irradiation of the wind material is crucial for the ionization structure : Since Kurucz flux distributions show a drastically reduced ionizing flux for wavelengths shortward of the Lyman edge, they give rise to systematically lower stages than in the black-body case. As a consequence, the force-multipliers are larger up to one order of magnitude.

In the first place, the magnitude of the force-multiplier is determined by the ionization structure present, rather than by the flux weighting factor [FORMULA]. As numerous test calculations have shown, the differences in M are only marginal if one uses either Planck [FORMULA] or Kurucz [FORMULA] flux weighting factors, if one assumes identical ionizing fluxes in both cases. This fact will turn out to be of importance for our further proceeding of calculating consistent force-multiplier parameters, described in the next section.

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Online publication: June 20, 2000