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Astron. Astrophys. 335, 1003-1008 (1998)

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5. Summary and conclusions

In the previous sections, inhomogeneous Wolf-Rayet type stellar winds were modeled in a first-order approximation, assuming that small-scale clumps are distributed with a constant volume filling factor within a void interclump space. Model calculations as well as analytical considerations showed that the main spectral features, i.e. the strength of the emission lines, are invariant if the clumping parameter D (i.e. the density enhancement of the clumps compared to a homogeneous model with the same mass-loss rate) is compensated by scaling down the mass-loss rate with the factor [FORMULA]. This holds in the same degree of approximation as the so-called transformation law for WR atmospheres and follows from the dominance of emission processes that depend on the square of the density.

Hence, to the first order the mass-loss rate is the only empirical parameter which is affected by the application of clumped models for spectral analyses. Note that the same clumping correction applies for empirical mass-loss rates derived from radio observations, as the free-free emission is a [FORMULA]-process as well. The good consistency between empirical mass-loss rates derived from the radio emission and the UV/optical/IR line spectrum, respectively, thus implies that the clump characteristics do not change very much from the line-forming regions close to the photosphere to the radio-emitting regions in hundreds of stellar radii.

When the clump density enhancement D is increased with a compensating decrease of the mass-loss rate (i.e. [FORMULA] = constant), the electron-scattering line wings become weaker. Along model series with same transformed radius but different combinations of mass-loss rate and radius (or luminosity), for which the main spectral features retain their strength approximately, the bigger stars have stronger electron scattering wings if the same clumping parameter is chosen.

These electron scattering wings can be used, in principle, to determine the adequate value of the clump density enhancement D. We select Wolf-Rayet stars of different spectral subclasses and compare their spectra with adequate models, varying the density enhancement D. In all cases, the homogeneous model can be definitely ruled out because it predicts electron scattering wings that are significantly stronger than observed. For the three WN stars considered, the line wings are in reasonable agreement if [FORMULA], while for [FORMULA] they are possibly to shallow, but still compatible with the observation within the error margin. In case of the WC example star, [FORMULA] fits best.

Adopting that [FORMULA] is a typical value for the density enhancement, the empirical mass-loss rates become smaller by factor [FORMULA] than obtained from the use of homogeneous models. This is in line with independent evidences. Moffat et al. (1994) performed wavelet analyses of emission line variations in WR spectra and derived a clump distribution which has the effect of [FORMULA] in our notation (Moffat & Robert 1994). For the WN+O binary system V 444 Cyg, the radio emission gives about three times higher mass-loss than the change of the orbital period (cf. St-Louis et al. 1993). Thus clumping with [FORMULA] would reduce the radio-derived rate to perfect agreement (although that close binary system might not be representative for single-star winds).

In Hamann & Koesterke (1998) we analyzed the nitrogen spectra of the Galactic WN stars. Now we claim that the resulting mass-loss rates have to be scaled down by at least 0.3 dex, while the other parameters are not affected. Adopting this factor of two, the average values of the "momentum ratio" [FORMULA] now become 4.5, 4.3 and 15 for the WNL, WNE-w and WNE-s spectral subclass, respectively. For our WC example (Br 43) with [FORMULA] the ratio is 17. These values are still above the single-scattering limit (i.e. unity), but no longer implausible for radiative acceleration with multiple-scattering effects (Lucy & Abbot 1993). Note that the non-radiative energy loss (Heger & Langer 1996) becomes less important when the mass-loss rates are smaller than previously thought. For the stellar evolution in the WR phase the consequences of smaller mass-loss should be examined.

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© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998