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Astron. Astrophys. 357, 233-240 (2000)

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4. Application to data

From the results of Sect. 3 it is clear that for a fixed number [FORMULA] of blobs per flow time, the mean level and variances of scattered light and polarisation scale up as the total mass loss rate [FORMULA] is increased, simply because there are more scatterers. On the other hand, the relative degree of variability and of polarimetric cancellation are mainly governed by the number of blobs lying within a few stellar radii - which is fixed by [FORMULA], the ejection rate of blobs per flow time, and by the velocity law [FORMULA] value. More distant blobs do not contribute much to the scattering (see Fig. 5). Due to increased angular cancellation, to achieve a prescribed mean polarisation, as [FORMULA] is increased a larger total number of blob electrons (and hence total mass) has to be emitted per second by the star. The effect of occultation is mainly to hide backward scattering electrons which contribute little to p but significantly to [FORMULA] so reducing [FORMULA]. When combined with the full 3-D treatment of blob distribution geometry, this in fact can bring results into compatibility with data for suitably chosen model parameters and can indeed be used as a means to infer wind parameters, as we now show.

The observable photometric and polarimetric variability quantities related to clumpy winds are essentially [FORMULA] and [FORMULA] plus [FORMULA] if the interstellar and ambient polarisations are either zero or known. (Mean scattered light [FORMULA] is not really an observable since it is difficult to distinguish from direct starlight). Because [FORMULA] primarily determines the absolute value of [FORMULA] while [FORMULA] mainly governs [FORMULA], we can use data on [FORMULA] and [FORMULA] to set limits on [FORMULA] and [FORMULA] using Figs. 7 to 10.

We thus ideally have a set of three observables [FORMULA] and the blobby wind model is largely controlled by three parameters [[FORMULA]] since we have found results to be insensitive to [FORMULA] and [FORMULA] over likely value ranges. It is thus of interest to see whether we can determine these parameters from the observables. We first note that since [FORMULA] all scale linearly with [FORMULA] for fixed [FORMULA] and [FORMULA], we can initially set [FORMULA] aside if we consider only the ratios [FORMULA] as previously defined and also the ratio [FORMULA] shown in Figs. 9 and 10 for various [FORMULA].

Typical observed values from Robert (1992) are [FORMULA], [FORMULA] so that [FORMULA] and [FORMULA]. We see from Fig. 9 that for small values of [FORMULA] it is only possible to match the observed value or [FORMULA] for very small [FORMULA] but these are excluded by the number of narrow features seen in WR emission lines, as discussed further below. For [FORMULA] or so, the [FORMULA] value can be matched for a wide range of [FORMULA] including larger values (greater than about 20) consistent with the emission features. Excluding low [FORMULA] on this basis we turn to [FORMULA] in Fig. 10, where we see that [FORMULA]1 excludes all [FORMULA] below about 1.5 and that for [FORMULA] we have to have [FORMULA]. These bounds on [FORMULA] provide an important confirmation of independent estimates from spectrometry. We note that the constraint from [FORMULA] is weaker than that from [FORMULA] in that the model values of [FORMULA] assumes that the observed [FORMULA] is solely due to the blobs, and that any constant interstellar or intrinsic polarisation (e.g. due to a flattened smooth wind) has been removed. A smaller value of [FORMULA] associated with the blobs alone would push our solution toward larger [FORMULA] and/or smaller [FORMULA] though the latter is quite tightly limited by the number of narrow emission line features discussed below. If we now return to the absolute value of [FORMULA] and again assume it is solely due to the blobs, we can estimate the [FORMULA] needed to achieve this [FORMULA] for the value or range of values estimated for [FORMULA] from [FORMULA] as discussed above.

[FIGURE] Fig. 10. A plot of [FORMULA] curves versus [FORMULA] for different velocity laws [FORMULA] but with the same parameters used in Fig. 9. The horizontal dotted line is the observed ratio. Velocity laws with [FORMULA] appear to be excluded. Comparison of the observed ratios [FORMULA] and [FORMULA] of Fig. 9 and this figure appear consistent with [FORMULA] and [FORMULA] in the range 20-50.

However, for a given [FORMULA], we also see that any value of [FORMULA] is consistent with the observed [FORMULA] but with increasing [FORMULA] the value of [FORMULA] needed to achieve [FORMULA] becomes unreasonably large both on physical grounds and to be consistent with the single scattering limitation on p. Specifically for [FORMULA] and [FORMULA], 100, and 400 we find respectively [FORMULA]/year, [FORMULA]/year, and [FORMULA]/year, so that the range [FORMULA] is again suggested as most plausible and in line with other estimates of WR star [FORMULA] values.

Another constraint on [FORMULA] comes from the emission line profile features produced by the blobs (see Robert 1992). A very high rate [FORMULA] will produce many narrow features which will blend to produce a smooth broad profile, like that from a smooth spherical wind (c.f., Brown et al. 1998) lacking the narrow features actually observed on top of a smooth profile. A very small [FORMULA] on the other hand would produce only a few narrow emission line features without the observed smooth underlying profile. To see whether the emission line profile for our estimated range of [FORMULA] resembles actual data, we have crudely modelled the line profile by taking each blob to emit at a total rate [FORMULA] for constant radial thickness, centred at a wavelength shift [FORMULA] and broadened with Gaussian spread which we chose to be [FORMULA] chosen rather arbitrarily to represent velocity turbulence and gradient effects - both much larger than thermal broadening. In Fig. 11, we show the profile at a random time for the case [FORMULA]. We see that about ten distinct narrow features are present at any random time, consistent with Robert's (1992) results. This is governed by the degree of blending resulting from our assumed narrow feature broadening, but shows that the line profiles observed are broadly consistent with [FORMULA] for our assumed smearing.

[FIGURE] Fig. 11. Narrow feature contributions from all blobs to a wind emission line profile (wavelength shift in velocity units). Parameters are [FORMULA]year, [FORMULA], [FORMULA] km s-1, [FORMULA], and [FORMULA]. Plot corresponds to a randomly chosen observational instant.

In summary our approach provides a valuable new means of studying blob ejection, mass loss rates, and also the blob velocity law in WR winds and should enable further insight into blob production processes. Other work in progress will address the relationship of these results to other hot wind signatures such as X-ray variability.

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Online publication: May 3, 2000