4. Application to data
From the results of Sect. 3 it is clear that for a fixed number of blobs per flow time, the mean level and variances of scattered light and polarisation scale up as the total mass loss rate 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 , the ejection rate of blobs per flow time, and by the velocity law 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 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 so reducing . 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 and plus if the interstellar and ambient polarisations are either zero or known. (Mean scattered light is not really an observable since it is difficult to distinguish from direct starlight). Because primarily determines the absolute value of while mainly governs , we can use data on and to set limits on and using Figs. 7 to 10.
We thus ideally have a set of three observables and the blobby wind model is largely controlled by three parameters  since we have found results to be insensitive to and 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 all scale linearly with for fixed and , we can initially set aside if we consider only the ratios as previously defined and also the ratio shown in Figs. 9 and 10 for various .
Typical observed values from Robert (1992) are , so that and . We see from Fig. 9 that for small values of it is only possible to match the observed value or for very small but these are excluded by the number of narrow features seen in WR emission lines, as discussed further below. For or so, the value can be matched for a wide range of including larger values (greater than about 20) consistent with the emission features. Excluding low on this basis we turn to in Fig. 10, where we see that 1 excludes all below about 1.5 and that for we have to have . These bounds on provide an important confirmation of independent estimates from spectrometry. We note that the constraint from is weaker than that from in that the model values of assumes that the observed 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 associated with the blobs alone would push our solution toward larger and/or smaller 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 and again assume it is solely due to the blobs, we can estimate the needed to achieve this for the value or range of values estimated for from as discussed above.
However, for a given , we also see that any value of is consistent with the observed but with increasing the value of needed to achieve becomes unreasonably large both on physical grounds and to be consistent with the single scattering limitation on p. Specifically for and , 100, and 400 we find respectively /year, /year, and /year, so that the range is again suggested as most plausible and in line with other estimates of WR star values.
Another constraint on comes from the emission line profile features produced by the blobs (see Robert 1992). A very high rate 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 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 resembles actual data, we have crudely modelled the line profile by taking each blob to emit at a total rate for constant radial thickness, centred at a wavelength shift and broadened with Gaussian spread which we chose to be 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 . 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 for our assumed smearing.
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.
© European Southern Observatory (ESO) 2000
Online publication: May 3, 2000