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

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2. Model of polarisation and scattering from blobs

We assume that the polarimetric and photometric variability of WR stars is due to localised mass loss density enhancements which are generated at random positions at the stellar surface and at random time intervals with a normal distribution of mean value [FORMULA]. The blobs are then taken to move radially outward with a velocity law, and constant solid angle [FORMULA] and radial thickness [FORMULA], the uniform electron density then decreasing as [FORMULA]. (This assumption of constant [FORMULA] and [FORMULA] has little effect on the results). The blobs thus have axisymmetric shapes and, on the assumption that they are not optically thick in the continuum, the results of Brown & McLean (1977) can be used to find the polarisation of a single blob as:

[EQUATION]

where

[EQUATION]

is a mean optical depth, and

[EQUATION]

is a "shape" factor. [FORMULA] is the blob axis inclination to the line of sight; [FORMULA], where [FORMULA] is the blob opening angle between the axis of symmetry and the direction of the scatterer seen from the center of the star; [FORMULA] is the electron number density in the blob; and [FORMULA] is the Thomson scattering cross-section. For the local reference frame [FORMULA] chosen, we let [FORMULA]. For constant solid angle [FORMULA] and radial extent [FORMULA], the electron density is assumed to vary only with radius inside the blob i.e. as [FORMULA]. In order to calculate the electron density in one blob, we can use the mass conservation law:

[EQUATION]

where [FORMULA] is the mass outflow rate within one blob, [FORMULA] is the mass density, and [FORMULA] is the blob velocity law which we adopt to be of the common form:

[EQUATION]

where [FORMULA] is the terminal wind speed, [FORMULA] is the radius of the WR star, b is a dimensionless parameter to ensure that the initial wind speed is non-zero, and [FORMULA] is a velocity law index. In our simulations we adopt [FORMULA]. We note that in using a [FORMULA]-law (5) for [FORMULA], it is in practice essential to use it in a form with a finite [FORMULA] at the wind base since for larger [FORMULA] values blobs would otherwise take an infinite time to reach finite [FORMULA]. This can be done either by setting [FORMULA] at [FORMULA] (which is effectively what we do through the parameter b) or by starting material at [FORMULA], with [FORMULA]. We experimented with the value of [FORMULA] and found that our results as a function of [FORMULA] were insensitive to [FORMULA] so long as it was small. This is a little surprising since small [FORMULA] causes a blob to dwell longer near the star, as does finite [FORMULA]. However, for very small [FORMULA] the effect of this is to delay the blob for many flow times very near [FORMULA]. This will increase the value of [FORMULA] (which is not measurable anyway) but not of [FORMULA] since the depolarisation factor is near zero. But, the travel time in this region is much longer than a flow time, so blobs there will affect the time variances very little. The effect of [FORMULA] on the dwell time, however, is felt further (around [FORMULA]) out where the speed and effect on variations is larger, and so dominates the effect of the [FORMULA] law parameters on the observables.

The blob electron density becomes

[EQUATION]

where [FORMULA] is the hydrogen mass and [FORMULA] is the mean particle weight per free electron.

To incorporate the finite star geometry, the correction factors [FORMULA] can be employed according to Cassinelli et al. (1987) and Brown et al. (1989), viz.

[EQUATION]

and

[EQUATION]

where [FORMULA], again [FORMULA] is the photospheric radius of the WR star, and [FORMULA] is the viewing inclination.

We now wish to combine Eqs. (1) to (8) to yield an expression for the polarisation from a single blob with the assumed geometry. Since the density does not vary with µ within the blob, the angle integrals in Eqs. (2) and (3) for [FORMULA] and [FORMULA] are straightforward. We also choose to rewrite the velocity in (5) as [FORMULA]. Combining, the polarisation expression now becomes

[EQUATION]

and the scattered light intensity as a fraction [FORMULA] of [FORMULA] in terms of Brown et al. (1995) is:

[EQUATION]

where [FORMULA].

These expressions are for a single blob. To deal with the presence of many blobs, we use the coordinate system [FORMULA] with oz along the line of sight. Then for each blob the "inclination" angle [FORMULA] is identical to the scattering angle [FORMULA] and the polar angle [FORMULA], while the polarization position angle [FORMULA] is just the coordinate component [FORMULA] (see Fig. 1). For a system of blobs labelled [FORMULA], the total scattered light fraction [FORMULA] and the net polarisation are as usual simply given by summing over j the Stokes intensity parameters [FORMULA], [FORMULA] of each to get the totals Q, U, then finding [FORMULA] and position angle [FORMULA]. Blobs enter the flow in random directions such that [FORMULA] is uniformly sampled in the interval -1 to [FORMULA] and [FORMULA] from 0 to [FORMULA]. The radii [FORMULA] are determined by time and the velocity law. The results are as follows, where it is to be understood that summations exclude all occulted blobs - i.e. blobs whose coordinates [FORMULA], [FORMULA], [FORMULA] satisfy [FORMULA] and [FORMULA].

[FIGURE] Fig. 1. Geometry of scattering from one blob. We employ a coordinate system [FORMULA] with oz along the line of sight. For each blob the "inclination" angle i is identical to the scattering angle [FORMULA] and the polar angle [FORMULA], while the polarisation position angle [FORMULA] is just the coordinate component [FORMULA].

The time averaged total mass loss rate in all blobs in the inhomogeneous but mean spherical "wind" is [FORMULA], where [FORMULA] is the mean blob ejection rate (s-1) and [FORMULA] is the number of electrons in one blob. Thus, if we fix [FORMULA] and increase [FORMULA] then there are more blobs in any given range of r but each of smaller [FORMULA] with the structure approaching spherical symmetry as [FORMULA]. Usually, we denote [FORMULA], the number of blobs ejected per flow time [FORMULA], as the measure of the blob ejection rate.

The system of equations governing the time varying polarisation and scattered light is then

[EQUATION]

and

[EQUATION]

Hence

[EQUATION]

and

[EQUATION]

On inspection of the above equations for polarisation and scattered light, we expect that, for given [FORMULA], [FORMULA], [FORMULA], and [FORMULA], results should depend mainly on the mean time intervals [FORMULA] between consecutive blobs as a fraction of the flow time [FORMULA] - i.e. on [FORMULA]. For fixed mass loss rate and flow time, if the blob generation rate is low, only a few blobs each of large density will be present near the star and these will dominate the p and [FORMULA] values. However for high generation rates, many low density blobs near the star will be controlling p and [FORMULA]. So the same total number of electrons is redistributed in different number of blobs, resulting in different statistical means and variances in the polarisation and scattered light fraction. For a fixed blob ejection and mass loss rate, the number of blobs in the inner radii near the star has a steady mean value and so therefore do the resulting polarisation, scattered intensity and their variances, but these values change with [FORMULA], [FORMULA], and [FORMULA]. So their observed values allow inference of the blob emission and flow parameters, as we now discuss.

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© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000
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