## 2. Model of polarisation and scattering from blobsWe 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 . The blobs are then taken to move radially outward with a velocity law, and constant solid angle and radial thickness , the uniform electron density then decreasing as . (This assumption of constant and 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: where is a mean optical depth, and is a "shape" factor. is the blob axis inclination to the line of sight; , where is the blob opening angle between the axis of symmetry and the direction of the scatterer seen from the center of the star; is the electron number density in the blob; and is the Thomson scattering cross-section. For the local reference frame chosen, we let . For constant solid angle and radial extent , the electron density is assumed to vary only with radius inside the blob i.e. as . In order to calculate the electron density in one blob, we can use the mass conservation law: where is the mass outflow rate within one blob, is the mass density, and is the blob velocity law which we adopt to be of the common form: where is the terminal wind speed,
is the radius of the WR star,
The blob electron density becomes where is the hydrogen mass and is the mean particle weight per free electron. To incorporate the finite star geometry, the correction factors can be employed according to Cassinelli et al. (1987) and Brown et al. (1989), viz. and where , again is the photospheric radius of the WR star, and 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 and the scattered light intensity as a fraction of in terms of Brown et al. (1995) is: where . These expressions are for a single blob. To deal with the presence
of many blobs, we use the coordinate system
with
The time averaged total mass loss rate in all blobs in the
inhomogeneous but mean spherical "wind" is
, where
is the mean blob ejection rate
(s The system of equations governing the time varying polarisation and scattered light is then and Hence and On inspection of the above equations for polarisation and scattered
light, we expect that, for given ,
, ,
and , results should depend mainly on
the mean time intervals between
consecutive blobs as a fraction of the flow time
- i.e. on
. 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
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