2. Polarisation of a parametric "blob" model
Polarisation resulting from electron scattering depends on moment integrals of the envelope electron density over the scattering volume (Brown & McLean 1977; Brown et al. 1978). Thus, for the present purpose of discussing the effect of polarisation of density changes on various scales, it will be sufficient to use a simple parametric model for the blob density. Analytic expressions for polarisation from Thomson scattering of stellar light are given by Brown & McLean (1977). These are based on the single scattering approximation, but work quite well at modest optical depths, especially for purposes of evaluating relative polarisations, rather than computing absolute values.
The Brown & McLean results apply to axisymmetric density structures only, but this will serve adequately to describe modifications of the polarisation resulting from changes in the wind electron density, either in the form of a plume or of a rotationally symmetric sector. In the former case, we take the plume to be axisymmetric about the central axis. We can also use this case to describe a localised "point" scattering region simply by giving the plume very small radial and angular extent. For the annular sector, the symmetry axis is that of the annulus. Brown & Mclean showed that the polarisation of any axisymmetric structure could be written as the product of an optical depth factor (scaling with the total number of electrons and the inverse square of the system size), an asphericity shape factor, and a viewing inclination factor . To the Brown & McLean expressions, we add the depolarisation correction factor derived by Cassinelli et al. (1987) and by Brown et al. (1989) for finite star size, although we do not account for envelope occultation by the stellar disk.
As illustrated in Fig. 1, our basic blob model is an axisymmetric density structure which is uniform over a range of co-latitude , in the interval with , and has an radial variation of density in the range of radii . For such a blob, the electron number density is taken as for a scale constant and R the stellar radius, hence the total number of electrons is
Using expressions from Brown & McLean (1977), the polarisation of the blob is given by
where the square root factor accounts for the depolarisation effects of the finite stellar size (Cassinelli et al. 1987). Evaluating the radial and latitudinal integrals and eliminating in favor of , the resulting blob polarisation becomes
and using a normalized radius , we define the scaled polarisation to be
The two functions g and f are conveniently defined to separate the dependence of the polarisation on angular extent and radial extent. The two functions are given by
The advantage of the scaled polarisation p is that a prescribed number of scattering electrons is maintained for any redistribution in x and µ. Hence the nett relative change in the scaled polarisation arising from redistribution is neatly given by the difference in p for the two respective structures with the same .
Here we derive expressions for the scaled polarisation from a number of specific simplified cases. These results are used in the following section to determine the scaled polarisation arising from redistribution of a fixed number of scatterers in geometries of interest to stellar winds. We consider the cases of scattering by a point blob, a conical cap, and a wedge sector (e.g., a disk):
© European Southern Observatory (ESO) 2000
Online publication: April 10, 2000