Astron. Astrophys. 356, 619-626 (2000)

## 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

 Fig. 1. The figure shows the geometry assumed throughout the paper. A conical region is defined to exist in the interval of angle from to and of radius from to . The conical volume is assumed azimuthally symmetric about the central axis Z, and we further assume the density of electrons to diminish with distance as . The cone is thus a volume of revolution formed by the wedge patch as shown.

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

Adopting as a constant scale parameter,

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

and

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):

• (a) Localised point scattering.

Setting and allowing in (2), the scaled polarisation from a point blob is

This has an extremum value at , which presents the maximum scaled polarisation resulting from any distribution of a prescribed number of electrons, thus providing a convenient benchmark used in subsequent discussion. The actual polarisation would be , hence is of the same order as . Note that we have adopted a convention for which a negative polarisation refers to a polarisation position angle perpendicular to the axis of symmetry; the position angle for positive polarisation is thus parallel to that axis.
• (b) Scattering by a conical cap.

Here we consider the case of part of a conical plume with opening angle and to give

For a conical cap of small radial extent, such that , the scaled polarisation reduces to

This latter case is of interest as representing a radially narrow density enhancement such as produced by a shock.
• (c) Scattering by a wedge sector.

Here we are seeking to describe the polarisation of an equatorial wedge sector of scatterers (as for example to approximate the density structure of a Wind Compressed Zone model from Ignace et al. 1996). We take , so that the scatterers exist only for latitudes , which gives

For a flat annular disk of finite radial width, the scaled polarisation is

whereas for a radially narrow annular rim with , but finite,

Note that in contrast to the conical cap of case (b), the polarisations and have opposite signs, as expected since the rim case is oriented symmetric to the equatorial plane, whereas the point case is taken to lie along the polar axis.

© European Southern Observatory (ESO) 2000

Online publication: April 10, 2000