3. Polarisations arising from matter redistribution
As suggested in Sect. 1, some wind density blobs might arise by local enhancements of stellar mass loss through the photosphere. These represent an absolute deviation from spherical symmetry (or other smooth distributions, like an ellipsoid), in the sense that there is an increase in mass along some ray paths from the star without compensating decreases along other ray paths. On the other hand, some blobs may arise from density redistribution within the wind above the photosphere. For example, firstly, a small random density enhancement may increase radiative cooling, so reducing pressure, and precipitating radiatively unstable collapse of a wind region into a cooler denser "condensation" in the wind. Secondly, in contrast to this intrinsically unstable process, dense local structures in the wind may arise through the formation of shocks via the line-driven instability mechanism (Lucy 1982; Owocki & Rybicki 1984). The shocks then "sweep up" mass along a region of constant solid angle, or "cone", in the wind. This scenario for wind shocks is almost certainly occurring in hot star winds as evidenced by UV resonance line profile variability (Gathier et al. 1981; Lamers et al. 1982; Massa et al. 1995; Kaper et al. 1999) and the multi-million degree temperature X-ray emission (Cassinelli & Swank 1983; Kudritzki et al. 1996; Feldmeier et al. 1997).
For conservative redistribution, no new electrons are added to the wind and the polarisation resulting from the dense "blob" is partially offset by the negative polarisation of the evacuated region (i.e., the polarisation of the total wind volume without the dense blob). The extent of this cancellation, which we shall hereafter refer to as the "cavity effect", depends on the specific redistribution geometry, so we next consider several particular cases using results from the preceding section.
Bear in mind that it is the change in scaled polarisation, not the actual polarisation itself, that we will be computing. The initial geometry is assumed spherical, which yields zero net polarisation. After redistribution the nett scaled polarisation is the difference between that arising from the compressed region and that of the spherical wind minus the cavity. The polarisation 1 would then be the scaled polarisation p multiplied by the scale factor from Eq. (4).
3.1. Collapse of a conical volume to a point
Consider the collapse of electrons in a conical element, as described by result (b) of Sect. 2, down to a point scattering region that lies within the element and on the cone's central axis. Outside this element, the wind remains unchanged. Note that use of the point blob as the geometry for redistributed wind material ensures that the scaled polarisation is maximized; any other redistribution geometry will produce smaller changes. The resulting nett scaled polarisation is
Fig. 2 plots the value of p in such a scenario for different values of and . The value of is fixed at unity (i.e., the stellar surface), and the results for p are plotted against the ratio of for the collapsed point to for the outer boundary of the original conical volume. Solid curves are for (narrow polar plume), short dashed for (the van Vleck angle), and long dashed for (hemisphere). The van Vleck (1925) angle of corresponds to the latitude at which the nett polarisation of a circular ring undergoes a change of sign; rings at higher latitudes have a "polar-like" polarisation (negative in our convention) and rings at lower latitudes are "equator-like" (positive polarisation). Consequently, we use a value of at the van Vleck angle as a canonical intermediate case between the extremes of a polar plume and hemisphere. The three curves beginning furthest left are for , and the other three are for . Note that because , all of the curves pass through zero at least once, hence there exists a position interior to the conical volume where the collapse of all the electrons to a dense point does not produce any change in the polarisation. This is the essence of Brown's (1994) discussion.
3.2. Radial compression of a conical volume
The shocked regions in a stellar wind can be represented by a "driven wave" as discussed by Hundhausen (1985) and MacFarlane & Cassinelli (1989). The density enhancement associated with a shock "contact surface" in the wind could be near the forward facing shock where the driven wave overtakes the material ahead of it. Or, it could be near the rearward shock where faster wind from behind collides with the driven wave. Both compression situations are possible because the density rise occurs where the radiative cooling of post-shock material is most rapid. This is illustrated in Figs. 12.3 and 12.4 in Lamers & Cassinelli (1999). Feldmeier et al. (1997) suggest that the shock structure responsible for the X-ray emission from O stars is perhaps more like the forward facing kind of MacFarlane & Cassinelli (1989) rather than the rearward shock type of Owocki & Rybicki (1984). Here we treat both cases individually.
For the evolution of a shock, the idea is that an instability results locally in cones of the wind, leading to an accumulation of mass into a cap geometry (either forward facing from overtaking upwind material or rearward facing from faster wind material that overtakes the shock). We assume the shock moves radially outward within the boundaries of the cone. Here we are mainly concerned to see how large the effect will be on polarisation of such sweeping of a fixed amount of material within the wind in a localised domain such as within a cone. We consider the case of a conical region of fixed opening angle. If we start with material in the interval and drive it into the interval included in , then the nett scaled polarisation that results is
3.2.1. Forward facing shock
We deal first with the case of a forward facing shock that we shall refer to as "snowploughing", for it sweeps up the preceding wind. The anticipated effect will be to drive most of the material to the outer boundary of the redistribution volume. To describe this, we define the radial width of the affected volume , the radial width of the redistributed volume , and set . The two functions of radius become
Of course, the Eqs. (18) and (19) are only valid if .
Note that in this expression, has the same sign as , meaning that it is the polarisation contribution of the evacuated region (i.e., of the wind with a cavity in it) and not of the dense snowploughed material that determines the polarisation position angle. Hence the position angle is parallel to the axis of the cone and not perpendicular to it, as one might expect from scattering by the dense shock. The fact that the cavity is of prime importance in determining the polarisation is a consequence of its being interior to the outward moving shock. If for example the mass concentration were instead inward at say , there would be a change of sign in the expression for , and the mass concentration would then be of prime importance in fixing the polarisation.
Fig. 3 shows as plotted against for four values of and 1.0. The scaled polarisation is normalized by the factor that describes the opening angle of the cone. The value of is zero for and has a tail at large L, as predicted in Eq. (21). The curves also show that can change sign as L changes. At large L, is positive indicating that the cavity is determining the nett scaled polarisation and position angle. However, at small L where the swept up cavity is small (on the order of or smaller), is negative, and it is the dense material and not the cavity that is dominating the polarisation.
In the most favorable case, the fractional change in polarisation (i.e., ) does not much exceed 0.1 at best. Thus for example, if all the cone electrons alone could produce a maximum of say 1% polarisation when optimally distributed, then the biggest polarisation which could result in a spherical wind by forward shock concentration of the electrons within a cone would be 0.1%. The radial extent over which is at least half this value is from to . The typical "flush time" for hot early type stellar winds is a few hours, hence polarimetric variability from the formation of shocks will be around 5-10% over a period of around 1 day if the shock forms near the base of the wind.
3.2.2. Rearward facing shock
Now we consider the opposite example of a rearward facing shock. In the case of the forward facing shock, a cavity was cleared out so that material was driven into a conical cap at the outward face of the cone. The cavity was allowed to stretch back to the star, which is not realistic since wind material should always be flowing out of the star and into the cone, however it allows maximisation of the scaled polarisation change. In the rearward shock case, we consider wind material to accumulate in a conical cap at the inward face of the cavity. Here we allow this dense cap to move outward instead of keeping it fixed at the wind base.
For the scaled polarisation, we note that the result derives essentially from the previous expressions (17)-(19), but with a slight modification. We can still use the definitions of L and , but now require that instead of . The expressions become
If the shock begins just at the photosphere, we allow at time . If we knew how and evolved with time, we could simply use Eqs. (23) and (24) with (17) to determine the time evolution of . However, existing observations do not provide such knowledge at this time, and best estimates would have to come from complicated time-dependent theoretical calculations. So as an illustrative example, we choose to model the shock as follows, letting
where and are characteristic flow times for the wind and shock. These expressions for and are linear with time and therefore assume that both the wind and shock are moving at constant speed over the region being considered. In this case, the separation L between and is also linear with time, and so we can invert Eq. (27) to eliminate t in favor of L, the radial length of the cavity, as the independent variable. The choice to use L makes for easier comparison with the previous case of a forward facing shock.
Fig. 4 shows how the scaled polarisation evolves with the length L. The lower axis is the logarithm of . The different curves are for different ratios of as indicated. As in Fig. 3, the vertical axis is normalized by the factor that accounts for the angular extent of the cone. Substantial values of require fairly low values of . This clearly must be the case, for if is near unity, the shock travels outward only slightly slower than the wind so that redistribution occurs over a relatively small volume (i.e., the radial width is small compared to ) resulting in little change of polarisation, as is seen for the case . Only when the wind is speeding away from a much slower shock will the redistribution volume be relatively large. In such cases can become as large as about 0.35 and as small as -0.40 for tending toward zero. In all cases the value of (a) can become zero, (b) is initially positive and then becoming negative, essentially opposite to the trend for a forward facing shock, and (c) shows a maximum for and minimum at occuring at greater L for smaller ratios of .
3.3. Collapse of an equatorial wedge sector to a disk
Another possible source of polarimetric variation could be changes in the oblateness factor of an equatorial wind density enhancement region. For example, using a wedge shaped envelope to approximate a Wind Compressed Zone structure that has been used to describe the axisymmetric density structure of a rotating wind (Ignace et al. 1996), the opening angle of the wedge could evolve with time. Ways for effecting such an evolution arise from variation in the stellar rotation speed , the wind acceleration to terminal speed , or the terminal speed itself . For a single star, the first of these (stellar rotation) will change on evolutionary time scales but not over the much shorter wind flush time. Mechanisms leading to for example stellar variability, such as pulsation, could possibly produce changes in the wind acceleration and terminal speed. However such changes might occur, the resulting modification in the polarisation can be described with our theory.
For a fixed interval of radius , a variation in the scaled polarisation results from changing the opening angle of the wedge from to , giving
Here the change in polarisation is positive for a wedge that becomes more compressed, but can be negative if the wedge were made less compressed. The polarisation is increased in the former case and decreased in the latter. Note that for wedges that are already quite flattened or disc-like (i.e., ), there will be little further increase of the polarisation resulting from any additional compression, since both and are tending to zero.
Alternatively, the compression might occur in radius, so that the wedge geometry in the interval of radius collapses to the interval for fixed. The resulting scaled polarisation is thus
Note that the radial dependence of this expression is the same as for radial compression of a conical volume in Sect. 3.2. Thus the results of that section apply here, the only difference being in the angular dependence.
© European Southern Observatory (ESO) 2000
Online publication: April 10, 2000