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

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3. Model results

1. By design, the model is in qualitative agreement with data showing photometric and polarimetric fluctuations about some mean values (see Robert 1992). Fig. 2 shows how typical polarisation, position angle, and scattered light fraction change with the total number of blobs emitted from the start - i.e with time, starting from ejection of the first blob, for the model with the indicated parameters. The observables all rise to steady mean values within the first few flow times, because thereafter the number of blobs near the star, which dominate the scattered light, becomes steady on average. Polarisation changes are also shown as a locus in the Q - U plane (see Fig. 2) which, as expected, shows no preferred direction, since the mean structure is spherical. In Fig. 3 we show "observational" time-smoothed results for the variations in mean polarisation and scattered light. We applied "boxcar" smoothing with a width of about one flow time (i.e., about 1 hour for the chosen star parameters). Standard deviations, [FORMULA], of these quantities are also plotted in Fig. 3. Fig. 4 is the ratio [FORMULA] versus total number of blobs N as time progresses.

[FIGURE] Fig. 2. The figures show respectively, instant by instant, model results versus number N of blobs emitted thus far for the following: upper left panel - polarisation p; upper right - scattered light [FORMULA]; lower left panel - polarisation position angle [FORMULA]; lower right panel - Q(N) versus U(N). Parameters used throughout are: [FORMULA]/year, [FORMULA], [FORMULA] km s-1, [FORMULA], [FORMULA], and [FORMULA].

[FIGURE] Fig. 3a-d. Based on the same data as Fig. 2, here we show smoothed results versus number of blobs N (increasing with time) for the following observables, with parameters as in Fig. 2: a mean polarisation [FORMULA]; b  variance of polarisation [FORMULA]; c  mean scattered light fraction [FORMULA]; d  variance of scattered light [FORMULA].

[FIGURE] Fig. 4. Ratio [FORMULA] of polarimetric versus photometric standard deviations versus number of blobs N which are for the same parameters as in Fig. 2. The steady mean value of [FORMULA] is about 0.07 for these parameters.

2. To confirm the dominance of inner blobs explicitly, we show the cumulative contributions of blobs to the polarisation and scattered light after a long time [FORMULA] in Fig. 5. We find that only the blobs which are close to the star give significant contributions. That is, once enough blobs have entered the flow so that the inner few stellar radii contain a mean steady state number of blobs, then steady mean values for p, [FORMULA], [FORMULA], and [FORMULA] are achieved, because the distant blobs are irrelevant and give only small contributions (see Fig. 5). The effect of an accelerating velocity law [FORMULA] is that a larger effective number of blobs lies near the star for a given [FORMULA]. Broadly speaking we expect that increasing [FORMULA] alone will increase mean values of p, [FORMULA] but will not change their relative [FORMULA], whereas varying [FORMULA] alone affects the variances. In the next section we discuss how these properties can be used to constrain [FORMULA], [FORMULA], and [FORMULA] from data.

[FIGURE] Fig. 5. Panels a and b show the values, with and without occultation, of the cumulative polarisation [FORMULA] at a certain moment due to all the blobs out to distance r from the star's center. Panels c and d similarly for the cumulative scattered light [FORMULA]. Model parameters are the same as Fig. 2. From these plots, it is clear that the blobs within a few [FORMULA] of the star dominate the observables. Fluctuations in p are mainly due to angular cancellation effects. Comparing occulted and unocculted cases, the latter have higher values of [FORMULA] as expected since more scatterers are visible, while p is somewhat increased since blobs nearly directly opposite the star have non-negligible polarisation.

3. In the Richardson et al. (1996) analysis, [FORMULA], [FORMULA], and [FORMULA] were taken as randomly distributed so that in effect it was assumed that [FORMULA] was constant. The total number of blobs N was taken to be a random variable given by a Poisson distribution with a mean value [FORMULA]. They found that no mean number of blobs, [FORMULA], could make [FORMULA] as small (0.05) as observed. But when we consider a [FORMULA] velocity law for the blob motion, and allow for occultation of blobs behind the star (whose back - scattered light has small values of polarisation but large values of [FORMULA]), we find using the same approach as they did, that results can match observed [FORMULA] for large enough [FORMULA] (see Fig. 6). (Note here we use the same definition for [FORMULA] as Moffat & Robert 1992). We conclude that Richardson et al's constant velocity assumption limited their ability to reproduce the observed [FORMULA]. In our modelling we treat the problem dynamically, following blob motion after ejection rather than the blob "snapshot" approach of Richardson et al. (1996), but the results from these two methods are consistent - our results essentially match theirs for [FORMULA] apart from minor differences due to their having a variation in blob numbers and sizes.

[FIGURE] Fig. 6. These figures were created similarly to the work of Richardson et al. (1996), considering blobs to have random distances but weighted for a [FORMULA] velocity law, the total number of blobs at any instant being random with Poisson mean value [FORMULA]. Parameters have the same values as in previous figures. They show that our dynamic model gives similar results to the purely statistical approach.

4. Results for "observables" as a function of [FORMULA] from our time-dependent [FORMULA] model are shown in Table 1 with occultation effects included, for the values of [FORMULA], [FORMULA], [FORMULA], [FORMULA], etc., indicated. In each case we ran our code for 5000 blobs in total and for the same value of [FORMULA] (equivalent to a cone semi-angle of roughly [FORMULA]). In Fig. 7 we show that results are relatively insensitve to [FORMULA] (unless we go to extreme values like 0 or [FORMULA] which yield zero polarisation by spherical symmetry). The same is also true of [FORMULA] which is the blob width. Fig. 8 shows the effects of varying [FORMULA]. As expected, all absolute values of scattered light parameters p, [FORMULA] just scale linearly with [FORMULA] while the ratios of p, [FORMULA] and of their variances [FORMULA], [FORMULA] are independent of [FORMULA]. The effect of occultation is not very large except for small [FORMULA]. The reason is that only a few blobs are near the star in the small [FORMULA] case, so that the occasional occultation of a blob can have a dramatic effect on the observables.


Table 1. Simulation results for finite star source with occultation and velocity law
The following parameters are employed in the simulations: [FORMULA]/year, [FORMULA], [FORMULA], [FORMULA], [FORMULA] km s-1, and [FORMULA].

[FIGURE] Fig. 7a-f. Effect on observables of varying the blob solid angle [FORMULA] for parameter values [FORMULA]year, [FORMULA], [FORMULA] km s-1, [FORMULA], and [FORMULA]. The points are for explicit model calculations.

[FIGURE] Fig. 8a-f. Effect on observables of varying [FORMULA]. Panel a shows that [FORMULA] corresponds to [FORMULA]/year for [FORMULA]. Panels e and f show that [FORMULA] and [FORMULA] are independent of [FORMULA]. Parameters used throughout are as follows: [FORMULA], [FORMULA], [FORMULA] km s-1, [FORMULA], and [FORMULA].

From Table 1, we can see that in the first three rows, the polarisation and scattered light exceed unity due to the high blob density and optical depth for small [FORMULA], so that our model violates its single scattering basis for very low blob ejection and high mass loss rates but is otherwise self-consistent.

5. Fig. 9 shows that for different [FORMULA], we get different [FORMULA] versus [FORMULA] curves with [FORMULA] going to near constant values for large [FORMULA]. The larger the [FORMULA] we have, the lower that [FORMULA] becomes. The reason is that large [FORMULA] implies more blobs near the star where p, [FORMULA] are small due to finite light source size. That is why Richardson et al. (1996) could not get [FORMULA] as small as observed using a constant velocity.

[FIGURE] Fig. 9. Figure shows [FORMULA] versus [FORMULA]. Here [FORMULA] and [FORMULA] is the flow time. The horizontal dotted line is the observed ratio. Parameters used throughout are as follows: [FORMULA]/year, [FORMULA], [FORMULA], [FORMULA] km s-1.

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Online publication: May 3, 2000