Up: Can interfero-polarimetry constrain extended

Subsections

# 4 Case of ellipsoidal envelopes

The major phenomenon producing intrinsic polarization in Be and shell stars is Thomson scattering of the starlight by an oblate rotationally symmetric envelope. Such envelopes are usually studied by photo- and spectro-polarimetry (Wood et al. 1993; Wood et al. 1996): the integrated degree of polarization is of the order of 1% and rarely exceeds 1.5% (Coyne 1976). We have modeled the observations of such objects (star + a flattened envelope) by interfero- polarimetry. We have considered electron distributions expressed by where a, b and c are the envelope dimensional parameters. a lies along (), b along () and c along () (Fig. 6). For axi-symmetric envelopes, a equals b. k is the exponent of the distribution law and is the envelope flattening. We aim at constraining , and k by interfero-polarimetry.

 Figure 6: Definition of the dimensional parameters of an ellipsoidal envelope. The system (, , ) related to the sight direction () is assumed to be coincident with the stellar system (x, y, z) defined in Fig. 1

## 4.1 Polarized visibilities

If we consider the polarized visibilities normalized to the natural one, their variations versus spatial frequency are too small (i.e. ) for determining k or . Moreover at small spatial frequencies (), the comparison between the normalized polarized visibilities do not enable to differentiate k except for (See discussion in Sect. 5.3) and for an accuracy of 0.1% on the visibility.

## 4.2 Equivalent uniform disks

The variations of and clearly depend upon k (Tables 2 and 3) and (Table 4). They are larger for an extended envelope (Tables 2 and 3) and for a large . Even for , these variations are too small for determining k or if we consider a typical error of 0.1 mas on angular diameters. In fact when k varies from 0 to 3, increases by 0.087 and by 8.1% (Table 2). When varies from 0.5 to 4, the variations of equals 0.029 and those of equals 2.81% (Table 4). The values have to be compared with error bars given in Sect. 2.1. (0.15 - 0.2 and 15 - 20% respectively). As shown for spherical envelopes, angular dimensions and thus can be determined from the natural visibility.

## 4.3 Degree of polarized visibility PV

The variations of PV versus spatial frequency have the same shapes for a variation of k as for a variation of (Fig. 7). Given a flattening , we can differentiate all the exponents k provided that the spatial frequency is small enough (), attains 10% and the accuracy on the visibility equals 0.1%. An accuracy of 1% on the visibility is enough to put a lower limit to k. The accuracy on k clearly depends on the envelope geometry but it is generally better than 0.5. Given an exponent k, using PV to determine leads to a poor accuracy ( for ). To constrain the envelope geometry it is clearly better to use the high angular resolution and the capacity of Earth-rotation synthesis of the interferometer (See Sect. 5.5).

 Figure 7: Degree of polarized visibility versus spatial frequency for an ellipsoidal envelope responsible for 10% of the total flux: top) for various exponents k of the electron distribution and for dimensional parameters of 10, 10 and 7.5 stellar radii respectively (), bottom) for various flattenings and for an uniform envelope (k=0)

Up: Can interfero-polarimetry constrain extended

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