Up: Can interfero-polarimetry constrain extended

Subsections

# 3 Case of spherical envelopes

We aim at determining three parameters of the envelope: its relative flux , its angular diameter (Eq. 11) and its radial distribution of electron expressed by n(r)=n0 .R* denotes the stellar radius, n0 the electron density at the stellar surface and k the exponent of the distribution. For this purpose, we study various interfero-polarimetric observables.

## 3.1 Polarized intensity maps and visibilities

For a spherical envelope, the three intensity maps are different (Figs. 2-4). Each polarized map is flattened along the polarizer direction. Its shape is identical whatever the polarizer direction and only its orientation varies. That's why classical polarimetry fails to detect any difference in such cases. The Fourier transforms of Figs. 2-4 provide polarized visibilities whose differences can be large () for small spatial frequencies (i.e. for equal to 1/(10 R*)). Such spatial frequencies correspond to large structures of the observed object and to short baselines from the ground (10 - 20 meters at 0.6 m for a stellar radius of 0.5 - 1 mas): the object is partially resolved. These differences between the polarized visibilities are obviously characteristic of the exponent k. But, in practice, these differences are greatly attenuated by the dominant flux contribution of the star (F*) and by its large visibility (since is often less than 1 mas, is close to 100%). Thus the polarized visibilities are not sufficient to strongly constrain our parameters (, , k) and a differential study is mandatory (See Sect. 3.3).

 Figure 2: Intensity map in natural light for an uniform scattering envelope of 10 stellar radius. The gap between two iso-contours equals 65 Intensity Units

 Figure 3: Intensity map in polarized light (with a linear polarizer perpendicular to the baseline) for an uniform scattering envelope of 10 stellar radius. The gap between two iso-contours equals 72 Intensity Units

 Figure 4: Intensity map in polarized light (with a linear polarizer parallel to the baseline) for an uniform scattering envelope of 10 stellar radius. The gap between two iso-contours equals 72 Intensity Unit

## 3.2 Equivalent uniform disks

For various configurations of (, , k) we fit the visibility curves versus spatial frequency from Eq. (11) (hypothesis of uniform disk). We compute the ratio of the polarized angular diameters and the relative variation of angular diameter with polarization . We check that the larger the exponent k, the smaller the angular diameter (Table 1). For when k varies from 0 to 3, increases by 0.051 and by 4.8%. Considering a typical error of 0.1 mas on the angular diameters, we obtain an error of 0.15 - 0.20 on and of 15 - 20% on . Therefore polarized angular diameters do not allow to differentiate k even for a high envelope flux of 30%. For more realistic smaller flux (See Sect. 5.3), the variations of and with k are clearly reduced and we conclude that and can be determined from the natural visibility curve. is obtained at large spatial frequencies (i.e. few tenths meters) where the envelope contribution is negligible. The resulting visibility is F*. (Eq. 11) with . can be determined by considering the envelope as a uniform disk (k=0). However this method can lead to an over-estimation of if k exceeds 2 (over-estimation of about 13% for k = 3 and ).

## 3.3 Degree of polarized visibility PV

The variations of PV versus spatial frequency have the same shapes whatever . They are also very small whatever the envelope radius . On the contrary, these curves are obviously characteristic of the exponent k. Provided that is large enough (i.e. ), we can disentangle all the exponents (0, 0.5, 1, 2, 3) if the accuracy on the visibility equals 0.1% (Fig. 5). An accuracy of 1% on the visibility is enough to put a lower limit to k. The accuracy on k is about 0.5 for and about 0.25 for .

 Figure 5: Degree of polarized visibility versus spatial frequency for an envelope of 10 stellar radii responsible for 15% of the total flux and for various exponent k of the electron distribution

Up: Can interfero-polarimetry constrain extended

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