Up: Can interferopolarimetry constrain extended
Subsections
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)=n_{0}
.R^{*} denotes the stellar radius, n_{0} the electron density
at the stellar surface and k the exponent
of the distribution. For this purpose, we study various interferopolarimetric observables.
For a spherical envelope, the three intensity maps are different
(Figs. 24).
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. 24 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 isocontours 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 isocontours
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 isocontours equals 72
Intensity Unit 
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 overestimation of if k exceeds 2 (overestimation
of about 13% for k = 3 and
).
Table 1:
Polarized angular diameters, ratios of polarized angular diameters and relative
variation of angular diameters with polarization for a spherical envelope of 10 stellar radii
responsible for 30% of the total flux and for various exponents k of the electron distribution

The variations of P_{V} 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 interferopolarimetry constrain extended
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