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
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.
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.
Table 2:
Polarized angular diameters, ratios of polarized angular diameters and relative
variation of angular diameters with polarization for an ellipsoidal envelope whose dimensional
parameters are respectively 10, 10 and 20 stellar radii () and responsible for 30% of the
total flux and for various exponents k of the electron distribution
Table 3:
Polarized angular diameters, ratios of polarized angular diameters and relative
variation of angular diameters with polarization for an ellipsoidal envelope whose dimensional
parameters are respectively 10, 10 and 7.5 stellar radii () and responsible for 30% of the
total flux and for various exponents k of the electron distribution
Table 4:
Polarized angular diameters, ratios of polarized angular diameters and relative
variation of angular diameters with polarization for an uniform
ellipsoidal envelope (k=0)
responsible for 30% of the total flux for various flattenings
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)