Up: Can interferopolarimetry constrain extended
Subsections
Since we wanted to know if interferopolarimetry is sensitive enough to detect polarization
effects in an hot envelope, we have considered the most favorable case (maximal polarization
rate) and evaluated the required accuracies (See Sect. 5.2). Thus
we have limited our study to a 90
axial inclination and pure single Thomson scattering by considering optically thin
approximations. In practice, because of high temperatures, hot envelopes are generally near
totally ionized and absorption effects are thus not important. But near total ionization also
implies multiple scattering which affects the polarization rate
(Brown & McLean 1977 and
Wood et al. 1996). So, since our formalism is general, it must be applied in a near future to any
axial inclination and to more realistic source functions (including opacities).
It appears difficult to use polarized angular diameters to determine the electron distribution
inside a circumstellar envelope whatever its shape. In fact it
requires an accuracy on angular
diameters of few as for a spherical envelope and of few tenths as
for an ellipsoidal one
whereas the best accuracy of the MARK III Interferometer
equals about 0.05 mas
(Mozurkewich et al. 1991) and that of the GI2T
Interferometer equals about mas
(Mourard et al. 1997). But, using the degree
of polarized visibility is less constraining since
whatever the shape of the envelope, an accuracy on visibility of 1% is enough to put a lower
limit to the exponent k and an accuracy of 0.1% allows to distinguish the exponents 0, 0.5, 1,
2, 3 if the envelope flux attains 15%. Note that an accuracy of 1% is expected with the
GI2T/REGAIN (Mourard et al. 1994) and an accuracy of 0.2% is obtained on the IOTA
interferometer equipped with the FLUOR beamcombiner (Perrin 1996) thanks to a realtime
photometric calibration explained by Roddier & Roddier (1976).
Our study shows that interferopolarimetry enables to determine matter distribution inside
circumstellar envelopes provided that their flux contributions attain 10  15%. Such
contributions are characteristic of Be stars in the visible spectrum: it has been predicted by
Poeckert & Marlborough (1978) and observed by
Stee et al. (1995) in interferometry. For B6B9
stars and even for A0 stars, such contributions are realistic whereas for O stars larger
contributions in the visible are expected. Interferopolarimetry can therefore be foreseen for
these hot objects. Our technique cannot be extended to weaker flux since it implies a too
severe accuracy on the visibility for the existing or planned instruments.
A signaltonoise ratio (SNR) of 150 is required to reach an accuracy of 1% and a SNR of
1500 is needed to reach an accuracy of 0.1%. For high visibilities (), the
SNR in the
photon limited case is given by (Percheron 1991  See also
Roddier & Léna 1984 for the
general expression):
 
(13) 
where V is the recorded visibility, the number of recorded
images, the number of
speckles and the average number of photoevents detected per elementary exposure
and per speckle.
Now is related to the visual magnitude M_{v} by (Roddier & Léna 1984):
 
(14) 
with the instrumental throughput (detector and
polarizer included), the exposure
time in seconds, the coherence area in m^{2} and
the optical bandwidth in nm.
For the GI2T/REGAIN we shall assume that the visibility losses due to instrumental
polarization attain 5% (RousseletPerraut 1996a),
m^{2}, ms and .
We take nm and an integration time of 15 min,
which leads to . The
limiting visual magnitudes in mono and multispeckle modes are given in Table 5. For
adaptive optics modes, and is replaced
by (Eq. 14) where s is the Strehl ratio
and R the telescope radius (Rousset et al. 1992). For
the GI2T/REGAIN s = 0.1 and R = 0.76 m. For
the VLTI, a Strehl ratio of 0.2 is expected in the visible and R = 0.9 m. All these values
show that a significant number of hot stars can be studied by the interferopolarimetric
technique.
Table 5:
Limiting visual magnitudes for various accuracies on
the visibility dV and various
configurations of mono and multispeckle mode

Our first observing run done with the GI2T
Interferometer (RousseletPerraut
et al. 1997a) on
the Be star Cassiopeiae has shown that: i) the instrumental
polarization calibration is crucial
and ii) the natural and polarized visibilities have to be
simultaneously recorded to calibrate the
temporal variation of the instrumental transfer function. The first
point implies an accurate
modeling and optimization of the instrumental design, especially
in terms of coatings
(RousseletPerraut et al. 1996a) as well as
the calibration of the residual instrumental
polarization. The second point requires the introduction of a
polarimeter inside the classical
focal instrumentation. Such an equipment aims at simultaneously
recording the polarized
visibilities as well as reducing the instrumental bias to achieve
the required accuracies (See Sect. 5.2). For the latter reason,
polarimetric focal equipments separating the polarized
interferograms must be advocated even for classical interferometers
such as the VLTI or the
Optical Very Large Array (OVLA, Labeyrie et al. 1987).
Thus the "classical" High Angular
Resolution observations (i.e. without polarimeter) would
not be biased by instrumental
polarization which dramatically degrades the fringe visibility at the combined focus (Rousselet
Perraut et al. 1996a). The principle of
the GI2T/REGAIN polarimeter is given in Fig. 8
(RousseletPerraut et al. 1996b).

Figure 8:
Optical design of the polarimetric focal instrumentation of the GI2T/REGAIN
Interferometer. The Wollaston prism can set two fixed
positions (0 and 180). The
wavequarter plates are oriented at 45 with respect to the Wollaston prism axes. The first one can
be removed and enables to measure the circularly polarized components. The second one is
fixed and lets optimize the global transmission of the interferometer (the grating of the
following spectrograph greatly polarizes the linearly polarized components). The polarimeter
can be removed for the measurements in natural light 
Our study enables us to define the main steps of an interferopolarimetric
observing run:
i) Observations at large baselines (several tenth meters). For these spatial frequencies, the
envelope contribution is very small and the visibility curve versus spatial frequency provides
the envelope flux (See Sects. 3.2
and Stee et al. (1995) for the example of the Be star
Cassiopeiae).
ii) Observations at short baselines (10  20 meters in the visible). For these spatial frequencies,
the curve of the degree of polarized visibility versus spatial frequency allows to determine the
exponent k of the electron distribution (See Sects. 3.3 and 4.3).
iii) Observations for various hour angles H. Given a baseline, observing at hour angles within
2  3 hours around the meridian transit enables Earthrotation synthesis: when H varies, the
orientation of the baseline projected on the sky varies and the spatial frequency too. Thus we
obtain a 2D information about the envelope geometry (angular
dimensions and flattening if
any). Such High Angular Resolution observations lead to accurate flattening determinations
(an accuracy of has been obtained by Quirrenbach
et al. (1993) with the MARK III
Interferometer).
For each step, "observations" obviously means that we record
the three visibilities (
without the polarimeter, V_{//} and with the polarimeter) for the studied object then for a
standard polarimetric calibrator. Note that an unresolved source can also be used as a
calibrator provided that its polarization degree remains less than 1%
(RousseletPerraut 1997c).
Up: Can interferopolarimetry constrain extended
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