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5 Discussion

5.1 Hypotheses of our formalism

Since we wanted to know if interfero-polarimetry 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$^\circ$ 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).

5.2 Required accuracies

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 $\mu$as for a spherical envelope and of few tenths $\mu$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 $0.1\, -\, 0.15$ 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 beam-combiner (Perrin 1996) thanks to a real-time photometric calibration explained by Roddier & Roddier (1976).

5.3 Astrophysical limitations

Our study shows that interfero-polarimetry 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 B6-B9 stars and even for A0 stars, such contributions are realistic whereas for O stars larger contributions in the visible are expected. Interfero-polarimetry 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 signal-to-noise 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 ($\approx 0.85$), the SNR in the photon limited case is given by (Percheron 1991 - See also Roddier & Léna 1984 for the general expression):
SNR=\frac{V}{2}\sqrt{N_{\rm i}}\sqrt{N_{\rm s}}\sqrt{\frac{\langle {N_{\rm ph}}\rangle}{2}}\end{displaymath} (13)
where V is the recorded visibility, $N_{\rm i}$ the number of recorded images, $N_{\rm s}$ the number of speckles and $\langle {N_{\rm ph}}\rangle$ the average number of photo-events detected per elementary exposure and per speckle.

Now $\langle {N_{\rm ph}}\rangle$ is related to the visual magnitude Mv by (Roddier & Léna 1984):
\langle {N_{\rm ph}}\rangle ={T_{\rm interf}}\ \tau \sigma \Delta\lambda\ 10^{8-0.4M_V}\end{displaymath} (14)
with ${T_{\rm interf}}$ the instrumental throughput (detector and polarizer included), $\tau$ the exposure time in seconds, $\sigma$ the coherence area in m2 and $\Delta\lambda$ the optical bandwidth in nm.

For the GI2T/REGAIN we shall assume that the visibility losses due to instrumental polarization attain 5% (Rousselet-Perraut 1996a), $\sigma =5\ 10^{-3}$ m2, $\tau=20$ ms and ${T_{\rm interf}}=0.5\%$. We take $\Delta\lambda = 5$ nm and an integration time of 15 min, which leads to $N_{\rm i} = 45000$. The limiting visual magnitudes in mono- and multi-speckle modes are given in Table 5. For adaptive optics modes, $N_{\rm s}= 1$ and $\sigma$ is replaced by $s\pi R^2$ (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 interfero-polarimetric technique.

Table 5: Limiting visual magnitudes for various accuracies on the visibility dV and various configurations of mono- and multi-speckle mode

\hline\\ d$V$\space & SNR & $N_{\rm s}$\space & Coheren...
 ...1$~m$^2$\space & 4.0 & Adaptive optics mode (case of VLTI)\\ \hline\end{tabular}

5.4 Instrumental device

Our first observing run done with the GI2T Interferometer (Rousselet-Perraut et al. 1997a) on the Be star $\gamma$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 (Rousselet-Perraut 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 (Rousselet-Perraut et al. 1996b).
\includegraphics[width=8.5cm]{6394f8.eps}\end{figure} Figure 8: Optical design of the polarimetric focal instrumentation of the GI2T/REGAIN Interferometer. The Wollaston prism can set two fixed positions (0$^\circ$ and 180$^\circ$). The wave-quarter plates are oriented at 45$^\circ$ 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

5.5 Observational requirements

Our study enables us to define the main steps of an interfero-polarimetric 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 $F_{\rm env}$ (See Sects. 3.2 and Stee et al. (1995) for the example of the Be star $\gamma$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 Earth-rotation synthesis: when H varies, the orientation of the baseline projected on the sky varies and the spatial frequency too. Thus we obtain a 2-D information about the envelope geometry (angular dimensions and flattening $\varepsilon$ if any). Such High Angular Resolution observations lead to accurate flattening determinations (an accuracy of $\pm 0.05$ 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 (${V_{\rm nat}}$ without the polarimeter, V// and ${V_{\perp}}$ 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% (Rousselet-Perraut 1997c).

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