3 Data reduction

The large dynamic ranges of the images considerably complicate the data reduction. In particular, the difficulty lies in separating the contribution of the light scattered in the circumstellar medium from the stellar light scattered in the Earth's atmosphere and the telescope. The reduction of the direct images was described in some detail in Paper I. Here we concentrate on the reduction of the polarisation data. Due to time limitations during the observing runs, we observed the objects at only three polarisation angles (0 $\hbox{$^\circ$ }$, 45 $\hbox{$^\circ$ }$, 90 $\hbox{$^\circ$ }$). The images were bias-subtracted, flatfield-corrected, and removed of cosmic ray hits using different IRAF tasks. For each target the exposures taken at a certain polarisation angle were aligned using stars in the FOV, and they were subsequently added to increase the S/N-ratio.

The measured Stokes parameters of the detected light ($I_{\rm m}$, $Q_{\rm m}$, $U_{\rm m}$) contain contributions from four components,

$$\left \{ \begin{array}{l} I_{\rm m}= I_{\rm\ast} + I_{\rm... ...U_{\rm sc} + U_{\rm sh} + U_{\rm bg}, \end{array} \right .$$ (1)

where ($I_{\ast}$, $Q_{\ast}$, $U_{\ast}$) are due to the direct stellar light, ( $I_{\rm sc}$, $Q_{\rm sc}$, $U_{\rm sc}$) are due to stellar light scattered in the interstellar medium, the Earth's atmosphere, and the telescope, ( $I_{\rm sh}$, $Q_{\rm sh}$, $U_{\rm sh}$) are the intrinsic values of the stellar light scattered by the circumstellar medium, and ( $I_{\rm bg}$, $Q_{\rm bg}$, $U_{\rm bg}$) are due to the sky background.

As we do not have polarimetric information on the target stars themselves, we have to assume that the photospheric stellar light is essentially unpolarised. The BVRI polarimetry study of a sample of ten carbon stars of different variability types done by Raveendran (1991) shows that the spatially unresolved light from these stars (and their envelopes) is only weakly polarised ($\lesssim$1%). This value sets a very low upper limit to the intrinsic polarisation of the stellar radiation, i.e., $Q_{\ast} \approx0$ and $U_{\ast} \approx0$.

The high galactic latitudes of R Scl and U Ant, together with them being relatively nearby, suggest that only negligible polarisation is introduced by the interstellar medium. It can be shown that the polarisation of the scattered stellar light in the Earth's atmosphere close to the star is equal to the intrinsic polarisation of the stellar light independent of the scattering mechanism considered, since the scattering takes place primarily through small angles (Le Borgne et al. 1986). Likewise, we expect the polarised flux due to stellar light scattered in the telescope to be very small. Therefore we can assume that $Q_{\rm sc} \approx0$ and $U_{\rm sc} \approx0$.

The observations were done in the absence of moonlight. Therefore, the sky background can be disregarded as a source of polarised light, i.e., we set $Q_{\rm bg}=0$ and $U_{\rm bg}=0$.

The fact that we could not observe polarimetric standards in any of the runs prevented us from having good estimates of the polarisation introduced by the telescopes and the optics. However, there is no evidence of such polarised light, in the regions where we detect scattered light towards our target stars, in the images of the template stars. This strongly supports our conclusion that polarised flux introduced by the telescope and the optics can be neglected, as well as the presence of a strong, polarised background flux.

Finally, the use of a coronograph eliminates the direct stellar light, i.e., $I_{\ast}=0$. Thus, we have

$$\left \{ \begin{array}{l} I_{\rm sh}= I_{\rm m} - I_{\rm ... ...Q_{\rm m} \\ U_{\rm sh}= U_{\rm m}, \end{array} \right .$$ (2)

where $I_{\rm bg}$ accounts for the sky background light, which must be subtracted during the reduction process.

Figure 1: Images showing the polarimetric information in the F77 filter of the light scattered in the circumstellar medium around R Scl. Upper left panel: measured normalized Stokes $q_{\rm m}$. Upper middle panel: measured normalized Stokes $u_{\rm m}$. Lower left panel: measured polarisation degree $p_{\rm m}$. Lower middle panel: brightness distribution of the scattered light $I_{\rm sh}$. Upper right panel: vector map showing the shell polarised intensity ( $P_{\rm sh}$) and polarisation angle ( $\theta _{\rm sh}$) averaged over square-boxes of 1 $\hbox{$.\!\!^{\prime\prime}$ }$3 (the total intensity image is shown as a grey contour). Lower right panel: AARP of the polarised intensity ( $P_{\rm sh}$; dash-dot line), total intensity ( $I_{\rm sh}$; solid line) and polarisation degree ( $p_{\rm sh}$; asterisks) of the scattered light. A fit of a step function, convolved with the seeing Gaussian, to the total intensity has been added (dotted line). The AARP of the CO( $J=3\rightarrow 2$) radio emission seen towards this star (Olofsson et al. 1996) is included for comparison (triangles). The CO peak value, which is reached inside the region probed by these observations, has been normalized to the plateau value of the fit to the total scattered intensity.

The polarised flux is expressed then as

$$P_{\rm m}= P_{\rm sh}= \left(Q_{\rm m}^2 + U_{\rm m}^2\right)^{1/2}.$$ (3)

We introduce the measured normalized Stokes parameters to emphasize the polarised scattered light. They, along with the measured polarisation degree, are given by

$$\left \{ \begin{array}{l} q_{\rm m}= Q_{\rm m}/I_{\rm m},... ... + U_{\rm m}^2\right)^{1/2}/I_{\rm m}, \end{array} \right .$$ (4)

and the intrinsic polarisation degree of the light scattered in the circumstellar medium is given by

$$p_{\rm sh}= \frac{\left(Q_{\rm sh}^2 + U_{\rm sh}^2\right)^... ...rm m}^2\right)^{1/2}}{I_{\rm m}-I_{\rm sc} - I_{\rm bg}}\cdot$$ (5)

The measured polarisation degree ($p_{\rm m}$) only provides a lower limit for the polarisation of the stellar light scattered in the circumstellar medium. The intrinsic polarisation degree ( $p_{\rm sh}$) gives the actual value for the polarisation but it must be cautioned that it relies on the more or less uncertain subtraction of the stellar PSF (see below). The polarisation angle is obtained from

$$\theta_{\rm sh}= \frac{1}{2} \arctan \left( \frac{U_{\rm m}}{Q_{\rm m}} \right)\cdot$$ (6)

In the case of observations taken at three polarisation angles, the measured Stokes parameters are derived from the set of images in the following way,

$$\left \{ \begin{array}{l} I_{\rm m}= {\rm Frame} (0\hbox{... ... {\rm Frame} (90\hbox{$^\circ$ }) . \end{array} \right .$$ (7)

The most arduous task in the data reduction, and probably the one introducing the largest uncertainties, is to obtain $I_{\rm sh}$. Apart from subtracting the sky background component ( $I_{\rm bg}$), it requires the subtraction of the direct stellar light scattered in the Earth's atmosphere ( $I_{\rm sc}$) using the PSF of the observed template star. This procedure is not straightforward, and it is discussed at some length in Paper I. We estimate that the location of the scattered emission, in particular the location of the shells, is not affected by the PSF subtraction. The unreduced data already allow to pinpoint the positions of the shells, confirming that the PSF subtraction procedure does not produce any displacement of the observed scattered light. The total intensity of the scattered emission is uncertain by $\approx$15% due to the PSF subtraction (see Fig. 1 in Paper I).

Figure 2: Same as Fig. 1 in the F59 filter.

The template star subtracted images showing the total intensity of the scattered light ( $I_{\rm sh}$) have been edited. Black spots of different sizes are used to blank out the central regions of the images where the mismatch of the target and template PSFs, along with the large intensity gradients, make it impossible to follow any scattered light. For homogeneity reasons, we have done this in the same way for all images shown in this paper. In most observations, the Lyot stop turned out to work inefficiently, yielding highly saturated stripes along two orthogonal directions in the total intensity images. We have lowered the effect of these stripes in the final images of $I_{\rm sh}$ by replacing them with azimuthally-averaged radial profiles with added Poisson-noise. Therefore, the scattered light detected in the various images is not reliable along these two directions.