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Astron. Astrophys. 357, 255-267 (2000)

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3. Reductions and results

3.1. Basic reduction

The data cubes at each position of the polarizer consists of M [FORMULA] 256[FORMULA]256 pixel frames, where M is given as the number of frames in Table 1. To produce a single image for each of the nine selected polarizer angles, the data were flat fielded using images obtained at the beginning of the night on the twilight sky with an identical set of polarizer angles. The data cubes were used to derive a bad pixel map as described by Ageorges & Walsh (1999) using a sky variation method. The sky from the offset position was subtracted separately from each of the M frames before combination into a single image for each polarizer angle. These reductions were performed with the dedicated adaptive optics reduction package `eclipse' (Devillard 1997). The data frames at position angle 157.5o were not used in any computations of polarization on account of the discrepancy noted by Ageorges & Walsh (1999). The reduced data then consisted of 2 sets of K band image pairs; 2 sets of H band image pairs; and 1 set of J band images; all with [FORMULA] Car displaced from the centre of the detector and 1 set of Kc filter images centred on [FORMULA] Car.

3.2. Registration of images

The rotation of the polarizer induces a small shift of upto 3 pixels in the position of the images (see Ageorges & Walsh 1999, Fig. 2) and coupled with the (intended) shifts of [FORMULA] Car across the SHARP II field, it is necessary to carefully align all images to a common centre in order to calculate precise colour or polarization maps for the whole of the Homunculus. On the J, H and K frames, the image of [FORMULA] Car was saturated (overflow of A-to-D converter). The centroids of all the J, H and K images were determined using a large radius (30 pixels = 1.5"); it was found that with such a large radius the centroid was not sensitive to the saturated core (typically a few pixel radius). Combined images of the coverage of the whole nebula, at each position of the polarizer, were formed by shifting, and rotating by 90o, each image pair (e.g. JA and JB which were taken consecutively) to a common centre and averaging the pixels in common. The rotation was required to produce astronomical orientation. Shifts were restricted to integer pixels, thus the alignment can have a maximum error of 0.050". The alignment procedure was carefully checked by examining the coincidence of features in the nebula and in the core when saturation was not severe (e.g. in the J band images in particular). Where there were repeats of the full combined image (Column 6 of Table 1), the two sets of images were averaged. The result was an image of dimensions 326[FORMULA]326 pixels (16.3[FORMULA]16.3") with no data in the top left and lower right corners. Fig. 1 shows the J, H, K and Kc total flux images (i.e. Stokes I) on a logarithmic scale. All images have identical scale and orientation. In the J, H and K images the saturation of the central source is indicated by the zero flagged pixels (region of radius 4 pixels about position of peak). There are a variety of artifacts: diffraction spikes along the principal axes caused by the secondary support; low level changes consequent on merging images (the overlap regions were used to scale the image pairs); a doughnut shaped feature caused by a hot pixel cluster which occurs at equal declination values at the extremity of the NW and SE lobes and near the rim of the NW lobe. These features, which are most apparent on the colour maps (Fig. 2), have not been masked out but are obviously not interpreted.

[FIGURE] Fig. 1. The final combined J (upper left), H (upper right), K (lower left) and Kc (lower right) total intensity (Stokes I) images of the Homunculus nebula are shown in logarithmic plots. The dynamic range in the plots is about 104 and north is at the top; east to the left. The 4 pixel radius region about the position of [FORMULA] Carinae containing saturated values has been set to zero in the J, H and K maps. The axes display the offset positions from [FORMULA] Carinae.

[FIGURE] Fig. 2. The logarithmic colour maps of J/H (left) and H/K (right) are shown for the Homunculus nebula. The plotted range is 2.5 mag. for J-H and and 1.5 for H-K. Masks have been applied to suppress the outer low signal level and the saturated core. The size and orientation are identical to Fig. 1.

3.3. Colour maps

`Colour' maps were made by ratioing the J, H and K images. A cut-off in the form of a mask was applied to each colour map in order to prevent division by small numbers and produces the sharp bulbous edges in the maps. Fig. 2 shows the J/H and H/K images on a logarithmic scale. On account of saturation the values over the core do not hold any colour information and have been set to zero. The range of valid ratio values are: (J-H) 0.02-2.5 mag.; (H-K) 0.02-1.5 mag.

3.4. Polarization maps

The linear polarization and position angle were calculated for the combined maps by fitting a cosine 2[FORMULA] curve to values at each point as a function of polarizer angle as fully described in Ageorges & Walsh (1999). The discrepant point at PA 157.5 was not included in these fits (Ageorges & Walsh 1999). The input images were binned to improve the signal-to-noise in the polarization determination at the expense of spatial resolution. In addition a cut-off in polarization signal-to-noise (i.e. [FORMULA]) was applied to exclude points with large errors, such as at the edges of the Homunculus. Fig. 3 shows the J, H and Kc band polarization vector maps superposed on logarithmic intensity contour maps to be directly compared to the images in Fig. 1. The data were binned into 4[FORMULA]4 pixels (0.2[FORMULA]0.2") before calculating the polarization; the polarization cut-off was set at errors of 2% for the J and Kc maps and 1.7% for the H band map.

[FIGURE] Fig. 3. The polarization vector maps of the combined J (upper left), H (upper right) and Kc (lower left) images of the Homunculus nebula are shown together with logarithmic contour of the total counts. The orientation is as Fig. 1 and the size of the polarization vectors are indicated in %.

3.5. Restoration of 2.15 µm images

As described in Ageorges & Walsh (1999) the Kc images were restored using the blind deconvolution algorithm IDAC (Jeffries & Christou 1993) to determine the PSF of the images. Only the central 128[FORMULA]128 pixel area was restored to save computer time and only the images at polarizer angles of 0, 45, 90 and 135o were employed. Fig. 4 shows the logarithmic total intensity map over the central 1[FORMULA]1" area resulting from restoring the four reduced images using the Richardson-Lucy algorithm (Lucy 1974) with the IDAC PSF; the resulting image was reconvolved with a Gaussian of 3 pixel FWHM (0.15") since this is about the expected diffraction limited resolution at this wavelength.

[FIGURE] Fig. 4. A logarithmic restored Kc image of the 1[FORMULA]1" region centred on [FORMULA] Carinae is shown in the upper plot. The image was restored with the Richardson-Lucy algorithm using the PSF from the IDAC deconvolution and the result was convolved with a Gaussian of 3 pixels width. The lower image shows the polarization vector map of the same region superimposed on a logarithmic contour map of the total count. The polarization vectors were calculated for 2[FORMULA]2 pixels binned. The positions of the Hofmann & Weigelt (1988) knots B, C and D are indicated on the vector map.

A polarization map was calculated from the four restored 1[FORMULA]1" images and is shown in Fig. 4 for direct comparison with the logarithmic image. The data were binned 2[FORMULA]2 pixels before calculating polarization and the polarization cut-off error was 4%. An attempt was made to calculate the polarization at the positions of the speckle knots, discovered by Weigelt & Ebersberger (1986) and Hofmann & Weigelt (1988), whose positions are shown in Fig. 4. Knot A is the central (assumed) point source whilst knots B, C and D are to the NW at offsets of 0.114 (B), 0.177 (C) and 0.211" (D); these offsets correspond to only 2.3, 3.5 and 4.2 pixels in the images. In the restored images no distinct knots could be discerned at these positions but it is clear from Fig. 4 that there is an apron of IR radiation in the NW direction strongly hinting on an area of elevated brightness in the vicinity of these knots.

Aperture photometry of the Weigelt et al. knots in a 2[FORMULA]2 pixel area was performed for the three sets of images - restored with IDAC, Richardson-Lucy restored with the IDAC PSF and Richardson-Lucy restored. All the restorations were convolved with a Gaussian of 3 pixel FWHM. For the three images the aperture polarization determinations showed that knot B could not be distinguished from knot A (identical polarization within errors). Knot C showed very differing results depending on the method (it lies on a diffraction spike); only for knot D could a fairly consistent value of polarization be determined. From the three methods a mean polarization of 18 [FORMULA] 7% and a position angle of 17 [FORMULA] 14o was derived for knot D. Given the position angle of knot D from knot A of 336o (Hofmann & Weigelt 1988), a position angle of the polarization vector of about 60o is expected. To reconcile this discrepancy, it is suggested that knot D may not be directly illuminated by [FORMULA] Carinae, i.e. there is multiple scattering within this core region which would not be too surprizing given the high (gas) densities (Davidson et al. 1997). The mean total intensity ratio knot A/knot D was 10:1, to be compared with the value of about 12:1 given by Hofmann & Weigelt (1988) for a wavelength of [FORMULA]8500 Å. It is justified to attempt polarimetry at these positions since the images of Morse et al. (1998, Fig. 5) and Ebbets et al. (1994) show no obvious indication that the knots have substantial proper motion. This supposition is partly supported by the low radial velocities measured by Davidson et al. (1997) who classify these knots as `compact slow' ejecta from [FORMULA] Car.

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Online publication: May 3, 2000