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5. Observations and data-analysis

We have seen in Sect. 2.1 that observations for at least two positions of the half-wave plate are required to determine the three unknown parameters namely, the total, intensity (I), fraction of light in linearly polarized condition (p), and position angle of the plane of polarization (tex2html_wrap_inline1257). However, the situation is not so simple in reality because - (i) the responsivity of the system to the two orthogonal polarization components may not be the same, and (ii) the responsivity of the CCD is a function of the position on its surface. Due to these effects the signals which are actually measured in the two images (tex2html_wrap_inline1453 and tex2html_wrap_inline1455) are given by
where tex2html_wrap_inline1457 and tex2html_wrap_inline1459 represent the effects mentioned above. In order to ensure that tex2html_wrap_inline1457 and tex2html_wrap_inline1459 do not change during observations of an object, the positions of the two images are kept fixed on the CCD. Further, as the analyzer is fixed with respect to the detector, the orientation of the ordinary and extraordinary polarizations also remain fixed with respect to it, independent of the polarization vector of the incident beam. Therefore the ratio of the factors tex2html_wrap_inline1459 and tex2html_wrap_inline1457 can be estimated as
by making use of the fact that a rotation of the half-wave plate by 45tex2html_wrap1285 simply leads to an interchange of the signals in the ordinary and extraordinary images. Now the actual ratio of the fluxes in the two images may be recovered as

This ratio is substituted in Eq. (2) and a cosine curve is fitted to the four values of tex2html_wrap_inline1361 obtained so as to make the best estimates of p and tex2html_wrap_inline1257.

Since CCDs exhibit both intrapixel as well as pixel to pixel sensitivity variations, ideally the images should be stable to within a small fraction of a pixel during exposures at different positions of the half-wave plate. But since this is difficult to achieve, it is advisable to dither the telescope pointing atleast by tex2html_wrap_inline1493 pixels along each axis and to get an FWHM of greater than 3 pixels.

Since the grid placed at the telescope focal plane covers almost half the field, several exposures at slightly different telescope orientations might be required to cover the entire object field. Alternatively, in the case of stellar fields with slowly changing (in intensity and polarization) background, the grid may be removed during observations, provided the field is not too crowded.

A polarimetry package has been developed for data analysis within the IRAF environment using a mixture of standard IRAF tasks, custom-made CL scripts and FORTRAN routines. PSF fitting tasks of the DAOPHOT package are used to determine accurately the centroids of the stellar images. The intensity estimates are, however made using aperture photometry covering a diameter greater than 2 FWHM so as to integrate more than 90% of the signal.

In order to keep a check on the errors, it is useful to take multiple exposures for each position of the half-wave plate. To ensure that the normalized Stoke's parameters follow a normal distribution as closely as possible, the signals measured at each position of the half-wave plate, should be averaged together before taking their ratios to give the normalized Stoke's parameters. However, this method has its demerits too. Firstly, it leaves the non-linear tex2html_wrap_inline1495 fitting technique with only one degree of freedom, to estimate the two parameters p and tex2html_wrap_inline1257 from the four values of tex2html_wrap_inline1361. Secondly, it does not allow the estimation of errors in the measurement of the Stoke's parameters, since there are not enough degrees of freedom. Therefore, it appears optimum to use a fitting technique which estimates p and tex2html_wrap_inline1257, from all the values of tex2html_wrap_inline1361 obtained from individual exposures.

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