5 Calibration of the polarimetric data

  In order to determine the source intrinsic polarization and its position angle, several corrections are necessary. The instrument possesses an instrumental polarization which must be vectorially subtracted from the measured polarization. The instrumental contribution is derived from the observation of unpolarized standards. The interstellar medium between the source and the observer also possesses an intrinsic polarization which needs to be corrected. The typical ISM polarization values are $\le$2% and can be neglected when observing high polarization sources. If the ISM polarization is not negligible, then it must be determined from measurements of stars in the neighbourhood of the source (see e.g. Vrba et al. 1976); alternatively the distance dependence of the ISM polarization must be determined from measurements of many stars. The zero point of the polarization position angle is checked by observing non variable polarized standards, or polarized sources with reliable measurements. The lattest offer an excellent check on the polarizing efficiency of the instrument (i.e. response to a 100% polarized source should be 100%).

  

Figure 6: J, H and K polarization maps of R Monocerotis and the brightest regions of the reflection nebulae NGC 2261. The contour maps shows the logarithm of the signal (polarized + unpolarized). The size of the polarization vectors is indicated and the orientation is north to the top and east left

5.1 Sky polarization

 In the optical during dark time the sky polarization is typically 3-4% (Scarrott, private communication). In the nights of our measurements, the sky polarization has been found to be consistent with zero within the error bars (typically $\leq$0.5%). Since it is the ratio of polarized intensity between the source and the sky that matters most and since the latter have carefully been subtracted (see Sect. 4.2), the sky contribution has been ignored in processing the data.

5.2 Instrumental calibration

 

5.2.1 Choice of the polarization calibrators

  Despite extensive polarization observations, there is a distinct lack of any such standards in the IR. The polarized reflection nebulae OH 0739-41 and NGC 2261 (illuminated by R Monocerotis) were observed because of their extensive IR polarization data (Heckert & Zeilik 1983 and Shure et al. 1995, for OH 0739-14, Minchin et al. 1991 and Close et al. 1997 for R Mon), although neither can be claimed as true, non-varying standards.

Since the observations are achieved using an adaptive optics system, the polarization standard could also be used as a PSF calibrator. Since the correction is optimized continuously, the resulting PSF is variable in time. Any point source observed as PSF calibrator needs to be close (< 10$^\circ$) to the target and be as similar as possible in terms of visible magnitude and spectral type, to ensure identical correction efficiency. Owing to the lack of polarization standards in the infrared, the polarization calibrators were chosen to be as close as possible to the source and bright enough to be used as reference for the wavefront sensor. In two cases, for HD 64299 and HD 95410, which have, respectively a B polarization of 0.151% (Turnshek et al. 1990) and a V polarization of 0.004% (Tinbergen 1979) it was assumed that the IR polarization is negligible, although no measurements exist at these wavelengths.

In reducing the data taken on 1996 March 02 it was found that the derived polarization for any source (even OH 0739-14) was consistent with zero polarization, and, in addition, did not exhibit the expected shift of image centroid with polarizer angle (Fig. 5). Either the photometric conditions were exceptionally poor (this is not borne out by large discrepancies between the 0 and 180$^\circ$ signal values - see Table 1) or, more probably, an instrumental problem, such as the polarizer not rotating to the requested angle, was present. The polarization information was therefore discarded for this night. However the K band image of $\eta$ Carinae had excellent spatial resolution and was retained (Walsh & Ageorges 1999).

5.2.2 ADONIS instrumental polarization

  For the unpolarized (actually low polarization) standards, the integrated counts within a circular aperture including all the flux from the star profile (radius typically 2'') above the sky background was measured for each angle of the polarizer and a $\cos(2 \theta)$ curve fitted to the data. Table 2 lists the results. HD 93737 has a measured V band polarization of 1.07% at position angle 122.4$^\circ$(Mathewson & Ford 1970). Given the typical shape of the interstellar extinction curve (the "Serkowski law'', see e.g. Whittet 1993), the probable values of the interstellar polarization for this star, assumed to have a typical Galactic interstellar extinction, are 0.5, 0.3 and 0.2% at J, H and K respectively. The position angle is usually similar between the visible and IR (see e.g. Whittet et al. 1994). For the purposes of computing the instrumental polarization it was assumed that the polarization was zero. The first two sets of data on HD 93737 on 1996 Mar. 02 (see Table 1) are not included on account of the problem with the data on that first night (see Sect. 5.2). In addition the first sequence of H band data on HD 93737 had poor photometry (see Table 1) and was not considered. There is a spread in the values indicating typical errors of $\pm$0.3% in linear polarization and $\pm$15$^\circ$ in position angle. Given the errors the J, H and K values are consistent with an instrumental polarization of 1.7%. Adopted values are listed in the last row of the Table 2. Given that only a single measurement was performed at $K_{\rm c}$,it is probably not significant that the instrumental polarization in this band is higher and that the position angle differs from the K band measurement.

  

Table 1: List of polarization sources observed. The exposure time ($T_{\rm exp}$) is given per frame

  

Table 2: Polarization of low polarization stars - instrumental polarization measurement

Once the instrumental polarization (intensity and angle) is determined this correction can be applied to the polarization maps point-by-point. Goodrich (1986, in Appendix) describes the application of the instrumental correction.

5.2.3 Position angle calibration

 

On producing polarization maps for the Homunculus nebula around $\eta$ Carinae, it was noticed that the polarization vectors did not point back to the position of $\eta$ Carinae. There is no reason for such a behaviour since it is known to be a reflection nebula. If the illumination were by an extended source then the offset should not be one of simple rotation. A novel method was used to determine the single offset required to align all the polarization vectors in a centrosymmetric pattern around the position of $\eta$ Carinae. A least squares problem was solved to minimize the impact parameter at the position of $\eta$ Carinae produced by the perpendiculars to all the polarization vectors in the Homunculus by application of a single rotation. A consistent value of 18 $\pm$ 1$^\circ$ was found for the J, H and K images. In order to verify that this was not an artifact of the $\eta$ Carinae nebula and the fact that the central point source was saturated, the 18$^\circ$ correction was applied to the polarization maps of NGC 2261. It was found that the vectors in the high polarization spur to the NE were well aligned with the direction expected for illumination by the peak of R Mon. Thus the calibration of the absolute position angle can be made without reference to a polarized standard.

  

Table 3: JHK Polarization of R Monocerotis in an 8'' aperture