Appendix A: Image alignment

   
A.1 Flux errors from misalignment

Consider two images which are not registered correctly and which have slightly different PSF widths. When performing photometry on these images, we assume that they are registered perfectly and have identical, known beam sizes. This amounts to making a flux measurement in a certain aperture in one image, but in a slightly offset aperture of slightly different size in the second image. The error is largest in the steepest gradients in the image, which are the flanks of the PSF of width $\sigma$, smoothed to the desired effective beam size $\sigma_{\rm eff}$, with $\sigma_{\rm eff}^2 = \sigma^2 + \sigma_{\rm smooth}^2$. Its magnitude can be assessed by considering the PSF as a Gaussian at given position and of given width, and as smoothing filter a second Gaussian slightly displaced from the PSF and of width slightly different from that achieving the desired effective beam area. The result of the performed wrong flux measurement is then proportional to the integral

$$\int\int_{-\infty}^{\infty} \exp \left(-\frac{x^2 + y^2}{2 \... ...{\rm smooth}} + \delta \sigma)^2} \right) {\rm d}x {\rm d}y,$$

where $\delta x$ is the offset of the aperture from the correct position and $\delta \sigma$ is the error in the determination of the PSF width. The correct measurement is obtained by setting $\delta x=0$ and $\delta \sigma=0$, and from this one obtains the fractional flux error as a function of the two errors. The fractional flux error $\Delta f$ for a misalignment is

  

$$% \Delta f 1 - {\rm e}^{\frac{\delta x^2}{2\sigma_{\rm smooth}^2}}$$ =

$$\approx \frac{\delta x^2}{2\sigma_{\rm smooth}^2} {\rm\ if\ } \frac{\delta x^2}{2\sigma_{\rm smooth}^2} \ll 1$$ (A.1)

$$\Leftrightarrow \frac{\delta x}{{FWHM_{\rm eff}}} = \sqrt{\frac{\Delta f}{4 \ln(2)}} \approx 0.6 \sqrt{\Delta f}.$$ (A.2)

Hence, the relative flux error is better than 5% if the misalignment $\delta x < 10\%$. Similarly, for a wrong PSF width, the fractional error is

$$% \Delta f = \frac{2\delta\sigma}{\sigma_{\rm smooth}} \frac... ...{\rm eff}} {\rm\ if\ } \delta\sigma \ll \sigma_{\rm smooth}.$$ (A.3)

This error is negligible if the desired effective PSF is much larger than the intrinsic PSF of the input images, as is the case here.

   
A.2 Refined image alignment

Figure A.1: The separation of a number of stars was measured both on a F336W (UV) and a F673W (red) image of R136. This plot shows the difference in separation between the frames, $\Delta S$, plotted against the separation on the "red'' image, S. The best-fitting straight line is shown. $\Delta S$ grows systematically with S, indicating a differing scale between the frames. The slope of the line is $(1.14\pm .04)\times 10^{-3}$, the intercept with the ordinate is $(0.3 \pm 8)\times 10^{-2}$. The slope is not changed within the errors by forcing the line to pass through the origin.

The two obvious ways to determine the relative shift between any two images are measuring the positions of point sources on the various images, or using the engineering files (also termed jitter files) provided as part of the observing data package. If the telescope pointing was known to be precise to better than 0 $.\!\!^{\prime\prime}$02, we could simply rely on the commanded shifts. The HST has a pointing repeatability within a single telescope visit of about 5 milli-arcsecond (mas). The offsets between individual exposures are accurate to 15 mas, leading to a total error of about 16 mas. This is already comparable to the demanded accuracy. The situation is expected to be worse for relating exposures in different visits, when the telescope has been pointing elsewhere in the mean time. Unfortunately, measuring the positions of only a few (four, in our case) astrometric reference stars does not immediately lead to accurate measurements of the telescope's pointing - especially since in our case, each star lies on a different chip. The undersampling of the telescope PSF by the WFPC2 pixels and the so-called sub-pixel scattering of the WFPC2 detectors lead to an additional scatter of the centroid positions of point sources, approximately uniformly distributed between +0.25 and -0.25 pixels and in excess of statistical uncertainties (Lallo 1998, priv. comm.). With only a small number of centroidable point sources available, the centroiding errors are of the same magnitude as the intrinsic pointing errors of the telescope. It is therefore worth considering the pointing error sources en détail to ensure that the alignment is at the required 0 $.\!\!^{\prime\prime}$02 level. Alignment errors can be caused by roll or pointing errors and less obviously by a scale difference between exposures using different filters in the same camera. The importance of the various alignment error sources can be estimated by considering the effect they have on the hot spot location if the quasar images are assumed to coincide. The hot spot is separated by 22 $^{\prime \prime }$ from the quasar, corresponding to about 480 PC pixels. The roll repeatability of the HST is at the 10 $^{\prime \prime }$ level. The engineering data provided with HST exposures record roll angle differences of about 6 $^{\prime \prime }$. This is well below the rotation of 3 $.\mkern-4mu^\prime$5 which would produce a 0.44 pixel difference over 480 pixels. The telescope roll differences can thus be neglected. The engineering files record the telescope pointing in three-second intervals and can be used to calculate the offsets. Their accuracy is only limited by the so-called "jitter'', vibrations due to thermal effects. The jitter was below 10 mas in all exposures, and below 5 mas in most. There is an additional uncertainty from the transformation between the telescope's focal plane and the detector: the location of a camera inside the telescope may change slightly over time (shifts, rotations, or both). This uncertainty is irrelevant for relative positions as long as the location and orientation of WFPC2 and the Fine Guidance Sensors (FGS, these perform the guiding observations) inside the telescope is stable, which is the case for the employed shifts of about 1 $^{\prime \prime }$ and for the timescales between the visits. Within a single visit, the engineering file information is used to obtain relative offsets, with a typical 5 mas error. The values differ from the commanded shifts by a few milli-arcseconds at most. All of these offsets are by an integer number of PC pixels. Between the various visits, the telescope has been pointing to a different part of the sky. One should therefore not assume that the relative shifts between various visits as determined from the jitter files are as accurate as shifts within one visit. We therefore measured the positions of the four astrometric reference stars on each of the short exposures (in fact, one of the stars is very faint in the UV, so the position was determined on a long exposure for this one). The shifts determined from the four point sources' positions have a typical standard deviation (accuracy) of 15 mas in each coordinate. On the three visits' sum images aligned this way, the scatter of the quasar image position is less than 5 mas and 7 mas in x and y, respectively. This means that although the measured shifts have a fairly large scatter, the resulting value is precise to about 10 mas. Finally, we note that the observed shifts of the stellar positions and those obtained from the jitter files agree to better than 0 $.\!\!^{\prime\prime}$02 in all cases, with an rms value of 0 $.\!\!^{\prime\prime}$01. There are, however, systematic differences between these and the commanded values. Hence, we do not blindly rely on the latter. Because of differential refraction in the MgF₂ field flattener windows employed in WFPC2 (Trauger et al. 1995), the pixel scales of images taken through the F622W and F300W filters differ by about 0.1%. This alone is enough to eat up the alignment error budget of 0.44 pixels over 480 pixels separation. The wavelength dependence was expected from ray-tracing studies of the WFPC2 optics. Its presence and magnitude were experimentally confirmed by comparing archival images of the star cluster R136 taken through similar filters as those employed in the present work (F336W and F672N) (Fig. A.1). The scale difference has to be removed before combining the images to a spectral index map. This was done in the following manner: the plate scale of each image was calculated using the parameters in Trauger et al. (1995). All images were resampled to a grid with pixel size of 0 $.\!\!^{\prime\prime}$0045548, which is one tenth of the average of the original scales, using bilinear interpolation. The result was then binned in blocks of $10\times 10$ pixels to a common pixel size of 0 $.\!\!^{\prime\prime}$045548. The scale of the two images is then identical to better than 1 part in 10000. The resampling required the use of the quasar image as common reference point between the two filters. The QSO's position can be determined to about 10-15 mas by centroiding routines on the unsaturated images.