5 Super-pixel light curves

 So far we have worked with elementary pixel light curves. Pixels which cover $1.21\hbox{$^{\prime\prime}$}\times 1.21\hbox{$^{\prime\prime}$}$ are much smaller than the typical seeing spot and receive on average only 20% of the flux of a star, whose centre lies in the pixel. A significant improvement on the light curves stability can be further achieved by considering super-pixel light curves. Super-pixels are constructed with a running window of $d_{\rm sp} \times d_{\rm sp}$ pixels, keeping as many super-pixels as there are pixels, and their flux is the sum of the $d_{\rm sp}^2$ pixel fluxes. These super-pixels have to be taken large enough to encompass most of the flux of a centred star, but not too large in order to avoid surrounding contaminants and dilution. As such, their size should be optimised for this dense star field given the seeing conditions.

Figure 9 illustrates the different super-pixel sizes that can be considered. The expected signal to noise (S/N) ratio is proportional to the ratio of the seeing fraction to the super-pixel size ($d_{\rm sp}$). Going from $1\times 1$ to $3\times 3$ super-pixels increases the seeing fraction by more than a factor 3. Then increasing the super-pixel size further increases the seeing fraction substantially less than the fluctuations of the sky background. Figure 10 displays the variation with seeing of the signal to noise ratio for different super-pixel sizes.

  

Figure 9: Example of a $19\hbox{$^{\prime\prime}$}\times 29\hbox{$^{\prime\prime}$}$ field on our data in blue, with 3 arcsec seeing. The grey scale gives the intensity in ADU

  

Figure 10: The signal to noise ratio expected for a single star, whose centroid lies in the central pixel, is given as a function of the seeing values for different super-pixel sizes. This assumes a circular Gaussian PSF

  

Figure 11: Flux stability achieved on super-pixel light curves in blue a) and in red b) for all the pixels of CCD 3 (before the seeing correction)

As discussed by Ansari et al. (1997), the alternative that consists in taking the average of the neighbouring pixels weighted with the PSF is not appropriate here, as it amplifies the fluctuations due to seeing variations.

Figure 11 shows the relative dispersion affecting the super-pixel fluxes for CCD 3: we measure in average 2.1% in blue and 1.6% in red, which corresponds to about twice the estimated level of photon noise (1.1% in blue and 0.7% in red). The comparison with Fig. 8 shows that the dispersion is reduced by a factor smaller than $\sqrt{ 9\, {\rm (pixels)}} = 3$ because of the correlation between neighbouring pixels. This stability can be improved even further by correcting for seeing variations.