6 Seeing correction

 Despite of the stability discussed above, fluctuations of super-pixel fluxes due to seeing variations are still present. For a star lying in the central pixel (of the $3\times 3$ super-pixel), on average 70% of the star flux enters on average the super-pixel for a Gaussian PSF, but this seeing fraction is correlated with the changing seeing. In this sub-section, we show that this correlation is linear and can be largely corrected for.

6.1 Correlation between flux and seeing

 Depending on their position with respect to the nearest star, super-pixel fluxes can significantly anti-correlate with the seeing if the super-pixel is in the seeing spot, or correlate if instead it lies in the tail of a star.

  

Figure 12: Histogram of the correlation coefficient $\rho$between the super-pixel flux and the seeing (before seeing correction)

  

Figure 13: Seeing correction applied to the flux of one super-pixel light curve anti-correlated with the seeing ($\rho^p =-0.89$). This corresponds to the super-pixel dominated by a centred resolved star, whose position on the CCD frame is labelled "A'' in Fig. 15. The error bars shown here and in the following figures are computed as described in Sect. 7

  

Figure 14: Variation of the seeing fraction of the star flux that enters a $3\times 3$ super-pixel as a function of seeing, for different star positions, given on the figures in pixel unit with respect to the centre of the super-pixel. Panel a) shows cases when the centre of the star lies within the super-pixel (anti-correlation). Panel b) shows the seeing fraction that could contribute from surrounding stars (correlation)

  

Figure 15: Different super-pixels labelled "A'', "B'', "C''. "D'' and "E'' corresponding to the different configurations discussed in the text

A correlation coefficient for each super-pixel p can be defined using the usual formula:  

$$\rho^p = \frac{\sum_n \left(\phi^p_n - \phi \right) \left(S_... ...\left(\phi^p_n - \phi \right)^2 \sum_n \left(S_n - S\right)^2}}$$ (7)

where $\phi$ and S are the mean values of the super-pixel flux $\phi^p_n$ and seeing S_n on night n. In Fig. 12, we show the distributions of correlation coefficients $\rho^p$ in blue (left) and in red (right) for each super-pixel p. These histograms look quite different in both colours but both distributions have a peak around $\rho \simeq -0.8$. This peak, which corresponds to the anti-correlation with seeing near the centre of resolved stars, is expected due to the large number of resolved stars. It is higher in red than in blue, which is consistent with the EROS colour-magnitude diagram where most detected stars have B-R > 0 (Renault 1996). The correlation with seeing expected for star tails ($\rho \gt 0$) is less apparent. However, a clear excess at high values of $\rho$ (around $\rho \simeq 0.6$) appears in red, again consistent with the EROS colour-magnitude diagram. Figure 13 gives an example of such a correlation. The upper left panel of Fig. 13 displays the scatter diagram of one super-pixel flux versus the seeing, corresponding to a correlation coefficient $\rho^p =-0.89$. Despite the intrinsic dispersion of the measurements (which could be large in particular when a temporal variation occurs), a linear relationship is observed. The bottom left panel displays the light curve of this super-pixel.

6.2 Correction

This seeing correction is aimed at eliminating the effect of the seeing variations and to obtain pixel light curves that can be described as the sum of a constant fraction of a centred star flux and the background (see Eq. 1). The variation of the super-pixel flux can be interpreted as a variation of the flux of this centred stars (see Eq. 2). However it is clear that the super-pixel flux contains the flux of several stars and that we are not doing stellar photometry, but rather super-pixel photometry.

The idea is to correct for the behaviour described in Sect. 6.1 using a linear expression:  

$${{\phi}^p_n}\vert_{\rm corrected} = \phi^p_n - \alpha^p \left({S_n - S}\right) ,$$ (8)

where $\alpha^p$ is the estimate of the slope for each super-pixel and ${{\phi}^p_n}\vert_{\rm corrected}$ is the corrected flux. In the following, $\phi^p_n$ will stand for this corrected flux.

Figure 14 shows that the seeing fraction of a given star varies linearly with seeing, and hence justifies this correction.

  

Figure 16: Seeing correction applied to the flux of a super-pixel light curve with no significant correlation with the seeing ($\rho^p =0.02$). The position of the super-pixel on the CCD frame is labelled "B'' in Fig. 15. The stars whose centres lie in the super-pixel are too dim to be resolved

  

Figure 17: Seeing correction applied to the flux of a super-pixel light curve strongly anti-correlated with the seeing ($\rho^p =-0.94$). The position of the super-pixel on the CCD frame is labelled "C'' in Fig. 15. This is a case of blending

  

Figure 18: Seeing correction applied to the flux of a super-pixel light curve strongly correlated with the seeing ($\rho^p =0.96$). The position of the super-pixel on the CCD frame is labelled "D'' in Fig. 15. The super-pixel flux is dominated by tails of surrounding stars. Centred stars are too dim to be resolved

  

Figure 19: Seeing correction applied to the flux of a super-pixel light curve not correlated with the seeing ($\rho^p =0.05$). The position of the super-pixel on the CCD frame is labelled "E'' in Fig. 15. Contributions due to the surrounding stars cancel each other

If several stars contribute to the super-pixel flux, their contribution will add up linearly, because the flux of the background $\phi_{\rm bg}$ (Eq. 1) can be written as:

$$\phi_{\rm bg}=\sum_{i=1}^{N^{\rm other}_{\rm stars}}f_i\phi_i+ \phi^{\rm sky}_{\rm bg},$$ (9)

where i refers to the stars whose fluxes enter the super-pixel, f_i is the seeing fraction of each of these stars, $\phi_i$ their flux, and $\phi^{\rm sky}_{\rm bg}$ the sky background flux that enters the super-pixel. The first term describes the blending and crowding components that can affect the pixel.

Different configurations can occur as shown in Fig. 15, and are discussed in the following. Firstly, if there is no significant contamination by surrounding stars, either (A in Fig. 15) the star flux is large compared to the noise that affects the super-pixel or not (B in Fig. 15). The effect of the seeing correction on super-pixels of type A and B is shown in Fig. 13 and Fig. 16 respectively. Secondly, if there is a significant contamination by surrounding stars, three cases must be considered:

• The centres of the surrounding stars lie in the super-pixel (C in Fig. 15; seeing correction in Fig. 17).
• The flux due to PSF wings of surrounding stars is larger than the contribution of the centred star we are interested in (D in Fig. 15; seeing correction in Fig. 18).
• The flux due to PSF wings is comparable with the centred star and their variation with the seeing cancel each other (E in Fig. 15; seeing correction in Fig. 19).

6.3 Importance of the seeing correction

  

Figure 20: Relative flux stability achieved on super-pixel light curves after seeing correction for all the pixels of CCD 3

  

Figure 21: Importance of the seeing correction. The ratio $\sigma^{\rm A}/\sigma^{\rm B}$ is displayed as a function of the correlation coefficient $\rho$ calculated before the seeing correction is applied, in blue a) and in red b). Data for $20\,000$ super-pixels are used

The correction described above significantly reduces the fluctuations due to seeing variations. Figure 20 displays the relative dispersion computed after this correction. With respect to the histograms presented in Fig. 11, this dispersion is reduced by 20% in blue and 10% in red, achieving a stability of 1.8% in blue and 1.3% in red, respectively 1.6 times the photon noise in blue and 1.9 in red. The improvement on the overall relative stability remains modest, because most light curves do not show a correlation with the seeing and do not need a correction. The importance of the seeing correction as a function of the correlation coefficient $\rho$ can be more precisely quantified. Figure 21 displays for both colours the ratio $\sigma^{\rm A}/\sigma^{\rm B}$, where $\sigma^A$ is the dispersion measured along the super-pixel light curves after the seeing correction, and $\sigma^{\rm B}$ the one measured before the correction, as a function of the initial correlation coefficient $\rho$. It can be shown that, if the slope $\alpha$ defined in Eq. (8) is measured with an error $\Delta \alpha$, then the following correlation is expected:

$$\begin{eqnarray} \left({\frac{\sigma_{\rm A}}{\sigma_{\rm B}}}\right)^2 = {1-\rh... ...}^2 \left(\frac{\sigma_{\rm S}}{\sigma_{\rm B}}\right)^2 \nonumber\end{eqnarray}$$

where ${\sigma_{\rm S}}$ is the dispersion of the seeing. This correlation shows that the stronger the correlation with seeing, the more important the seeing correction is. The dispersion of the measurements can be reduced up to 40% for very correlated light curves. The limitation of this correction comes from the errors $\Delta a$ which explain why most points are slightly above this envelope. When $\vert \rho \vert < 0.15$, most points in fact lie above 1, in which case the "correction" worthens things. Therefore we do not apply the correction to light curves with $\vert \rho \vert < 0.15$.

As the seeing is randomly distributed in time, the above correction will not induce artificial variations that could be mistaken for a microlensing event or a variable star.

One can wonder however what happens to the super-pixel flux when the flux of the contributing star varies. In this case, the slope a of the correlation between the flux and the seeing does change, thus resulting in a lower correlation coefficient. In extreme cases, when the correction coefficient is small ($\vert \rho \vert < 0.15$), the correction is thus not appropriate and not applied.

6.4 Residual systematic effects

 The seeing correction is empirical, and can be sensitive to bad seeing determination due to inhomogeneous seeing across the image or a (slightly) elongated PSF. Part of these problems is certainly due to the atmospheric dispersion, as mentioned by Tomaney & Crotts (1996). This phenomenon correlates with air mass, and affects stars with different colour differently. This is a serious problem for pixel monitoring as we do not know the colours of unresolved stars.

  

Figure 22: Possible systematics. Panel a) displays the variations of the air mass towards the LMC for each individual images. Panel b) shows the small-amplitude variations of the angle of rotation of the PSF measured on the composite images

Figure 22a displays the air mass towards the LMC as a function of time for the images studied (before the averaging procedure), and shows, besides a quite large dispersion of air mass during the night, a slow increase with time. All the measurements have an air mass larger than 1.3, and half of them have an air mass larger than 1.6, producing non negligible atmospheric prism effects because of the large passband of the filters. According to Filippenko (1982), photons at the extreme wavelengths of our filters would spread over 0.73 to 2.75 arcsec in blue depending on the air mass, and over 0.34 to 1.17 arcsec in red.

While the PSF can be well approximated by a Gaussian (residuals $\simeq\!3$%), a more careful study shows that the PSF is elongated with $\langle \sigma_b / \sigma_a \rangle \simeq 0.7$, where $\sigma_b$ and $\sigma_a$ are the dispersions along the minor and major axis of the ellipse. However the fact that the PSF is elongated does not affect the efficacy of the seeing correction: on the one hand, for the central part of the stars, a similar seeing fraction enters the super-pixel for a given seeing value; whereas on the other hand, for pixels dominated by the tails of neighbouring stars, the correlation of the flux with seeing will be slightly different, but the principle remains the same. As the PSF function rotates up to 20$^\circ$ during the period of observation (see Fig. 22b), this could affect the super-pixels whose content is dominated by the tail of one star and could produce spurious variations correlated with the angle of rotation. Fortunately, this rotation is small and we estimate that even in this unfavourable case it cannot produce fluctuations of the super-pixel flux larger than 3%, which can be disturbing when close to bright stars. We expect this will produce the kind of trends that can be observed in the bottom right panel of Figs. 17 and 19. However this cannot mimic any microlensing-like variation.

We reach a level of stability close to photon noise, and this stability can be expressed in terms of detectable changes in magnitude: taking into account a typical seeing fraction f=0.8 for a super-pixel, and assuming a total background characterised by a surface magnitude $\mu_B \simeq 20$ in blue and $\mu_R \simeq 19$ in red, stellar variability will be detected 5 $\sigma$ above the noise if the star magnitude gets brighter than 20 in blue and 19 in red at maximum. With the Pixel Method, our ability to detect a luminosity variation is not hindered by star crowding as we do not require to resolve the star, whereas for star monitoring, the sample of monitored stars is far from complete down to magnitude 20. Although the dispersion measured along the light curves gives a good estimate of the overall stability, we can refine it further and provide an error bar for each super-pixel flux.