5 Summary

  This section is restricted, as Sect. 3, to instruments with pixel size small compared to the PSF.

5.1 Planning the observation

If one knows that the target will be piled-up because it exceeds the threshold of Sect. 3.2, the count rate must be maximized (because one is interested in time variations, or in weak spectral features), and there is nothing of interest in the rest of the CCD, then it is better to select a window on the CCD to reduce the frame integration time (directly linked to the window size) down to the level when pile-up is not a problem, and/or to use a spectral filter to cut-off low-energy photons. It is also advised to take strong steps to avoid pile-up if the source is extended and its shape not known.

If, on the other hand, the source is not expected to be bright, then it is not worth reducing the CCD window simply as a precaution. The above methods will allow use of the data at times when the source is bright. It can also be applied if the count rate is too high even in window mode and the guest observer does not want to lose spatial information (as would be the case by using timing mode).

It is also not worth taking special precautions to avoid pile-up if the bright source is not the target, except if it is so close that the target sits on the wings of the bright source.

5.2 Analysing the data

  The first thing to do is to check for the occurrence of pile-up in the data, by systematically checking the local count rate (/frame) of monopixel events (averaged over PSF sized areas, to reduce statistical fluctuations) against the threshold defined in Sect. 3.2, equating the flux loss of Eq. (7) to 1% or, more generally, to the precision required. The average over time must be performed in sky coordinates. If the threshold is exceeded then pile-up must be taken into account.

The second thing to do is to estimate the pattern distribution. One should not use the observed distribution (corrupted by pile-up), but take the energy spectrum S₁(E) of single events only, then deduce the pattern distribution in the data $\overline{\alpha_i}$from that $\alpha_i(E)$ calibrated as a function of energy, according to  

$$\overline{\alpha_i} = \frac{\int_E \: S_1(E) / \alpha_1(E) \... ...\rm d}E} {\int_E \: S_1(E) / \alpha_1(E) \: {\rm d}E}\cdot \\$$ (10)

The third thing to do is to check whether the source is point-like for the instrument. This may be obvious if it is known to be a compact source. If nothing is known, then from Eq. (6) (Fig. 6) it is possible to estimate the incoming flux from the measured count rate in the hypothesis of a point source ($\gamma_1$ being deduced from the $\overline{\alpha_i}$ as in (1)). Comparing then the observed radial profile with the theoretical one for a source at that flux - the integrand of Eq. (6) - it is possible to check the point source hypothesis. If the source is extended then see Sect. 4.5.

Equation (6) may be used over any spatial domain, and can be generalized to split events using the formulae in Appendix A. However it can be used reliably only over the full spectral domain. All detected charge patterns contribute to pile-up, whatever their energy. In particular any spectral selection on events done after the detection process (in the course of the data analysis or even on-board) will not erase its effects. Once the overall input source flux (/frame) is known, and the spectrum is estimated from $S_1(E) / \alpha_1(E)$,a simulation may be carried out to quantify the spectral effects of pile-up. An analytical estimate is also possible. This will be the subject of a future paper.

Acknowledgements

It is a pleasure to acknowledge numerous discussions about pile-up with J.L. Sauvageot and P. Ferrando. I am also indebted to J. Osborne and to the referees M. Popp and L. Strueder, who suggested many ways to make the paper clearer.