*Astron. Astrophys. Suppl. Ser.* **135**, 371-381

**J. Ballet**

DSM/DAPNIA/SAp, Bât. 709, CEA-Saclay,
F-91191 Gif-sur-Yvette, France

e-mail: jballet@Cea.fr

Received February 12; accepted December 4, 1998

This paper presents a statistical and analytical analysis of the pile-up phenomenon on X-ray CCD detectors, whereby two incoming X-rays are counted as one. The probability of measuring configurations involving 1, 2, 3 or 4 pixels is written down for a uniform incoming flux, thus allowing the computation of the flux loss (fraction of photons rejected because of pile-up) and of the pile-up fraction (fraction of events with wrong energy).

For detectors with pixels which are small compared to
the point spread function (PSF)
of the telescope (such as *XMM/EPIC-MOS* or *ASCA/SIS*)
the formulae can be readily
integrated over space to predict and account for pile-up on the total flux
of point sources. It is shown that if only single events are selected
the total pile-up fraction can never get very high for usual PSF shapes.
The main effect is a loss of efficiency which is perfectly quantifiable.
It is concluded that
taking extreme care to avoid pile-up by adapting the instrument setting
(for example restricting the useful area of the CCD) may not be so
important after all, although it remains necessary if it is important
to collect as many photons as possible from the source.

An extension to the theory is proposed for detectors
with pixels comparable to the PSF width (such as *AXAF/ACIS*), and for
extended sources.

**Key words: **methods: analytical --
methods: statistical --
X-rays: general

- 1 Introduction
- 2 Patterns and local pile-up probability
- 3 Pile-up for point sources
- 4 Complications
- 5 Summary
- Appendix A: Pile-up probability for split events
- Appendix B: Asymptotic expressions for measured count rates
- Appendix C: Example of application to real data
- References

Copyright The European Southern Observatory (ESO)