If the half width at half maximum (HWHM) of the PSF is safely larger
than the pixel size (as for *XMM/EPIC-MOS* for instance),
then it is possible to use Eqs. (2-3)
locally, and estimate pile-up at any point in the PSF.
Note that the formulae for monopixels, which are the most important
as is shown next, depend on the incoming X-ray
flux at distance pixel at most.
For *XMM/EPIC-MOS* it is expected that the HWHM will be about 4 pixels.
Therefore for a source centred on a pixel, the flux (per pixel)
in the eight neighbouring
pixels will be about 10% lower than that in the central pixel,
so that formulae (1-3)
will overestimate pile-up there.
But the central pixel receives no more than 1% of the total source flux,
and for other pixels the problem is much less severe.
The opposite case of pixels comparable to the HWHM of the PSF
is discussed in Sect. 4.4.

Figure 4:
Expected radial distribution of the monopixel count rate
(solid line), as opposed to the distribution one would get
in the absence of pile-up (dashed line)
and that of clean (not piled-up) monopixels (dotted line).
The point spread function (PSF) follows a King profile
(g(r) defined in (B1)) with
and r = 5 (diameter at half maximum = 7.2,
at half encircled energy = 13.5 pixels).
The total input source flux is 100 photons/frame.
The pattern distribution is [0.778, 0.195, 0.014, 0.013].
The global flux loss is 61.42%.
The global pile-up rate is 1.63%
_{0} |

Figure 4 shows an example of the expected count rate
for an axisymmetric PSF with radial profile representative
of the *XMM/EPIC-MOS*, a low-energy pattern distribution
(the relevant pattern distribution is that averaged over energy,
which is dominated by the more numerous low-energy photons)
and a large source flux.
What is important to note is that even though the flux loss is large
(61.42%) as expected for such a bright source, the pile-up fraction
remains very modest at 1.63%. This is because the probability of
a true monopixel pile-up is much less (only 1 pixel target area)
than that of destroying the monopixel (8 pixels target area for other
monopixels, and more for bi-, tri- and quadripixels).
Adding the diagonal monopixels
(Eqs. A3-A4) reduces
the flux loss to 58.56%, but increases the pile-up fraction to 2.02%.
Counting as good events monopixels touching some other pattern
by a corner (Eq. (A5)) reduces further
the flux loss to 51.80%, while increasing the pile-up fraction to 2.76%.
Note that the figures given here are for integration to 1000 pixels distance,
but they do not change a lot if it is performed to 50 pixels (10 *r _{0}*) only
(4.5% of the flux falls beyond 50 pixels).

Figure 5:
Same as Fig. 4 for bipixel events,
using formulae (16-17).
The global flux loss is 57.15%.
The global pile-up rate is 21.30%, much larger than for monopixels
because the clean bipixel events are dominated near the center
by adjacent monopixel events counted as one bipixel event |

Figure 5 shows the same plot for bipixels. In this case the flux loss is comparable (57%), but the pile-up fraction is much higher (21%). This is because for this (typical) pattern distribution there are four times more monopixels than bipixels, thus leading to a large fraction of adjacent monopixels being counted as bipixels. The figures are somewhat similar for tripixels and quadripixels.

For a less extreme source flux, such as 10 photons/frame, the flux loss and pile-up rate for monopixels are 18.40% and 0.74%, respectively. Including diagonal events, the figures are 14.45% and 0.86%. For bipixels the flux loss is 12.65% and the pile-up rate 12.43%.

The main conclusion is that flux loss and transfer of single events to bipixels or larger is the dominant effect of pile-up. For all spectral applications when pile-up is suspected to have occurred it is much safer to select only monopixel events. Diagonal events may be used, but never total up a large fraction of the true monopixels. However, rejecting all split events incurs a severe loss of effective area at high energy (a factor of 2), where the number of events may not be large even though the number of low-energy photons exceeds the pile-up limit.

After integrating the local rates from equations
(2-3) over space
(any spatial domain is allowed, but here I consider the full space),
one obtains the count rate of clean (not piled-up) monopixel events and
the full expected monopixel count rate *M _{1}*
as a function of the incoming source flux.
I call the integral source flux (/frame) , to avoid confusion
with the local flux (/pixel/frame) .

(5) | ||

(6) |

This has two important consequences. First, because the expected count rate is a strictly increasing function of the input source flux it is possible to estimate quite precisely the source flux from the measured count rate, and model the piled-up PSF, similarly to "curve of growth'' analysis for absorption lines. Secondly, because the pile-up fraction among monopixel events remains small, a spectrum restricted to monopixels is not hopelessly corrupted, even for very bright sources. The detailed study of the spectral perturbations induced by pile-up is deferred to a later paper.

It is also useful to know how pile-up behaves at low flux, allowing one to estimate when it can be safely ignored. Developing Eqs. (6) and (5) over to order 2, one easily obtains for the flux loss and the pile-up rate on single events:

(7) | ||

(8) |

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