HEAD 2002 Pileup Talk  

Overview of the Pileup Model


The Standard Model

The "standard" model has been used for many years for the analysis of X-ray data. The model is based upon the equation (Davis 2001a)
C(h)=(Nt)\int R(h,E)A(E)s(E),
which relates the number of observed events C(h) in a pulse-height channel h to a spectral photon distribution s(E). The quantity R(h,E) represents the mapping from photon energy E to pulse-height h by the CCD, and A(E) represents the effective area of the telescope. The functions R(h,E) and A(E) are commonly known as the RMF and ARF, respectively The total exposure time of the observation is given by Nt where N specifies the number of CCD exposures, or frames, and t represents the frame time.

Note that in this model, the N and t occur together as a product. Within the standard model, errors in N or t affect only the normalization of the fitted spectral model s(E) and not its shape. In contrast, within the pileup model, errors in N and t affect both the shape and the normalization of the fitted model s(E).

In the integral equation above, the count rate increases linearly with the incident source flux. This is illustrated in the following figure:

Figure showing linear nature of the standard model

The Pileup Model

The CCD pileup model as implemented in isis is assumed to apply to an extraction region consisting of two subregions: a central region where pileup is assumed to take place, and an outer region assumed to be free of pileup. Furthermore, the pileup region is assumed to be subdivided into n subregions. Let f represent the fraction of flux falling into the pileup region. Then as shown by Davis 2001b, the number of counts expected in the full extraction region is given by
C_(h) & = & (Nt)(1-f)\int dE \;R(h,E) A(E) s(E) \\
& & + Nn
e^{-(t/\bar{g}_0) \int dE\; A(E) fs(E)/n}
\sum_{p=1}^{\infty} \alpha^{p-1}
\int dE \; R(h, E)
\frac{[t A(E)fs(E)/n]^{*p}}{p!},
where the quantity  α represents the effects of grade migration. See Davis 2001b for the details.

It is important to note that the number of frames N and the frame time t do not always occur together. This means that unlike the standard model, the total number of detected counts depends not just on the total exposure time Nt, but also on the frame time t and number of frames N separately. For this reason, it is important to specify the exposure time and frame time accurately for the pileup model to work.

In the pileup model, the predicted count rate is a highly non-linear function of the source flux. An indication of this can be seen from the following figure which shows how the count rate varies with the incident flux in the pileup model:

Figure showing flux vs counts in pileup model
Here, for low fluxes, the pileup model and the standard model agree. However, as the flux increases, pileup effects start to become noticeable and at some point the count rate turns over and starts decreasing. This effect is quite noticible in observations of bright sources where the "core" of the PSF is "burned" out. At that point, the count rate is dominated by counts in the wings of the PSF.

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This page was last updated May 31, 2002 by John E. Davis. To comment on it or the material presented here, send email to davis@space.mit.edu.
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