Single Pixel Pileup

As an imaging x-ray telescope, AXAF focuses a point source onto the CCD. Since the point response function FWHM is comparable to the size of one pixel, pileup from a strong point source is a particular problem.

All pileup cross sections presented so far are evaluated in terms of detected x-rays per quadrant per frame. However, it is often useful to describe pileup in units of x-rays per pixel per frame. The factors are translated multiplying $\epsilon$ by $256 \times 1024 =2.62\times10^5$. Thus, if R_nopileup=incident x-ray rate per pixel per frame for a monochromatic beam, then to first approximation the detected rate R_nopileup is given by the transcendental equation

$$R_{pileup} = R_{no pileup} \exp\left(2.62\times10^5R_{no pileup}\epsilon\right)$$ (4.18)

From another viewpoint, pileup in a single pixel with a monochromatic source is easily understood as a simple Poisson process. That is, if the probability for an x-ray to interact with a pixel during one exposure is $\lambda$, then the probability that N x-rays interact during one exposure is

$$P_N=\frac{\lambda^Ne^{-\lambda}}{n!}$$ (4.19)

Consider an entire observation of N_frames frames. If there was no pileup, the total incident flux of $N_i = N_{frames}\lambda$ x-rays would be detected. The effect of pileup is manifest as several (or many) x-rays which add during one exposure. Let

$$N_0 = N_{nframes}e^{-\lambda} \equiv {\rm number~of~frames~with~no~interaction}.$$ (4.20)

Combining these two equations yields,

$$\frac{N_{i}}{N_{frames}}=- \log\left(\frac{N_0}{N_{frames}}\right)$$ (4.21)

For example, if out of 1000 frames, x-rays from a point source are present in 100 frames, then

$$N_i=-1000 \log(900/1000)=105$$ (4.22)

Thus, pileup accounted for an apparent reduction in the incident flux of 5%.

This method differs from the first method developed in this chapter by treating every incident x-ray as a valid event, including the coincidence of two or many x-rays. This method has use in high pileup situations where Eqn. 4.16 presents ambiguity. For example, the shape of the graphs in Fig. 4.22 show that for a given value of the detected flux, there are two possibilities for the incident flux. Without additional information it is impossible to discriminate between the low and high flux solution. Although most astrophysical point sources should not be so strong, the ambiguity of strong sources can be resolved by application of the method described above.