The XMM/EPIC-MOS Flight Model 1 was calibrated at the Panter facility in January 1998 behind the XMM mirrors (Sembay 1998), giving data with the spatial distribution of a point source in flight. Several runs were performed with the aim of calibrating pile-up. I have used the data taken with the Fe L source at varying beam intensity (runs 01343 to 01346), and the night-long run at low intensity (01351), in full-frame mode. The centering was the same in all these runs.
The data was cleaned as thoroughly as possible of bad pixels and background (using energy selection). The observed pattern distribution (outside the core, in order to be free of pile-up contamination) was [0.8834, 0.1040, 0.0061, 0.0065]. Patterns more complicated than quadripixels were negligible. The idea is not here to analyse in any detail the data itself, but simply to illustrate the relevance of the model to real data.
However this direct application was not very satisfactory because it amplified the statistical fluctuations in the data (about 10% in the core). Since in (2) should be the expectation value (not the measured value), the pixels above average were over-corrected and those below average under-corrected. The next idea was to smooth the data with a boxcar average of pixels, prior to solving (2).
This was not acceptable either, because it resulted in smoothing the PSF itself with the same boxcar average, leading to 10% or so errors. The final solution (actually very close to solving the problem with large pixels, as presented in Sect. 4.4), was to proceed in two steps. In a first step, Eq. (2) was solved on the smoothed data, resulting in a smoothed PSF . In a second step, (2) was solved after replacing by in the suppression term, resulting in
where d is the raw data. This allows to keep the original resolution while not over-correcting for pile-up. Of course the statistical fluctuations are still there, but there is no way to get rid of them without a model for the PSF.
|Figure C1: Comparison between data obtained at Panter with the XMM/EPIC-MOS in January 1998 at high intensity (run 01346) and the pile-up model according to (9). The data (crosses with error bars) and the model (shown as a dashed line) were limited to a band 21 pixels wide along Y (centered on the source), and projected onto the X axis. The source was centered at 100. The total incoming flux was estimated to be 359 photons/frame (the measured count rate was 90 cts/frame). The PSF used in the model was derived as explained in C.2|
between 10 and 30 pixels, the PSF (from run 01351) was also smoothed with a boxcar average of width 3 pixels to reduce its fluctuations (no true feature is expected at that distance at the pixel scale).
The comparison was performed for monopixels only. The events from runs 01343 to 01346 were projected onto an image and divided by the number of frames. The spatial domain was limited to a window of pixels centered on the maximum. For each run Eq. (6) was solved for the incoming flux in the window, using the PSF constructed as described in C.2. Then the radial profiles, and profiles projected onto the X and Y axes, were compared with those expected from the pile-up model applied locally.
The comparison was fully satisfactory. It indicated that the fit was significantly better when one used Eqs. (9) than (2), and Fig. C1, illustrating the comparison, was obtained using (9). It is not clear, however, whether the improvement is due to the fact that even for XMM/EPIC-MOS the pixels are not small enough to allow using (2) with a high precision, or that the statistical fluctuations in the model PSF lead to the wrong pile-up corrections (as described in C.2). Data with higher statistical significance in the PSF core would be required to decide.
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