Appendix C: Example of application to real data

 

C.1 Presentation of the data

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.

C.2 Construction of the Point Spread Function

  This was not so easy for three reasons. Firstly there were clear deviations in the data from the ideal King profile. Secondly even the low intensity run could not be considered totally free of pile-up. Thirdly the number of counts in the wings of the low intensity run was inadequate to model the high intensity data. The problems were tackled in the following way:

• There were clear deviations in the data both from a simple King profile (Sembay 1998) and from the hypothesis of axial symmetry. Rather than try and invent a complicated analytical model, I decided to use the 2-D data directly.
• After normalizing the data of run 01351 to units of cts/pixel/frame, the total count rate in the low intensity run was 1.6 cts/frame. The local count rate in the central pixel was larger than 0.01 cts/pixel/frame. The flux loss due to pile-up was therefore larger than 10% in the core, and had to be accounted for. The idea was to solve Eq. (2) for $\lambda$ at all pixels separately. For count rates lower than 0.01 (everywhere except the core), the correction was small and a second order development was used instead.

  However this direct application was not very satisfactory because it amplified the statistical fluctuations in the data (about 10% in the core). Since $\mu_1$ 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 $3\times 3$ 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 $\overline{\lambda}$. In a second step, (2) was solved after replacing $\lambda$ by $\overline{\lambda}$ in the suppression term, resulting in

  $$\lambda = \frac{\ln(1 + d \: \mathrm e^{\gamma_1 \overline{\lambda}})} {\alpha_1}$$

  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.
• With the statistics at hand, the resulting PSF was very noisy. In particular the run at highest intensity (01346) had about 10 times more cts/pixel in the wings than the long low intensity run. In order to have the best PSF possible, I used data from run 01346 in the wings (distance larger than 30 pixels) where pile-up was not too large. This data was corrected for pile-up in the way described above, and smoothed with a boxcar average of variable width (from 3 pixels at distance 30 pixels to 17 at distance 100 and above). It was normalized to have the same integral in the wings as the PSF derived from the low intensity run. At distance
    

  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).

C.3 Model comparison with piled-up data

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 $201 \times 201$ pixels centered on the maximum. For each run Eq. (6) was solved for the incoming flux $\Lambda$ 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.