7.8 examples

7.8.1 Introduction

In this section I present some estimates of the optical binning for som e instruments. The ideal situation would be to use the original software that generates the response matrices for these instruments, in order to derive first the optimal binning and then to produce the response matrix in the formats as proposed here. Instead I take here the faster approach to use spectra and responses in OGIP format and to derive the rebinned spectra and responses by interpolation. This is less accurate than the direct method mentioned above, but is sufficiently accurate for the present purposes. A drawback is however that given a ”solid” square response matrix, it is difficult if not impossible to extract from that data the different instrumental components that contribute to the matrix, like for example the main diagonal and the split-events for a CCD detector. If we were able to do that (as the instrument-specific software should be able to do) we could reach an even larger compression of the response matrices.

I consider here data from three instruments, with low, medium and high resolution, and work out the optimal binning. The instruments are the Rosat PSPC, ASCA SIS and XMM RGS detectors.

7.8.2 Rosat PSPC

For the Rosat PSPC data I used the spectrum of the cluster of galaxies A 2199 as extracted in the central 1′. The spectrum was provided by J. Mittaz and had a total count rate of 1.2 counts/s, with an exposure time of 8600 s. The PSPC has a low spectral resolution, as can be seen from fig. 7.7. The number of resolution elements is only 6.7. After rebinning, from the original 256 data channels only 15 remain! This shows that for Rosat PSPC data analysis rebinning is absolutely necessary.

The optimum resolution of the model energy grid below 0.6 keV and above 2 keV is dominated by the effective area curvature (upper right panel of fig. 7.7). It is also seen that the original PSPC model energy grid has a rather peculiar, variable bin size, with sometimes a few very narrow bins, apparently created in the neighbourhood of instrumental edges. This causes the ”spikes” in the number of model bins to be merged. Note that this number is a quantity valid for only a single bin; if e.g. a thousand bins need to be merged, this means in practice that only the next few bins are merged, since these next bins will have a more decent value of the number of bins to be merged.

Table 7.13 summarizes the gain in bin size as obtained after optimum binning of the data and response matrix. The reduction is quite impressive.

Figure 7.7: Binning of the Rosat PSPC response. Left panel, from top to bottom: instrumental FWHM, effective area A(E), and effective number of events N_r within one resolution element. Right panel, top frame: optimum model energy bin size in units of the FWHM, eqn. (7.61) as the solid line, with the contribution from the effective area eqn. (7.60) as the upper dotted line, the contribution from the model binning eqn. (7.48) as the lower dotted line; middle frame: optimum model energy bin size in keV (solid line) and original bin size in the OGIP-format matrix (dotted line); bottom frame: number of original model energy bins to be taken together.

7.8.3 ASCA SIS0

For the ASCA SIS0 data I used the spectrum of the supernova remnant Cas A. The spectrum was provided by J. Vink and had a total count rate of 52.7 counts/s, with an exposure time of 6000 s. There are 86.2 resolution elements in the spectrum, which extended from 0.35 to 12.56 keV. The results are summarized in fig. 7.8. In the effective number of events per data channel (lower left panel) clearly the strong spectral lines of Si, S and Fe can be seen. The present modelling shows that a slightly smaller bin size is needed near those lines (right panels).

Contrary to the PSPC case, for the SIS detector the optimum resolution of the model energy grid is dominated verywhere by the requirements for the model binning, not the effective area curvature (upper right panel of fig. 7.8). This is due to the higher spectral resolution as compared to the PSPC. Only near the instrumental edges below 1 keV a small contribution from the effective area exists.

The original OGIP-matrix uses a constant model energy grid bin size of 10 eV, as can be seen from the middle-right panel of fig. 7.8. It can be seen from that panel that below 3 keV that binning is much too coarse; one should take into account that the OGIP-matrices are used in the approximation of all flux coinciding with the bin centroid; for that approximation however the optimum bin size would be even a factor of ~10 smaller! On the other hand, the ASCA model grid is over-sampled at high energies.

Table 7.14 summarizes the gain in bin size as obtained after optimum binning of the data and response matrix. The reduction is again quite large. It should be noted however that since at low energies the model energy grid binning of the original OGIP data is undersampled, so that the final number of model bins in the optimum case of direct response generation in the new format might be somewhat larger. On the other hand, decomposing the matrix into its physical components reduces again the number of matrix elements considerably, since most of the non-zero matrix elements are caused by the low-resolution split-events that need not a very fine energy mesh.

Figure 7.8: Binning of the ASCA SIS response. Panels as for the Rosat PSPC data.

7.8.4 XMM RGS data

I applied the procedures of this paper also to a simulated RGS spectrum of Capella. The original onboard 3×3pixel binning produces 3068 data channels. The OGIP response matrix used is the one provided by C. Erd, which was calculated for a smooth grid of 9300 model energy bins. The calculation is done for a single RGS. The spectrum has a count rate of 4.0 counts/s, with an exposure time of 80000 s.

After rebinning, there remain only 1420 new data channels. Most of these new data channels contain 2 old data channels, only at the boundaries of the detector 3 old bins are taken together. The number of resolution elements is 611. After rebinning, from the original 3068 data channels 1420 remain.

The optimum model binning is completely dominated by the requirements for the model binning. Effective area effects can be neglected (fig. 7.9). The 8 discrete features in several frames of this figure are artefacts due to the gaps between the 9 CCD detectors of each RGS. This is due to the fact that we started with a pre-fab OGIP type response matrix.

Table 7.15 summarizes the gain in bin size as obtained after optimum binning of the data and response matrix. At first sight the reduction appears not to be too impressive. This is however illusionary. The ”classical” OGIP response matrix that was used as the basis for my calculations is heavily under-sampled. Would it have been sampled properly, then the model energy grid should have a size of a few hundred thousand bins! Since this causes the response matrix to be massive and disk consuming, the undersampled matrix was created; this matrix can be safely used for spectral simulations, but is insufficient for flight data analysis. It is exactly for that reason that the curren theoretical framework has been set up!

Figure 7.9: Binning of the XMM RGS response. Panels as for the Rosat PSPC data.