SPEX User's Manual - SRON

5.5 The proposed response matrix 103
Detector regions
The observed count rate spectra are extracted in practice in different regions of the detector. It is
necessary here to distinguish clearly the (sky) sectors and (detector) regions. A detector region for the
XMM EPIC camera would be for example a rectangular box, spanning a certain number of pixels in the
x- and y-directions. It may also be a circular or annular extraction region centered around a particular
pixel of the detector, or whatever spatial filter is desired. For the XMM RGS it could be a specific
”banana” part of the detector-coordinate CCD pulse-height plane (z, c).
Note that the detector regions need not to coincide with the sky sectors, neither should their number to be
equal. A good example of this is again the example of an AGN superimposed upon a cluster of galaxies.
The sky sector corresponding to the AGN is simply a point, while, for a finite instrumental psf, its
extraction region at the detector is for example a circular region centered around the pixel corresponding
to the sky position of the source.
Also, one could observe the same source with a number of different instruments and analyse the data
simultaneously. In this case one would have only one sky sector but more detector regions, namely one
for each participating instrument.
Consequences for the response
In all the cases mentioned above, where there is either more than one sky sector or more than one
detector region involved, it is necessary to generate the response contribution for each combination of
sky sector - detector region. In the spectral analysis, for each sky sector the model photon spectrum is
calculated, and all these model spectra are convolved with the relevant response contributions in order to
predict the count spectra for all detector regions. Each response contribution for a sky sector - detector
region combination itself may consist again of different response components, as outlined in the previous
subsection.
Combining all this, the total response matrix then will consist of a list of components, each component
corresponding to a particular sky sector and detector region. For example, let us assume that the RGS
has two response contributions, one corresponding to the core of the lsf, and the other to the scattering
wings. Let us assume that this instrument observes a triple star where the instrument cannot resolve
two of the three components. The total response for this configuration then consists of 12 components,
namely 3 sky sectors (assuming each star has its own characteristic spectrum) times two detector regions (a
spectrum extracted around the two unresolved stars and one around the other star) times two instrumental
contributions (the lsf core and scattering wings).
5.5.3 A response component
We have shown in the previous section how for the combination of a particular source configuration and
instrumental properties response matrix components can be defined. In this subsection we investigate
how to build such a matrix component.
Let us first return to our discussion on the optimum model energy grid binning. For high-resolution
instruments the factor E j /FWHM in (5.60) is by definition large, and hence for most energies the requirements
for the binning are driven by the model accuracy, not by the effective area accuracy. One
might ask whether the approximation of a constant effective area within a model bin instead of the linear
approximation (5.54) would not be sufficient in that case. The reason is the following.
In order to acquire the high accuracy, we need to convolve the model spectrum for the bin, approximated
as a δ-function centered around E a , with the instrument response. In most cases we cannot do this
convolution analytically, so we have to make approximations. From our expressions for the observed
count spectrum s(c), eqns. 5.35) and (5.36), it can be easily derived that the number of counts or count
rate for data channel i is given by
∫c i2 ∫
S i = dc
c i1
dER(c, E)f(E) (5.62)