SPEX User's Manual - SRON

96 Response matrices
bin j of this model grid we define a lower and upper bin boundary E 1j and E 2j as well as a bin centroid
E j and a bin width ∆E j , where of course E j is the average of E 1j and E 2j , and ∆E j = E 2j - E 1j .
Provided that the bin size of the model energy bins is sufficiently smaller than the spectral resolution
of the instrument, the summation approximation to eqn. (5.35) is in general sufficiently accurate. The
response function R(c, E) has therefore been replaced by a response matrix R ij , where the first index i
denotes the data channel, and the second index j the model energy bin number.
We have explicitly:
S i = ∑ j
R ij F j , (5.36)
where now S i is the observed count rate in counts/s for data channel i and F j is the model spectrum
(photons m −2 s −1 ) for model energy bin j.
This basic scheme is widely used and has been implemented e.g. in the XSPEC and SPEX (version 1.10
and below) packages.
5.4.2 Evaluation of the model spectrum
The model spectrum F j can be evaluated in two ways. First, the model can be evaluated at the bin
centroid E j , taking essentially
F j = f(E j )∆E j . (5.37)
This is appropriate for smooth continuum models like blackbody radiation, power laws etc. For line-like
emission it is more appropriate to integrate the line flux within the bin analytically, and taking
F j =
E 2j ∫
E 1j
f(E)dE. (5.38)
Now a sometimes serious flaw occurs in most spectral analysis codes. The observed count spectrum S i
is evaluated straightforward using eqn. (5.36). Hereby it is assumed that all photons in the model bin j
have exactly the energy E j . This has to be done since all information on the energy distribution within
the bin is lost and F j is essentially only the total number of photons in the bin. Why is this a problem?
It should not be a problem if the model bin width ∆E j is sufficiently small. However, this is (most often)
not the case.
Let us take an example. The standard response matrix for the ASCA SIS detector uses a uniform
model grid with a bin size of 10eV. At a photon energy of 1keV, the spectral resolution (FWHM) of
the instrument is about 50eV, hence the line centroid of an isolated narrow line feature containing N
counts can be determined with a statistical uncertainty of 50/2.35 √ N eV (we assumed here for simplicity
a Gaussian instrument response, for which the FWHM is approximately 2.35σ). Thus, for a moderately
strong line of say 400 counts, the line centroid can be determined in principle with an accuracy of 1eV,
ten times better than the step size of the model energy grid. If the true line centroid is close to the
boundary of the energy bin, there is a mis-match (shift) of 5eV between the observed count spectrum
and the predicted count spectrum, at about the 5σ significance level! If there are more of these lines in
the spectrum, it is not excluded that never a satisfactory fit (in a χ 2 sense) is obtained, even in those
cases where we know the true source spectrum and have a perfectly calibrated instrument! The problem
becomes even more worrysome if e.g. detailed line-centroiding is done in order to derive velocity fields,
like has been done e.g. for the Cas A supernova remnant.
A simple way to resolve these problems is just to increase the number of model bins. This robust method
always works, but at the expense of a lot of computing time. For CCD-resolution spectra this is perhaps
not a problem, but with the increased spectral resolution and sensitivity of the grating spectrometers of
AXAF and XMM this becomes cumbersome.
For example, with a spectral resolution of 0.04Å over the 5–38Å band, the RGS detector of XMM will
have a bandwidth of 825 resolution elements, which for a Gaussian response function translates into 1940