SPEX User's Manual - SRON

5.4 Model binning 97
times σ. Several of the strongest sources to be observed by this instrument will have more than 10000
counts in the strongest spectral lines (example: Capella), and this makes it necessary to have a model
energy grid bin size of about σ/ √ (10000). The total number of model bins is then 194000.
Most of the computing time in thermal plasma models stems from the evaluation of the continuum. The
reason is that e.g. the recombination continuum has to be evaluated for all energies, for all relevant
ions and for all relevant electronic subshells of each ion. On the other hand, the line power needs to be
evaluated only once for each line, regardless of the number of energy bins. Therefore the computing time
is approximately proportional to the number of bins. Going from ASCA-SIS resolution (1180 bins) to
RGS resolution (194000 bins) therefore implies more than a factor of 160 increase in computer time. And
then it can be anticipated that due to the higher spectral resolution of the RGS more complex spectral
models are needed in order to explain the observed spectra than are needed for ASCA, with more free
parameters, which also leads to additional computing time. And finally, the response matrices of the RGS
become extremely large. The RGS spectral response is approximately Gaussian in its core, but contains
extended scattering wings due to the gratings, up to a distance of 0.5Å from the line center. Therefore,
for each energy the response matrix is significantly non-zero for at least some 100 data channels, giving
rise to a response matrix of 20 million non-zero elements. In this way the convolution of the model
spectrum with the instrument response becomes also a heavy burden.
For the AXAF-LETG grating the situation is even worse, given the wider energy range of that instrument
as compared to the XMM-RGS.
Fortunately there are more sophisticated ways to evaluate the spectrum and convolve it with the instrument
response, as is shown in the next subsection.
5.4.3 Binning the model spectrum
We have seen in the previous subsection that the ”classical” way to evaluate model spectra is to calculate
the total number of photons in each model energy bin, and to act as if all flux is at the center of the
bin. In practice, this implies that the ”true” spectrum f(E) within the bin is replaced by its zeroth order
approximation f 0 (E), given by
f 0 (E) = Nδ(E − E j ), (5.39)
where N is the total number of photons in the bin, E j the bin centroid as defined in the previous section
and δ is the delta-function.
In this section we assume for simplicity that the instrument response is Gaussian, centered at the true
photon energy and with a standard deviation σ. Many instruments have line spread functions with
Gaussian cores. For instrument responses with extended wings (e.g. a Lorentz profile) the model binning
is a less important problem, since in the wings all spectral details are washed out, and only the lsf core
is important. For a Gaussian profile, the FWHM of the lsf is given by
FWHM = √ ln(256)σ ≃ 2.35σ. (5.40)
How can we measure the error introduced by using the approximation (5.39)? Here we will compare
the cumulative probability distribution function (cdf) of the true bin spectrum, convolved with the
instrumental lsf, with the cdf of the approximation (5.39) convolved with the instrumental lsf. The
comparison is done using a Kolmogorov-Smirnov test, which is amongst the most powerful tests in
detecting the difference between probability distributions, and which we used successfully in the section
on data binning. We denote the lsf-convolved cdf of the true spectrum by φ(c) and that of f 0 by φ 0 (c).
By definition, φ(−∞) = 0 and φ(∞) = 1, and similar for φ 0 .
Similar to section 5.3, we denote the maximum absolute difference between φ and φ 0 by δ. Again,
λ k ≡ √ Nδ should be kept sufficently small as compared to the expected statistical fluctuations √ ND,
where D is given again by eqn. (5.21). Following section 5.3 further, we divide the entire energy range
into R resolution elements, in each of which a KS test is performed. For each of the resolution elements,
we determine the maximum value for λ k from (5.26)–(5.28), and using (5.34) we find for a given number
of photons N r in the resolution element the maximum allowed value for δ in the same resolution element.