SPEX User's Manual - SRON

5.5 The proposed response matrix 101
The bin width constraint derived here depends upon the dimensionless curvature of the effective area
A/Ej 2A′′ . In most parts of the energy range this will be a number of order unity or less. Since the
second prefactor E j /FWHM is by definition the resolution of the instrument, we see by comparing (5.58)
with (5.47) that in general (5.47) gives the most severe constraint upon the bin width, unless either the
resolution gets small (what happens e.g. for the Rosat PSPC detector at low energies), or the effective
area curvature gets large (what may happen e.g. near the exponential cut-offs caused by filters).
Large effective area curvature due to the presence of exponential cut-offs is usually not a very serious
problem, since these cut-offs also cause the count rate to be very low and hence weaken the binning
requirements. Of course, discrete edges in the effective area should always be avoided in the sence that
edges should always coincide with bin boundaries.
In practice, it is a little complicated to estimate from e.g. a look-up table of the effective area its curvature,
although this is not impossible. As a simplification for order of magnitude estimates we can use the case
where A(E) = A 0 e bE locally, which after differentiation yields
√
8A
Ej 2 = √ 8 d lnE
A′′ d lnA . (5.59)
Inserting this into (5.58), we obtain
5.4.6 Final remarks
∆E
FWHM < √ 8
( d lnE
)(
d lnA
E j
FWHM
)
λ(R) 0.5 N −0.25 . (5.60)
In the previous two subsections we have given the constraints for determining the optimal model energy
grid. Combining both requirements (5.48) and (5.60) we obtain the following optimum bin size:
∆E
FWHM = 1
1
w 1
+ 1
(5.61)
w a
where w 1 and w a are the values of ∆E/FWHM as calculated using (5.48) and (5.60), respectively.
This choice of model binning ensures that no significant errors are made either due to inaccuracies in the
model or effective area for flux distributions within the model bins that have E a ≠ E j .
5.5 The proposed response matrix
5.5.1 Dividing the response into components
When a response matrix is stored on disk, usually only the non-zero matrix elements are stored and
used. This is done in order to save both disk space and computational time. The procedure as used in
XSPEC and the older versions of SPEX is then that for each model energy bin j the relevant column of
the response matrix is subdivided into groups. The groups are the continuous pieces of the column where
the response is non-zero. In general these groups are stored in a specific order: starting from the lowest
energy, all groups of a single energy are given before turning to the next higher photon energy.
This is not the optimal situation neither with respect to disk storage nor with respect to computational
efficiency, as is illustrated by the following example. For the XMM/RGS, the response consists of a narrow
gaussian-like core with in addition a broad scattering component due to the gratings. The FWHM of
the scattering component is typically 10 times larger than that of the core of the response. As a result,
if the response would be saved as a ”classical” matrix, we would end up with one response group per
energy, namely the combined core and wings response, since these contributions have overlap, namely in
the region of the gaussian-like core. As a result, the response becomes large, being a significant fraction
of the product of the total number of model energy bins times the total number of data channels. This
is not necessary, since the scattering contribution with its ten times larger width needs to be specified
only on a model energy grid with ten times fewer bins, as compared to the gaussian-like core. Thus,
by separating out the core and the scattering contribution, the total size of the response matrix can be