*Astron. Astrophys. 349, 11-28 (1999)*
## Appendix A: systematic errors on the shape of the spectrum
To illustrate the effect of systematic errors for determination of
energies and effective areas in more detail, we consider here the
following model. For simplicity, we consider a detector consisting of
a single Cherenkov telescope, which is located inside a uniformly
illuminated Cherenkov light pool of area
. To good approximation, the light
yield *I* (in photons per m^{2}) is proportional to the
shower energy,
neglecting logarithmic corrections. The response of the telescope
is now characterized by two quantities, namely the total intensity
*Q* detected in the image (usually the *size* , measured
e.g. in units of ADC channels, or, with a conversion factor, in units
of `photoelectrons'), and the signal *V* induced in the highest
pixel(s), which is fed into the trigger circuitry. Both *Q* and
*V* should be proportional to *I*,
but they are influenced by rather different factors. While both
and
include the mirror reflectivity, the
PMT quantum efficiency, and the PMT gain, *V* is to a much higher
degree sensitive to the point spread function of the mirror, and to
the shape of the signal generated by the
PMT ^{2}. The
effective detection area is given by
where *P* is the trigger probability for a given value of
*V*. The fact that triggering and energy determination are not
based on identical quantities is the key origin of systematic errors,
in particular in the threshold region.
In the analysis of data, values ,
and
are assumed for these constants,
usually based on Monte Carlo simulations. The values may differ from
the true values due to imperfections in the parameterization of the
atmosphere () or of the optics and
electronics of the telescope (,
). The reconstructed energy
of a shower of true energy *E*
is then, in the absence of fluctuations,
Based on Eqs. 1 and 2 and using
the rate of events with measured energy
is then
In the analysis, the Monte Carlo simulated rate
is used to evaluate the effective
area
resulting in a reconstructed flux
Incorrect constants used in the simulation may hence result in 1) a
factor *f* modifying the energy scale, but not the shape of the
spectra, and 2) a change of the shape of the spectra in particular in
the threshold region, where varies
steeply with *V*. The scale factor *f* cannot be determined
from IACT data alone; some external reference is required. The second
effect - the distortion of the spectra - disappears provided that
, i.e.,
, assuming that the simulation
correctly accounts for the statistical fluctuations determining the
shape of . This condition can be
checked internally within the data set. In particular, one should
compare the distribution in *Q* for a given value of *V* (or
for a fixed range in *V*, ) in
the data and in the simulation (see Sect. 4.3 and Fig. 5). If
differs between data and
simulation, the distribution of *Q* in the threshold region will
be different. This comparison also tests the simulation of the shape
of .
If the thresholds in *Q* are shifted by a factor
in the data relative to the
simulation, this implies that is
off by the same factor, resulting in the flux error given by
Eq. 4.
© European Southern Observatory (ESO) 1999
Online publication: August 25, 1999
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