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Astron. Astrophys. 349, 11-28 (1999)

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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 [FORMULA]. To good approximation, the light yield I (in photons per m2) is proportional to the shower energy,

[EQUATION]

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,

[EQUATION]

but they are influenced by rather different factors. While both [FORMULA] and [FORMULA] 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

[EQUATION]

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 [FORMULA], [FORMULA] and [FORMULA] 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 ([FORMULA]) or of the optics and electronics of the telescope ([FORMULA], [FORMULA]). The reconstructed energy [FORMULA] of a shower of true energy E is then, in the absence of fluctuations,

[EQUATION]

Based on Eqs. 1 and 2 and using [FORMULA] the rate of events with measured energy [FORMULA] is then

[EQUATION]

In the analysis, the Monte Carlo simulated rate [FORMULA] is used to evaluate the effective area

[EQUATION]

resulting in a reconstructed flux

[EQUATION]

[EQUATION]

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 [FORMULA] 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 [FORMULA], i.e., [FORMULA], assuming that the simulation correctly accounts for the statistical fluctuations determining the shape of [FORMULA]. 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, [FORMULA]) in the data and in the simulation (see Sect. 4.3 and Fig. 5). If [FORMULA] 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 [FORMULA].

If the thresholds in Q are shifted by a factor [FORMULA] in the data relative to the simulation, this implies that [FORMULA] is off by the same factor, resulting in the flux error given by Eq. 4.

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© European Southern Observatory (ESO) 1999

Online publication: August 25, 1999
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