Astron. Astrophys. 359, 682-694 (2000) 7. Further studies of systematic uncertainties; analysis methods independent of absoluteThe trend for an enrichment in heavy elements above the knee - which Fig. 10 suggest - is not significant within our errors. In this Sect. we elucidate this fact further, and explore what it would take to detect significantly a modest trend for an enrichment in heavy elements - as expected e.g. in a diffusion model of the knee (see Introduction) - with the present techniques. The agreement between the distribution shape at low energies - predicted assuming a composition at low energies which is not very different from the one obtained by direct experiments - and the data (see Fig. 9) is satisfactory. This is an argument in favour of a correct MC simulation. To explore the effect of our systematic error in slope (Sect. 5.4) Fig. 12 shows the results derived with an initial assumption of slope changed by 5% from its preferred value for energy-reconstruction method 3 and 4. It becomes clear that not only the mean but even the overall apparent "trend" may change, e.g. for method 3 with slope increased by 5% the composition appears to become lighter from the knee up to the penultimate bin. It should be stressed that none of the discussed "trends" is significant within our statistical errors.
The deviation of the penetration depth at the highest energy point from a constant elongation rate discussed in Sect. 6.5 is of the order of disagreements between different Monte Carlo codes at this energy(Heck et al. 1998). Therefore the possibility remains open that this deviation is due to a change in cascade characteristics not reliably modelled by the Monte Carlo, rather than to an enrichment in heavy elements. As it is also quite similar in size to our systematic error in slope, an origin in the HEGRA experiment for this deviation is also difficult to rule out. Previous conference publications (Plaga et al. 1995; Cortina et al. 1997b; Cortina et al. 1998) are superseded by the present results - differences are mainly due to a more sophisticated amplifier calibration and the simpler energy reconstruction for the present analysis. In the last two of these publications, we tried to lessen the dependence of our composition result on the correct absolute values by allowing a free "shift parameter"; the fit to the was then performed with two free parameters: the ratio of light to all nuclei and an overall shift in penetration depth of all MC distributions. In this way the result is mainly determined by the shape of the distributions (in first order its width, i.e. RMS value). This width depends only weakly on a systematic uncertainty in the determination of slope , relative to the expected difference of a purely light or heavy composition. Fig. 13 displays the result of such an attempt in a two dimensional plot showing the reduced for various "fraction of light nuclei" - "shift parameter" combinations for the data lowest and highest in energy. The shift is varied in an interval 30 g/cm^{2}, estimated from the likely systematic uncertainty of our detector and the Monte Carlo code. While in the low energy bin small (p+)/all ratios lead to unsatisfactory values for all shift values, in the highest energy bin - well above the knee - practically all fractions give acceptable values for appropriate shifts as seen in Fig. 13. The reason for this behaviour is that - given the small number of events in the high-energy bin - the distribution can be fitted both with the relatively broad predominantly light composition shifted to larger depths in the atmosphere and a mixed heavy/ light composition (where the difference in mean penetration depth of the heavy and light component contributes to the total width) shifted to small penetration depth. We have to conclude that it is not reliably possible to determine the composition based mainly on the width of the distribution. We found in numerical experiments that with this method, and assuming a Monte-Carlo simulation describing the experimental data well, together with a statistics increased by about a factor of 100 (which is difficult but not impossible to reach in future experiments) it will just be possible to reach the desired precision of 10% mentioned in the introduction on a 2 level beyond the knee.
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