## 5. The analysis methodsMonte Carlo simulations revealed that the distance
between detector and shower maximum
(defined as the point in the shower development with the maximal
number of charged particles) can be reconstructed independently of the
primary mass with the shape parameter From this relation the distance to the shower maximum is determined
with a resolution (i.e. root mean square (RMS) of a
(true)-(reconstructed)
distribution) ranging from 40 g/cm ## 5.1. Energy reconstruction
Methods have been developed to reconstruct the primary CR energy from
the scintillator and AIROBICC data One reason is that the methods described below are based on
physically transparent properties of air-showers inferred from the
Monte-Carlo simulations. Whether these properties really hold, is
tested to some degree using different energy estimators with different
biases and comparing the obtained results. These consistency checks
are an important advantage over more refined and complete methods when
it is doubtful how well the Monte-Carlo simulation describes the data.
The other reason is that the Monte Carlo statistics at the highest
energy is still rather limited and mean shower properties are inferred
with higher certainty than a complete response matrix. The mass
independent energy reconstruction methods will be applied to the data
in a forthcoming publication together with a discussion of the
influence of different EAS simulations. The energy estimators used in
this paper and described below are based on
and Using and Here were obtained from the discrete Monte Carlo data as 0.965, -2.545 (0.890, -2.010) for protons (iron). is the shower size at the maximum of shower development and is inferred from as: with given as (0.57833, -85.146, 6181.8, -714054) for all primary nuclei. This procedure is valid because the shape of the shower development is only weakly dependent on the mass of the CR nucleus A, especially after the shower maximum (Lindner 1998a). Only the fraction of the total energy fed into the electromagnetic cascade depends on A for a given energy per nucleus. The comparison of the results assuming initially proton and iron primaries is a consistency check for the dependence of shower size at the maximum of shower development on the energy per nucleon. Alternatively the energy is reconstructed from the AIROBICC data alone (method 3 (4) with the assumption of proton (iron) primaries). Here it turns out that is a good estimator of the energy contained in the electromagnetic EAS cascades. From simulations the relation was derived, where the coefficients
Naturally (because the fraction of the primary energy deposited in electromagnetic cascades depends on the energy per nucleon of the primary particle) the mean of the calculated energy is only correct for the assumed particle type (Fig. 2). The biases shown in Fig. 2 have to be corrected for, to derive the real energy spectrum and CR mass composition from our measurements (see Sect. 5.2, 5.3). In order to check that our final results do not depend on the assumed primary-particle mass, we shall always compare the results based on the four energy reconstruction methods below.
The distribution of the reconstructed energy compared to the
simulated energy is shown for examples in Fig. 3. Note that the energy
reconstruction from alone shows
Gaussian distributions while the energy obtained from
and
In all analyses below we bin the data in six equidistant energy
intervals from log ## 5.2. Chemical composition
The composition of CR is determined by analysing the EAS penetration
depth () distributions in intervals
of the reconstructed primary energy. Information is contained in the
differences of the mean values for
different primaries (protons penetrate about 100-130 g/cm We perform an analysis which uses both of these parameters in one fitting procedure. As the error from such an analysis turns out to be already quite large, we do not perform an analysis based on mean penetration depth alone. An analysis based mainly on the fluctuation of penetration depths is discussed in Sect. 7. The present data are not sensitive enough on the chemical composition to allow a analysis with several mass groups; therefore we restrict ourselves to a determination of the fraction of light nuclei (protons and helium) by fitting the expected to the measured distributions. To define the MC expectations for light nuclei, the generated distributions for primary protons and helium nuclei are added with weights of 40% and 60% (the ratio derived from direct measurements at energies around 100 TeV (Wiebel-Sooth et al. 1998)). The distribution of heavier nuclei is constructed analogously by summing 65% oxygen and 35% iron induced EAS. Variations in this ratio at higher energies are possible and are an additional potential source for systematic errors that is not further considered below. The spectrally weighted Monte-Carlo data are fitted to the measured
penetration-depth distributions for each of the four
energy-reconstruction methods. Because spectrally weighted Monte-Carlo
data were available only for the energy bins log Due to the primary dependent energy-reconstruction method the results for the "fraction of light nuclei" (abbreviated "(p + )/all" below) are biased. The results for these fits in the chosen energy bins are shown for method 3 in Fig. 9. The obtained (p + )/all ratios are then corrected for the A dependent bias which is illustrated in Fig. 2. The correction can be described as a single overall factor for the (p + )/all ratio for each energy bin - rather than a transformation of the penetration depth distribution - to a good approximation because of the independence of our energy reconstruction methods of as discussed in Sect. 5.1. These correction factors were derived from spectrally weighted Monte-Carlo data via determining the true (p + )/all in the Monte Carlo that yields the fitted biased (p + )/all in the given reconstructed-energy bin. In this way the ratio of biased to true (p + )/all at the true mean energy of the Monte-Carlo showers in the energy bin for a given energy reconstruction method is obtained. As an illustration the correction factors for the case of energy reconstruction method 3 are shown in Table 2.
For the spectral weighing of the Monte-Carlo sampling a primary-spectrum as obtained from low-energy measurements (Wiebel-Sooth et al. 1998) with a power law index of =-2.67 and a "knee" at 3.4 PeV with a change in the power-law index to =-3.1 was assumed. An iterative repetition of this procedure with the energy spectrum as inferred below from the present data is possible. However, it was found that the contribution to the systematic error introduced by not performing the iterations is negligible for the initial parameters chosen. Two Monte-Carlo samples were used for bias corrections in this work, the Monte-Carlo sample with events continuously distributed in energy, mentioned in Sect. 3, and a "toy Monte-Carlo sample" with unlimited statistics, which was created by randomly choosing all measured parameters (like reconstructed energy, etc.) of a shower with a given true primary energy from one dimensional distributions inferred from the Monte Carlo sample with discrete energies. It was checked that the corrections obtained with these samples are very similar in energy regions where the continuously distributed Monte-Carlo data were available. ## 5.3. Energy spectra, elongation diagrams and penetration depth fluctuationsEnergy spectra obtained with the four energy-reconstruction methods were corrected for the A dependent bias by dividing the flux values in bins with true and reconstructed energy in the Monte-Carlo samples. The chemical composition as determined with the methods in the previous Sect. is used. These factors were applied to the flux in each energy bin when going from reconstructed (Fig. 6) to true energy (Fig. 7).
as a function of true energy is obtained if the mean is plotted at the mean true energy of the events in a given reconstructed-energy bin, as calculated with the measured chemical composition. This procedure leads to correct results as long as the elongation rate of different nuclei is identical; this is fulfilled to a good approximation for all hadron generators. The RMS of the shower penetration depth distributions were directly calculated from the distributions calculated with a given energy-reconstruction method, i.e. no procedure to remove the bias was applied. These results were compared with RMS values from Monte Carlo data treated in the same way. ## 5.4. Experimental statistical and systematic uncertaintiesFor the energy spectrum the statistical uncertainties correspond to the square root of the energy-bin contents N for the energy spectrum and the mean divided by for the penetration depth. In all other cases statistical errors were obtained by changing the fit parameter from its best-fit value until the increases by 1. In case of best fit 's in excess of 1.5 the best fit value of the fit parameter was increased until doubled. Systematic uncertainties of the Monte-Carlo simulation of hadronic
air-showers - estimated by using different hadronic Monte-Carlo
generators - will be considered in a forthcoming paper. Here we
concentrate on experimental uncertainties related to the For the chemical composition, the energy spectrum and the variation
of with energy (elongation rate),
the systematic error was evaluated by changing all
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