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Astron. Astrophys. 359, 682-694 (2000)

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5. The analysis methods

Monte Carlo simulations revealed that the distance [FORMULA] 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 slope of the lateral Cherenkov light density distribution:

[EQUATION]

From this relation the distance to the shower maximum is determined with a resolution (i.e. root mean square (RMS) of a [FORMULA](true)-[FORMULA](reconstructed) distribution) ranging from 40 g/cm2 at 300 TeV to 20 g/cm2 at 10 PeV (including all detector effects but no systematic error in the mean [FORMULA]). The most important technical improvement in the data presented here to previous experiments is that these values are distinctly smaller than the width of natural shower fluctuations of proton induced showers in the atmosphere (see below Table 3). This makes the shape of the penetration-depth distribution a sensitive parameter for the chemical composition (see Sect. 7). Simple geometrical relations permit to infer [FORMULA], the depth of the shower maximum in the atmosphere, from [FORMULA]. Relation 2 is only weakly energy dependent (Lindner 1998a). This dependence is neglected here.

5.1. Energy reconstruction

Methods have been developed to reconstruct the primary CR energy from the scintillator and AIROBICC data independently of the primary mass with an accuracy better than 35% (Lindner 1998a; Cortina et al. 1997a; Cortina 1998). However, these methods lead to a relatively strong correlation between reconstructed energy and [FORMULA] (showers with a maximum position that fluctuated to smaller values compared to the mean [FORMULA], are reconstructed with higher energies). In order to infer the chemical composition, a careful modelation of the response function between the variables [FORMULA] (or [FORMULA]) and slope on the one hand and energy and composition (or penetration depth) on the other hand (e.g. via two-dimensional regularised unfolding) is then necessary. Such procedures have been employed in some analyses of HEGRA data (Wiebel-Sooth 1998; Kornmayer 1999). Two reasons lead us to prefer to circumvent the mentioned problem with the use of two simpler energy estimators here.

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 [FORMULA] and slope , or [FORMULA]. Both estimators are used under the assumption that all primary CR are either protons or iron nuclei. These extreme assumptions lead to a bias which then has to be corrected for.

Using [FORMULA] and slope the energy of the primary cosmic-ray nucleus is reconstructed in two basic steps here (Lindner 1998b; Cortina et al. 1997a; Plaga et al. 1995): first slope (a measure of the distance to the shower maximum) is combined with [FORMULA] to estimate the number of particles in the shower maximum which is proportional to the energy contained in the electromagnetic component of the EAS. In the second step a specific primary mass is assumed; with the assumption of primary proton (iron) we denote the methods as 1 (2). This allows to calculate the primary energy from the energy deposited in the electromagnetic component. The following relation was used in our analysis:

[EQUATION]

Here [FORMULA] were obtained from the discrete Monte Carlo data as 0.965, -2.545 (0.890, -2.010) for protons (iron). [FORMULA] is the shower size at the maximum of shower development and is inferred from [FORMULA] as:

[EQUATION]

with [FORMULA] 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 [FORMULA] is a good estimator of the energy contained in the electromagnetic EAS cascades. From simulations the relation

[EQUATION]

was derived, where the coefficients a and b are given as 0.958,-1.810 (0.840,-1.061) for primary protons (iron). slope (the parameter used to estimate the primary mass composition, see next Sect. ) is not involved in this energy reconstruction. For ease of reference the four energy-reconstruction methods are summarised in Table 1. The agreement of analyses based on [FORMULA] and [FORMULA] is a consistency test for the accurate description of the longitudinal shower development by the Monte Carlo simulation.


[TABLE]

Table 1. Summary of the energy reconstruction methods 1-4 as discussed in the text.


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.

[FIGURE] Fig. 2. The bias of the energy reconstruction as a function of primary energy for different primary masses. Shown is the ratio of the reconstructed energy with method 3, divided by the true energy (from the Monte-Carlo simulation). Very similar results are obtained from [FORMULA] and slope . The lines show fits used for convolution procedures to determine the final results (see text).

The distribution of the reconstructed energy compared to the simulated energy is shown for examples in Fig. 3. Note that the energy reconstruction from [FORMULA] alone shows Gaussian distributions while the energy obtained from [FORMULA] and slope exhibits tails to high values which have to be taken into account properly in the analyses. Fig. 4 shows the relative energy resolution achieved for different primary particles and energy reconstruction methods 1 and 3. If [FORMULA] is involved in the energy reconstruction the energy resolution is limited by the experimental accuracy of the shower size determination at the detector level. Due to the smaller [FORMULA] of iron compared to proton induced EAS the accuracy of the energy reconstruction for iron showers is a little worse than for proton showers. The energy resolution obtained from [FORMULA] is mainly determined by fluctuations in the shower development (being larger for proton than iron induced showers) and could not be decreased much by improving the detector.

[FIGURE] Fig. 3. Distribution of the ratio of reconstructed to MC generated energy (300 TeV) for two primaries (left distribution: iron nuclei, right distribution: protons) and the two different energy reconstruction methods discussed in the text. Each distribution is normalised to the same area.

[FIGURE] Fig. 4. The energy resolution obtained for different primary nuclei as a function of the generated MC energy. [FORMULA] in brackets denotes an energy reconstruction that combines the measured shower size at detector level and slope , [FORMULA] denotes the results obtained from the Cherenkov light density alone. The energy reconstruction was always performed assuming that the primaries are protons (referred to as methods 1 (stars in the figure) and 3 (dots in the figure) in the text; this fact is symbolised by the subscript "p" on the energy). The "proton" (filled symbols) "iron" (open symbols) after the colon indicate the primary for which the energy resolution was determined. The lines show fits used for convolution procedures to determine the final results (see text).

In all analyses below we bin the data in six equidistant energy intervals from log10 [FORMULA] [TeV]=2.5 to log10 [FORMULA] [TeV]=4.0 (see e.g. Fig. 9). Event samples defined to contain events in a certain reconstructed-energy interval for the four energy-reconstruction methods then contain events with different true primary energies. It should always be kept in mind that these four samples are not independent because they are all based on the same total data sample.

5.2. Chemical composition

The composition of CR is determined by analysing the EAS penetration depth ([FORMULA]) distributions in intervals of the reconstructed primary energy. Information is contained in the differences of the mean [FORMULA] values for different primaries (protons penetrate about 100-130 g/cm2 deeper than iron in the energy range considered here) and also in the the different fluctuations of the shower maxima position. Including experimental resolution we obtained RMS([FORMULA] and RMS([FORMULA] at 1 PeV). The RMS values of the depth distributions of Monte-Carlo events slightly decrease with rising energy, an effect that is partly due to an improving measurement of slope .

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 [FORMULA] 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 log10 [FORMULA] = 2.5-2.75 and 3.5-3.75 the energy bins between 2.5-3.25 (3.25-4) were fitted with the former (latter) distribution. The MC events used in energy intervals other than the two for which the simulations were done, were shifted in the mean penetration depth according to the elongation rate of the various elements. To avoid any systematic uncertainties related to imperfect parameterisations of the MC distributions and to take into account the statistical uncertainty of the simulated event sample we directly fit the MC generated distributions to the experimental data.

Due to the primary dependent energy-reconstruction method the results for the "fraction of light nuclei" (abbreviated "(p + [FORMULA])/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 + [FORMULA])/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 + [FORMULA])/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 [FORMULA] as discussed in Sect. 5.1. These correction factors were derived from spectrally weighted Monte-Carlo data via determining the true (p + [FORMULA])/all in the Monte Carlo that yields the fitted biased (p + [FORMULA])/all in the given reconstructed-energy bin. In this way the ratio of biased to true (p + [FORMULA])/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.


[TABLE]

Table 2. The fraction of light elements (uncorrected) and correction factor for the A dependent bias with energy reconstruction method 3. The uncorrected ratio has to be multiplied by this factor to yield the unbiased ratio. The energy intervals are specified as the logarithm to the base of ten in units of TeV.


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 [FORMULA]=-2.67 and a "knee" at 3.4 PeV with a change in the power-law index to [FORMULA]=-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, [FORMULA] 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 fluctuations

Energy 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).

[FIGURE] Fig. 5. The differential shower-size spectrum and "light-density at 90 m core distance (L90)" spectrum. The values used for the construction of these spectra, were employed for the energy reconstruction. The full lines indicate the best fit in the range 5.3-6.8 and 4.5-6.1 for [FORMULA] and [FORMULA] respectively. Two power laws which meet in a single flux value ("the knee") were assumed. The best-fit power law indices are (-2.35/-2.92) and (-2.61/-3.13) (before/after) the knee at a position of log10([FORMULA])/log10([FORMULA] = 5.99/5.64 for the shower size/light density respectively. The energy scales were derived under the assumption that the primaries are all protons, resp. iron nuclei for shower size (upper scales) and light density (lower scales).

[FIGURE] Fig. 6. The integral energy spectra obtained with the four energy reconstruction methods denoted with different symbols. Filled square: Method 1, open square: method 2, filled triangle: method 3, open triangle: method 4. The biases of the energy reconstruction have not been corrected for.

[FIGURE] Fig. 7. Integral cosmic-ray spectrum corrected for the A dependent bias. The shaded area denotes the systematic error. Symbols are the same as in the previous figure.

[FORMULA] as a function of true energy is obtained if the mean [FORMULA] 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 uncertainties

For the energy spectrum the statistical uncertainties correspond to the square root of the energy-bin contents N for the energy spectrum and the mean [FORMULA] divided by [FORMULA] for the penetration depth. In all other cases statistical errors were obtained by changing the fit parameter from its best-fit value until the [FORMULA] increases by 1. In case of best fit [FORMULA]'s in excess of 1.5 the best fit value of the fit parameter was increased until [FORMULA] 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 slope reconstruction. These are contributions from remaining uncertainties in the characteristics of the AIROBICC amplifier (3% uncertainty for slope ) and non-perfect knowledge of the layer structure and the light absorption of the atmosphere above the detector (4% and 2%). Models of the atmosphere have been carefully checked using the large statistics of photon induced air showers which were registered with the HEGRA imaging air Cherenkov telescopes in 1997 (Konopelko et al. 1999). Added in quadrature the systematic uncertainty of slope amounts to 5%. The mean [FORMULA] for 300 TeV proton (iron) induced showers is then determined with an uncertainty in the absolute values of 20 (13) g/cm2. The uncertainties for different primaries are strongly correlated.

For the chemical composition, the energy spectrum and the variation of [FORMULA] with energy (elongation rate), the systematic error was evaluated by changing all slopes by 5% (the systematic error of this parameter) up or down. The whole analysis, including energy reconstruction, was then repeated and the deviation of the results thus obtained to the original ones was taken as the systematic error (errors beyond the tick mark in Figs. 10, 11 and shaded bands in Figs. 7,8). The shaded band in Fig. 11 is obtained by varying the best-fit composition within its total systematic and statistical error. In the case of the elongation rate the systematic error was found to be dominated by the differences in the four energy-reconstruction methods, this is dicussed in detail in the Results Sect. 6.4. In case of the RMS of the penetration depths, the spectral fit parameters (knee position, power-law indices) and the elongation rate, the systematic error was estimated as the sample standard deviation of the best fit parameters obtained with the four energy-reconstruction methods.

[FIGURE] Fig. 8. The differential CR energy spectrum obtained with four energy reconstruction methods. Symbols are the same as in Fig. 6. The light shaded region represents systematical uncertainty. The "star" with vertical error bars shows the uncertainty of the HEGRA data originating from the 10[FORMULA] systematic uncertainty of the absolute energy scale that we estimate from the uncertainty in the determination of the absolute [FORMULA] scale. The full line is the best fit as described in the text.

[FIGURE] Fig. 9. The fit of MC expectations for light and heavier nuclei to the measured shower maximum depth distribution in the analysed reconstructed-energy intervals for method 3. The numbers in the upper left corner are the logarithms to the base of ten of the energy-bin boundaries in TeV. The full dots mark the experimental data with statistical errors, the crosses with error bars are the fitted MC distribution (where the errors correspond to the MC statistics). The two components fitted to the data are shown as dark shaded (large penetrations depths, light nuclei) and light shaded (heavy nuclei) histograms. Details of the procedure are described in the text.

[FIGURE] Fig. 10. The corrected fraction of light nuclei determined with four different energy reconstruction methods (see text for details). Each data point is plotted at the true mean energy of the events used to infer the mean [FORMULA]. The error bars are statistical and systematic error added in quadrature (up to the tick-mark: only statistical error). The statistical errors are correlated due to the use of an identical Monte-Carlo sample in the first and last three energy intervals. The shaded band shows the allowed region between a polynomial fit to the upper and lower ends of the error bars.

[FIGURE] Fig. 11. The mean shower maximum depth as a function of energy using QGSJET simulations to model the EAS development. To obtain an unbiased elongation plot each data point is plotted at the true mean energy of the events used to infer the mean [FORMULA]. The shaded region indicates the region expected from our best fit composition within its total error. Errors are statistical and systematical errors added in quadrature, up to the tick mark only statistical. Up to the highest energies the systematical error dominates.

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Online publication: July 7, 2000
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