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

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6. Results

In this Sect. the methods explained in Sect. 5 are applied to the data set discussed in Sect. 4.

6.1. [FORMULA] and [FORMULA] spectra

Fig. 5 shows the [FORMULA] and [FORMULA] spectra. These spectra display a relatively sharp knee at values consistent with a primary energy for the knee as determined below.

6.2. Energy spectra

Fig. 6 displays the integral energy spectra uncorrected for an A dependent bias obtained with the four reconstruction methods. The differences in absolute normalisation and spectral slope originate from the different mass dependent biases. After the correction of the chemical bias the integral spectra are similar (Fig. 7). This nontrivial fact is in favour of the internal consistency of data analysed here; a longitudinal shower development different from the one predicted by the Monte Carlo or errors in the calibration of [FORMULA] and [FORMULA] could have spoiled the agreement of spectra obtained with different energy reconstructions. The differential energy spectrum is shown in Fig. 8. A steepening of the energy spectrum is visible around an energy of 4 PeV. There seems to be no "fine structure" in the energy spectrum around the knee in excess of 20 [FORMULA]. Apparent structure with smaller amplitudes that appears in the spectrum reconstructed with a given energy-reconstruction method is not reproduced with other methods. This is expected due to the A dependent bias of our energy-reconstruction methods (see Fig. 2). Note that with these methods, a potential structure in the energy spectrum consisting of different nuclei is smeared out. If two different power laws, smoothly connected at the knee (corresponding to a "sharp" knee), are fitted to the differential spectra we obtain:

  • a "knee" position of

    E(Knee) = [FORMULA] PeV,

  • a spectral index of [FORMULA] below and [FORMULA] above the knee.

The reduced [FORMULA] values of the fits to the differential spectrum (12 d.o.f.) were 6.75, 4.03, 3.53 and 1.47 with energy reconstruction methods 1 to 4 respectively. Some of these values are much larger than one. It is then difficult to specify a statistical error; we specify the statistical errors for method 4 that has a marginally acceptable reduced [FORMULA] value. The large [FORMULA] values for the analysis with energy-reconstruction methods 1-3 can be interpreted as an argument in favour of a knee not absolutely sharp in energy. However, the fact that one of the fits is acceptable on the 90 [FORMULA] confidence level means that we cannot reject the hypothesis of a "sharp" knee (two power laws with no transition region) within our systematic errors. The large statistical error on the knee position further indicates that we cannot reject the hypothesis of a spectrum without a knee in the limited energy range of this analysis with high significance.

The spectral index for the spectrum below the knee is consistent with direct measurements at lower energies (Wiebel-Sooth et al. 1998) and a recent Cherenkov-light based determination of the spectral index in the TeV range (Aharonian et al. 1999); there is therefore no evidence for any change in spectral index from the TeV range right up to the knee.

6.3. Composition of CR

The fraction of light nuclei as a function of reconstructed energy - obtained from the fits to the measured penetration-depth distributions (see Sect. 5.2)- is presented in Fig. 10. At energies below the knee the composition is mixed and consistent with direct measurements around 100 TeV, namely ([FORMULA])/all = 0.54 [FORMULA] 0.08 (Watson 1997). The data points seem to indicate a gradual enrichment of heavy elements above about 1 PeV though the error bars are large (remember that there are only six independent data points). We will discuss in Sect. 7 how reliable the qualitative conclusion of a gradual enrichment in heavy elements is within our systematic errors. The data rule out a predominantly light composition at all energies and does not give evidence for a drastic change of composition at the knee.

6.4. Elongation rate

Fig. 11 shows the corrected mean shower maximum depth as a function of energy. A least-squares fit to the [FORMULA] values as a function of energy, using only the statistical errors,


yields an elongation rate ER=78.3 [FORMULA] 1.0 (stat) [FORMULA] 6.2 (syst) g/cm2 and mean depth parameter ERB=243.1 [FORMULA] 2.6 (stat) [FORMULA] 15.7 (syst) g/cm2. The specified mean values and statistical errors are the mean of fit values with the four energy-reconstruction methods. The systematic error is estimated as the standard deviation of the mean values inferred with the four energy-reconstruction methods. The systematic error introduced by the systematic uncertainty in slope is smaller (about 3 and 14 g/cm2 for ER and ERB respectively). The reduced [FORMULA] values of the fit to relation (6) (4 d.o.f.) are very large (6.6,9.2,17.2,23.5) for energy-reconstruction methods 1-4, i.e the systematic errors dominate over the rather small statistical errors for the mean [FORMULA]. Therefore the specified estimates of the statistical errors obtained with the procedure explained in Sect. 5.4 have to be treated with caution. The data point at the highest energy lies about 20 g/cm2 higher in the atmosphere than expected for a constant elongation rate.

These results are not in contradiction with previous measurements in this energy range (Wdowczyk 1994; Turver 1992). This elongation rate, and also the absolute [FORMULA], is consistent with data at higher energies, obtained mainly by the Yakutsk and Fly's Eye collaboration (Watson 1997). A constant elongation rate of [FORMULA] 73 g/cm2 from 300 TeV up to 107 TeV (dotted fit line in the summary diagram 10 in Watson 1997) is an intriguing hypothesis which is not in contradiction with our data.

6.5. Fluctuation of shower penetration depth

The RMS of the penetration depth distributions - calculated in reconstructed-energy bins, i.e. biased in favour of the light component of CR especially at low energies - is shown in Table 3. It does not show any obvious trend towards a heavy composition. Therefore the fact that the composition at the highest energy seems to be heavy with all energy reconstruction methods (Fig. 10) is mainly determined by the fact that the [FORMULA] in the highest energy bin lies about 20 g/cm2 below a constant elongation rate.


Table 3. The RMS of the penetration depth distributions [g/cm2] as a function of reconstructed energy (given in the same units as in Table 1) in the data and spectral Monte-Carlo sample. Given is the value inferred for the energy bins as defined with energy-reconstruction method 3, i.e. the specified values contain an A dependent bias. The first error is statistical and the second systematic (due to the systematic error in slope). For the numbers from Monte-Carlo simulations only a statistical error is given. "Mixed composition" represents the expectation for our best-fit chemical composition. Based on numerical experiments the statistical error was taken as the inferred value divided by [FORMULA] rather than half of that, as would be correct for Gaussians, due to the non-Gaussian tails of the distribution. The comparison between experimental data and Monte Carlo simulations shows no trend towards a heavy composition at the higher energies.

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000