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We present a general method for estimating the number of particles impinging on a segmented counter or, in general, on a counter with sub-units. We account for unresolved particles, i.e., the effect of two or more particles hitting the same sub-unit almost simultaneously. To achieve full time resolution we account for the dead time that occurs after the first time-bin of a particle signal. This general counting method can be applied to counting muons in existing detectors like the Underground Muon Detector of the Pierre Auger Observatory. We therefore use the latter as a study case to test the performance of our method and to compare it to other methods from literature. Our method proves to perform with little bias, and also provides an estimate of the number of particles as a function of time (as seen by the detector) to a single time-bin resolution. In this context, the new method can be useful for reconstructing parameters sensitive to cosmic ray mass, which are key to unveiling the origin of cosmic rays., Comment: Accepted for publication in EPJ C

Multiple experiments reported evidences of a muon deficit in air shower simulations with respect to data, which increases with the primary energy. In this work, we study the muon deficit using measurements of the muon density at $1000\,$m from the shower axis obtained by the Akeno Giant Air Shower Array (AGASA). The selected events have reconstructed energies in the range $18.83\,\leq\,\log_{10}(E_{R}/\textrm{eV})\,\leq\,19.46$ and zenith angles $\theta\leq 36^{\circ}$. We compare these muon density measurements to proton, iron, and mixed composition scenarios, obtained by using the high-energy hadronic interaction models EPOS-LHC, QGSJetII-04, and Sibyll2.3c. We find that AGASA data are compatible with a heavier composition, lying above the predictions of the mixed composition scenarios. The average muon density divided by the energy in AGASA data is greater than in the mixed composition scenarios by a factor of $1.49\pm0.11\,\textrm{(stat)}\pm^{0.49}_{0.30}\,\textrm{(syst)}$, $1.54\pm0.12\,\textrm{(stat)}\pm^{0.50}_{0.31}\,\textrm{(syst)}$, and $1.66\pm0.13\,\textrm{(stat)} \pm ^{0.54}_{0.34}\,\textrm{(syst)}$ for EPOS-LHC, Sibyll2.3c, and QGSJetII-04, respectively. We interpret this as further evidence of a muon deficit in air shower simulations at the highest energies., Comment: Proceedings of the 37th International Cosmic Ray Conference (ICRC 2021)

Recently, several experiments reported a muon deficit in air shower simulations with respect to the data. This problem can be studied using an estimator that quantifies the relative muon content of the data with respect to those of proton and iron Monte Carlo air shower simulations. We analyze two estimators. The first one, based on the logarithm of the mean of the muon content, is built from experimental considerations. It is ideal for comparing results from different experiments as it is independent of the detector resolution. The second estimator is based on the mean of the logarithm of the muon content, which implies that it depends on shower-to-shower fluctuations. It is linked to the mean-logarithmic mass $\left \langle \ln A \right \rangle$ through the Heitler-Matthews model. We study the properties of the estimators and their biases considering the knowns and unknowns of typical experiments. Furthermore, we study these effects in measurements of the muon density at $1000\,$m from the shower axis obtained by the Akeno Giant Air Shower Array (AGASA). Finally, we report the estimates of the relative muon content of the AGASA data, which support a muon deficit in simulations. These estimates constitute valuable additional information of the muon content of air showers at the highest energies., Comment: Proceedings of the 37th International Cosmic Ray Conference (ICRC 2021)

In this work, direct measurements of the muon density at $1000\,\textrm{m}$ from the shower axis obtained by the Akeno Giant Air Shower Array (AGASA) are analysed. The selected events have zenith angles $\theta \leq 36^{\textrm{o}}$ and reconstructed energies in the range $18.83\,\leq\,\log_{10}(E_{R}/\textrm{eV})\,\leq\,19.46$. These are compared to the predictions corresponding to proton, iron, and mixed composition scenarios obtained by using the high-energy hadronic interaction models EPOS-LHC, QGSJetII-04, and Sibyll2.3c. The mass fractions of the mixed composition scenarios are taken from the fits to the depth of the shower maximum distributions performed by the Pierre Auger Collaboration. The cross-calibrated energy scale from the Spectrum Working Group [D. Ivanov, for the Pierre Auger Collaboration and the Telescope Array Collaboration, PoS(ICRC2017) 498 (2017)] is used to combine results from different experiments. The analysis shows that the AGASA data are compatible with a heavier composition with respect to the one predicted by the mixed composition scenarios. Interpreting this as a muon deficit in air shower simulations, the incompatibility is quantified. The muon density obtained from AGASA data is greater than that of the mixed composition scenarios by a factor of $1.49\pm0.11\,\textrm{(stat)}\pm0.18\,\textrm{(syst)}$, $1.54\pm0.12\,\textrm{(stat)}\pm0.18\,\textrm{(syst)}$, and $1.66\pm0.13\,\textrm{(stat)}\pm0.20\,\textrm{(syst)}$ for EPOS-LHC, Sibyll2.3c, and QGSJetII-04, respectively., Comment: 11 pages, 7 figures, to be published in Physical Review D