Estimation of the number of muons with muon counters

Abstract

The origin and nature of the cosmic rays is still uncertain. However, a big progress has been achieved in recent years due to the good quality data provided by current and recent cosmic-rays observatories. The cosmic ray flux decreases very fast with energy in such a way that for energies $\gtrsim {10}^{15}$ eV, the study of these very energetic particles is performed by using ground based detectors. These detectors are able to detect the atmospheric air showers generated by the cosmic rays as a consequence of their interactions with the molecules of the Earth’s atmosphere. One of the most important observables that can help to understand the origin of the cosmic rays is the composition profile as a function of primary energy. Since the primary particle cannot be observed directly, its chemical composition has to be inferred from parameters of the showers that are very sensitive to the primary mass. The two parameters more sensitive to the composition of the primary are the atmospheric depth of the shower maximum and the muon content of the showers. Past and current cosmic-rays observatories have been using muon counters with the main purpose of measuring the muon content of the showers. Motivated by this fact, in this work we study in detail the estimation of the number of muons that hit a muon counter, which is limited by the number of segments of the counters and by the pile-up effect. We consider as study cases muon counters with segmentation corresponding to the underground muon detectors of the Pierre Auger Observatory that are currently taking data, and the one corresponding to the muon counters of the AGASA Observatory, which stopped taking data in 2004.

Introduction

The cosmic ray energy spectrum extends over several orders of magnitude in energy. The highest primary energies observed at present are of the order of ${10}^{20}$ eV. Above $\sim {10}^{15}$ eV the cosmic ray flux is so small that these very energetic particles are studied by means of ground-based detectors, which are able to detect the atmospheric air showers generated by the cosmic ray interactions with the molecules of the atmosphere. Since the primary particle is not observed directly, its energy, arrival direction, and chemical composition have to be inferred from the shower information obtained by the detectors.

The origin of the cosmic rays is still unknown. The three main observables used to study their nature are: The energy spectrum, the distribution of their arrival directions, and the chemical composition. The composition of the primary particle has to be inferred from different properties of the showers. The most sensitive parameters to the primary mass are the atmospheric depth at which the shower reaches its maximum development and the muon content of the showers [1], [2], [3]. In general, the muon density at a given distance to the shower axis is used in composition analyses.

The composition profile as a function of energy is very important to understand several aspects of the cosmic-ray physics. In particular, at the highest energies the composition information plays an important role to find the transition between the galactic and extragalactic components of the cosmic rays [4], [5] and to elucidate the origin of the suppression observed at $\sim {10}^{19.7}$ eV [6]. The composition analyses are subject to large systematic uncertainties originated by the lack of knowledge of the hadronic interactions at the highest energies (see for instance [7]). The composition is determined by comparing experimental data with simulations of the atmospheric showers and the detectors (when it corresponds). The showers are simulated by using high-energy hadronic interaction models that extrapolate low-energy accelerator data to the highest energies. This practice introduces large systematic uncertainties even when models updated using the Large Hadron Collider data are considered. Moreover, experimental evidence has been found recently about a muon deficit in shower simulations [8], [9]. Even though this is an important limitation for composition analyses, it is expected that mass-sensitive parameters, obtained with the next generation of high-energy hadronic interaction models, present smaller differences allowing for a reduction of the systematic uncertainties introduced by those models.

Past and current cosmic-rays experiments have been measuring muons by using different types of detectors [9]. A particular class of detector is the muon counter. This type of detectors has been used in the past in the Akeno Giant Air Shower Array (AGASA) [10] and at present in the Pierre Auger Observatory [11]. The muon counters are designed to count muons through a segmented detector. The segments of the Auger muon counters are scintillator bars whereas the segments of the AGASA muon counters were proportional counters. The limitation to measure a given number of incident muons is given by the number of segments of the counters.

In general, the principle of operation of the muon counters is based on a binary logic in which each channel of the electronics, associated to a given segment of the detector, is able to differentiate between a state in which the signal is larger than a given threshold level and the one in which it is smaller. The threshold level is chosen in such a way that almost all muons can be identified, i.e. the efficiency of each segment is close to $100\%$. For the case in which the signal is larger than the threshold level, the segment is said to be on and otherwise off. The time structure of the signal corresponding to one muon limits the time interval in which it is possible to identify single muons. This leads to the definition of a time interval, usually called inhibition window, in which it is decided whether a given segment is on or off. As a consequence, if one or more muons hit the same segment in a time interval of the order of the one corresponding to the inhibition window, the segment is tagged as on, losing the information about the number of muons that hit that segment. Therefore, when a given number of muons hit a muon counter in a time interval of the order of the one corresponding to the inhibition window, a number $k$ of segments on is obtained. If the number of incident muons is much smaller than the number of segments, the random variable $k$ is close to the number of muons that hit the counter. However, if the number of incident muons is close to the number of segments, the variable $k$ becomes much smaller than the number of incident muons. This effect is known as pile up [1].

In this work, we find an analytic expression for the distribution function of $k,$ the number of segments on, given the number of incident muons, $n_{\mu }$. In this case, $k$ is the random variable and $n_{\mu }$ is taken as a parameter of the distribution. The expressions for the mean value and the variance of $k$ are inferred by using the new formula, these expressions are equal to the ones obtained in Ref. [1] by using a different approach. We also study how to estimate the parameter $n_{\mu }$ from measured values of $k$ and how to obtain a confidence interval. These studies are done for 192 segments, which correspond to the total number of segments of the Auger muon counters and 50 segments that correspond to the number of segments of muon counters used in AGASA.

It is worth mentioning that the main purpose of the muon counters is the reconstruction of the muon lateral distribution function (MLDF), i.e. the muon density as a function of the distance to the shower axis, which is proportional to the mean value of $n_{\mu }$. Even though the estimation of the mean value of $n_{\mu },$ which is studied in Ref. [12], is closely related to the determination of the MLDF, the estimation of $n_{\mu }$ is also important for different types of applications. In particular, it is important for the method used to reconstruct the MLDF developed in [1], in which an estimator of the number of muon that hit a given muon counter is inserted in the Poisson likelihood that approximates the exact likelihood in a given range of $n_{\mu }$. Also, the estimation of $n_{\mu }$ is necessary to obtain the calibration curve of the integrator (a complementary acquisition mode that muon counter can have), which is given by the mapping of the number of incident muons into the integrated signal [11]. The estimation of $n_{\mu }$ is also relevant for studies related to the signal fluctuations and systematic uncertainties performed by using twin muon counters [13].