Ultra-high energy cosmic rays generate extensive air showers in Earth's atmosphere. A standard approach to reconstruct the energy of an ultra-high energy cosmic rays is to sample the lateral profile of the particle density on the ground of the air shower with an array of surface detectors. For cosmic rays with large inclinations, this reconstruction is based on a model of the lateral profile of the muon density observed on the ground, which is fitted to the observed muon densities in individual surface detectors. The best models for this task are derived from detailed Monte-Carlo simulations of the air shower development. We present a phenomenological parametrization scheme which allows to derive a model of the average lateral profile of the muon density directly from a fit to a set of individual Monte-Carlo simulated air showers. The model reproduces the detailed simulations with a high precision. As an example, we generate a muon density model which is valid in the energy range 1e18 eV < E < 1e20 eV and the zenith angle range 60 deg < theta < 90 deg. We will further demonstrate a way to speed up the simulation of such muon profiles by three orders of magnitude, if only the muons in the shower are of interest.
Deep Dive into A phenomenological model of the muon density profile on the ground of very inclined air showers.
Ultra-high energy cosmic rays generate extensive air showers in Earth’s atmosphere. A standard approach to reconstruct the energy of an ultra-high energy cosmic rays is to sample the lateral profile of the particle density on the ground of the air shower with an array of surface detectors. For cosmic rays with large inclinations, this reconstruction is based on a model of the lateral profile of the muon density observed on the ground, which is fitted to the observed muon densities in individual surface detectors. The best models for this task are derived from detailed Monte-Carlo simulations of the air shower development. We present a phenomenological parametrization scheme which allows to derive a model of the average lateral profile of the muon density directly from a fit to a set of individual Monte-Carlo simulated air showers. The model reproduces the detailed simulations with a high precision. As an example, we generate a muon density model which is valid in the energy range 1e18
Ultra-high energy cosmic rays (UHECRs) are cosmic rays with energies above 1 EeV = 10 18 eV. They have been under study for several decades, still their origin is not well known. Information about the origin is encoded in the energy spectrum [1], the mass composition [2] and a possible anisotropy of their arrival directions [3], which can be measured experimentally.
The huge energy and the low flux (roughly 1 particle per km 2 per year above 10 EeV) make a direct measurement of the momentum and energy of a UHECR with balloon or satellite experiments unfeasible. Instead, they are observed indirectly with ground-based detectors that use Earth’s atmosphere as a large calorimeter. The interactions of the UHECR with atmospheric nuclei generate an extensive particle shower which is sampled by these detectors.
One realization of such a detector is a large surface array of particle counters. The array samples the lateral profile of the particle density of the shower which consist mainly of photons, electrons, and muons. The arrival direction of the UHECR can be obtained rather directly from the measured arrival time of the shower front in individual particle counters. The reconstruction of the energy E of the cosmic ray is more complex and requires to fit a model of the lateral particle distribution around the shower axis to the measured particle counts.
Most surface array experiments concentrate on showers with inclinations less than 60 • . For such showers, the particle distribution is radially symmetric in good approximation and well described by comparably simple empirical models of the NKG-type [4,5]. At larger inclinations, the effect of the geomagnetic field on the particle distribution cannot be neglected. The symmetry becomes broken and the NKG-type models fail to describe the particle distribution.
In an ideal surface array, these very inclined showers constitute 25 % of the number of arriving events. Recovering them yields a significant gain in the event statistics of the experiment. The ability to reconstruct very inclined showers also increases the field of view of the detector and thus the total observable region of the sky. This is particularly relevant for anisotropy searches.
It was first demonstrated with the Water-Cherenkov detectors of the Haverah Park experiment [6] that the energy E of cosmic rays at large zenith angles θ > 60 • can be derived from the total number of muons N µ which arrive at the ground [7,8]. The same approach is now used by the Pierre Auger Observatory [9]. The number of muons N µ on the ground is obtained from a fit of a model of the average lateral profile of the muon density 1 n µ on the ground to the measured signals. This fit exploits the following factorization
whereas N µ is the number of muons on the ground which depends only on the energy E, mass A, and inclination θ of the cosmic ray, while f µ is a normalized lateral profile of the muon density which depends only on the ground coordinates (x, y) and the shower direction (θ, φ). The normalized profile f µ also depends on the properties of the observation site like the ground altitude, the geomagnetic field and the atmosphere, but those are considered fixed here. In simple terms, the factorization says that the shape of the lateral profile remains the same for all showers arriving from a certain direction in very good approximation, while its amplitude carries all the information about the energy E and mass A of the cosmic ray. This invariance of the profile shape is called shower universality.
The universality is very useful, because f µ is too complex to be fitted from the data sampled by the surface array on an event-by-event basis. It has to be modeled. However, if f µ is predicted by a model, the reconstruction of N µ reduces to a fit of three parameters: the two intersection coordinates of the shower axis with the ground and the amplitude N µ .
This reconstruction approach even works if the particle counters of the surface detector cannot distinguish between different species of charged particles. Very inclined air showers arrive in a very late stage of their development on the ground, where the only remaining electromagnetic particles in the shower are generated by the muons themselves, mostly via decay. Therefore, signals generated by such old showers remain proportional to the local muon density n µ , because the electromagnetic particles only enhance the signal by a constant factor in first approximation [10].
Models of the normalized profile f µ are currently derived from detailed Monte-Carlo simulations of extensive air showers, which represent best our current theoretical knowledge. However, it is not feasible to make a Monte-Carlo simulation of f µ for every possible shower direction (θ, φ), simply because these simulations consume considerable computing time and and storage space. The established solution to this issue is a semi-analytical model of f µ , which is based on detailed simulation
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