Reinterpreting the development of extensive air showers initiated by nuclei and photons
Ultra-high energy cosmic rays (UHECRs) interacting with the atmosphere generate extensive air showers (EAS) of secondary particles. The depth corresponding to the maximum development of the shower, X{sub max}, is a well-known observable for determining the nature of the primary cosmic ray which initiated the cascade process. In this paper, we present an empirical model to describe the distribution of X{sub max} for EAS initiated by nuclei, in the energy range from 10{sup 17} eV up to 10{sup 21} eV, and by photons, in the energy range from 10{sup 17} eV up to 10{sup 19.6} eV. Our model adopts the generalized Gumbel distribution motivated by the relationship between the generalized Gumbel statistics and the distribution of the sum of non-identically distributed variables in dissipative stochastic systems. We provide an analytical expression for describing the X{sub max} distribution for photons and for nuclei, and for their first two statistical moments, namely (X{sub max}) and σ{sup 2}(X{sub max}). The impact of the hadronic interaction model is investigated in detail, even in the case of the most up-to-date models accounting for LHC observations. We also briefly discuss the differences with a more classical approach and an application to the experimental data based on information theory.