On the parametrization of the distributions of depth of shower maximum of ultra-high energy extensive air showers

The distribution of depth in which a cosmic ray air shower reaches its maximum number of particles (Xmax) is studied and parametrized. Three functions are studied for proton, carbon, silicon, and iron primary particles with energies ranging from 1017 eV to 1020 eV for three hadronic interaction models: EPOS-LHC, QGSJETII.04, and SIBYLL2.3C. The function which best describes the Xmax distribution of a mixed composition is also studied. A very large number of simulated showers and a detailed analysis procedure are used to guarantee negligible effects of undersampling and of fitting in the final results. For the first time, a comparison of several functions is presented under the same assumption with the intention of selecting within them the best functional form to describe the Xmax distribution. The Generalized Gumbel distribution is shown to be the best option among the options for a general description of all cases. The Log-normal distribution is also a good choice for some cases while the Exponentially Modified Gaussian distribution has shown to be the worst choice in almost all cases studied. All three functions are parametrized as a function of energy and primary mass. © 2019 Elsevier B.V.