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

## TL;DR

This study evaluates different statistical functions to accurately model the distribution of the depth of maximum particle count in ultra-high energy cosmic ray air showers, identifying the Generalized Gumbel as the most effective.

## Contribution

It provides the first comprehensive comparison of multiple functions for modeling $X_{max}$ distributions across various primaries, energies, and hadronic models, with detailed parametrizations.

## Key findings

- Generalized Gumbel best describes $X_{max}$ distribution.
- Log-normal is a good alternative in some cases.
- Exponentially Modified Gaussian performs poorly in most cases.

## Abstract

The distribution of depth in which a cosmic ray air shower reaches its maximum number of particles ($X_{max}$) is studied and parametrized. Three functions are studied for proton, carbon, silicon, and iron primary particles with energies ranging from $10^{17}$ eV to $10^{20}$ eV for three hadronic interaction models: EPOS-LHC, QGSJetII.04, and Sibyll2.3c. The function which best describes the $X_{max}$ distribution of a mixed composition is also studied. A very large number of simulated showers and a detailed analysis procedure were 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 the best functional form to describe the $X_{max}$ distribution. The Generalized Gumbel distribution is shown to be the best option 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.