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.

中文翻译：

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超高能大风淋雨淋最大深度分布参数化研究

研究并参数化宇宙射线风淋达到其最大粒子数 ($X_{max}$) 时的深度分布。研究了三种强子相互作用模型的质子、碳、硅和铁初级粒子的三个函数，其能量范围从 $10^{17}$ eV 到 $10^{20}$ eV：EPOS-LHC、QGSJetII.04 和 Sibyll2 .3c。还研究了最能描述混合组合的 $X_{max}$ 分布的函数。大量的模拟淋浴和详细的分析程序被用来保证在最终结果中的欠采样和拟合的影响可以忽略不计。第一次在相同的假设下比较了几个函数，目的是选择最好的函数形式来描述 $X_{max}$ 分布。广义 Gumbel 分布被证明是对所有情况进行一般描述的最佳选择。对数正态分布在某些情况下也是一个不错的选择，而指数修正高斯分布在几乎所有研究的情况下都被证明是最差的选择。所有三个函数都被参数化为能量和初级质量的函数。