The calculation of charged particles density in Extensive Air Showers (EAS) that reach the earth’s surface is described through estimating the lateral distribution function (LDF) at very high energies of different primary particles. The simulation of LDF is performed through the simulator of air shower which is called AIR-shower Extended Simulations (AIRES) system (2.8.4a version). The LDF simulation is performed for different charged particles like the muons, electron-positron pair production and all charged particles as well as gamma rays at very high energies (1016, 1018 and 1019) eV. The influence of the primary energies, primary particles (such as the proton and the iron nuclei) and the zenith angle (θ) on the charged particles LDF that generated in the EAS has been taken into the account. The calculation of charged particles LDF is fulfilled using “Nishimura-Kamata-Greisen” (NKG) function. The LDF using NKG function is compared with that simulated using AIRES system and gave a good agreement at high energies for (electron and positron) secondary particles, which were initiated by the primary proton.

The simulation analysis of the Extensive Air Showers (EAS) was executed by exploring the longitudinal development employing the AIRES system (version 19.04.00) for several hadronic interaction models (SIBYLL, QGSJET, and EPOS) for high energies. The simulation was performed for different high energies (10 17 , 10 18 , and 10 19 ) eV and two dissimilar primary particles, proton as well iron nuclei, with several zenith angles values (0 o , 10 o , and 30 o ). The shower size of longitudinal development was parameterized using the sigmoidal function (Boltzmann model) and gave a new four parameters as functions of the primary energy between the energy extent (10 17 -10 19 ) eV. The comparison among the acquired results data (the parameterized number of shower particles) along with the experimental results (Pierre Auger experiment) had offered a fascinating matching for the primary proton at the fixed primary energy 10 19 eV for vertical EAS showers.

The importance of investigating ultra-high energy cosmic ray particles interactions was investigated in this work. Different hadronic interaction models such as (SIBYLL, EPOS, and QGSJET) were used air showers simulation AIRES program (version 19.04.00). Also, the shower size of Extensive Air Showers (EAS) was calculated by estimating the longitudinal development. Moreover, the longitudinal development simulation of the two primary particles (iron nuclei and proton) was performed, taking into account their primary energies effect as well the zenith angle for charged particles that produced in the EAS, with energies (1017 and 1019) eV. At such extremely high energies (1017 and 1019) eV, new parameters as a function of the primary energy were obtained by fitting the longitudinal development curves of EAS using the Lorentz function. Comparison of the results showed a good agreement between the values obtained from the parameterized longitudinal development using Lorentz function with experimental results by Pierre Auger EAS observatory as well the simulated results by Sciutto for the primaries iron nuclei as well proton, with the electrons and the charged muons secondary particles of high energies.