Simulation study of the correlation ($X_{max}^{\mu }$, $N^{\mu }$) in view of obtaining information on primary mass of the UHECRs

In this paper we study, using Monte Carlo simulations, the possibility to discriminate the mass of the Ultra High Energy Cosmic Rays (UHECRs) by combining information obtained from the maximum $X_{max}^{\mu }$ of the muon production rate longitudinal profile of Extensive Air Showers (EAS) and the number of muons, $N^{\mu }$, which hit an array of detectors located in the horizontal plane. We investigate the sensitivity of the 2D distribution $X_{max}^{\mu }$ versus $N^{\mu }$ to the mass of the primary particle generating the air shower. To this purpose we analyze a set of CORSIKA showers induced by protons and iron nuclei at energies of ${10}^{19}$eV and ${10}^{20}$eV, at five angles of incidence, $0^{\circ }$, ${37}^{\circ }$, ${48}^{\circ }$, ${55}^{\circ }$ and ${60}^{\circ }$. Using the simulations we obtain the 2D Probability Functions $Prob(X_{max}^{\mu },N^{\mu }\mathrm{\mid }p)$ and $Prob(X_{max}^{\mu},N^{\mu} \ |
Fe)$ which give the probability that a shower induced by a proton or iron nucleus contributes to a specific point on the plane ($X_{max}^{\mu }$, $N^{\mu }$). Then we construct the probability functions $Prob(p\ |
X_{max}^{\mu},N^{\mu})$and$Prob(Fe \ | \ X_{max}^{\mu},N^{\mu})$ which give the probability that a certain point on the plane ($X_{max}^{\mu }$, $N^{\mu }$) corresponds to a shower initiated by a proton or an iron nucleus, respectively. Finally, a test of this procedure using a Bayesian approach, confirms an improved accuracy of the primary mass estimation in comparison with the results obtained using only the $X_{max}^{\mu }$ distributions.