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The number of muons in an air shower is a strong indicator of the mass of theprimary particle and increases with a small power of the cosmic ray mass by the$\beta$-exponent, $N_{\mu} \sim A^{(1-\beta)}$. This behaviour can be explainedin terms of the Heitler-Matthews model of hadronic air showers. In this paper,we present a method for calculating $\beta$ from the Heitler-Matthews model.The method has been successfully verified with a series of simulated eventsobserved by the Pierre Auger Observatory at $10^{19}$ eV. To follow realmeasurements of the mass composition at this energy, the generated sampleconsists of a certain fraction of events produced with p, He, N and Fe primaryenergies. Since hadronic interactions at the highest energies can differ fromthose observed at energies reached by terrestrial accelerators, we generate amock data set with $\beta =0.92$ (the canonical value) and $\beta =0.96$ (amore exotic scenario). The method can be applied to measured events todetermine the muon signal for each primary particle as well as the muon scalingfactor and the $\beta$-exponent. Determining the $\beta$-exponent caneffectively constrain the parameters that govern hadronic interactions and helpsolve the so-called muon problem, where hadronic interaction models predict toofew muons relative to observed events. In this paper, we lay the foundation forthe future analysis of measured data from the Pierre Auger Observatory with asimulation study.

Method for calculation of the beta exponent from the Heitler-Matthews model of hadronic air showers