Method for calculation of the beta exponent from the Heitler-Matthews model of hadronic air showers
The number of muons in an air shower is a strong indicator of the mass of the primary 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 explained in 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 events observed by the Pierre Auger Observatory at $10^{19}$ eV. To follow real measurements of the mass composition at this energy, the generated sample consists of a certain fraction of events produced with p, He, N and Fe primary energies. Since hadronic interactions at the highest energies can differ from those observed at energies reached by terrestrial accelerators, we generate a mock data set with $\beta =0.92$ (the canonical value) and $\beta =0.96$ (a more exotic scenario). The method can be applied to measured events to determine the muon signal for each primary particle as well as the muon scaling factor and the $\beta$-exponent. Determining the $\beta$-exponent can effectively constrain the parameters that govern hadronic interactions and help solve the so-called muon problem, where hadronic interaction models predict too few muons relative to observed events. In this paper, we lay the foundation for the future analysis of measured data from the Pierre Auger Observatory with a simulation study.