Hadron diffusion equations with energy-dependent interaction mean free paths and inelasticities are solved using the Mellin transform. Instead of using operators on the finite difference terms, the Mellin transformed equations are expanded in a Taylor series to a first-order partial differential equation in atmospheric depth t and in transform parameter s. These equations are then solved by the method of characteristics. The hadron fluxes (nucleon and pion) in real space are evaluated by the method of residues. For the case of a regularized power law primary spectrum, these hadron fluxes are given by simple residues and one, never before mentioned, essential residue. Comparisons of our solutions are made with the nucleon flux measured at sea level and with the hadron fluxes measured at t = 840 g cm−2 and at sea level. The integral hadron flux is also compared with experimental data from the Mt Fuji Collaboration (t = 650 g cm−2). The agreement between all of them is very good in general, better than 90%.
The diffusion equation of cosmic-ray nucleons is exactly integrated using
the successive approximation method for a general distribution of the
primary component, and taking into account the rising nucleon-air
cross sections with energy. The interaction probability law for the
nucleon in the atmosphere is obtained as a consequence of the respective
diffusion equation. If the nucleon-air cross sections rise logarithmically,
this probability law assumes a binomial form, and for the constant cross
section, it is purely Poissonian. The well known approximate solution is
compared with our exact solution. It is found that the former always gives
a nucleon number greater than ours by, for example, 15-25% in the
energy region 30-10 000 GeV at sea level in the case of the mean
inelasticity ⟨κ⟩ = 0.60. It is also shown that a fairly
accurate description of nucleon flux at sea level (1030 g cm-2) and hadron intensities at 840 g cm-2 and at
1030 g cm-2 are obtained with ⟨κ⟩
varying between 0.55 and 0.60.
Hadron diffusion equations are solved using an alternative analytical method based on depth-like ordered exponential operators, similar to those used by Feynman. With this method, these equations are solvable for any form of the primary spectrum (an improvement compared with other methods). The muon fluxes generated by these hadronic showers are then obtained for zenith angles covering 0°–89°. A comparison of our calculations for the vertical and horizontal muon fluxes with experimental data and with another theoretical calculation is made. The agreement between them is in general very good, greater than 90%.
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