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Analytical description of the distributions of primary and secondary cosmogenic particles

In this work, we present an analytical description of the energy distributions of primary and secondary cosmogenic particles on Earth in terms of parameters having clear physical meaning. A modified power law is assumed for energy distributions, incorporating terms such as energy loss/decay, which are effective at low energies, and a source term, which is dominant at high energies. The parametrizations of the momentum distribution of primary protons and helium have been obtained including energy loss term. For muons, both the energy loss and decay terms have been included. It is shown analytically that zenith angle distributions is given by $\cos^{n-1}θ$ in terms of energy index $n$ and the presence of decay term does not affect it. The analytical function describes the muon momentum distribution data at different altitudes and zenith angles. The same form is also applied to describe the atmospheric muon and electron-type neutrino distributions simulated at various sites. The presented analytical functions provide an excellent description of all kinds of cosmogenic particles.