Intermediate models for longitudinal profiles of cosmic showers

Abstract

Cosmic rays impacting on the atmosphere cause particle-showers. Several descriptions exist for the evolution of the shower size along the atmospheric depth. The well known functions for shower profiles, Greisen, Gaisser-Hillas and 'Gaussian in Age', are intimately connected in that they all are approximate solutions of versions of the Rossi and Greisen diffusion equations. The mathematical connection will be demonstrated by means of two simple models for the longitudinal electromagnetic shower profile. Both models can be regarded either as a generalization of the Heitler model or as a simplification of the diffusion model of Rossi and Greisen. These models are far closer to reality than the Heitler model, while they are not as close to reality as the model of Rossi and Greisen. Therefore, they will be referred to as intermediate models. For each intermediate model the evolution of the shower is governed by either a single differential equation or a single integro-differential equation. The approximate solution of the differential equation is a Gaisser-Hillas function and can be adjusted such that it almost matches the Greisen profile. The approximate solution of the integro-differential equation is a 'Gaussian in Age' function. The corresponding profile is, after suitable adjustment, in excellent agreement with the Greisen profile. The analysis also leads to an alternative functional form for the age parameter.

Spallation backgrounds in Super-Kamiokande are made in muon-induced showers

“…Compared to the naive Heitler model, where all electrons stop in one radiation length after shower maximum, the tails of realistic showers are long. This arises from two types of fluc- tuations in showers: the distances particles travel before splitting obey an exponential distribution, and secondary particles do not always split the energy equally [61]. These fluctuations give a distribution to the particle energies in the shower at a given depth, instead of all particles having the same energy at the same location.…”

A branching model for hadronic air showers

“…We utilized the idea of the extended Heitler-Matthews model [4,5] to calculate the muon production in the extended air showers. Despite the simplicity of the model we were able to reproduce MPD profiles for proton and iron primaries.…”

“…The shape of the electromagnetic component can be well reproduced by the intermediate shower model that modifies the original Heitler model, as explained in Ref. [5]. In this study, we perform a similar alteration for the extended Heitler-Matthews model.…”

Alternative description of particle shower longitudinal profile

“…The efficiency of four-parametric parametrization (1) is in its applicability to a wide range of energies and primary nuclei. However, the observed correlations between * Electronic address: samvel@icecube.wisc.edu parameters result in a loss of the physical meaning of X 0 and λ [7] and reduce the range of effective atmosphere depths for Eq. (1).…”