Intermediate models for longitudinal profiles of cosmic showers

TL;DR

This paper introduces intermediate models for cosmic ray shower profiles, bridging simple Heitler models and complex diffusion models, and demonstrates their mathematical connections and practical accuracy in describing shower evolution.

Contribution

It presents two new intermediate models for cosmic shower profiles that are mathematically connected to existing functions and closely match observed profiles, improving understanding of shower evolution.

Findings

The models are governed by single differential or integro-differential equations.

Solutions include Gaisser-Hillas and Gaussian in Age functions.

Profiles closely match the Greisen profile after adjustments.

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.