Precision of analytical approximations in calculations of Atmospheric Leptons

TL;DR

This paper assesses the accuracy of simple analytical formulas for calculating atmospheric lepton fluxes by comparing them with detailed numerical models, identifying their limitations and suitable application conditions.

Contribution

It systematically evaluates the applicability of analytic approximations in atmospheric lepton calculations using the MCEq code and compares different hadronic interaction models.

Findings

Analytic approximations are accurate within specific phase space regions.

Errors in simple formulas are smaller than differences among hadronic models in certain conditions.

The study delineates the conditions where approximations are reliably used.

Abstract

We use the Matrix Cascade Equation code (MCEq) to evaluate the range of applicability of simple analytic approximations parameterized by spectrum-weighted moments and power-law spectral indices that vary slowly with energy. We compare spectra of leptons as a function of zenith angle and energy between MCEq and the analytic approximation. We also compare fluxes obtained with different models of hadronic interactions. The goal is to quantify the effects of the approximations inherent in the simpler formulas in order to determine their limitations and the conditions under which they may be used. Specifically we look for the range of phase space for which the errors in the approximate formulas are smaller than the differences among several different hadronic interaction models. Potential applications include the muon charge ratio, the fraction of prompt leptons from decay of charm and seasonal variations of muons and neutrinos.