Precision of analytical approximations in calculations of Atmospheric Leptons

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.