  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.
