Secondary electron spectrum from 30 GeV to 10 TeV in the upper atmosphere

In the previous paper [1], we have evaluated the secondary electron spectrum in the upper atmosphere using our direct measurements of atmospheric gamma-ray spectrum in the 30 GeV- 10 TeV energy range. We have solved the simultaneous equations of electron and gamma-ray spectrum using the cascade shower theory combined by the gamma-ray production rate in the atmosphere. In this paper we present more precise solutions taking into account some nuclear decay modes, particularly, Dalitz decay mode which produces pair electrons directly from neutral pions, contributing about 10% to the secondary spectrum. The analytic results solved exactly are compared with those of the Monte Carlo simulation(MC), and the secondary electron/gammaray ratio shows the good agreement with each other. The contribution of Dalitz electrons increases the importance at the higher altitude, and cannot be negligible in the TeV electron balloon measurements. Our observed atmospheric gamma-ray spectrum is 20% higher than the results of MC as indicated previously [2], so that the estimates of secondary electrons become higher than 20% than those of MC. Although the discrepancy still remains, for the correction of primary electron spectrum we use the secondary spectrum obtained from the measured gamma-ray spectrum, because it does not include uncertainties of nuclear interaction models of MC.