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Atmospheric Electron Spectrum above 30 GeV at the high altitude

We have observed the primary electron spectrum from 30 GeV to 3 TeV using emulsion chambers flown by balloons at the top of the atmosphere, for the purpose of exploring the origin of cosmic rays in the Galaxy. The atmospheric gamma rays have been simultaneously observed in the 30 GeV $\sim$ 8 TeV energy range. In this paper, we estimate the atmospheric electron spectrum in the upper atmosphere ($<$ 10 ${\rm g/cm^2}$) from our observed gamma-ray spectrum using the electromagnetic shower theory in order to derive the primary cosmic-ray electron spectrum. The transport equations of the electron and gamma-ray spectrum are analytically solved and the results are compared with those of Monte Carlo simulation (MC). Since we used the observed atmospheric gamma rays as the source of atmospheric electrons, our solutions are free from ambiguities on the primary cosmic-ray nuclear spectra and nuclear interaction models included in MC. In the energy range above several hundred GeV, the Dalitz electrons produced directly from neutral pions contribute around 10 percent to the whole atmospheric electron spectrum at the depth of 4 ${\rm g/cm^2}$, which increases in importance at the higher altitude and cannot be ignored in TeV electron balloon observations.