Energy spectrum of tau leptons induced by the high energy Earth-skimming neutrinos

We present a semianalytic calculation of the tau-lepton flux emerging from the Earth induced by incident high energy neutrinos interacting inside the Earth for ${10}^{5}l~{E}_{\ensuremath{\nu}}/\mathrm{GeV}l~{10}^{10}.$ We obtain results for the energy dependence of the tau-lepton flux coming from the Earth-skimming neutrinos, because of the neutrino-nucleon charged-current scattering as well as the resonant ${\overline{\ensuremath{\nu}}}_{e}{e}^{\ensuremath{-}}$ scattering. We illustrate our results for several anticipated high energy astrophysical neutrino sources such as the active galactic nuclei, the gamma-ray bursts, and the Greisen-Zatsepin-Kuzmin neutrino fluxes. The tau-lepton fluxes resulting from rock-skimming and ocean-skimming neutrinos are compared. Such comparisons can render useful information about the spectral indices of incident neutrino fluxes.