Transport equations of cosmic neutrinos passing through Earth and secondary ν μ fluxes

The convergence of the iterative solutions of the transport equations of cosmic muon and tau neutrinos propagating through Earth is studied and analyzed. For achieving a fast convergence of the iterative solutions of the coupled transport equations of ${\ensuremath{\nu}}_{\ensuremath{\tau}}$, ${\overline{\ensuremath{\nu}}}_{\ensuremath{\tau}}$ and the associated ${\ensuremath{\tau}}^{\ifmmode\pm\else\textpm\fi{}}$ fluxes, a new semianalytic input algorithm is presented where the peculiar $\ensuremath{\tau}$-decay contributions are implemented already in the initial zeroth order input. Furthermore, the common single transport equation for muon neutrinos is generalized by taking into account the contributions of secondary ${\ensuremath{\nu}}_{\ensuremath{\mu}}$ and ${\overline{\ensuremath{\nu}}}_{\ensuremath{\mu}}$ fluxes due to the prompt $\ensuremath{\tau}$-decay $\ensuremath{\tau}\ensuremath{\rightarrow}{\ensuremath{\nu}}_{\ensuremath{\mu}}$ initiated by the associated tau flux. Differential and total nadir-angle-integrated upward-going ${\ensuremath{\mu}}^{\ensuremath{-}}+{\ensuremath{\mu}}^{+}$ event rates are presented for underground neutrino telescopes and compared with the muon rates initiated by the primary ${\ensuremath{\nu}}_{\ensuremath{\mu}}$, ${\ensuremath{\nu}}_{\ensuremath{\tau}}$, and $\ensuremath{\tau}$ fluxes.