of charged pions ($\pi^{\pm}\to e \nu_e\,\nu_\mu\,\bar\nu_\mu $) as $$\begin{aligned}
&&L_{\nu} \approx 8\times 10^{33}\, \left(
\frac{f_{\nu}}{0.15}\right) \left(\frac{\eta_p}{0.1}\right)\,
\left(\frac{n}{1 \cmc}\right)^2 \times  \nonumber \\
&&
\left(\frac{S}{10^{38} \cm2}\right)\left(\frac{u_s}{5,000\,{\mathrm{km\,s}^{-1}}}\right)^3 \left(\frac{\tau_c}{10^{10}\, \rm{s} }\right)\,
\ergs ,
\label{eq:Lnu}\end{aligned}$$ where $f_{\nu}$ is the fraction of energy in the inelastic [$p\!-\!p$]{}–collisions which is deposited in the high-energy neutrinos, $S$ is the cross section of the colliding flows, and $\tau_c$ is the confinement time for protons in the emission region where the target density is $r_s \geq 4$ times the ambient density $n$ due to the shock compression, $r_s$.

In Eq. (\[eq:Lnu\]), we take the inelastic [$p\!-\!p$]{} collision cross section to be $\sim 70$mb above a proton energy of 10 PeV. Then the proton cooling time, $n\,t_{pp}$, can be estimated as $n\,t_{pp} \approx 2.5 \cdot 10^{14}$ s assuming that two inelastic collisions are needed to convert most of the proton energy into secondaries. The fraction of the proton energy deposited into neutrinos of all types, $f_{\nu} \sim$ 0.15, was derived using both the analytical parameterizations for the inelastic [$p\!-\!p$]{}–collisions presented by [@kelner2006; @cross_sect_PRD14] and the [*GEANT4*]{} package simulations. At the energies of photons and neutrinos considered in Figs. \[fig:spectra\_src\], \[fig:gam1m\] and \[fig:NeuOnly\], the neutrino fluxes calculated with the [@kelner2006] or [@cross_sect_PRD14] 