Consider the muon neutrinos generated by the decays of the mesons produced by the collisions of cosmic rays in the atmosphere (‘atmospheric neutrinos’). Their energy spectrum at the surface of the Earth extends over several orders of magnitudes,decreasing with energy roughly as E _ν ^-3 and with important dependence onthe angle to the zenith. At Eν ≈1 GeV their flux around the zenith is approximatelyΦ_νμ ≃ 130 m^-2s^-1sr^-1GeV^-1. The Super-Kamiokande detector is a 22.5 kt fiducial mass water Cherenkov detector in the Kamioka underground observatory. Muon neutrinos (and antineutrinos, but we will not consider them) are mainly detected via their CC interactions on ¹⁶O nuclei ν_μ+ ¹⁶O →μ^-+ X. Assume σ(ν_μ¹⁶O)’10^-42m², θ₂₃=45^°, θ₁₃= °, Δm² = 2500m_eV².

1. How many interactions per year will happen induced by muon neutrinos arriving with directions within ΔΩ = 1 sr around the zenith in 1 GeV energy interval? (For the purpose of this problem, assume, unrealistically, all quantities to be constant in these intervals.)

2. What is the fraction of surviving muon neutrinos coming vertically upwards?

3. What is the fraction of surviving muon neutrinos incoming at 90° to the zenith?