Astroparticle Physics

17.7 Chapter 7 3574. N(> E,R) = A(aR) −γ , see (7.15) ,NR =−γAa−γ R −(γ +1) ,NN=−γR−1 R =−γ RR ,traditionally R = 100 g/cm 2 .With γ = 2 (exponent of the integral sea-level spectrum):∣ N∣∣∣ ∣ N ∣ = S200 g/cm2N ∣ = = 2 × 10 −2 , e.g., for 100 m w.e.R5. ϑ r.m.s. = 13.6MeV √ x(1 + 0.038 ln x )βcp X 0 X 0≈ 4.87 × 10 −3 (1 + 0.27) ≈ 6.19 × 10 −3 ≈ 0.35 ◦ .6. The lateral separation is caused bya) transverse momenta in the primary interaction,b, c) multiple scattering in the air and rock.a) p T ≈ 350 MeV/c ,ϑ = p Tp≈350 MeV/c100 GeV/c = 3.5 × 10−3 ,average displacement:x 1 ≈ 70 m .b) Multiple-scattering angle in air,(see [2])x 1 = ϑh,whereh is the production height (20 km),ϑ air ≈ 6.90 × 10 −4 (1 + 0.123) ≈ 7.75 × 10 −4 ≈ 0.044 ◦ ,x 2 = 15.5m.c) Multiple scattering in rock,x 3 ≈ 0.8m.√⇒ x = x1 2 + x2 2 + x2 3 ≈ 72 m .7. Essentially only those geomagnetic latitudes are affected where the geomagnetic cutoffexceeds the atmospheric cutoff. This concerns latitudes between 0 ◦ and approximately50 ◦ . The average latitude can be worked out from〈λ〉 =∫ 50 ◦0 ◦ λ cos 4 λ dλ∫ 50 ◦0 ◦ cos 4 λ dλ ≈ 18.4◦corresponding to an average geomagnetic cutoff of 〈E〉 ≈ 12 GeV. For zero field allparticles with E ≥ 2 GeV have a chance to reach sea level. Their intensity is (ε =E/GeV)