Astroparticle Physics

17.6 Chapter 6 3493. a) The average column density traversed by primary cosmic rays is ≈ 6g/cm 2 (seeSect. 6.1). The interaction rate Φ is related to the cross section by, see (3.57),Φ = σN A .Because of Φ[(6g/cm 2 ) −1 ]≈0.1 and assuming that the collision partners are nucleons,one getsσ ≈ 0.1 × (6g/cm 2 ) −1 /N A ≈ 1.66 × 10 −25 cm 2 = 166 mbarn .b) If the iron–proton fragmentation cross section is ≈ 170 mb, the cross sectionfor iron–air collisions can be estimated to be σ frag (iron–air) ≈ 170 mb × A α ,where A ≈ 0.8 A N + 0.2 A O = 11.2 + 3.2 = 14.4. With α ≈ 0.75 onegets σ frag (iron–air) ≈ 1.26 b. The probability to survive to sea level is P =exp (−σN A d/A) = 1.3 × 10 −23 , where it has been used that the thickness of theatmosphere is d ≈ 1000 g/cm 2 .Sect. 6.21. The neutrino flux φ ν is given by the number of fusion processes 4p → 4 He + 2e + + 2ν etimes 2 neutrinos per reaction chain:φ ν =solar constantenergy gain per reaction chain × 21400 W/m 2≈28.3MeV× 1.6 × 10 −13 J/MeV × 2 ≈ 6.2 × 1010 cm −2 s −1 .2. (q να + q e −) 2 = (m α + m νe ) 2 , α = µ, τ ;assuming m να to be small (≪ m e ,m µ ,m τ ) one gets2E να m e + m 2 e = m2 α ⇒ E να = m2 α − m2 e⇒2m eα = µ : E νµ = 10.92 GeV , α = τ : E ντ = 3.09 TeV ;since solar neutrinos cannot convert into such high-energy neutrinos, the proposed reactionscannot be induced.3. a) The interaction rate isR = σ N N A dAφ νwhere σ N is the cross section per nucleon, N A = 6.022 × 10 23 g −1 is the Avogadroconstant, d the area density of the target, A the target area, and φ ν the solar neutrinoflux. With d ≈ 15gcm −2 , A = 180 × 30 cm 2 , φ ν ≈ 7 × 10 10 cm −2 s −1 ,andσ N = 10 −45 cm 2 one gets R = 3.41 × 10 −6 s −1 = 107 a −1 .b) A typical energy of solar neutrinos is 100 keV, i.e., 50 keV are transferred to theelectron. Consequently, the total annual energy transfer to the electrons isE = 107 × 50 keV = 5.35 MeV = 0.86 × 10 −12 J .