Astroparticle Physics

344 17 Solutions2. The relative energy resolution is determined by the fluctuations of the number N ofproduced particles. If W is the energy required for the production of a particle (pair),the relative energy resolution isEE= NN= √NN = 1 √N=√WE ,since N = E/W. HereE and E are the energy of the particle and its uncertainty,⎧⎪ a) 10%E ⎨E = b) 5.5%.⎪ c) 1.9% ⎩d) 3.2 × 10 −4∫ 0∫dE E3. R =E dE/dx = dE0 a + bE = 1 b ln(1 + b a E) .The numerical calculation givesR ≈()14.4 × 10 −6 ln 4.4 × 10−61 + 10 52g≈ 45 193gcm2 or R = 181 m of rock taking into account that ϱ rock = 2.5g/cm 3 .4. The Cherenkov angle θ C is given by the relationcos θ C = 1nβ .From the relation between the momentum p and β:p = γm 0 βc ⇒ β = pγm 0 candcos θ C = γm 0cnp .cm 2Thennp cos θ Cm 0 c√= Em 0 c 2 = c p 2 + m 2 0 c2m 0 c 2and(np cos θ C ) 2 = p 2 + m 2 0 c2 ⇒ p 2 (n 2 cos 2 θ C − 1) = m 2 0 c2 .Finally,m 0 = p c√n 2 cos 2 θ C − 1 .