Astroparticle Physics

17.13 Chapter 13 373(p ′ W −m Wm W + m ep W) 2= p 2 WCentral collisionm Wm 2 e(m W + m e ) 2 .p W ′ − p W =±p Wm W + m esince only the negative sign is physically meaningful.E = 1 ( )2 m W(v1 2 − v′2 1 ) = Ekin W 1 − v′2 1v12 = EWkinm e≈ 1GeV× 2 × 10 −5 = 20 keV .3. Classical Fermi gas of neutrinos:m W + m e⇒ p ′ W = m W − m em W + m ep W ,(1 −( ) )mW − m 2 em W + m eE F = p2= ¯h 2 k 2, k – wave vector .2m ν 2m νIn a quantized Fermi gas there is one k vector per 2π/L if one assumes that the neutrinogas is contained in a cube of side L. Number of states (at T = 0) for 2 spin states:N = 243 πk3 F(2π/L) 3 = V 13π 2 k3 F ⇒ k F = (3π 2 n) 1/3with n = N/V particle density,⇒ E F = ¯h 22m (3π 2 n) 2/3 .Relativistic Fermi gas:E F = p F c = ¯hk F c = ¯hc(3π 2 n) 1/3assuming relativistic neutrinos one would get, e.g., for 300 neutrinos per cm 3 .¯hc = 197.3MeVfm,E F = ¯hc (3π 2 × 300) 1/3 cm −1 = 197.3 × 10 6 eV × 10 −13 cm × 20.71 cm −1≈ 409 µeV .4. Since v ≪ c is expected, one can use classical kinematics:12 mv2 = G mM √2GM⇒ v = ⇒RR√2 × 6.67 × 10v ≈−11 m 3 kg −1 s −2 × 10 10 × 2 × 10 30 kg3 × 10 3 × 3.086 × 10 16 ≈ 1.7 × 10 5 m/s ,mβ ≈ 5.66 × 10 −4 .