Astro 160: The Physics of Stars

for the proton-He interaction we findr c = e2 Z 1 Z 2E 0≈ 1.44 × 10 −10 cmthus the probability is given byE G ≈ 3.15 MeVP ≈ 5.8 × 10 −18(b) What energy E would be required for i) the two 4 He nuclei, ii) the proton and the 4 He nucleus, andiii) two 12 C nuclei to have the same probability of penetrating the Coulomb barrier as the two protons?For particles with energies equal to the mean thermal energy of the plasma, what temperatures do thesecorrespond to?Since we know thatfor the He-He interaction we findfor the proton-He interaction we findE =E G(lnP) 2E He−He ≈ 0.125 MeVE p−He ≈ .013 MeVfor the carbon-carbon interaction the Gamow energy is given byE G ≈ 2592π 2 α 2 6m p c 2 ≈ 7.7 GeVand thuswe know thatE c−c ≈ 31 MeVT ≈ E k bsoT He−He ≈ 1.45 × 10 9 KT p−He ≈ 1.5 × 10 8 KT c−c ≈ 3.6 × 10 11 KProblem # 5Calculations of nuclear reaction rates are done in the center of mass (COM) frame, so it is useful toremember a few results about the COM. Consider two particles of mass m 1 and m 2 with positions x 1 andx 2 and velocities v 1 and v 2 .(a) .What is the velocity of the COM?We kbnow that the center of mass is given bycom = m 1r 1 + m 2 r 2m 1 + m 224