Binding energies of different elements

Quantum Tunnelling Coulomb Potential: E c = Z A Z B e 2 / 4πε 0 r Tunnelling probability: P tunnel  exp(-(E G /E) 0.5 ) Gamow Energy: E G = (π  Z A Z B ) 2 2m r c 2 m r = m A m B /(m A +m B )  = e 2 / (4πε 0 ћc)  1/137

Energy-dependent fusion rates Dashed line: Boltzmann factor, P Boltz  exp(-E/kT) Dot-dash line: Tunnelling factor, P tun  exp(-(E G /E) 0.5 Solid line: Product gives the reaction rate, which peaks at about E 0 = (kT/2) 2/3 E G 1/3 Sun: E G = 493keV; T = 1.6x10 7 K  kT = 1.4keV Hence E 0  6.2 keV (  4.4 kT)

The PP chain: the main branch 9x

The PP chain ~85% in Sun ~15% in Sun ~0.02% in Sun

The CNO Cycle Slowest reaction in this case: 14 N + p  15 O + γ. 14 N lives for ~ 5x10 8 yr. Abundances: 12 C ~ 4%, 13 C ~ 1%, 14 N ~ 95%, 15 N ~ 0.004%

Recap Equation of continuity: dM(r)/dr = 4 π r 2 ρ Equation of hydrostatic equilibrium: dP/dr = -G M(r) ρ / r 2 Equation of energy generation: dL/dr = 4 π r 2 ρ ε where ε = ε pp + ε CNO is energy generation rate per kg. ε = ε 0 ρT α where α~4 for pp chain and α~17-20 for CNO cycle.

Discussion: random walk Consider tossing a coin N times (where N is large). It comes as “heads” N H times and “tails” N T times. Let D = N H – N T. What do you expect the distribution of possible values of D to look like (centre, shape)? How do you expect the distribution of D to change with N? Would you ever expect D to get larger than 100? If so, for what sort of value of N?

Random Walk Photon scattered N times:  = ř 1 + ř 2 + ř 3 + ř 4 +……+ ř N Mean position after N scatterings: = ……+ = 0 But, mean average distance travelled |  | comes from .  = |  | 2, hence |  | = ( .  ) 0.5 .  = (ř 1 + ř 2 + ř 3 + ř 4 +……+ ř N ).(ř 1 + ř 2 + ř 3 + ř 4 +……+ ř N ) = (ř 1.ř 1 + ř 2.ř 2 + ř 3.ř 3 +……+ ř N.ř N ) + (ř 1.ř 2 + ř 1.ř 3 + ř 1.ř 4 +……+ ř 2.ř 1 + ř 2.ř 3 + ř 2.ř 4 + …… ř 3.ř 1 + ř 3.ř 2 + ř 3.ř 4 + ……) =  (ř i.ř i ) = Nl 2  |  | = (√N) l } =0