Binding energies of different elements

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Binding energies of different elements
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Quantum Tunnelling

• Coulomb Potential
• Ec ZAZBe2 / 4pe0r
• Tunnelling probability
• Ptunnel ? exp(-(EG/E)0.5)

Gamow Energy EG (p?ZAZB)2 2mrc2 mr
mAmB/(mAmB) ? e2 / (4pe0?c) ? 1/137
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Energy-dependent fusion rates

• Dashed line Boltzmann factor, PBoltz ?
  exp(-E/kT)
• Dot-dash line Tunnelling factor, Ptun ?
  exp(-(EG/E)0.5
• Solid line Product gives the reaction rate,
  which peaks at about E0 (kT/2)2/3 EG1/3

Sun EG 493keV T 1.6x107 K ? kT
1.4keV Hence E0 ? 6.2 keV (? 4.4 kT)
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The PP chain the main branch
9x
9x
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The PP chain

• 85 in Sun
• 15 in Sun
• 0.02 in Sun

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The CNO Cycle

• Slowest reaction in this case 14N p ? 15O ?.
• 14N lives for 5x108 yr.
• Abundances 12C 4, 13C 1, 14N 95, 15N
  0.004

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Recap

• Equation of continuity
• dM(r)/dr 4 p r2 ?
• Equation of hydrostatic equilibrium
• dP/dr -G M(r) ?
  / r2
• Equation of energy generation
• dL/dr 4 p r2 ? e
• where e epp eCNO is energy generation rate
  per kg.
• e e0?Ta
• where a4 for pp chain and a17-20 for CNO cycle.

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Discussion random walk

• Consider tossing a coin N times (where N is
  large). It
• comes as heads NH times and tails NT times.
• Let D NH NT.
• 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?

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Random Walk

• Photon scattered N times
• r1 r2 r3 r4 rN
• Mean position after N scatterings
• lt?gt ltr1gt ltr2gt ltr3gt ltrNgt 0
• But, mean average distance travelled ? comes
  from
• ?.? ?2, hence ? (?.?)0.5
• ?.? (r1 r2 r3 r4 rN).(r1 r2 r3
  r4 rN)
• (r1.r1 r2.r2 r3.r3 rN.rN)
• (r1.r2 r1.r3 r1.r4 r2.r1
  r2.r3 r2.r4 r3.r1 r3.r2 r3.r4 )
• ? (ri.ri) Nl2
• ? ? (vN) l

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