Lecture 11

1. Lecture 11 Energy transport

2. Review: Nuclear energy • If each reaction releases an energy L,the amount of energy released per unit mass is just • The sum over all reactions gives the nuclear reaction contribution to e in our fifth fundamental equation:

3. Proton-proton chain (PPI) • The net reaction is: • But each of the above reactions occurs at its own rate. The first step is the slowest because it requires a proton to change into a neutron: • This occurs via the weak force. The rate of this reaction determines the rate of Helium production

4. Proton-proton chain (PPII and PPIII) • Alternatively, helium-3 can react with helium-4 directly: • In the Sun, this reaction occurs 31% of the time; PPI occurs 69% of the time. • Yet another route is via the collision between a proton and the beryllium-7 nucleus • This reaction only occurs 0.3% of the time in the Sun.

5. The PP chain • The nuclear energy generation rate for the PP chain, including all three branches: • Near T~1.5x107 K (i.e. the central temperature of the Sun):

6. Example • If we imagine a core containing 10% of the Sun’s mass, composed entirely of hydrogen (X=1), calculate the total energy produced by the PP reaction.

7. The CNO cycle • There is a second, independent cycle in which carbon, nitrogen and oxygen act as catalysts. The main branch (accounting for 99.6% of CNO reactions) is: • at T~1.5x107 K

8. Helium collisions • As hydrogen is converted into helium, the mean molecular weight increases. • To keep the star in approximate pressure equilibrium, the density and temperature of the core must rise • Recall that the temperature at which quantum tunneling becomes possible is: • As H burning progresses, the temperature increases and eventually He burning becomes possible

9. The triple-alpha process • The burning of helium occurs via the triple alpha process: • The intermediate product 8-beryllium is very unstable, and will decay if not immediately struck by another Helium. Thus, this is almost a 3-body interaction • Note the very strong temperature dependence. A 10% increase in T increases the energy generation by a factor 50.

10. Nucleosynthesis • At the temperatures conducive to helium burning, other reactions can take place by the capturing of a-particles (He atoms).

11. Nucleosynthesis • The binding energy per nucleon describes the stability of a nucleus. It is easier to break up a nucleus with a low binding energy.

12. Break

13. Summary • We have now established four important equations: • Hydrostatic equilibrium: • Mass conservation: • Equation of state: • Energy production • There are 5 variables (P,r,Mr, T and Lr) and 4 equations. To solve the stellar structure we will need to know something about the energy transportation.

14. Energy transport • Radiation: the photons carry the energy as they move through the star, and are absorbed at a rate that depends on the opacity. • Convection: buoyant, hot mass will rise • Conduction: collisions between particles transfer kinetic energy of particles. This is usually not important because gas densities are too low.

15. Radiation transport • When we considered the properties of radiation, we found an equation relating the pressure gradient to the radiative flux: • From this we can derive an expression for the temperature gradient, assuming a blackbody. • In regions of high opacity, or high radiative flux, the temperature gradient must be steep to transport the energy outward.