Post Main-Sequence Evolution: The Red Giant Branch

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  When we last saw the Sun it was evolving up the red giant branch (also known as First Ascent Giant Branch) with an ever-larger He core that is contracting and heating the surrounding H-fusion shell.

• So where is this heading?

  Eventually the core stops contracting for a Strange Reason .

  You might expect the core to heat up to 10⁸ K where He fusion would start and produce the thermal pressure required to balance gravity and then work in the thermostat mode. But, this is wrong

• The Red Giant core is supported by Electron Degeneracy.

  This comes about because you can only squeeze electrons so close together and no further because of the Pauli Exclusion Principle.

• The Red Giant core is made up of mostly ionized He so it is a gas of nuclei and e^-.

  All subatomic particles have various properties characterized by "quantum numbers". One is "spin". The quantum world gets split into two flavors of particles according to their spin:

  1. "spin 1/2" particles are called Fermions (e^- are Fermions).

  2. "spin 1" particles are called Bosons (neutrons and He nuclei are Bosons)
• Fermions all obey a law called the Pauli Exclusion Principle:
  You can have at most 2 fermions occupying a volume of phase space density.

  ΔxΔyΔzΔp_xΔp_yΔp_z = Quantum Phase Space Volume

  Here, the x, y, z refer to position and p_x etc. refer to momentum.

• At a given temperature (this means give the mean speed of the e^-), as the density goes up (packing those e^- closer and closer together), the e^- gas starts to run out phase space in which to put e^-. The inability to increase the density adds up to a form of supporting pressure called Electron Degeneracy Pressure.

• Electron Degeneracy Pressure is what balances gravity in the RGB star core and EDP has a strange property in that it is more-or-less independent of temperature.

• Remember the usual Gas Law:

  This provides a saftey value:

• The Usual Gas Law as a Safety Value:
  1. Suppose the He fusion started in the core. This would release some heat, the temperature and pressure would go up, the core expands and cools off and the fusion rate decreases a little bit.

  2. Alternatively suppose the fusion rates go down. Then gravity starts to compress the core, the Temperature goes up, fusion rates go up and the core re-expands.

• But, because EDP is independent of temperature, when He fusion starts, it is like a bomb going off.
  1. Core Temperature reaches 10⁸K.

  2. A special form of He fusion starts where ³He Carbon and there is an enormous release of energy and increase in the core temperature.

  3. Because the pressure in the core is due to e^- degeneracy and is T independent, the core does NOT expand and cool, but rather there is a temperature runaway - this is the Helium Flash.

• Although it sounds exciting, the Helium Flash does not add up to all that much. It runs for a few hours until the temperature reaches 10⁹ and at this temperature there is lots of extra phase space available to the electrons and the degeneracy is "lifted". The result is a small hiccup and the internal structure now looks like a version of a main-sequence star with core helium fusion. The new equilibrium puts the star in a spot on the H-R Diagram called the "horizontal branch".

Horizontal Branch Stars and Helium Fusion

There is a problem with ⁴He₂ + ⁴He₂ ⁸Be₄ because this reaction is endothermic. It does not release energy, but rather requires energy for it to happen.

So, He and He atoms DO fuse to Be sometimes, but the Be decays in ~10^-12 seconds. So, you need to have the temperature and density be high enough so that before this decay happens another He collides with the Be to produce Carbon. This is called the Triple Alpha reaction and it is the energy source for horizontal branch stars.

• Horizontal Branch Stars have a structure that is analogous to main-sequence stars.

  

• The fusion rates are running fast and the structure of the HB stars changes relatively quickly. The carbon core starts to shrink and release energy, the He fusion moves into a shell (now we have a "double shell" source) and this starts to look like the situation when a main-sequence star became a Red Giant.

• This time, the star ascends the Asymptotic Giant Branch.

• For a 1 M_Sun star, it never gets hot enough to fuse carbon together. Now the star is running out of fuel for good

• The triple-alpha reaction is extremely sensitive to temperature: Triple-Alpha rate propto Temp³⁰ which leads to instabilities called Helium shell flashes.