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My understanding of the transition from hydrogen shell burning to helium flash point:

The hydrogen shell burning occurs because the hydrogen in the suns core during the main sequence has been used up, gravity wins at the core and it contracts. Temperature goes up and there is a region around the core where the temperature is large enough to start fusion in the hydrogen in the shell. The core continues to contract because the increase in pressure and resistance to gravity from the shell isnt large enough to stop the contraction. The helium begins to fuse. There is a big flash because the helium fuses uncontrollably.

The question:

Why does the helium fuse uncontrollably? Also as the core contracts, shouldn't larger and larger regions around the core start to fuse hydrogen? I.e shouldn't the shell actually increase in size as the sun goes through the hydrogen shell burning phase.

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    $\begingroup$ Can't the answer be found in Wikipedia? $\endgroup$ – my2cts Apr 23 at 19:17
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    $\begingroup$ Probably, but the problem with Wikipedia is that understanding it’s physics articles pretty much requires you to already know the subject it’s talking about in significant depth $\endgroup$ – Ubaid Hassan Apr 23 at 20:36
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Stars with an initial mass below about 2.1 solar masses will form a helium core with a high degree of electron degeneracy. Above this core is a shell of fusing hydrogen that deposits more helium onto the core. The core contracts as it becomes more massive; it is also heated by the fusing shell that surrounds it. Eventually a temperature is reached that is large enough to ignite the helium.

The heat capacity of the degenerate electrons in the core is very low, yet they contribute most of the gas pressure. Most of the fusion energy, at least initially goes into raising the temperature of the helium ions in the core and this does not (initially) increase the electron degeneracy pressure, which isn't very temperature sensitive. This allows the helium fusion rate to increase very rapidly, and is known as the "helium flash".

Your second question is puzzling. The approach to the helium flash is an ascent of the red giant branch. The luminosity of the star is increasing because more and more power is being produced in the hydrogen-burning shell as a result of increasing temperature and more hydrogen being brought up to fusion temperatures.

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