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I have a question regarding the above mentioned. When a star have a mass of about 3-8 it does not go through the so-called helium flash phase, but instead just run along as nothing had happened, turn on helium fusion, and in the end, end up becoming a white dwarf.

Lower mass stars though, do get into the helium flash phase, and this is because of electron degeneracy, if I'm not mistaken? But I'm not sure I understand electron degeneracy all that well.

If the star has mass enough, the gravitational potential will be enough to increase the temperature for helium fusion to start. But if not, it will not get hot enough, at least for a while. So in order for helium fusion to begin, it needs to be hot enough, but for a lower mass star, the gravitational potential is not large enough, so in order to reach this temperature, it would need to compress even more. So before that happens, degeneracy takes over?

So what makes it come out of degeneracy? If it cannot compress anymore, due to the degeneracy, how do the core temperature increase in order for helium fusion to happen, and from that make the helium flash? Is it just a steady increase from the still ongoing hydrogen shell burning that makes this happen, or...?

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The division between He flash and no He flash is more like $2M_{\odot}$.

The core of a low-mass star doesn't come out of degeneracy until the He flash is underway; until then, the partial degeneracy increases as more He ash is dumped onto the core and both density and temperature continue to increase (because the number of mass units per particle increases). Degeneracy cannot halt the contraction of the core, because the more massive the partially degenerate core is, the smaller it gets. In addition any cooling is inhibited because it is surrounded by a burning shell of hydrogen.

Here is a diagram of the central density vs central temperature for stars of 1.3 and 2.1M⊙ leaving the main sequence and igniting He in their cores - at point A for the lower-mass star, which undergoes a helium flash. The main sequence is to the bottom-left of the diagram. The diagram is from a thesis by Gautschy (2012). The two green lines mark first (left) where partial degeneracy is felt in the core, then on the right (only reached by the lower mass stars) where a high level of degeneracy is reached.

Note that the central density monotonically increases from the main sequence until He is ignited (the star is already a red giant at this point). There is a brief period of roughly constant temperature in the core, which I think coincides with peak H shell burning, but once the He ash stars to build up, then so does the density and temperature.

Approach to He flash

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