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I understand that for low-mass stars the helium flash occurs due to their degenerate helium cores. Thus the answer to this question is probably that more massive stars do not have a degenerate core, but I do not understand why they wouldn't. Due to their increased mass, I'd assume their central pressure is even higher, so surely their cores should also be degenerate?

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After H burning has finished, the he mass of the He core gradually increases, as does its density and temperature.

Low-mass stars have denser cores when they reach a temperature at which He is ignited. The density is high enough that the electrons in the core are degenerate. In such conditions the heat from the nuclear reactions goes almost exclusively into raising the temperature of the He ions, but almost none goes to the degenerate electrons which have a very low heat capacity, yet dominate the pressure. This leads to a runaway increase in the nuclear fusion rate.

In higher mass stars the core is less dense when it reaches the He ignition temperature. This temperature is almost exactly the same as the ignition temperature for lower mass stars because of the very strong temperature dependence of the triple alpha reaction. The level of degeneracy depends on the ratio of density to temperature (it does not directly depend on the pressure).

At the lower densities in the higher mass stars, the electrons are not degenerate when He ignites and the heat capacity of the electron gas is higher than the ions (because there are more electrons than ions). In this case the heat from the nuclear reactions can effectively raise the pressure, doing work which expands the core and regulates the nuclear reaction rate.

See also https://physics.stackexchange.com/questions/174801/why-is-the-release-of-energy-during-the-he-flash-in-stars-almost-explosive

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More massive stars indeed have higher pressures, but what's key is that they also have higher temperatures. After leaving the main sequence, they reach core temperatures of a few times $\sim10^8$ Kelvin while their densities remain at something like $\sim10^4$ g cm$^{-3}$ - which, you can check, is within the nondegenerate regime. Therefore, helium fusion begins comparatively gently, rather than with the explosion helium flash characteristic of evolved Sun-like stars.

Central density tends to decrease with increasing mass, and central temperature tends to increase with increasing mass. It's not surprising that there's a mass cutoff above which temperature clearly wins out, at around $1.5M_{\odot}$-ish; similarly, it's not surprising that there's a mass cutoff below which a star will never reach core temperatures required for helium fusion.

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Massive stars do not undergo helium flash because they have core temperatures high enough to prevent the helium core from becoming electron-degenerate. Check here for some more information.

Therefore, the star can burn helium in a smooth transition, instead of undergoing helium flash. In more details, the massive star's core heats up past the helium burning threshold, preventing a degenerate helium core from forming. I hope this helps.

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