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A star that initially produces energy through the $pp$ Chain will eventually exhaust its Hydrogen leading to a core contraction in order to increase the temperature to generate again nuclear reactions. Since the $pp$ Chain has a temperature threshold smaller than the $\rm CNO$ cycle: Why is then the H-burning shell a $\rm CNO$-cycle one and not a $pp$-Chain one?

If the answer is: Because the temperature corresponds to the $\rm CNO$ cycle and not to the $pp$ Chain; then why the contraction does not halt when the temperature surrounding the Helium core is enough to fuse Hydrogen through the $pp$ Chain?

Looking forward to your comments!

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At the end of core hydrogen burning in a low-mass star you have an inert He core that is essentially isothermal and hotter than the temperatures required for H-burning. H-burning starts at the edge of the core and this is hot enough for the CNO cycle to dominate.

Why does core contraction halt? No energy is generated in the core so it will keep contracting and getting hotter (by the virial theorem) until the energy losses outward are balanced by inward flux from the shell burning. But this doesn't happen until the burning rate is high (higher than on the main sequence - the luminosity of subgiants and giants is higher than for main sequence stars) and occurs in a relatively thin shell (because of the temperature gradient). Hence the temperature of this shell needs to be higher than the core of a main sequence star of the same mass and so the H-burning occurs via the CNO cycle.

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