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Question

It requires more heat to fuse helium than to fuse hydrogen, yet when a star has depleted its supply hydrogen it cools down! How does the star warm up again enough to fuse helium in its core?

Answer 1

Helium doesn't start fusing

After about ten billion years, a main sequence star has converted approximately 10% of its hydrogen to helium. Although this might seem as though it could still undergo hydrogen fusion for another 90 billion years, this is not the case. Remember that there are immense pressures at the core of stars, and it is only because of these pressures that the fusion can occur -- in a fixed volume, increased pressure leads to increased heat. Outside of the range of pressures there is still mostly hydrogen, but it cannot be used because the pressures are not high enough to initiate fusion.

The helium core is not hot nor dense enough to fuse to create energy, so the outward pressure is stopped, and gravity takes over again. Gravity will contract the star, and eventually a shell of hydrogen around the helium core will become hot enough to fuse H -> He. This shell will produce more energy than the previous hydrogen core phase did, so the luminosity will rise. Not all of the energy will escape, though, and it will go into expanding the star. This expansion will result in a surface temperature drop. The star will be in the subgiant star, and the cooler surface will have changed from yellow to orange-red. This cooling is due to the energy spreading over a larger surface area, so each unit of area receiving less energy.

The helium "ash" from the hydrogen fusion in the shell will effectively fall onto the core, which will result in the star continuing to contract to maintain pressure to hold up the star. Once the mass in the core is approximately 8% of the sun (the Schonberg-Chandrasekhar Limit), the density will be so great that the core will no longer act as a perfect gas, and it will become degenerate.

Now the core will be held up by the Pauli Exclusion Principle, AKA it will be supported by electron degeneracy. This phase will still have the hydrogen burning shell, but the star's outer layers will continue to expand, causing it to cool. This begins the star's Red Giant Ascent.