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Core Fusion Leading to Expansion
Stars, being composed largely of hydrogen, operate via the process of nuclear fusion. The nuclei of hydrogen atoms combine under the intense pressure and heat of the star's core, releasing a great deal of energy (this is responsible for the star's energy output) and creating a helium nuclei instead. The energy of the fusion reaction helps keep the core from collapsing.
Helium is heavier than hydrogen, and thus it is more difficult for helium itself to be fused; when stars run out of hydrogen to fuse in their core, the absence of fusion to keep the core inflated causes it to contract, generating even more heat and pressure. As a result, fusion begins to occur outside of the core itself.
Rate of Fusion
When hydrogen fusion spreads to the outer layers of the star, it expands dramatically; when Earth's sun enters this phase, it will get much larger, perhaps even engulfing the Earth itself. While the core stops fusing hydrogen as a star gets bigger, because this fusion is occurring in so many other places, it means the overall rate has increased dramatically. For this reason, while stars in the main sequence may burn for billions of years, stars in their final expansion phase exhaust their remaining hydrogen fuel supply in only a few million years.
Color and Luminosity
Because so much more fusion is occurring in a star as it expands, the total luminosity of the star -- how bright it is -- grows as a star gets bigger. However, this energy is being radiated over a far larger surface area; as a result, the surface temperature lowers, meaning that the light emitted by the star shifts from yellow or white to red. For this reason, stars in this part of their life cycle are sometimes known as "red giants."
Helium Flash
Fusion generates outward pressure on the star, keeping it large; when this begins to slow down, the core contracts further and further until eventually the only thing keeping the atoms themselves separate are fundamental quantum mechanical principles. Eventually, the core is hot enough to begin fusing helium; for average-sized stars like our own sun, this process occurs rapidly, in a "helium flash" that reinflates the core and leads to a series of expansion and contraction cycles that cause the star to blow off large amounts of its matter. At the conclusion of this cycle, little is left of the original star but a non-fusable core and a shell of gas known as a "nebula."