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Red Giants

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This is my first post, so forgive me if this is a stupid question (most of mine are) but I do not understand red giants. A normal star is in equilibrium between radiation pressure pushing out from nuclear reactions in its core, and gravity pulling in. When a normal star, such as our own sun, runs out of fuel, the outward pressure from radiation necessarily stops. Gravity doesn't, of course, so the sucker shrinks. Except it doesn't. It expands into red giant first, flings off lots of stuff, and only then becomes a white dwarf. Dumb star.

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