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Is it true that when a star 'runs out' of its fuel, it hasn't really used up all its hydrogen, it still has hydrogen left, but the hydrogen is not at its center where the fusion happens. If something crashed into it rearranging its content a bit, could its life be extended?

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The outer shell of the star contains still a lot of hydrogen, after running out of fuel.

Elements other than hydrogen are denser than hydrogen. This makes it difficult to get hydrogen to the core of a star, or to the region of hydrogen-burning at least, after the core has been burnt to helium or heavier elements.

But the life of a star can be extended in some way, e.g. by adding hydrogen to a white dwarf. This can trigger burning of carbon with a supernova Ia as result.

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