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They do. Each time their cores run out of the current fuel, they undergo gravitational collapse, heat up and start to fuse the next element in the sequence. Each time this happens the core gets hotter and the surface of the star expands (making red giants).

Eventually the element Iron (Fe) is made in the core and at this point the star dies because you can not fuse Iron into anything heavier and get energy out - all elements heavier than Iron absorb energy as they are made.

Thus as soon as a ball of Iron is made in the star's core there is no energy there to fight the pull of gravity and the core shrinks to become a superdense ball of neutrons or if the star is very massive a black hole. As this happens the star explodes as a supernova.

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Q: Why don't high mass stars rekindle after collapsing?
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