The answer depends on what you mean by “existence.” Astronomers frequently talk about stars being “born” and “dying”; what they really mean is “starting nuclear fusion” and “ending nuclear fusion.” Fusing hydrogen into helium and helium into other, heavier chemical elements is what makes a star “alive.” (By this definition, I guess I really haven’t lived.) Of course, most stars leave behind fascinating corpses that will endure indefinitely — white dwarfs, neutron stars, or black holes. Whether you think of these remnant objects as stars is a matter of semantics.
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