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For a start, the Sun emits energy; the laws of Thermodynamics state that that can't go on forever. Beyond that, astronomers know quite a deal about what goes on in stars such as the Sun; specifically, hydrogen is converted to helium, in a process known as fusion. After a while (a few billion years in a star with the mass of our Sun), the star will run out of its fuel (the hydrogen). Helium can then be converted to heavier elements, but this doesn't last forever, either.

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Q: How do astronomers know that the sun has a finite lifespan?
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