What would happen if the Sun somehow vanished? Earth would notice nothing for about 8 minutes and 20 seconds. Then both sunlight and the Sun's gravitational influence would disappear from Earth at essentially the same time.
Earth would leave its solar orbit, photosynthesis would stop, the planet would cool dramatically, and the oceans would begin freezing from the surface downward. But geothermal energy could preserve liquid water — and perhaps engineered habitats — beneath the frozen surface.
This is a thought experiment: under known physics, the Sun cannot simply vanish.
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