This question carries with it immense complications owing to the Sun’s influence on the Solar System, so we’ll take a hypothetical look at it first from the perspective of the motion of bodies in the Solar System.
Anything in orbit around the Sun is basically moving fast enough to be in a constant state of freefall towards it. If the planets, for example, suddenly stopped moving they would immediately be pulled towards the Sun and fall into it.
So, let’s imagine that, due to some unknown reason, our Sun suddenly vanished. In this instance all the planets, asteroids, comets and whatever else would retain their forward motion. Therefore, rather than fall towards the now nonexistent Sun, they would fly off in a straight line into space.
What happens next is anyone’s guess. Some of the planets might gravitationally interact with one another, especially as the inner planets are moving faster than the outer planets and would potentially ‘catch up’ to them. Otherwise, the likelihood is all the objects would continue to move in a straight line through space out of the Solar System, and it would be many thousands of years before they came into the vicinity of another system or object.
Of course, for us on Earth, the consequences would be pretty dire. On the plus side, our planet retains heat rather well, so we wouldn’t freeze to death instantly. Also, as light from the Sun takes eight and a half minutes to reach us, we’d have a final few moments of glorious sunshine before our planet was bathed in darkness.
Those on the night side wouldn’t notice much difference until, a few seconds after day-dwellers were thrust into darkness, the Moon suddenly disappeared as it no longer had the Sun’s light to reflect. The planets in the sky would follow suit, disappearing one by one as the wave of darkness reached them.
Eventually, though, the lack of the Sun’s radiation would leave us pretty chilly. Just think about how much colder it is at night time rather than the day, but imagine that same temperature drop constantly occurring.
Within days the world would be a hundred or so degrees below freezing, and within weeks it’d be just 50 or so degrees above absolute zero. The atmosphere itself would also freeze and fall to Earth, leaving us exposed to the harsh radiation travelling through space.
Life as we know it would have to adapt to survive to our new frozen Earth, and it’s likely only microorganisms beneath the surface could survive thanks to heat from the core. For humans, we’d probably have to pool together and build a few nuclear fusion reactors in order to last a while.
All in all, it probably wouldn’t be that preferable for the Sun to disappear but, if it did, at least our days wouldn’t be numbered immediately. Just, you know, a little beyond immediately.
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