What would happen if the sun disappeared?

Initially nothing.

T = 0: Sun miraculously disappears.
T = 3.2 minutes Mercury suddenly gets slung into outer space at 47.87 km/s as if thrown by a slingshot. It's direction depends on it's location at T=3.2
T = 6 minutes. Venus suffers the same fate as Mercury and is slung into outer space.
T = 8.33 minutes. Everything on Earth goes dark. Earth is slung into outer space at 29.78 km/s - direction unknown.

What the Moon does is unknown - will it continue to orbit the Earth or will it it do something else? The Sun only applies about 0.0006g to the Earth, so the Moon will probably continue to revolve around the Earth as if nothing has happned.

T=10. Certain panic in daylight zones.

Anything after t=10 is just speculation.

  • Day 1, satellites will fail Panic, looting, killings - plenty of people revisiting churches, synagogues or whatever.
  • Day 2, diesel and all fuels will freeze. Burst pipes - water, gas overhead power lines will fail - ice accumulation. Winds will drop or stop completely as there is now no difference in temperature. Oceans/seas will begin to freeze and the levels will drop as ice is accumulated around the polar regions.
  • Day 3. All surface water would have frozen. Most people frozen dead in their homes.
  • Day 4. Nobody can venture outside. All non nuclear power stations have now ceased working.
  • Day 5. Nuclear power stations cease working as electrical systems mal function.
  • Day 6. Seas frozen. Ice sheets everywhere.
  • Day 7. Total nothingness.

At the end of the day it won't matter even if you have a heated
nuclear bunker as all you will be doing is just extending your
ultimate doom.

I suppose, if warning was given, it would be possible to build
underground cities, warmed by thermal or nuclear power, living off
cultivated plants and animals - a sort of Sci-Fi Noahs ark. Maybe
hoping that eventually the Earth will be drawn into another stars
gravity, but then can you guarantee it will be in the right orbit?

I prefer to let the Sun be.

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