It would take eight minutes for anything to happen to Earth. We’d still see the sun for eight minutes and still continue in our orbit for eight minutes.
After that, everything would go dark and we’d continue in a straight line in whatever direction we were headed at the time. Then the sun’s heat gradually dies away and everything on the surface of the earth freezes and anyone who wants to survive would have to dig deep underground and live off geothermal energy.
Another thing to keep in mind, on top of what others are saying: The same thing would happen to every other planet, comet and asteroid in the solar system. Depending on when this event occurs, we could see other bodies headed out on their own trajectories.
Every day, Jupiter would look like it was getting farther and farther away.
Or…
Every day, Jupiter would look like it was getting closer and closer…
8min after disappearance, Earth would stop elliptic orbit and proceed to go pretty much straight ahead, as other planets. Darkness would fall at that moment as well. Moon would go dark, 0-2s after Sun disappears. Other planets would go dark in the following minutes and hours.
Earth would freeze over quite soon, and surface temperature would fall to something like -200 celcius. Under Earths surface you would however still find warmth as nuclear fission happening in the mantle would keep interior of our planet warm and core molten hot for millions of years more.
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