I believe we'd still be warm for those 8 minutes. We have an 8 minute grace period before having to do anything, then enough time to add sweaters faster than earth cools.
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it's because light takes 8 minutes to get from the sun to us. and since gravity also travels at the speed of light our orbit would only change after 8 minutes.
That's boring.
Can we just have the reverse, like a "When Day Breaks" scenario? At least its fun.
The real question is if the earth becomes a rogue planet or if Jupiter captures most/all of the remaining solar system. Jupiter is technically a failed star, so could it finally get it's glowup from being the sun's understudy and keep us all together until we fall into the gravitational well of a new star?
If the sun just disappears, I doubt even having another sun would keep everything from flying off to fuck knows where. Jupiter, by comparison, is beyond hope. The Barycenter is far from Jupiter.
Only two ways to find out. Time to fire up universe sandbox, cause I'm fresh out of the ability to delete the sun in the production environment.
For further reading, see Galaxias by Stephen Baxter.