One proposal I've seen involves putting asteroids into an orbit such that their gravity pulls the Earth outwards a tiny bit each time they pass. Over an extended period of time (millions of years?) we could move the Earth outwards from the sun before the sun expands and cooks it. Obviously the Earth would pull the asteroid inward as well so we would have to adjust the asteroid's orbit after each pass. One proposal I've seen would be use Venus to keep the asteroid in a stable orbit. Essentially each pass of the Earth would slow the asteroid down a little bit, but each pass by Venus would speed it back up again, essentially pulling the Earth away from the sun while pulling Venus closer to the sun. In theory we could set up a system like this and just let it do its thing over a very long time, the change would be imperceptible over a human lifetime.
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