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There are different froms of this and it depends on your goal.

For the 1-5 billion year range your issue is the Earth overheating because Solar output is increasing by about 10% per billion years. There are three basic ways of doing this:

1. Putting something between the EArth and the Sun to reduce the amount of radiation that hits the Earth. This is by far the easiest;

2. Move the Earth to an orbit further from the Sun; or

3. Removing mass from the Sun to extend its life or reduce its output. This is the hardest.

So how do you move the Earth to a different orbit? Well that's pretty easy actually. I mean "easy" in the sense that it requires no new hypsics, no negative energy or negative mass. It just ruses gravity.

The Earth interacts with any mass. A sufficiently sized mass will attract the Earth. So you just fly large bodies of mass past the Earth to slowly nudge it in the direction you want. The Earth is relatively heavy (~10^24kg) so the energy requiremetns are still immense but it can be done slowly (and probably should be).



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