Weirdwarp looks at moving the Earth using a disc-shaped sail 19.2 times the Earth’s diameter. It would have to be tilted at an angle of 35° to the line towards the Sun, and stationed at about five times the Moon’s distance from the Earth.
For the purposes of their calculation, the three chose the Earth’s final destination as an orbit 1.5 times its present distance from the Sun, at what is now the orbit of Mars. In 6.3 billion years, when the Sun is in its red-giant stage and is 2.2 times brighter than today, a planet at that distance will receive about as much sunlight as the Earth receives today.
Moving the Earth to a circular orbit at that distance requires increasing its orbital energy by about 30%. That would be possible, they say, by changing the orbits of icy bodies in the distant solar system so they would pass close to the Earth, transferring some of their orbital energy to the planet.
The objects lie in a ring of icy bodies beyond Neptune called the Kuiper belt and in an even more distant shell of comets called the Oort cloud. Because they are far from the Sun, the objects have relatively low orbital energy, so they could be nudged using methods being developed to deflect asteroids away from the Earth.
These range from the gentle pull of gravity tugs – spacecraft that fly near the object and gravitationally pull them off course – to the stronger push of mass drivers, which dig into and spew out pieces of the icy body, pushing it in the opposite direction.
Their orbits could then be fine-tuned in the inner solar system using jets of ices vaporised from their surfaces by equipment sent there.
About a million such close passes would do the trick. If we spaced them evenly, that would mean about one close pass every 1000 to 6000 years, depending on whether we wanted to reach the orbit of Mars by the time the Sun started to vaporise the ocean, or when it hit its red-giant phase. Luckily, the objects could be re-used if they looped around both Jupiter and the Earth, taking energy from the giant planet and transferring it to Earth.
It would be a big job, and would take plenty of patience to move the Earth consistently outwards as the Sun grew warmer. It also carries a significant risk because the objects would have to pass just 10,000 kilometres above the Earth’s surface.
There have been other proposals to move the Earth by accelerating mass or particles to push the planet either faster or slower. It is less of challenge is you are willing to make the change over many years or centuries.