It would have to be rockets. My napkin says 2B starship boosters and 250K tons of propellant per second should give us around 1 second per year.
[Spoiler]
They are good reasons. Conspiracy theorists are able to persuade almost everybody that the reasons were bullshit, an excuse to seize power or something, and so the few who still insist this was necessary and mustn't stop are executed. Almost immediately after those executions, Mother Nature proves them right. So that leaves everybody: Guilty of having murdered their saviours and with no choice but to carry on with the very plan they had insisted was bogus...
There is a Chinese movie but I'm kinda surprised no Hollywood studio got themselves a rights deal and made a US-friendly movie where the Sun conveniently blows up slightly earlier and our heroes are vindicated and everybody agrees they were right all along.
In the real world their scheme is doomed as it would strip the atmosphere off of the Earth, but since they were planning to leave the solar system the atmosphere was going to freeze to a solid anyway so maybe it didn't really matter. To be honest I thought the entire scheme was an extremely elaborate cover for the fact the lead ship was the actual ark and everyone else was just plain doomed. This would have mirrored one of the themes in the 3 Body Problem where interstellar space travel is strictly forbidden until everybody can participate at once. The "launch the Earth as a spaceship" concept was so poorly thought out I thought it had to be a fraud in the story, and our somewhat dense protagonists just didn't catch on.
(oh and it wouldn't be strong enough to affect the Earth's rotation)