Physicist: Almost prohibitively!
Ever since the Moon entered the scene 4.5 billion years ago, it’s been slowly drifting away. Initially it was around 15 times closer and bigger across in the sky. The effects that push it away decrease rapidly with distance, so the Moon climbed most of the distance to its current orbit early, but even today it gains about 4 cm per year.
The tides raised by the Moon are made up of a lot of water and that water, like all matter, creates gravity. The Earth turns in the same direction that the Moon orbits and, since water doesn’t instantly slosh back to where it’s supposed to be, the tidal bulge is dragged a little ahead of the Moon. The nearly insignificant gravity generated by that extra water pulls the Moon forward and speeds it up. The effect is especially small, since a nearly identical bulge appears on the opposite side of the Earth, pulling in the wrong direction. The difference is that the forward water bulge is slightly closer.
If you’re in orbit and you start moving faster, your orbit gets higher (ironically you end up slowing down overall, just higher and slower). At the same time that the tidal bulge pulls on the Moon, the Moon pulls on that bulge, slowing the Earth.
So while the Moon drifts away, days on Earth get longer. According to coral records, as recently as half a billion years ago the day was a mere 22 hours. Corals aren’t inherently clerical, but they do have both daily and yearly growth cycles with corresponding growth rings. And 4.5 billion years ago, before the Moon had siphoned off so much of our rotational inertia, a day on Earth was a mere 6 hours. Give or take (there wasn’t any coral at the time to obsessively write it down). So the Moon is drifting away and in exchange we Earthlings get both lunch and brunch.
A retro-reflector is a clever set of three mutually perpendicular mirrors that reflect any light back toward where it came from, regardless of where that is. Basically, if someone shoots you with a laser, wearing a suit covered in retro-reflectors is the fastest way to get revenge. By bouncing lasers off of any of five sets of retro-reflectors left on the Moon between 1969 and 1973 by the USA and USSR, we can determine the distance to the Moon to within millimeters and have found that, at present, the Moon is drifting away at a sedate 4 cm per year.
Given enough time, the Moon would eventually escape entirely. Luckily, in about five billion years the Sun will swell up and destroy both the Earth and Moon before that happens. So: nothing to worry about.
However! If we’d like to continue being the only planet to have both annular and total solar eclipses (the only objective measure of planetary exceptionalism), we need to act fast in the next several million years to keep the Moon from getting any farther away.
The Moon is “rolling down a hill”, energetically speaking, and this total needs to be expended every year (effectively) forever to keep the Moon from “rolling” any farther away. Unlike practically every other question about planetary-scale engineering, “Project Sisyphus” is not impossible! Merely so expensive and pointless that it may as well be.
Anything that moves in a circle has angular momentum. Something a Moon with mass m orbiting at a radius of R around a planet with mass M in a universe with gravitational constant G will have an angular momentum of:
As r gets bigger (as the Moon moves away), L increases. To hold the Moon in place we’d need to counteract that increase and keep the angular momentum of the Moon constant. To figure out how much L changes over small distances (and 4 cm counts), we just find the differential. This is what calculus is good at: finding tiny changes in one thing, dL, given tiny changes in something else, dr.
Ion drives have exhaust velocities much higher than the escape velocity of the Moon, which is good; otherwise they wouldn’t be rockets moving the Moon, so much as fountains decorating the Moon. They’re also the most efficient type of space propulsion, maxing out around 80% efficiency. The most powerful use about 100 kW of power and are capable of generating about 5N of force (about 1 pound).
One of these drives, pushing for a year, could add or subtract
from the Moon’s orbital angular momentum.
So we could counter the Moon’s annual energy gain with a mere 24.5 million of our most powerful ion drives, each coupled with (at least) a 50 m by 50 m array of solar cells and batteries to supply the 100kW they need (including inefficiencies and the lack of Sun at night). Powering those drives to keep the Moon in place only requires a modest one sixth of the total energy generated by people every year (as of now). So if we really, really felt like doing it, this is the sort of “problem” that could be “solved” with little more than the greatest collective effort in human history.
If you lived on the Moon, you could build a statue pointing at the Earth and you’d never have to change it, because the Moon’s orientation is fixed with respect to the Earth. If that statue is pointing straight up, it’s in the middle of the justifiably-named “near side”. For exactly the same reason, you could build a statue pointing in the direction the Moon is moving as it orbits. If the statue were pointing straight up, it would be in the middle of the west side, near Mare Orientale.
If we put all of those ion drives, their batteries, and solar panels in Mare Orientale, they would cover about 75% of it. Combined with all of the access roads and support structures, there’d be scarcely a single stone left unturned.
Each of those ion drives would need on the order of a ton of gas every year (argon and xenon are popular choices) to use as propellant. This is the stuff that would literally be thrown in the direction the Moon is moving in order to slow it down. Summed over the millions of drives, the gas continuously fired from the Moon by Project Sisyphus would be on par with the peak loss of gas in comets as they pass through the inner solar system. That is: on a clear night we would be able to see the gas as a ghostly comet tail extending, in a very straight band, from the left side of the Moon (from the right side for our southern hemisphere readers. G’day and dia bom!).
We can harvest enough argon from our atmosphere (it’s about 1% of the air you’re breathing now) to provide propellant for the next 2 billion years, so we’ll have time to figure out another option before we exhaust the supply. The Falcon Heavy could carry enough argon for about a dozen or two of the 24.5 million drives, so we’d need to launch several thousand of them to the Moon every day until we find a better system.
Compared to that, actually building and maintaining all of the infrastructure on the Moon would be difficult. It would be a lot easier to harvest everything (steel for structures, silicon for solar cells, etc.) we need from the Moon directly. So the damage wouldn’t be restricted to just Mare Orientale; there would be deep scars all over the Moon from where we tore up the place for mining.
So the Moon is drifting away, meaning that in a few hundred million years we won’t have total eclipses and the tide will be slightly more mild. With an achievable, but mind-boggling, effort we could stop the Moon from drifting away and ensure that our descendants can, in several hundred million years, continue to appreciate a slightly wider variety of astronomical phenomena (assuming eyes are still in style). In ten million years our barely-human descendants will curse us for handing this chore down to them, and in a hundred million years their definitely-not-human descendants will shake their cyber-tentacles at us in fury.
But you guys. I think we should try it.