Surprisingly, we can - but only for a short time. It loses its atmosphere quickly. Geoffery Landis looked into this. The Moon loses its hydrogen and helium very quickly, on a timescale of fifteen minutes on the sunlit side, just because the gravity isn't strong enough to hold them in place. But it loses heavier gases like nitrogen and oxygen over time periods of thousands of years, longer than most civilizations last.

When the atmosphere is very thin, it loses nitrogen and oxygen much more quickly, as a result of the atoms getting ionized and then swept away by electric fields associated with the solar wind. They get lost in about 100 days. But if we can thicken the atmosphere up enough, this mechanism is no longer significant (the number of atoms ionized is the same or even increased a bit, but it's a tiny fraction of the atmosphere), and then the atmosphere will last for thousands of years. See his Air Pollution on the Moon for details. His main focus there is the adverse effect of the atmosphere created as a byproduct of industry on the Moon, leading to degradation of the valuable high vacuum there. But he does briefly touch on terraforming at the end.

Then Gregory Benford, the hard science fiction writer, thinks we can terraform it, in this rather intriguing article: A Terraformed Moon Would Be an Awful Lot Like Florida.

He envisions hitting the Moon with a hundred comets the size of Halley's comet - at the same time spinning it up so it has a day of 60 hours.(I'm not sure of his calculation, it seems that the number of comets needed is more like 10,000, see below).

Presumably you would keep adding new comets to it - but if he is right that 100 comets are enough - then just adding one new comet a century would keep it going - and you could do that without harm to the citizens of the Moon by breaking the comet up into tiny pieces before it impacts onto the lunar atmosphere. So what would it look like if we could terraform it? Well here is an artist's impression of a terraformed Moon.

I prefer this to ideas for terraforming Mars, because there is no life on the Moon to be impacted by it. Also the Moon is close to Earth, and it is clear that it has to maintain a high technology to keep it terraformed, so if both Earth and Moon have high technology, they can work together. The Moon is at a fixed distance, and we can set up some easy way of getting back and forth with space elevators or space tether systems. And use the same factories - and exchange materials from one to the other easily etc. It is similar to the idea of a very hugeStanford Torus, to terraform the Moon. It turns a lifeless though very large region into a habitable area.

With Mars, then if it can be terraformed, it's on the thousands of years timescale, with lots to go wrong. It may seem like a new earth but if you can terraform it as quickly as that, it needs mega technology to stay terraformed, and it might unterraform as quickly as it terraformed.

• PLANTS HAVE TO WORK SIX TIMES HARDER - TO MAINTAIN THE OXYGEN IN AN ATMOSPHERE SIX TIMES THE MASS PER SQUARE METER

On Mars the plants have to work roughly three times harder than on Earth so it needs about three times as much oxygen to achieve the same partial pressurs on the surface. For the Moon the plants have to work six times harder, but they get about double the levels of sunlight they get on Mars for photosynthesis.

If you work it out in detail, there is very little between them in this respect

Detailed calculation: to achieve an Earth normal 10 tons per square meter of atmosphere pressure in lunar gravity - you need 60 tons per square meter of gas in mass. The plants need to maintain 12.7 tons of oxygen per square meter (2.095*9.807/1.622) instead of the 2..095 tons per square meter of oxygen for Earth. On Mars they need to maintain 5.54 tons per square meter (2.095*9.807/3.711). So they need to create 2.29 times as much oxygen on the Moon. Earth (and so the Moon) gets 2.25 times as much sunlight as Mars. So there isn't much in it.

• CHECKING THE CALCULATION - 100 COMETS - OR 10,000 COMETS??

Let's check his Halley's comet calculation, calculation indented and I'll include all the steps in detail to make it easy to check.

With a radius of the Moon of 1737.4 km I make the surface area of the Moon 4×π×1737.42 = 37,932,328 square kms. For an Earth pressure atmosphere we need (9.807/1.622)×10 tons per square meter, or around 60 tons per square meter, and multiply also by 10^6 for the number of square meters per square kilometer, that's 37,932,328 × 106×60 = 2.28×10^15 tons or 2.28 quadrillion tons. Halley's comet is 242.5 billion tons.

So you would need around 2.28 quadrillion/242.5 billion or 9,402 of Halley's comet.

Geoffrey Landis assumes a one psi atmosphere, of pure oxygen, at the Armstrong limit so about 6.9% and works out the total mass needed as two hundred trillion tons or about 825 Halley comets..But he works that out as 50 to 100 Halley comets so must be assuming a larger mass for Halley's comet of two to four trillion tons - it's an early paper from 1990 so I think he is just using older data for Halley's comet.

So, I think Gregory Benford's 100 Halley comets probably comes from Geoffrey Landis's paper. - it means 100 comets the size of Halley but with an older figure for the mass of Halley. And both are assuming the thinnest atomsphere a human can breathe without the moisture lining their lungs boiling.

Approaching it another way, our 2.28 quadrillion tons of atmosphere for an Earth normal atmosphere corresponds to around 2.28 million cubic kilometers of ice assuming average density of 1 (there are a billion tons to a cubic kilometer of water). Or assuming a density of 0.532 tons / cubic meter (same as comet 67p) that's 4.3 million cubic kilometers

So solving for radius, then you get
π×r^3×4/3 =4.3 ×10^6

So r = cube root(4.3×106×3/(4×π))
= 100 km approx.

So in short, it seems that you need more like 10,000 copies of Halley's comet, or you could hit the Moon with a comet of about 200 km in diameter - or larger if the density is less than 0.532, less if it is more than 0.532. If you did that, you'd have enough material for an instant atmosphere. That is if it is all potential atmosphere, but of course a lot would be water, perhaps 80% which you'd need to convert to atmosphere somehow, perhaps split the hydrogen and oxygen to create an oxygen atmosphere.

If you aim is to make a CO2 atmosphere, then assuming it is 80% water, then you'd need

r = cube root(5×3.83×106×3/(4×π))

= 166 km approx. Or 332 km in diameter.

If the aim is nitrogen, with 0.5% of the comet made of nitrogen, and needing 78% of the atmosphere as nitrogen, you are talking about cube root((100/0.5)×0.78×3.83×10^6×3/(4×π)) or 522 kim in radius, so about 1044 km in diameter. There would then be plenty of water, and carbon dioxide.

Once we can move large comets easily from the outer to the inner soar system, this could be possible selecting a large comet of a suitable composition. You'd have a lot of water as well which would be useful.

• BUFFER GAS - SUCH AS NITROGEN, NEEDED FOR A BREATHABLE ATMOSPHERE

Then - for a breathable atmosphere - then you need to have a buffer gas, which on Earth is nitrogen (CO2 is poisonous to humans in large concentrations). You can have a thinner pure oxygen atmosphere with no buffer gas, but this is a fire risk (as we found out in practice with theApollo 1 disaster), so not likely to be used for terraforming or large scale habitats, though it is used for spacesuits as it reduces the pressure inside the suit so makes them more flexible and easier to use and the fire risk can be managed in a spacesuit by using fireproof materials. It's not really feasible though to make a terraformed Moon in its entirety fire resistant.

So most of that weight needs to be nitrogen - unless you have some alternative buffer gas. Halley's comet has hardly any ammonia (NH3). As for Kuiper belt objects, their interior composition is highly varied from rocky all the way to solid ice,
The compositions of Kuiper belt objects - but I can't find much about the ammonia and nitrogen abundances inside the objects (rather than on the surface). There are some meteorites also that are rich in nitrates. But finding enough nitrogen might be a problem if that's our buffer gas, seems to me. Titan has a dense nitrogen atmosphere, and is larger than our Moon and has an Earth pressure atmosphere so it has enough nitrogen, it's just in the wrong place and inaccessible from Earth. It doesn't seem practical to transport its atmosphere to the Moon. Also, it's unique and interesting in its own right, as the only moon of its type in our solar system.

So, it seems that we depend on comets. If a comet is only 0.4% ammonia, or nitrogen etc., you need nearly 200 times as many comets for the nitrogen, so two million copies of Halley's comet. Or a comet 6.3 times larger turning our 200 km comet into a 1,260 km diameter dwarf planet.

Maybe you have to hunt around - there are lots of Kuiper belt objects, we only have discovered a tiny fraction of them and maybe one of them has lots of nitrogen? For all we know, maybe when we expand the search, maybe we find tens of thousands of nitrogen rich Kuiper belt objects the size of Halley's Comet - far too small to spot from Earth with existing telescopes? We can't really do a decent calculation here unless someone has a good idea of a source with a well known nitrogen rich composition.

If we find one, then we have to move it into the inner solar system and hit the Moon (gently) before the nitrogen rich ammonia (or more difficult, nitrogen ice) gets a chance to evaporate. If it is a large body and we move it into the inner solar system quickly, this seems feasible, without going into details of the calculation. It would be a bit like storing ice through the summer which they used to do in high latitudes before freezers and refrigerators. Or for that matter, with the level of technology we are imagining here, we could just cover the comet with a reflective layer to keep it cool for the journey.

• REACTION OF OXYGEN WITH THE LUNAR CRUST

Another problem - if you want an oxygen rich atmosphere - well the Earth had reduced iron and it took millions of years to oxidize it before we managed to get an oxygen rich atmosphere. Basically, all the reduced iron has to rust. A process that has already happened on Earth, and on Mars (it's the reason the surface is red) - but not yet on the Moon.

So - same is likely to happen on the Moon. Maybe we can speed it up but for a long time all the oxygen we create will get absorbed by the lunar crust through chemical reactions, as happened on Earth in the early stages of the Great Oxygenation Event

"The upper few kilometers of the lunar surface contain several times 1018 kg of iron(II) which in the presence of water would readily react with oxygen to form iron(III). Such an amount of iron(II) could easily absorb all of the oxygen in the Earth atmosphere.
 
"A large fraction of the Moons crust consists of oxides of calcium, magnesium, and iron(II), which in the presence of water would react to form hydroxides that would (partly) dissolve in the forming seas to create a poisonously alkaline fluid, with pH 10--11. If enough oxygen were available to oxidize the dissolved iron(II) hydroxides, insoluble iron(III) hydroxides would precipitate on the sea floors and shores, creating vast quantities of slightly poisonous, orange mud. Such reactions would be violent and fast in the upper part of the crust, but their rate would decrease with increasing depth. The oxidizing, hydration, and other processes would continue for ages. In the meantime oxygen and other pressures would not be stable. Most of important all: the absorption of such enormous amounts of oxygen, water, by the upper part of the crust of the Moon would make the rocks expand by perhaps as much as ten percent or more. One can wonder if such expansion would be a tranquil process. It could create strong quakes for possibly many thousands of years. "
from: An Atmosphere for the Moon

So, the upshot of all this is, the terraforming the Moon may well be possible with future technology which may not be that far away, especially with nuclear fusion or such like. But I think it may be a little harder than Greg Benford suggests in his article.

Terraformed lunar far side by Ittiz.

If I've got the figures right here, you need 10,000 Halley comets, or a giant comet 200 km in diameter. If you need to supply nitrogen as a buffer gas from comets with the same composition as Halley, you need two million copies of Halley's comet, or a larger dwarf planet perhaps up to 1,260 km diameter depending on how much nitrogen it has in its composition. After that, you would have many issues with reaction of the water with the dry lunar surface. And then the plants would have to work six times harder than on Earth to produce the same partial pressure of oxygen.

Do be sure to correct me if I have made any mistakes here!

Of course many of these issues would also turn up for Mars, and if you compare it with Mars it doesn't seem so bad.

Mars would need similarly huge amounts of nitrogen for instance, less per surface area but more in absolute terms.

Comparison of mass of nitrogen needed for Mars and the Moon: the Mars surface area is 144.8 km² and for the Moon, 37.9 km². To get the same atmospheric pressure, the Moon has to have 2.29 times as much mass per square meter than Mars. So the amount of mass needed for Mars is 144.8/(37.9* 2.29) so Mars needs 1.67 times the mass for the Moon. Or about 3.34 million Halley comets to supply it with nitrogen, unless it is available indigenously.

The plants have to work six times harder just as for the Moon. The atmosphere lasts longer on Mars, but it's not a permanent feature without megaengineering - it will disappear over millions of years timescales. On Mars you need to have global mirrors or greenhouse gases and still supply some volatiles with comets, for the Moon you don't need to compensate for reduced sunlight but need a constant input of volatiles.

New carbon cycles have to be set in place in both cases to return carbon to the atmosphere and these have to be based on novel principles as Earth's cycles won't work in the same way, especially the long term conversion of limestone back to CO2 as a result of subduction due to continental drift won't work on the Moon or Mars. As for the Moon, Mars also has deserts which are extremely dry and will take up much of the water if water is added to the planet (though it doesn't have the problem of oxygen reacting with the surface materials). And so on.

So - the Moon might not be so bad if you compare it to Mars, maybe you could terraform it a bit faster as a smaller object needing less total mass, and it doesn't need any supplemented sunlight or greenhouse gases. But it seems an impractically mega project even so with present day technology.

To last longer than a few thousand years, it would need constant maintenance in the form of extra volatiles from comets, but the same is true for ideas of terraforming Mars they need constant maintenance in the form of orbiting mirrors or greenhouse gases and need some resupply of volatiles as well to keep it terraformed (if it worked). In the case of Mars that's because the planet is too cold to remain habitable without orbiting mirrors or greenhouse gases, while in the cases of the Moon it is because its gravity can't hold onto its atmosphere. Mars loses its atmosphere also, but on much longer timescales.

However, it's not really that much different.

The timescales are similar too for creating the atmospheres. To create an oxygen rich atmosphere on Mars means sequestering out all the carbon assuming there is enough CO2 to make an Earth density atmosphere whihch most think there isn't (at most enough for 10%) and that process would take around 100,000 years using photosynthesis, as a result of which Mars would of course cool down even further without CO2 to warm it up so need more greenhouse gases or orbital mirrors.

I'm not suggesting we terraform either. I don't think we are anywhere near the stage where it makes much sense to attempt terraforming, a trillion dollars a year project that you have to commit to for thousands of years, whether it is for the Moon or for Mars or anywhere else. We find it hard to commit to a space project for a few decades and a few billion dollars a year. That's apart from planetary protection issues. And as well, we just don't know anything like enough about how ecosystems work, getting unpleasant surprises with "toy ecosystems" the size of Biosphere II, and not able to make even tiny changes to the atmosphere of Earth. If we could make a 0.01% change in the amount of CO2 in the atmosphere the global warming crisis would be over right away.

However, just as a matter of the physics. the Moon can in principle be terraformed though with many issues you'd have to sort out. But the same is true for Mars. I don't see them as that much different actually. Not with present day ideas of terraforming. We would need to understand this all in a lot more detail than we do now to see which is best, if either can be terraformed in practice

Terraforming the Moon is another of those topics that doesn't seem to have a lot of attention in the academic literature. But apart from Greg Benford's article, here are forum discussions which are a good source of ideas, though of course not peer reviewed:

• Issues concerning the very difficult terraformation of Luna [Archive]
• Terraforming the Moon - Your opinion, please (Page 1) / Terraformation / New Mars Forums

And then the An Atmosphere for the Moon and there's the Universe Today's HOW DO WE TERRAFORM THE MOON?

However there is another solution:

PARATERRAFORMING THE MOON

Paraterraforming means covering the surface with habitats, eventually domed cities and eventually the entire surface covered in habitats. Eventually they could merge together to make a kind of a sky to hold the atmosphere in - with lots of partitions for safety.

That needs far less atmosphere - because instead of 60 tons of mass per square meter to supply atmospheric pressure - you just have as much air as is needed to fill your greenhouses, which may have heights measured in meters. Even if the greenhouses are a hundred meters high, that's 122.5 kilograms per square meter instead of 60 tons per square meter of air, a huge saving (using density of air of 1.225 kg / m2)

Can a complex closed ecosystem work with a shallower atmosphere like that? They thought it might with Biosphere II but it proved harder than expected, still there doesn't seem to be any major reasons why not. You still have the problem also of oxygen and chemical reactions. But maybe at the same time that you enclose them from above, you can insulate them from the subsurface as well so they don't lose their oxygen through chemical reactions with the lunar soil. One way to do that would be to turn the surface into glass, though you might instead want to use bulldozers to remove a few meters depth of regolith, turn the layer below into glass then replace the regolith to use as soil. This is quite reminiscent of Biosphere II where one of the main reasons it failed was because of chemical reactions involving the concrete the habitat was made of.

This has many advantages

• You need much less by way of air, because it only needs to cover the surface to a depth of meters rather than kilometers.
• You ned much less water, because you can use lined ponds, and generally, insulate surface habitats from the ground, rather than just pour the water onto the surface which is covered to considerable depth with dry regolith over dry rock. Drier even than the Sahara sands.
• Insulating habitats from the ground would also prevent those issues of moon quakes.
• You can reduce losses of water and air. As I said above, a 1 kg loss per person per day would be a thousand tons lost per day. That could be reduced to almost zero by using ionic fluids based liquid airlocks, or other techniques to make sure that almost no water or air is lost when you go in and out, and much improved recycling. Same also if the entire surface is covered in
• If you have really good technology here, to keep air and water inside your habitats, you may even be able to retain parts of the Moon as hard vacuum for factories and processes that need vacuum, or the pristine original surfaces for scientific study.
• You can start by living in the caves, which if they are as large as theory suggests, may be huge O'Neil colony sized habitats already set up for you. Then move up to domed cities in craters. There's a natural progression there on the Moon which could gradually lead to paraterraforming in the future.

I can't find an illustration of a paraterraformed Moon but the same idea can be used for large asteroids and small moons too, if you can tolerate the low gravity. Here is an artist's impression of a paraterraformed Phobos by Ittiz

MOON COMPARED WITH MARS AND FREE FLIGHT STANFORD TORUS ETC

Has a major advantage over Mars that it is close to Earth so easier for the Moon and Earth to support each other technologically, rescue missions in early stages etc.

Plus - that it has similar orbit to Earth, many similarities such as tides (from the Earth) - same length of year - and same amount of light from the sun.

What we don't know is the gravity prescription - how much gravity do humans need for health? If we can be healthy in lunar gravity then that would be encouraging. But if not - you do have the option of continually spinning habitats - or sleeping in a train that runs around a track, or habitats on tracks etc. Or depending on spin tolerance and gravity prescription, personal small arm centrifuges for sleep or for a few hours a day.

MORE RESEARCH NEEDED

I think - that making a Stanford Torus is likely to work better than either - though we haven't yet done the basic research needed to know for sure.

However - I don't see also how a Mars colony can keep it free of Earth life and the main reason to go to Mars right now is to learn about Martian life if it exists. So that would seem to rule out Mars, for now at least - at least Mars surface - free flying habitats in Mars orbit using materials from Deimos might be compatible with planetary protection and also very useful for exploring Mars via telepresence.

DO WE WANT TO TERRAFORM?

Do we want to terraform or colonize anywhere? What's the reason for doing it?

For most of the reasons I can think of, a Stanford Torus works better, small, self contained, and you can experiment with many different ways of doing things, variable gravity so you can set it to full Earth g, even more, or to far less Mars, Lunar or even less if you want - and if things go wrong can just start again.

Seems the best way to begin given that we know nothing about terraforming and have no experience of what it might do.

WHAT ABOUT LONG DISTANT FUTURE?

As you might see from some of my other answers, I have a few questions about whether we should colonize the galaxy at all - - and - if we start to colonize our solar system - can we avoid also colonizing the galaxy long term?

But can a galaxy filled with a hundred billion trillion humans work, and would it be good for other ETs or ourselves or would we be the monsters in the galaxy who spoil it for everyone?

So - I have these long term issues with it all.

Tend to urge caution - go slowly, find out as much as we can - and is no urgency about it.

NO IMMINENT THREAT

Earth is not faced by any imminent threat, not a natural one for sure - yes in a few hundred million years, but for the next few thousands and millions of years we don't need to work on imminent plans to leave Earth.

The creatures who may need to escape Earth would be so far into the future they have surely evolved to new species. It's far enough into the future for humans to evolve all the way from the first multi-cellular lifeforms a second time.

PERHAPS IN DISTANT FUTURE?

But - maybe some time in the not so distant future we will understand the universe far better, and our capabilities better, maybe even also encountered other ETs - we might learn a lot from them about what is possible or not and what can go wrong. Maybe spot other technological ETs that have attempted terraforming and find out what happened to them. Maybe spot other galaxies that have been colonized (if ours has not) - and see what happened to them.

Or learn about exoplanets and have experience of closed habitats in Stanford Toruses etc.

Maybe that will give us the knowledge we need to do big things like terraforming planets safely and with full understanding of how it works and its implications for the future.

And who knows how soon that future might come, but personally I don't think we are there quite yet :).