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This question already has an answer here:

I suppose that the kind of atmosphere a planet is able to maintain is a function of its gravity, the amount of solar energy it receives, and the atomic weight of gasses. Meaning, Jupiter could have a hydrogen atmosphere, but not the Earth, which has to content itself with an atmosphere of Nitrogen/Oxygen/Carbonic Dioxide. Would the Moon (as in, Earth's natural satellite) be able to retain a stable atmosphere of, say, Argon? Uranium hexafluoride?

What would the mathematical formula be?

Edited to explain why I think this is not a duplicate of "How to blanket moon with a permanent atmosphere?": The older question seems to have implied the idea of a respirable atmosphere ("how can we blanket Moon with lasting atmosphere similar to Earth's" - bolding mine). As I think the references to argon and uranium hexafluoride make clear, this is a question, not about how to blanket the Moon with an Earth-like atmosphere, but about what kind of atmosphere the Moon could conceivably hold, even though very different from Earth and consequently not breathable for Earth-like life forms. (The main idea would be, how to give the Moon atmospheric pressure so that breaches in oxygen-nitrogen domes would be reparable without the risk of people being sucked into the vacuum outside.)

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marked as duplicate by JDługosz, Brythan, Hohmannfan, James, Aify May 28 '16 at 20:37

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Thus has been explained many times already. Your specific question is a subset, mentioned as a detail in any moon-atmosphere Answer. $\endgroup$ – JDługosz May 28 '16 at 1:14
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The question is a bit more complex than just a mathematical formula, and should probably be asked on the physics SE, or the space exploration SE, instead of worldbuilding.

The main problem is - how long do you want to keep your atmosphere? If you gave the moon an earth-like atmosphere right now, it wouldn't lose it in a couple of years. Your humans could live quite well there, for a while. And the earth, as well, is losing parts of its atmosphere - mainly hydrogen and helium - and at the same time, it's picking up cosmic debris, which consists at least partially of water and silicates, providing "new" oxygen and to a lesser extend hydrogen. Still, over all, the earth seems to lose weight every year.

But just like a stone you throw will eventually fall back to earth, an oxygen molecule trying to escape will "fall back" to the atmosphere as well, unless it reaches escape velocity. Once a cosmic ray hits your molecule, and accelerates it beyond that, it's going to be lost to earth.

Now, the escape velocity on earth is $11.2 km/s$, and on the moon, just $2.4 km/s$. Which means you need about $4.6$ times the speed, on earth. But the formula for energy is $E=\frac{1}{2}mv^2$, which means to achieve $4.6$ times the speed, you need $4.6^2=21.8$ times the energy. Or, to accelerate a given molecule to escape velocity on the moon, you need less than $\frac{1}{20}$ the energy, compared to earth.

And since the mass, in that formula, is linear, while the velocity is squared, you need to increase the mass much more if you want to balance the lower velocity "requirement" - you need to increase the weight of your atmoshphere molecule by a factor of 20 to make it need the same energy to leave the planet. Oxygen ($O_2$) has a molar mass of approx. 32, Argon has 40, Uraniumhexafluoride ($UF_6$) 244. But, to need the same amount of energy to accelerate to escape velocity, your "moon atmosphere" molecule needs to have a molar weight of $20*32 - 640$. (Note that Argon is a bad example here. Most gaseous forms of elements have molecules consisting of 2 atoms; noble gases do not. Which is why i multiplied the atomic weight by 2 for Oxygen, but not for Argon).

In other words, the energy needed to remove a $UF_6$ molecule from the moon is just a bit more than a third of the energy needed to remove a $O_2$ molecule from earth. So, a $UF_6$ atmosphere on the moon would probably last a lot shorter than a Oxygen atmosphere on earth; Nitrogen is slightly lighter, but not that much, so those 2 are comparable.

The situation is actually worse than "one third the energy - 3 times the loss". The cosmic particle that removes your molecule from the moon will just accelerate the molecule on the earth. You need a second particle hit the earth molecule in a short time frame after the first, from the generally same direction, to push it over the threshold. Another reason for the moon to lose its atmosphere faster.

And a problem with $UF_6$ - in many cases, your cosmic ray won't leave your molecule intact. Cosmic rays break up $O_2$ molecules to single atoms in the upper earth atmosphere all the time, where they form ozone. I wasn't able to find the bonding energy in $UF_6$, but i assume there's a good chance that in your moon atmosphere, it'll get separated into uranium and fluoride, witht the fluoride light enough to leave the planet, and metallic uranium remaining.

And i didn't even start to take the moon's missing magnetic field into account - while most particles of the solar wind get deflected from earth's atmosphere, this is not the case on the moon. More particles hitting your atmosphere - higher loss.

So, no, you won't be able to give the moon a stable long term atmosphere, no matter which gas you use.

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  • $\begingroup$ There are calculations, an atmosphere like on the Earth would escape the Moon in some ten millions of years. This is why it is long gone, but if we could somehow replenish it (for example, by extracting the oxygen content of the silicate and oxide minerals), it would work. The only problem that it would require thousands times of the energy production of the current humanity. Btw, an UF6 atmosphere would be an atmosphere, but... :-) (Uranium is one of the most poisonous elements, not because of its radioactivity, which is low, but because of its "normal" biochemical behavior). $\endgroup$ – Gray Sheep May 28 '16 at 0:48
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I think the moon could maintain a permanent atmosphere, with a major tech investment. Did anyone see Spaceballs? Seriously, the idea came to me that a planetary canopy could be feasible. Basically a high strength fabric could be inflated by an atmosphere over vast distances across a lunar mare for the testing phase, then extended to encircling the moon completely at a height sufficient to avoid scraping mountaintops - or perhaps anchored to them, providing some kind of opening(s) could be closed in between arrivals and departures. It could be a floating solar collector as well as protector, maybe even designed with the ability to simulate a 24 hour day/night light cycle (The amount of solar energy microwaved to collectors on the ground could be used for mining resources and the generation of rocket fuels to help payback the investment.) Maintenance and upgrading the canopy would be one of the moon colony's priorities, accessed by specialized aircraft or suspension cable cars.

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We could maintain a normal, around 0.2 atm oxygene atmosphere relative easily.

As the other answers explained, the escape velocity from the Moon is much lower as from the Earth, but it is still many times more as the mean velocity of the molecules or the air. Although it will be going very fast. Compared to the escape of the atmosphere of the Earth, whose characteristic time is in billions of years, the atmosphere of the Moon would escape in some tens of millions of years.

This is why it is long gone. But if we could somehow replenish it, some tens of millions of years would be quite enough for us.

Fortunately, the soil of the Moon contains a lot of oxygene - it is mainly from oxide and silicate minerals. There is another problem: it is very strongly bound, extracting the oxygene, for example, from aluminium trioxide (one of the most dense component of the Moon soil) would require to have high-technology, 2000K ovens and tremendous energy need.

On the Moon, there is essentially no nitrogen, no carbon and no hydrogene.

The lack of N2 is not a big problem - it is inert, only the organic materials use it but it is a quite little quantity in planetary sizes.

The lack of carbon is a bigger problem, but probably there is a lot carbonate minerals deeply in its soil. It should be regularly mined.

The lack of hydrogen is the biggest problem. Although there is a little bit of ice in the poles of the Moon, it is far too few to support a sustainable ecology. It should be transferred from another planet (moon). I think the simplest solution would be to shot it there with rail guns from an icy moon with low escape velocity.

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