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Let's say I've got an Earth-like planet with no atmosphere: it's just a barren ball of rock. I want to live there, but I don't like domes, so instead I'm just going to dig a big hole and let gravity keep the air in.

How deep a hole do I need?

According to a chart I found, the density of the atmosphere drops to pretty much zero by about 50km, at the top of the stratosphere. But 'pretty much zero' is not zero; the mesosphere beyond that extends up to about 80km and while vanishingly thin is responsible for dealing with most meteors.

If my hole is a mere 50km deep, then, some of my air is going to diffuse out of the hole and onto the planet's surface. But the surface of my planet is largely flat; there's nowhere for the air to go, so it's just going to hang around and form a dynamic equilibrium. (Unlike, say, if I built a 50km wall and tried to keep the air inside. Air would leak over the top of the wall, fall down into the vacuum on the other side, and be lost forever. Which is why the Ringworld had walls 1000km high.)

So I don't really know how shallow a hole I can get away with. I can replace the air, but I would like it to go without maintenance for at least small geological timescales. Any advice before I start up the earth-moving equipment?

(Yes, it's SF worldbuilding.)

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For what it's worth: Mars's Hellas Planitia is 23km below datum and has 30km-high walls, and the atmospheric pressure at the surface at the bottom is 1155 Pa; normal atmospheric pressure is only 610 Pa. Not good enough for humans either way. :( –  Mark Beadles Feb 21 '12 at 0:53
    
Can we assume that the planet has enough gravity to keep an atmosphere of the composition you're aiming for? (I guess that would be part of the definition of "Earth-like"...) –  David Z Feb 21 '12 at 1:16
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If the planet is Earth-like in its geology, digging a hole deeper than 50km is going to be problematic since you'd be up against the mantle. –  Mark Beadles Feb 21 '12 at 1:25
    
This is equivalent to constructing a shell around a planet at some arbitrary altitude. The pressure at the bottom of your hole will remain steady as long as the air pressure at the top is the same as the 'natural' pressure for that altitude.By which I mean if you start with some global specific (low) pressure at the top before you start digging, the pressure at the bottom of the hole will simply be the pressure you would get if the atmosphere was as much thicker as your hole is deep. –  Benjamin Franz Feb 25 '12 at 20:51
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up vote 3 down vote accepted

Even a normal planet doesn't permanently lock its atmosphere: a little bit of it is creeping out all the time. The air molecules are distributed according to a Maxwell-Boltzmann distribution, which falls off to zero exponentially. A small fraction of that air will always be above escape velocity and will disappear into space. The distribution of air re-thermalizes, and thus another fraction is lost to space. The fraction that is above escape velocity depends on the mass of the molecule: it's appreciable for helium on Earth (popped balloons are gone forever).

For your deep well, you'd have to consider the shape of the Maxwell-Boltzmann distribution and the variation of pressure with altitude (and include a non-Earth "g"). Frame the problem in terms of the amount of loss that you're comfortable with--- something so small that it won't be missed or can easily be replenished.

Someone who's actually engineering this might also want to chill the upper layer of gas with some kind of large-scale air conditioning. That would reduce the loss so that the hole wouldn't need to be as deep. Maybe a greenhouse effect could be useful to keep the upper layer cold and the lower layer warm. After all, who needs to see the sun?

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Unfortunate that convective currents would be working against your goals when cooling the top portion of the column. –  AlanSE Feb 21 '12 at 22:53
    
Thinking about this a bit more sensibly, I realise my question is actually unanswerable. A better question would be: 'How fast does my well lose air?' I think I can work out the maximum rate by multiplying the surface area which I lose air across by the average speed of an air molecule as it passes that surface. If I assume an arbitrary large planet of very low density (giving it a very shallow gravity gradient) then escape velocity is very high, so I only have to worry about losing air sideways over the surface; so my surface of loss is a fairly short cylinder around the well opening. –  David Given Feb 23 '12 at 0:23
    
When I'm less tired I'll go look up how to calculate the height of a column of air at constant pressure, which will let me estimate my escape surface area, and throw some numbers at it. That'll let me estimate the maximum rate of loss. The actual rate of loss will be less than this, of course, because there's lost air getting in the way... anyway, thanks very much! –  David Given Feb 23 '12 at 0:25
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