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It’s the 26th century, and scientists on a barren planet have decided to make the Alpha Centauri Book of World Records by conducting the galaxy’s largest social experiment.

These scientists construct a huge dome, 80 kilometres wide and 2 kilometres in height. This dome contains a habitable environment with an earth-like atmosphere and ecosystem, (though the gravity is only 1/3 that of earth). The dome has a planetarium-like display of the sun and stars as seen from Earth, so that a person standing inside would (apart from the lower gravity) think they were on the earth.

These scientists then seal a number of volunteers in this habitat, having wiped their memories and reduced them to a 15th century level of technology. Their goal is to observe the development of this Society, and eventually to open it up to tourists once they judge it to be stable.

Unfortunately, after a few generations, the inhabitants realise they are in a dome. Not knowing better, they come to the conclusion that this is the legendary Firmament, and that their world was created by God to afford them. So, with textbook human logic, the inhabitants decide to mine through the walls of the dome in order to behold God’s creation from outside.

After a while, they break through. Chaos! The planet outside the dome is a barren wasteland devoid of any atmosphere, and the air inside the dome is being sucked out.

How long have these dome-dwellers got before the air within the dome is entirely drained away entirely?

Edit

The hole is roughly 2.5 metres wide.

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    $\begingroup$ How big of a hole do they make? $\endgroup$ Feb 9, 2023 at 22:45
  • $\begingroup$ See the above Edit $\endgroup$
    – user98816
    Feb 9, 2023 at 22:55
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    $\begingroup$ Is the air in the dome constantly recirculated, or being quietly replenished with modern tech? The latter seems more likely, since 80km seems awfully small to maintain a closed ecosystem. $\endgroup$
    – jdunlop
    Feb 9, 2023 at 23:12
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    $\begingroup$ Seems like a big hole to make before learning it was a Bad Idea. Did they dynamite the wall or something? $\endgroup$ Feb 10, 2023 at 0:38
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    $\begingroup$ Yeah, I can't imagine them chiseling their way through the wall, breaking through and pulling their tool out to find an airless void sucking air through the resulting hole, and deciding to enlarge that hole to 2.5 meters across. Also, it seems like even with the stated technology level they should decide pretty quickly that sealing the hole up is a good idea. I also find it odd that it took multiple generations for them to realize they're in a dome that takes only 2-3 days to walk across. $\endgroup$ Feb 10, 2023 at 3:48

2 Answers 2

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If you're asking for "entirely" you'll get an answer you probably don't want because the rate of leakage is proportional to the pressure differential. The more air leaks the slower it'll go so the answer is basically "forever".

You can get a better answer by choosing a pressure to stop at instead of a complete vacuum.

Using the formula derived here: http://www.geoffreylandis.com/higgins.html (along with their stopping point of 0.5 atm)

We get $t = 0.005 (V/A) Ln[pi/pf] \approx 7{\small,}000{\small,}000{\small,}000s \approx 222$ years.

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This question is less sensible than you think

First of all, please remember to do your homework. It took me a few seconds to find an online calculator for the volume of a spherical cap (aka, dome).

Second, if the vacuum of space worked like you're thinking, planets wouldn't have atmospheres. They'd be sucked away. A barren planet without an atmosphere doesn't cause suction (which is what you're thinking of). In other words, what we're really dealing with is the expansion rate of a gas.

It's worth noting that when Hollywood shows "explosive decompression," they're basically lying through their teeth — kinda. A small amount of gas (e.g., in the cargo hold of a ship) will quickly expand into the vast volume of space. The higher the initial pressure and the lower the initial volume, the more "explosive" it will be. To see this basic truth in action, consider what happens to a paint ball gun as the gas cartridge becomes depleted.

To make things worse, while the initial pressure of your atmosphere will produce the equivalent of a suction force (in this case, it's a pushing force rather than a pulling force), as the pressure drops, the expansion of the atmosphere into the void outside the hole slows. At a rough guess, the first 60% of the atmospheric volume would go quite quickly (maybe an hour or so), but the last 40% would take forever because there's literally nothing sucking it out the hole.

And "sucking" is the word to think about. There's nothing pulling the atmosphere out the hole. Atmosphere is just another kind of "dust" that's milling about around the surface of the planet. Drop a big glob of atmosphere inside the gravity well of a barren planet and it will eventually disperse around the planet. And that brings us to an important point....

You need to realize that while the atmospheric pressure and volume inside the dome will eventually drop to the point where everything dies (for a number of reasons, like the inability to retain heat), it doesn't actually completely evacuate the dome. In fact, thanks to the presence of the dome, it won't completely evacuate to zero atmosphere. There will always be some trace floating around. And that means the question is really complicated.

Because the real question is, "how long would it take for the atmosphere inside the dome to deplete to the point where there's no longer the ability to support the inhabitant's lives?"

And the reason it's complicated is because there's more involved than just the ability to bring enough oxygen to the lungs of your inhabitants. There's the moisture in the air that keeps the lungs, mouth, and esophagus operable. There's the ability to retain heat. There's the need to provide enough pressure to keep problems like the bends from occuring.

There's actually a lot of things that can go wrong. And it's likely that one or more of them will go wrong a long time before the air could be said to have "drained away entirely."

Honestly, the odds are your dome's inhabitants will begin suffocating within hours. But there will be air in the dome long after their dead.

TL;DR

  • The lack of an atmosphere (what we often call "the vacuum of space") is not a pump pulling atmosphere away from anything. There's a huge difference between using a pump to create vacuum pressure and the natural expansion rate of gas into a volume of lower pressure.

  • As air pressure decreases, the ability to provide oxygen for breating, moisture for biological operation, and heat for survival, all decrease with it.

  • Your inhabitants will be dead long before the air is "entirely depleted." In fact, because of the way gases work on a planet without air pressure, there technically won't ever be a time when it's "entirely depleted."

  • Your inhabitants will begin dying, if not be completely dead, within hours. But it will take centuries for the air to reachit's final equilibrium on the planet, which we can't really calculate unless you provide us with the planet's gravity, surface area, and the energy exposure from its star.

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  • $\begingroup$ An hour or so? For that much air? At that rate, it would be like a hurricane. The people, and all the loose stuff, would be pulled out the hole with the tremendous volume of air. And, of course, would most likely seal the hole back up with debris. $\endgroup$ Feb 10, 2023 at 14:09
  • $\begingroup$ It's not about the difference in atmospheric pressure, it is all about the volume of air that can reasonably transit the hole given the pressure differential. How much air can you squeeze through a 2.5 m tunnel at one atmosphere pressure differential? Try it with a bicycle pump. No matter how hard you push, no matter what the pressure differential, only so much air can go through the nozzle in any given time period. $\endgroup$ Feb 10, 2023 at 14:47
  • $\begingroup$ FWIW if you use the formula I linked in my answer it'll take ~3 years to drop to 0.99 atm, which would barely be noticeable (aka the same pressure decrease as going up 300ft in elevation). $\endgroup$
    – Leo Adberg
    Feb 10, 2023 at 15:59
  • $\begingroup$ @JustinThymetheSecond But that's the problem, there isn't a pressure differential per-se. There's nothing creating a vacuum on the other side, there's simply the absense of atmosphere. The moment the first burst of air leaves the tunnel, the pressure "outside the tunnel" is equal to the pressure inside the dome. As the air settles outside (not slow, assuredly), the pressure between the two equalizes. Note, though, that my measurement is not the time to empty the dome of air. It is only the time until inhabitants believably start dying. $\endgroup$
    – JBH
    Feb 11, 2023 at 0:22
  • $\begingroup$ @LeoAdberg I'm not defending my assertion of hours. It was a guestimate because it's just more math that the OP didn't take the time to look up. Before responding to your comment I did a quick online check. There are online gas flow through a pipe calculators all over the internet. You're undoubtadly right - but it's something the OP could have found out for him/her self quickly. The purpose of my post was to point out the fallacy of waiting until all the air was gone ... if it ever was completely gone. $\endgroup$
    – JBH
    Feb 11, 2023 at 0:25

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