How long would it take to deflate the Earth's atmosphere out into space?

My roommate and I were in a heated debate that lead us to read your post about the ability to survive the end of Portal 2. However, our question is slightly different. Suppose the same kind of portal was created on Earth’s surface to the Moon’s, how long would it take for the Earth’s air supply to be released through the portal into space?
 At higher and higher altitudes, the Earth's atmosphere becomes so thin that it essentially ceases to exist. Gradually, the atmospheric halo fades into the blackness of space. This astronaut photograph captured on July 20, 2006, shows a nearly translucent moon emerging from behind the halo. Image credit:  NASA

At higher and higher altitudes, the Earth's atmosphere becomes so thin that it essentially ceases to exist. Gradually, the atmospheric halo fades into the blackness of space. This astronaut photograph captured on July 20, 2006, shows a nearly translucent moon emerging from behind the halo. Image credit: NASA

If any of you haven’t seen the previous Portal 2 post, I’d recommend having a look at it here, because I’m going to pull some numbers from it. I’m also going to make some slightly unphysical assumptions, but the results of those assumptions is that we’re going to calculate a lower limit to the amount of time it would take to bleed the atmosphere dry. In a world where portals actually worked, it would almost definitely take longer, for reasons we’ll go over later.

Our scenario is thus: we have opened a portal between the surface of the Earth and the Moon, as in the end of Portal 2. Effectively, we’re opening a window between the surface of the Earth and a pretty hard vacuum. The dramatic pressure difference here produces a tremendous, faster than the speed of sound, wind, as we worked out in that previous post. Presumably, if you left that portal open for a long time, you would reduce the amount of atmosphere left on the Earth. In the game, this portal is only open for about 30 seconds, but what if we left it permanently open?

The first thing I’m going to assume is that the whole atmosphere of the Earth is entirely at the same pressure (which it is not). Down at the surface where we humans live, the atmosphere is pretty compressed, and so we have an ambient atmospheric pressure of 1 atmosphere. (Yep. That’s the unit.) 1 atmosphere is equivalent to about 14.7 pounds per square inch, or psi. However, the further up away from the surface you go, the more diffuse the atmosphere gets, and both the density of atoms and the atmospheric pressure drops. If the density of the atmosphere drops, the wind speed through our window will also drop, because it’s the difference in pressure on the two sides of our window that drives the wind speed. By assuming that I can compress down the upper layers of the atmosphere so that the air on Earth is at a constant 14.7 psi, then the wind speed will stay at its fastest, and bleed the atmosphere out into Moon space as fast as possible.

 A setting, waning crescent moon amid the thin line of Earth's atmosphere. Image credit:  NASA

A setting, waning crescent moon amid the thin line of Earth's atmosphere. Image credit: NASA

If you compress the atmosphere down, it would fit in a sphere 1999 km across, which then has a volume of 4.19 x 10^18 cubic meters. This...is a big number. How fast can we drop it to zero?

I will have a reasonable guess that the portal itself is about five feet tall by three feet wide - it seems a bit shorter than Chell in game, and wide enough for her to fit through. If we assume that it’s rectangular instead of an oval, the math is nicer, so I’m going to square up the portal dimensions at about 1.5 meters high by 1 meter wide. This gives a portal area of 1.5 square meters. This is key, because with the area of the window, and the wind speed, we can figure out the volume of air lost every second. At 411 meters per second, our speed from the older post, that means that after one second, a bit of air will have traveled 411 meters.

Every second, we’re going to lose about 617 cubic meters of high pressure Earth atmosphere into the space surrounding the Moon. We know how much we have to lose, so from here we can sort out how many seconds it would take to get the total volume of the Earth’s atmosphere out through our portal. As you can probably guess by the 18 zeros following the total volume of the Earth’s atmosphere, it’s going to be a lot of seconds.  In fact, it’s so many seconds that seconds are not a useful unit even a little bit. Converting into years is a little better.

It would take 215 million years.

 Most ISS images are nadir, in which the center point of the image is directly beneath the lens of the camera, but this one is not. This highly oblique image of northwestern African captures the curvature of the Earth and shows its atmosphere. Image credit:  NASA/JPL/UCSD/JSC

Most ISS images are nadir, in which the center point of the image is directly beneath the lens of the camera, but this one is not. This highly oblique image of northwestern African captures the curvature of the Earth and shows its atmosphere. Image credit: NASA/JPL/UCSD/JSC

And remember, this is assuming that the wind speed stays the same the whole time, which it would not in real life. The other thing we’re assuming is that none of this gas will hang around the moon and increase the atmospheric pressure around the Moon. That would also start to balance out the pressure difference, slowing the wind speed down and making this take even longer. The moon historically is not very good at holding onto an atmosphere, so this would likely be a temporary arrangement, but millions of years is not very long for astronomical things, and it’s possible the lunar atmosphere could hang around long enough to slow down our wind. The estimates for the atmosphere around the young moon is that it would have stuck around for 70 million years or so - shorter than our fueling time, but long enough that we could expect it to hang around for a while, before we’re able to finish emptying the Earth’s atmosphere into outer space.

In reality, there would likely be an equilibrium point reached, where both the Moon’s newfound atmosphere and the Earth’s freshly drained atmosphere would find themselves at the same pressure, and the wind, having gradually slowed, would come to a stop, with only the vaguest breeze from the Earthward side as the Sun gradually stripped the atmosphere from around the Moon.


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