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. 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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Could you really have survived the end of Portal 2?

Could Chell really have survived her trip to the moon at the end of Portal 2?

Let’s review exactly what happens to our protagonist at the end of Portal 2. (If you have not played this game and are planning to, stop, go play it, and then come back. We’ll wait. Everyone has the above achievement? Good.)

So. Wheatley booby trapped the Stalemate Resolution Button with a bomb, blasting Chell through a metal grill, back into the main chamber and off her feet. Chell was meant to have been killed by this blast and wasn’t, but we can safely assume she was not feeling 100% after being bodily flung through a metal grate. Chell then puts a portal on the moon, as one does. This portal then sucks pretty much everything that isn’t bolted down out of the room and onto the moon, including Chell, who has the presence of mind to grab onto Wheatley’s handles before she gets pulled through the portal. Chell then spends 30 seconds clinging on to Wheatley until GLaDOS takes over control, bashes Wheatley off the system, and hauls Chell back through to Earth by her wrist with a metal clamp, and the portal is shut down. Can a person survive this?

There are a few concerns here; firstly, the moon has no air. Chell, fortunately, will not need to worry about the air to breathe - she’s only about two feet away from the portal, and there’s lots of air rushing past her. Nor would she need to worry about the average temperature on the moon being -53 C (-63 F), again because she’s surrounded by a vast rushing of room temperature air, and all the wind chill equations I could find indicate that room temperature air (starting at 70F) will only get down to 60F or so. So we’ll assume that she’s perfectly capable of breathing while on the moon. Even if there wasn’t, Chell could have survived the vacuum of space for about a minute - but we’ll assume that she won’t pass out due to a lack of air, since there’s so much of it nearby.

Chell’s main problem is going to be the wind speed. There’s a handy equation called the Ensewiler formula to convert a pressure difference into a wind speed, which is written out as P= 0.002496 v^2. P is the pressure difference in pounds per square foot, and the velocity comes out in miles per hour. The moon portal is a window between standard air pressure (1 atmosphere, or 2116.216 pounds per square feet) and the moon, which has no atmosphere, and thus no atmospheric pressure at all. So if you put in a difference of one atmosphere into the Ensewiler equation, we get a velocity of 920 miles per hour.

920 mph is kind of a problem. This is 1480 km/h, or 411 m/s, and faster than the speed of sound. For some context, the fastest recorded wind speeds on Earth are 253 mph (recorded in Tropical Cyclone Olivia, which hit Australia in 1996) and an F5 tornado that hit Oklahoma in 1999, which clocked in at 302 mph. Felix Baumgartner’s jump from the edge of space got him to clock in at 843 miles per hour before he pulled his parachute, but he was wearing a massive protective suit, specifically designed to keep him safe, and he crossed the sound barrier quite high in the atmosphere, where the air was not very dense. Chell has a tank top. However, the shock from entering a 920 mph wind wouldn’t have killed her immediately - a shock wave from a bomb only becomes lethal at about 1500 mph. It’s still not doing her any great favors, but she wouldn’t have died instantly.

Wheatley’s cabling would have to bear quite a lot of tension - if they were 3 cm x 3 cm cables, holding a weight of 135 pounds (my assumption for the weight of Chell + the weight of the core) should have been fine for steel cabling. However, those cables are also holding up 15,154 Newtons of force pulling Chell in the other direction. A Newton is defined as the amount of force required to push a kilogram one meter in one second. This number of 15,154 Newtons is assuming that the gravity of the moon would have partially counteracted the insane drag from the wind, but since the force of the wind is so huge, and the force of gravity on the moon is so weak, the moon’s gravity doesn’t really help out much. Even if Chell were on the Earth, the Earth’s gravity would only lessen the total force by 500 Newtons. Steel cabling of 3 cm x 3 cm should still be able to hold up to this wind, even through this is a lot of force. But it does mean that Chell genuinely would have been blown perpendicular to the surface of the moon by the wind - 920 mph is well above the terminal velocity for a human on Earth.

A bigger issue would have been Chell’s ability to hold on to Wheatley’s convenient handles. The average person has grip strength of approximately 500 Netwons of force. If the drag force from 920 mph is 15,154 Newtons in the other direction, 500 Newtons is not going to be enough to keep ahold of a bar against that pressure. 500 Newtons will keep you holding on to something in the face of 182 mph (293 km/h) winds, but nothing more than that. No matter how much Chell would have tried to cling to the handles, the sheer force of the wind would have blown her away.

This is assuming, of course, that Chell would be able to keep her hands gripping the bar. In actuality, being dragged out of that portal would be very much like being thrown out of a jet travelling at 900 mph, and told to hang on to a trapeze bar. The first thing that actually would have happened is that both of her shoulders would have dislocated. Shoulders dislocate with 325 Newtons of force, well below the force the wind is placing on her. If your shoulder is dislocated, there’s a very low probability that you’re going to be able to hold on to much of anything, since the major nerves travelling down the arm are being stretched a lot more than they usually are, and nerves don’t like being stretched. While we’re thinking about dislocations, if any part of Chell’s legs had been twisted at all by the wind, it’s very likely she could have also dislocated one or both knees or ankles.

Even worse, if any of those boxes flying out of GLaDOS’ chamber weighed about 10 pounds and happened to collide with Chell, they would inflict about 1860 Newtons of force on her, which is more than enough to break a finger bone. A five pound object would cause a wrist fracture similar to the kind of injuries that boxers can get.

There is a story about a fighter jet pilot who ejected from his aircraft at 800 mph. His shoulder was dislocated, his knee was dislocated, his other ankle was broken, and all the capillaries in his face were broken from the force of the wind - his head swelled up to the size of a basketball, and his lips swelled enormously to the point where he had a hard time moving them. He survived, but his clothes were torn to bits, and he was in the hospital and undergoing physical therapy for six months afterwards before he felt back to normal.

Chell’s wind is 120 mph stronger, and she wasn’t immediately slowing down after her trip to the moon - she was in 920 mph winds for 30 seconds, not the 3 seconds of Capt. Udell. It’s probably safe to assume that the capillary breakage suffered by Capt. Udell would also pose a problem for Chell, meaning that by the time she got back to Earth, her face would have been one enormous bruise, and she’d have been nigh unrecognizable from swelling in a matter of minutes.

In all likelihood, the most reasonable thing that happened was that after she was dragged back into GLaDOS’ chamber by her wrist (which almost certainly would have broken her wrist, if it hadn’t already been broken by passing debris) is that Chell promptly passed out. However, she would have needed extensive medical care and certainly morphine-strength painkillers to deal with her injuries, instead of being back up on her feet after a few minutes.

What of Wheatley in this scenario? Well, rather tragically for the Space Core, 920 mph is not enough to escape the gravitational pull of the moon. Escape velocity for the moon is 2.5 km/s - 920 mph corresponds to 0.411 km/s, six times too slow to escape. We can calculate how long Wheatley would spend flying in space before crashing down to the surface of the moon - since he was flung out perpendicular to the surface, he’d plop back down exactly where the portal was. If we assume that Wheatley was sped up to wind speed, he has 8 and a half minutes of flight (reaching a grand old height of 52 km above the surface of the moon) before crash-landing on the moon again at 920 miles per hour. Since the Space Core left the portal about 10 seconds before Wheatley was released, the space core would land about 10 seconds before Wheatley, pretty close to in the same place. Assuming neither of them smash upon impact, Space Core would then undoubtably drive Wheatley insane by repeating “I’m a moon base! Moon base! Spaaaaaaaace” ad infinitum.

Something here unclear? Have your own space question? Feel free to ask!