How fast could you suck up the atmosphere [closed]

Question

So kind of a strange question, but if i had a 1000 foot wide hose with endless storage space, whats the quickest possible time it could be used to suck up the entirety of the Earth's atmosphere.

Edit: Taking into consideration Knzhou's comment, assume I can manipulate gravity, bundle up the atmosphere, and push it through that 1000 ft wide hole. Is there a limit to how fast it could go?

Edit2: Assume it is being pushed through at half the speed of light.

P.S. if anyone is curious why I'm so hung up on this, I've been planning out a science fiction novel where humanity figured out how to control gravity and this deals with the workings of a terraformer.

Answer 1

The first version of the question asked about just sucking up the atmosphere with a hose, without manipulating gravity. I'll answer that first, since I think it's a very good question that many — including my previous self — get wrong.

No manipulation of gravity

They did this with a giant vacuum cleaner in the movie Spaceballs. But as knzhou comments, it is fundamentally impossible. The reason that the atmosphere stays where it is, is there there is hydrostatic equilibrium: Gravity tries to pull the air molecules down, but pressure builds up and prevents it from collapsing altogether. Whether or not you build a 500 km long hose, doens't change that.

A vacuum cleaner works by creating a lower pressure $P$ inside than outside. But in this case the gravitational potential $\Phi$ is the same inside and outside. In the case of the atmosphere, $P$ is lower in space, but $\Phi$ is lower closer to Earth.

Here a drawing that may help understand. At a given height, $P$ and $\Phi$ is the same inside and outside the hose

Fiercely manipulating gravity

Your first edit assumes that gravity can be manipulated arbitrarily. In that case, there is no limit to how fast the atmosphere can be sucked out. Just create an Alcubierre drive. Essentially this works by constructing a metric of space such that there is gradient that can, in principle, be arbitrarily large. Although the air molecules don't move through space faster than light locally, as seen from "outside" the speed can be faster than $c$.

You'd have to think carefully about how exactly you do this without simultaneously tearing Earth apart.

Moderately manipulating gravity

Your second edit assumes a maximum speed of $v=c/2$. In that case the answer is simply given by the distance from the antipode of the hose (since air on the other side of the Earth from the hose needs to travel around Earth), plus the distance from ground to space. Assuming 500 km for the latter, that distance is roughly $d = 20,\!500\,\mathrm{km}$, so the time is $t=d/c=0.07$ seconds.

Answer 2

I believe that despite the difficulties you'd have getting the atmosphere to rush into a tube, that would be the far easier of the two problems you'd have. The bigger problem is dealing with all that air. Where would it go if not back to earth?

Since we're talking about science fiction, I propose the following technology to make it work. Suppose you could open a portal of some sort. You could say, simply open a portal on earth and another on mars and transport all the air. The atmosphere from the earth would be sucked out due to the vaccuum and spit out on mars.

Of course portals in of themselves would violate several rules of physics (specifically the law of conservation of energy), so it would be more realistic if the cost of keeping the portals opened required a level of energy at least greater than the energy that would be gained. In other words, the energy cost of moving a ball from the ground to the top of the eiffel tower would be the potential energy difference of the ball dropping that height. While it would take energy to move air into space, potential energy is lost when reentering the gravitational pull of mars, meaning the net energy change is marginally less. It would still require a great deal of energy, but it would also probably be a very efficient way of removing the earth's atmosphere.