Planetary atmosphere interfacing with near vacuum - now put it in a jar

Question

Celestial bodies that have atmospheres have a gravity sufficient to hold the atmosphere to the body. This atmosphere interfaces with the near vacuum of space yet is held to the body by gravity.

I was asked a question that I need help with.

Why can't a sealed jar be filled with half air and half vacuum on Earth and have the air settle to the bottom of the jar in response to Earth's gravity?

Answer 1

It is entirely possible to make such a jar, just very impractical.

If we have air in hydrostatic equilibrium the pressure is equal to the weight of the air above, then at height $h$ the pressure is $$P(h)= P(0) \exp(-h/H)$$ where $$H=\frac{kT}{mg}$$ is the scale height. On Earth it is typically about 8 km but the temperature dependence of course complicates things in the upper atmosphere. If everything has the same temperature the above equations will work even better, and this is the case for the jar. But we need jars that are either many kilometres tall, or very low temperature.

Lower the temperature enough and the air condenses out as a liquid at the bottom. There will not be a perfect vacuum above since some molecules will randomly leave the liquid and bounce around before settling. Just before the nitrogen condenses at 77 K the scale height has been reduced by a factor of 3.5. So even for near-condensed air the jar needs to be kilometres tall to have a noticeable pressure difference.

Answer 2

It is not a silly question. The point is that air consists of molecules wizzing around at speed, and the speeds at sea level are such that even under the influence of gravity the molecules will easily wizz to the upper parts of the jar, so there will be negligible 'slump' of the air towards the bottom. The slumping would occur if you could extract enough energy out of the jar so that the average speed of the molecules is reduced to the point at which the effects of gravity dominate their motion- indeed eventually the air would liquefy.

Taking the atmosphere as a whole, the distances are sufficiently great that an air molecule wizzing upwards at high speed would eventually be decelerated to the point at with it stops and falls down again.

So the answer in a nutshell is that jars are too small to allow the deceleration due to gravity to have any appreciable affect on the speed of the air molecules.

Answer 3

Sure, but only if earth had a gravitational field comparable to that of a neutron star (and your jar would need to be fairly indestructible). In this article, it is stated that the Chandra X-Ray Observatory has discovered$^1$ a neutron star with an atmosphere with a height of about $10\ \text cm$. Yes, ten centimeters. So, if you filled your jar with air and put it on the surface of the star, most of it will be vacuum (when I say vacuum, I mean really low pressure as you would expect in space) with the air on the bottom (probably in another phase). The materials in this atmosphere would be vastly different to air on earth and under huge pressure, far greater than air on earth (see image below: "thin carbon atmosphere").

---

EDIT: Added the above text and image below that was found during a search of something related, and found it very interesting, so I have added it to this answer.

---

But to answer your exact question, you should first note that gases that are in the highest part of earth's atmosphere have small kinetic energies (usually not enough to escape into space) that are far less than the kinetic energy of gases close to the surface of the earth. The average kinetic energy of gas atoms/molecules close to the surface of earth are far greater and certainly not small enough to stay at the bottom of the jar without bouncing to the top.

Also note that technically, a jar with a gas inside will have a slightly greater pressure at the bottom than the top due to earth's gravity. But this pressure difference will be so small that it’s virtually undetectable.

$^1$ From the actual NASA press release:

Answer 4

You have a misperception here:

This atmosphere interfaces with the near vacuum of space yet is held to the body by gravity.

There's no place that atmosphere interfaces with vacuum. At each point, the ratio of density of air above is only imperceptibly smaller than that of below. If you go up far enough, the density is half what is at sea level. Go up far enough past that, and it's half of that (so one fourth the atmosphere at sea level). Keep going and it eventually is half of that. And so on. If you halve it enough, it becomes near vacuum. But at each point in the process, the density is, locally, near constant.