What does "rarely" mean in NASA's statement: "technically referred to as an exosphere because it’s so thin, its atoms rarely collide."?

Question

The following statement is from this article:

The behavior of a dense atmosphere is driven by collisions between its atoms and molecules. However, the moon's atmosphere is technically referred to as an exosphere because it’s so thin, its atoms rarely collide.

However, it doesn't explain what is meant by "rarely collide". I find this strange, especially since the word 'technically' is used. There doesn't seem to be anything technical about such a vague definition. I mean, someone might think that the atoms of the atmosphere rarely collide at an altitude of 400 km above the earth's surface. So he or she might think that that must be exosphere, but it's not. It's the thermosphere.

The following statement is from this article:

However the moon's atmosphere is so thin, atoms and molecules almost never collide. Instead, they are free to follow arcing paths determined by the energy they received from the processes described above and by the gravitational pull of the moon.

Much the same thing in other words. I would have thought that between collisions atoms and molecules follow arcing paths whenever there is gravity. The higher the pressure the shorter the arcs, but arcs nevertheless. So this doesn't clarify anything for me.

So my question is: what does it mean to say that "its atoms rarely collide"? Does it mean they normally bounce off the moon rather than each other?

Answer 1

Any given gas molecule goes a long way on average without colliding with another gas molecule. The mean free path for a gas molecule is on the order

$$ \lambda\sim \frac{k_BT}{d^2p} $$

where $k_B$ is the Boltzmann constant, $T$ is the temperature, $d$ is the kinetic diameter of the molecule in question and $p$ is the pressure. Plugging in some rough values for the lunar environment gives

$$ \lambda\sim 10^5~\rm km $$

as a back of the envelope estimate. The moon's circumference is only $\sim10^4~\rm km$, so most gas molecules probably escape the moon's atmosphere or collide with the moon without ever colliding with another gas molecule.

Answer 2

The phrasing is a reference to some of the assumptions made in the physics of gasses. For gasses at a pressure level we are used to, like on the surface of earth, the mean free path length is very tiny (about 68nm at STP). This means that a gas molecule undergoes a great many random collisions as it travels a distance. By the central limit theorem, this makes the behavior of the gas very predictable in a statistical sense. Many of our laws are formulated on these assumptions.

In rarefied gasses, such as the atmosphere of the moon, mean path lengths can be much longer. At $2\cdot10^{-12}$ Torr, that atmosphere qualifies as "ultra high vacuum" with mean path lengths on the order of $10^5$ km!

This means many assumptions we make about how gasses work require more complex physics to model them. In our atmosphere, it is very reasonable to assume that the velocity of a particle is almost uniformly distributed, and they travel in almost straight lines. On the moon, we must account for the fact that particles have been under the influence of gravity for quite some time between collisions.

The practical differences show up in high vacuum chambers as well. Turbo-pumps look a lot like fans, but they aren't designed to create pressure gradients. The idea of a "pressure gradient" is not as meaningful in such vacuums. Instead, the fan blades act more like baseball bats, knocking the gas atoms away.

Answer 3

"Rarely" doesn't have a black-and-white, sharply-delineated definition, because there is no non-arbitrary cutoff for when things "almost never" happen (unless you're a mathematician - then "almost never happen" means "happens with probability $0$"; that's not the case here, though!).

The key point is that, the thinner a gas gets, the less and less likely it is for its molecules (including single atoms as "molecules") to collide with each other, because they're farther apart and so it is easy for an atom moving in any given direction to miss a lot of atoms before it finally finds one to collide with - if it ever does at all.

The reason though why this is important is that the usual "thermal" distribution of molecular speeds in a gas - the Maxwell-Boltzmann distribution - comes about by repeated collisions of the gas molecules with each other. Note that if you think about the ideal gas model, there is no a priori reason that the molecules can't have any old speed distribution whatsoever: so that we model a gas as having this distribution must come from some sort of assumption. That assumption is collisions, and that assumption is what starts to fail here. In a sense, the gas starts to lack a well-defined "temperature" under generic circumstances.