Gases and escape velocities

[Elfrae]

The Earth's atmosphere is lacking in gases like hydrogen and helium because at a given temperature, a 'significant fraction' of hydrogen or helium particles have speeds greater than Earth's escape speed, while gases like oxygen only have a 'negligible fraction' of the total number of particles moving at these speeds.

What kind of fraction is 'significant' ? The light gases have been lost over the lifetime of the planet so I suppose it is still quite a small fraction of particles which are traveling faster than the escape velocity, but what kind of fraction of the gas molecules would be classed as 'significant' ?

 

[StatMechGuy]

You could always just look at the Boltzmann distribution and figure out the average velocities of particles and then do some numerical thing or another to figure out what percentage has the right escape velocity.

 

[Elfrae]

Well I did try that, but the percentages I got were so small that I wondered whether I'd done the calculations right...

I found that:

(1.01e-32)% of oxygen atoms (yes I mean atoms not molecules, as in the Earth's upper atmosphere),

(8.99e-7)% of helium atoms,

and 0.00016% of hydrogen atoms,

have speeds greater than the Earth's escape speed.

From these results I would say that a percentage like 0.0000001% is a significant enough fraction of gas particles for them to escape the atmosphere. That seems pretty small to me! But maybe that is a reasonable result?

 

[StatMechGuy]

First of all you have to double all the masses for hydrogen and oxygen and such since they come in diatomic form.

But once you do that, think about it this way: at any given moment that fraction of atoms can escape. So do the calculations and figure out how this translates into loss of atoms as a function of time (hint: I believe it should be exponential). Now obviously, atmospheric dynamics are slightly more complicated than this, but nevertheless it gives a good estimate.

 

[RandallBSmith]

at any given moment that fraction of atoms can escape. So do the calculations and figure out how this translates into loss of atoms as a function of time (hint: I believe it should be exponential).

Assuming this thread is still of interest after lo these many months ... I Googled upon this posting because I was musing along similar lines. My further thoughts to get the time constant needed in the above suggestion: Imagine painting those molecules that are beyond escape velocity red. Now a red atom might hit another atom and slow down, but assume for the moment the reds have such a large mean free path that they all escape. This leaves a Boltzmann distribution with a piece missing at the high end. How long before the non-reds re-equilibrate so that there is a new batch of molecules ready to leave?

Also, re-examining the assumption of the large mean free path: A red molecule might TRY to escape, only to be thwarted by a collision, in which the high energy is now shared with something else, and the two molecules end up with sub-escape speeds. That is to say, due to collisions, it isn't sufficient to have escape velocity in order to escape. How might one calculate what fraction of red molecules actually do escape? Perhaps as a crude approximation you could model the distribution of free paths and find out what fraction exceed some arbitrary large value.

I realize all I have done is (hopefully) refine the question, but maybe that's a useful next step. I also bet this is a well known calculation and a more productive route would be to do a more complete search of the literature.