I'm looking at Jeans (thermal) escape of hydrogen from the early Earth's atmosphere. I understand how to calculate the rate in (g cm^-2 s^-1) using the number density, average particle velocity, and integrating the tail of the boltzmann distribution, but those units are frustrating me.
How would I convert to mass per second? Multiply by the surface area of the atmosphere? This seems to give me a mass-loss rate that is too low for the Earth (that is, it would seem that the Earth should still have hydrogen).
Answer
I think your calculation might be correct, because the loss of hydrogen isn't accurately modeled by Jeans escape.
I remember in my statistical mechanics class we had the same problem as homework, and a lot of people got the answer "it should all be gone by now", but that was because they erroneously used an atomic mass of 1 (instead of 2 for $H_2$ molecules). At least one person used the atomic mass of 2 and got a quite different answer like "most of it should still be here". Then the professor actually emailed the author of the problem and explained, and he replied back something like "Well I'll be darned!".
So if you calculate the Jeans escape rate for $H_2$ molecules and assume nothing else is going on, you get that most of them should still be here. However, it's easy to think of other effects that could make it disappear faster. For one thing, UV light (and to some extent solar wind / cosmic rays) ionizes $H_2$ and dissociates it to $H$ in the upper atmosphere, which justifies using an atomic mass of 1. I think just as important, however, is the fact that the outermost wisps of the atmosphere are constantly being "blown away" by the solar wind.
(Of course, it's also possible that you just made a mistake, but if you get an answer that's sort of in-the-ballpark in order of magnitude, but still not nearly enough to explain the current lack of $H_2$ in the atmosphere, this is probably what's going on.)
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