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I read this in my class textbook, and learnt the reason of absence of Hydrogen from atmosphere and Maxwell distribution curve from Physics stack exchange. But I am unsure of why should the average velocity be greater than 1/6 times escape velocity. Also why do we use average velocity instead of root mean square velocity?

Edit : average speed > 1/6 the escape velocity. The book is given by our coaching for astronomy Olympiads.

Please refer this question: Why doesn't hydrogen gas exist in Earth's atmosphere?

Vardaan Srivastava
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  • which gas? What is the timescale for its production through geological or biological processes? What fraction of the atmosphere has 6x the average speed? What factor of that is at an altitude such that the mean-free-path + altitude is "in space"? Is the average per species or per molecule? – JEB Mar 06 '23 at 04:15
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    This question seems confused about whether average velocity is 6 times escape velocity, or escape velocity is 6 times average velocity. Please provide a proper citation to what you are reading, including the title and authors of the textbook, where in the textbook it is found, and a quotation from that textbook. – D.W. Mar 06 '23 at 06:47
  • @D.W. I was confused at this point as well , but my intuition says it must be $V_{avg}*6>=V_{escape}$ because the gases are trapped in the planet and must posses some greater velocity to escape the planet's gravitational field – Naveen V Mar 06 '23 at 07:37
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    Also, average velocity is zero. Presumably you mean average speed? – Sten Mar 06 '23 at 08:57
  • @Sten Can you elaborate on this , I know that vector sum as velocity of molecules going in forward and opposite directions turn out to be $0$ but that's a mere approximation right ? – Naveen V Mar 06 '23 at 12:06

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A gas molecule travelling faster than the escape velocity will only escape if it doesn't hit anything to slow it down. However, gas molecules are constantly colliding, so the even if a molecule/atom achieves escape velocity, the chance of it maintaining that velocity all the way to the edge of the atmosphere is tiny.

I vaguely remember a similar claim in one of my courses that the figure 6 x escape velocity is just an approximate rule of thumb to estimate whether the gas has enough energy to escape the planet/atmosphere at a high enough rate, such that that the planet cannot hold on to it over long timescales. As it's just a very rough estimate, it doesn't matter whether it's the average or the RMS velocity.

Rustony
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  • To your first point: That's why you usually compute $v_{\rm esc}$ at the exobase, where the mean-free-path towards collisions is larger than the gas scale height - essentially a free-streaming limit.

    To your second point, that's correct where this number 6 is coming from: The escape rates coming from different values of $v_{\rm thermal}/v_{\rm esc}$ are compared to the total atmospheric mass reservoir in a popular calculation, for one value of the atmospheric mass. For any other planet, this number would vary wildly.

     – AtmosphericPrisonEscape Mar 06 '23 at 15:55
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Probably some confusion or unclear text source. Any gas molecule that is faster than the escape speed will escape. There is nothing about the number six to see here.

rfl
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