<blockquote>
<p>I didn't really understand where did this &quot;one-sixth of the escape velocity&quot; figure come up from.</p>
</blockquote>
<p>The &quot;one-sixth of the escape velocity&quot; a ballpark number. As a general rule, three sigma events (events that are three standard deviations from the mean) happen all the time, while twelve sigma events are so very, very rare that they can essentially be ruled out. Six sigma events are rare but do happen. In a logarithmic sense, six is halfway between three and twelve.</p>
<p>A bit less hand-waving, <a href="http://geosci.uchicago.edu/%7Ekite/doc/Catling_and_Kasting_ch_5.pdf" rel="nofollow noreferrer">it can be shown</a> that the Jean's escape flux (which is what your text is addressing) is proportional to <span class="math-container">$v_s \, (1+\left(v_e/v_s\right)^2)\,\exp\left({-\,\left(v_e/v_s\right)^2}\right)$</span> , where <span class="math-container">$v_s$</span> is the expected value of the magnitude of the three-dimensional velocity of the gas at the top of the exobase and <span class="math-container">$v_e$</span> is the escape velocity at the top of the exobase. That factor of <span class="math-container">$\exp\left({-\,\left(v_e/v_s\right)^2}\right)$</span> means that Jean's escape is extremely sensitive to changes in <span class="math-container">$v_s$</span>.</p>

CC BY-SA 4.0
2020-06-29T08:40:57.550Z
2020-06-29T08:40:57.550Z
David Hammen
<p><a href="https://books.google.co.in/books?id=WA9ST5S_2v0C&amp;printsec=frontcover&amp;dq=An+Introduction+to+Our+Dynamic+Planet&amp;hl=en&amp;sa=X&amp;ved=2ahUKEwiOwOmggKLqAhVH4zgGHW_pCYoQ6AEwAHoECAAQAg#v=onepage&amp;q=An%20Introduction%20to%20Our%20Dynamic%20Planet&amp;f=false" rel="nofollow noreferrer">An Introduction to Our Dynamic Planet</a> under section 2.4 &quot;The origins of the atmosphere and hydrosphere&quot;(Page 80) says on the topic of gases escaping from our atmosphere as:</p>
<blockquote>
<p>Different gases have different molecular masses, so their average velocities are
different at a given temperature. In order for a planetary body to retain a particular
gas in its atmosphere for a period of time of the same order as the age of the Solar
System, the average velocity of the molecules in the gas should be less than about
one-sixth of the escape velocity. (If the average velocity exceeds one-sixth of the
escape velocity, a significant proportion of molecules will be moving faster, and will
be lost.) This condition is achieved on only a few planets and satellites.</p>
</blockquote>
<p>I didn't really understand where did this &quot; one-sixth of the escape velocity&quot; figure come up from. Is there some rigorous mathematical derivation or physical significance of this figure? In the previous page, the book had introduced the concept of how the velocities of molecules for a gas follow the <a href="https://en.wikipedia.org/wiki/Normal_distribution" rel="nofollow noreferrer">normal distribution</a>, and only those molecules which have velocities greater than or equal to the <a href="https://en.wikipedia.org/wiki/Escape_velocity" rel="nofollow noreferrer">escape velocity</a> of a planet can leave the surface(the formula for which was also shown). But I wasn't able to link this with the paragraph mentioned above. Any derivation/logic for this one-sixth figure would be welcome</p>

<atmosphere><geochemistry>
Relation between molecular mass and retention of a gas on a planet
Yusuf Hasan
# Relation between molecular mass and retention of a gas on a planet

(Asked by: username_0 on 2020-06-27)

[An Introduction to Our Dynamic Planet](https://books.google.co.in/books?id=WA9ST5S_2v0C&printsec=frontcover&dq=An+Introduction+to+Our+Dynamic+Planet&hl=en&sa=X&ved=2ahUKEwiOwOmggKLqAhVH4zgGHW_pCYoQ6AEwAHoECAAQAg#v=onepage&q=An%20Introduction%20to%20Our%20Dynamic%20Planet&f=false) under section 2.4 "The origins of the atmosphere and hydrosphere"(Page 80) says on the topic of gases escaping from our atmosphere as:

> Different gases have different molecular masses, so their average velocities are
> different at a given temperature. In order for a planetary body to retain a particular
> gas in its atmosphere for a period of time of the same order as the age of the Solar
> System, the average velocity of the molecules in the gas should be less than about
> one-sixth of the escape velocity. (If the average velocity exceeds one-sixth of the
> escape velocity, a significant proportion of molecules will be moving faster, and will
> be lost.) This condition is achieved on only a few planets and satellites.

I didn't really understand where did this " one-sixth of the escape velocity" figure come up from. Is there some rigorous mathematical derivation or physical significance of this figure? In the previous page, the book had introduced the concept of how the velocities of molecules for a gas follow the [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution), and only those molecules which have velocities greater than or equal to the [escape velocity](https://en.wikipedia.org/wiki/Escape_velocity) of a planet can leave the surface(the formula for which was also shown). But I wasn't able to link this with the paragraph mentioned above. Any derivation/logic for this one-sixth figure would be welcome

**Tags:** atmosphere, geochemistry


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## 1 answers

### Answer by username_1 on 2020-06-29. Score: 2 | Accepted answer

> I didn't really understand where did this "one-sixth of the escape velocity" figure come up from.

The "one-sixth of the escape velocity" a ballpark number. As a general rule, three sigma events (events that are three standard deviations from the mean) happen all the time, while twelve sigma events are so very, very rare that they can essentially be ruled out. Six sigma events are rare but do happen. In a logarithmic sense, six is halfway between three and twelve.

A bit less hand-waving, [it can be shown](http://geosci.uchicago.edu/%7Ekite/doc/Catling_and_Kasting_ch_5.pdf) that the Jean's escape flux (which is what your text is addressing) is proportional to $v\_s \, (1+\left(v\_e/v\_s\right)^2)\,\exp\left({-\,\left(v\_e/v\_s\right)^2}\right)$ , where $v\_s$ is the expected value of the magnitude of the three-dimensional velocity of the gas at the top of the exobase and $v\_e$ is the escape velocity at the top of the exobase. That factor of $\exp\left({-\,\left(v\_e/v\_s\right)^2}\right)$ means that Jean's escape is extremely sensitive to changes in $v\_s$.


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