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The wikipedia article on atmospheric escape has this chart:

atmospheric escape chart

It all seems plausible enough... for a body to retain an atmosphere of a gas, the velocity of gas molecules in its exosphere must be mostly below escape velocity, implying either plenty of gravity, or a sufficiently cold atmosphere. The solar system bodies end up in appropriate places, and their atmosphere is kinda described OK. It'll do as a handwavy overview.

What I'm having trouble with is how the values of the gas "escape velocities" were calculated. The hydrogen line looks to have a value of about ~18km/s at Earth's temperature, for example. The wikimedia page includes a reference to [these lecture notes3], which mention the Maxwell-Boltzmann speed distribution and come up with this, by working out the temperature at which the most probable speed in the distribution is equal to the planetary escape velocity:

\begin{align*} \sqrt{\frac{2kT}{m}} &= \sqrt{\frac{2GM}{R}} \\ T_{esc} &= \frac{GMm}{kR} \end{align*}

(where k is the Boltzmann constant, T is the atmospheric temperature, M is the mass of the planet, m is the mass of the molecule we're interested in and G is the gravitational constant)

I can't get any meaningful value of T_esc that looks anything like the lines plotted on that chart, and having the average of that speed distribution greater than escape velocity seems to require ridiculously high temperatures... it seems like an entirely wrong way to go about the problem.

On the reasonable assumption that I don't know what I'm doing, can someone tell me how a chart like this could be created? Is it even possible to handwave a very approximate answer to "will Jeans escape deplete this gas in under a billion years, give or take" just by supplying surface temperature and escape velocity? Was it actually done by more nuanced application of the Maxwell-Boltzmann distribution, and neither the original author of the lecture notes nor the creator of the chart thought to mention this?

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    $\begingroup$ I would start by calculating the actual escape velocity of each planet and then seeing what thermal speeds those correspond to (i.e., the equivalent temperature of a Maxwellian gas) for a baseline check. The escape speed of Earth is ~11 km/s whereas the sound speed at STP is ~343 m/s, i.e., over 30 times smaller than the escape speed. $\endgroup$ – honeste_vivere Feb 17 at 21:12
  • $\begingroup$ @honeste_vivere I've had a go at that; I'm not entirely convinced by my workings, but for eg. monatomic hydrogen on Earth I don't see any figures coming out that resemble the 18-20km/s I see on the chart. I get average speeds in the exosphere of ~4-5km/s and ~97% molecules below escape velocity, which seems plausible, so I don't think I'm completely wrong. I just don't understand where the figures shown on the chart came from. $\endgroup$ – Starfish Prime Feb 17 at 21:23
  • $\begingroup$ I think what it's showing is the escape speeds and thermal speeds but it's badly labeled. That is, the translucent bands correspond to rough thermal speed ranges for the various elements/molecules based upon the temperature of the horizontal axis. The planet location indicates the rough/mean atmospheric temperature and escape speed. $\endgroup$ – honeste_vivere Feb 17 at 22:06
  • $\begingroup$ Wait, no, I stand correct. The most probable thermal speed of hydrogen at 100 K is only ~1285 m/s, so now I am thoroughly confused as to what the bands represent... $\endgroup$ – honeste_vivere Feb 17 at 22:12
  • $\begingroup$ @honeste_vivere with a bit more searching I've found a different (and somewhat more plausible) version of the chart, and it looks like the author of the prettier chart may have just scaled everything up inappropriately. $\endgroup$ – Starfish Prime Feb 18 at 9:46
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Note only a small percentage of the hydrogen escape specifically the ones with the higher velocities. The leftover distribution after these particle escape is no longer a boltzmann distribution and is therefore not in thermal equilibrium. It will change gradually to try to become a boltzmann distribution again. This then allows more particle to be faster and this cycle acts as a slow leak over time I believe at an exponentially decaying speed. This is likely what the bands are trying to show i.e. that a planet just cold may still have these gasses for a decent amount of time.

This is called the jeans escape process.

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I've located this suspiciously similar looking chart from chapter 5 of Atmospheric Evolution on Inhabited and Lifeless Worlds, which is available in PDF form here.

Jeans Escape chart

It has a number of important differences:

  • the temperature is the temperature of the exobase, not the surface (except where no appreciable atmosphere exists, like on the moon, where the surface temperature is used instead).
  • the speeds plotted along the gas lines appear to be the peak of the Maxwell-Boltzmann speed distribution of the relevant molecular form of the gasses... H2 at 1000K has a most probable speed of ~2.9km/s, which looks to be about the value shown on the graph.
  • the relevant rule of thumb, where the speeds of the gas molecules exceed one sixth of the escape velocity being a indicator of Jeans escape depleting the gas over less than geological timescales, is clearly stated.

I'm still none the wiser as to the source of the numbers on the original, prettier chart, but it seems like this is a more correct presentation of the same information. It is possible that the author of the new chart, when they changed the y-axis scale from one-sixth escape velocity to just escape velocity, they scaled up the gas lines by a factor of 6 too (or forgot to add an appropriate scale for gas speeds on the right of the chart). That might explain the source of error.

(I'll leave this answer up but unaccepted for a while, in case someone is able to cast any light on the original chart)

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