The wikipedia article on atmospheric escape had this chart:
(this was removed from the page as a result of this question in August 2022)
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 wikipedia page included a reference to these AST121 lecture notes which included a very similar diagram, though with no explanation as to how it was derived. The wikimedia.org page referenced these lecture notes (dead link, no alternative source found), 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?