Relation between molecular mass and retention of a gas on a planet

Question

An Introduction to Our Dynamic Planet 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, and only those molecules which have velocities greater than or equal to the 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

Answer 1

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 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 $v_s$.