Eli5 why is it a rule of thumb used by astrophysicists that a gas will escape from a planet’a atmosphere in 10^8 years if the speed of its molecules is one sixth of the escape velocity?

Question

It was mentioned in one of my classes exercises

In: 1

Answer 1

Gases don’t have a single fixed speed. Instead, the speeds of the particles in a gas follow a distribution called the [Maxwell-Boltzmann distribution](https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution). A particle can escape the planet’s gravity if it is far enough to the right hand side of this distribution that it reaches the planet’s escape velocity, which it can do even if the *average* speed in the gas is much lower than that. This is essentially the same process by which liquid particles can evaporate far below the boiling point of the liquid in question.

Since the shape of the Maxwell-Boltzmann distribution is known, we can use it to figure out what rate particles escape at. And it (presumably – I haven’t done the math here myself) turns out that ~1 in 10^8 molecules escapes per year at 1/6 of escape velocity.

(Strictly speaking this distribution is for an *ideal* gas, and atmospheric gases aren’t quite ideal, but they’re close enough.)