Planetary retention of atmospheres

[rieman zeta]

To all: By memory I have seen in the literature the factor 1/6 when analytically discussing the retention of a planetary atmosphere (more correctly a particular specie of molecule in the atmosphere).


The relation would go something like this:

Atmospheric mean thermal speed of molecule < 1/6 escape speed of the planet.

How is this factor calculated?

I am familiar with the Maxwell distribution of velocites and its high end tail.
Thanks
Rieman Zeta

 

[SpaceTiger]

How is this factor calculated?

The basic idea is that the particles, given enough time, will settle into a Maxwell-Boltzmann distribution. This distribution will always have some particles with speeds greater than the escape speed and mean free paths that take them to infinity (effectively). Once these particles are lost, the distribution is no longer Maxwell-Boltzmann (the high-velocity tail is missing), so there won't be anymore escaping. However, the atmosphere will eventually thermalize again and return to a MB distribution, after which time more particles will be lost. This process will repeat throughout the Earth's lifetime and if there are no particles of a particular species left after that time, then you won't have them in the atmosphere.

Basically, you just need to calculate the thermal timescale (relaxation time of the particles) and fraction of the MB distribution with speeds greater than the escape speed. Since heavier particles will thermalize to lower average speeds, it will be easier for atmospheres to retain them. This is why, despite it being the most abundant element in the universe, there is very little hydrogen in our atmosphere.

Note that this analysis neglects input from outside sources, such as volcanoes. This can be very important for some species.

 

[rieman zeta]

Sir: "Basically, you just need to calculate the thermal timescale (relaxation time of
the particles) and fraction of the MB distribution with speeds greater than the
escape speed. Since heavier particles will thermalize to lower average speeds,
it will be easier for atmospheres to retain them. This is why, despite it being
the most abundant element in the universe, there is very little hydrogen in our
atmosphere."
This puts some teeth into an analysis. thank you.
Do you have a web or other reference showing an actual analysis.
zeta rieman

 

[SpaceTiger]

Do you have a web or other reference showing an actual analysis.

Unfortunately not. I remember doing the calculation in one of my undergrad classes, but I no longer have the notes.