A mathematical description of the physics behind Aurora?

[dykuma]

TL;DR Summary

How do you calculate the chance that a molecule (let's say O2 or N2) will emit light at a specific wavelength?

Maybe a bit of an odd question (not really sure where it would belong on this site to be honest), but I was wondering if anyone can explain, or at least knows of a source that explains in a quantitative way, the physics behind aurora?

Now I've seen websites like this that discuss conceptually how an aurora occures. The basic premise behind the phenominon seems to be a tight balancing act between the number of gas particles (air pressure), the chance that the molecule will get enough energy to emit light at a specific wavelength, the time it takes for the excited molecule to emit that energy as light (between seconds for green light, and minutes for red light), and the chances of something quenching the reaction (collisions with other molecules stealing the energy before it could be emitted as light).

I'm familiar with the Stern-Volmer equation (which might not be helpful here) and the concept of quenching, where the quenchers here seems to be atmospheric pressure and the quantity of the excitable molecule. What I want, more or less, is a way to calculate the chance of a molecule emitting light at a given wavelength at a given altitude. Specifically, I'm interested in a mathematical way to explain the colors of lighting (blue at lower altitudes and red at higher altitudes due to N2) and the colors of aurora (green at lower altitudes, red at higher altitudes due to O2). However, I can't find much on calculating something like this. Does anyone know anything that can help me?

 

[davenn]

Summary:: How do you calculate the chance that a molecule (let's say O2 or N2) will emit light at a specific wavelength?

, I'm interested in a mathematical way to explain the colors of lighting (blue at lower altitudes and red at higher altitudes due to N2) and the colors of aurora (green at lower altitudes, red at higher altitudes due to O2).

Blue is only rarely be produced. In the 100's of aurora I have seen and/or photo'ed I don't think I have ever seen blue.
Green is the most common colour with red above it. Yellow would be the 3rd most common colour, mainly produced by mixing.

I have no way of describing it mathematically, maybe look for some articles neon shop signs, the process is pretty much the same.

 

[dykuma]

Blue is only rarely be produced. In the 100's of aurora I have seen and/or photo'ed I don't think I have ever seen blue.

I should have clarified, I was asking about the colors/wavelengths of light of excited gasses (N2 and O2) as a function of altitude. Lightning at lower altitudes is blue, where light from excited nitrogen is dominate, and I was considering the excitation due to lower and upper atmospheric lightning to be roughly the same as aurora, though obviously both are initiated by very different processes (usually).

Anyway, I have a rough understanding of how atoms with electrons level emit light, or how vibrational modes of molecules do the same. My question was more directed at a more statistical approach to figure out the idea altitudes for various emissions. Using your neon sign analogy, if I were to excite a column of air, what I'm looking for it a way to determine what colors are present at a certain altitude for a given molecule.

 

[Baluncore]

My question was more directed at a more statistical approach to figure out the idea altitudes for various emissions.

Start here; https://en.wikipedia.org/wiki/Aurora#Colors_and_wavelengths_of_auroral_light