Refractive Index of Air Depending on Temperature How does the refractive index of air rely on the temperature?  Is there a theoretical derivation of it?
 A: The refractive index of air is easy, because air is a dilute gas with a very small refractive index, which is given by:
$$
n = 1-\sum n_i \delta_i(k)
$$
for small wavenumbers k. The $n_i$ are the number density for each species of molecule, and $\delta_i$ is the contribution to the index from this molecular species. You can just use N2 and O2 to get a good enough fit, and include CO2 and H2O for a better fit.
In the ideal gas limit, which is nearly perfect for air, $n={P\over kT}$. If you double the pressure, you double the deviation from 1. If you double the temperature, you halve the deviation from one, because all the components go with the same ideal gas law:
So the formula for the long-wavelength index of air is
$$
\boxed{n(P,T) = 1 + .000293 \times {P\over P_0}{T_0\over T}}
$$
Where $P_0$ is atmospheric pressure, and $T_0$ is the standard temperature of 300K. and this is essentially exact for all practical purposes, the corrections are negligible away from oxygen/nitrogen/water/CO2 resonances, and any deviation from the formula will be due to varying humidity.
The actual contributions $\delta_i$ requires the forward scattering amplitude for light on a diatomic molecule. This is just beyond what you can do with pencil and paper, but it is within the reach of simulations.
To read about the relation between the refractive index and forward scattering, see Feynman, Richard P.; Acta Physica Polonica 24, 697 (1963).
A: The general form of such dependence is known for many different types of substance, but the exact values are not theoretically derivable as far as I know. What you are looking for is the temperature-dependent Sellmeier equation, but the constants of all Sellmeier equations for any substance are always fit to experimental data.
This is a very good overview of all the available work in this area. Reading this, it looks like Jones' 1981 paper (freely available, since it was American government work at what was then called the National Bureau of Standards) contains a formula for the refractive index of air depending on temperature and pressure, among other things, although it doesn't appear to take the form of the Sellmeier equation.
A: There is an online calculator whose parameters are not just air temperature, but also wavelength, air pressure, humidity, and CO2 content:
https://emtoolbox.nist.gov/Wavelength/Ciddor.asp
Another formula without CO2 content parameter:
https://emtoolbox.nist.gov/Wavelength/Edlen.asp
The page https://aty.sdsu.edu/explain/atmos_refr/air_refr.html (thanks to @ptomato's answer) explains the empirical research behind these and several other formulas. (It doesn't provide any theoretical derivations though.) It also explains why the original Edlén formulas should not be used anymore. Note that the link given above uses a modified Edlén formula, which the authors explain here: https://emtoolbox.nist.gov/Wavelength/Documentation.asp (Also without theoretical derivations though.)
