The following problem was part of a homework for my Cosmology class:
Compare the probability of interstellar scattering of photons of yellow light (5000 angstroms) and 50 micron infra-red light.
The only explanation my professor provided was: sigma is inversely proportional to lambda (this was written on the board)
In other words the scattering cross section is inversely proportional to the wavelength. (this is my understanding of what was on the board)
However, the units do not match up. Cross section is an area (distance squared) and wavelength is distance. I know that in astronomy we often make 'radical' approximations, but a mismatch on the units seems to be an inconsistency far beyond the "within an order of magnitude" tolerance level.
On a purely conceptual level, it would make sense that the probability of scattering would be inversely proportional to some power of the wavelength, because radio waves (large wavelength) pass through brick walls (object is much smaller than wavelength) while visible light (small wavelength) is stopped by brick walls (object is much larger than wavelength).
I guess my question is: what is the reasoning behind sigma is inversely proportional to wavelength? Was my interpretation of those symbols correct? If so, why don't the units match up? Are there any other ways to solve or conceptualize this problem?