Why is cross section inversely proportional to wavelength for interstellar scattering?

Question

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?

Answer 1

The main mechanism for the scattering of light by the interstellar medium is Mie scattering. This mechanism dominates when the scattering bodies are comparable to the wavelength of the light.

Scattering by bodies much smaller than the wavelength of the light is fairly simple to model, and is given by the Rayleigh formula. In this regime the scattering is proportional to $\lambda^{-4}$. Likewise, scattering by bodies much larger than the wavelength of the light is also simple to model because it's just dependent on the physical area of the scattering bodies and it's independant of $\lambda$.

Unfortunately the regime in between, described by Mie scattering, is fiendishly complicated to calculate. However since the wavelength dependance goes from $\lambda^{-4}$ to $\lambda^0$ as the particle size increases, in between the two limits we expect it to depend on a power of $\lambda$ somewhere in between $-4$ and $0$. Your professor is suggesting an approximate equation:

$$ \sigma \propto \lambda^{-1} $$

that applies in the wavelength range being considered. I don't think there's anything fundamental about this relationship. It's just that if you calculate the Mie scattering for the particles sizes in the interstellar medium, and the range of wavelengths from visible to near IR, then the $\lambda^{-1}$ dependance is a good approximation.