Radiative transfer is one of the most fundamentally important concepts in astrophysics. The quantification of how light passes through an absorbing or emitting material lets us study a wide range of physical phenomena, from the structure of planetary atmospheres to cosmology and the big bang.

The fundamental equation of radiative transfer is very simple:

d(Iν)/ds = -ανIν + jν,


and the subscript ν implies that all terms are functions of the frequency of the light.

This is a very simple differential equation, but it can contain some very complicated physics when you start to consider all the sources of absorption and emission within a given system, as well as the frequency dependence of the terms. Here are two very simple cases.

Absorption only

Suppose we have a light source, embedded in a uniform cloud of absorbers (think of a light in a thick fog, where the water droplets act as the absorbers). In this case, we set j to zero. If we integrate the differential equation, we find that the intensity of light as a function of distance from the light bulb decays exponentially:

I = I0 × e(-α × s)

So, as you move farther and farther away from the light source, it gets fainter and fainter. This is why you may not see a pair of oncoming headlights in a dense fog until they are very close to you.

Emission only

Suppose we now have a thin plasma, which emits light constantly and evenly, without absorbing it. In this case, α is zero, and j is a constant, so integration of the differential equation results in light intensity which increases linearly with the size of the emitting region (the integral of the path length ds):

I = j × s

In this case, the larger the emitting region, the brighter it is. Imagine you could chop up a fluorescent light (the long, overhead ones) into short pieces which could each give off a little light. The more of them you stack together, the brighter the light gets.

Other cases

The two examples above are extremely simple ones, and real life is more complicated. For one, both absorption and emission are nearly always functions of the light frequency. For example, when you are in a fog, you may not be able to see very well, but your radio will still work -- radio waves aren't (much) affected by fog. Emission mechanisms are also frequency dependent (for example, in black body radiation or emission lines).

Another thing I left out entirely is scattering, where photons are not absorbed or emitted but just change direction. Even worse, scattering is not only dependent upon the property of the scattering material, but also upon what direction you happen to be looking at the source from, and (especially) what the frequency of the light happens to be.

Finally, I also assumed that the absorption and emission coefficients are independent of position, and are thus constants in the integral over the path length. This is almost never the case. Often the density of a gas will change along the path, as will the temperature, so both the emission and absorption coefficients have to be included in the integral. This is one reason why modeling of stellar and planetary atmospheres and interiors requires numerical rather than analytic solutions.

The excellent (but overpriced) book Radiative Processes in Astrophysics by George Rybicki and Alan Lightman goes into the physics of radiative transfer in exquisite, gory detail, so students and other interested persons can look there. The older (but more reasonably priced) book Radiative Transfer by Subrahmanyan Chandrasekhar is also a good resource, as is Frank Shu's The Physics of Astrophysics, Volume 1: Radiation.

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