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(I_{ν})/ds = -α_{ν}I_{ν} +
j_{ν},

where

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 = I_{0} × 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*.