# Radiative Transfer Equation

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## 1 The Fundamental Equation of Radiative Transfer

The fundamental equation of radiative transfer is governed by emission and extinction. Extinction is brought about by absorption (which changes photon energy) or by scattering (which does not). Examples of scattering are Thomson scattering of light off of cold electrons, Rayleigh scattering in the atmosphere, and Line scattering (reemission in a different direction). An example of absorption is photoionization (where a photon ionizes an atom, say by knocking off an electron).

### 1.1 Absorption

Let’s say radiation passes through a region of absorption/scattering on its way to us. Then:

where is the *extinction* coefficient (units of ). We may compute a couple different ways:

Solving for intensity:

where is the *optical depth* at .

Optical depth is often computed as:

where , the *column density*, is in and is the # of extinguishers per unit area. Similarly,

where is the mass surface density and is the density-weighted extinction coefficient.

The *Mean Free Path* is given by: . Thus:

That is, the optical depth is the number of mean-free-paths deep a medium is. For Poisson processes, the probability of absorption is given by:

Therefore:

### 1.2 Emission

If is the emissivity, then the contribution of the emissivity of a medium to the flux is:

### 1.3 Emission and Extinction Together

(Fundamental Equation of Transfer)

It is often convenient to express this in terms of optical depth. Dividing by and recognizing :

where is a “source function”. In general,

There is a formal solution for . Let’s define and . Then:

If is constant with , then:

That second term on the righthand side can be approximated as for , since self-absorption is negligible. Similarly, for , it may be approximated as . The source function is everything. It has both the absorption and emission coefficients embedded in it.