Radiation is the transport of energy via the absorption and re-emission of photons. When the primary mechanism of energy transport in an atmosphere is radiation, the temperature-pressure profile is governed by the equations of radiative transport. The change in intensity, dI, due to absorption and emission within a cloud of gas is equal to the difference in intensity between emitted and absorbed radiation. We define j as the emission coefficient due to scattering and to thermal excitation, and as the mass extinction coefficient along a small increment in pathlength ds.

Absorption (including stimulated emission, which we can think of as negative absorption), and scattering both contribute to the extinction: = + , where is the mass absorption coefficient and is the mass scattering coefficient.

We define z to be the direction of the normal to the surface of a planet, and to be the angle between s and z. It follows that ds = sec dz. Introducing the variable x = cos , we define the optical depth as follows. is simply the integral along the vertical pathway of the extinction.

The change in intensity can be written as

where S is defined as the ratio of j over . For the case = 0 (x = 1), this simplifies to the following. When S does not vary with optical depth, it can be extracted from the integrand (second line). For the general case of arbitrary , we would replace by / x (the slant optical depth).

If >> 1, then I = S and the intensity of the emission is completely determined by the source function. If << 1, then I = I (o) and the intensity of the radiation is defined by the incident radiation.