The equation of radiative transfer is given by

,

where

is the specific intensity (red line),

is the gas density,

is the opacity or absorption coefficient, and

is the emission coefficient. The equation describes how incident radiation is affected along a path length

. We define the source function

as well as the optical depth

:

and can rewrite the equation of radiative transfer in terms of

:

.

The formal solution for the specific intensity at a given frequency

for a zero angle of incidence (plane-parallel) is

.

We assume a constant source function

in this Demonstration. The controls let you vary these normalized quantities and show examples where the environment is both optically thin (

) and optically thick (

). The horizontal blue line indicates that for optically thick cases the specific intensity tends toward the constant source function

. For optically thin cases, the specific intensity can be approximated by the green line as

.

In-depth studies of these relations can be found in [1] and [2].

[1] B. W. Carroll and D. A. Ostlie,

*An Introduction to Modern Astrophysics,* New York: Addison–Wesley, 1996.

[2] G. B. Rybicki and A. P. Lightman,

*Radiative Processes in Astrophysics*, New York: Wiley–Interscience, 1979.