Details

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].

References

[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.