Computation of Radiative Transfer

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

The equation of radiative transfer describes the propagation of radiation and the effects of emission, absorption, and scattering through a medium. This Demonstration shows the simple case of an initial intensity through a volume of gas with no scattering, constant opacity, gas density, and source function intensity. The differential equation is solved and the intensity as a result of the radiative transfer through the gas volume is computed as a function of optical depth. The intensity approaches the source function intensity for optically thick cases () and the initial intensity of the background source for optically thin cases ().

Contributed by: Brian Kent (March 2012)
Open content licensed under CC BY-NC-SA



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.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.