Rayleigh-Jeans Distribution

The equation for blackbody radiation (the Planck equation) is:

F lambda = 2 h c² lambda ^-5 / (exp(hc/k lambda T) - 1).

or, combining the constants:

F lambda = c₁ lambda ^-5 / exp(c₂ / lambda T) - 1),

where c₁ = 2 hc² = 3.7419 × 10^-5 erg cm² s^-1 [ lambda in cm]
and c₂ = hc/k = 1.4288 cm °K.

On the red side of the distribution (when lambda T >> 1) the denominator approaches zero as x => 0 and e^x => 1. For small values of the exponent, the series expansion for the exponential can be used as an excellent approximation of the value of the exponential.

e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + . . .

But when x is very small, all terms beyond the first in x can be ignored, allowing us to use:

e^x = 1 + x

Making this substitution, we obtain:

F lambda = c₁ lambda ^-5 / (1 + c₂ / lambda T - 1),

or:

F lambda = (c₁/c₂) T lambda ^-4.

This expression for the blackbody distribution on the red side of the maximum of the Planck curve is called the Rayleigh-Jeans Distribution.

Note: The notation exp(x) is another way of writing e^x. In this case I used this form because of the complexity of the term in the exponent, the difficulty of properly creating it in HTML (the language used to write these pages), and the relatively low resolution of computer screens.

Introduction to Blackbody radiation
Wien Distribution
Wien Displacement Law
Stefan-Boltzmann Law
Constants

This VRML + HTML package on Blackbody emission was constructed by Karen M. Strom. Contact your local instructor for help if it is used as part of your class.