Black-body radiation

Planck's blackbody equation describes the spectral exitance of an ideal blackbody.

$M(\lambda ,T)[{\frac {W}{m^{2}m}}]={\frac {2\pi hc^{2}}{\lambda ^{5}(\exp ^{\frac {hc}{\lambda KT}}-1)}}$

where:

Symbol	Units	Description
$\lambda$	$[m]$	Input wavelength
$T$	$[K]$	Input temperature
$h=6.6261\times 10^{-34}$	$[J*s]$	Planck's constant
$c=2.9979\times 10^{8}$	$[{\frac {m}{sec}}]$	Speed of light in vacuum
$k=1.3807\times 10^{-23}$	$[erg*K]$	Boltzmann constant

Note that the input $\lambda$ is in meters and that the output is a spectral irradiance in $[W/m^{2}*m]$. Omitting the $\pi$ term from the numerator gives the blackbody emission in terms of radiance, with units $[W/m^{2}*sr*m]$ where "sr" is steradians. There is a different formulation of the Planck equation in terms of frequency.

Taking the first derivative leads to the wavelength with maximum exitance. This is known as the Wien Displacement Law.

A closed form solution exists for the integral of the Planck blackbody equation over the entire spectrum. This is the Stefan-Boltzmann equation. In general, there is no known closed-form solution for the definite integral of the Planck blackbody equation; numerical integration techniques must be used.

The relationship between the ideal blackbody exitance and the actual exitance of a surface is given by emissivity.

An ideal blackbody at 300K (~30 Celsius) has a peak emission 9.66 microns. It has virtually no self-emission before 2.5 microns, hence self-emission is typically associated with the "thermal" regions of the EM spectrum. However, the Sun has a peak emission around 0.49 microns which is in the visible region of spectrum.

The Planck equation has a single maximum. The wavelength with peak exitance becomes smaller as temperature increases. The total exitance increases with temperature.