Formula

Thermal radiation power of a black body per unit of area, unit of solid angle and unit of frequency ν is given by Planck's law as:

or

where β is a constant.

This formula mathematically follows from calculation of spectral distribution of energy in quantized electromagnetic field which is in complete thermal equilibrium with the radiating object.

Integrating the above equation over ν the power output given by the Stefan–Boltzmann law is obtained, as:

where the constant of proportionality σ is the Stefan–Boltzmann constant and A is the radiating surface area.

Further, the wavelength , for which the emission intensity is highest, is given by Wien's Law as:

For surfaces which are not black bodies, one has to consider the (generally frequency dependent) emissivity factor ε(υ). This factor has to be multiplied with the radiation spectrum formula before integration. If it is taken as a constant, the resulting formula for the power output can be written in a way that contains ε as a factor:

This type of theoretical model, with frequency-independent emissivity lower than that of a perfect black body, is often known as a gray body. For frequency-dependent emissivity, the solution for the integrated power depends on the functional form of the dependence, though in general there is no simple expression for it. Practically speaking, if the emissivity of the body is roughly constant around the peak emission wavelength, the gray body model tends to work fairly well since the weight of the curve around the peak emission tends to dominate the integral.

Constants

Definitions of constants used in the above equations:

	Planck's constant	6.626 0693(11)×10−34 J·s = 4.135 667 43(35)×10−15 eV·s
	Wien's displacement constant	2.897 7685(51)×10−3 m·K
	Boltzmann constant	1.380 6505(24)×10−23 J·K−1 = 8.617 343(15)×10−5 eV·K−1
	Stefan–Boltzmann constant	5.670 400(40)×10−8 W·m−2·K−4
	Speed of light	299,792,458 m·s−1

Variables

Definitions of variables, with example values:

	Temperature	Average surface temperature on Earth = 288 K
	Surface area	Acuboid = 2ab + 2bc + 2ac; Acylinder = 2π·r(h + r); Asphere = 4π·r2