## Black Body Emission Plancks Radiation Law Stefans

By definition, a black body is an ideal physical body, isolated, consisting of a medium that is in thermodynamical equilibrium, and with a unique equilibrium temperature. It is a perfect absorber and an ideal emitter. A black body's radiation field is isotropic, and depends solely on the temperature. The spectral distribution of the radiation intensity is given by the Planck function, which gives the monochromatic brightness at a frequency v of the black body as a function of its temperature T :

where v: frequency in Hz h: Planck constant = 6.62620 x 10~34J.s c: velocity of light = 2.9979 x 108 m.s-1 k: Boltzmann constant = 1.38 x 10~23 J.K-1

The Planck function may be drawn for various values of T (Fig. A.1). This function reaches a maximum, which depends on the temperature. So each temperature of the black body may be associated with a colour (the wavelength at the peak emission) and, conversely, by determining the maximum emission of a black body, it is possible to determine what is known as the black-body temperature.

By integrating the Planck function over all frequencies and in all directions, it is possible to determine the total power (or flux) emitted by a black body at temperature T. This relationship is known as the Stefan-Boltzmann law:

where o = Stefan's constant = 5.66956 x 10~8 W.m~2.K~4.

Conversely, for every source emitting a flux F (as measured by a bolometer, for example), we may derive a temperature known as the 'effective temperature' Tiff, obtained from F by use of Stefan's law.

A direct application of Stefan's law is the calculation of the temperature Teff of a planet in radiative equilibrium, of radius ^pl, with a mean albedo (reflection coefficient) A, lying at a distance D from a star that emits a flux S. The equality