Stefan–Boltzmann law
For an ideal absorber/emitter or black body, the Stefan–Boltzmann law states that the total energy radiated per unit surface area per unit time (also known as the radiant exitance) is directly proportional to the fourth power of the black body's temperature, T:
It has the value In the general case, the Stefan–Boltzmann law for radiant exitance takes the form:
Matter that does not absorb all incident radiation emits less total energy than a black body.
[3]: 71 In the more general (and realistic) case, the spectral emissivity depends on wavelength.
Wavelength- and subwavelength-scale particles,[5] metamaterials,[6] and other nanostructures[7] are not subject to ray-optical limits and may be designed to have an emissivity greater than 1.
is recommended to denote radiant exitance; a superscript circle (°) indicates a term relate to a black body.
[2] (A subscript "e" is added when it is important to distinguish the energetic (radiometric) quantity radiant exitance,
, from the analogous human vision (photometric) quantity, luminous exitance, denoted
[12][13][14][15] The proportionality to the fourth power of the absolute temperature was deduced by Josef Stefan (1835–1893) in 1877 on the basis of Tyndall's experimental measurements, in the article Über die Beziehung zwischen der Wärmestrahlung und der Temperatur (On the relationship between thermal radiation and temperature) in the Bulletins from the sessions of the Vienna Academy of Sciences.
[16] A derivation of the law from theoretical considerations was presented by Ludwig Boltzmann (1844–1906) in 1884, drawing upon the work of Adolfo Bartoli.
[17] Bartoli in 1876 had derived the existence of radiation pressure from the principles of thermodynamics.
[21] The numerical value of the Stefan–Boltzmann constant is different in other systems of units, as shown in the table below.
[23] He inferred from the data of Jacques-Louis Soret (1827–1890)[24] that the energy flux density from the Sun is 29 times greater than the energy flux density of a certain warmed metal lamella (a thin plate).
A round lamella was placed at such a distance from the measuring device that it would be seen at the same angular diameter as the Sun.
Stefan surmised that 1/3 of the energy flux from the Sun is absorbed by the Earth's atmosphere, so he took for the correct Sun's energy flux a value 3/2 times greater than Soret's value, namely 29 × 3/2 = 43.5.
[26][27] Pouillet also took just half the value of the Sun's correct energy flux.
The temperature of stars other than the Sun can be approximated using a similar means by treating the emitted energy as a black body radiation.
where L is the luminosity, σ is the Stefan–Boltzmann constant, R is the stellar radius and T is the effective temperature.
The law is also met in the thermodynamics of black holes in so-called Hawking radiation.
This gives an effective temperature of 6 °C on the surface of the Earth, assuming that it perfectly absorbs all emission falling on it and has no atmosphere.
The Earth has an albedo of 0.3, meaning that 30% of the solar radiation that hits the planet gets scattered back into space without absorption.
Another interesting question is to ask what the temperature of a blackbody surface on the earth would be assuming that it reaches equilibrium with the sunlight falling on it.
We can think of the earth's surface as "trying" to reach equilibrium temperature during the day, but being cooled by the atmosphere, and "trying" to reach equilibrium with starlight and possibly moonlight at night, but being warmed by the atmosphere.
The fact that the energy density of the box containing radiation is proportional to
[32][15] This derivation uses the relation between the radiation pressure p and the internal energy density
where the factor 1/3 comes from the projection of the momentum transfer onto the normal to the wall of the container.
The law can be derived by considering a small flat black body surface radiating out into a half-sphere.
The intensity of the light emitted from the blackbody surface is given by Planck's law,
is the Gamma function), giving the result that, for a perfect blackbody surface:
Finally, this proof started out only considering a small flat surface.
