The Stefan–Boltzmann law, also known as Stefan's law, describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature. It is named for Josef Stefan, who empirically derived the relationship, and Ludwig Boltzmann who derived the law theoretically.
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, :
The constant of proportionality, , is called the Stefan–Boltzmann constant. It has the value
In the general case, the Stefan–Boltzmann law for radiant exitance takes the form:
where is the emissivity of the surface emitting the radiation. The emissivity is generally between zero and one. An emissivity of one corresponds to a black body.
Detailed explanation The radiant exitance (previously called radiant emittance), , has dimensions of energy flux (energy per unit time per unit area), and the SI units of measure are joules per second per square metre (J⋅s⋅m), or equivalently, watts per square metre (W⋅m). The SI unit for absolute temperature, , is the kelvin (K).
To find the total power, , radiated from an object, multiply the radiant exitance by the object's surface area, :
Matter that does not absorb all incident radiation emits less total energy than a black body. Emissions are reduced by a factor , where the emissivity, , is a material property which, for most matter, satisfies . Emissivity can in general depend on wavelength, direction, and polarization. However, the emissivity which appears in the non-directional form of the Stefan–Boltzmann law is the hemispherical total emissivity, which reflects emissions as totaled over all wavelengths, directions, and polarizations.
The form of the Stefan–Boltzmann law that includes emissivity is applicable to all matter, provided that matter is in a state of local thermodynamic equilibrium (LTE) so that its temperature is well-defined. (This is a trivial conclusion, since the emissivity, , is defined to be the quantity that makes this equation valid. What is non-trivial is the proposition that , which is a consequence of Kirchhoff's law of thermal radiation.)
A so-called grey body is a body for which the spectral emissivity is independent of wavelength, so that the total emissivity, , is a constant. In the more general (and realistic) case, the spectral emissivity depends on wavelength. The total emissivity, as applicable to the Stefan–Boltzmann law, may be calculated as a weighted average of the spectral emissivity, with the blackbody emission spectrum serving as the weighting function. It follows that if the spectral emissivity depends on wavelength then the total emissivity depends on the temperature, i.e., . However, if the dependence on wavelength is small, then the dependence on temperature will be small as well.
Wavelength- and subwavelength-scale particles, metamaterials, and other nanostructures are not subject to ray-optical limits and may be designed to have an emissivity greater than 1.
In national and international standards documents, the symbol is recommended to denote radiant exitance; a superscript circle (°) indicates a term relative to a black body. (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 .) In common usage, the symbol used for radiant exitance (often called radiant emittance) varies among different texts and in different fields.
The Stefan–Boltzmann law may be expressed as a formula for radiance as a function of temperature. Radiance is measured in watts per square metre per steradian (W⋅m⋅sr). The Stefan–Boltzmann law for the radiance of a black body is:
The Stefan–Boltzmann law expressed as a formula for radiation energy density is:
where is the speed of light.
History In 1864, John Tyndall presented measurements of the infrared emission by a platinum filament and the corresponding color of the filament. ; see p. 333.In his physics textbook of 1875, Adolph Wüllner quoted Tyndall's results and then added estimates of the temperature that corresponded to the platinum filament's color:From : "Wie aus gleich zu besprechenden Versuchen von Draper hervorgeht, … also fast um das 12fache zu." (As follows from the experiments of Draper, which will be discussed shortly, a temperature of about 525°[C] corresponds to the weak red glow; a [temperature] of about 1200°[C], to the full white glow. Thus, while the temperature climbed only somewhat more than double, the intensity of the radiation increased from 10.4 to 122; thus, almost 12-fold.) 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.Stefan stated : "Zuerst will ich hier die Bemerkung anführen, … die Wärmestrahlung der vierten Potenz der absoluten Temperatur proportional anzunehmen." (First of all, I want to point out here the observation which Wüllner, in his textbook, added to the report of Tyndall's experiments on the radiation of a platinum wire that was brought to glowing by an electric current, because this observation first caused me to suppose that thermal radiation is proportional to the fourth power of the absolute temperature.)
A derivation of the law from theoretical considerations was presented by Ludwig Boltzmann (1844–1906) in 1884, drawing upon the work of Adolfo Bartoli. Bartoli in 1876 had derived the existence of radiation pressure from the principles of thermodynamics. Following Bartoli, Boltzmann considered an ideal heat engine using electromagnetic radiation instead of an ideal gas as working matter.
The law was almost immediately experimentally verified. Heinrich Weber in 1888 pointed out deviations at higher temperatures, but perfect accuracy within measurement uncertainties was confirmed up to temperatures of 1535 K by 1897. The law, including the theoretical prediction of the Stefan–Boltzmann constant as a function of the speed of light, the Boltzmann constant and the Planck constant, is a direct consequence of Planck's law as formulated in 1900.
Stefan–Boltzmann constant
The Stefan–Boltzmann constant, , is derived from other known physical constants:
where is the Boltzmann constant, the is the Planck constant (with the reduced Planck constant), and is the speed of light in vacuum.
As of the 2019 revision of the SI, which establishes exact fixed values for , , and , the Stefan–Boltzmann constant is exactly:
Thus,
Prior to this, the value of was calculated from the measured value of the gas constant.
The numerical value of the Stefan–Boltzmann constant is different in other systems of units, as shown in the table below.
+ Stefan–Boltzmann constant, σ Context Value Units SI W⋅m⋅K CGS erg⋅cm⋅s⋅KUS customary units BTU⋅hr⋅ft⋅°RThermochemistry cal⋅cm−2⋅day−1⋅K−4
Examples Temperature of the Sun
With his law, Stefan also determined the temperature of the Sun's surface. He inferred from the data of Jacques-Louis Soret (1827–1890) 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. Soret estimated the temperature of the lamella to be approximately 1900 °C to 2000 °C. 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.
Precise measurements of atmospheric absorption were not made until 1888 and 1904. The temperature Stefan obtained was a median value of previous ones, 1950 °C and the absolute thermodynamic one 2200 K. As 2.574 = 43.5, it follows from the law that the temperature of the Sun is 2.57 times greater than the temperature of the lamella, so Stefan got a value of 5430 °C or 5700 K. This was the first sensible value for the temperature of the Sun. Before this, values ranging from as low as 1800 °C to as high as On p. 505, the Scottish physicist John James Waterston estimated that the temperature of the sun's surface could be 12,880,000°. were claimed. The lower value of 1800 °C was determined by Claude Pouillet (1790–1868) in 1838 using the Dulong–Petit law. On p. 36, Pouillet estimates the sun's temperature: " … cette température pourrait être de 1761° … " ( … this temperature [i.e., of the Sun] could be 1761° … )English translation: Pouillet also took just half the value of the Sun's correct energy flux.
Temperature of stars 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. So:
where is the luminosity, is the Stefan–Boltzmann constant, is the stellar radius and is the effective temperature. This formula can then be rearranged to calculate the temperature:
or alternatively the radius:
The same formulae can also be simplified to compute the parameters relative to the Sun:
where is the solar radius, and so forth. They can also be rewritten in terms of the surface area A and radiant exitance :
where and
With the Stefan–Boltzmann law, astronomers can easily infer the radii of stars. The law is also met in the thermodynamics of black holes in so-called Hawking radiation.
Effective temperature of the Earth Similarly we can calculate the effective temperature of the Earth T⊕ by equating the energy received from the Sun and the energy radiated by the Earth, under the black-body approximation (Earth's own production of energy being small enough to be negligible). The luminosity of the Sun, L☉, is given by:
At Earth, this energy is passing through a sphere with a radius of a0, the distance between the Earth and the Sun, and the irradiance (received power per unit area) is given by
The Earth has a radius of R⊕, and therefore has a cross-section of . The radiant flux (i.e. solar power) absorbed by the Earth is thus given by:
Because the Stefan–Boltzmann law uses a fourth power, it has a stabilizing effect on the exchange and the flux emitted by Earth tends to be equal to the flux absorbed, close to the steady state where:
T⊕ can then be found:
where T☉ is the temperature of the Sun, R☉ the radius of the Sun, and a0 is the distance between the Earth and the Sun. 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. The effect of albedo on temperature can be approximated by assuming that the energy absorbed is multiplied by 0.7, but that the planet still radiates as a black body (the latter by definition of effective temperature, which is what we are calculating). This approximation reduces the temperature by a factor of 0.71/4, giving .
The above temperature is Earth's as seen from space, not ground temperature but an average over all emitting bodies of Earth from surface to high altitude. Because of the greenhouse effect, the Earth's actual average surface temperature is about , which is higher than the effective temperature, and even higher than the temperature that a black body would have.