The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature T:
A body that does not absorb all incident radiation (sometimes known as a grey body) emits less total energy than a black body and is characterized by an emissivity, :
The 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, or equivalently, watts per square metre. The SI unit for absolute temperature T is the kelvin. is the emissivity of the grey body; if it is a perfect blackbody, . In the still more general (and realistic) case, the emissivity depends on the wavelength, .
To find the total power radiated from an object, multiply by its surface area, :
In 1864, John Tyndall presented measurements of the infrared emission by a platinum filament and the corresponding color of the filament. 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.
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 Planck's constant, is a direct consequence of Planck's law as formulated in 1900.
As of the 2019 redefinition of SI base units, which fixes the values of the Boltzmann constant k, the Planck constant h, and the speed of light c, the Stefan–Boltzmann constant is exactly
- σ = 5.67037441918442945397099673188923087584012297029130...×10−8 W/(m2K4).
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 ⅓ 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 13,000,000 °C were claimed. The lower value of 1800 °C was determined by Claude Pouillet (1790–1868) in 1838 using the Dulong–Petit law. Pouillet also took just half the value of the Sun's correct energy flux.
Temperature of stars
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 emittance :
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 255 K (−18 °C).
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 288 K (15 °C), which is higher than the 255 K effective temperature, and even higher than the 279 K temperature that a black body would have.
In the above discussion, we have assumed that the whole surface of the earth is at one temperature. 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. This of course depends on the angle of the sun on the surface and on how much air the sunlight has gone through. When the sun is at the zenith and the surface is horizontal, the irradiance can be as high as 1120 W/m2. The Stefan–Boltzmann law then gives a temperature of
or 102 °C. (Above the atmosphere, the result is even higher: 394 K.) 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.
Thermodynamic derivation of the energy density
The fact that the energy density of the box containing radiation is proportional to can be derived using thermodynamics. This derivation uses the relation between the radiation pressure p and the internal energy density , a relation that can be shown using the form of the electromagnetic stress–energy tensor. This relation is:
Now, from the fundamental thermodynamic relation
we obtain the following expression, after dividing by and fixing :
The last equality comes from the following Maxwell relation:
From the definition of energy density it follows that
where the energy density of radiation only depends on the temperature, therefore
Now, the equality
after substitution of Meanwhile, the pressure is the rate of momentum change per unit area. Since the momentum of a photon is the same as the energy divided by the speed of light,
where the 1/3 factor comes from the projection of the momentum transfer onto the normal to the wall of the container.
Since the partial derivative can be expressed as a relationship between only and (if one isolates it on one side of the equality), the partial derivative can be replaced by the ordinary derivative. After separating the differentials the equality becomes
which leads immediately to , with as some constant of integration.
Derivation from Planck's law
The law can be derived by considering a small flat black body surface radiating out into a half-sphere. This derivation uses spherical coordinates, with θ as the zenith angle and φ as the azimuthal angle; and the small flat blackbody surface lies on the xy-plane, where θ = π/2.
The intensity of the light emitted from the blackbody surface is given by Planck's law :
- is the amount of power per unit surface area per unit solid angle per unit frequency emitted at a frequency by a black body at temperature T.
- is Planck's constant
- is the speed of light, and
- is Boltzmann's constant.
The Stefan–Boltzmann law gives the power emitted per unit area of the emitting body,
Note that the cosine appears because black bodies are Lambertian (i.e. they obey Lambert's cosine law), meaning that the intensity observed along the sphere will be the actual intensity times the cosine of the zenith angle. To derive the Stefan–Boltzmann law, we must integrate over the half-sphere and integrate from 0 to ∞.
Then we plug in for I:
To evaluate this integral, do a substitution,
The integral on the right is standard and goes by many names: it is a particular case of a Bose–Einstein integral, the polylogarithm, or the Riemann zeta function . The value of the integral is (where is the Gamma function), giving the result that, for a perfect blackbody surface:
Finally, this proof started out only considering a small flat surface. However, any differentiable surface can be approximated by a collection of small flat surfaces. So long as the geometry of the surface does not cause the blackbody to reabsorb its own radiation, the total energy radiated is just the sum of the energies radiated by each surface; and the total surface area is just the sum of the areas of each surface—so this law holds for all convex blackbodies, too, so long as the surface has the same temperature throughout. The law extends to radiation from non-convex bodies by using the fact that the convex hull of a black body radiates as though it were itself a black body.
The total energy density U can be similarly calculated, except the integration is over the whole sphere and there is no cosine, and the energy flux (U c) should be divided by the velocity c to give the energy density U:
Thus, in total:
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- Stefan stated (Stefan, 1879), p. 421: "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.)
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- English translation: Pouillet (1838) "Memoir on the solar heat, on the radiating and absorbing powers of atmospheric air, and on the temperature of space" in: Taylor, Richard, ed. (1846) Scientific Memoirs, Selected from the Transactions of Foreign Academies of Science and Learned Societies, and from Foreign Journals. vol. 4. London, England: Richard and John E. Taylor. pp. 44–90; see pp. 55–56.
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