Stefan–Boltzmann law

The Stefan–Boltzmann law, also known as Stefan's law, states that the total energy radiated per unit surface area of a black body in unit time (known variously as the black-body irradiance, energy flux density, radiant flux, or the emissive power), j^*, is directly proportional to the fourth power of the black body's thermodynamic temperature T (also called absolute temperature):

Failed to parse (Missing texvc executable; please see math/README to configure.): j^{\star} = \sigma T^{4}

A more general case is of a grey body, the one that doesn't absorb or emit the full amount of radiative flux. Instead, it radiates a portion of it, characterized by its emissivity, Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon

Failed to parse (Missing texvc executable; please see math/README to configure.): j^{\star} = \epsilon\sigma T^{4}

The irradiance j^* has dimensions of power density (energy per time per square distance), and the SI units of measure are joules per second per square meter, or equivalently, watts per square meter. The SI unit for absolute temperature T is the kelvin. Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon is the emissivity of the gray body; if it is a perfect blackbody, Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon=1 . Still in more general (and realistic) case, the emissivity depends on the wavelength, Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon=\epsilon(\lambda) .

To find the total absolute power of energy radiated for an object we have to take into account the surface area, A(in m²):

Failed to parse (Missing texvc executable; please see math/README to configure.): P= A j^{\star} = A \epsilon\sigma T^{4}

The constant of proportionality σ, called the Stefan–Boltzmann constant or Stefan's constant, is non-fundamental in the sense that it derives from other known constants of nature. The value of the constant is

Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma=\frac{2\pi^5 k^4}{15c^2h^3}= 5.670 400 \times 10^{-8} \textrm{J\,s}^{-1}\textrm{m}^{-2}\textrm{K}^{-4}.

where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum. Thus at 100 K the energy flux density is 5.67 W/m², at 1000 K 56,700 W/m², etc.

The Stefan–Boltzmann law is an example of a power law.

The law was deduced by Jožef Stefan (1835-1893) in 1879 on the basis of experimental measurements made by John Tyndall and was derived from theoretical considerations, using thermodynamics, by Ludwig Boltzmann (1844-1906) in 1884. Boltzmann treated a certain ideal heat engine with the light as a working matter instead of the gas. This law is the only physical law of nature named after a Slovene physicist. The law is valid only for ideal black objects, the perfect radiators, called black bodies. Stefan published this law on March 20 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.

Derivation of the Stefan–Boltzmann law

Integration of intensity derivation

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 φ as the zenith angle and θ as the azimuthal angle; and the small flat blackbody surface lies on the xy-plane, where φ = ^π/₂.

The intensity of the light emitted from the blackbody surface is given by Planck's law as:

Failed to parse (Missing texvc executable; please see math/README to configure.): I(\nu,T) =\frac{2 h\nu^{3}}{c^2}\frac{1}{ e^{\frac{h\nu}{kT}}-1}.

To restate the meaning of I: the quantity Failed to parse (Missing texvc executable; please see math/README to configure.): I(\nu,T) ~A ~d\nu ~d\Omega

is the power radiated by a surface of area A through a solid angle dΩ in the frequency range (ν, ν+dν). To derive the Stefan–Boltzmann law, we must integrate Ω over the half-sphere and integrate ν from 0 to ∞. Furthermore, because of Lambert's cosine law, the intensity observed along the sphere will be the actual intensity times the cosine of the zenith angle φ, and in spherical coordinates, dΩ = sin(φ) dφ dθ. On the other hand, since j* was the power radiated per blackbody surface area, the factor of A divides out. The whole integral, then, is:

Failed to parse (Missing texvc executable; please see math/README to configure.): j^{\star} = \frac{2 h}{c^2} ~ \int_0^\infty \!d\nu \int_0^{2\pi} \!d\theta \int_0^{\pi/2}\!d\phi ~\frac{\nu^3}{ e^{\frac{h\nu}{kT}}-1} \cos(\phi) \sin(\phi)

The integral with respect to θ can be done immediately; it's just 2π. The integral with respect to φ can also be done by observing that sin(φ)cos(φ) = 0.5 sin(2φ); it yields 1/2. The remainder requires a u-substitution given by ν = u k T / h, dν = du k T / h. Substituting throughout gives:

Failed to parse (Missing texvc executable; please see math/README to configure.): j^{\star} = \frac{2 \pi k^4 T^4 }{c^2 h^3} ~ \int_0^\infty \!d u ~\frac{u^3}{ e^u - 1}

The integral on the right can be done in a number of ways (one is included in this article's appendix) -- its answer is π⁴/15, giving the result that, for a perfect blackbody surface:

Failed to parse (Missing texvc executable; please see math/README to configure.): j^{\star} = \sigma T^4 ~, ~~ \sigma = \frac{2 \pi^5 k^4 }{15 c^2 h^3}

Finally, this proof started out only considering a small flat surface. However, any differentiable surface can be approximated by a bunch 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.

Thermodynamic derivation

The fact that the energy density of the box containing radiation is proportional to Failed to parse (Missing texvc executable; please see math/README to configure.): T^{4}

can be derived using thermodynamics. It follows from classical electrodynamics that the radiation pressure Failed to parse (Missing texvc executable; please see math/README to configure.): P
is related to the internal energy density:

Failed to parse (Missing texvc executable; please see math/README to configure.): P=\frac{u}{3}

The total internal energy of the box containing radiation can thus be written as:

Failed to parse (Missing texvc executable; please see math/README to configure.): U=3PV\,

Inserting this in the fundamental law of thermodynamics

Failed to parse (Missing texvc executable; please see math/README to configure.): dU=T dS - P dV\,

yields the equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): dS=4\frac{P}{T}dV + 3\frac{V}{T}dP

We can now use this equation to derive a Maxwell relation. From the above equation it can be seen that:

Failed to parse (Missing texvc executable; please see math/README to configure.): \left(\frac{\partial S}{\partial V}\right)_{P}=4\frac{P}{T}

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \left(\frac{\partial S}{\partial P}\right)_{V}=3\frac{V}{T}

The symmetry of second derivatives of Failed to parse (Missing texvc executable; please see math/README to configure.): S

w.r.t. Failed to parse (Missing texvc executable; please see math/README to configure.): P
and Failed to parse (Missing texvc executable; please see math/README to configure.): V
then implies:

Failed to parse (Missing texvc executable; please see math/README to configure.): 4\left(\frac{\partial \left(P/T\right)}{\partial P}\right)_{V}= 3\left(\frac{\partial \left(V/T\right)}{\partial V}\right)_{P}

Because the pressure is proportional to the internal energy density it depends only on the temperature and not on the volume. In the derivative on the r.h.s. the temperature is thus a constant. Evaluating the derivatives gives the differential equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{P}\frac{dP}{dT}=\frac{4}{T}

This implies that

Failed to parse (Missing texvc executable; please see math/README to configure.): u=3P \propto T^{4}

Examples

Temperature of the Sun

With his law Stefan also determined the temperature of the Sun's surface. He learned from the data of Charles Soret (1854–1904) that the energy flux density from the Sun is 29 times greater than the energy flux density of a warmed metal lamella. A round lamella was placed at such a distance from the measuring device that it would be seen at the same angle 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, 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 a lamella, so Stefan got a value of 5430 °C or 5700 K (modern value is 5780 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 Servais Mathias Pouillet (1790-1868) in 1838 using the Dulong-Petit law. Pouilett also took just half the value of the Sun's correct energy flux. Perhaps this result reminded Stefan that the Dulong-Petit law could break down at large temperatures.

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.^[1][2] So:

Failed to parse (Missing texvc executable; please see math/README to configure.): L = 4 \pi R^2 \sigma T_{e}^4

where L is the luminosity, σ is the Stefan–Boltzmann constant, R is the stellar radius and T is the effective temperature. This same formula can be used to compute the approximate radius of a main sequence star relative to the sun:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{R}{R_\bigodot} \approx \left ( \frac{T_\bigodot}{T} \right )^{2} \cdot \sqrt{\frac{L}{L_\bigodot}}

where Failed to parse (Missing texvc executable; please see math/README to configure.): R_\bigodot , is the solar radius, and so forth.

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.

Temperature of the Earth

Similarly we can calculate the effective temperature of the Earth T_E by equating the energy received from the Sun and the energy transmitted by the Earth, under the black-body approximation:

where T_S is the temperature of the Sun, r_S the radius of the Sun, and a₀ is the distance between the Earth and the Sun. Thus resulting in an effective temperature of 6°C on the surface of the Earth.

The above derivation is a rough approximation only, as it assumes the Earth is a perfect blackbody. The same equilibrium planetary temperature would result if the planet's emissivity and absorptivity were reduced by some constant fraction at all wavelengths, since the incoming and outgoing powers would still match at the same temperature (this equilibrium temperature would no longer fit the definition of effective temperature, however).

The real Earth does not have this "gray-body" property. The terrestrial albedo is such that about 30% of incident solar radiation is reflected back into space; taking the reduced energy from the sun into account and computing the temperature of a black-body radiator that would emit that much energy back into space yields an "effective temperature", consistent with the definition of that concept, of about 255 K.^[3] However, compared to the 30% reflection of the Sun's energy, a much larger fraction of long-wave radiation from the surface of the earth is absorbed or reflected in the atmosphere instead of being radiated away, by greenhouse gases, namely water vapor, carbon dioxide and methane.^[4][5] Since the emissivity (weighted more in the longer wavelengths where the Earth radiates), is reduced more than than the absorptivity (weighted more in the shorter wavelengths of the Sun's radiation), the equilibrium temperature is higher than the simple black-body calculation estimates, not lower. The Earth's actual average surface temperature is about 288 K, rather than 279 K, as a result; global warming is an increase in this equilibrium temperature due to human-caused additions to the greenhouse gasses.

Appendix

In one of the above derivations, the following integral appeared:

Failed to parse (Missing texvc executable; please see math/README to configure.): J=\int_{0}^{\infty}\frac{x^{3}}{\exp\left(x\right)-1}dx

There are a number of ways to do this integration; a simple one is given in the appendix of the Planck's law article. This appendix does the integral by contour integration. Consider the function:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(k)=\int_{0}^{\infty}\frac{\sin\left(kx\right)}{\exp\left(x\right)-1}dx

Using the Taylor expansion of the sine function, it should be evident that the coefficient of the k³ term would be exactly -J/6. By expanding both sides in powers of Failed to parse (Missing texvc executable; please see math/README to configure.): k , we see that Failed to parse (Missing texvc executable; please see math/README to configure.): J

is minus 6 times the coefficient of Failed to parse (Missing texvc executable; please see math/README to configure.): k^3
of the series expansion of Failed to parse (Missing texvc executable; please see math/README to configure.): f(k)

. So, if we can find a closed form for f(k), its Taylor expansion will give J.

In turn, sin(x) is the imaginary part of e^ix, so we can restate this as:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(k)=\lim_{\epsilon\rightarrow 0}~\mbox{Im}~\int_{\epsilon}^{\infty}\frac{\exp\left(ikx\right)}{\exp\left(x\right)-1}dx

To evaluate the integral in this equation we consider the contour integral:

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_{C(\epsilon, R)}\frac{\exp\left(ikz\right)}{\exp\left(z\right)-1}dz

where Failed to parse (Missing texvc executable; please see math/README to configure.): C(\epsilon,R)

is the contour from Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon
to Failed to parse (Missing texvc executable; please see math/README to configure.): R

, then to Failed to parse (Missing texvc executable; please see math/README to configure.): R+2\pi i , then to Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon+2\pi i , then we go to the point Failed to parse (Missing texvc executable; please see math/README to configure.): 2\pi i - \epsilon i , avoiding the pole at Failed to parse (Missing texvc executable; please see math/README to configure.): 2\pi i

by taking a clockwise quarter circle with radius Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon
and center Failed to parse (Missing texvc executable; please see math/README to configure.): 2\pi i

. From there we go to Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon i , and finally we return to Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon , avoiding the pole at zero by taking a clockwise quarter circle with radius Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon

and center zero.

Because there are no poles in the integration contour we have:

Failed to parse (Missing texvc executable; please see math/README to configure.): \oint_{C(\epsilon, R)}\frac{\exp\left(ikz\right)}{\exp\left(z\right)-1}dz=0

We now take the limit Failed to parse (Missing texvc executable; please see math/README to configure.): R\rightarrow\infty . In this limit the contribution from the segment from Failed to parse (Missing texvc executable; please see math/README to configure.): R

to Failed to parse (Missing texvc executable; please see math/README to configure.): R+2\pi i
tends to zero. Taking together the integrations over the segments from Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon
to Failed to parse (Missing texvc executable; please see math/README to configure.): R
and from Failed to parse (Missing texvc executable; please see math/README to configure.): R+2\pi i
to Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon+2\pi i
and using the fact that the integrations over clockwise quarter circles about simple poles are given by minus Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{i \pi}{2}
times the residues at the poles we find:

Failed to parse (Missing texvc executable; please see math/README to configure.): \left[1-\exp\left(-2\pi k\right) \right]\int_{\epsilon}^{\infty}\frac{\exp\left(ikx\right)}{\exp\left(x\right)-1} dx= i \int_{\epsilon}^{2\pi-\epsilon}\frac{\exp\left(-ky\right)}{\exp\left(iy\right)-1}dy+i\frac{\pi}{2}\left[1+\exp\left(-2\pi k\right)\right]\mbox{ (1)}

The left hand side is the sum of the integral from Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon

to Failed to parse (Missing texvc executable; please see math/README to configure.): R
and from Failed to parse (Missing texvc executable; please see math/README to configure.): R+2 \pi i
to Failed to parse (Missing texvc executable; please see math/README to configure.): 2 \pi i + \epsilon

. We can rewrite the integrand of the integral on the r.h.s. as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{\exp\left(iy\right)-1} = \frac{\exp\left(-i\frac{y}{2}\right)}{\exp\left(i\frac{y}{2}\right)-\exp\left(-i\frac{y}{2}\right)}=\frac{1}{2i}\frac{\exp\left(-i\frac{y}{2}\right)}{\sin\left(\frac{y}{2}\right)}

If we now take the imaginary part of both sides of Eq. (1) and take the limit Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon\rightarrow 0

we find:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(k) = -\frac{1}{2k} + \frac{\pi}{2}\coth\left(\pi k\right)

after using the relation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \coth\left(x\right) = \frac{1+\exp\left( 2x\right)}{1 - \exp\left( 2x \right)}

.

Using that the series expansion of Failed to parse (Missing texvc executable; please see math/README to configure.): \coth(x)

is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \coth(x)= \frac{1}{x}+\frac{1}{3}x-\frac{1}{45}x^{3}\ldots

we see that the coefficient of Failed to parse (Missing texvc executable; please see math/README to configure.): k^{3}

of the series expansion of Failed to parse (Missing texvc executable; please see math/README to configure.): f(k)
is Failed to parse (Missing texvc executable; please see math/README to configure.): -\frac{\pi^{4}}{90}

. This then implies that Failed to parse (Missing texvc executable; please see math/README to configure.): J = \frac{\pi^{4}}{15}

and the result

Failed to parse (Missing texvc executable; please see math/README to configure.): j^{\star}=\frac{2\pi^{5} k^{4}}{15 h^{3}c^{2}}T^{4}

follows.

See also

• Wien's displacement law
• Rayleigh-Jeans law
• Zero-dimensional models
• Temperature relation between a planet and its star

References

• Stefan, J.: Über die Beziehung zwischen der Wärmestrahlung und der Temperatur, in: Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften, Bd. 79 (Wien 1879), S. 391-428.
• Boltzmann, L.: Ableitung des Stefan'schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electromagnetischen Lichttheorie, in: Annalen der Physik und Chemie, Bd. 22 (1884), S. 291-294ca:Llei de Stefan-Boltzmann

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