- Stefan–Boltzmann law
The

**Stefan–Boltzmann law**, also known as**Stefan's law**, states that the totalenergy radiated per unit surfacearea of ablack body in unittime (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'sthermodynamic temperature "T" (also called**absolute temperature**)::$j^\{star\}\; =\; sigma\; T^\{4\}.$

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 itsemissivity , $epsilon$::$j^\{star\}\; =\; epsilonsigma\; T^\{4\}.$

The irradiance "j"

^{*}has dimensions of energy flux (energy per time per area), and theSI units of measure arejoule s per second per square meter, or equivalently,watt s per square meter. The SI unit for absolute temperature "T" is thekelvin . "$epsilon$" is theemissivity of the grey body; if it is a perfect blackbody, $epsilon=1$. Still in more general (and realistic) case, the emissivity depends on the wavelength, $epsilon=epsilon(lambda)$.To find the total absolute power of

energy radiated for an object we have to take into account the surface area, A(in m^{2})::$P=\; A\; j^\{star\}\; =\; A\; epsilonsigma\; T^\{4\}.$

The

constant of proportionality σ, called theStefan–Boltzmann constant or**Stefan's constant**, is non-fundamental in the sense that it derives from other knownconstants of nature . The value of the constant is:$sigma=frac\{2pi^5\; k^4\}\{15c^2h^3\}=\; 5.670\; 400\; imes\; 10^\{-8\}\; extrm\{J,s\}^\{-1\}\; extrm\{m\}^\{-2\}\; extrm\{K\}^\{-4\}$

where k is the

Boltzmann constant , h isPlanck's constant , and c is the speed of light in a vacuum. Thus at 100 K the energy flux density is 5.67 W/m^{2}, at 1000 K 56,700 W/m^{2}, 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, usingthermodynamics , byLudwig Boltzmann (1844-1906) in 1884. Boltzmann treated a certain idealheat engine with thelight as a working matter instead of the gas. This law is the only physical law of nature named after a Slovenephysicist . The law is valid only for ideal black objects, the perfect radiators, called black bodies. Stefan published this law onMarch 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 usesspherical 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:: $I(\; u,T)\; =frac\{2\; h\; u^\{3\{c^2\}frac\{1\}\{\; e^\{frac\{h\; u\}\{kT-1\}.$

To restate the meaning of I: the quantity $I(\; u,T)\; ~A\; ~d\; u\; ~dOmega$ 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 ofLambert'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:: $j^\{star\}\; =\; frac\{2\; h\}\{c^2\}\; ~\; int\_0^infty\; !d\; u\; int\_0^\{2pi\}\; !d\; heta\; int\_0^\{pi/2\}!dphi~frac\{\; u^3\}\{\; e^\{frac\{h\; u\}\{kT-1\}\; cos(phi)\; sin(phi).$

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(2"φ"); 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:

: $j^\{star\}\; =\; frac\{2\; pi\; k^4\; T^4\; \}\{c^2\; h^3\}\; ~\; int\_0^infty\; !d\; u\; ~frac\{u^3\}\{\; e^u\; -\; 1\}.$

The integral on the right can be done in a number of ways (one is included in this article's appendix) -- its answer is π

^{4}/15, giving the result that, for a perfect blackbody surface:: $j^\{star\}\; =\; sigma\; T^4\; ~,\; ~~\; sigma\; =\; frac\{2\; pi^5\; k^4\; \}\{15\; c^2\; h^3\}.$

An alternative form of the Stefan–Boltzmann constant, more fundamental to physics::$sigma\; =\; frac\{pi^2\; k^4\}\{60\; hbar^3\; c^2\}$

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 $T^\{4\}$ can be derived using thermodynamics. It follows from classical electrodynamics that the radiation pressure $P$ is related to the internal energy density:

:$P=frac\{u\}\{3\}$

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

:$U=3PV,$

Inserting this in the

fundamental thermodynamic relation :$dU=T\; dS\; -\; P\; dV,$

yields the equation:

:$dS=4frac\{P\}\{T\}dV\; +\; 3frac\{V\}\{T\}dP$

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

:$left(frac\{partial\; S\}\{partial\; V\}\; ight)\_\{P\}=4frac\{P\}\{T\}$

and

:$left(frac\{partial\; S\}\{partial\; P\}\; ight)\_\{V\}=3frac\{V\}\{T\}$

The

symmetry of second derivatives of $S$ w.r.t. $P$ and $V$ then implies::$4left(frac\{partial\; left(P/T\; ight)\}\{partial\; P\}\; ight)\_\{V\}=\; 3left(frac\{partial\; left(V/T\; ight)\}\{partial\; V\}\; ight)\_\{P\}$

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:

:$frac\{1\}\{P\}frac\{dP\}\{dT\}=frac\{4\}\{T\}$

This implies that

:$u=3P\; propto\; T^\{4\}$

**Examples****Temperature of the Sun**With his law Stefan also determined the temperature of the

Sun 's surface. He learned from the data ofCharles 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 theEarth'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 2.57

^{4}= 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 byClaude Servais Mathias Pouillet (1790-1868) in 1838 using theDulong-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

star s other than the Sun can be approximated using a similar means by treating the emitted energy as ablack body radiation. [*cite web | url = http://www.uclan.ac.uk/facs/science/physastr/x99/PAM98/UCert/Ch04/4_4cst~1.htm | title = Stefan–Boltzmann Law | publisher = University of Central Lancashire | accessdate = 2006-08-13*] cite web | url = http://outreach.atnf.csiro.au/education/senior/astrophysics/photometry_luminosity.html | title = Luminosity of Stars | publisher = Australian Telescope Outreach and Education | accessdate = 2006-08-13 ] So:: $L\; =\; 4\; pi\; R^2\; sigma\; T\_\{e\}^4$

where

**L**is theluminosity ,**σ**is theStefan–Boltzmann constant ,**R**is the stellar radius and**T**is theeffective temperature . This same formula can be used to compute the approximate radius of a main sequence star relative to the sun:: $frac\{R\}\{R\_igodot\}\; approx\; left\; (\; frac\{T\_igodot\}\{T\}\; ight\; )^\{2\}\; cdot\; sqrt\{frac\{L\}\{L\_igodot$

where $R\_igodot$, is the

solar radius , and so forth.With the Stefan–Boltzmann law,

astronomer s can easily infer the radii of stars. The law is also met in the thermodynamics ofblack hole s in so calledHawking 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 "a"_{0}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 "grey-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. [*cite book | title = The CRC Handbook of Thermal Engineering | author = Frank Kreith | publisher = CRC Press/Springer | year = 2000 | isbn = 3540663495 | url = http://books.google.com/books?id=Xuc7dYB-iKMC&pg=PT447&dq=earth+%22effective+temperature%22+6&lr=&as_brr=3&ei=s3SeR-2qKJHIsQOtmKSbCg&sig=WjwotG5FQk5qOZn1z6JwhnfifS8*] 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, bygreenhouse gases , namelywater vapor ,carbon dioxide andmethane . [*P. K. Das, " [*] cite book | author=Cole, George H. A.; Woolfson, Michael M.*http://www.ias.ac.in/resonance/Mar1996/pdf/Mar1996p54-65.pdf The Earth's Changing Climate*] ", Resonance. Vol. 1. No. 3. pp. 54-65, 1996

title=Planetary Science: The Science of Planets Around Stars (1st ed.)

publisher=Institute of Physics Publishing | year=2002 | id=ISBN 0-7503-0815-X | pages = 36–37, 380–382 | url = http://books.google.com/books?id=Bgsy66mJ5mYC&pg=RA3-PA382&dq=black-body+emissivity+greenhouse+intitle:Planetary-Science+inauthor:cole&lr=&as_brr=0&ei=LrSOR9OYA4uotAP2ifyPBw&sig=mYO6KVgqvmYvdOu5-_qnQWCuVZk ] 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. As a result, the Earth's actual average surface temperature is about 288 K, rather than 279 K.Global warming is an increase in this equilibrium temperature due to human-caused additions to the quantity of greenhouse gases in the atmosphere.**Appendix**In one of the above derivations, the following integral appeared:

:$J=int\_\{0\}^\{infty\}frac\{x^\{3\{expleft(x\; ight)-1\}dx$

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:

:$f(k)=int\_\{0\}^\{infty\}frac\{sinleft(kx\; ight)\}\{expleft(x\; ight)-1\}dx$

Using the

Taylor expansion of the sine function, it should be evident that the coefficient of the "k"^{3}term would be exactly -"J"/6.By expanding both sides in powers of $k$, we see that $J$ is minus 6 times the coefficient of $k^3$ of the series expansion of $f(k)$. So, if we can find a closed form for "f"("k"), itsTaylor expansion will give J.In turn, sin(x) is the imaginary part of e

^{ix}, so we can restate this as::$f(k)=lim\_\{epsilon\; ightarrow\; 0\}~mbox\{Im\}~int\_\{epsilon\}^\{infty\}frac\{expleft(ikx\; ight)\}\{expleft(x\; ight)-1\}dx$

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

:$oint\_\{C(epsilon,\; R)\}frac\{expleft(ikz\; ight)\}\{expleft(z\; ight)-1\}dz$

where $C(epsilon,R)$ is the contour from $epsilon$ to $R$, then to $R+2pi\; i$, then to $epsilon+2pi\; i$, then we go to the point $2pi\; i\; -\; epsilon\; i$, avoiding the pole at $2pi\; i$ by taking a clockwise quarter circle with radius $epsilon$ and center $2pi\; i$. From there we go to $epsilon\; i$, and finally we return to $epsilon$, avoiding the pole at zero by taking a clockwise quarter circle with radius $epsilon$ and center zero.

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

:$oint\_\{C(epsilon,\; R)\}frac\{expleft(ikz\; ight)\}\{expleft(z\; ight)-1\}dz=0$

We now take the limit $R\; ightarrowinfty$. In this limit the contribution from the segment from $R$ to $R+2pi\; i$ tends to zero. Taking together the integrations over the segments from $epsilon$ to $R$ and from $R+2pi\; i$ to $epsilon+2pi\; i$ and using the fact that the integrations over clockwise quarter circles about

simple pole s are given by minus $frac\{i\; pi\}\{2\}$ times the residues at the poles we find::$left\; [1-expleft(-2pi\; k\; ight)\; ight]\; int\_\{epsilon\}^\{infty\}frac\{expleft(ikx\; ight)\}\{expleft(x\; ight)-1\}\; dx=\; i\; int\_\{epsilon\}^\{2pi-epsilon\}frac\{expleft(-ky\; ight)\}\{expleft(iy\; ight)-1\}dy+ifrac\{pi\}\{2\}left\; [1+expleft(-2pi\; k\; ight)\; ight]\; mbox\{\; (1)\}$

The left hand side is the sum of the integral from $epsilon$ to $R$ and from $R+2\; pi\; i$ to $2\; pi\; i\; +\; epsilon$. We can rewrite the integrand of the integral on the r.h.s. as follows:

:$frac\{1\}\{expleft(iy\; ight)-1\}\; =\; frac\{expleft(-ifrac\{y\}\{2\}\; ight)\}\{expleft(ifrac\{y\}\{2\}\; ight)-expleft(-ifrac\{y\}\{2\}\; ight)\}=frac\{1\}\{2i\}frac\{expleft(-ifrac\{y\}\{2\}\; ight)\}\{sinleft(frac\{y\}\{2\}\; ight)\}$

If we now take the imaginary part of both sides of Eq. (1) and take the limit $epsilon\; ightarrow\; 0$ we find:

:$f(k)\; =\; -frac\{1\}\{2k\}\; +\; frac\{pi\}\{2\}cothleft(pi\; k\; ight)$

after using the relation:

:$cothleft(x\; ight)\; =\; frac\{1+expleft(\; 2x\; ight)\}\{1\; -\; expleft(\; 2x\; ight)\}$.

Using that the series expansion of $coth(x)$ is given by:

:$coth(x)=\; frac\{1\}\{x\}+frac\{1\}\{3\}x-frac\{1\}\{45\}x^\{3\}ldots$

we see that the coefficient of $k^\{3\}$ of the series expansion of $f(k)$ is $-frac\{pi^\{4\{90\}$. This then implies that $J\; =\; frac\{pi^\{4\{15\}$ and the result

:$j^\{star\}=frac\{2pi^\{5\}\; k^\{4\{15\; h^\{3\}c^\{2T^\{4\}$follows.

**See also***

Wien's displacement law

*Rayleigh-Jeans law

*Zero-dimensional models

*Black body **Notes****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. 291-294

*Wikimedia Foundation.
2010.*