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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):

<math> j^{\star} = \epsilon\sigma T^{4}</math>

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. <math>\epsilon</math> is the emissivity of the blackbody; if it is a perfect blackbody, <math>\epsilon=1</math>.

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

<math>

\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}. </math>

where k is Boltzmann constant. 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 discovered experimentally by Jožef Stefan (1835-1893) in 1879 and derived theoretically, 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.

Contents 1 Derivation of the Stefan-Boltzmann law 1.1 Integration of intensity derivation 1.2 Thermodynamic derivation 2 Examples 2.1 Temperature of the Sun 2.2 Temperature of stars 2.3 Temperature of the Earth 3 Criticism of the law 4 Appendix 5 See also 6 References

[edit] Derivation of the Stefan-Boltzmann law

[edit] Integration of intensity derivation

The Stefan-Boltzmann law can be easily derived by integrating the emitted intensity from the surface of a black body given by Planck's law of black body radiation over the half-sphere into which it is emitted, and over all frequencies.

<math>

j^{\star}=\int_0^\infty \!d\nu \int_{\Omega_0} d\Omega~I(\nu,T) \cos(\theta) </math>

where Ω₀ is the half-sphere into which the radiation is emitted, and <math>I(\nu,T)d\nu</math> is the amount of energy emitted by a black body at temperature T per unit surface per unit time per unit solid angle in the frequency range <math>[\nu,\nu+d\nu]</math>. The cosine factor is included because the black body is a perfect Lambertian radiator. Using dΩ= sin(θ) dθdφ and integrating yields:

<math>

j^{\star}=\int_0^\infty \!d\nu \int_0^{2\pi} \!d\phi \int_0^{\pi/2}\!d\theta ~I(\nu,T) \cos(\theta)\sin(\theta)=\frac{2\pi^5 k^4}{15c^2h^3}\,T^4 </math>

(See appendix for the solution of this integral)

[edit] Thermodynamic derivation

The fact that the energy density of the box containing radiation is proportional to <math>T^{4}</math> can be derived using thermodynamics. It follows from classical electrodynamics that the radiation pressure <math>P</math> is related to the internal energy density:

<math>P=\frac{u}{3}</math>

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

<math>U=3PV\,</math>

Inserting this in the fundamental law of thermodynamics

<math>dU=T dS - P dV\,</math>

yields the equation:

<math>dS=4\frac{P}{T}dV + 3\frac{V}{T}dP</math>

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

<math>\left(\frac{\partial S}{\partial V}\right)_{P}=4\frac{P}{T}</math>

and

<math>\left(\frac{\partial S}{\partial P}\right)_{V}=3\frac{V}{T}</math>

The symmetry of second derivatives of <math>S</math> w.r.t. <math>P</math> and <math>V</math> then implies:

<math>4\left(\frac{\partial \left(P/T\right)}{\partial P}\right)_{V}= 3\left(\frac{\partial \left(V/T\right)}{\partial V}\right)_{P}</math>

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:

<math>\frac{1}{P}\frac{dP}{dT}=\frac{4}{T}</math>

This implies that

<math>u=3P \propto T^{4} </math>

[edit] Examples

[edit] 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. If we collect the Sun's light with a lens, we can warm a solid to much higher temperature than 1800 °C.

[edit] 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.<ref>Stefan-Boltzmann Law (English). University of Central Lancashire. Retrieved on 2006-08-13.</ref><ref name="luminosity">Luminosity of Stars (English). Australian Telescope Outreach and Education. Retrieved on 2006-08-13.</ref> So:

<math>L = 4 \pi R^2 \sigma T_{e}^4 </math>

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:

<math>\frac{R}{R_\bigodot} \approx \left ( \frac{T_\bigodot}{T} \right )^{2} \cdot \sqrt{\frac{L}{L_\bigodot}}</math>

where <math>R_\bigodot</math>, 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.

[edit] Temperature of the Earth

Similarly we can calculate the temperature of the Earth T_E by equating the energy received from the Sun and the energy transmitted by the Earth:

<math> T_E \, </math>	<math> = T_S \sqrt{r_S\over 2 a_0 } \; </math>
	<math> = 5780 \; {\rm K} \times \sqrt{696 \times 10^{6} \; {\rm m} \over 2 \times 149.59787066 \times 10^{9} \; {\rm m} } </math>
	<math> = 278.776 \; {\rm K} \; , </math>

where T_S is the temperature of the Sun, r_S the radius of the Sun and a₀ astronomical unit, giving 6°C.

Summarizing: the surface of the Sun is 21 times as hot as that of the Earth, therefore it emits 190,000 times as much energy per square metre. The distance from the Sun to the Earth is 215 times the radius of the Sun, reducing the energy per square metre by a factor 46,000. Taking into account that the cross-section of a sphere is 1/4 of its surface area, we see that there is equilibrium (342 W per m² surface area, 1,370 W per m² cross-sectional area).

This shows roughly why T ~ 300 K is the temperature of our world. The slightest change of the distance from the Sun might change the average Earth's temperature.

[edit] Criticism of the law

Image:Unbalanced scales.svg

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Some physicists have criticised Stefan for using a theoretically unsound method to determine the law. It is true that he was helped by some fortunate coincidences, but this does not mean that he found the law blindly.

[edit] Appendix

The spectral intensity is given by

<math>I(\nu,T) =\frac{2h\nu^{3}}{c^2}\frac{1}{ e^{\frac{h\nu}{kT}}-1}</math>

The integrations over the angles <math>\theta</math> and <math>\phi</math> yields a factor <math>\pi</math>. Making the substitution <math>\nu=\frac{k T}{h} x</math> in the remaining integration over the frequency <math>\nu</math> gives

<math>j^{\star}=\frac{2\pi k^{4}T^{4}}{h^{3}c^{2}} J

</math>

where <math>J</math> is given by

<math>J=\int_{0}^{\infty}\frac{x^{3}}{\exp\left(x\right)-1}dx </math>

A simple way to calculate this integral is given in the appendix of the article Planck's law of black body radiation. Here we give a different derivation. Consider the function:

<math>f(k)=\int_{0}^{\infty}\frac{\sin\left(kx\right)}{\exp\left(x\right)-1}dx </math>

By expanding both sides in powers of <math>k</math>, we see that <math>J</math> is minus 6 times the coefficient of <math>k^{3}</math> of the series expansion of <math>f(k)</math>. The function <math>f(k)</math> can be written as

<math>

f(k)=\lim_{\epsilon\rightarrow 0}\mbox{Im}\int_{\epsilon}^{\infty}\frac{\exp\left(ikx\right)}{\exp\left(x\right)-1}dx </math>

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

<math>

\oint_{C(\epsilon, R)}\frac{\exp\left(ikz\right)}{\exp\left(z\right)-1}dz </math>

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

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

<math>

\oint_{C(\epsilon, R)}\frac{\exp\left(ikz\right)}{\exp\left(z\right)-1}dz=0 </math>

We now take the limit <math>R\rightarrow\infty</math>. In this limit the contribution from the segment from <math>R</math> to <math>R+2\pi i</math> tends to zero. Taking together the integrations over the segments from <math>\epsilon</math> to <math>R</math> and from <math>R+2\pi i</math> to <math>\epsilon+2\pi i</math> and using the fact that the integrations over the clockwise quarter circles are given by minus <math>\frac{\pi}{2}</math> times the residues at the poles we find:

<math>

\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)} </math>

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

<math>

\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)} </math>

If we now take the imaginary part of both sides of Eq. (1) and take the limit <math>\epsilon\rightarrow 0</math> we find:

<math>f(k) = -\frac{1}{2k} + \frac{\pi}{2}\coth\left(\pi k\right)

</math>

Using that the series expansion of <math>\coth(x)</math> is given by:

<math>

\coth(x)= \frac{1}{x}+\frac{1}{3}x-\frac{1}{45}x^{3}\ldots </math>

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

<math>j^{\star}=\frac{2\pi^{5} k^{4}}{15 h^{3}c^{2}}T^{4}

</math> follows.

[edit] See also

• Wien's displacement law
• Rayleigh-Jeans law

[edit] 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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