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Stefan-Boltzmann law

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

$j^{\star} = \sigma T^{4}$

The non-fundamental constant of proportionality is called the Stefan-Boltzmann constant or the Stefan's constant σ. Its value is

$\sigma=\frac{2\pi^5 k^4}{15c^2h^3}= 5.670 400(40) \times 10^{-8} \textrm{J\,s}^{-1}\textrm{m}^{-2}\textrm{K}^{-4}.$

Thus at 100 K the energy flux density is 5.67 W/m2, at 1000 K 56.7 kW/m2, etc.

The law was experimentally discovered by Jožef Stefan (1835-1893) in 1879 and theoretically derived in the frame of the 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 blackbodies. 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

Derivation of the Stefan-Boltzmann law

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.

$j^{\star}=\int_0^\infty \!d\nu \int_{\Omega_0} d\Omega~I_\nu \cos(\theta)$

where Ω0 is the half-sphere into which the radiation is emitted, and Iν is the amount of the black body emitted energy per unit surface per unit time per unit solid angle. The cosine factor is included because the black body is a perfect Lambertian radiator. Using dΩ= sin(θ) dθdφ and integrating yields:

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

(See polylogarithms for the solution of this Bose integral over frequency)

Temperature of the Sun

With his law Stefan also determined the temperature of the Sun's surface. He learnt from the data of Charles Soret (18541904) 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 circa 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, 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 from circa 1800 °C to 13,000,000 °C were claimed. The first 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.

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

Examples

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.

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

 TE $= T_S \sqrt{r_S\over 2 a_0 } \;$ $= 5780 \; {\rm K} \times \sqrt{696 \times 10^{6} \; {\rm m} \over 2 \times 149.59787066 \times 10^{9} \; {\rm m} }$ $= 278.7755970 \; {\rm K} \; ,$

where TS is the temperature of the Sun, rS 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 or atmospheric conditions might change the average Earth's temperature.

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.