Stefan-Boltzmann law

Stefan-Boltzmann law (also Stefan's law) states that the total energy radiated per unit surface area of a blackbody in unit time (blackbody 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:

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

The non-fundamental constant of proportionality is called the Stefan-Boltzmann constant or the Stefan's constant σ. Its value is 5.670 400(40) × 10^-8 J s^-1 m^-2 K^-4. 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 the nature named after one Slovene physicist. Today we can derive the law from the Planck's law of black body radiation:

<math> j^{\star} = \int_{0}^{\infty} \left( {dj^{\star}\over d\lambda} \right) d\lambda </math>

and 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 (About the relation between heat equilibrium and temperature) in the Bulletins from the sessions of the Vienna Academy of Sciences.

Temperature of the Sun

With his law Stefan also determined the temperature of the Sun's surface. He leant on the datum of Charles Soret[?] (1854–1904) that the energy flux density from the Sun is 29-times greater from 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 a lamella to be circa 1900 °C to 2000 °C. Stefan surmised that 1/3 of the energy flux from the Sun is retained by the Earth's atmosphere, so he took for the correct Sun energy flux 3/2 greater value, namely 29 × 3/2 = 43.5. The precise measurements of the athmospheric absorption wasn't made until 1888 and 1904. The temperature, Stefan has taken, was a median value of previous ones, 1950 °C and the absolute thermodynamic one 2200 K. As it is 2.57⁴ = 43.5, from the law folows 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 (today value is 5780 K). This was the first sensible value for the temperature of the Sun. Before him they alleged values from circa 1800 °C to 13,000,000 °C. The first value of 1800 °C was determined by Claude Servais Mathias Pouillet (1790-1868) in 1838 with the Dulong-Petit law. Pouilett also took just a half value of the correct Sun energy flux. Perhaps this result reminded Stefan that Dulong-Petit law could break down at large temperatures. If we collect Sun's light with the 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 Stefan-Boltzmann law astronomers can easily infer the radii of stars. The law is also met in the thermodynamics of black holes. Similarly we can calculate the temperature of the Earth T_E:

<math> T_E = T_S \sqrt{r_S\over 2 a_0 } \; =

5780 \times \sqrt{696 \times 10^{6}\over 2 \times 149.59787066 \times 10^{9} } = 278.7755970 \; {\rm K} \; , </math>

where T_S is the temperature of the Sun, r_S the radius of the Sun and a₀ astronomical unit and we get 6 °C, so our Sun is about 964-times hotter than the Earth. 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 reproached Stefan that his way of determining the law was quite shaky. We would make him a great injustice if we would think that he has found the law blindly. Many happy coincidences has influenced on his determination as it usually happens by many important discoveries.