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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 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:

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:

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 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. Similarly we can calculate the temperature of the Earth TE:

where TS is the temperature of the Sun, rS 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 criticized 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.