Stefan-Boltzmann law 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): 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. ε is the emissivity of the blackbody; if it is a perfect blackbody, ε = ١. 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 where k is Boltzmann constant. Thus at ١٠٠ K the energy flux density is ۵ ۶٧ W/m ٢, at ١٠٠٠ K ۵۶ ٧٠٠ W/m ٢, etc. The Stefan-Boltzmann law is an example of a power law. The law was discovered experimentally by Jožef Stefan (١٨٣۵-١٨٩٣) in ١٨٧٩ and derived theoretically, using thermodynamics, by Ludwig Boltzmann (١٨۴۴-١٩٠۶) in ١٨٨۴. 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 ٢٠ 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. 1
Derivation of the Stefan-Boltzmann law 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. where Ω ٠ is the half-sphere into which the radiation is emitted, and I(ν,T)dν 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 [ν,ν + dν]. The cosine factor is included because the black body is a perfect Lambertian radiator. Using dω= sin(θ) dθdφ and integrating yields: (See appendix for the solution of this integral) Thermodynamic derivation The fact that the energy density of the box containing radiation is proportional to T ۴ can be derived using thermodynamics. It follows from classical electrodynamics that the radiation pressure P is related to the internal energy density: The total internal energy of the box containing radiation can thus be written as: Inserting this in the fundamental law of thermodynamics yields the equation: 2
We can now use this equation to derive a Maxwell relation. From the above equation it can be seen that: and The symmetry of second derivatives of S w.r.t. P and V then implies: 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: This implies that Examples 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 (١٨۵۴ ١٩٠۴) that the energy flux density from the Sun is ٢٩ 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 ١٩٠٠ C to ٢٠٠٠ C. Stefan surmised that ⅓ of the energy flux from the 3
Sun is absorbed by the Earth's atmosphere, so he took for the correct Sun's energy flux a value ٣/٢ times greater, namely ٢٩ ٣/٢ = ۴٣ ۵. Precise measurements of atmospheric absorption were not made until ١٨٨٨ and ١٩٠۴. The temperature Stefan obtained was a median value of previous ones, ١٩۵٠ C and the absolute thermodynamic one ٢٢٠٠ K. As ٢ ۵٧ ۴ = ۴٣ ۵, it follows from the law that the temperature of the Sun is ٢ ۵٧ times greater than the temperature of a lamella, so Stefan got a value of ۵۴٣٠ C or ۵٧٠٠ K (modern value is ۵٧٨٠ K). This was the first sensible value for the temperature of the Sun. Before this, values ranging from as low as ١٨٠٠ C to as high as ١٣ ٠٠٠ ٠٠٠ C were claimed. The lower value of ١٨٠٠ C was determined by Claude Servais Mathias Pouillet (١٧٩٠-١٨۶٨) in ١٨٣٨ 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. 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. [١][٢] So: 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: where, 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. 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: 4
where T S is the temperature of the Sun, r S the radius of the Sun and a ٠ astronomical unit, giving ۶ C. Summarizing: the surface of the Sun is ٢١ times as hot as that of the Earth, therefore it emits ١٩٠ ٠٠٠ times as much energy per square metre. The distance from the Sun to the Earth is ٢١۵ times the radius of the Sun, reducing the energy per square metre by a factor ۴۶ ٠٠٠. Taking into account that the cross-section of a sphere is ١/۴ of its surface area, we see that there is equilibrium (٣۴٢ W per m ٢ surface area, ١ ٣٧٠ W per m ٢ cross-sectional area). This shows roughly why T ~ ٣٠٠ K is the temperature of our world. The slightest change of the distance from the Sun might change the average Earth's temperature. 5