- Stefan–Boltzmann law
The Stefan–Boltzmann law, also known as Stefan's law, states that the total
energyradiated per unit surface areaof a black bodyin 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):
A more general case is of a
grey body, the one that doesn't absorb or emit the full amount of radiative flux. Instead, it radiates a portion of it, characterized by its emissivity, :
The irradiance "j"* has dimensions of energy flux (energy per time per area), and the
SIunits 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. "" is the emissivityof the grey body; if it is a perfect blackbody, . Still in more general (and realistic) case, the emissivity depends on the wavelength, .
To find the total absolute power of
energyradiated for an object we have to take into account the surface area, A(in m2):
constant of proportionalityσ, called the Stefan–Boltzmann constantor 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 the
Boltzmann constant, h is Planck's constant, and c is the speed of light in a vacuum. 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
The law was deduced by Jožef Stefan (1835-1893) in 1879 on the basis of experimental measurements made by
John Tyndalland was derived from theoretical considerations, using thermodynamics, by Ludwig Boltzmann(1844-1906) in 1884. Boltzmann treated a certain ideal heat enginewith the lightas 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 20in 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.
Derivation of the Stefan–Boltzmann law
Integration of intensity derivation
The law can be derived by considering a small flat
black bodysurface radiating out into a half-sphere. This derivation uses spherical coordinates, with "φ" as the zenith angle and "θ" as the azimuthal angle; and the small flat blackbody surface lies on the xy-plane, where "φ" = π/2.
The intensity of the light emitted from the blackbody surface is given by
To restate the meaning of I: the quantity is the power radiated by a surface of area A through a
solid angle"dΩ" in the frequency range ("ν", "ν"+"dν"). To derive the Stefan–Boltzmann law, we must integrate "Ω" over the half-sphere and integrate "ν" from 0 to ∞. Furthermore, because of Lambert's cosine law, the intensity observed along the sphere will be the actual intensity times the cosine of the zenith angle "φ", and in spherical coordinates, "dΩ" = sin("φ") "dφ dθ". On the other hand, since "j"* was the power radiated per blackbody surface area, the factor of A divides out. The whole integral, then, is:
The integral with respect to θ can be done immediately; it's just 2π. The integral with respect to φ can also be done by observing that sin("φ")cos("φ") = 0.5 sin(2"φ"); it yields 1/2. The remainder requires a "u"-substitution given by "ν" = "u k T" / "h", "dν" = "du k T / h". Substituting throughout gives:
The integral on the right can be done in a number of ways (one is included in this article's appendix) -- its answer is π4/15, giving the result that, for a perfect blackbody surface:
An alternative form of the Stefan–Boltzmann constant, more fundamental to physics::
Finally, this proof started out only considering a small flat surface. However, any
differentiablesurface can be approximated by a bunch of small flat surfaces. So long as the geometry of the surface does not cause the blackbody to reabsorb its own radiation, the total energy radiated is just the sum of the energies radiated by each surface; and the total surface area is just the sum of the areas of each surface -- so this law holds for all convex blackbodies, too, so long as the surface has the same temperature throughout.
The fact that the energy density of the box containing radiation is proportional to can be derived using thermodynamics. It follows from classical electrodynamics that the radiation pressure 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 thermodynamic relation
yields the equation:
We can now use this equation to derive a Maxwell relation. From the above equation it can be seen that:
symmetry of second derivativesof w.r.t. and 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
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.
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 bodyradiation. [cite web | url = http://www.uclan.ac.uk/facs/science/physastr/x99/PAM98/UCert/Ch04/4_4cst~1.htm | title = Stefan–Boltzmann Law | publisher = University of Central Lancashire | accessdate = 2006-08-13 ] cite web | url = http://outreach.atnf.csiro.au/education/senior/astrophysics/photometry_luminosity.html | title = Luminosity of Stars | publisher = Australian Telescope Outreach and Education | accessdate = 2006-08-13 ] 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
effective temperatureof the Earth "T"E by equating the energy received from the Sun and the energy transmitted by the Earth, under the black-body approximation:
where "T"S is the temperature of the Sun, "r"S the radius of the Sun, and "a"0 is the distance between the Earth and the Sun. Thus resulting in an effective temperature of 6°C on the surface of the Earth.
The above derivation is a rough approximation only, as it assumes the Earth is a perfect blackbody. The same equilibrium planetary temperature would result if the planet's emissivity and absorptivity were reduced by some constant fraction at all wavelengths, since the incoming and outgoing powers would still match at the same temperature (this equilibrium temperature would no longer fit the definition of effective temperature, however).
The real Earth does not have this "grey-body" property. The terrestrial
albedois such that about 30% of incident solar radiation is reflected back into space; taking the reduced energy from the sun into account and computing the temperature of a black-body radiator that would emit that much energy back into space yields an "effective temperature", consistent with the definition of that concept, of about 255 K. [cite book | title = The CRC Handbook of Thermal Engineering | author = Frank Kreith | publisher = CRC Press/Springer | year = 2000 | isbn = 3540663495 | url = http://books.google.com/books?id=Xuc7dYB-iKMC&pg=PT447&dq=earth+%22effective+temperature%22+6&lr=&as_brr=3&ei=s3SeR-2qKJHIsQOtmKSbCg&sig=WjwotG5FQk5qOZn1z6JwhnfifS8 ] However, compared to the 30% reflection of the Sun's energy, a much larger fraction of long-wave radiation from the surface of the earth is absorbed or reflected in the atmosphere instead of being radiated away, by greenhouse gases, namely water vapor, carbon dioxideand methane. [P. K. Das, " [http://www.ias.ac.in/resonance/Mar1996/pdf/Mar1996p54-65.pdf The Earth's Changing Climate] ", Resonance. Vol. 1. No. 3. pp. 54-65, 1996] cite book | author=Cole, George H. A.; Woolfson, Michael M.
title=Planetary Science: The Science of Planets Around Stars (1st ed.)
publisher=Institute of Physics Publishing | year=2002 | id=ISBN 0-7503-0815-X | pages = 36–37, 380–382 | url = http://books.google.com/books?id=Bgsy66mJ5mYC&pg=RA3-PA382&dq=black-body+emissivity+greenhouse+intitle:Planetary-Science+inauthor:cole&lr=&as_brr=0&ei=LrSOR9OYA4uotAP2ifyPBw&sig=mYO6KVgqvmYvdOu5-_qnQWCuVZk ] Since the emissivity (weighted more in the longer wavelengths where the Earth radiates) is reduced more than than the absorptivity (weighted more in the shorter wavelengths of the Sun's radiation), the equilibrium temperature is higher than the simple black-body calculation estimates, not lower. As a result, the Earth's actual average surface temperature is about 288 K, rather than 279 K.
Global warmingis an increase in this equilibrium temperature due to human-caused additions to the quantity of greenhouse gases in the atmosphere.
In one of the above derivations, the following integral appeared:
There are a number of ways to do this integration; a simple one is given in the appendix of the Planck's law article. This appendix does the integral by contour integration. Consider the function:
Taylor expansionof the sine function, it should be evident that the coefficient of the "k"3 term would be exactly -"J"/6.By expanding both sides in powers of , we see that is minus 6 times the coefficient of of the series expansion of . So, if we can find a closed form for "f"("k"), its Taylor expansionwill give J.
In turn, sin(x) is the imaginary part of eix, so we can restate this as:
To evaluate the integral in this equation we consider the contour integral:
where is the contour from to , then to , then to , then we go to the point , avoiding the pole at by taking a clockwise quarter circle with radius and center . From there we go to , and finally we return to , avoiding the pole at zero by taking a clockwise quarter circle with radius and center zero.
Because there are no poles in the integration contour we have:
We now take the limit . In this limit the contribution from the segment from to tends to zero. Taking together the integrations over the segments from to and from to and using the fact that the integrations over clockwise quarter circles about
simple poles are given by minus times the residues at the poles we find:
The left hand side is the sum of the integral from to and from to . We can rewrite the integrand of the integral on the r.h.s. as follows:
If we now take the imaginary part of both sides of Eq. (1) and take the limit we find:
after using the relation:
Using that the series expansion of is given by:
we see that the coefficient of of the series expansion of is . This then implies that and the result
Wien's displacement law
* 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. 291-294
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