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Thursday, May 21, 2015

Black body


From Wikipedia, the free encyclopedia


As the temperature of a black body decreases, its intensity also decreases and its peak moves to longer wavelengths. Shown for comparison is the classical Rayleigh–Jeans law and its ultraviolet catastrophe.

A black body (also, blackbody) is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. A white body is one with a "rough surface [that] reflects all incident rays completely and uniformly in all directions."[1]

A black body in thermal equilibrium (that is, at a constant temperature) emits electromagnetic radiation called black-body radiation. The radiation is emitted according to Planck's law, meaning that it has a spectrum that is determined by the temperature alone (see figure at right), not by the body's shape or composition.

A black body in thermal equilibrium has two notable properties:[2]
  1. It is an ideal emitter: at every frequency, it emits as much energy as – or more energy than – any other body at the same temperature.
  2. It is a diffuse emitter: the energy is radiated isotropically, independent of direction.
An approximate realization of a black surface is a hole in the wall of a large enclosure (see below). Any light entering the hole is reflected indefinitely or absorbed inside and is unlikely to re-emerge, making the hole a nearly perfect absorber. The radiation confined in such an enclosure may or may not be in thermal equilibrium, depending upon the nature of the walls and the other contents of the enclosure.[3][4]

Real materials emit energy at a fraction—called the emissivity—of black-body energy levels. By definition, a black body in thermal equilibrium has an emissivity of ε = 1.0. A source with lower emissivity independent of frequency often is referred to as a gray body.[5][6] Construction of black bodies with emissivity as close to one as possible remains a topic of current interest.[7]

In astronomy, the radiation from stars and planets is sometimes characterized in terms of an effective temperature, the temperature of a black body that would emit the same total flux of electromagnetic energy.

Definition

The idea of a black body originally was introduced by Gustav Kirchhoff in 1860 as follows:

...the supposition that bodies can be imagined which, for infinitely small thicknesses, completely absorb all incident rays, and neither reflect nor transmit any. I shall call such bodies perfectly black, or, more briefly, black bodies.[8]

A more modern definition drops the reference to "infinitely small thicknesses":[9]

An ideal body is now defined, called a blackbody. A blackbody allows all incident radiation to pass into it (no reflected energy) and internally absorbs all the incident radiation (no energy transmitted through the body). This is true for radiation of all wavelengths and for all angles of incidence. Hence the blackbody is a perfect absorber for all incident radiation. [10]

Idealizations

This section describes some concepts developed in connection with black bodies.

An approximate realization of a black body as a tiny hole in an insulated enclosure.

Cavity with a hole

A widely used model of a black surface is a small hole in a cavity with walls that are opaque to radiation.[10]
Radiation incident on the hole will pass into the cavity, and is very unlikely to be re-emitted if the cavity is large. The hole is not quite a perfect black surface — in particular, if the wavelength of the incident radiation is longer than the diameter of the hole, part will be reflected. Similarly, even in perfect thermal equilibrium, the radiation inside a finite-sized cavity will not have an ideal Planck spectrum for wavelengths comparable to or larger than the size of the cavity.[11]

Suppose the cavity is held at a fixed temperature T and the radiation trapped inside the enclosure is at thermal equilibrium with the enclosure. The hole in the enclosure will allow some radiation to escape. If the hole is small, radiation passing in and out of the hole has negligible effect upon the equilibrium of the radiation inside the cavity. This escaping radiation will approximate black-body radiation that exhibits a distribution in energy characteristic of the temperature T and does not depend upon the properties of the cavity or the hole, at least for wavelengths smaller than the size of the hole.[11] See the figure in the Introduction for the spectrum as a function of the frequency of the radiation, which is related to the energy of the radiation by the equation E=hf, with E = energy, h = Planck's constant, f = frequency.

At any given time the radiation in the cavity may not be in thermal equilibrium, but the second law of thermodynamics states that if left undisturbed it will eventually reach equilibrium,[12] although the time it takes to do so may be very long.[13] Typically, equilibrium is reached by continual absorption and emission of radiation by material in the cavity or its walls.[3][4][14][15] Radiation entering the cavity will be "thermalized"; by this mechanism: the energy will be redistributed until the ensemble of photons achieves a Planck distribution. The time taken for thermalization is much faster with condensed matter present than with rarefied matter such as a dilute gas.
At temperatures below billions of Kelvin, direct photon–photon interactions[16] are usually negligible compared to interactions with matter.[17] Photons are an example of an interacting boson gas,[18] and as described by the H-theorem,[19] under very general conditions any interacting boson gas will approach thermal equilibrium.

Transmission, absorption, and reflection

A body's behavior with regard to thermal radiation is characterized by its transmission τ, absorption α, and reflection ρ.

The boundary of a body forms an interface with its surroundings, and this interface may be rough or smooth. A nonreflecting interface separating regions with different refractive indices must be rough, because the laws of reflection and refraction governed by the Fresnel equations for a smooth interface require a reflected ray when the refractive indices of the material and its surroundings differ.[20] A few idealized types of behavior are given particular names:

An opaque body is one that transmits none of the radiation that reaches it, although some may be reflected.[21][22]
That is, τ=0 and α+ρ=1

A transparent body is one that transmits all the radiation that reaches it. That is, τ=1 and α=ρ=0.

A gray body is one where α, ρ and τ are uniform for all wavelengths. This term also is used to mean a body for which α is temperature and wavelength independent.

A white body is one for which all incident radiation is reflected uniformly in all directions: τ=0, α=0, and ρ=1.

For a black body, τ=0, α=1, and ρ=0. Planck offers a theoretical model for perfectly black bodies, which he noted do not exist in nature: besides their opaque interior, they have interfaces that are perfectly transmitting and non-reflective.[23]

Kirchhoff's perfect black bodies

Kirchhoff in 1860 introduced the theoretical concept of a perfect black body with a completely absorbing surface layer of infinitely small thickness, but Planck noted some severe restrictions upon this idea. Planck noted three requirements upon a black body: the body must (i) allow radiation to enter but not reflect; (ii) possess a minimum thickness adequate to absorb the incident radiation and prevent its re-emission; (iii) satisfy severe limitations upon scattering to prevent radiation from entering and bouncing back out. As a consequence, Kirchhoff's perfect black bodies that absorb all the radiation that falls on them cannot be realized in an infinitely thin surface layer, and impose conditions upon scattering of the light within the black body that are difficult to satisfy.[24][25]

Realizations

A realization of a black body is a real world, physical embodiment. Here are a few.

Cavity with a hole

In 1898, Otto Lummer and Ferdinand Kurlbaum published an account of their cavity radiation source.[26] Their design has been used largely unchanged for radiation measurements to the present day. It was a hole in the wall of a platinum box, divided by diaphragms, with its interior blackened with iron oxide. It was an important ingredient for the progressively improved measurements that led to the discovery of Planck's law.[27][28] A version described in 1901 had its interior blackened with a mixture of chromium, nickel, and cobalt oxides.[29]

Near-black materials

There is interest in blackbody-like materials for camouflage and radar-absorbent materials for radar invisibility.[30][31] They also have application as solar energy collectors, and infrared thermal detectors. As a perfect emitter of radiation, a hot material with black body behavior would create an efficient infrared heater, particularly in space or in a vacuum where convective heating is unavailable.[32] They are also useful in telescopes and cameras as anti-reflection surfaces to reduce stray light, and to gather information about objects in high-contrast areas (for example, observation of planets in orbit around their stars), where blackbody-like materials absorb light that comes from the wrong sources.

It has long been known that a lamp-black coating will make a body nearly black. An improvement on lamp-black is found in manufactured carbon nanotubes. Nano-porous materials can achieve refractive indices nearly that of vacuum, in one case obtaining average reflectance of 0.045%.[7][33] In 2009, a team of Japanese scientists created a material called nanoblack which is close to an ideal black body, based on vertically aligned single-walled carbon nanotubes. This absorbs between 98% and 99% of the incoming light in the spectral range from the ultra-violet to the far-infrared regions.[32]

Another example of a nearly perfect black material is super black, made by chemically etching a nickel–phosphorus alloy.[34]

Stars and planets

An idealized view of the cross-section of a star. The photosphere contains photons of light nearly in thermal equilibrium, and some escape into space as near-black-body radiation.

A star or planet often is modeled as a black body, and electromagnetic radiation emitted from these bodies as black-body radiation. The figure shows a highly schematic cross-section to illustrate the idea. The photosphere of the star, where the emitted light is generated, is idealized as a layer within which the photons of light interact with the material in the photosphere and achieve a common temperature T that is maintained over a long period of time.
Some photons escape and are emitted into space, but the energy they carry away is replaced by energy from within the star, so that the temperature of the photosphere is nearly steady. Changes in the core lead to changes in the supply of energy to the photosphere, but such changes are slow on the time scale of interest here. Assuming these circumstances can be realized, the outer layer of the star is somewhat analogous to the example of an enclosure with a small hole in it, with the hole replaced by the limited transmission into space at the outside of the photosphere.
With all these assumptions in place, the star emits black-body radiation at the temperature of the photosphere.[35]

Effective temperature of a black body compared with the B-V and U-B color index of main sequence and super giant stars in what is called a color-color diagram.[36]

Using this model the effective temperature of stars is estimated, defined as the temperature of a black body that yields the same surface flux of energy as the star. If a star were a black body, the same effective temperature would result from any region of the spectrum. For example, comparisons in the B (blue) or V (visible) range lead to the so-called B-V color index, which increases the redder the star,[37] with the Sun having an index of +0.648 ± 0.006.[38] Combining the U (ultraviolet) and the B indices leads to the U-B index, which becomes more negative the hotter the star and the more the UV radiation. Assuming the Sun is a type G2 V star, its U-B index is +0.12.[39] The two indices for two types of stars are compared in the figure with the effective surface temperature of the stars assuming they are black bodies. It can be seen that there is only a rough correlation. For example, for a given B-V index from the blue-visible region of the spectrum., the curves for both types of star lie below the corresponding black-body U-B index that includes the ultraviolet spectrum, showing that both types of star emit less ultraviolet light than a black body with the same B-V index. It is perhaps surprising that they fit a black body curve as well as they do, considering that stars have greatly different temperatures at different depths.[40] For example, the Sun has an effective temperature of 5780 K,[41] which can be compared to the temperature of the photosphere of the Sun (the region generating the light), which ranges from about 5000 K at its outer boundary with the chromosphere to about 9500 K at its inner boundary with the convection zone approximately 500 km (310 mi) deep.[42]

Black holes

A black hole is a region of spacetime from which nothing escapes. Around a black hole there is a mathematically defined surface called an event horizon that marks the point of no return. It is called "black" because it absorbs all the light that hits the horizon, reflecting nothing, making it almost an ideal black body[43] (radiation with a wavelength equal to or larger than the radius of the hole may not be absorbed, so black holes are not perfect black bodies).[44] Physicists believe that to an outside observer, black holes have a non-zero temperature and emit radiation with a nearly perfect black-body spectrum, ultimately evaporating.[45] The mechanism for this emission is related to vacuum fluctuations in which a virtual pair of particles is separated by the gravity of the hole, one member being sucked into the hole, and the other being emitted.[46] The energy distribution of emission is described by Planck's law with a temperature T:
T=\frac {\hbar c^3}{8\pi Gk_BM} \ ,
where c is the speed of light, ℏ is the reduced Planck constant, kB is Boltzmann's constant, G is the gravitational constant and M is the mass of the black hole.[47] These predictions have not yet been tested either observationally or experimentally.[48]

Cosmic microwave background radiation

The big bang theory is based upon the cosmological principle, which states that on large scales the Universe is homogeneous and isotropic. According to theory, the Universe approximately a second after its formation was a near-ideal black body in thermal equilibrium at a temperature above 1010 K. The temperature decreased as the Universe expanded and the matter and radiation in it cooled. The cosmic microwave background radiation observed today is "the most perfect black body ever measured in nature".[49] It has a nearly ideal Planck spectrum at a temperature of about 2.7K. It departs from the perfect isotropy of true black-body radiation by an observed anisotropy that varies with angle on the sky only to about one part in 100,000.

Radiative cooling

The integration of Planck's law over all frequencies provides the total energy per unit of time per unit of surface area radiated by a black body maintained at a temperature T, and is known as the Stefan–Boltzmann law:
P/A = \sigma T^4 \ ,
where σ is the Stefan–Boltzmann constant, σ ≈ 5.67 × 10−8 W/(m2K4).[50] To remain in thermal equilibrium at constant temperature T, the black body must absorb or internally generate this amount of power P over the given area A.

The cooling of a body due to thermal radiation is often approximated using the Stefan–Boltzmann law supplemented with a "gray body" emissivity ε ≤ 1 (P/A = εσT4). The rate of decrease of the temperature of the emitting body can be estimated from the power radiated and the body's heat capacity.[51] This approach is a simplification that ignores details of the mechanisms behind heat redistribution (which may include changing composition, phase transitions or restructuring of the body) that occur within the body while it cools, and assumes that at each moment in time the body is characterized by a single temperature. It also ignores other possible complications, such as changes in the emissivity with temperature,[52][53] and the role of other accompanying forms of energy emission, for example, emission of particles like neutrinos.[54]

If a hot emitting body is assumed to follow the Stefan–Boltzmann law and its power emission P and temperature T is known, this law can be used to estimate the dimensions of the emitting object, because the total emitted power is proportional to the area of the emitting surface. In this way it was found that X-ray bursts observed by astronomers originated in neutron stars with a radius of about 10 km, rather than black holes as originally conjectured.[55] It should be noted that an accurate estimate of size requires some knowledge of the emissivity, particularly its spectral and angular dependence.[56]
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CO2 Emissions, Levels, and a Simple Moldel of AGW

Some 150 years ago, atmospheric CO2 level was about 280 PPM.  Now it is 400PPM, the global temperature is measured around 1 degree C higher.  This is not mere observation; it follows from (a version of) the Arrhenius law (http://www.ams.org/notices/201010/rtx101001278p.pdf)

delta T = ~3 * ln(C(current)/C(initial))

where delta T is the temperature rise, ln is the natural logarithm (base e), C(current) and C(initial) the 400 and 280 PPM CO2 levels just mention, and ~3 the constant needed to convert the ratio to delta T.  Plugging the 400 and 280 PPM measurement into this equation does indeed a yield of 1 degree C.

If one looks at the data for CO2 emissions, levels, and the first derivative of those levels:













Figure 1 -- CO2 emissions 1900 - 2010 (http://www.epa.gov/climatechange/ghgemissions/global.html)














Figure 2 -- CO2 detailed level increments 1959 - 2014 (http://www.epa.gov/climatechange/ghgemissions/global.html)














Figure 3 -- First derivative of CO2 detailed level increments 1960 - 2014 (http://www.epa.gov/climatechange/ghgemissions/global.html),

we see that, from the second chart, that from around 1960 - 2014, the average CO2 rise is about 1.5 PPM per years, somewhat greater from the first ~100 years), and that the first derivative is almost exactly 0.  This means CO2 has been rising in a perfectly straight line for over the last 60+ years.

If we project this increase out to 2100, CO2 levels will be about 529 PPM, and the resulting rise in temperature will be = 3 * ln(529/400) = 0.84 C.  However, most likely CO2 emissions will scale back dramatically during this century; if the levels are only 0.5 * 129 + 400 = 465 PPM, the resulting T will equal 0.45 C.  On the other hand, should yearly increment levels increase to 2 PPM (highly unlikely), the temperature should be about 1.5 C.

Given that a significant increase is highly unlikely, we are probably looking at a temperature increase somewhere between 0.5 and 1.0 degrees C.  Incidentally, I am assuming the same degree of methane and albedo changes as part of the temperature record (i.e., no "tipping points"), so those don't affect the results.  Frankly, for such small changes in Earth's temperature (even a total of 2.5K out of 288K is < 1%, the assumption of no dramatic changes is reasonable.
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Stefan–Boltzmann law


From Wikipedia, the free encyclopedia


Graph of a function of total emitted energy of a black body j^{\star} proportional to its thermodynamic temperature T\,. In blue is a total energy according to the Wien approximation, j^{\star}_{W} = j^{\star} / \zeta(4) \approx 0.924 \, \sigma T^{4} \!\,

The Stefan–Boltzmann law, also known as Stefan's law, describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant exitance or emissive power), j^{\star}, is directly proportional to the fourth power of the black body's thermodynamic temperature T:
j^{\star} = \sigma T^{4}.
The constant of proportionality σ, called the Stefan–Boltzmann constant or Stefan's constant, derives from other known constants of nature. The value of the constant is
\sigma=\frac{2\pi^5 k^4}{15c^2h^3}= 5.670373 \times 10^{-8}\, \mathrm{W\, m^{-2}K^{-4}},
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 is 5.67 W/m2, at 1000 K 56,700 W/m2, etc. The radiance (watts per square metre per steradian) is given by
L = \frac{j^{\star}}\pi = \frac\sigma\pi T^{4}.
A body that does not absorb all incident radiation (sometimes known as a grey body) emits less total energy than a black body and is characterized by an emissivity, \varepsilon < 1:
j^{\star} = \varepsilon\sigma T^{4}.
The irradiance j^{\star} has dimensions of energy flux (energy per time per area), and the SI units of measure are joules per second per square metre, or equivalently, watts per square metre. The SI unit for absolute temperature T is the kelvin. \varepsilon is the emissivity of the grey body; if it is a perfect blackbody, \varepsilon=1. In the still more general (and realistic) case, the emissivity depends on the wavelength, \varepsilon=\varepsilon(\lambda).

To find the total power radiated from an object, multiply by its surface area, A:
P= A j^{\star} = A \varepsilon\sigma T^{4}.
Metamaterials may be designed to exceed the Stefan–Boltzmann law.[1]

History

The law was deduced by Jožef Stefan (1835–1893) in 1879 on the basis of experimental measurements made by John Tyndall and was derived from theoretical considerations, using thermodynamics, by Ludwig Boltzmann (1844–1906) in 1884. Boltzmann considered a certain ideal heat engine with light as a working matter instead of gas. The law is highly accurate only for ideal black objects, the perfect radiators, called black bodies; it works as a good approximation for most "grey" bodies. Stefan published this law 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.

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 (1854–1904) that the energy flux density from the Sun is 29 times greater than the energy flux density of a certain warmed metal lamella (a thin plate). 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 than Soret's value, 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 the lamella, so Stefan got a value of 5430 °C or 5700 K (the modern value is 5778 K[2]). 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. Pouillet also took just half the value of the Sun's correct energy flux.

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.[3] So:
L = 4 \pi R^2 \sigma T_{e}^4
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:
\frac{R}{R_\odot} \approx \left ( \frac{T_\odot}{T} \right )^{2} \cdot \sqrt{\frac{L}{L_\odot}}
where R_\odot, 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 temperature of the Earth TE by equating the energy received from the Sun and the energy radiated by the Earth, under the black-body approximation. The amount of power, ES, emitted by the Sun is given by:
E_S = 4\pi r_S^2 \sigma T_S^4
At Earth, this energy is passing through a sphere with a radius of a0, the distance between the Earth and the Sun, and the energy passing through each square metre of the sphere is given by
E_{a_0} = \frac{E_S}{4\pi a_0^2}
The Earth has a radius of rE, and therefore has a cross-section of \pi r_E^2. The amount of solar power absorbed by the Earth is thus given by:
E_{abs} = \pi r_E^2 \times E_{a_0} :
Assuming the exchange is in a steady state, the amount of energy emitted by Earth must equal the amount absorbed, and so:
\begin{align} 4\pi r_E^2 \sigma T_E^4 &= \pi r_E^2 \times E_{a_0} \\ &= \pi r_E^2 \times \frac{4\pi r_S^2\sigma T_S^4}{4\pi a_0^2} \\ \end{align}
TE can then be found:
\begin{align} T_E^4 &= \frac{r_S^2 T_S^4}{4 a_0^2} \\ T_E &= T_S \times \sqrt\frac{r_S}{2 a_0} \\ & = 5780 \; {\rm K} \times \sqrt{696 \times 10^{6} \; {\rm m} \over 2 \times 149.598 \times 10^{9} \; {\rm m} } \\ & \approx 279 \; {\rm K} \end{align}
where TS is the temperature of the Sun, rS the radius of the Sun, and a0 is the distance between the Earth and the Sun. This gives an effective temperature of 6 °C on the surface of the Earth, assuming that it perfectly absorbs all emission falling on it and has no atmosphere.

The Earth has an albedo of 0.3, meaning that 30% of the solar radiation that hits the planet gets scattered back into space without absorption. The effect of albedo on temperature can be approximated by assuming that the energy absorbed is multiplied by 0.7, but that the planet still radiates as a black body (the latter by definition of effective temperature, which is what we are calculating). This approximation reduces the temperature by a factor of 0.71/4, giving 255 K (−18 °C).[4][5]

However, long-wave radiation from the surface of the earth is partially absorbed and re-radiated back down by greenhouse gases, namely water vapor, carbon dioxide and methane.[6][7] Since the emissivity with greenhouse effect (weighted more in the longer wavelengths where the Earth radiates) is reduced more than the absorptivity (weighted more in the shorter wavelengths of the Sun's radiation) is reduced, the equilibrium temperature is higher than the simple black-body calculation estimates. As a result, the Earth's actual average surface temperature is about 288 K (15 °C), which is higher than the 255 K effective temperature, and even higher than the 279 K temperature that a black body would have.

Origination

Thermodynamic derivation of the energy density

[8]
The fact that the energy density of the box containing radiation is proportional to T^{4} can be derived using thermodynamics. It follows from the Maxwell stress tensor of classical electrodynamics that the radiation pressure p is related to the internal energy density u:
p = \frac{u}{3}.
From the fundamental thermodynamic relation
dU = T dS - p dV ,
we obtain the following expression, after dividing by dV and fixing T  :
\left(\frac{\partial U}{\partial V}\right)_{T} = T \left(\frac{\partial S}{\partial V}\right)_{T} - p = T \left(\frac{\partial p}{\partial T}\right)_{V} - p .
The last equality comes from the following Maxwell relation:
\left(\frac{\partial S}{\partial V}\right)_{T} = \left(\frac{\partial p}{\partial T}\right)_{V} .
From the definition of energy density it follows that
u = \frac{U}{V}
and[why?]
\left(\frac{\partial U}{\partial V}\right)_{T} = u .
Now, the equality
\left(\frac{\partial U}{\partial V}\right)_{T} = T \left(\frac{\partial p}{\partial T}\right)_{V} - p ,
after substitution of \left(\frac{\partial U}{\partial V}\right)_{T} and p for the corresponding expressions, can be written as
u = \frac{T}{3} \left(\frac{\partial u}{\partial T}\right)_{V} - \frac{u}{3} .
Since the partial derivative \left(\frac{\partial u}{\partial T}\right)_{V} can be expressed as a relationship between only u and T (if one isolates it on one side of the equality), the partial derivative can be replaced by the ordinary derivative. After separating the differentials the equality becomes
\frac{du}{4u} = \frac{dT}{T} ,
which leads immediately to u = A T^4 , with A as some constant of integration.

Stefan–Boltzmann's law in n-dimensional space

It can be shown that the radiation pressure in n-dimensional space is given by
P=\frac{u}{n}[citation needed]
So in n-dimensional space,[why?]
T dS= (n+1)P dV + n V dP\,
So,[why?]
\frac{1}{P}\frac{dP}{dT}=\frac{(n+1)}{T}
yielding
P \propto T^{n+1}
or
u \propto T^{n+1}
implying
\frac{dQ}{dt} \propto T^{n+1}
The same result is obtained as the integral over frequency of Planck's law for n-dimensional space, albeit with a different value for the Stefan-Boltzmann constant at each dimension. In general the constant is
\sigma=\frac{1}{p(n)} \frac{\pi^{\frac{n}{2}}}{\Gamma(1+\frac{n}{2})} \frac{1}{c^{n-1}} \frac{n(n-1)}{h^{n}} k^{(n+1)} \Gamma(n+1) \zeta(n+1)[citation needed]
where \zeta(x) is Riemann's zeta function and p(n) is a certain function of n, with p(3)=4.

Derivation from Planck's law

The law can be derived by considering a small flat black body surface 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 Planck's law :
I(\nu,T) =\frac{2 h\nu^{3}}{c^2}\frac{1}{ e^{\frac{h\nu}{kT}}-1}.
where
The quantity I(\nu,T) ~A ~d\nu ~d\Omega is the power radiated by a surface of area A through a solid angle dΩ in the frequency range between ν and ν + dν.

The Stefan–Boltzmann law gives the power emitted per unit area of the emitting body,
\frac{P}{A} = \int_0^\infty I(\nu,T) d\nu \int d\Omega \,
To derive the Stefan–Boltzmann law, we must integrate Ω over the half-sphere and integrate ν from 0 to ∞. Furthermore, because black bodies are Lambertian (i.e. they obey 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.
\begin{align} \frac{P}{A} & = \int_0^\infty I(\nu,T) \, d\nu \int_0^{2\pi} \, d\theta \int_0^{\pi/2} \cos \phi \sin \phi \, d\phi \\ & = \pi \int_0^\infty I(\nu,T) \, d\nu \end{align}
Then we plug in for I:
\frac{P}{A} = \frac{2 \pi h}{c^2} \int_0^\infty \frac{\nu^3}{ e^{\frac{h\nu}{kT}}-1} d\nu \,
To do this integral, do a substitution,
u = \frac{h \nu}{k T} \,

du = \frac{h}{k T} \, d\nu
which gives:
\frac{P}{A} = \frac{2 \pi h }{c^2} \left(\frac{k T}{h} \right)^4 \int_0^\infty \frac{u^3}{ e^u - 1} \, du.
The integral on the right can be done in a number of ways (one is included in this article's appendix) – its answer is \frac{\pi^4}{15} , giving the result that, for a perfect blackbody surface:
j^\star = \sigma T^4 ~, ~~ \sigma = \frac{2 \pi^5 k^4 }{15 c^2 h^3} = \frac{\pi^2 k^4}{60 \hbar^3 c^2}.
Finally, this proof started out only considering a small flat surface. However, any differentiable surface 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 law extends to radiation from non-convex bodies by using the fact that the convex hull of a black body radiates as though it were itself a black body.

Appendix

In one of the above derivations, the following integral appeared:
J=\int_0^\infty \frac{x^{3}}{\exp\left(x\right)-1} \, dx = \Gamma(4)\,\mathrm{Li}_4(1) = 6\,\mathrm{Li}_4(1) = 6 \zeta(4)
where \mathrm{Li}_s(z) is the polylogarithm function and \zeta(z) is the Riemann zeta function. If the polylogarithm function and the Riemann zeta function are not available for calculation, 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:
f(k) = \int_0^\infty \frac{\sin\left(kx\right)}{\exp\left(x\right)-1} \, dx.
Using the Taylor expansion of the sine function, it should be evident that the coefficient of the k3 term would be exactly -J/6. By expanding both sides in powers of k, we see that J is minus 6 times the coefficient of k^3 of the series expansion of f(k). So, if we can find a closed form for f(k), its Taylor expansion will give J.
In turn, sin(x) is the imaginary part of eix, so we can restate this as:
f(k)=\lim_{\varepsilon\rightarrow 0}~\text{Im}~\int_\varepsilon^\infty \frac{\exp\left(ikx\right)}{\exp\left(x\right)-1} \, dx.
To evaluate the integral in this equation we consider the contour integral:
\oint_{C(\varepsilon, R)}\frac{\exp\left(ikz\right)}{\exp\left(z\right)-1} \, dz
where C(\varepsilon,R) is the contour from \varepsilon to R, then to R+2\pi i, then to \varepsilon+2\pi i, then we go to the point 2\pi i - \varepsilon i, avoiding the pole at 2\pi i by taking a clockwise quarter circle with radius \varepsilon and center 2\pi i. From there we go to \varepsilon i, and finally we return to \varepsilon, avoiding the pole at zero by taking a clockwise quarter circle with radius \varepsilon and center zero.

Integration contour

Because there are no poles in the integration contour we have:
\oint_{C(\varepsilon, R)}\frac{\exp\left(ikz\right)}{\exp\left(z\right)-1} \, dz = 0.
We now take the limit R\rightarrow\infty. In this limit the contribution from the segment from R to R+2\pi i tends to zero. Taking together the integrations over the segments from \varepsilon to R and from R+2\pi i to \varepsilon+2\pi i and using the fact that the integrations over clockwise quarter circles withradius \varepsilon about simple poles are given up to order \varepsilon by minus \textstyle \frac{i \pi}{2} times the residues at the poles we find:
\left[1-\exp\left(-2\pi k\right) \right]\int_\varepsilon^\infty \frac{\exp\left(ikx\right)}{\exp\left(x\right)-1} \, dx = i \int_\varepsilon^{2\pi-\varepsilon} \frac{\exp\left(-ky\right)}{\exp\left(iy\right)-1} \, dy + i\frac{\pi}{2}\left[1 + \exp \left(-2\pi k\right)\right] + \mathcal{O} \left(\varepsilon\right) \qquad \text{ (1)}
The left hand side is the sum of the integral from \varepsilon to R and from R+2 \pi i to 2 \pi i + \varepsilon. We can rewrite the integrand of the integral on the r.h.s. as follows:
\frac{1}{\exp\left(iy\right)-1} = \frac{\exp\left(-i\frac{y}{2}\right)}{\exp \left(i \frac{y}{2}\right) - \exp\left(-i\frac{y}{2}\right)} = \frac{1}{2i} \frac{\exp\left(-i\frac{y}{2}\right)}{\sin\left(\frac{y}{2}\right)}
If we now take the imaginary part of both sides of Eq. (1) and take the limit \varepsilon\rightarrow 0 we find:
f(k) = -\frac{1}{2k} + \frac{\pi}{2}\coth\left(\pi k\right)
after using the relation:
\coth\left(x\right) = \frac{1+\exp\left( -2x\right)}{1 - \exp\left( -2x \right)}.
Using that the series expansion of \coth(x) is given by:
\coth(x)= \frac{1}{x}+\frac{1}{3}x-\frac{1}{45}x^{3} + \cdots
we see that the coefficient of k^3 of the series expansion of f(k) is \textstyle -\frac{\pi^4}{90}. This then implies that \textstyle J = \frac{\pi^4}{15} and the result
j^\star = \frac{2\pi^5 k^4}{15 h^3 c^2} T^4
follows.
at
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