# Black body

File:Blackbody-lg.png
As the temperature decreases, the peak of the black-body radiation curve moves to lower intensities and longer wavelengths. The black-body radiation graph is also compared with the classical model of Rayleigh and Jeans.
File:PlanckianLocus.png
The color (chromaticity) of black-body radiation depends on the temperature of the black body; the locus of such colors, shown here in CIE 1931 x,y space, is known as the Planckian locus.

In physics, a black body is an idealized object that absorbs all electromagnetic radiation that falls on it. No electromagnetic radiation passes through it and none is reflected. Because no light (visible electromagnetic radiation) is reflected or transmitted, the object appears black when it is cold. However, a black body emits a temperature-dependent spectrum of light. This thermal radiation from a black body is termed black-body radiation.[1]

At room temperature, black bodies emit mostly infrared light, but as the temperature increases past a few hundred degrees Celsius, black bodies start to emit visible wavelengths, from red, through orange, yellow, and white before ending up at blue, beyond which the emission includes increasing amounts of ultraviolet.

The term "black body" was introduced by Gustav Kirchhoff in 1860.

Black-body emission gives insight into the thermal equilibrium state of a continuous field. In classical physics, each different Fourier mode in thermal equilibrium should have the same energy, leading to the theory of ultraviolet catastrophe that there would be an infinite amount of energy in any continuous field. Black bodies could test the properties of thermal equilibrium because they emit radiation which is distributed thermally. Studying the laws of the black body historically led to quantum mechanics.

## Explanation

In the laboratory, black-body radiation is approximated by the radiation from a small hole entrance to a large cavity, a hohlraum. (this technique leads to the alternative term cavity radiation) Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped, in which process it is nearly certain to be absorbed. This occurs regardless of the wavelength of the radiation entering (as long as it is small compared to the hole). The hole, then, is a close approximation of a theoretical black body and, if the cavity is heated, the spectrum of the hole's radiation (i.e., the amount of light emitted from the hole at each wavelength) will be continuous, and will not depend on the material in the cavity (compare with emission spectrum). By a theorem proved by Kirchhoff, this curve depends only on the temperature of the cavity walls.[2]

Calculating this curve was a major challenge in theoretical physics during the late nineteenth century. The problem was finally solved in 1901 by Max Planck as Planck's law of black-body radiation.[3] By making changes to Wien's radiation law (not to be confused with Wien's displacement law) consistent with thermodynamics and electromagnetism, he found a mathematical formula fitting the experimental data in a satisfactory way. To find a physical interpretation for this formula, Planck had then to assume that the energy of the oscillators in the cavity was quantized (i.e., integer multiples of some quantity). Einstein built on this idea and proposed the quantization of electromagnetic radiation itself in 1905 to explain the photoelectric effect. These theoretical advances eventually resulted in the superseding of classical electromagnetism by quantum electrodynamics. Today, these quanta are called photons and the black-body cavity may be thought of as containing a gas of photons. In addition, it led to the development of quantum probability distributions, called Fermi-Dirac statistics and Bose-Einstein statistics, each applicable to a different class of particle, which are used in quantum mechanics instead of the classical distributions. See also fermion and boson.

File:Pahoehoe toe.jpg
The temperature of a Pāhoehoe lava flow can be estimated by observing its color. The result agrees well with the measured temperatures of lava flows at about 1,000 to 1,200 °C.

The wavelength at which the radiation is strongest is given by Wien's displacement law, and the overall power emitted per unit area is given by the Stefan-Boltzmann law. So, as temperature increases, the glow color changes from red to yellow to white to blue. Even as the peak wavelength moves into the ultra-violet, enough radiation continues to be emitted in the blue wavelengths that the body will continue to appear blue. It will never become invisible — indeed, the radiation of visible light increases monotonically with temperature.[4]

The radiance or observed intensity is not a function of direction. Therefore a black body is a perfect Lambertian radiator.

Real objects never behave as full-ideal black bodies, and instead the emitted radiation at a given frequency is a fraction of what the ideal emission would be. The emissivity of a material specifies how well a real body radiates energy as compared with a black body. This emissivity depends on factors such as temperature, emission angle, and wavelength. However, it is typical in engineering to assume that a surface's spectral emissivity and absorptivity do not depend on wavelength, so that the emissivity is a constant. This is known as the grey body assumption.

Due to the rapid fall-off of emitted photons with decreasing energy, a black body at room temperature (300 K) with 1 m² of surface area emits a visible photon every thousand years or so, which is negligible for most purposes.

When dealing with non-black surfaces, the deviations from ideal black-body behavior are determined by both the geometrical structure and the chemical composition, and follow Kirchhoff's Law: emissivity equals absorptivity, so that an object that does not absorb all incident light will also emit less radiation than an ideal black body.

File:WMAP 2008.png
WMAP image of the cosmic microwave background radiation anisotropy. It has the most precise thermal emission spectrum known and corresponds to a temperature of 2.725 kelvin (K) with an emission peak at 160.2 GHz.

In astronomy, objects such as stars are frequently regarded as black bodies, though this is often a poor approximation. An almost perfect black-body spectrum is exhibited by the cosmic microwave background radiation. Hawking radiation is the hypothetical black-body radiation emitted by black holes.

### Black body simulators

Although a black body is a theoretical object (i.e. emissivity (e) = 1.0), common applications define a source of infrared radiation as a black body when the object approaches an emissivity of 1.0, (typically e = .99 or better). A source of infrared radiation less than .99 is referred to as a greybody.[5] Applications for black body simulators typically include the testing and calibration of infrared systems and infrared sensor equipment.

Super black is an example of such a material, made from a nickel-phosphorus alloy. More recently, a team of Japanese scientists discovered a material even closer to a black body, based on single-walled carbon nanotubes (SWNTs), which absorbs between 97% and 99% of the wavelengths of the light that hits it.[6]

## Equations governing black bodies

### Planck's law of black-body radiation

Main article: Planck's law
$I(\nu,T)d\nu = \frac{2 h\nu^{3}}{c^2}\frac{1}{e^{\frac{h\nu}{kT}}-1}\, d\nu$

where

### Wien's displacement law

The relationship between the temperature T of a black body, and wavelength $\lambda_{max}$ at which the intensity of the radiation it produces is at a maximum is

• $T \lambda_\mathrm{max} = 2.898... \times 10^6 \ \mathrm{nm \ K}. \,$

The nanometer is a convenient unit of measure for optical wavelengths. Note that 1 nanometer is equivalent to 10−9 meters.

### Stefan–Boltzmann law

This law states that amount of thermal radiations emitted per second per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature. The total energy radiated per unit area per unit time $j^{\star}$ (in watts per square meter) by a black body is related to its temperature T (in kelvins) and the Stefan–Boltzmann constant $\sigma=5.67 x 10^{-8}Wm^{-2}K^{-4}$ as follows:

$j^{\star} = \sigma T^4.\,$

## Radiation emitted by a human body

 229px 284px Much of a person's energy is radiated away in the form of infrared energy. Some materials are transparent to infrared light, while opaque to visible light (note the plastic bag). Other materials are transparent to visible light, while opaque or reflective to the infrared (note the man's glasses).

Black-body laws can be applied to human beings. For example, some of a person's energy is radiated away in the form of electromagnetic radiation, most of which is infrared.

The net power radiated is the difference between the power emitted and the power absorbed:

$P_{net}=P_{emit}-P_{absorb}.$

Applying the Stefan–Boltzmann law,

$P_{net}=A\sigma \epsilon \left( T^4 - T_{0}^4 \right) \,$.

The total surface area of an adult is about 2 m², and the mid- and far-infrared emissivity of skin and most clothing is near unity, as it is for most nonmetallic surfaces.[7][8] Skin temperature is about 33°C,[9] but clothing reduces the surface temperature to about 28°C when the ambient temperature is 20°C.[10] Hence, the net radiative heat loss is about

$P_{net} = 100 \ \mathrm{W} \,$.

The total energy radiated in one day is about 9 MJ (Mega joules), or 2000 kcal (food calories). Basal metabolic rate for a 40-year-old male is about 35 kcal/(m²·h),[11] which is equivalent to 1700 kcal per day assuming the same 2 m² area. However, the mean metabolic rate of sedentary adults is about 50% to 70% greater than their basal rate.[12]

There are other important thermal loss mechanisms, including convection and evaporation. Conduction is negligible since the Nusselt number is much greater than unity. Evaporation (perspiration) is only required if radiation and convection are insufficient to maintain a steady state temperature. Free convection rates are comparable, albeit somewhat lower, than radiative rates.[13] Thus, radiation accounts for about 2/3 of thermal energy loss in cool, still air. Given the approximate nature of many of the assumptions, this can only be taken as a crude estimate. Ambient air motion, causing forced convection, or evaporation reduces the relative importance of radiation as a thermal loss mechanism.

Also, applying Wien's Law to humans, one finds that the peak wavelength of light emitted by a person is

$\lambda_{peak} = \frac{2.898\times 10^6 \ \mathrm{K} \cdot \mathrm{nm}}{305 \ \mathrm{K}} = 9500 \ \mathrm{nm} \,$.

This is why thermal imaging devices designed for human subjects are most sensitive to 7000–14000 nanometers wavelength.

## Temperature relation between a planet and its star

Here is an application of black-body laws to determine the black body temperature of a planet. The surface may be warmer due to the greenhouse effect.[14]

### Factors

File:Erbe.gif
Earth's longwave thermal radiation intensity, from clouds, atmosphere and ground

The temperature of a planet depends on a few factors:

For the inner planets, incident and emitted radiation have the most significant impact on temperature. This derivation is concerned mainly with that.

### Assumptions

If we assume the following:

1. The Sun and the Earth both radiate as spherical black bodies.
2. The Earth is in thermal equilibrium.

then we can derive a formula for the relationship between the Earth's temperature and the Sun's surface temperature.

### Derivation

To begin, we use the Stefan–Boltzmann law to find the total power (energy/second) the Sun is emitting:

The Earth only has an absorbing area equal to a two dimensional circle, rather than the surface of a sphere.
$P_{S emt} = \left( \sigma T_{S}^4 \right) \left( 4 \pi R_{S}^2 \right) \qquad \qquad (1)$
where
$\sigma \,$ is the Stefan–Boltzmann constant,
$T_S \,$ is the surface temperature of the Sun, and
$R_S \,$ is the radius of the Sun.

The Sun emits that power equally in all directions. Because of this, the Earth is hit with only a tiny fraction of it. This is the power from the Sun that the Earth absorbs:

$P_{E abs} = P_{S emt} (1-\alpha) \left( \frac{\pi R_{E}^2}{4 \pi D^2} \right) \qquad \qquad (2)$
where
$R_{E} \,$ is the radius of the Earth and
$D \,$ is the astronomical unit, the distance between the Sun and the Earth.
$\alpha \$ is the albedo of Earth.

Even though the earth only absorbs as a circular area $\pi R^2$, it emits equally in all directions as a sphere:

$P_{E emt} = \left( \sigma T_{E}^4 \right) \left( 4 \pi R_{E}^2 \right) \qquad \qquad (3)$
where $T_{E}$ is the black body temperature of the earth.

Now, our second assumption was that the earth is in thermal equilibrium, so the power absorbed must equal the power emitted:

$P_{E abs} = P_{E emt}\,$
So plug in equations 1, 2, and 3 into this and we get
$\left( \sigma T_{S}^4 \right) \left( 4 \pi R_{S}^2 \right) (1-\alpha) \left( \frac{\pi R_{E}^2}{4 \pi D^2} \right) = \left( \sigma T_{E}^4 \right) \left( 4 \pi R_{E}^2 \right).\,$

Many factors cancel from both sides and this equation can be greatly simplified.

### The result

After canceling of factors, the final result is

 $T_{E} = T_{S}\sqrt{\frac{\sqrt{1-\alpha}R_{S}}{2 D}}$ where $T_E \,$ is the blackbody temperature of the Earth. $T_S \,$ is the surface temperature of the Sun, $R_S \,$ is the radius of the Sun, $D \,$ is the distance between the Sun and the Earth, $\alpha$ is the albedo of the Earth.

In other words, given the assumptions made, the temperature of Earth depends only on the surface temperature of the Sun, the radius of the Sun, the distance between Earth and the Sun and the albedo of Earth.

### Temperature of Earth

If we substitute in the measured values for the Sun,

$T_{S} = 5778 \ \mathrm{K},$
$R_{S} = 6.96 \times 10^8 \ \mathrm{m},$
$D = 1.496 \times 10^{11} \ \mathrm{m},$
$\alpha = 0.367 \$

we'll find the effective temperature of the Earth to be

$T_E = 248.573 \ \mathrm{K}$

This is the black body temperature that would cause the same amount of energy emission, as measured from space, while the surface temperature is higher due to the greenhouse effect.

Estimates of the Earth's average albedo vary in the range 0.3–0.4, resulting in different estimated effective temperatures. Estimates are often based on the solar constant (total insolation power density) rather than the temperature, size, and distance of the sun. For example, using 0.4 for albedo, and an insolation of 1400 Wm-2), one obtains an effective temperature of about 245 K.[15] Similarly using albedo 0.3 and solar constant of 1372 Wm-2), one obtains an effective temperature of 255 K. [16][17]

## Doppler effect for a moving black body

The Doppler effect is the well known phenomenon describing how observed frequencies of light are "shifted" when a light source is moving relative to the observer. If f is the emitted frequency of a monochromatic light source, it will appear to have frequency f' if it is moving relative to the observer :

$f' = f \frac{1}{\sqrt{1-v^2/c^2}} (1 - \frac{v}{c} \cos \theta)$

where v is the velocity of the source in the observer's rest frame, θ is the angle between the velocity vector and the observer-source direction, and c is the speed of light.[18] This is the fully relativistic formula, and can be simplified for the special cases of objects moving directly towards ( θ = π) or away ( θ = 0) from the observer, and for speeds much less than c.

To calculate the spectrum of a moving black body, then, it seems straightforward to simply apply this formula to each frequency of the blackbody spectrum. However, simply scaling each frequency like this is not enough. We also have to account for the finite size of the viewing aperture, because the solid angle receiving the light also undergoes a Lorentz transformation. (We can subsequently allow the aperture to be arbitrarily small, and the source arbitrarily far, but this cannot be ignored at the outset.) When this effect is included, it is found that a black body at temperature T that is receding with velocity v appears to have a spectrum identical to a stationary black body at temperature T' , given by:[19]

$T' = T \frac{1}{\sqrt{1-v^2/c^2}} (1 - \frac{v}{c} \cos \theta)$

For the case of a source moving directly towards or away from the observer, this reduces to

$T' = T \sqrt{\frac{c-v}{c+v}}$

Here v > 0 indicates a receding source, and v < 0 indicates an approaching source.

This is an important effect in astronomy, where the velocities of stars and galaxies can reach significant fractions of c. An example is found in the cosmic microwave background radiation, which exhibits a dipole anisotropy from the Earth's motion relative to this blackbody radiation field.

## References

1. When used as a compound adjective, the term is typically hyphenated, as in "black-body radiation", or combined into one word, as in "blackbody radiation". The hyphenated and one-word forms should not generally be used as nouns.
2. Huang, Kerson (1967). Statistical Mechanics. New York: John Wiley & Sons.
3. Planck, Max (1901). "On the Law of Distribution of Energy in the Normal Spectrum" (HTML). Annalen der Physik 4: 553.
4. Landau, L. D.; E. M. Lifshitz (1996). Statistical Physics, 3rd Edition Part 1, Oxford: Butterworth-Heinemann.
6. Kohei Mizuno, Juntaro Ishii, Hideo Kishida, Yuhei Hayamizu, Satoshi Yasuda, Don N. Futaba, Motoo Yumura, and (March 30, 2009). A black body absorber from vertically aligned single-walled carbon nanotubes.. Proceedings of the National Academy of Sciences.
7. Infrared Services. Emissivity Values for Common Materials. Retrieved on 2007-06-24.
8. Omega Engineering. Emissivity of Common Materials. Retrieved on 2007-06-24.
9. Farzana, Abanty (2001). Temperature of a Healthy Human (Skin Temperature). The Physics Factbook. Retrieved on 2007-06-24.
10. Harris J, Benedict F (1918). "A Biometric Study of Human Basal Metabolism.". Proc Natl Acad Sci USA 4 (12): 370–3. doi:10.1073/pnas.4.12.370. PMID 16576330.
11. Levine, J (2004). "Nonexercise activity thermogenesis (NEAT): environment and biology". Am J Physiol Endocrinol Metab 286: E675–E685. doi:10.1152/ajpendo.00562.2003. PMID 15102614.
12. DrPhysics.com. Heat Transfer and the Human Body. Retrieved on 2007-06-24.
13. Cole, George H. A.; Woolfson, Michael M. (2002). Planetary Science: The Science of Planets Around Stars (1st ed.). Institute of Physics Publishing, 36–37, 380–382. ISBN 0-7503-0815-X.
14. Michael D. Papagiannis (1972). Space physics and space astronomy. Taylor & Francis, 10–11. ISBN 9780677040004.
15. Willem Jozef Meine Martens and Jan Rotmans (1999). Climate Change an Integrated Perspective. Springer, 52–55. ISBN 9780792359968.
16. F. Selsis (2004). "The Prebiotic Atmosphere of the Earth", in Pascale Ehrenfreund et al.: Astrobiology: Future Perspectives. Springer, 279–280. ISBN 9781402025877.
17. The Doppler Effect, T. P. Gill, Logos Press, 1965
18. McKinley, John M., "Relativistic transformations of light power", Am. J. Phys. 47 (7), Jul 1979

### Other textbooks

• Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics, 2nd, W. H. Freeman Company. ISBN 0716710889.
• Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics, 4th, W. H. Freeman. ISBN 0716743450.