# Black body

File:Blackbodygraph.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 that preceded it.

In physics, a black body is an object that absorbs all electromagnetic radiation that falls onto it. No radiation passes through it and none is reflected. Despite the name, black bodies are not actually black as they radiate energy as well. How much electromagnetic radiation they give off just depends on their temperature. Black bodies below around 700K produce very little radiation at visible wavelengths and appear black. Black bodies above this temperature however, start to produce radiation at visible wavelengths starting at red, going through orange, yellow and white before ending up at blue as the temperature increases.

The term "black body" was introduced by Gustav Kirchhoff in 1862. The light emitted by a black body is called black-body radiationTemplate:Fn.

## Details

In the laboratory, the closest thing to black-body radiation is the radiation from a small hole entrance to a larger cavity. Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped and is almost certain to be absorbed by the walls in the process, regardless of what they are made of or the wavelength of the radiation (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.

Calculating this curve was a long-standing challenge that was solved in 1900 by Max Planck as Planck's law of black-body radiation. To calculate the curve, Planck had to assume that electromagnetic radiation could propagate only in discrete packets, or quanta. This idea was later used by Einstein in 1905 to explain the photoelectric effect. These theoretical advances eventually resulted in the replacement of classical electromagnetism by quantum mechanics. Today, these quanta are called photons.

It was later realized that the observed spectrum of black-body radiation can not be explained by classical electromagnetism and statistical mechanics: these predict infinite brightness at high frequencies (i.e. low wavelength), a prediction often called the ultraviolet catastrophe.

File:Pahoehoe toe.jpg
The temperature of a Pahoehoe lava flow can be approximated by merely observing its colour. The result agrees nicely 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 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.

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, a typical engineering assumption is 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.

File:Incandescent flashlight spectrum.gif
The spectrum of an incandescent bulb in a typical flashlight immediately reveals the mechanism of incandescent lighting to be blackbody radiation. Here, the filament temperature appears to be about 3200 kelvins due to a peak emittance of around 630 nanometers.

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.

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 black-body radiation emitted by black holes.

## Equations governing black bodies

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

Main article: Planck's law of black-body radiation
$\displaystyle I(\nu) = \frac{2h\nu^{3}}{c^2}\frac{1}{\exp({h\nu}/kT)-1}$

where

• $\displaystyle I(\nu)d\nu \,$ is the amount of energy per unit surface per unit time per unit solid angle emitted in the frequency range between ν and ν+dν;
• $\displaystyle T \,$ is the temperature of the black body;
• $\displaystyle h \,$ is Planck's constant;
• $\displaystyle c \,$ is the speed of light; and
• $\displaystyle k \,$ is Boltzmann's constant.

### Wien's displacement law

Main article: Wien's displacement law

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

$\displaystyle T \lambda_\mbox{max} = 0.002\ 898... K\cdot m\ \mathbf{(kelvin\ meters)}.\,$

### Stefan-Boltzmann law

Main article: Stefan-Boltzmann law

The total energy radiated per unit area per unit time ($\displaystyle j^{\star}$ ) by a black body is related to its temperature T and the Stefan-Boltzmann constant $\displaystyle \sigma$ as follows:

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

## Temperature relation between a planet and its star

Here is an application of black-body laws. It is a rough derivation that gives an order of magnitude answer. See p380-382 of Planetary Science, for further discussion.

### Assumptions

The surface temperature of a planet depends on a few factors:

• Incident radiation (from the sun, for example)
• The Albedo effect (the fraction of light a planet reflects)
• The greenhouse effect (for planets with an atmosphere)
• Energy generated interally by a planet itself (This is more important for planets like Jupiter)

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

If we assume the following:

1. The Sun and the Earth both radiate as spherical black bodies in thermal equilibrium with themselves.
2. The Earth absorbs all the solar energy that it intercepts from the Sun.

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

### Derivation

To begin, let's use the Stefan-Boltzmann law to find the total power (energy/second) the Sun is emitting:

$\displaystyle P_{S emt} = \left( \sigma T_{S}^4 \right) \left( 4 \pi R_{S}^2 \right) \qquad \qquad (1)$
where
$\displaystyle \sigma \,$ is the Stefan-boltzmann constant,
$\displaystyle T_S \,$ is the surface temperature of the Sun, and
$\displaystyle R_S \,$ 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:

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

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

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

Now, in the first assumption the earth is in thermal equilibrium, so the power absorbed must equal the power emitted:

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

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

### The result

After canceling of factors, the final result is

 $\displaystyle \frac{T_{S}^4 R_{S}^2}{4 D^2} = T_{E}^4$ where $\displaystyle T_S \,$ is the surface temperature of the Sun, $\displaystyle R_S \,$ is the radius of the Sun, $\displaystyle D \,$ is the distance between the Sun and the Earth, and $\displaystyle T_E \,$ is the average surface temperature of the Earth.

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

### Temperature of the Sun

If we plug in the measured values for Earth,

$\displaystyle T_{E} \approx 14 \ \mathbf{{}^\circ C} = 287 \ \mathbf{K},$
$\displaystyle R_{S} = 6.96 \times 10^8 \ \mathbf{m},$
$\displaystyle D = 1.5 \times 10^{11} \ \mathbf{m},$

we'll find the surface temperature of the Sun to be

$\displaystyle T_{S} \approx 5960 \ \mathbf{K}.$

This isn't too far off from a more accurate measure of 5780 kelvins.

### Temperature of the Sun using Mars data

If we plug in the measured values for Mars instead of Earth,

$\displaystyle T_{mars} \approx 210 \ \mathbf{K},$
$\displaystyle D = 2.28 \times 10^{11} \ \mathbf{m},$

we'll find the surface temperature of the Sun to be

$\displaystyle T_{S} \approx 5380 \ \mathbf{K}.$

## Footnotes

Template:Fnb 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, however.

## References

• Cole, George H. A.; Woolfson, Michael M. (2002). Planetary Science: The Science of Planets Around Stars (1st ed.), Institute of Physics Publishing. ISBN 075030815X.
• Kroemer, Herbert; Kittle, Charles (1980). Thermal Physics (2nd ed.), W. H. Freeman Company. ISBN 0716710889.
• Planck, Max, "On the Law of Distribution of Energy in the Normal Spectrum". Annalen der Physik, vol. 4, p. 553 ff (1901).
• Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.), W. H. Freeman. ISBN 0716743450.