Black-body radiation is the thermal electromagnetic radiation emitted from a body in thermodynamic equilibrium with its environment. A black body is an idealized opaque and non-reflective body. The radiation emitted is a continuous spectrum over all possible radiation wavelengths that depends only on the body's temperature. A perfectly-insulated enclosure which is in thermal equilibrium internally contains black-body radiation and will emit it through a hole made in its wall, provided the hole is small enough to have a negligible effect upon the equilibrium. The thermal radiation spontaneously emitted by many ordinary objects can be approximated as black-body radiation. Of particular importance, although planets and stars (including the Earth and Sun) are neither in thermal equilibrium with their surroundings nor perfect black bodies, black-body radiation is still a good first approximation for the energy they emit. A black body at room temperature (23 °C (296 K; 73 °F)) radiates mostly in the infrared spectrum, which cannot be perceived by the human eye, but can be sensed by some reptiles. As the object increases in temperature to about 500 °C (773 K; 932 °F), the emission spectrum gets stronger and extends into the human visual range, and the object appears dull red. As its temperature increases further, it emits more and more orange, yellow, green, and then blue light (and ultimately beyond violet, ultraviolet).
The term black body was introduced by Gustav Kirchhoff in 1860. Black-body radiation is also called thermal radiation, cavity radiation, complete radiation or temperature radiation.
Theory
Spectrum
Black-body radiation has a characteristic, continuous frequency spectrum that depends only on the body's temperature, called the Planck spectrum or Planck's law. The spectrum is peaked at a characteristic frequency that shifts to higher frequencies with increasing temperature, and at room temperature most of the emission is in the infrared region of the electromagnetic spectrum. As the temperature increases past about 500 degrees Celsius, black bodies start to emit significant amounts of visible light. Viewed in the dark by the human eye, the first faint glow appears as a "ghostly" grey (the visible light is actually red, but low intensity light activates only the eye's grey-level sensors). With rising temperature, the glow becomes visible even when there is some background surrounding light: first as a dull red, then yellow, and eventually a "dazzling bluish-white" as the temperature rises. When the body appears white, it is emitting a substantial fraction of its energy as ultraviolet radiation. The Sun, with an effective temperature of approximately 5800 K, is an approximate black body with an emission spectrum peaked in the central, yellow-green part of the visible spectrum, but with significant power in the ultraviolet as well. Black-body radiation provides insight into the thermodynamic equilibrium state of cavity radiation.
Black body
Most types of matter emit electromagnetic radiation while at a temperature above absolute zero. The radiation represents a conversion of a body's internal energy into electromagnetic energy, and is therefore called thermal radiation. It is a spontaneous process of radiative distribution of entropy.
Conversely, all normal matter absorbs electromagnetic radiation to some degree. An object that absorbs all radiation falling on it, at all wavelengths, is called a black body. When a black body is at a uniform temperature, its emission has a characteristic frequency distribution that depends on the temperature. Its emission is called black-body radiation. The concept of the black body is an idealization, as perfect black bodies do not exist in nature. Graphite and lamp black have emissivities greater than 0.95, and are good approximations to an ideal black material. Experimentally, black-body radiation may be established best as the steady state equilibrium radiation inside a cavity within a rigid body, at a uniform temperature, that is entirely opaque and is only partly reflective. A closed box with walls of graphite at a constant temperature with a small hole on one side produces a good approximation to ideal black-body radiation emanating from the opening. Black-body radiation has the unique absolutely stable distribution of radiative intensity that can persist in thermodynamic equilibrium inside a cavity. In equilibrium, for each frequency, the intensity of radiation which is emitted and reflected from a body relative to other frequencies (that is, the net amount of radiation leaving its surface, called the spectral radiance) is determined solely by the equilibrium temperature and does not depend upon the shape, material or structure of the body. For a black body (a perfect absorber) there is no reflected radiation, and so the spectral radiance is entirely due to emission. In addition, a black body is a diffuse emitter (its emission is independent of direction). Black-body radiation becomes a visible glow of light if the temperature of the object is high enough. The Draper point is the temperature at which all solids glow a dim red, about 525 °C (800 K; 980 °F). At 1000 K, a small opening in the wall of a large uniformly heated opaque-walled cavity (such as an oven), viewed from outside, looks red; at 6000 K, it looks white. No matter how the oven is constructed, or of what material, as long as it is built so that almost all light entering is absorbed by its walls, it will contain a good approximation to black-body radiation. The spectrum, and therefore color, of the light that comes out will be a function of the cavity temperature alone. A graph of the spectral radiation intensity plotted versus frequency(or wavelength) is called the black-body curve. Different curves are obtained by varying the temperature.
When the body looks black, the absorption is obvious: the amount of light absorbed is all the light that hits the surface. When the black body is much bigger than the wavelength, the light energy absorbed is strictly proportional to the black-body curve. When the black body is small, so that its size is comparable to the wavelength of light, the absorption is modified, because a small object is not an efficient absorber of light of long wavelength, but the principle of strict equality of emission and absorption is always upheld in a condition of thermodynamic equilibrium. In the laboratory, black-body radiation is approximated by the radiation from a small hole in a large cavity, a hohlraum, in an entirely opaque body that is only partly reflective, that is maintained at a constant temperature. 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. Absorption 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 (that is, the amount of light emitted from the hole at each wavelength) will be continuous, and will depend only on the temperature and the fact that the walls are opaque and at least partly absorptive, but not on the particular material of which they are built nor on the material in the cavity (compare with emission spectrum). 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 gray body assumption.
With non-black surfaces, the deviations from ideal black-body behavior are determined by both the surface structure, such as roughness or granularity, and the chemical composition. On a "per wavelength" basis, real objects in states of local thermodynamic equilibrium still 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; the incomplete absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the surface of the 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 the hypothetical black-body radiation emitted by black holes, at a temperature that depends on the mass, charge, and spin of the hole. If this prediction is correct, black holes will very gradually shrink and evaporate over time as they lose mass by the emission of photons and other particles. A black body radiates energy at all frequencies, but its intensity rapidly tends to zero at high frequencies (short wavelengths). For example, a black body at room temperature (300 K) with one square meter of surface area will emit a photon in the visible range (390–750 nm) at an average rate of one photon every 41 seconds, meaning that, for most practical purposes, such a black body does not emit in the visible range. The study of the laws of black bodies and the failure of classical physics to describe them helped establish the foundations of quantum mechanics.
Additional explanations According to the classical theory of radiation, if each Fourier mode of the equilibrium radiation (in an otherwise empty cavity with perfectly reflective walls) is considered as a degree of freedom capable of exchanging energy, then, according to the equipartition theorem of classical physics, there would be an equal amount of energy in each mode. Since there are an infinite number of modes, this would imply infinite heat capacity, as well as a nonphysical (i.e. not real) spectrum of emitted radiation that grows without bound with increasing frequency, predicting infinite emission power. The problem is known as the ultraviolet catastrophe. Moreover, the classical theory cannot explain the experimentally observed peak in emission spectra (see also Wien's law). Instead, in the quantum treatment of this problem, the numbers of the energy modes are quantized, attenuating the spectrum at high frequency in agreement with experimental observation and resolving the catastrophe. The modes that had more energy than the thermal energy of the substance itself were not considered, and because of quantization modes having infinitesimally little energy were excluded. Thus for shorter wavelengths very few modes (having energy more than h ν {\displaystyle h\nu } ) were allowed, supporting the data that the energy emitted is reduced for wavelengths less than the wavelength of the observed peak of emission. Notice that there are two factors responsible for the shape of the graph, which can be seen as working opposite to one another. Firstly, shorter wavelengths have a larger number of modes associated with them. This accounts for the increase in spectral radiance as one moves from the longest wavelengths towards the peak at relatively shorter wavelengths. Secondly, though, at shorter wavelengths more energy is needed to reach the threshold level to occupy each mode: the more energy needed to excite the mode, the lower the probability that this mode will be occupied. As the wavelength decreases, the probability of exciting the mode becomes exceedingly small, leading to fewer of these modes being occupied: this accounts for the decrease in spectral radiance at very short wavelengths, left of the peak. Combined, they give the characteristic graph. Calculating the black-body curve was a major challenge in theoretical physics during the late nineteenth century. The problem was solved in 1901 by Max Planck in the formalism now known as Planck's law of black-body radiation. 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 expression fitting the experimental data satisfactorily. Planck had to assume that the energy of the oscillators in the cavity was quantized, which is to say that it existed in 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. These quanta were called photons and the black-body cavity was 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 particles, fermions and bosons. 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. As the peak wavelength moves into the ultra-violet and further on, a tail of the spectrum will remain in the visible range and even will increase its intensity, appearing blue. It will never become invisible—indeed, the radiation of visible light increases monotonically with temperature. The Stefan–Boltzmann law says that the total radiant heat power emitted from a surface of a black body is proportional to the fourth power of its absolute temperature. The law was formulated by Josef Stefan in 1879 and later derived by Ludwig Boltzmann. The formula E = σT4 is given, where E is the radiant heat emitted from a unit of area per unit time (power emitted from a unit area), T is the absolute temperature, and σ = 5.670367×10−8 W·m−2⋅K−4 is the Stefan–Boltzmann constant.
Equations
Planck's law of black-body radiation
Planck's law states that
B ν ( T ) = 2 h ν 3 c 2 ( e h ν k T − 1 ) , {\displaystyle B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}\left(e^{\frac {h\nu }{kT}}-1\right)}}\ ,}
where
For a black body surface, the spectral radiance density (defined per unit of area normal to the propagation) is independent of the angle θ {\displaystyle \theta } of emission with respect to the normal. However, this means that, following Lambert's cosine law, B ν ( T ) cos θ {\displaystyle B_{\nu }(T)\cos \theta } is the radiance density per unit area of emitting surface as the surface area involved in generating the radiance is increased by a factor 1 / cos θ {\displaystyle 1/\cos \theta } with respect to an area normal to the propagation direction. At oblique angles, the solid angle spans involved do get smaller, resulting in lower aggregate intensities. The emitted energy flux density or irradiance B ν ( T , E ) {\displaystyle B_{\nu }(T,E)} is related to the photon flux density b ν ( T , E ) {\displaystyle b_{\nu }(T,E)} through
B ν ( T , E ) = E ⋅ b ν ( T , E ) {\displaystyle B_{\nu }(T,E)=E\cdot b_{\nu }(T,E)}
The same equation in terms of ω {\displaystyle \omega } will be different:
B ω ( T ) = ℏ ω 3 4 π 3 c 2 ( e ℏ ω k T − 1 ) . {\displaystyle B_{\omega }(T)={\frac {\hbar \omega ^{3}}{4\pi ^{3}c^{2}\left(e^{\frac {\hbar \omega }{kT}}-1\right)}}\ .}
Wien's displacement law
Wien's displacement law shows how the spectrum of black-body radiation at any temperature is related to the spectrum at any other temperature. If we know the shape of the spectrum at one temperature, we can calculate the shape at any other temperature. Spectral intensity can be expressed as a function of wavelength or of frequency. A consequence of Wien's displacement law is that the wavelength at which the intensity per unit wavelength of the radiation produced by a black body has a local maximum or peak, λ p e a k {\displaystyle \lambda _{\mathsf {peak}}} , is a function only of the temperature:
λ p e a k = b T , {\displaystyle \lambda _{\mathsf {peak}}={\frac {b}{T}}\ ,}
where the constant b {\displaystyle b} , known as Wien's displacement constant, is equal to
b = h c k [ 5 + W 0 ( − 5 e − 5 ) ] ≈ 2.897771955 × 10 − 3 m K , {\displaystyle b={\frac {hc}{k\left[5+W_{0}(-5e^{-5})\right]}}\approx 2.897771955\times 10^{-3}\ {\mathsf {mK}},}
and W 0 {\displaystyle W_{0}} is the Lambert W function. At a typical room temperature of 293 K (20 °C), the maximum intensity is at 9.9 μm . Planck's law was also stated above as a function of frequency. The intensity maximum for this is given by
ν peak = T × 5.879... × 10 10 H z / K . {\displaystyle \nu _{\text{peak}}=T\times 5.879...\times 10^{10}\mathrm {Hz} /\mathrm {K} ~.}
In unitless form, the maximum occurs when e x ( 1 − 1 3 x ) = 1 {\displaystyle e^{x}\left(1-{\tfrac {1}{3}}x\right)=1} , where x = h ν k T {\displaystyle x={\tfrac {h\nu }{kT}}} . The approximate numerical solution is x ≈ 2.82 {\displaystyle x\approx 2.82} . At a typical room temperature of 293 K (20 °C), the maximum intensity is for ν = {\displaystyle \nu =} 17 THz.
Stefan–Boltzmann law
By integrating B ν ( T ) cos ( θ ) {\displaystyle B_{\nu }(T)\cos(\theta )} over the frequency the radiance (luminosity) L {\displaystyle L} (units: [power] / ( [area] × [solid angle] ) ) is
L = ∫ 0 ∞ B ν ( T ) cos ( θ ) d ν = 2 π 5 k 4 T 4 15 c 2 h 3 cos ( θ ) π = σ T 4 cos ( θ ) π {\displaystyle L=\int _{0}^{\infty }B_{\nu }(T)\cos(\theta )\ \mathrm {d} \nu ={\frac {2\pi ^{5}k^{4}T^{4}}{15c^{2}h^{3}}}{\frac {\cos(\theta )}{\pi }}=\sigma T^{4}{\frac {\cos(\theta )}{\pi }}}
by using ∫ 0 ∞ x 3 e x − 1 d x = π 4 15 {\displaystyle \int _{0}^{\infty }{\frac {x^{3}}{e^{x}-1}}\ \mathrm {d} x={\frac {\pi ^{4}}{15}}} with x ≡ h ν k T {\displaystyle x\equiv {\frac {h\nu }{kT}}} and with σ ≡ 2 π 5 k 4 15 c 2 h 3 ≈ 5.670373 × 10 − 8 W m 2 K 4 {\displaystyle \sigma \equiv {\frac {2\pi ^{5}k^{4}}{15c^{2}h^{3}}}\approx 5.670373\times 10^{-8}\mathrm {\frac {W}{m^{2}K^{4}}} } being the Stefan–Boltzmann constant. On a side note, at a distance d ^ {\displaystyle {\hat {d}}} , the intensity d I {\displaystyle \mathrm {d} I} per area d A {\displaystyle \mathrm {d} A} of radiating surface is the useful expression
d I = σ T 4 cos θ π d ^ 2 d A {\displaystyle \mathrm {d} I=\sigma T^{4}{\frac {\cos \theta }{\pi {\hat {d}}^{2}}}\ \mathrm {d} A}
when the receiving surface is perpendicular to the radiation. By subsequently integrating L {\displaystyle L} over the solid angle Ω {\displaystyle \Omega } for all azimuthal angle (0 to 2 π {\displaystyle 2\pi } ) and polar angle θ {\displaystyle \theta } from 0 to 1 2 π , {\displaystyle {\tfrac {1}{2}}\pi ,} we arrive at the Stefan–Boltzmann law: the power j ⋆ {\displaystyle j^{\star }} emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature:
j ⋆ = σ T 4 , {\displaystyle j^{\star }=\sigma T^{4},}
We used
∫ cos θ d Ω = ∫ 0 2 π ∫ 0 π / 2 cos θ sin θ d θ d ϕ = π . {\displaystyle \int \cos \theta \ \mathrm {d} \Omega =\int _{0}^{2\pi }\int _{0}^{\pi /2}\cos \theta \sin \theta d\theta \ \mathrm {d} \phi =\pi ~.}
Applications
Human-body emission
The human body radiates energy as infrared light. The net power radiated is the difference between the power emitted and the power absorbed:
P net = P emit − P absorb . {\displaystyle P_{\text{net}}=P_{\text{emit}}-P_{\text{absorb}}.}
Applying the Stefan–Boltzmann law,
P net = A σ ε ( T 4 − T 0 4 ) , {\displaystyle P_{\text{net}}=A\sigma \varepsilon \left(T^{4}-T_{0}^{4}\right),}
where A and T are the body surface area and temperature, σ {\displaystyle \sigma } is the Stefan–Boltzmann constant (See above), ε {\displaystyle \varepsilon } is the emissivity, and T0 is the ambient temperature. 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. Skin temperature is about 33 °C, but clothing reduces the surface temperature to about 28 °C when the ambient temperature is 20 °C. Hence, the net radiative heat loss is about
P net = P emit − P absorb = 100 W . {\displaystyle P_{\text{net}}=P_{\text{emit}}-P_{\text{absorb}}=\mathrm {100~W} .}
The total energy radiated in one day is about 8 MJ, or 2000 kcal (food calories). Basal metabolic rate for a 40-year-old male is about 35 kcal/(m2·h), 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. There are other important thermal loss mechanisms, including convection and evaporation. Conduction is negligible – the Nusselt number is much greater than unity. Evaporation by perspiration is only required if radiation and convection are insufficient to maintain a steady-state temperature (but evaporation from the lungs occurs regardless). Free-convection rates are comparable, albeit somewhat lower, than radiative rates. Thus, radiation accounts for about two-thirds 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. Application of Wien's law to human-body emission results in a peak wavelength of
λ peak = 2.898 × 10 − 3 K ⋅ m 305 K = 9.50 μ m . {\displaystyle \lambda _{\text{peak}}=\mathrm {\frac {2.898\times 10^{-3}~K\cdot m}{305~K}} =\mathrm {9.50~\mu m} .}
For this reason, thermal imaging devices for human subjects are most sensitive in the 7–14 micrometer range.
Temperature relation between a planet and its star
The black-body law may be used to estimate the temperature of a planet orbiting its sun.
The temperature of a planet depends on several factors:
Incident radiation from its star Emitted radiation of the planet (for example, Earth's infrared glow) The albedo effect causing a fraction of light to be reflected by the planet The greenhouse effect for planets with an atmosphere Energy generated internally by a planet itself due to radioactive decay, tidal heating, and adiabatic contraction due to cooling. The analysis only considers the Sun's heat for a planet in a Solar System. The Stefan–Boltzmann law gives the total power (energy/second) that the Sun emits:
where The Sun emits that power equally in all directions. Because of this, the planet is hit with only a tiny fraction of it. The power from the Sun that strikes the planet (at the top of the atmosphere) is:
where Because of its high temperature, the Sun emits to a large extent in the ultraviolet and visible (UV-Vis) frequency range. In this frequency range, the planet reflects a fraction α {\displaystyle \alpha } of this energy where α {\displaystyle \alpha } is the albedo or reflectance of the planet in the UV-Vis range. In other words, the planet absorbs a fraction 1 − α {\displaystyle 1-\alpha } of the Sun's light, and reflects the rest. The power absorbed by the planet and its atmosphere is then:
Even though the planet only absorbs as a circular area π R 2 {\displaystyle \pi R^{2}} , it emits in all directions; the spherical surface area being 4 π R 2 {\displaystyle 4\pi R^{2}} . If the planet were a perfect black body, it would emit according to the Stefan–Boltzmann law
where T E {\displaystyle T_{\rm {E}}} is the temperature of the planet. This temperature, calculated for the case of the planet acting as a black body by setting P a b s = P e m t b b {\displaystyle P_{\rm {abs}}=P_{\rm {emt\,bb}}} , is known as the effective temperature. The actual temperature of the planet will likely be different, depending on its surface and atmospheric properties. Ignoring the atmosphere and greenhouse effect, the planet, since it is at a much lower temperature than the Sun, emits mostly in the infrared (IR) portion of the spectrum. In this frequency range, it emits ϵ ¯ {\displaystyle {\overline {\epsilon }}} of the radiation that a black body would emit where ϵ ¯ {\displaystyle {\overline {\epsilon }}} is the average emissivity in the IR range. The power emitted by the planet is then:
For a body in radiative exchange equilibrium with its surroundings, the rate at which it emits radiant energy is equal to the rate at which it absorbs it:
Substituting the expressions for solar and planet power in equations 1–6 and simplifying yields the estimated temperature of the planet, ignoring greenhouse effect, TP:
In other words, given the assumptions made, the temperature of a planet depends only on the surface temperature of the Sun, the radius of the Sun, the distance between the planet and the Sun, the albedo and the IR emissivity of the planet. Notice that a gray (flat spectrum) ball where ( 1 − α ) = ε ¯ {\displaystyle (1-\alpha )={\overline {\varepsilon }}} comes to the same temperature as a black body no matter how dark or light gray.
Effective temperature of Earth Substituting the measured values for the Sun and Earth yields:
T S = 5772 K , {\displaystyle T_{\rm {S}}=5772\ \mathrm {K} ,}
R S = 6.957 × 10 8 m , {\displaystyle R_{\rm {S}}=6.957\times 10^{8}\ \mathrm {m} ,}
D = 1.496 × 10 11 m , {\displaystyle D=1.496\times 10^{11}\ \mathrm {m} ,}
α = 0.309 {\displaystyle \alpha =0.309\ }
With the average emissivity ε ¯ {\displaystyle {\overline {\varepsilon }}} set to unit