Black-body radiation

Spectrum s judge workpiece temperatures by the colour of the glow. 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. However, graphite and lamp black, with emissivities greater than 0.95, are good approximations to a black material. Experimentally, black-body radiation may be established best as the ultimately stable steady state equilibrium radiation in a cavity in a rigid body, at a uniform temperature, that is entirely opaque and is only partly reflective. Black-body radiation has the unique absolutely stable distribution of radiative intensity that can persist in thermodynamic equilibrium in a cavity. 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 . At , a small opening in the wall of a large uniformly heated opaque-walled cavity (such as an oven), viewed from outside, looks red; at , 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. lava flow can be estimated by observing its color. The result agrees well with other measurements of temperatures of lava flows at about . When the body is black, the absorption is obvious: the amount of light absorbed is all the light that hits the surface. For a black body much bigger than the wavelength, the light energy absorbed at any wavelength λ per unit time is strictly proportional to the black-body curve. This means that the black-body curve is the amount of light energy emitted by a black body, which justifies the name. This is the condition for the applicability of Kirchhoff's law of thermal radiation: the black-body curve is characteristic of thermal light, which depends only on the temperature of the walls of the cavity, provided that the walls of the cavity are completely opaque and are not very reflective, and that the cavity is in thermodynamic equilibrium. 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. image (2012) of the cosmic microwave background radiation across the universe. 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 () 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 \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 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 is the Stefan–Boltzmann constant. ==Equations==

Planck's law of black-body radiation Planck's law states that 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 \ \theta\ of emission with respect to the normal. However, this means that, following Lambert's cosine law, \ 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 \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_\nu(T,E)\ , is related to the photon flux density \ b_\nu(T,E)\ through B_\nu(T,E) = E \cdot b_\nu(T,E) The same equation in terms of \omega will be different: B_\omega(T) = \frac{ \hbar\ \omega^3 }{ 4\ \pi^3 c^2 \left( e^{\frac{\hbar \omega}{k T}} - 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, \ \lambda_\mathsf{peak}\ , is a function only of the temperature: \lambda_\mathsf{peak} = \frac{b}{T}\ , where the constant \ b\ , known as Wien's displacement constant, is equal to b = \frac{ h\ c }{\ k \left[\ 5 + W_0(-5e^{-5})\ \right]\ } \approx 2.897771955\times 10^{-3} \ \mathsf{m\ K}\ , and \ W_0\ is the Lambert W function. At a typical room temperature of 293 K (20 °C), the maximum intensity is at  . Planck's law was also stated above as a function of frequency. The intensity maximum for this is given by \nu_\text{peak} = T \times 5.879... \times 10^{10} \ \mathrm{Hz}/\mathrm{K} ~. In unitless form, the maximum occurs when \ e^x \left(1 - \tfrac{1}{3} x \right) = 1\ , where \ x = \tfrac{ h\nu }{ kT } ~. The approximate numerical solution is \ x \approx 2.82 ~. At a typical room temperature of 293 K (20 °C), the maximum intensity is for Stefan–Boltzmann law By integrating \ B_\nu(T)\cos(\theta)\ over the frequency the radiance (luminosity) \ L\ units: [power] / ( [area] × [solid angle] ) is L = \int_0^\infty B_\nu(T) \cos(\theta)\ \mathrm{d}\nu = \frac{\ 2\ \pi^5 k^4 T^4}{\ 15\ c^2 h^3}\ \frac{\cos(\theta)}{\pi} = \sigma\ T^4\ \frac{\cos(\theta)}{\pi} by using \ \int_0^\infty \frac{x^3}{e^x - 1}\ \mathrm{d}x = \frac{\ \pi^4\ }{ 15 }\ with \ x \equiv \frac{\ h\ \nu\ }{k\ T}\ and with \ \sigma \equiv \frac{\ 2 \pi^5\ k^4 }{\ 15\ c^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 \ \hat{d}\ , the intensity \ \mathrm{d}I\ per area \ \mathrm{d}A\ of radiating surface is the useful expression \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\ over the solid angle \ \Omega\ for all azimuthal angle ( to \ 2\pi\ ) and polar angle \ \theta\ from to \ \tfrac{1}{2}\pi\ , we arrive at the Stefan–Boltzmann law: the power \ 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^\star = \sigma\ T^4\ , We used \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_\text{net} = P_\text{emit} - P_\text{absorb}. Applying the Stefan–Boltzmann law, P_\text{net} = A \sigma \varepsilon \left( T^4 - T_0^4 \right), where and are the body surface area and temperature, \sigma is the Stefan–Boltzmann constant (See above), \varepsilon is the emissivity, and is the ambient temperature. The total surface area of an adult is about , 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, Hence, the net radiative heat loss is about 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. {{NumBlk||P_{\rm abs} = P_{\rm emt} |}} Substituting the expressions for solar and planet power in equations 1–6 and simplifying yields the estimated temperature of the planet, ignoring greenhouse effect, : {{NumBlk||T_P = T_S\sqrt{\frac{R_S\sqrt{\frac{1-\alpha}{\overline{\varepsilon}}}}{2D}} |}} 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 - \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_{\rm S} = 5772 \ \mathrm{K}, • R_{\rm S} = 6.957 \times 10^8 \ \mathrm{m}, With the average emissivity \overline{\varepsilon} set to unity, the effective temperature of the Earth is: T_{\rm E} = 254.356\ \mathrm{K} or −18.8 °C. This is the temperature of the Earth if it radiated as a perfect black body in the infrared, assuming an unchanging albedo and ignoring greenhouse effects (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrums). The Earth in fact radiates not quite as a perfect black body in the infrared which will raise the estimated temperature a few degrees above the effective temperature. If we wish to estimate what the temperature of the Earth would be if it had no atmosphere, then we could take the albedo and emissivity of the Moon as a good estimate. The albedo and emissivity of the Moon are about 0.1054 and 0.95 respectively, yielding an estimated temperature of about 1.36 °C. 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 W m−2, one obtains an effective temperature of about 245 K. It is a "snapshot" of the radiation at the time of decoupling between matter and radiation in the early universe. Prior to this time, most matter in the universe was in the form of an ionized plasma in thermal, though not full thermodynamic, equilibrium with radiation. At very high temperatures (above 1010 K; such temperatures existed in the very early universe), where the thermal motion separates protons and neutrons in spite of the strong nuclear forces, electron-positron pairs appear and disappear spontaneously and are in thermal equilibrium with electromagnetic radiation. These particles form a part of the black body spectrum, in addition to the electromagnetic radiation. Light bulb Tungsten filament lights have a continuous black body spectrum with a colour temperature of around , and also emits considerable energy in the infrared range. Modern-day fluorescent and LED lights, which are more efficient, do not have a continuous black body emission spectrum, rather emitting directly, or using combinations of phosphors that emit multiple narrow spectrums. ==History==

In query 6 of Isaac Newton's Opticks, he states that "Do not black Bodies conceive heat more easily from Light than those of other Colours do, by reason that the Light falling on them is not reflected outwards, but enters into the Bodies, and is often reflected and refracted within them, until it be stifled and lost?", thereby introducing the notion of a black body. In his first memoir, Augustin-Jean Fresnel (1788–1827) responded to a view he extracted from a French translation of Newton's Opticks. He says that Newton imagined particles of light traversing space uninhibited by the caloric medium filling it, and refutes this view (never actually held by Newton) by saying that a black body under illumination would increase indefinitely in heat. Balfour Stewart In 1858, Balfour Stewart described his experiments on the thermal radiative emissive and absorptive powers of polished plates of various substances, compared with the powers of lamp-black surfaces, at the same temperature. Stewart chose lamp-black surfaces as his reference because of various previous experimental findings, especially those of Pierre Prevost and of John Leslie. He wrote, "Lamp-black, which absorbs all the rays that fall upon it, and therefore possesses the greatest possible absorbing power, will possess also the greatest possible radiating power." Stewart's statement assumed a general principle: that there exists a body or surface that has the greatest possible absorbing and radiative power for every wavelength and equilibrium temperature. Stewart was concerned with selective thermal radiation, which he investigated using plates which selectively radiated and absorbed different wavelengths. He discussed the experiments in terms of rays which could be reflected and refracted, and which obeyed the Stokes-Helmholtz reciprocity principle. His research did not consider that properties of rays are dependent on wavelength, and he did not use tools such as prisms or diffraction gratings. His work was quantitative within these constraints. He made his measurements in a room temperature environment, and quickly so as to catch his bodies in a condition near the thermal equilibrium in which they had been prepared. Gustav Kirchhoff In 1859, Gustav Robert Kirchhoff reported the coincidence of the wavelengths of spectrally resolved lines of absorption and emission of visible light. Importantly for thermal physics, he also observed that bright lines or dark lines were apparent depending on the temperature difference between emitter and absorber. Kirchhoff then went on to consider some bodies that emit and absorb heat radiation, in an opaque enclosure or cavity, in equilibrium at a temperature . Here is used a notation different from Kirchhoff's. Here, the emitting power denotes a dimensioned quantity, the total radiation emitted by a body labeled by index at temperature . The total absorption ratio of that body is dimensionless, the ratio of absorbed to incident radiation in the cavity at temperature . (In contrast with Balfour Stewart's, Kirchhoff's definition of his absorption ratio did not refer in particular to a lamp-black surface as the source of the incident radiation.) Thus the ratio of emitting power to absorptivity is a dimensioned quantity, with the dimensions of emitting power, because is dimensionless. Also here the wavelength-specific emitting power of the body at temperature is denoted by and the wavelength-specific absorption ratio by . Again, the ratio of emitting power to absorptivity is a dimensioned quantity, with the dimensions of emitting power. In a second report made in 1859, Kirchhoff announced a new general principle or law for which he offered a theoretical and mathematical proof, though he did not offer quantitative measurements of radiation powers. His theoretical proof was and still is considered by some writers to be invalid. His principle, however, has endured: it was that for heat rays of the same wavelength, in equilibrium at a given temperature, the wavelength-specific ratio of emitting power to absorptivity has one and the same common value for all bodies that emit and absorb at that wavelength. In symbols, the law stated that the wavelength-specific ratio has one and the same value for all bodies. In this report there was no mention of black bodies. In 1860, still not knowing of Stewart's measurements for selected qualities of radiation, Kirchhoff pointed out that it was long established experimentally that for total heat radiation emitted and absorbed by a body in equilibrium, the dimensioned total radiation ratio has one and the same value common to all bodies. Again without measurements of radiative powers or other new experimental data, Kirchhoff then offered a fresh theoretical proof of his new principle of the universality of the value of the wavelength-specific ratio at thermal equilibrium. His fresh theoretical proof was and still is considered by some writers to be invalid. Kirchhoff's proof considered an arbitrary non-ideal body labeled as well as various perfect black bodies labeled . It required that the bodies be kept in a cavity in thermal equilibrium at temperature . His proof intended to show that the ratio was independent of the nature of the non-ideal body, however partly transparent or partly reflective it was. His proof first argued that for wavelength and at temperature , at thermal equilibrium, all perfectly black bodies of the same size and shape have the one and the same common value of emissive power , with the dimensions of power. His proof noted that the dimensionless wavelength-specific absorptivity of a perfectly black body is by definition exactly 1. Then for a perfectly black body, the wavelength-specific ratio of emissive power to absorptivity is again just , with the dimensions of power. Kirchhoff considered thermal equilibrium with the arbitrary non-ideal body, and with a perfectly black body of the same size and shape, in place in his cavity in equilibrium at temperature . He argued that the flows of heat radiation must be the same in each case. Thus he argued that at thermal equilibrium the ratio was equal to , which may now be denoted . is a continuous function, dependent only on at fixed temperature , and an increasing function of at fixed wavelength . It vanishes at low temperatures for visible wavelengths, which does not depend on the nature of the arbitrary non-ideal body (Geometrical factors, taken into detailed account by Kirchhoff, have been ignored in the foregoing). Thus Kirchhoff's law of thermal radiation can be stated: For any material at all, radiating and absorbing in thermodynamic equilibrium at any given temperature , for every wavelength , the ratio of emissive power to absorptivity has one universal value, which is characteristic of a perfect black body, and is an emissive power which we here represent by . (For our notation , Kirchhoff's original notation was simply .) Kirchhoff announced that the determination of the function was a problem of the highest importance, though he recognized that there would be experimental difficulties to be overcome. He supposed that like other functions that do not depend on the properties of individual bodies, it would be a simple function. Occasionally by historians that function has been called "Kirchhoff's (emission, universal) function," though its precise mathematical form would not be known for another forty years, till it was discovered by Planck in 1900. The theoretical proof for Kirchhoff's universality principle was worked on and debated by various physicists over the same time, and later. According to Helge Kragh, "Quantum theory owes its origin to the study of thermal radiation, in particular to the "blackbody" radiation that Robert Kirchhoff had first defined in 1859–1860." ==Doppler effect==

The relativistic Doppler effect causes a shift in the frequency f of light originating from a source that is moving in relation to the observer, so that the wave is observed to have frequency f': f' = f \frac{1 - \frac{v}{c} \cos \theta}{\sqrt{1-v^2/c^2}}, 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 measured in the reference frame of the source, and c is the speed of light. This 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. Through Planck's law the temperature spectrum of a black body is proportionally related to the frequency of light and one may substitute the temperature (T) for the frequency in this equation. 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 black-body radiation field. ==See also==