Spectral distribution blackbodies

Investigations on the emission properties of INSs started quite a long time ago, mainly in connection with the X-ray emission from PSRs. In the seventies it was a common wisdom that the radiation emitted by INSs comes directly from their solid crust and is very close to a blackbody. Lenzen and Trumper (1978) and Brinkmann (1980) were the first to address in detail the issue of the spectral distribution of INS surface emission. Their main result was that... 

This is Planck s famous radiation law, which predicts a spectral energy density, p , of the thermal radiation that is fully consistent with the experiments. Figure 2.1 shows the spectral distribution of the energy density p for two different temperatures. As deduced from Equation (2.2), the thermal radiation (also called blackbody radiation) from different bodies at a given temperature shows the same spectral shape. In expression (2.2), represents the energy per unit time per unit area per frequency interval emitted from a blackbody at temperature T. Upon integration over all frequencies, the total energy flux (in units of W m ) - that is, Atot = /o°° Pv Av - yields... 

Let us now assume that our two-level system is placed in a blackbody cavity whose walls are kept at a constant temperature T. Once thermal equilibrium has been reached, we can consider that our system is immersed in a thermal cavity where an electromagnetic energy density has been estabhshed. The spectral distribution Pa of this energy density is given by Planck s formula ... 

At low pressure, the only interactions of the ion with its surroundings are through the exchange of photons with the surrounding walls. This is described by the three processes of absorption, induced emission, and spontaneous emission (whose rates are related by the Einstein coefficient equations). In the circumstances of interest here, the radiation illuminating the ions is the blackbody spectrum at the temperature of the surrounding walls, whose intensity and spectral distribution are given by the Planck blackbody formula. At ordinary temperatures, this is almost entirely infrared radiation, and near room temperature the most intense radiation is near 1000 cm". ... 

Outside the atmosphere, the solar flux approximates blackbody emission at 5770 K. However, light absorption or scattering by atmospheric constituents modifies the spectral distribution. The attenuation due to the presence of various naturally occurring atmospheric constituents is shown by the hatched areas in Fig. 3.12. 

When an object is heated, it emits radiation—it glows. Even at room temperature, objects radiate at infrared frequencies. Imagine a hollow sphere whose inside surface is perfectly black. That is, the surface absorbs all radiation striking it. If the sphere is at constant temperature, it must emit as much radiation as it absorbs. If a small hole were made in the wall, we would observe that the escaping radiation has a continuous spectral distribution. The object is called a blackbody, and the radiation is called blackbody radiation. Emission from real objects such as the tungsten filament of a light bulb resembles that from an ideal blackbody. 

Spectral distribution of blackbody radiation. The family of curves is called the Planck distribution after Max Planck, who derived the law governing blackbody radiation. Note that both axes are logarithmic. 

Planck distribution Equation giving the spectral distribution of blackbody radiation ... 

The sun s total radiation output is approximately equivalent to that of a blackbody at 10,350°R (5750 K). However, its maximum intensity occurs at a wavelength that corresponds to a temperature of 11,070°R (6150 K) as given hy Wien s displacement law. A figure plotting solar irradiance versus spectral distribution of solar energy is given in Fig. 9. See also Solar Energy. 

The measurement of the spectral distribution of solar radiation outside the atmosphere and the subsequent association of this spectral distribution with the spectral distribution of radiation in a blackbody cavity has, I believe, biased the attempts to characterize the actual radiation in the atmosphere to an undue extent. Figure 1 indicates typical spectral distributions of radiation in the atmosphere as compared to that of solar radiation outside the atmosphere. Outside the atmosphere m 0 and if the flux is directly through m 1. If slanted at and angle from the zenith angle 90, then m is approximately 1/cos 60. 

The processes of scattering and absorption of radiation in the atmosphere so significantly alter the spectral distribution that any similarity to extra terrestrial radiation is almost coincidental. Experiments with radiation between surfaces have shown that blackbody radiation theory can be extended successfully to many radiation heat transfer situations. In these situations the strict equilibrium requirements of the initial model have so far not proved to be necessary for practical designs. Most importantly the concept of temperature has proved useful in non-equilibrium radiation flux situations(3). 

In dealing with problems of solar radiation, as opposed to blackbody radiation, the effect of the solid angle in which the radiation is confined has been examined (2-4) by considering the volume density of photons to be reduced. Landsberg(6) considers dilute radiation in the sense that the spectral distribution is retained but the radiation density is reduced. This leads to defining the temperature of a spectral component as... 

The Planck theory of blackbody radiation provides a first approximation to the spectral distribution, or intensity as a function of wavelength, for the sun. The black-body theory is based upon a "perfect" radiator with a uniform composition, and states that the spectral distribution of energy is a strong function of wavelength and is pro portional to the temperature (in units of absolute temperature, or Kelvin), and several fundamental constants. Spectral radiant exitance (radiant flux per unit area) is de fined as ... 

Spectral distribution of solar radiation just outside the alinosphere, at the surface of the carlli on a typical day, and comparison with blackbody radiation at 5780 K. 

For a blackbody, the spectral distribution of hemispherical emissive power (integrated over the hemispherical solid angle) follows Plank s law ... 

Figure 7.3a represents the Planck distribution for blackbody spectral emissive power with E-i p / as a function of XT. The band fraction of emitted energy in the region from 0 to XT is equal to the shaded area, which is expressed as and shown in Figure 7.3b. About a quarter of the emitted energy is at wavelengths shorter than nd nearly 95% of the emitted energy is distributed between and The spectral distribution of solar radiation can be...

III a = e Incident radiation has a spectral distribution proportional to that of a blackbody at the temperature of the surface, or ai = are independent of wavelength... 

Figure 3.19 Spectral distribution ofthe intensity of blackbody radiation as a function of frequency for several temperatures. The dashed line is the prediction of classical physics. (Reprinted with permission from University Science Books.) °...

The spectral distribution of radiation as a function of wavelength for three different temperature stars. A star is one of the closest approximations to a perfect blackbody radiator. [Reproduced from http //en.wikipedia.org/wiki /Black-body radiation (accessed October 17, 2013).]... 

Figure 2.8 shows the comparative spectral distribution of energy emitted by a blackbody, a graybody and a non-graybody, all at the same temperature. For gray-body measurements a simple emissivity correction can usually be implemented... 

Figure 2.8 Spectral distributions of a blackbody, graybody, and non-graybody.

Blackbody Radiators. According to the well-known Planck distribution law, the spectral distribution of energy emitted by an ideal blackbody is determined solely by the temperature of the radiating element. Figure 2-6 shows this distribution for a source at 1500°K,... 

Figure 2-6. Spectral distribution of energy emitted by a 1500°K blackbody source.

As the temperature of the blackbody increases, its colour changes as evident in Figures 5 and 6. The colour of a blackbody is thus uniquely dependent on its temperature. Real sources such as tungsten lamps that have spectral distributions approximating those of blackbodies can be assigned a colour temperature at which the spectrum most nearly overlays that of the blackbody. Outside the Earth s atmosphere, the Sun can, for example, be approximated as a blackbody at 6200 K, which is, then, the Sun s colour temperature. 

FIGURE 5.17 Blackbody radiation spectral distributions superimposed on the transmittance curve for a 3.8 mm thick PET sample. (Data from Shelby, 1991.)... 

Measurements of the total emission from a small hole in a heated cavity showed thermal radiation to be proportional to the fourth power of the cavity temperature (Stefan, 1879) Boltzmaim (1884) derived this power law from thermodynamic considerations. Nine years later, Wien (1893) found that the product of the wavelength at the radiation maximum and the cavity temperature was the same for a wide range of temperatures he also proposed an exponential radiation law, which was in good agreement with available measurements at short wavelengths (Wien, 1896). Shortly thereafter, Lummer and Pringsheim (1897,1899) made fairly precise measurements of blackbody emission between 100 °C and 1300 °C. By the end of the nineteenth century an extensive set of experimental evidence was available on the spectral distribution and temperature dependence of blackbody radiation. 

To obtain the spectral distribution of thermal emission emerging at velocity c from a blackbody into a steradian, the energy density must be multiplied by c/djr, yielding... 

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