The Planck function

nr is the real part and i the imaginary part of the complex refractive index, n. Since the intensity is proportional to the square of the amplitude, the reflectivity, r, is given by the equation 

Maxwell s equations describe the propagation of electromagnetic radiation as waves within the framework of classical physics however, they do not describe emission phenomena. The search for the law that defines the energy distribution of radiation from a small hole in a large isothermal cavity gave rise to quantum theory. The function that describes the frequency distribution of blackbody radiation was the first result of that new theory (Planck, 1900,1901). 

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. 

The energy density in this expression increases with the square of frequency, contrary to common experience that shows that blackbodies at a few hundred degrees 

Planck realized that the equipartition law, which assigns equal energy to each standing wave, could not be valid he also realized that the roll-off in the energy distribution at high frequencies could be obtained with the assumption that the energy of a harmonic oscillator cannot take on any value, as is assumed in the classical equipartition law, but that it is quantized a harmonic oscillator can absorb and emit energy only in finite steps. 

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Consider continuous radiation with specific intensity I incident normally on a uniform slab with a source function 5 = Bv(Tex) unit volume per unit solid angle to the volume absorption coefficient Kp and is equal to the Planck function Bv of an excitation temperature Tcx obtained by force-fitting the ratio of upper to lower state atomic level populations to the Boltzmann formula, Eq. (3.4). For the interstellar medium at optical and UV wavelengths, effectively S = 0. 

A form of the curve of growth more relevant to stellar (as opposed to interstellar) absorption lines is derived from work by E. A. Milne, A. S. Eddington, M. Min-naert, D. H. Menzel and A. Unsold. In the Milne-Eddington model of a stellar photosphere, the continuum source function (equated to the Planck function in the LTE approximation) increases linearly with continuum optical depth rA and there is a selective absorption i]K, in the line, where rj(Av), the ratio of selective to continuous absorption, is a constant independent of depth given by... 

The function G 4- T occurs so often in thermodynamics that we call it the Planck function. 

Thus, the Planck function is a temperamre-normalized Gibbs function. 

APPLICATION OF THE GIBBS FUNCTION AND THE PLANCK FUNCTION TO SOME PHASE CHANGES... 

Compute the change in the Gibbs function and the change in the Planck function for this allotropic transition at 25°C. 

We will adopt this statement as the working form of the third law of thermodynamics. This statement is the most convenient formulation for making calculations of changes in the Gibbs function or the Planck function. Nevertheless, more elegant formulations have been suggested based on statistical thermodynamic theory [5]. 

Consider an enclosure of dimensions large compared with any wavelengths under consideration, which is opaque but otherwise arbitrary in shape and composition (Fig. 4.11). If the enclosure is maintained at a constant absolute temperature T, the equilibrium radiation field will be isotropic, homogeneous, and unpolarized (see Reif, 1965, p. 373 et seq. for a good discussion of equilibrium radiation in an enclosure). At any point the amount of radiant energy per unit frequency interval, confined to a unit solid angle about any direction, which crosses a unit area normal to this direction in unit time is given by the Planck function... 

In the case of local thermal equilibrium B(a, x) is equal to the Planck function. 

ORM assumes that the atmosphere is in local thermodynamic equilibrium this means that the temperature of the Boltzmann distribution is equal to the kinetic temperature and that the source function in Eq. (4) is equal to the Planck function at the local kinetic temperature. This LTE model is expected to be valid at the lower altitudes where kinetic collisions are frequent. In the stratosphere and mesosphere excitation mechanisms such as photochemical processes and solar pumping, combined with the lower collision relaxation rates make possible that many of the vibrational levels of atmospheric constituents responsible for infrared emissions have excitation temperatures which differ from the local kinetic temperature. It has been found [18] that many C02 bands are strongly affected by non-LTE. However, since the handling of Non-LTE would severely increase the retrieval computing time, it was decided to select only microwindows that are in thermodynamic equilibrium to avoid Non-LTE calculations in the forward model. 

Figure 2 is a plot of the low resolution ETR spectrum compared with the Planck function for a blackbody with a temperature of 6000 Kelvin. The differences in the infrared, beyond 1000 nanometers are small. The larger differences in the shortwave length region are due to the absorption of radiation by the constituents of the solar composition, resulting in the "lines" observed by Fraunhofer and named after him. 

The radiant flux

thermal radiation source through a spectrometer is calculated by multiplying the spectral radiance by the spectral optical conductance, the square of the bandwidth of the spectrometer, and the transmission factor of the entire system (Eq, 3.1-9). Fig. 3.3-1 shows the Planck function according to Eq. 3.3-3. The absorption properties of non-black body radiators can be described by the Bouguer-Lambert-Beer law ... 

The diagram of the Planck function, Fig. 3.3-1, demonstrates further that for smaller absolute wavenumbers there is a strong increase of the thermal emission. 

Where n is the refractive index, a is the absorption coefficient from the Lambert-Beer law and JE is the Planck function specifying the spectral hemispherical em/ssivity of a black body ... 

Here, MXs(X,T) is the Planck function according to (5.50). The emissivity e(T) is the only material function of a grey Lambert radiator all four emissivities are equal and the same as the four absorptivities ... 

Bound-bound absorption is very important in stellar atmospheres and stellar envelopes, where many millions of lines, particularly from iron-group elements, have been shown to have a profound influence on stellar models. The frequencies which dominate the opacity must correspond roughly to the peak of the Planck function, i.e. vmax/T = b (Wien s law again). Most atomic transitions have wavelengths longer than 100 A, so for T > 5 x 106K, bound-bound absorption becomes less important. 

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