Planck intensity function

As in energy representation the fundamental thermodynamic equation in entropy representation (3) may also be subjected to Legendre transformation to generate a series of characteristic functions designated as Massieu-Planck (MP) functions, m. The index m denotes the number of intensive parameters introduced as independent variables, i.e. 

The connection between these functions and the microphysical properties of the layers can be found by returning to the original formal solutions of the transfer equation, Eqs. (2.2.3) and (2.2.4). If the layer is thin enough, the Planck intensity B is essentially constant throughout, and the diffuse field becomes vanishingly small. Hence term (4) in Eq. (2.2.3), and term (3) in Eq. (2.2.4), suffice to describe the emitted radiation fields in the upward and downward directions, respectively. Letting a>o and B be independent of r (valid for a sufficiently thin layer), the solution for Eq. (2.2.3) becomes... 

The atmosphere is very transparent at200 cm and almost opaque at the line center at400 cm . The corresponding contribution functions are relatively narrow, and their peaks are located at the surface and at the top of the atmosphere, respectively. As a result the calculated intensities of the spectrum at these two wavenumbers are almost equal to the Planck intensities associated with the temperature (T = 150 K) at the bottom and top of the atmosphere. 

The contribution function for v = 393 cm is associated with a peak spectral intensity at this wavenumber, and also has a maximum at z = 60 km, the altitude at which the temperature profile is maximum. However, the function itself is rather broad, and fairly large contributions to the outgoing intensity at 393 cm arise from a moderate range of altitudes centered about 60 km, over which the temperature is less than maximum. Consequently, the spectral intensity at 393 cm is only slightly greater than the Planck intensity for T = 180 K, as indicated in Fig. 4.2.5a, rather than that for T = 210 K, as implied by Fig. 4.2.4. 

A minimum in the spectrum occurs at v = 367 cm implying the associated weighting function is maximum near z = 30 km, where the temperature (and hence the Planck intensity) has a minimum. Thus the weighting function and Planck intensity tend to counteract each other, and their product results in the broad, double-peaked contribution function shown in Fig. 4.2.6. In this case the concept of an effective emission level has little meaning, since there exists a broad altitude range over which individual levels contribute about equally to the outgoing intensity. This phenomenon is characteristic of temperature minima... 

In summary, the spectral intensity at a given wavenumber can, in certain spectral regions, be closely associated with the Planck intensity of the atmosphere at a given effective emission level Zeff- In other spectral regions, especially near spectral minima, the association is not as close. To the extent that Zeff is meaningful, the emission properties of this level are governed by the optical properties and cross sections of the particles and molecules present at this level, as well as the temperature profile. The sharper the contribution function associated with Zeff, the better defined this level is. 

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. 

LI The Planck Distribution of Black-body Radiation. The Planck relationship between the energy of the photon and the frequency of monochromatic light leads to the equation of distribution of the intensity of light as a function of frequency (or wavelength)... 

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 ... 

In conclusion, it may be mentioned in addition that the Boltzmann constant h, which by definition is the quotient of the gas constant R by Avogadro s number, can be also measured directly by determining the spectral distribution of intensity in the radiation emitted by a black body. The function which expresses the intensity in terms of the frequency and the temperature involves only two universal constants, k and h, the first of w hich is Boltzmann s constant the second is called Planck s constant, and is the fundamental constant of the quantum theory (Chap. VII, 1, p. 185). 

Blackbody radiation is achieved in an isothermal enclosure or cavity under thermodynamic equilibrium, as shown in Figure 7.4a. A uniform and isotropic radiation field is formed inside the enclosure. The total or spectral irradiation on any surface inside the enclosure is diffuse and identical to that of the blackbody emissive power. The spectral intensity is the same in all directions and is a function of X and T given by Planck s law. If there is an aperture with an area much smaller compared with that of the cavity (see Figure 7.4b), X the radiation field may be assumed unchanged and the outgoing radiation approximates that of blackbody emission. All radiation incident on the aperture is completely absorbed as a consequence of reflection within the enclosure. Blackbody cavities are used for measurements of radiant power and radiative properties, and for calibration of radiation thermometers (RTs) traceable to the International Temperature Scale of 1990 (ITS-90) [5]. 

Figure 4.2. In the limit the bar graph becomes a curve, the graph of /(A) vs. A, where /(A) = lim t >0 AF/AA = df/dy, essentially intensity of radiation vs. wavelength. Planck s efforts to find the function /(A) led to the quantum theory.

Fig. 8.17b. Normalized fluorescence intensities of 212 gM calcofluor at A.em - 435 nm as a function of the sial ic acids concentration. The intensities were corrected for the dilution and for the inner filter effect. Sources of figures 8.16 and 8.17 Albani, J. R., Sillen A., Plancke, Y. D., Coddeville, B. and Engelborghs, Y. 2000. Carbohydr. Res. 327, 333-340.

Planck wanted to understand black body radiation. The black body may be modeled by a box, with a small hole (shown in Fig. 1.1). We heat the box up, wait for the system to reach a stationary state (at a fixed temperature), and see what kind of electromagnetic radiation (intensity as a function of frequency) comes out of the hole. In 1900, Rayleigh and Jeans tried to apply classical mechanics to this problem, and they calculated correctly that the black body would emit the electromagnetic radiation with a distribution of frequencies. However,... 

---