The equation of transfer

Because radiation tends to be modified when it interacts with matter, it is possible to infer certain physical properties of planetary atmospheres and surfaces by studying their reflected and emitted radiation. Although these modifications are macroscopic in nature (they are manifested over an extended volume), their origins are contained in the processes of absorption, scattering, and emission of radiant energy on a microscopic scale. A quantitative assessment of the relation between these interactions and the resulting radiation field is known as the theory of radiative transfer. It is the purpose of this section to develop the equation central to this theory. 

The specific intensity, also known as the spectral radiance, has been defined by Eq. (1.8.3). Another parameter of interest is the phase function for single scattering, p(cos 0), which describes the angular distribution of radiation scattered once through the angle 0. If represents the fraction of energy per unit time incident on dy in the direction (ft, (j ) that is either absorbed or scattered in all directions, and AEv is that fraction of scattered into the direction (ijl, (p ) contained in the solid angle dco, then p(cos 0) is defined by 

If the albedo for single scattering, mq, is defined to be the ratio of radiant power scattered in all directions to that extinguished (absorbed plus scattered), we have, from Eiqs. (2.1.1) and (2.1.3), 

Two points of view are possible in describing radiation-matter interactions on a microscopic scale, hi the Lagrangian point of view the movements of individual 

Consider the photons of wavenumber v interacting with dV to be classified according to the interactions they undergo as well as upon the intrinsic characteristics of the photons themselves. We restrict ourselves to one field at a time, which in essence is the same as restricting our attention to one photon of this field at a time. Thus the field from the Lagrangian point of view (in a looser sense of the phrase) is followed. 

---

Equation (3.19) gives a first approximation to the temperature structure of an atmosphere in radiative equilibrium, and departures from greyness can also be treated approximately by defining a suitable mean absorption coefficient (see Chapter 5). The emergent monochromatic intensity at an angle 9 to the normal (relevant to some point on the solar disk) is also found by integrating the equation of transfer (3.11) ... 

The equations of transfer do not incorporate time explicitly, but all local variations can be transformed into temporal variations via dt = dr / c, where c is the mean velocity of light in the scattering medium. As a consequence of scattering, the incident as well as the emitted photons show TOF dispersion in spatially extended samples. On diffuse... 

Sykes, J. B. (1951). Approximate Integration of the Equation of Transfer. Monthly Notes Royal Astronomical Society, 111, 377. 

The Maxwellian molecules are useful in exploratory calculations in which a differentiable potential function is needed. For these molecules [66] the in-termolecular force between pairs at a distance r apart is of the form where K is a constant. Adopting this particular potential representation the solution of the equation of transfer reduces to a feasible problem, thereby Maxwell [66] obtained analytical expressions for the transport coefficients as mentioned earlier. 

The formal solution of the equation of transfer is obtained by integration along a given path from the point s = 0 (Fig. 5.11) ... 

On the other hand, if we can treat the system as being made up of elements, then we must identify the system as independent or dependent. In theory, all systems are dependent, but if the deviation from the independent theory is not large, the assumption of independent scattering should be made. The range of validity of this assumption can be approximately set at OX > 0.5 and e > 0.95. If the problem lies in the independent range, then the properties of the bed can be readily calculated, and the equation of transfer can be solved. 

Under great following simplifications the force of shift disappears from the equation of transfer. As a result, the expression for the viscosity coefficient accordingly to Eyring becomes as follows... 

Our aim in this chapter is to develop the mathematical formalism that serves as the foundation for all our analyses involving the radiation field in sufficient depth to be essentially autonomous, though our indebtedness to some of the procedures developed by Chandrasekhar (1950) is obvious. The equation of transfer is derived in Section 2.1, and formal solutions are found in Section 2.2. Very general techniques for solving the transfer equation numerically are developed in Section 2.3. Though... 

This is the equation of transfer for an arbitrary, monochromatic field of radiation. In practice this field consists of a diffuse component that originates from thermal emission of the atmosphere and the planetary surface, as well as a component (both diffuse and direct) that originates from the Sun. The former component dominates in the middle and far infrared, whereas the latter component is the sole contributor... 

The discussions of the equation of transfer and the solution of this equation in Chapter 2 rest entirely on concepts of classical physics. Such treatment was possible because we considered a large number of photons interacting with a volume element that, although it was assumed to be small, was still of sufficient size to contain a large number of individual molecules. But with the assumption of many photons acting on many molecules we have only postponed the need to introduce quantum theory. Single photons do interact with individual atoms and molecules. The optical depth, r (v), depends on the absorption coefficients of the matter present, which must fully reflect quantum mechanical concepts. The role of quantum physics in the derivation of the Planck function has already been discussed in Section 1.7. Both the optical depth and the Planck function appear in the radiative transfer equation (2.1.47). 

From this example it is clear that some combination of thermal structure and emitting gas abimdance can be inferred from observed spectra. As we have seen, a qualitative picture can often be obtained simply by inspecting a display of the spectral data. A more quantitative assessment requires solutions of the equation of transfer. First, however, it is necessary to examine how instrumental effects modify the appearance of planetary spectra. This will be discussed in the next chapter. 

The equation of transfer governing the infrared radiation field is found from Eq. (2.5.10) to be... 

Once the emission and absorption coefficients of the excited gas are specified as functions of frequency and position, the equation of transfer, equation (10.5), may in principle be solved to obtain the intensity at any point. 

---