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2 Figure 1: Ahrens, Chapter 2 2

4 The Planck Function Determined experimentally, the intensity of radiation emitted by a blackbody is c 1 λ 5 B λ = (6) π(e c 2/λT 1) where c 1 = W m 2 and c 2 = mk. Theoretical justification of this empirical relationship led to the development of the theory of quantum physics. Wien s Displacement Law Differentiating Equation 6 and setting the derivative equal to zero, gives the wavelength of peak emission for a blackbody at temperature T (HW6). λ m = 2897 (7) T where T in K and λ m in µm. An important consequence of Wien displacement law is the fact that solar radiation is concentrated in the visible and near-infrared parts of the spectrum, while radiation emitted by the planets and their atmospheres is largely confined to the infrared. The nearly complete absence of overlap between the curves justifies dealing with solar and planetary radiation separately in many problems of radiative transfer. Stefan-Boltzmann Law The black body flux density obtained by integrating the Planck function πb λ over all wavelengths. F = σt 4 (8) Where σ is the Stefan-Boltzmann constant equal to W m 2 K 4 4

6 N 2 and O 2, the two most abundant gases in the atmosphere, are transparent in the infrared. (d) Rotational transitions a molecule changes rotational states. These occur in the far infrared and microwave portion of the spectrum. They can occur at the same time as vibrational transitions. Figure 4.7. The three are related by α λ + R λ + T λ = 1. For a black body α λ = 1. 5a. Other Definitions Total Mass Extinction Coefficient If a beam of intensity I λ becomes I λ + di λ upon traversing a distance ds in its direction of propagation through a medium of density ρ, then the reduction of intensity due to extinction (which could be absorption, reflection, scattering, diffraction, refraction etc.) is di λ = k λt ρi λ ds (9) where k λt is the total mass extinction coefficient and has units [m 2 kg 1 ]. We define the optical depth τ λ in terms of the total mass extinction coefficient as follows: τ λ = s 0 k λt ρds Mass Absorpion Coefficient If a beam of intensity I λ becomes I λ + di λ upon traversing a distance ds in its direction of propagation through a medium of density ρ, then the reduction of intensity due to absorption is: di λ = k λa ρi λ ds (10) where k λa is the mass absorption coefficient and has units [m 2 kg 1 ]. We define the absorption optical depth τ λa as τ λa = s 0 k λaρds Mass Scattering Coefficient If a beam of intensity I λ becomes I λ + di λ upon traversing a distance ds in its direction of propagation through a medium of density ρ, then the reduction of intensity due to scattering is: di λ = k λs ρi λ ds (11) where k λs is the mass scattering coefficient and has units [m 2 kg 1 ]. We define the scattering optical depth τ λs as τ λs = s 0 k λsρds It follows that k λt = k λa + k λs 6

7 5b. Kirchoff s Law It can be shown that the radiation emitted by a given material is a function of temperature and wavelength only. Consider an opaque, hollow enclosure with zero transmissivity into which is placed a slab of finite thickness. In general, this slab will reflect, absorb and transmit parts of the incident radiation. In addition, it will emit radiation itself. We now allow the enclosure and the slab to reach thermodynamic equilibrium, such that the slab and the enclosure walls are the same temperature. Under this condition, the flow of energy in all directions must be the same. In thermodynamic equilibrium, the amount entering the slab must exactly equal the amount leaving, or there would be a net flow of heat to or from the walls, into or out of the slab. Since the slab and the walls are in thermodynamic equilibrium, this would constitute a violation of the Second Law of Thermodynamics. Therefore, the balance equation is: I λ R λ I λ = T λ I λ + E λ (12) Where E λ is the emitted radiance in the same direction as I λ. But T λ I λ = I λ (1 α λ R λ ) since α λ + R λ + T λ = 1. Therefore, I λ (1 R λ ) = I λ (1 α λ R λ ) + E λ (13) Thus, E λ α λ I λ = 0 or E λ = α λ I λ Thus, inside of an opaque, hollow enclosure in thermodynamic equilibrium, the amount emitted by the slab equals the amount absorbed by the slab. We now imagine our enclosure to be replaced by a different one, constructed from a different material, and again allow it to come into thermodynamic equilibrium with the same slab and at the same temperature as before. Consequently, the slab emission will be the same as before, since it depends only on temperature and wavelength, neither of which has been changed. Similarly, the slab absorption will not change because the slab material is the same. Thus we have: E λ = α λ I λ (14) Where I λ is the incident radiation on the slab in the new enclosure, thus it follows that I λ = I λ.thus, the radiation within an opaque, hollow enclosure is independent of the material from which the walls are made. Re-writing the above E equation we see λ = I α l ambda λb = f(t, λ) only and I λb is the radiance inside an opaque hollow enclosure at temperature T and wavelength λ. 7

8 This result is known as Kirchhoff s Law, which states that The ratio of the emission to the fractional absorptivity of a slab of any material in a state of thermodynamic equilibrium and at wavelength λ is equal to a constant. We may now define the fractional emissivity ɛ λ as the ratio of the radiation emitted at the wavelength λ to that within a hollow enclosure at the same temperature or: ɛ λ = E λ I λb (15) From this definition, we see that E λ = ɛ λ I λb. But from Kirchoff s Law it then follows: ɛ λ = α λ (16) Or the fractional emissivity equals the fractional absorptivity Kirchoff s Law is fundamental to further development of the subject of radiative transfer, and is frequently applied in a variety of applications. Recalling that it is strictly valid only under conditions of thermodynamic equilibrium, it is nevertheless generally assumed to be valid for atmospheric problems even though the atmosphere is not strictly in thermodynamic equilibrium. We may now carry this thought experiment one step further. Let s replace this slab by an ideal black body such that, by definition, it completely absorbs all radiation falling on it. Inside the hollow enclosure then, the radiation leaving the black body slab consists entirely of radiation emitted by the slab. The equilibrium condition becomes I λb = E λ (17) leading to the important conclusion that the radiation flowing in any direction within the hollow enclosure in thermodynamic equilibrium is equal to the energy emitted in the same direction as an ideal black body. Such radiation is called black body radiation, and from our earlier arguments is isotropic or equal in all directions. 8

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