Beyond the material medium of the atmosphere, heat is transferred across the vacuum of space by electromagnetic radiation. In fact, radiation is the only way heat can cross a vacuum and this radiative transfer of heat is governed by the Stefan-Boltzmann Law. As we shall see, this is critical to calculating body temperature from heat entering an otherwise thermally isolated body. It also dictates the temperature of the ideal greenhouse. However, as the Stefan-Boltzmann Law concerns radiation emitted, we must first extend this law to relate temperature to incident radiation. This is achieved by applying the the principle of equal absorptivity and emissivity best known as “Kirchhoff’s Law”.
“Kirchhoff’s Law” can be used to simplify the Stefan-Boltzmann Equation (Boltzmann, 1884) yielding a form that is surprisingly elegant. The significance of Kirchhoff’s Law lies in the fact that emissivity not only constrains the proportion absorbed, but the readiness with which the body may emit (Kirchhoff, 1859; Kirchhoff, 1860, translated by Guthrie, 1860). Thus as emissivity decreases for the same emission of radiation, the temperature rises. However, given a constant incident radiation, the proportion by which temperature is raised by lack of emissivity is balanced by the reduced proportion of absorbed radiation. Substituting incident radiation multiplied by emissivity for emitted radiation in the Stefan-Boltzmann Equation arises the following way:
Where:
Wb = Radiation (heat flux) in Wm-2 emitted by the body in question if it is a perfect black body
Wi = Radiation (heat flux) in Wm-2 incident upon the body in question
We = Radiation (heat flux) in Wm-2 emitted by the body in question
T = Absolute Temperature in ºK of the body in question
ε = Emissivity = Absorption / (Absorption + Reflection) of the body in question
σ = Stefan’s Constant = 0.000000056704
Wm = Mean incidence of radiation over the entire surface of the body in Wm-2
Ax = Mean cross-sectional area of radiation incident on the body in m2
At = Topographical area of the body in m2
Wb = σT4 Stefan’s Law relating black body radiation to temperature (Stefan, 1879)
We = εWb Emissivity is the proportion of hypothetical black body radiation emitted
Wb = Wi And at thermal equilibrium, black body radiation is equal to incident radiation
We = εWi Ergo emissivity is also the proportion of incident radiation emitted
We = σεT4 As the Stefan-Boltzmann Equation (Boltzmann, 1884) elaborates on emitted radiation:
εWi = σεT4
Thus a body’s temperature response to incident radiation is entirely independent of emissivity, such that
Wi = σT4
This is confusing because it looks just like Stefan’s Law for black bodies. However, as the radiation in question is not the body’s emitted radiation as used by Stefan (1879), but is instead the incident radiation, it applies not only to black bodies but in general – as shown by the simple derivation. However, this case is strictly for omnidirectional radiation, which is only incident when all the radiation is diffuse or scattered. Radiation from a given source is directional and when the source is distant, the radiation is measured in a plane perpendicular to incidence. As a body is a three dimensional object with a much larger surface area than the area across which incident radiation falls, the emitted radiation of a body is always correspondingly lower in intensity then the incident radiation. As the area of incidence is less than the area of emission, we must further modify our equation so:
WiAx/At = σT4
Wm = WiAx/At
Wm = σT4
As you can see, the temperature of a body in constant incident radiation cannot be raised by compositional changes, and solely depends on the intensity of the radiation. This confirms the duplication of energy and to some degree, the perpetuum mobile inherent in the “Greenhouse Effect.”
Article credits to http://greenhouse.geologist-1011.net/