A simple radiation model

Let us put aside the spatial variation of solar radiation for a while and try to obtain an estimate of the average surface temperature on Earth, given the average solar radiation coming in at the top of the atmosphere. Solar radiation causes Earth's surface to warm and emit its own radiation back to space, and the balance between incoming and outgoing radiation determines the average temperature of Earth's surface and of the atmosphere. To calculate the temperature, we need to know a few pieces of physics; in particular, we need to know how much radiation a body emits as a function of its temperature and the wavelength of the radiation.

A blackbody is a body that absorbs and emits electromagnetic radiation with perfect efficiency. Thus, all the radiation—and therefore all the visible light—that falls upon it is absorbed. So, unless the body is emitting its own visible radiation, the body will appear black. Now, unless it has a temperature of absolute zero (0 K), the blackbody emits radiation and, as we might expect, the amount of this radiation increases with temperature, although not linearly. The amount, in fact, increases at the fourth power of the absolute temperature; that is, the flux of radiation emitted by the body per unit area varies as

where v = 5.67 X 10-8 Wm-2 K-4 is the Stefan-Boltzmann constant. The presence of a fourth power means that the radiation increases very rapidly with temperature. As a concrete illustration, let us suppose that Earth is a black-body with a temperature of — 18°C, or 255 K (which is a temperature representative of places high in the atmosphere). The energy flux per unit area is F = v X 2554 = 240 Wm2. The sun, by contrast, has a surface temperature of about 6,000 K, and so the radiation it emits is Fsun = v X 6,0004 = 7.3 X 107 Wm2. Thus, although the sun's surface is only about 24 times hotter than Earth, it emits about 300,000 times as much radiation per unit area. It is, of course, the sun's radiation that makes life on Earth possible.

A blackbody emits radiation over a range of wave numbers, but the peak intensity occurs at a wavelength that is inversely proportional to the temperature; this is known as Wien's law. That is, m = T, (1.5)

where Apeak is the wavelength at peak intensity, b is a constant, and T is the temperature. For T in Kelvin and mpeak in meters, b = 2.898 X 10-3 mK. From equation 1.5, it is evident that not only does the sun emit more radiation than Earth, it also emits it at a shorter wavelength. With T = 6,000 K, as for the sun, we find Apeak = 0.483 X 10-6m or about 0.5 nm. Electromagnetic radiation at this wavelength is visible; that is (no surprise), the peak of the sun's radiation is in the form of visible light. (This fact is no surprise because eyes have evolved to become sensitive to the wavelength of the radiation that comes from the sun.) On the other hand, the radiation that Earth emits (at 255 K) occurs at Apeak = 1.1 X 10-5m, which is infrared radiation, also called longwave radiation. The importance of this difference lies in the fact that the molecules in Earth's atmosphere are able to absorb infrared radiation quite efficiently but they are fairly transparent to solar radiation; this difference gives us the greenhouse effect, which we will come to soon.

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