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Determining the Temperature of a Planet
--2nd
Approximation
If we recognize that only a fraction of incident energy, given by (1-A)
where A is albedo, is absorbed by Earth, a second approximation
(Figure 4)
of the Earth's temperature gives 256 K or -17°C (1.6°F) as the
effective temperature. This is even further from the observed average
surface temperature of 283 K, or 15°C (59°F) than the first
approximation.
A more refined approximation is required to obtain a temperature for Earth that is close to the observed value. The factor not accounted for in the previous approximations is the effect of the atmosphere, for which we need to understand the relation of radiating temperature to dominant spectral wavelength.
Figure 5 gives a schematic representation of the radiated energy flux emitted by the sun (left-hand curve) radiating at about 6,000 K and the earth (right-hand curve) radiating at about 300 K. Note that the wavelength of radiated energy is 0.5 microns (0.5 x 10-6 m) for the sun and 10 microns for the earth. The sun radiates visible energy and the earth radiates infrared energy. The relationship between temperature and maximum radiating wavelength is given by Wien's Displacement Law, which is
where T is radiating temperature in K and lm is wavelength in microns. More information and simulations are available online.
An object or gas will absorb energy differently at different wavelengths, depending on the atomic structure of its molecules. At visible wavelengths, dark-appearing objects absorb more energy than light colored objects. An object that appears red to our eyes receives visible light from the sun or other source and absorbs all wavelengths except red, which is reflected. A black object absorbs almost all energy at all wavelengths.
A gas will absorb some fraction of radiant energy incident on it. Figure 6 gives the absorptivity of various gases, ranging from 0 to 1.0, for various wavelengths of radiant energy in the visible and infrared spectrum. The sun, radiating most strongly at wavelengths around its maximum at about 0.5 microns, has essentially all of its energy absorbed by ozone below 0.3 microns (ultraviolet light), but none of the atmospheric constituents absorb very much in the "visible window" between 0.3 and 0.7 microns.
Energy from the earth, on the other hand, radiates over a range of wavelengths centered on about 10 microns, which, according to the absorption graph, is a region where energy is absorbed strongly by water vapor (H2O) and carbon dioxide (CO2) and, at certain wavelengths, by methane (CH4), nitrous oxide (N2O), oxygen (O2) and ozone (O3). The graph at the bottom of the figure gives the aggregate absorptivity for all gases in the atmosphere.
