Blackbody radiation represents the upper limit to the amount of radiation that a real substance may emit at a given temperature. At any given wavelength λ, emissivity is defined as the ratio of the actual emitted radiance, Rλ, to that from an ideal blackbody, Bλ,

Emissivity is a measure of how strongly a body radiates at a given wavelength; it ranges between zero and one for all real substances. A gray body is defined as a substance whose emissivity is independent of wavelength. In the atmosphere, clouds and gases have emissivities that vary rapidly with wavelength. The ocean surface has near-unit emissivity in the visible regions.

For a body in local thermodynamic equilibrium, the amount of thermal energy emitted must be equal to the energy absorbed; otherwise, the body would heat up or cool down in time, contrary to the assumption of equilibrium. In an absorbing and emitting medium in which Iλ is the incident spectral radiance, the emitted spectral radiance Rλ is given by

where aλ represents the absorptance at a given wavelength. If the source of radiation is in thermal equilibrium with the absorbing medium, then

so that

This is often referred to as Kirchhoff’s Law. In qualitative terms, it states that materials that are strong absorbers at a given wavelength are also strong emitters at that wavelength; similarly, weak absorbers are weak emitters.

Planetary albedo is defined as the fraction of the total incident solar irradiance, S, that is reflected back into space. Radiation balance then requires that the absorbed solar irradiance is given by

The factor of one-fourth arises because the cross sectional area of the earth disc to solar radiation, πr2, is one-fourth the earth radiating surface, 4πr2. Recalling that S = 1380 Wm-2, if the earth albedo is 30 percent, then E = 241 Wm-2.

Scattering is a physical process by which a particle in the path of an electromagnetic wave continuously abstracts energy from the incident wave and re-radiates that energy in all directions. Therefore, the particle may be thought of as a point source of the scattered energy. In the atmosphere, the particles responsible for scattering cover the sizes from gas molecules (~10-8 cm) to large raindrops and hail particles (~1 cm). The relative intensity of the scattering pattern depends strongly on the ratio of particle size to wavelength of the incident wave. If scattering is isotropic, the scattering pattern is symmetric about the direction of the incident wave. A small anisotropic particle tends to scatter light equally into the forward and backward directions. When the particle becomes larger, the scattered energy is increasingly concentrated in the forward directions. Distribution of the scattered energy involving spherical and certain symmetrical particles may be quantitatively determined by means of the electromagnetic wave theory. When particles are much smaller than the incident wavelength, the scattering is called Rayleigh scattering. For particles whose sizes are comparable to or larger than the wavelength, the scattering is customarily referred to as Mie scattering.

Rayleigh scattering indicates that the intensity scattered by air molecules in a specific direction is inversely proportional to the fourth power of the wavelength. A large portion of solar energy, lying between the blue-to-red portion of the visible spectrum, is Rayleigh scattered by the atmosphere. Blue light (λ ≈ 0.425 μm) has a shorter wavelength than red light (λ ≈ 0.650 μm). Consequently, blue light scatters about 5.5 more than red light. It is apparent that the λ-4 law causes more of the blue light to be scattered than the red, the green, and the yellow; and, so the sky, when viewed away from the sun’s disk, appears blue. Moreover, since the molecular density decreases drastically with height, it is anticipated that the sky should gradually darken to become completely black in outer space in directions away from the sun. And the sun itself should appear whiter and brighter with increasing height. As the sun approaches the horizon (at sunset or sunrise), sunlight travels through more air molecules; and, therefore, more and more blue light and light with shorter wavelengths are scattered out of the beam of light, and the luminous sun shows a deeper red colour than at the zenith.

Larger particles in the atmosphere, such as aerosols, cloud droplets, and ice crystals, also scatter sunlight and produce many fascinating optical phenomena. However, their single scattering properties are less wavelength selective and depend largely upon the particle size. As a result of this, clouds in the atmosphere generally appear white instead of blue. In a cloudy atmosphere, the sky appears blue diluted with white scattered light, resulting in a less pure blue sky than would have been expected just from Rayleigh scattering. Scattering by a spherical particle of arbitrary size has been treated exactly by Mie in 1908 by means of solving the electromagnetic wave equation derived from the fundamental Maxwell equations.

It is possible to formulate an expression analogous to absorption for dsλ, the fraction of parallel beam radiation that is scattered when passing downward through a layer of infinitesimal thickness, namely

where K is a dimensionless coefficient and A is the cross- sectional area that the particles in a unit volume present to the beam of incident radiation. If all the particles which the beam encounters in its passage through the differential layer were projected onto a plane perpendicular to the incident beam, the product A sec φ dz would represent the fractional area occupied by the particles. Thus, K plays the role of a scattering area coefficient which measures the ratio of the effective scattering cross section of the particles to their geometric cross section. In the idealized case of scattering by spherical particles of uniform radius r, the scattering area coefficient K can be prescribed on the basis of theory. It is convenient to express K as a function of a dimensionless size parameter α = 2πr/λ, which is a measure of the size of the particles in comparison to the wavelength of the incident radiation. Fig. shows a plot of α as a function of r and λ. The scattering area coefficient K depends not only upon the size parameter, but also upon the index of refraction of the particles responsible for the scattering. Fig. shows K as a function of two widely differing refractive indices. When α 1, we have Rayleigh scattering (K~α4); and between .1 and 50, we are in the Mie regime. For α>50, the angular distribution of scattered radiation can be described by the principles of geometric optics. The scattering of visible radiation by cloud droplets, raindrops, and ice particles falls within this regime and produces a number of distinctive optical phenomena such as rainbows, halos, and so forth. It should be noted that the scattering of terrestrial radiation in the atmosphere is of secondary importance compared to absorption and emission of terrestrial radiation.