## Top TOA and Bottom BOA of the Atmosphere Solar Radiation and Atmospheric Transmission

Several of the quantities encountered when studying the earth's short wave radiation balance are easily computed. Understanding these relationships can give the quantitatively minded reader more confidence about solar radiation and its trends.

Black-body radiation is described by the Stefan Boltzman equation, that is,

where B is radiant flux density emitted from a black body of temperature T, and s is the Stefan Bolzmann constant, 5.67 x 10-8 Wm 2K 4. Taking the sun's average surface temperature to be 5800 K, calculating solar output for a sphere of solar radius, 6.96 x 108 m, and irradiating a large sphere whose radius is the earth sun distance, or one astronomical unit (1.5 x 1011 m), the radiant flux density reaching a surface normal to the sun's rays on the earth before it is influenced by the atmosphere, that is, the extra-terrestrial 'solar constant', is 1380 Wm 2, which is very close to the currently accepted value of 1366 Wm 2 . The latter varies during the year by about 3.3% due to eccentricity of the earth's orbit. As long as the solar surface temperature and composition doesn't change, the yearly average will be constant. In fact, the solar constant has varied by much less than 1% over the past few centuries [3,4]. The ratio of the area of a sphere to that of a circle of the same radius is 4, so the mean solar radiant energy reaching the TOA is 342 Wm 2.

TOA (or extra-terrestrial) solar radiation on a plane parallel to the surface varies with the solar zenith angle, that is, the angle between the vertical and the solar vector. Calculation of solar angles and TOA solar radiation is straightforward and given elsewhere [1,5 7]. TOA values are used to compare with BOA measurements in order to determine atmospheric absorption of radiation, for example, atmospheric transmission and turbidity and aerosol optical depth.