cos e
FIGURE 3.18 Typical light ray striking a thin layer of air in the atmosphere (adapted from Madronich, 1987).
dz cos e
FIGURE 3.18 Typical light ray striking a thin layer of air in the atmosphere (adapted from Madronich, 1987).
the portion of the surface perpendicular to the incoming beam is 2x = (da)cos 9. Thus the net photons (or energy), dP in a wavelength interval d\ that originates in a solid angle dw and at an angle 6 to the normal and crosses a small surface area, da, in time dt is given by dP = L(\, 9, 0)cos 9 dadoi did A. (DD)
The irradiance E( A), which is directly measured by flat-plate devices, is by definition the total number of photons per unit surface area, time, and wavelength. Thus
The actinic flux F(\) is the total incident light intensity integrated over all solid angles, given by
Thus the irradiance, E( A), and the actinic flux, F( A), differ by the factor cos 6. Only for 6 = 0°, i.e., for a parallel beam of light perpendicular to the surface, are the irradiance and flux equal.
We now need to convert den into terms involving the spherical coordinates 0 and cf>. As shown in Fig. 3.19a, a given solid angle m traces an area a on the surface of a sphere of radius r. When a = r2, the solid angle w is by definition 1 sr (sr = steradian). For the more general case of a surface area a subtended by the solid angle w, or m (steradians) = a/r2
As shown in Fig. 3.19b, for small changes in the angles 9 and 4>, there is a change in the surface area, da, on a sphere of radius r given by da = (r sin 0 d$)(rd0) = r2 sin 9 d9 d(f), i.e., dw = da/r2 = sin 9 d0 dcf).
Combining Eqs. (FF) and (HH), the actinic flux becomes
Similarly, the irradiance E( A) is given by E( A) = f L( A, 9, (f>)cos 9 dw
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