If short-wavelength light is preferentially scattered out of direct sunlight, long-wavelength light must be preferentially transmitted in the direction of sunlight. Transmission is exponential if multiple scattering is negligible (see Sec. 5.2):
where L is the radiance in the direction of the sun, L0 is that of sunlight outside the atmosphere, and t is the optical thickness along the line of sight. If the wavelength dependence of t follows Rayeigh's scattering law, transmitted sunlight is reddened, comparatively richer at the long-wavelength end of the visible spectrum than the incident light. But to say that sunlight is reddened is not to say that it is red. The perceived color can be yellow, orange, or red depending on the magnitude of the optical thickness. Equation (8.20) applies to the radiance only in the direction of the sun. Yet oranges and reds can be seen in other directions because reddened sunlight illuminates scatterers that are not on the line of sight to the sun. A striking example of this is a horizon sky tinged with oranges and pinks in the direction opposite the sun.
In an atmosphere free of all particles the optical thickness along a path from the sun, even on or below the horizon, is not sufficient to give perceptually red transmitted light. Although selective scattering by molecules yields a blue sky, reds are not possible in a molecular atmosphere, only yellows and oranges. Although this can be proven by the kind of colorimetric analysis in Section 4.3, Nature itself provides the proof. On exceptionally clear days the horizon sky at sunrise or sunset may be tinged with yellow or orange but not red.
The color and brightness of the sun changes as it arcs across the sky because the optical thickness along the line of sight to it changes with solar zenith angle ©. If Earth were flat, as some still aver, the transmitted solar radiance would be
This equation is a good approximation except near the horizon. On a flat Earth, the optical thickness is infinite for horizon paths. On a spherical Earth, all optical thicknesses are finite although much larger for horizon than for vertical paths (Fig. 8.4).
Variations on the theme of reds and oranges at sunrise and sunset can be seen even when the sun is overhead. The radiance at an observer an optical distance t from a horizon cloud is the sum of transmitted cloudlight and airlight:
which is an extension of Eq. (8.3). If the cloud is approximated as an isotropic reflector with reflectivity R and illuminated at an angle $ from the normal to it, the cloud geometrical factor Gc is QsR cos $. If Gc > G the observed radiance is redder than the incident radiance, but if
Gc < G the observed radiance is bluer than the incident radiance. Thus distant horizon clouds can be reddish if they are bright or bluish if they are dark.
Underlying Eq. (8.22) is the implicit assumption that the line of sight is uniformly illuminated by sunlight. The first term in this equation is airlight; the second is transmitted cloud-light. Suppose, however, that the line of sight is shadowed from direct sunlight by clouds that do not occlude the distant clouds. This may reduce the first term in Eq. (8.22) so that the second term dominates. Thus under a partly overcast sky, distant horizon clouds may be reddish even when the sun is high in the sky.
Small particles affect the color of the low sun out of proportion to their normal optical thickness because they are concentrated more toward the surface. The scale height for molecules is about 8 km whereas that for particles is typically 1-2 km. Subject to the approximations underlying Eq. (8.7), the ratio of the tangential (horizon) optical thickness for particles Ttp to that for molecules Ttm is where m denotes molecules and p particles. Because of the incoherence of scattering by atmospheric molecules and particles, scattering coefficients are additive, and hence so are optical thicknesses. Even for equal normal optical thicknesses, the tangential optical thickness for particles is more than twice that for molecules. As we noted previously, molecules by themselves cannot give red sunsets and sunrises. Molecules need the help of small particles, and for a fixed normal optical thickness for particles, their tangential optical thickness is greater the more they are concentrated near the surface.
For equal normal optical thicknesses, particles also disproportionately increase the rate at which transmitted radiance decreases with angle near the horizon:
which follows from Eq. (8.7) and Eq. (8.20) with t = Tm + tp. The rate of decrease because of particles can be so great that the color of the setting sun varies across its diameter, from yellow at its top, to red at its bottom.
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