Selective scattering by molecules is necessary but not sufficient for a blue sky. The atmosphere also must be optically thin, at least for most zenith angles. The blackness of space as a backdrop is taken for granted but also is necessary, as Leonardo recognized. Figure 8.2 shows the normal scattering optical thickness versus wavelength for the Standard Atmosphere.

400 450 500 550 600

Wavelength (nm)

Figure 8.2: Normal scattering optical thickness for the Standard Atmosphere. From Penndorf (1957).

400 450 500 550 600

Wavelength (nm)

Figure 8.2: Normal scattering optical thickness for the Standard Atmosphere. From Penndorf (1957).

From the two-stream approximation [Eq. (5.66)] with g = 0 (molecular scattering) the diffuse downward irradiance D^ of overhead skylight at the surface is

D 1 , where F0 is the incident irradiance and Tn is the normal optical thickness of the atmosphere. For Tn C 1 this approximates to

Equations (8.1) and (8.2) are for a black underlying surface (zero reflectivity). What about the other extreme, a white underlying surface (a reflectivity of 1)? We can answer this by solving Eq. (5.49) subject to equal downward and upward irradiances at the surface. But it is better for our souls (i.e., our physical intuition) if we guess that it must be approximately twice that given by Eq. (8.2) because the air is illuminated by two approximately equal sources: direct and reflected sunlight. From Fig. 8.2 it follows that the condition for the validity of Eq. (8.2) is satisfied. Thus the spectrum of skylight for a molecular atmosphere should be the solar spectrum modulated by Rayleigh's scattering law, as indeed it is for the overhead sky (Fig. 8.1). What about the other extreme, the horizon sky? To answer this leads us to consider airlight.

The only real distinction between airlight and skylight is that the backdrop for airlight can be finite objects at a finite distance, whereas the backdrop for skylight is nearly empty, boundless space. Because of airlight, light scattered by all the molecules and particles along the line of sight from observer to object, even an intrinsically black object is luminous. Consider a horizontally uniform line of sight uniformly illuminated by sunlight (Fig. 8.3). Denote by L0 the solar radiance illuminating the line of sight. The irradiance in the direction of the sun is

Ax x

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Figure 8.3: An observer looking at a distant black object nevertheless receives some light because of scattering by all the molecules and particles along the line of sight.

L0Qs, where is the solid angle subtended by the sun. Assume negligible reflection by the ground. The light scattered toward the observer per unit solid angle by all the molecules and particles in a small volume AAx at a distance x is therefore L0Qs(3AxAp(ê), where (3 is the scattering coefficient and p(ê) is the probability per unit solid angle of scattering in the direction ê. Divide by A, which is perpendicular to the line of sight, to obtain the contribution to the radiance from AAx. This is the radiance scattered at x toward the observer, which has to be multiplied by the transmissivity exp(-(x) to obtain the fraction of this radiance received by the observer. The total radiance as a consequence of scattering by everything along a line of sight between the observer and a black object at a distance d is the integral where t = [3d is the optical thickness along the path d and the two geometrical factors are lumped into a single factor G = Qsp(&). Underlying Eq. (8.3) is the assumption that light scattered out of the line of sight is not scattered again in this direction, which is a good assumption if the optical thickness in directions lateral to the line of sight is small (which it is for the clear atmosphere but not for fog).

Only in the limit t ^ 0 is L = 0 and a black object seen to be black. For t c 1, L « L0Gt. In a purely molecular atmosphere t varies with wavelength according to Rayleigh's law, and hence the distant black object is perceived to be bluish. As t increases so does L but not proportionately: the longer the path, the greater the number of scatterers, but also the greater the attenuation. The limiting value of L ([d > 1) is L0G, and the radiance spectrum is that of the source illumination on the line of sight regardless of the wavelength dependence of [ . This result ought to put an end to blather about the white horizon sky infallibly signaling scattering by "big particles."

Although the molecular optical thickness in the visible of Earth's atmosphere is small along a radial path, this is no longer true for paths near or along the horizon. The optical

Angle (Degrees)

Figure 8.4: Scattering optical thickness of a pure molecular atmosphere with scale height 8 km on Earth relative to the normal optical thickness for a range of zenith angles near the horizon. The solid line is the uniform atmosphere approximation; the dashed line is for an exponentially decreasing scattering coefficient.

thickness along any path is an integral:

For a path from the surface making a constant zenith angle © with the vertical direction in an atmosphere with an exponentially decreasing density of scatterers, Eq. (8.4) is where R is Earth's radius, H the scale height for molecular number density (i.e., the rate at which number density decreases exponentially with height), and the scattering coefficient at sea level. For a radial (normal) path (© = 0) Eq. (8.5) can be integrated to obtain

Thus the normal optical thickness of an atmosphere in which the number density of scatterers decreases exponentially with height is the same as that for a uniform atmosphere of finite thickness H.

Although Eq. (8.5) cannot be integrated analytically for arbitrary zenith angle, the uniform, finite atmosphere approximation

is surprisingly good right down to the horizon (© = n/2), as shown in Fig. 8.4. Taking the exponential decrease of molecular number density into account yields an optical thickness at most 10% lower. A flat Earth is one with infinite R, for which Eq. (8.7) yields the expected relation

The tangential (horizon) optical thickness (© = n/22) from Eq. (8.7) is to good approximation

because 2R/H > 1. For R = 6400 km and H = 8 km, Tt = 40t„.

The variation of brightness and color of dark objects with distance was called aerial perspective by Leonardo. By means of it we unconsciously estimate distances to objects of unknown size, such as mountains. Aerial perspective is similar to the variations of color and brightness of the sky with zenith angle. Although the optical thickness along a horizon path is not infinite, it is sufficiently large (Figs. 8.2 and 8.4) that GL0 is a good approximation for the radiance of the horizon sky. For isotropic scattering, a condition almost satisfied by molecules (see Sec. 7.3), G is about 10~5, the ratio of the solid angle subtended by the sun to the solid angle of all directions (4n). Thus the horizon sky is not nearly so bright as direct sunlight.

Unlike in the milk experiment described in Section 5.2, what one observes when looking at the horizon sky is not (much) multiply scattered light. Both the whiteness of milk and that of the horizon sky have their origins in multiple scattering but manifested in different ways. Milk is white because it is weakly absorbing and optically thick, and hence all components of incident white light are multiply scattered to the observer even though the violet and blue components traverse a shorter average path in the milk than the orange and red components. White horizon light is that which has escaped being multiply scattered, although multiple scattering is why this light is white (strictly, has the spectrum of the source). More light at the short-wavelength end of the spectrum than at the long-wavelength end is scattered toward the observer, as evidenced by 3 in Eq. (8.3). But long-wavelength light has the greater probability of being transmitted to the observer without being scattered out of the line of sight, as evidenced by exp(-3^) in Eq. (8.3). For a sufficiently long optical path, these two processes compensate, resulting in a horizon radiance that of the source.

With Eq. (8.3) in hand we can make a stab at estimating the ratio of the horizon radiance to the zenith (overhead) radiance. If we take the incident sunlight to be nearly directly overhead the horizon (tangential) radiance is approximately

Lt = LoQsP(90°){1 - exp(-Tt)} « Lo^sp(90°) (8.10)

and the zenith radiance is approximately

Ln = Lo^sp(0°){1 - exp(-Tn)} « Lo^sp(0°)Tn, (8.11)

where P is the phase function for molecular scattering and Lo is the radiance outside the atmosphere. All we need is the ratio of phase functions for the two scattering directions,

Figure 8.5: Measured ratio (solid line) of the horizon radiance to the radiance directly overhead with the sun high in the sky on a clear day in State College, Pennsylvania. The dotted line is this ratio predicted by simple theory for a pure molecular atmosphere.

Wavelength (nm)

Figure 8.5: Measured ratio (solid line) of the horizon radiance to the radiance directly overhead with the sun high in the sky on a clear day in State College, Pennsylvania. The dotted line is this ratio predicted by simple theory for a pure molecular atmosphere.

Figure 8.6: Measured ratio of the spectral radiance of magnesium oxide (MgO) powder illuminated by daylight to the spectral radiance of the horizon sky on a clear day in State College, Pennsylvania.

Wavelength (nm)

Figure 8.6: Measured ratio of the spectral radiance of magnesium oxide (MgO) powder illuminated by daylight to the spectral radiance of the horizon sky on a clear day in State College, Pennsylvania.

which we get from Eq. (7.117). The result is irk ,SJ2)

Although attenuation of sunlight illuminating the line of sight is neglected in Eqs. (8.10) and (8.11), when attenuation is included Eq. (8.12) is unchanged. Also p(0°) does not mean that

500 550 600

Wavelength (nm)

Figure 8.7: Spectra of overhead skylight from the two-stream theory for a molecular atmosphere with the present optical thickness (solid line), 10 times this thickness (dashed line), and 40 times this thickness (dotted line).

500 550 600

Wavelength (nm)

Figure 8.7: Spectra of overhead skylight from the two-stream theory for a molecular atmosphere with the present optical thickness (solid line), 10 times this thickness (dashed line), and 40 times this thickness (dotted line).

the sun is directly overhead and the line of sight is directly toward the sun but rather that the sun is high in the sky and the scattering angle is, say, less than 10-20°. Evidence for the validity of Eq. (8.12) is shown in Fig. 8.5, the ratio of measured radiances of the horizon and overhead skies, with the sun high in the sky, on a clear day. Agreement between measured ratios and those calculated with Eq. (8.12) using the normal optical thickness in Fig. 8.2 is surprisingly good. Moreover, the disagreement at longer wavelengths is in the expected direction: the normal optical thickness is almost always greater than that for a pure molecular atmosphere even in a very clean environment.

The optical thickness through the atmosphere along a horizon path is essentially infinite even in clear air. The source of illumination of this path is sunlight. The optical thickness of a large cumulus cloud is also essentially infinite, the source of illumination for which is also sunlight. Yet the radiance of the brightest cumulus cloud is larger, by roughly a factor of four, than that of the clear horizon sky. This is shown in Fig. 8.6, the ratio of the measured spectral radiance of magnesium oxide powder, which simulates a thick cloud, illuminated by daylight (i.e., direct sunlight and skylight) to the radiance of the horizon.

Although Eq. (8.12) is strictly valid only for small « 1) normal optical thicknesses, it does suggest that with increasing optical thickness the gradient (in angle) of skylight radiance should decrease. And, in fact, this is what is observed: on murky days the sky is more nearly uniformly bright. Moreover, Eq. (8.12) also suggests that as one ascends in the atmosphere, and hence rn of everything above one's elevation decreases, the gradient of skylight should increase; this can be observed from an airplane.

It follows from the plot of Eq. (8.1) in Fig. 5.13 and the molecular optical thickness spectrum that a blue sky is not inevitable. For optical thicknesses less than about 2.2, skylight irradiance relative to the solar irradiance increases with increasing optical thickness. Because the optical thickness of the molecular atmosphere increases with decreasing wavelength, and over the visible spectrum is less than about 0.36 (Fig. 8.2), skylight irradiance increases with decreasing wavelength. But for optical thicknesses greater than about 2.2, skylight irradiance decreases with increasing optical thickness. The smallest molecular optical thickness in the visible is about 0.04 (at 700 nm). Thus if the atmosphere were about 50 times thicker skylight irradiance would decrease with decreasing wavelength. Figure 8.7 shows calculated spectra of the zenith sky over black ground for a molecular atmosphere with the present normal optical thickness as well as for hypothetical atmospheres 10 and 40 times thicker. What we take to be inevitable is accidental: if Earth's atmosphere were much thicker the sky would not only be brighter, its color would be quite different from what it is now.

By showing that the blue sky is not inevitable, we hope to have given you a taste for thinking the unthinkable. We emphasized that the white horizon sky is not a consequence of "big particles" but occurs in a purely molecular atmosphere. Now we are going to turn this on its head and assert that "big particles" not only are not necessary for a white horizon sky, they can make it bluer than it would be otherwise.

Equation (8.3) for airlight has the same form for an atmosphere populated by molecules and particles because scattering coefficients and optical thicknesses are additive. For sufficiently large total optical thickness along a horizon path, the horizon radiance is

where p is the weighted average phase function for molecules (to) and particles (p):

To understand the observable consequences of Eqs. (8.13) and (8.14) consider a few limiting cases. Suppose that both scattering toward the observer and total scattering are dominated by particles (ftppp > /3mpm and /3P > 3m):

If the particles are big in the sense that angular scattering by them is independent of wavelength for scattering angles tf of interest, the airlight spectrum is white (i.e., that of the source L0). No surprise here.

Now suppose that both scattering toward the observer and total scattering are dominated by molecules (3mPm > 3pPp and 3m > 3p):

Again the radiance is that of the source because the molecular phase function is to good approximation independent of wavelength. Equation (8.14) also predicts a white horizon if the particles are sufficiently small that scattering by them has the same wavelength dependence as molecular scattering.

If molecules dominate total scattering whereas particles dominate scattering toward the observer (3m > 3p and 3pPp > 3mPm)

Here the horizon radiance is inversely related to the molecular scattering coefficient, and hence the airlight is reddish. This is a variation on the theme of distant reddish clouds discussed in the following section on colors at sunrise and sunset. Particles distributed along the entire line of sight play the same role as localized clouds.

When we consider the converse of the previous limiting case, namely total scattering dominated by particles but scattering toward the observer dominated by molecules (3p > 3m and 3mPm > 3ppp), we obtain a surprising result:

For this example, the horizon airlight is bluish. To understand this perhaps contra-intuitive result we return to the example of a pure molecular atmosphere for which the horizon sky is white even though the scatterers are selective. Scattering of sunlight toward the observer favors light at the short wavelength end of the visible spectrum. If this light were transmitted without attenuation, the airlight would be bluish. But it is impossible for molecules to scatter light toward the observer without also scattering some of this light out of the line of sight. This selective attenuation of light scattered toward the observer favors the long wavelength end of the spectrum. For a sufficiently long optical path selective scattering toward the observer is exactly balanced by selective scattering out of the line of sight.

Now we can better understand why big particles can, contrary to what might be expected, make the horizon sky bluer than it would otherwise be. Given the assumptions underlying Eq. (8.18) we can write the airlight radiance as

The factor 3mpm is the wavelength-dependent scattering toward the observer and is the same everywhere along the line of sight. The integral is an attenuation function; the exponential term in the integral is the probability that light scattered at x toward the observer will not be scattered again in traversing this distance. Although light scattered at all points on the line of sight contributes to the radiance, most of the contribution comes from scattering at distances less than about 3/(3m + 3p), which is approximately 3/3P if 3P > 3m. Over such distances, however, molecular scattering does not greatly redden the transmitted light (i.e., exp{-3mx} « 1 for x < 3/3p). Thus the color balance is not restored by attenuation as it was for a pure molecular atmosphere.

In his famous book, The Nature of Light and Color in the Open Air, Marcel Minnaert notes that "to this day there are scientists who do not consider the problem of the blue sky as being definitively solved...On very exceptional days, occurring perhaps not even once a year, the sky is beautifully blue right down to the horizon. Observations on days like these should be carefully recorded and described...for according to the theory of scattering, such a phenomenon is impossible: with layers of such thickness, the air ought to appear white." Yet the "theory of scattering" [Eq. (8.18)] does show why the sky can be beautifully blue right down to the horizon, although Minnaert was correct in saying that this is "exceptional." The concentration of particles has to be high enough that total scattering is dominated by them, but sufficiently low that differential scattering is dominated by molecules. This is possible

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