SCATTERING OF LIGHT IN THE ATMOSPHERE

I. I. Tikhanovskii.

§ 1. Introduction.

In 1910 Lord Rayleigh1, in his lecture “The Colour of the Sea and Sky,” delivered at the Royal Institution, touching on the question of the origin of the light of the sky, said the following: “As regards the light of the sky, the theory which ascribes its origin to light scattering caused by the presence of the tiniest particles, most of which are smaller than a light wave, may now be regarded as generally accepted. It explains, at least in a first approximation, both the polarization and the colour of the light of the sky... There remains one interesting question still unresolved: on what kind of small particles, which scatter chiefly the short waves, does the azure of the sky depend? It cannot be denied that small particles of salts and other suspended substances, among which must be included the germs of the plant world, play a certain role...; but, apparently, it may be considered established that the air molecules themselves are capable of scattering blue light to a sufficient degree to give the sky its azure appearance.”

These few words of a first-rate scientist give a quite precise characterization of the state of the question of atmospheric light scattering at that time, i.e. 16 years ago. Of undoubted interest is a characterization of the present state of this question, which is of great interest not only for geophysics and meteorology, but also for pure physics. In particular, very interesting are attempts to construct a theory of atmospheric light scattering that goes beyond the first approximation, and attempts to resolve the question of the nature of the particles that scatter light in the atmosphere.

To give, as far as possible, an objective account of these questions is the purpose of the present article.

§ 2. The Simplest Theory of the Scattering of Light in the Terrestrial Atmosphere

Let us consider the simplest theory of the scattering of light in the terrestrial atmosphere, which assumes that the solar (or lunar) light entering the atmosphere is scattered in it by air molecules, which are here taken to be isotropic1. The theory further assumes that these molecules scatter only the light falling on them directly from the sun, and does not take into account the circumstance that the molecules may also scatter light falling on them from other molecules (double and, in general, multiple scattering of light), and light coming from the earth’s surface.

Considering the question from the point of view of the electron theory, we assume that each molecule contains a certain number of electrons, each of which oscillates under the action of the incident wave about its position of equilibrium. The strength of the electric \((E)\) and magnetic \((H)\) fields produced by an oscillating electron at a distance \(r\) from the electron in a direction making an angle \(\theta\) with the direction of the rectilinear oscillations of the electron is determined, as is known, by the formula

\[ E = H = \frac{e j \sin \theta}{c^2 r}, \tag{1} \]

where \(e\) is the charge of the electron, \(j\) its acceleration at the given moment, and \(c\) the velocity of light. If the value of \(j\) corresponds to the instant of time \(t\), then the values \(E\) and \(H\), determined by formula (1), correspond, of course, to the instant \(t + \frac{r}{c}\). The (mean) intensity of the light scattered by the electron at the point determined by the coordinates \(r\) and \(\theta\) will be

\[ I = \frac{e^2 \overline{j^2}\sin^2\theta}{4\pi c^3 r^2}, \]

where \(\overline{j^2}\) is the time mean of the quantity \(j^2\). Let a sinusoidal wave of period \(T\) fall upon the electron; under its action the electron is brought into forced sinusoidal oscillations of the same period \(T\). Then, denoting by \(s\) the amplitude of the electron’s oscillations, we may write the preceding formula in the following form:

\[ I = \frac{2\pi^3 e^2 (e s)^2 \sin^2\theta}{r^2 \lambda^4}, \tag{2} \]

where \(\lambda\) is the length of the light wave.

Let, further, a wave, unpolarized, fall along the axis \(X\), in the direction indicated by the arrow, upon an electron situated at the point \(O\).

light, having at the point \(O\) amplitudes of the components of the electric-field intensity \(E_y\) and \(E_z\). Let us consider light scattered in the direction \(OP\), lying in the plane \(XOY\) and forming an angle \(\varphi\) with the direction of the incident rays. Then, applying formula (2) to each of the components of the electron oscillation along the axes \(Y\) and \(Z\), we obtain for the components of the intensity of the scattered light, polarized respectively in the planes \(POX\) and \(POZ\), the expressions

\[ I_1=\frac{2\pi^3 c(ez)^2}{r^2\lambda^4},\qquad I_2=\frac{2\pi^3 c(ey)}{r^2\lambda^4}\cos^2\varphi. \tag{3} \]

Fig. 1.

Passing from a single electron to a system of electrons of a molecule and observing that the electrons of one and the same molecule oscillate coherently if the wavelength is large in comparison with the distances of the intramolecular electrons (for rays of the visible part of the spectrum this is the case), we must add the amplitudes of the oscillating electrons, i.e. replace in formulas (3) \((ez)\) and \((ey)\) by \(\sum ez\) and \(\sum ey\), where the summation is carried out over all electrons of the molecule; for brevity, denote \(\sum ez\) and \(\sum ey\) respectively by \(\xi\) and \(\eta\). Let us note the following. We have made the assumption that in the presence of the components \(E_z\) and \(E_y\) there will exist components of the electric displacement only along the same axes \(Z\) and \(Y\). This assumption is valid, as we shall see later, only in the case of isotropic molecules.

If, further, from scattering by one molecule we pass to scattering by a system of molecules contained in a unit volume, then, because of the incoherence of the oscillations of the electrons of the individual molecules, we must in this case, in summing, add not amplitudes but energies. We must, consequently, instead of \(\xi^2\) and \(\eta^2\), take \(L\xi_m^2\) and \(L\eta_m^2\), where \(L\) is the number of molecules in a unit volume, and \(\xi_m^2\) and \(\eta_m^2\) are the mean values of \(\xi^2\) and \(\eta^2\) for all molecules. But, owing to the isotropy of the molecules, the orientations in space of all molecules are equivalent, and therefore instead of \(\xi_m^2\) and \(\eta_m^2\) one may take \(\xi^2\) and \(\eta^2\) for any one molecule. Formulas (3) then take the form

\[ I_1=\frac{2\pi^3 cL\xi^2}{r^2\lambda^4},\qquad I_2=\frac{2\pi^3 cL\eta^2}{r^2\lambda^4}\cos^2\varphi. \tag{4} \]

But \(L\xi\) and \(L\eta\) are the components of the electric moment of a unit volume along the corresponding axes, and therefore, for a sufficiently rarefied gas, such as atmospheric air,

\[ L\xi=\frac{n^2-1}{4\pi}E_z,\qquad L\eta=\frac{n^2-1}{4\pi}E_y, \tag{5} \]

where \(n\) is the refractive index of air.

Substituting the obtained values for \(\zeta\) and \(\eta\) into formulas (4), we obtain, introducing the intensity of the incident light,

\[ I_0=\frac{c}{4\pi}E_z^2=\frac{c}{4\pi}E_y^2, \]

the final formulas

\[ I_1=I_0\frac{\pi^2(n^2-1)^2}{2r^2\lambda^4 L},\qquad I_2=I_0\frac{\pi^2(n^2-1)^2}{2r^2\lambda^4 L}\cos^2\varphi. \tag{6} \]

From these formulas we obtain expressions for the intensity of the light scattered by a unit volume,

\[ I=I_1+I_2=I_0\frac{\pi^2(n^2-1)^2}{r^2\lambda^4 L}\frac{1+\cos^2\varphi}{2} \tag{7} \]

and for the magnitude of the polarization of the scattered light

\[ P=\frac{I_1-I_2}{I_1+I_2}=\frac{\sin^2\varphi}{2-\sin^2\varphi}. \tag{8} \]

Formulas (7) and (8) give, in the first approximation, an explanation of the color and polarization of skylight. Namely, formula (7), containing the factor \(\dfrac{1}{\lambda^4}\), thereby explains the blue color of the sky. Formula (8) gives, first of all, a theoretical explanation of the fact of the polarization of the sky; further, according to this formula, the maximum of polarization must be situated at \(90^\circ\) from the sun, decreasing from there both in the direction toward the sun and away from the sun; at \(90^\circ\) from the sun the polarization should, according to formula (8), be complete; in the direction toward the sun and toward the antisolar point it should be equal to zero.

§ 3. Relation of Rayleigh’s theory to reality.

We now turn to the question of the extent to which the theory set forth, belonging to Rayleigh, agrees with the results of atmospheric-optical investigations.

As regards formula (7), spectrophotometric investigations of skylight have, as is known, given entirely satisfactory confirmation of the law of inverse proportionality to \(\lambda^4\). The question of whether the molecules of air play the role of the scattering particles of the atmosphere was resolved, proceeding from formula (7), in the following way.

As a consequence of the scattering of light by the particles of the atmosphere, sunlight, as it passes through the atmosphere, must undergo attenuation; moreover, the attenuation of energy through scattering is, as is known,

the name extinction. If a layer of the atmosphere \(dx\) causes a loss of energy equal to \(dI\), then

\[ dI=-Ikdx, \tag{9} \]

where \(k\) is the absorption coefficient, in the present case the extinction coefficient. Let us derive, on the basis of Rayleigh’s theory, an expression for the extinction coefficient.

The amount of energy scattered in some direction by one molecule will, according to formula (7), be

\[ I=I_0\frac{\pi^2(n^2-1)^2}{r^2\lambda^4L^2}\frac{1+\cos^2\varphi}{2}. \]

Let us calculate the total amount of energy scattered by a molecule, multiplying the right-hand side by \(2\pi r^2\sin\varphi\,d\varphi\) and taking the integral with respect to \(\varphi\) from \(0\) to \(\pi\). We then obtain

\[ I_0\frac{8\pi^3}{3\lambda^4}\frac{(n^2-1)^2}{L^2}. \]

Since a layer of the atmosphere of thickness \(dx\) and cross-section \(1\ \mathrm{cm}^2\) contains \(Ldx\) particles, the loss of energy in passing through a layer of thickness \(dx\) will be

\[ dI=-I_0\frac{8\pi^3}{3\lambda^4}\frac{(n^2-1)^2}{L}\,dx. \]

Comparing this expression with (9), we obtain the following expression for the extinction coefficient:

\[ k=\frac{8\pi^3}{3\lambda^4}\frac{(n^2-1)^2}{L}. \tag{10} \]

In 1909 Schuster1, making use of careful measurements of the coefficient of absorption of light in the atmosphere, carried out by Abbot on Mount Wilson and in Washington, showed that for clear days both on Mount Wilson and in Washington the observed values of the absorption coefficient agree completely with those which can be obtained by calculation from formula (10), if in it \(L\) is taken to be the number of molecules in \(1\ \mathrm{cm}^3\) of gas under normal conditions. Thus the nature of the particles causing the scattering of light in the atmosphere was revealed: these particles proved to be air molecules.

It would seem that, after this entirely satisfactory confirmation of the law \(1/\lambda^4\) and the brilliant, as it were, elucidation of the kind of particles causing atmospheric light scattering, one could consider

the theory of this scattering developed beyond the first approximation. However, this proves to be far from the case if one turns in more detail to Rayleigh’s theory on the basis of investigations of the polarization of skylight.

Although the polarization of the sky does indeed have its greatest value at \(90^\circ\) from the sun, the polarization there is not complete: it does not exceed \(850\)¹) (expressing the magnitude of the polarization in promilles, i.e. in thousandths of unity). Further, whereas according to Rayleigh’s theory the plane of polarization of points of the solar vertical must everywhere coincide with the plane of this vertical (see § 2)—the so-called positive polarization—observations have established, in the solar vertical near the sun and the antisolar point over a range of several tens of degrees, the existence of negative polarization, i.e. the plane of polarization of the light emitted by these regions of the solar vertical is perpendicular to the plane of the solar vertical. As a result of the existence of negative polarization, in the solar vertical, at the places of transition from positive to negative polarization, we have the so-called neutral points (the points of Arago, Babinet, Brewster), emitting unpolarized light. Finally, whereas from formula (8) there follows the independence of the magnitude of the polarization from the wavelength, observations have ascertained a dependence of the magnitude of the polarization of skylight on the length of the light wave (the so-called dispersion of polarization); as a result of this dispersion of polarization, the two components of polarized skylight have a different spectral composition, and consequently also a different color (the so-called polychroism of the polarization of skylight).

Rayleigh’s theory is thus only a first, and moreover rather crude, approximation to the actual picture of the phenomena of the polarization of the sky. We therefore proceed to the consideration of attempts to construct more exact theories of the polarization of skylight, and consequently theories of atmospheric light scattering in general.

§ 4. DOUBLE SCATTERING OF LIGHT IN THE ATMOSPHERE.

One of the factors which must undoubtedly be taken into account by a theory of atmospheric light scattering is double scattering of light, i.e. scattering by particles of light that has already been scattered once. The possible role of this factor in the atmosphere was pointed out already by Rayleigh; later Soret and Gurban made attempts to take into account the influence of this factor on the phenomena of the polarization of the sky. In the most complete

¹) The highest value of the magnitude of sky polarization ever observed, so far as I know, is \(817\); this value was observed by me on Ai-Petri (\(1200\) m) in Crimea in 1925.

Furthermore, a theory of the polarization of the sky, taking account of double scattering, was developed in 1914 by Ahlgrimm1.

Let us consider the simplest case, when the sun is on the horizon and we study the light coming from the zenith. Let us take a system of coordinates at whose center the observer is located, the \(X\)-axis directed toward the sun, and the \(Z\)-axis toward the zenith. Applying to this case formula (6), we have (for single scattering of light) only one component of the intensity of the scattered light, with oscillations along the \(y\)-axis, equal to

\[ I'_y = I_0 \frac{\pi^2(n^2 - 1)^2}{2r^2\lambda^4 L} \]

and the magnitude of the polarization

\[ p = 1000. \]

Proceeding from the Rayleigh theory of scattering by isotropic molecules, Ahlgrimm’s theory gives the following expressions for the intensity components, as the result of double scattering alone,

\[ I''_y = I_0 \frac{\pi^2(n^2 - 1)^2}{2r^2\lambda^4 L}\,Q, \qquad I''_x = I_0 \frac{\pi^2(n^2 - 1)^2}{2r^2\lambda^4 L}\,M, \tag{11} \]

where

\[ Q = 0.26877, \qquad M = 0.07814. \]

The combined action of single and double diffusion of light gives the intensity components

\[ I'_y + I''_y \qquad \text{and} \qquad I''_x, \]

with the aid of which the following value is obtained for the magnitude of the polarization

\[ p = 884. \]

This value agrees more or less with the results of measurements of the magnitude of polarization at the zenith, which, as was already indicated above, reaches the value 847.

Let us note further that Ahlgrimm’s theory also makes it possible to explain the existence of negative polarization near the sun and the antisolar point, and consequently also of neutral points; moreover, for the distances of these points from the sun or from the antisolar point, values are obtained that agree more or less with the observed ones. Finally, since in secondarily scattered light the rays with short wavelength will predominate more than in primarily scattered light, the spectral composition of the components \(I'_y + I''_y\) and \(I''_x\) will be different; hence

there follows a difference in polarization for rays of different wavelength, the polarization having to decrease with decreasing wavelength, and a difference in the colors of the two components of such a character that the second component—\(I''_x\)—must have a more saturated blue color than the first \(I'_y + I''_y\). Observations made by me \(^{1}\) in various places in the Crimea on the polarization of the sky in different parts of the spectrum and on the color of the polarization components confirm this: with the purest atmosphere the maximum of polarization occurs in the red part of the spectrum, decreasing with decreasing wavelength; the second, smaller, component of the skylight has, with the purest atmosphere, a deeper blue color than the first, larger one.

§ 5. Pro and contra of double scattering in the atmosphere.

It might have seemed possible to stop at the theory of atmospheric light scattering that takes account of primary and secondary scattering, as a theory giving a satisfactory second approximation to reality, and to devote oneself to its further development. However, a few years after the appearance of Al’grim’s theory, the question of the role of double scattering in the atmosphere took a completely unexpected direction.

Let us note, first of all, that the second approximation of the theory of atmospheric light scattering could be obtained if one takes into account the presence in the atmosphere of large particles, i.e., particles whose dimensions are not small in comparison with the wavelength of light. Thus, Rayleigh had already pointed out that the presence of large particles in the atmosphere (dust, etc.) is capable of explaining the incomplete polarization at \(90^\circ\) from the sun. The theory of the optics of turbid media consisting of particles not small in comparison with the wavelength of light was developed, as is known, for the case of spherical particles, in 1908 by Mie \(^{2}\).

In 1919 Schirmann \(^{3}\), developing Mie’s theory, showed that, by admitting the presence of large particles in the atmosphere, one can explain not only the incomplete polarization at \(90^\circ\) from the sun, but also negative polarization, neutral points, dispersion, and the polychroism of polarization. However, for a qualitative, and still more for a quantitative, explanation of the indicated phenomena, knowledge of the sizes and other physical constants of the scattering particles is necessary; Schirmann, in doing so, took the position of completely denying any significant role of double scattering of light in the atmosphere. In this he relied on the investigation of Maxwell-Garnett, who, starting from the theo-

\(^{1}\) Tichanowsky. Meteor. Zeitschr. 43, 288 (1926).

\(^{2}\) Mie. Ann. d. Phys., 25, 377 (1908).

\(^{3}\) Schirmann. Ann. d. Phys., 59, 493 (1919); 61, 195 (1920); Meteor. Zeitschr., 37, 12 (1920).

SCATTERING OF LIGHT IN THE ATMOSPHERE

Lorenz, showed that the intensity of doubly scattered light is very small if the scattering particles do not lie very close to one another. According to Schirmann, in the atmosphere we have precisely such a rarefied turbid medium that double scattering in it plays no appreciable role.

In 1925 there appeared a paper by W. Milch1, devoted to the investigation of the influence of large particles in the atmosphere on the phenomena of polarization of the sky, where Milch completely neglects double scattering for the following reasons. Investigating the distribution of energy in the spectrum of light doubly scattered according to Rayleigh’s law, Milch comes to the conclusion that if, in singly scattered light, the maximum of intensity falls at the wavelength \(455\ \mu\mu\), then in doubly scattered light it falls at \(375\ \mu\mu\), i.e. it will be situated in the ultraviolet part of the spectrum. For us it remains unclear why from this, in Milch’s opinion, it follows that doubly scattered light for the visible part of the spectrum will be so insignificant that it cannot be detected by the eye.

As for the arguments recently advanced in favor of a significant role of double scattering in the atmosphere, we shall first of all note the following circumstance. Double scattering of light may be negligible under laboratory conditions, but it may be quite significant in the atmosphere, owing to its great extent. This opinion was expressed by Bothe2 and F. Exner3.

Further, in favor of an undoubtedly significant effect of double scattering in the atmosphere, for example, the following facts speak.

The blue haze enveloping mountain ranges can often still be visible even after sunset, although the part of the atmosphere which lies between the observer and the mountains is no longer illuminated directly by the rays of the sun (Exner3).

The values of the coefficient of transparency of the atmosphere obtained from sighting distant objects turn out to be comparatively high; this may be explained by the fact that in this case, owing to the large number of scattering particles between the object and the observer, the latter send into the observer’s eye a comparatively large amount of light falling on them from the side (Jensen4).

A strong argument in favor of double scattering of light in the atmosphere is the agreement, noted above, of Al’grim’s theory with the results of investigations of the polarization of skylight5.

§ 6. Scattering of Light by Anisotropic Molecules.

We shall return somewhat later to the question raised in the preceding paragraph concerning the role of large particles in atmospheric scattering of light, and for the present we shall consider one circumstance whose inclusion makes it possible to construct a theory of the polarization of the sky more consistent with reality than the Rayleigh theory set forth in § 2.

When, several years ago, a number of investigators discovered, under laboratory conditions, the scattering of light by pure (uncontaminated) gases and measured the degree of polarization of the scattered light, it turned out, as is known, that the degree of polarization at \(90^\circ\) to the direction of the incident rays is not complete. Since here this incompleteness of polarization could be ascribed neither to double scattering (since in the case, for example, of argon the observed polarization was complete), nor to the presence of extraneous large particles, it became necessary to ascribe to gas molecules, generally speaking, a certain anisotropy. After this the necessity will become clear of revising the theory set forth in § 2 and of generalizing it to the case of anisotropic molecules.

We proceed to an exposition of the theory of the scattering of light by anisotropic molecules, following the works of Cabannes \(^{1}\).

We shall now restrict ourselves only to considering light scattered perpendicular to the direction of the incident rays. We have, therefore, a gaseous medium consisting of anisotropic molecules oriented at random. Let this medium be illuminated by unpolarized light incident along the \(x\)-axis. Consider the light scattered in the direction of the \(y\)-axis. Let the components of the electric-field strength of the incident wave have amplitudes \(E_z\) and \(E_y\). We shall first consider the action of the component \(E_z\).

For one molecule we have, using the first of formulas (3), the following components of intensity

\[ I'_z=\frac{2\pi^3 c \zeta^2}{r^2\lambda^4}, \qquad I'_x=\frac{2\pi^3 c \xi^2}{r^2\lambda^4}. \]

Denoting by \(I_z\) the component of the intensity of the incident light having oscillations along the \(z\)-axis, we have

\[ I_z=\frac{c}{8\pi}E_z^2. \]

\(^{1}\) Cabannes. Annales de phys., 15, 5 (1921); the theory of the scattering of light by anisotropic molecules was also developed by Born: Verhandl. d. Deutsch. Phys. Ges. 19, 43 (1917); 20, 16 (1918), and Rayleigh, Phil. Mag. 35, 373 (1918).

SCATTERING OF LIGHT IN THE ATMOSPHERE

Dividing each of the two preceding ones by this equality, we obtain

\[ I'_z=I_z\,\frac{16\pi^4\zeta^2}{r^2\lambda^4 E_z^2},\qquad I'_x=I_z\,\frac{16\pi^4\xi^2}{r^2\lambda^4 E_z^2}. \]

Passing from one molecule to \(L\) molecules contained in a unit volume, the preceding formulas then become the following:

\[ I'_z=I_z\,\frac{16\pi^4 L\zeta_m^2}{r^2\lambda^4 E_z^2},\qquad I'_x=I_z\,\frac{16\pi^4 L\xi_m^2}{r^2\lambda^4 E_z^2}, \tag{12} \]

where \(\zeta_m^2\) and \(\xi_m^2\) are the mean values of \(\zeta^2\) and \(\xi^2\) for all molecules. If we denote the projections of the amplitude of the electric field intensity \(E_z\) on the three axes of the anisotropic molecule by \(U, V, W\), and similarly those of the electric moment by \(\Sigma ea, \Sigma eb, \Sigma ec\), we have

\[ \Sigma ea=AU,\qquad \Sigma eb=BV,\qquad \Sigma ec=CW, \]

where \(A, B, C\) are certain coefficients that may be called the polarization coefficients of the molecule. The difference of these coefficients \(A, B, C\) characterizes the anisotropy of the molecule; for an isotropic molecule \(A=B=C\).

The quantities \(\zeta_m^2\) and \(\xi_m^2\) can be expressed in terms of \(A, B, C\), namely

\[ \frac{\xi_m^2}{E_z^2}=\frac{1}{15}\left(\Sigma A^2-\Sigma BC\right),\qquad \frac{\zeta_m^2}{E_z^2}=\frac{1}{15}\left(3\Sigma A^2+2\Sigma BC\right), \tag{13} \]

where

\[ \Sigma A^2=A^2+B^2+C^2,\qquad \Sigma BC=BC+CA+AB. \]

We do not give the derivation of formulas (13), referring the reader to the above-mentioned paper by Cabannes.

Formulas (12), by means of formulas (13), take the form

\[ I'_x=I_z\,\frac{16\pi^4 L}{15\,r^2\lambda^4}\left(\Sigma A^2-\Sigma BC\right),\qquad I'_z=I_z\,\frac{16\pi^4 L}{15\,r^2\lambda^4}\left(3\Sigma A^2+2\Sigma BC\right). \]

Let us now take into account the existence, in the incident wave, in addition to the component \(E_z\), also of the component \(E_y\). This component, if to it we apply the same reasoning that we applied to the component \(E_z\), will produce in the ray scattered in the direction \(y\) the following components of intensity:

along the \(x\)-axis—a component of the same magnitude as that produced by \(E_z\), i.e. \(I'_x\), since the vectors \(E_z\) and \(E_y\) are symmetric with respect to the \(x\)-axis;

along the \(z\)-axis—a component also equal to \(I'_x\), since the axes \(x\) and \(y\) are symmetric with respect to the vector \(E_y\).

Thus the components of the intensity of the scattered light, if the intensity of the incident unpolarized light is denoted by \(I_0\), will be

\[ I_1=I'_z+I'_x=I_0\,\frac{16}{15}\,\frac{\pi^2L}{r^2\lambda^4}\, \frac{4\Sigma A^2+\Sigma BC}{2} \quad \text{(vibr. along the } z\text{-axis)} \tag{14} \]

\[ I_2=I'_x+I'_x=I_0\,\frac{16}{15}\,\frac{\pi^2L}{r^2\lambda^4}\, \left(\Sigma A^2-\Sigma BC\right) \quad \text{(vibr. along the } x\text{-axis)} \]

We shall replace the quantities \(\Sigma A^2\) and \(\Sigma BC\) entering into these two formulas by two others, more convenient. From (14) we have

\[ \frac{I_2}{I_1}= \frac{2(\Sigma A^2-\Sigma BC)}{4\Sigma A^2+\Sigma BC} =\rho. \tag{15} \]

Let us now make one remark concerning formulas (14) and (15). For the case of isotropic molecules \(\Sigma A^2=\Sigma BC\), and consequently \(I_2=0\). Hence the component of the vibrations along the \(x\)-axis, arising in the scattered ray, owes its origin exclusively to the anisotropy of the molecules. Since \(\rho\) is therefore a measure of the anisotropy of the molecules, insofar as this anisotropy manifests itself in the purely optical phenomenon of light scattering, we shall call the quantity \(\rho\) the coefficient of optical anisotropy of the molecule. This will be one of the quantities through which we shall express \(\Sigma A^2\) and \(\Sigma BC\). The second quantity will be the refractive index of the medium—\(n\).

Formula (5) for the case of anisotropic molecules may be written in the following form:

\[ \frac{\xi_m}{E_z}=\frac{n^2-1}{4\pi L}. \]

The quantity \(\xi_m\) can be expressed through \(A, B, C\), namely

\[ \frac{\xi_m}{E_z}=\frac{\Sigma A}{3}. \]

where \(\Sigma A=A+B+C\); a derivation of this relation is given in the paper of Cabannes. The last two formulas give:

\[ \frac{n^2-1}{4\pi L}=\frac{\Sigma A}{3}. \tag{16} \]

Using relations (15) and (16), we express \(\Sigma A^2\) and \(\Sigma BC\) through \(\rho\) and \(n\), after which formulas (14) reduce to the following form:

\[ I_1=I_0\,\frac{3\pi^2(n^2-1)^2}{r^2\lambda^4L(6-7\rho)},\qquad I_2=I_0\,\frac{3\pi^2(n^2-1)^2\rho}{r^2\lambda^4L(6-7\rho)}. \tag{17} \]

The formulas (17) give the final expressions for the components of the intensity of light scattered by a unit volume of gas with anisotropic molecules in a direction perpendicular to the direction of the incident rays.

For the magnitude of the polarization we have

\[ p=\frac{1-\rho}{1+\rho}, \]

or, in the case of air, if for \(\rho\) we take the value obtained by Cabanes from laboratory measurements, equal to 0.040, then

\[ p=923. \]

The anisotropy of air molecules, therefore, is capable of explaining incomplete polarization at \(90^\circ\) from the sun. However, the value of the polarization just obtained is considerably higher than the observed values of the polarization of the sky. Consequently, the theory of single scattering of light by anisotropic molecules still does not agree sufficiently with the data of measurements of the polarization of the sky. But let us also take into account double scattering, while considering the molecules of the air, in accounting for the latter, as isotropic. Then from formulas (17) and (11) we obtain:

\[ p=\frac{6(1-\rho)+(Q-M)(6-7\rho)}{6(1+\rho)+(Q+M)(6-7\rho)}. \tag{18} \]

Substituting here \(\rho=0.04\) and the above-indicated values of \(Q\) and \(M\), we have:

\[ p=833. \]

We have obtained a value of the polarization that fully coincides with the observed values and is even somewhat lower than some of the highest observed values of the magnitude of the polarization of the sky.

§ 7. The role of foreign particles of the atmosphere in the scattering of light.

We shall now consider the question of the role of foreign particles of the atmosphere in the scattering of light, in particular—in the phenomena of atmospheric absorption and polarization.

In 1918 there appeared a study by Fowle1, devoted to clarifying the question of the role of air molecules and foreign particles in the atmospheric absorption of light. The latter, as is known, is characte-

is characterized by the magnitude of the coefficient of atmospheric absorption \((k)\) or the coefficient of atmospheric transparency \((p)\), which enter Bouguer’s law,

\[ I=Ae^{-km}=Ap^m, \]

where \(I\) is the intensity of solar radiation that has passed through \(m\) “atmospheres” (for 1 atmosphere one conventionally takes the mass of the atmosphere in the zenith direction at an atmospheric pressure of \(760\) mm), and \(A\) is the solar constant.

Fowle used values of the coefficient of atmospheric transparency obtained at Mount Wilson \((1700\ \text{m})\) during the period from 1910 to 1916 for wavelengths from \(350\ \mu\mu\) to \(574\ \mu\mu\), i.e. for a region free from selective absorption. Fowle decomposed the coefficient of transparency for rays of a given wavelength \((p)\) into two parts, of which one is due to the molecules of dry air \((p_a)\), and the other to the water vapor present in the air \((p_w)\). Let \(w\) denote the so-called “precipitable water,” i.e. the height (in cm) of that layer of water on the earth’s surface which would be obtained if all the water vapor over the place of observation fell out as precipitation; \(p_w\) is referred to \(w=1\). The three coefficients \(p\), \(p_a\), \(p_w\) are connected with one another by the relation:

\[ p=p_a\cdot p_w^w \]

or

\[ \lg p=\log p_a+w\log p_w . \]

With the aid of this formula, from the data \(p\) and \(w\), Fowle found graphically the values of \(p_a\) and \(p_w\). Considering the values of \(p_a\) and \(p_w\) obtained in this way only “rough,” i.e. as a first approximation, Fowle subjected them to further “refining.” For this purpose he used the following expression for the absorption coefficient of a dry atmosphere \(k\), where it is clear that \(e^{-k}=p_a\):

\[ k=\frac{32}{3}\left\{\frac{\pi^3(n-1)^2h}{LN^4}+bh\right\}\frac{H}{H_o}+D, \tag{19} \]

where \(h\) is the height of a homogeneous atmosphere, \(H\) the observed atmospheric pressure, \(H_o\) the normal pressure, \(b\) a coefficient characterizing the amount of energy absorbed and converted into heat, this coefficient being, in the spectral region under consideration (absence of selective absorption), close to zero, and \(D\) an absorption coefficient due to the presence chiefly of dry haze, this haze being assumed to consist of particles so large that \(D\) does not depend on wavelength; the remaining quantities have the same meaning as in formula (10), of which formula (19) is in fact a generalization. Taking in equation (19)

SCATTERING OF LIGHT IN THE ATMOSPHERE

For known \(L\) and \(D\), Fowle obtains, solving this equation by the method of least squares, the following value for Loschmidt’s number

\[ L=(2.72\pm 0.01)\cdot 10^{19}. \]

Introducing the correction for the anisotropy of the molecules, this number, according to Cabannes, must be increased by \(8.5\%\); thus,

\[ L=2.95\cdot 10^{19}, \]

which agrees with the results of laboratory measurements of \(L\), which gave for this quantity the values, for example, \(2.71\cdot 10^{19}\) (Millikan) and \(3.06\cdot 10^{19}\) (Perrin). For the transparency coefficient of dry haze \(p_d\), equal to \(e^D\), for the period 1910–1911 the value \(0.995\) was obtained, which shows that only \(0.5\%\) of the energy coming from the sun is absorbed by dust. In 1912, owing to the eruption of the volcano Katmai, this absorption of energy by dust particles on hazy days reached \(25\%\); in 1913 it fell to \(2.6\%\), in 1914–1915 to \(1\%\), and in 1916 rose again to \(3.2\%\).

As regards the transparency coefficient caused by the presence of water vapor in the atmosphere, this coefficient, referred to \(1\ \mathrm{cm}\) of precipitated water, proved, for the wavelength interval from \(350\ \mu\mu\) to \(570\ \mu\mu\), to lie within the limits from \(0.93\) to \(0.98\). If the precipitated water (in \(\mathrm{mm}\)) is calculated by Humphreys’ formula1

\[ w=2.0\,e, \]

where \(e\) is the absolute humidity at the earth’s surface, expressed in \(\mathrm{mm}\) of mercury, then, according to Fowle, at an absolute humidity of, for example, \(5\ \mathrm{mm}\), water vapor in a column of one atmosphere causes, for the above-mentioned wavelengths, an absorption of energy from \(2\%\) to \(7\%\).

In 1922 Linke2 proposed the following method of estimating the role in the absorption of light of foreign particles in the atmosphere.

Using Bouguer’s formula:

\[ I=Ap^m, \]

we ascribe the observed changes of \(I\), under the condition of constant \(A\), to changes of the transparency coefficient \(p\), considering, for a given height of the sun \((h)\) and atmospheric pressure \((H)\), the mass of atmosphere traversed by the solar rays to be unchanged. However, formally, with equal right we may, considering \(p\) constant, ascribe the changes of \(I\) observed at one and the same \(h\) and \(H\) to changes of \(m\). This second

point of view underlies Linke’s concept of the so-called turbidity coefficient—a factor characterizing the degree of turbidity of the atmosphere.

Let us suppose that we have an absolutely pure atmosphere, i.e. one devoid of dust and water vapor; let us denote its coefficient of transparency by \(q\), and the mass traversed by the rays of the sun and computed from \(h\) and \(H\), by \(m\). Let us further imagine that, owing to the presence of water vapor and dust in the atmosphere (and also to its optical inhomogeneity), under the same \(h\) and \(H\) and the same \(q\), we have not the mass \(m\), but the mass \(mT\), i.e. the number of absolutely pure atmospheres increased \(T\)-fold. We then have

\[ I=Aq^{mT}. \]

The quantity \(T\) is called the turbidity coefficient (Trübungsfaktor) and may, according to Linke, be interpreted in the following way: the turbidity coefficient is numerically equal to the number of pure atmospheres whose combined action would produce the same attenuation of radiation as the given turbid atmosphere. Without entering into further details\(^{1}\) of this very interesting question, we shall cite only a few values of the turbidity coefficient for various places on the globe: Davos—1.76, Slutsk—1.88, Feodosia—2.31, Baku—2.43; the smallest value of \(T\), so far as I know, was observed in Feodosia—1.30, and at Laquiaca (the plateau of Bolivia at an altitude of 3,600 m)—1.36; the greatest, naturally in the absence of clouds in front of the sun, in the North Sea—4.30.

From the numbers just given it is evident that, in a more or less turbid atmosphere, foreign particles may play a very considerable role in the atmospheric absorption of light. To such conditions of the atmosphere there also belongs, one may think, the result obtained by G. I. Pokrovskii,\(^{2}\) namely that in the atmosphere the so-called Mie effect is extremely strongly expressed, testifying to the significant role of large particles in the atmospheric scattering of light.

As for the influence of foreign particles of the atmosphere on the phenomena of polarization of the sky, this influence, from the quantitative side, was investigated, apparently, only by the author of these lines,\(^{3}\) who carried out in 1925 in Simferopol and in 1925–1926 on Ai-Petri a series of parallel measurements of the magnitude of the polarization of the zenith at a solar altitude of \(0^\circ\), the number of dust particles (with an Aitken dust counter), and the absolute humid-

\(^{1}\) See, for example, my article “Turbidity of the Atmosphere in General, and in Feodosia in Particular,” in the Ten-Day Bulletin of the Hydrometeorological Service of the Black Seas, 1926.

\(^{2}\) Pokrowski. Zeitschr. f. Phys., 34, 49 (1925).

\(^{3}\) Tichanowsky. Meteor. Zeitschr., 43, 154 (1926).

[[unclear: beginning of sentence]] The greatest number of series of measurements (20) was obtained on Ai-Petri. We therefore confine ourselves here to conclusions only from the Ai-Petri measurements.

From all 20 series of measurements, by the formula

\[ p=a+\frac{\Delta p}{\Delta n}\,n, \]

where \(p\) is the magnitude of the polarization (in promilles), \(n\) is the number of dust particles (in thousands per \(\mathrm{cm}^3\)), the quantity \(\frac{\Delta p}{\Delta n}\) was determined by the method of least squares; using the value obtained,

\[ \frac{\Delta p}{\Delta n}=-10.9, \]

all measured values of the magnitude of the polarization were reduced to a number of dust particles equal to 0. Using these reduced values of the polarization, by the formula

\[ p=p_0+\frac{\Delta p}{\Delta e}\,e, \]

where \(e\) is the absolute humidity in mm of mercury column, the quantities \(p_0\) and \(\frac{\Delta p}{\Delta e}\) were calculated by the method of least squares, and it turned out that

\[ \frac{\Delta p}{\Delta e}=-2.8, \]

\[ p_0=860. \]

Let us suppose that the extrapolation used may be regarded as admissible. Let us further assume that, if \(n=0\) and \(e=0\) at the point of observation on the earth’s surface, then it is free of dust and water vapor, and all that part of the atmosphere which determines the magnitude of the polarization at the given point is likewise free of them; then \(p_0\) may be regarded as the magnitude of the polarization of an absolutely dry and absolutely dust-free atmosphere.

We then arrive at the conclusion that the influence of dust and water vapor in the atmosphere on the magnitude of the polarization is very small in a sufficiently clean atmosphere (Ai-Petri); that the observed incompleteness of polarization can in no way be attributed solely to the presence of foreign particles in the atmosphere; and that, finally, the value of the polarization obtained under certain assumptions for an absolutely clean atmosphere, 860, agrees sufficiently with the value (833) which is obtained if, in the first approximation, allowance is made also for the anisotropy of the scattering molecules and for secondary scattering. We may expect that with further development of Al’grim’s theory (for example, its extension to the case of anisotropic molecules) we shall obtain still greater agreement.

If one uses the value 86%, as the magnitude of the polarization of a pure atmosphere, then, substituting it into formula (18), we can calculate the coefficient of optical anisotropy of the molecule \(\rho\). We obtain

\[ \rho = 0.019, \]

whereas Cabannes obtained the value 0.040, Strutt\(^1\)—0.042 (1st series) and 0.050 (2nd series), Raman and Rao\(^2\)—0.044.

Measurements of the polarization of the light of the sky thus, first, confirm the anisotropy of air molecules and, secondly, give for the coefficient of anisotropy a value of the same order as that obtained from laboratory measurements. Again, a more exact theory of the polarization of the sky and a more reliable determination of the magnitude of the polarization of the sky of an absolutely pure atmosphere will probably make it possible to obtain better agreement between the results of laboratory and atmospheric measurements. It would be very important, it seems to me, to try to obtain the magnitude of the polarization of an absolutely pure atmosphere by studying the dependence of the magnitude of the polarization on the coefficient of transparency of the atmosphere and calculating, by extrapolation, the magnitude of the polarization for the value of the coefficient of transparency corresponding to an absolutely pure atmosphere.

§ 8. Conclusion.

Summing up the progress made in the field of research on atmospheric scattering of light, we may, recalling the words of Rayleigh cited at the beginning of this review, say the following.

The theory which ascribes the origin of the light of the sky to the scattering of light by the molecules of air may now be regarded as generally accepted. This theory, taking into account the anisotropy of air molecules and the double scattering of light in the atmosphere, may be qualified as a second approximation to reality. It owes its development to a considerable degree to investigations of the polarization of the sky, which more insistently demanded and still demand a sufficiently elaborated and refined theory than did spectrophotometric investigations of the light of the sky and investigations of the absorption of light in the atmosphere. Foreign particles of the atmosphere (not molecules of air), when the latter is in a sufficiently normal state, play a very insignificant role in the phenomena of atmospheric light scattering, except perhaps for the more or less noticeable role of water vapor in phenomena of absorption of light. Further development of the theory of atmospheric light—

\(^1\) Strutt (Rayleigh). Proc. Roy. Soc. XCV, 155; XCVII, 435; XCVIII, 57.

\(^2\) Raman and Rao. Phil. Mag., 46, 426 (1923).

scattering, it would seem, should proceed in the direction of a more exact accounting for the two factors indicated above—the anisotropy of the molecules and double scattering, i.e., in the direction of a more detailed development of the theory of an absolutely pure atmosphere. For a successful comparison of the conclusions of this theory with observational data, it is necessary to “cleanse” the atmosphere of the extraneous particles “polluting” it, deriving from the observations a sufficiently precise and reliable picture of the phenomena of light scattering in an absolutely pure atmosphere; this picture may also be used for the purposes of pure physics¹).

¹) Note added in proof. At the present time the author, having processed the Ai-Petri observations by another, more refined method, has obtained a somewhat different value for the polarization of the sky in an absolutely pure atmosphere, namely [[unclear: value]]. On the other hand, the author has developed a more refined theory of the polarization of the sky than that expressed by formula (18). This theory and the value of the polarization, 85%, gave for the coefficient of anisotropy of air molecules the value 0.016. I. T.

1. Humphreys. Bull. of the Mount Weather observatory. IV, 1912. ↩↩↩↩↩↩↩

2. Linke. Beiträge z. Phys. d. freien Atm. 10, 91 (1922); Linke und Boda. Meteor. Zeitschr., 39, 161 (1922). ↩↩

3. Pernter-Exner. Meteor. Optik, 1922, p. 692. ↩↩

4. Jensen. Himmelswelt, 35, 180 (1925). ↩

5. See my article “Spielt die sekundäre Diffusion in der Atmosphäre eine bedeutende Rolle in der Himmelspolarisation?” in Meteor. Zeitschr. for 1926. ↩