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Sky

The Colour of the Sky

It is a matter of common observation that the blue of the sky is highly variable, even on days that are free fromaclouds. The colour usually deepens toward the zenith and also with the elevation of the observer. It is evident that the normal blue is more or less diluted with extraneous white light, having its origin in reflections from the grosser particles of foreign matter with which the air is usually charged. Closely associated with the colour is the polarization of the light from the sky. This takes place in a plane passing through the sun, and attains a maximum about 90° therefrom. Under favourable conditions more than half the light is polarized.

As to the origin of the normal blue, very discrepant views have been held. Some writers, even of good reputation, have held that the blue is the true body colour of the air, or of some ingredient in it such as ozone. It is a sufficient answer to remark that on this theory the blue would reach its maximum development in the colour of the setting sun. It should be evident that what we have first to explain is the fact that we receive any light from the sky at all. Were the atmosphere non-existent or absolutely transparent, the sky would necessarily be black. There must be something capable of reflecting light in the wider sense of that term.

A theory that has received much support in the past attributes the reflections to thin bubbles of water, similar to soap-bubbles, in which form vapour was supposed to condense. According to it, sky blue would be the blue of the first order in Newton's scale. The theory was developed by R. Clausius (Pogg. Ann. vols. 72, 76, 88), who regarded it as meeting the requirements of the case. It must be noticed, however, that the angle of maximum polarization would be about 76° instead of 90°.

Apart from the difficulty of seeing how the bubbles could arise, there is a formidable objection, mentioned by E. W. Briicke (Pogg. Ann. 88, 363), that the blue of the sky is a much richer colour than the blue of the first order. Briicke also brought forward an experiment of great importance, in which he showed that gum mastic, precipitated from an alcoholic solution poured into a large quantity of water, scatters light of a blue tint. He remarks that it is impossible to suppose that the particles of mastic are in the form of bubbles. Another point of great importance is well brought out in the experiments of John Tyndall (Phil. Mag. (4), 1 37, 388) upon clouds precipitated by the chemical action of light. Whenever the particles are sufficiently fine, the light emitted laterally is blue in colour and, in a direction perpendicular to the incident beam, is completely polarized. About the colour there can be no prima facie difficulty; for, as soon as the question is raised, it is seen that the standard of linear dimension, with reference to which the particles are called small, is the wave-length of light, and that a given set of particles would (on any conceivable view as to their mode of action) produce a continually increasing disturbance as we pass along the spectrum towards the more refrangible end.

On the other hand, that the direction of complete polarization should be independent of the refracting power of the matter composing the cloud has been considered mysterious. Of course, on the theory of thin plates, this direction would be determined. by Brewster's law; but, if the particles of foreign matter are small in all their dimensions, the circumstances are materially different from those under which Brewster's law is applicable.

The investigation of this question upon the elastic solid theory will depend upon how we suppose the solid to vary from one optical medium to another. The slower propagation of light in gas or water than in air or vacuum may be attributed to a greater density, or to a less rigidity, in the former case; or we may adopt the more complicated supposition that both these quantities vary, subject only to the condition which restricts the ratio of velocities to equality with the known refractive index. It will presently appear that the original hypothesis of Fresnel, that the rigidity remains the same in both media, is the only one that can be reconciled with the facts; and we will therefore investigate upon this basis the nature of the secondary waves dispersed by small particles.

Conceive a beam of plane polarized light to move among a number of particles, all small compared with any of the wavelengths. According to our hypothesis, the foreign matter may be supposed to load the aether, so as to increase its inertia without altering its resistance to distortion. If the particles were away, the wave would pass on unbroken and no light would be emitted laterally. Even with the particles retarding the motion of the aether, the same will be true if, to counterbalance the increased inertia, suitable forces are caused to act on the aether at all points where the inertia is altered. These forces have the same period and direction as the undisturbed luminous vibrations themselves. The light actually emitted laterally is thus the same as would be caused by forces exactly the opposite of these acting on the medium otherwise free from disturbance, and it only remains to see what the effect of such force would be.

On account of the smallness of the particles, the forces acting throughout the volume of any individual particle are all of the same intensity and direction, and may be considered as a whole. The determination of the motion in the aether, due to the action of a periodic force at a given point, is discussed in the article Diffraction Of Light (§ Ii). Before applying the solution to a mathematical investigation of the present question, it may be well to consider the matter for a few moments from a more general point of view.

In the first place, there is necessarily a complete symmetry round the direction of the force. The disturbance, consisting of transverse vibrations, is propagated outwards in all directions from the centre; and, in consequence of the symmetry, the direction of vibration in any ray lies in the plane containing the ray and the axis of symmetry; that is to say, the direction of vibration in the scattered or diffracted ray makes with the direction of vibration in the incident or primary ray the least possible angle. The symmetry also requires that the intensity of the scattered light should vanish for the ray which would be propagated along the axis; for there is nothing to distinguish one direction transverse to the ray from another. The application of this is obvious. Suppose, for distinctness of statement, that the primary ray is vertical, and that the plane of vibration is that of the meridian. The intensity of the light scattered by a small particle is constant, and a maximum, for rays which lie in the vertical plane running east and west, while there is no scattered ray along the north and south line. If the primary ray is unpolarized, the light scattered north and south is entirely due to that component which vibrates east and west, and is therefore perfectly polarized, the direction of its vibration being also east and west. Similarly any other ray scattered horizontally is perfectly polarized, and the vibration is performed in the horizontal plane. In other directions the polarization becomes less and less complete as we approach the vertical.

The observed facts as to polarization are thus readily explained, and the general law connecting the intensity of the scattered light with the wave-length follows almost as easily from considerations of dimensions. The object is to compare the intensities of the incident and scattered light, for these will clearly be proportional. The number (i) expressing the ratio of the two amplitudes is a function of the following quantities: - (T) the volume of the disturbing particle; (r) the distance of the point under consideration from it; (A) the wave-length; (b) the velocity of propagation of light; (D) and (D') the original and altered densities: of which the first three depend only upon space, the fourth on space and time, while the fifth and sixth introduce the consideration of mass. Other elements of the problem there are none, except mere numbers and angles, which do not depend upon the fundamental measurements of space, time and mass. Since the ratio (i), whose expression we seek, is of no dimensions in mass, it follows at once that D and D' occur only under the form D: D', which is a simple number and may therefore be disregarded. It remains to find how i varies with T, r, A, b. Now, of these quantities, b is the only one depending on time; and therefore, as i is of no dimensions in time, b cannot occur in its expression. Moreover, since the same amount of energy is propagated across all spheres concentric with the particle, we recognize that i varies as r. It is equally evident that i varies as T, and therefore that it must be proportional to T/Xr, T being of three dimensions in space. In passing from one part of the spectrum to another, A is the only quantity which varies, and we have the important law: - When light is scattered by particles which are very small compared with any of the wave-lengths, the ratio of the amplitudes of the vibrations of the scattered and incident lights varies inversely as the square of the wave-length, and the ratio of intensities as the inverse fourth power.

The light scattered from small particles is of a much richer blue than the blue of the first order as reflected from a very thin plate. From the general theory (see Interference Of Light, § 8), or by the method of dimensions, it is easy to prove that in the latter case the intensity varies as A2, instead of A-4.

The principle of energy makes it clear that the light emitted laterally is not a new creation, but only diverted from the main stream. If I represent the intensity of the primary light after traversing a thickness x of the turbid medium, we have dI = - hIA-4dx, where h is a constant independent of A. On integration, log(I/Io) = - hA4 x.. (1) if Io correspond to x = o, - a law altogether similar to that of absorption, and showing how the light tends to become yellow and finally red as the thickness of the medium increases (Phil. Mag., 1871, 4 1, pp. 107, 274).

Sir William Abney has found that the above law agrees remarkably well with his observations on the transmission of light through water in which particles of mastic are suspended (Prot. Roy. Soc., 1886).

We may now investigate the mathematical expression for the disturbance propagated in any direction from a small particle upon which a beam of light strikes. Let the particle be at the origin of coordinates, and let the expression for the primary vibration be =sin (nt - kx). .. . (2) The acceleration of the element at the origin is - n 2 sin nt; so that the force which would have to be applied to the parts where the density is D' (instead of D), in order that the waves might pass on undisturbed, is, per unit of volume, (D' - D)n 2 sin nt. To obtain the total force which must be supposed to act, the factor T (representing the volume of the particle) must be introduced. The opposite of this, conceived to act at the origin, would give the same disturbance as is actually caused by the presence of the particle. Thus by equation (18) of § II of the article Diffraction Of Light, the secondary disturbance is expressed by D' - D n 2 Tsin sin (nt - kr) D 47rb2 r _ D' - D irTsin O sin (nt - kr) (3)1 The preceding investigation is based upon the assumption that in passing from one medium to another the rigidity of the aether does not change. If we forego this assumption, the question is 1 In strictness the force must be supposed to act upon the medium in its actual condition, whereas in (18), previously cited, the medium is supposed to be absolutely uniform. It is not difficult to prove that (3) remains unaltered, when this circumstance is taken into account; and it is evident in any case that a correction would depend upon the square of (D' - D).

necessarily more complicated; but, on the supposition that the changes of rigidity (AN) and of density (AD) are relatively small, the results are fairly simple. If the primary wave be represented by = e-ikx the component rotations in the secondary wave are '1'3= P (- AN y) N r2 ' cwi= r x D y N 'y)' lw2=P ( - AD + 6,N z2 - x2 ' D r N r2 where ik3T e-ikr _ P - 4 r The expression for the resultant rotation in the general case would be rather complicated, and is not needed for our purpose. It is easily seen to be about an axis perpendicular to the scattered ray (x, y, z), inasmuch as _ _ x&Ji+y02+z03 Let us consider the more special case of a ray scattered normally to the incident ray, so that x=o. We have 62=W12+W22+032=P2 (N) 2 Y 2+P2 (A I - 3) 2 If AN, AD be both finite, we learn from (7) that there is no direction perpendicular to the primary (polarized) ray in which the secondary light vanishes. Now experiment tells us plainly that there is such a direction, and therefore we are driven to the conclusion that either AN or AD must vanish.

The consequences of supposing AN to be zero have already been traced. They agree very well with experiment, and require us to suppose that the vibrations are perpendicular to the plane of polarization. So far as (7) is concerned the alternative supposition that AD vanishes would answer equally well, if we suppose the vibrations to be executed in the plane of polarization; but let us now revert to (5), which gives w 3 = _ PAN y z - = + PAN xy _ PAN z 2 - x2 8 N r 2 N r2' W 2 + N r2 (8) from 0 along which there is no scattered light, - two along the axis According to these equations there would be, in all, six directions of y normal to the original ray, and four (y z = =x) at angles of 45° with that ray. So long as the particles are small no such vanishing of light in oblique directions is observed, and we are thus led to the conclusion that the hypothesis of a finite AN and of vibrations in the plane of polarization cannot be reconciled with the facts. No form of the elastic solid theory is admissible except that in which the vibrations are supposed to be perpendicular to the plane of polarization, and the difference between one medium and another to be a difference of density only (Phil. Mag., 1871, 41, p 447).

It is of interest to pursue the applications of equation (3) so as to connect the intensity of the scattered and transmitted light with the number and size of the particles (see Phil. Mag., 18 99, 47, p. 375). In order to find the whole emission of energy from one particle (T), we have to integrate the square of (3) over the surface of a sphere of radius r. The element of area being 22rr 2 sin 04, , we have f:2 l 2x r2 si n 2 d ?=gam, r so that the energy emitted from T is represented by 87r3 (D, - D) 2 T2 (9) D2 x4' on such a scale that the energy of the primary wave is unity per unit of wave-front area.

The above relates to a single particle. If there be n similar particles per unit volume, the energy emitted from a stratum of thickness dx and of unit area is found from (9) by the introduction of the factor ndx. Since there is no waste of energy upon the whole, this represents the loss of energy in the primary wave. Accordingly, if E be the energy of the primary wave, dE 87-2n (D' - D) 2 T2 E dx 3 D2%4 ' whence E = Eoe-hx (II) where h = 8?r 2 n (D' - D)2T2 3 D2 x 4, (12) If we had a sufficiently complete expression for the scattered light, we might investigate (12) somewhat more directly by considering the resultant of the primary vibration and of the secondary vibrations which travel in the same direction. If, however, we apply this process to (3), we find that it fails to lead us to (12), though it furnishes another result of interest. The combination of the secondary waves which travel in the direction in question have this peculiarity: that the phases are no more distributed at random. The intensity of the secondary light is no longer to be arrived at by addition of individual intensities, but must be calculated with consideration of the particular phases involved. If we consider a number of particles which all lie upon a primary ray, we see that the phases of the secondary vibrations which issue along this line are all the same.

The actual calculation follows a similar course to that by which Huygens's conception of the resolution of a wave into components corresponding to the various parts of the wave-front is usually verified (see Diffraction Of Light). Consider the particles which occupy a thin stratum dx perpendicular to the primary ray x. Let P AP (fig. be this stratum, and 0 the point where the vibration is to be estimated. If AP =p, the element of volume is dx2xpdp, and the number A of particles to be found in it is deduced by the introduction of the factor n. Moreover, if OP = r, and AO=x, then r 2 =x 2 + p2, and pdp=rdr. The resultant at 0 of all the secondary vibrations which issue from the stratum dx is by (3), with sin ¢ equal to unity, ndx f ? D' D D y? cos ? (b t r)2 d or ndxD 'D D sin A (bt - x) To this is to be added the expression for the primary wave itself, supposed to advance undisturbed, viz. cos(21r/X)(bt - x), and the resultant will then represent the whole actual disturbance at 0 as modified by the particles in the stratum dx. It appears, therefore, that to the order of approximation afforded by (3), the effect of the particles in dx is to modify the phase, but not the intensity, of the light which passes them. If this be represented by Cos - x (bt - x - S),. . (14) S is the retardation due to the particles, and we have I =nTdx(D' - D)/2D. ... (15) Ifµ be the refractive index of the medium as modified by the particles, that of the original medium being taken as unity, then 6=(µ - I)dx, and µ - I =nT(D' - D)/2D.. .

If µ' denote the refractive index of the material composing the particles regarded as continuous, D'/D =µ' 2, and µ = 2 nT 0 2 ' 2 - I), (17) reducing to µ - I =nT(µ' - i),.. (18) in the case when CV - can be regarded as small.

It is only in the latter case that the formulae of the elastic solid accepted, the results are more complicated, in that when (µ' - is are to replace (D' - D)/D by 3(K' - K)/(K'-1-2K), where K, K' are theory are applicable to light. On the electric theory, now generally not small, the scattered ray depends upon the shape and not merely upon the volume of the small obstacle. In the case of spheres, we the dielectric constants proper to the medium and to the obstacle respectively (Phil. Mag., 1881, 12, p. 98); so that instead of (17) 3nT I= µ 12 µ,2+I On the same suppositions is replaced by 2 µ'2 2 T2 h = 247r n (. (20) I On either theory h= 3 2g3 (µ1) 2 /3 nX4,. (21) a formula giving the coefficient of transmission in terms of the refraction, and of the number of particles per unit volume. As Lord Kelvin has shown (Baltimore Lectures, p. 304, 1904) (16) may also be obtained by the consideration of the mean density of the altered medium.

Let us now imagine what degree of transparency of air is admitted by its molecular constituents, viz. in the absence of all foreign (4) (5). (6) (io) FIG. I.

(19) matter. We may take X =6 X I o' 5 cms., ,u - I =0.0003; whence from (21) we obtain as the distance x, equal to I/h, which light must travel in order to undergo alteration in the ratio e : 1, x=4 4XIo 13 Xn (22) The completion of the calculation requires a knowledge of the value of n, the number of molecules in unit volume under standard conditions, which, according to Avogadro's law, is the same for all gases. Maxwell estimated I. 9 X Io 19, but modern work suggests a higher number, such as 4.3XTo o (H. A. Wilson, Phil. Mag., 1903; see A. Schuster, Theory of Optics, § 178). If we substitute the latter value in (22) we find X Io 6 cm. =190 kilometres.

Although Mount Everest appears fairly bright at 100 miles' distance, as seen from the neighbourhood of Darjeeling, we cannot suppose that the atmosphere is as transparent as is implied in the above numbers; and, of course, this is not to be expected, since there is certainly suspended matter to be reckoned with. Perhaps the best data for a comparison are those afforded by the varying brightness of stars at different altitudes. P. Bouguer and others estimate about o8 for the transmission of light through the entire atmosphere from a star in the zenith. This corresponds to 8.3 kilometres of air at standard pressure. At this rate the transmission through 190 kilometres would be (. 8) 23 or oo06 in place of e1 - or 0.37. Or again if we inquire what, according to (21), would be the transmission through 8.3 kilometres, we find 1 - o044 =o956.

The general conclusion would appear to be that, while as seen from the earth's surface much of the light from the sky is due to comparatively gross suspended matter, yet an appreciable proportion is attributable to the molecules of air themselves, and that at high elevations where the blue is purer, the latter part may become predominant.

For a further discussion founded upon the observations of Q. Majorana and A. Sella, reference may be made to Lord Kelvin's Baltimore Lectures, p. 317, where a higher estimate of the value of n is favoured. It may be remarked that it is only the constant part of sky-light that can be due to detached molecules. Ordinary observation of the landscape shows that there is another part, highly variable from day to day, and due to suspended matter, much of which is fine enough to scatter light of blue quality.

The experiments of Tyndall upon precipitated clouds have been already referred to. So long as the precipitated particles are very fine, the light dispersed in a perpendicular direction is sky-blue and fully polarized. At a further stage of their growth the particles disperse in the perpendicular direction a light which is no longer fully polarized. When quenched as far as possible by rotation of a nicol prism, it exhibits a residue of a more intense blue colour; and further it is found that the direction of the most nearly complete polarization becomes inclined to the direction of the primary rays.

A discussion of these and other questions upon the basis of the electromagnetic theory of light is given in the Phil. Mag., 1881, 12, p. 81. Here we must be content with a statement of some of the results. So long as the particles are supposed to be very small and to differ little from their environment in optical properties, there is little difference between the electric and the elastic solid theories, and the results expressing the character of the scattered light are equivalent to (5). Whatever may be the shape or size of the particles, there is no scattered light in a direction parallel to the primary electric displacements. In order to render an 'account of Tyndall's "residual blue" it is necessary to pursue the approximation further, taking for simplicity the case of spherical shape. We learn that the light dispersed in the direction of primary vibration is not only of higher order in the difference of optical quality, but is also of order k 2 c 2 in comparison with that dispersed in other directions, where c is the radius of the sphere, and k=21r/X as before. The incident light being white, the intensity of the component colours scattered in this direction varies as the inverse eighth power of the wave-length, so that the resultant light is a rich blue.

As regards the polarization of the dispersed light as dependent on the angle at which it is emitted, we find that although, when terms of the second order are included, the scattered light no longer vanishes in the same direction as before, the peculiarity is not lost but merely transferred to another direction. The angle 0 through which the displacement occurs is measured backwards, i.e. towards the incident ray, and its value is given by _ OK k2.2 8 K ?5, being the difference of specific inductive capacities. Experiments upon this subject are not difficult. In a darkened room a beam of sunlight (or electric light) is concentrated by a large lens of 2 or 3 ft. focus; and in the path of the light is placed a glass beaker containing a dilute solution of sodium thiosulphate (hyposulphite of soda). On the addition, well stirred, of a small quantity of dilute sulphuric acid, a precipitate of sulphur slowly forms, and during its growth manifests exceedingly well the phenomena under consideration. The more dilute the solutions, the slower is the progress of the precipitation. A strength such that there is a delay of 4 or 5 minutes before any effect is apparent will be found suitable, but no great nicety of adjustment is necessary.

In the optical examination we may, if we prefer it, polarize the primary light; but it is usually more convenient to analyse the scattered light. In the early stages of the precipitation the polarization is complete in a perpendicular direction, and incomplete in oblique directions. After an interval the polarization begins to be incomplete in the perpendicular direction, the light which reaches the eye when the nicol is set to minimum transmission being of a beautiful blue, much richer than anything that can be seen in the earlier stages. This is the moment to examine whether there is a more complete polarization in a direction somewhat oblique; and it is found that with 0 positive there is, in fact, a direction of more complete polarization, while with 0 negative the polarization is more imperfect than in the perpendicular direction itself.

The polarization in a distinctly oblique direction, however, is not perfect, a feature for which more than one reas9n may be put forward. In the first place, with a given size of particles, the direction of complete polarization indicated by (23) is a function of the colour of the light, the value of 0 being 3 or 4 times as large for the violet as for the red end of the spectrum. The experiment is, in fact, much improved by passing the primary light through a coloured glass. Not only is the oblique direction of maximum polarization more definite and the polarization itself more complete, but the observation is easier than with white light in consequence of the uniformity in the colour of the light scattered in various directions. If we begin with a blue glass, we may observe the gradually increasing obliquity of the direction of maximum polarization; and then by exchanging the blue glass for a red one, we may revert to the original condition of things, and observe the transition from perpendicularity to obliquity over again. The change in the wave-length of the light has the same effect in this respect as a change in the size of the particles, and the comparison gives curious information as to the rate of growth.

But even with homogeneous light it would be unreasonable to expect an oblique direction of perfect polarization. So long as the particles are all very small in comparison with the wave-length, there is complete polarization in the perpendicular direction; but when the size is such that obliquity sets in, the degree of obliquity will vary with the size of the particles, and the polarization will be complete only on the very unlikely condition that the size is the same for them all. It must not be forgotten, too, that a very moderate increase of dimensions may carry the particles beyond the reach of our approximations.

The fact that at this stage the polarization is a maximum, when the angle through which the light is turned exceeds a right angle, is the more worthy of note, as the opposite result would probably have been expected. By Brewster's law (see Polarization of light) this angle in the case of regular reflection from a plate is less than a right angle; so that not only is the law of polarization for a very small particle different from that applicable to a plate, but the first effect of an increase of size is to augment the difference.

The simple theory of the dispersion of light by small particles suffices to explain not only the blue of the zenith, but the comparative absence of small wave-lengths from the direct solar rays, and the brilliant orange and red coloration of the setting sun and of the clouds illuminated by his rays. The hyposulphite experiment here again affords an excellent illustration. But we must not expect a simple theory to cover all the facts. It is obvious that the aerial particles are illuminated not only by the direct solar rays, but also by light dispersed from other parts of the atmosphere and from the earth's surface. On this and other accounts the coloration of the sky is highly variable. The transition from blue to orange or red at sunset is usually through green, but exceptional conditions may easily disturb the normal state of things. The brilliant sunset effects observed in Europe after the Krakatoa eruption may naturally be attributed to dust of unusual quality or quantity in the upper regions of the atmosphere (see DusT).

Related to abnormalities of colour we may expect to find corresponding polarization effects. Of this nature are the neutral points, where the polarization changes character, observed by F. J. D. Arago, J. Babinet and Sir D. Brewster, for an account of which reference may be made to E. Mascart, Traite d'optique. The normal polarization at the zenith, as dependent upon the position of the sun, was the foundation of Sir C. Wheatstone's polar clock. (R.)