Atmospheric Optics - Western Michigan Universityhomepages.wmich.edu/~korista/atmospheric_optics.pdf · planations of the blue sky requiring the presence of water in the atmosphere:

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Atmospheric Optics

Craig F. BohrenPennsylvania State University, Department of Meteorology, University Park,Pennsylvania, USAPhone: (814) 466-6264; Fax: (814) 865-3663; e-mail: [email protected]

AbstractColors of the sky and colored displays in the sky are mostly a consequence of selectivescattering by molecules or particles, absorption usually being irrelevant. Molecularscattering selective by wavelength – incident sunlight of some wavelengths beingscattered more than others – but the same in any direction at all wavelengths givesrise to the blue of the sky and the red of sunsets and sunrises. Scattering by particlesselective by direction – different in different directions at a given wavelength – gives riseto rainbows, coronas, iridescent clouds, the glory, sun dogs, halos, and other ice-crystaldisplays. The size distribution of these particles and their shapes determine what isobserved, water droplets and ice crystals, for example, resulting in distinct displays.

To understand the variation and color and brightness of the sky as well as thebrightness of clouds requires coming to grips with multiple scattering: scatterers inan ensemble are illuminated by incident sunlight and by the scattered light fromeach other. The optical properties of an ensemble are not necessarily those of itsindividual members.

Mirages are a consequence of the spatial variation of coherent scattering (refraction)by air molecules, whereas the green flash owes its existence to both coherent scatteringby molecules and incoherent scattering by molecules and particles.

Keywordssky colors; mirages; green flash; coronas; rainbows; the glory; sun dogs; halos; visibility.

1 Introduction 542 Color and Brightness of Molecular Atmosphere 552.1 A Brief History 55

54 Atmospheric Optics

2.2 Molecular Scattering and the Blue of the Sky 572.3 Spectrum and Color of Skylight 582.4 Variation of Sky Color and Brightness 592.5 Sunrise and Sunset 623 Polarization of Light in a Molecular Atmosphere 633.1 The Nature of Polarized Light 633.2 Polarization by Molecular Scattering 644 Scattering by Particles 664.1 The Salient Differences between Particles and Molecules:

Magnitude of Scattering 664.2 The Salient Differences between Particles and Molecules:

Wavelength Dependence of Scattering 674.3 The Salient Differences between Particles and Molecules:

Angular Dependence of Scattering 684.4 The Salient Differences between Particles and Molecules:

Degree of Polarization of Scattered Light 694.5 The Salient Differences between Particles and Molecules:

Vertical Distributions 705 Atmospheric Visibility 716 Atmospheric Refraction 736.1 Physical Origins of Refraction 736.2 Terrestrial Mirages 736.3 Extraterrestrial Mirages 766.4 The Green Flash 777 Scattering by Single Water Droplets 787.1 Coronas and Iridescent Clouds 787.2 Rainbows 807.3 The Glory 828 Scattering by Single Ice Crystals 838.1 Sun Dogs and Halos 839 Clouds 869.1 Cloud Optical Thickness 869.2 Givers and Takers of Light 87

Glossary 89References 90Further Reading 90

1Introduction

Atmospheric optics is nearly synonymouswith light scattering, the only restrictionsbeing that the scatterers inhabit the

atmosphere and the primary source oftheir illumination is the sun. Essentiallyall light we see is scattered light, even thatdirectly from the sun. When we say thatsuch light is unscattered we really meanthat it is scattered in the forward direction;

Atmospheric Optics 55

hence it is as if it were unscattered.Scattered light is radiation from matterexcited by an external source. When thesource vanishes, so does the scattered light,as distinguished from light emitted bymatter, which persists in the absence ofexternal sources.

Atmospheric scatterers are either mole-cules or particles. A particle is an aggrega-tion of sufficiently many molecules thatit can be ascribed macroscopic proper-ties such as temperature and refractiveindex. There is no canonical number ofmolecules that must unite to form abona fide particle. Two molecules clearlydo not a quorum make, but what about10, 100, 1000? The particle size corre-sponding to the largest of these numbersis about 10−3 µm. Particles this smallof water substance would evaporate sorapidly that they could not exist long underconditions normally found in the atmo-sphere. As a practical matter, therefore,we need not worry unduly about scatterersin the shadow region between moleculeand particle.

A property of great relevance to scat-tering problems is coherence, both of thearray of scatterers and of the incident light.At visible wavelengths, air is an array ofincoherent scatterers: the radiant powerscattered by N molecules is N times thatscattered by one (except in the forwarddirection). But when water vapor in aircondenses, an incoherent array is trans-formed into a coherent array: uncorrelatedwater molecules become part of a singleentity. Although a single droplet is a coher-ent array, a cloud of droplets taken togetheris incoherent.

Sunlight is incoherent but not in anabsolute sense. Its lateral coherence lengthis tens of micrometers, which is whywe can observe what are essentiallyinterference patterns (e.g., coronas and

glories) resulting from illumination ofcloud droplets by sunlight.

This article begins with the colorand brightness of a purely molecu-lar atmosphere, including their variationacross the vault of the sky. This nat-urally leads to the state of polarizationof skylight. Because the atmosphere israrely, if ever, entirely free of particles,the general characteristics of scatteringby particles follow, setting the stagefor a discussion of atmospheric visibil-ity.

Atmospheric refraction usually sits byitself, unjustly isolated from all those at-mospheric phenomena embraced by theterm scattering. Yet refraction is anothermanifestation of scattering, coherent scat-tering in the sense that phase differencescannot be ignored.

Scattering by single water droplets andice crystals, each discussed in turn, yieldsfeasts for the eye as well as the mind. Thecurtain closes on the optical propertiesof clouds.

2Color and Brightness of MolecularAtmosphere

2.1A Brief History

Edward Nichols began his 1908 presiden-tial address to the New York meeting of theAmerican Physical Society as follows: ‘‘Inasking your attention to-day, even briefly,to the consideration of the present state ofour knowledge concerning the color of thesky it may be truly said that I am invitingyou to leave the thronged thoroughfaresof our science for some quiet side streetwhere little is going on and you may evensuspect that I am coaxing you into some


blind alley, the inhabitants of which belongto the dead past.’’

Despite this depreciatory statement,hoary with age, correct and completeexplanations of the color of the sky stillare hard to find. Indeed, all the faultyexplanations lead active lives: the bluesky is the reflection of the blue sea;it is caused by water, either vapor ordroplets or both; it is caused by dust.The true cause of the blue sky is notdifficult to understand, requiring only abit of critical thought stimulated by beliefin the inherent fascination of all naturalphenomena, even those made familiar byeveryday occurrence.

Our contemplative prehistoric ancestorsno doubt speculated on the origin of theblue sky, their musings having vanishedinto it. Yet it is curious that Aristotle, themost prolific speculator of early recordedhistory, makes no mention of it in hisMeteorologica even though he deliveredpronouncements on rainbows, halos, andmock suns and realized that ‘‘the sunlooks red when seen through mist orsmoke.’’ Historical discussions of the bluesky sometimes cite Leonardo as the first tocomment intelligently on the blue of thesky, although this reflects a European bias.If history were to be written by a supremelydisinterested observer, Arab philosopherswould likely be given more credit forhaving had profound insights into theworkings of nature many centuries beforetheir European counterparts descendedfrom the trees. Indeed, Moller [1] beginshis brief history of the blue sky withJakub Ibn Ishak Al Kindi (800–870), whoexplained it as ‘‘a mixture of the darknessof the night with the light of the dust andhaze particles in the air illuminated bythe sun.’’

Leonardo was a keen observer of light innature even if his explanations sometimes

fell short of the mark. Yet his hypothesisthat ‘‘the blueness we see in the atmo-sphere is not intrinsic color, but is causedby warm vapor evaporated in minute andinsensible atoms on which the solar raysfall, rendering them luminous against theinfinite darkness of the fiery sphere whichlies beyond and includes it’’ would, withminor changes, stand critical scrutiny to-day. If we set aside Leonardo as sui generis,scientific attempts to unravel the origins ofthe blue sky may be said to have begun withNewton, that towering pioneer of optics,who, in time-honored fashion, reduced itto what he already had considered: inter-ference colors in thin films. Almost twocenturies elapsed before more pieces inthe puzzle were contributed by the exper-imental investigations of von Brucke andTyndall on light scattering by suspensionsof particles. Around the same time Clau-sius added his bit in the form of a theorythat scattering by minute bubbles causesthe blueness of the sky. A better theorywas not long in coming. It is associatedwith a man known to the world as LordRayleigh even though he was born JohnWilliam Strutt.

Rayleigh’s paper of 1871 marks thebeginning of a satisfactory explanationof the blue sky. His scattering law, thekey to the blue sky, is perhaps themost famous result ever obtained bydimensional analysis. Rayleigh argued thatthe field Es scattered by a particle smallcompared with the light illuminating itis proportional to its volume V andto the incident field Ei. Radiant energyconservation requires that the scatteredfield diminish inversely as the distancer from the particle so that the scatteredpower diminishes as the square of r. Tomake this proportionality dimensionallyhomogeneous requires the inverse squareof a quantity with the dimensions of


length. The only plausible physical variableat hand is the wavelength of the incidentlight, which leads to

Es ∝ EiV

rλ2 . (1)

When the field is squared to ob-tain the scattered power, the result isRayleigh’s inverse fourth-power law. Thislaw is really only an often – but notalways – very good approximation. Miss-ing from it are dimensionless proper-ties of the particle such as its refractiveindex, which itself depends on wave-length. Because of this dispersion, there-fore, nothing scatters exactly as the inversefourth power.

Rayleigh’s 1871 paper did not give thecomplete explanation of the color andpolarization of skylight. What he did thatwas not done by his predecessors was togive a law of scattering, which could beused to test quantitatively the hypothesisthat selective scattering by atmosphericparticles could transform white sunlightinto blue skylight. But as far as givingthe agent responsible for the blue sky isconcerned, Rayleigh did not go essentiallybeyond Newton and Tyndall, who invokedparticles. Rayleigh was circumspect aboutthe nature of these particles, settling onsalt as the most likely candidate. It was notuntil 1899 that he published the capstoneto his work on skylight, arguing that airmolecules themselves were the source ofthe blue sky. Tyndall cannot be giventhe credit for this because he consideredair to be optically empty: when purgedof all particles it scatters no light. Thiserroneous conclusion was a result of thesmall scale of his laboratory experiments.On the scale of the atmosphere, sufficientlight is scattered by air molecules to bereadily observable.

2.2Molecular Scattering and the Blue of the Sky

Our illustrious predecessors all gave ex-planations of the blue sky requiring thepresence of water in the atmosphere:Leonardo’s ‘‘evaporated warm vapor,’’Newton’s ‘‘Globules of water,’’ Clausius’sbubbles. Small wonder, then, that waterstill is invoked as the cause of the bluesky. Yet a cause of something is that with-out which it would not occur, and the skywould be no less blue if the atmospherewere free of water.

A possible physical reason for attributingthe blue sky to water vapor is that, becauseof selective absorption, liquid water (andice) is blue upon transmission of whitelight over distances of order meters. Yetif all the water in the atmosphere at anyinstant were to be compressed into a liquid,the result would be a layer about 1 cm thick,which is not sufficient to transform whitelight into blue by selective absorption.

Water vapor does not compensate forits hundredfold lower abundance thannitrogen and oxygen by greater scatteringper molecule. Indeed, scattering of visiblelight by a water molecule is slightly lessthan that by either nitrogen or oxygen.

Scattering by atmospheric moleculesdoes not obey Rayleigh’s inverse fourth-power law exactly. A least-squares fit overthe visible spectrum from 400 to 700 nmof the molecular scattering coefficient of sea-level air tabulated by Penndorf [2] yields aninverse 4.089th-power scattering law.

The molecular scattering coefficient β,which plays important roles in followingsections, may be written

β = Nσs, (2)

where N is the number of moleculesper unit volume and σs, the scatteringcross section (an average because air is


a mixture) per molecule, approximatelyobeys Rayleigh’s law. The form of thisexpression betrays the incoherence ofscattering by atmospheric molecules. Theinverse of β is interpreted as the scatteringmean free path, the average distance aphoton must travel before being scattered.

To say that the sky is blue becauseof Rayleigh scattering, as is sometimesdone, is to confuse an agent with alaw. Moreover, as Young [3] pointed out,the term Rayleigh scattering has manymeanings. Particles small compared withthe wavelength scatter according to thesame law as do molecules. Both canbe said to be Rayleigh scatterers, butonly molecules are necessary for the bluesky. Particles, even small ones, generallydiminish the vividness of the blue sky.

Fluctuations are sometimes trumpetedas the ‘‘real’’ cause of the blue sky. Pre-sumably, this stems from the fluctuationtheory of light scattering by media in whichthe scatterers are separated by distancessmall compared with the wavelength. Inthis theory, which is associated with Ein-stein and Smoluchowski, matter is takento be continuous but characterized by arefractive index that is a random functionof position. Einstein [4] stated that ‘‘it isremarkable that our theory does not makedirect use of the assumption of a discretedistribution of matter.’’ That is, he cir-cumvented a difficulty but realized it couldhave been met head on, as Zimm [5] didyears later.

The blue sky is really caused by scat-tering by molecules – to be more precise,scattering by bound electrons: free elec-trons do not scatter selectively. Because airmolecules are separated by distances smallcompared with the wavelengths of visiblelight, it is not obvious that the power scat-tered by such molecules can be added. Yetif they are completely uncorrelated, as in

an ideal gas (to good approximation theatmosphere is an ideal gas), scattering byN molecules is N times scattering by one.This is the only sense in which the blue skycan be attributed to scattering by fluctua-tions. Perfectly homogeneous matter doesnot exist. As stated pithily by Planck, ‘‘achemically pure substance may be spo-ken of as a vacuum made turbid by thepresence of molecules.’’

2.3Spectrum and Color of Skylight

What is the spectrum of skylight? What isits color? These are two different questions.Answering the first answers the secondbut not the reverse. Knowing the color ofskylight we cannot uniquely determine itsspectrum because of metamerism: A givenperceived color can in general be obtainedin an indefinite number of ways.

Skylight is not blue (itself an impre-cise term) in an absolute sense. Whenthe visible spectrum of sunlight outsidethe earth’s atmosphere is modulated byRayleigh’s scattering law, the result is aspectrum of scattered light that is nei-ther solely blue nor even peaked in theblue (Fig. 1). Although blue does not pre-dominate spectrally, it does predominateperceptually. We perceive the sky to beblue even though skylight contains light ofall wavelengths.

Any source of light may be looked uponas a mixture of white light and light ofa single wavelength called the dominantwavelength. The purity of the source isthe relative amount of the monochromaticcomponent in the mixture. The dominantwavelength of sunlight scattered accordingto Rayleigh’s law is about 475 nm, whichlies solidly in the blue if we take thisto mean light with wavelengths between450 and 490 nm. The purity of this


Fig. 1 Rayleigh’s scattering law (dots), thespectrum of sunlight outside the Earth’satmosphere (dashes), and the product of the two(solid curve). The solar spectrum is taken fromThekaekara, M. P., Drummond, A. J. (1971), Nat.Phys. Sci. 229, 6–9 [6]

scattered light, about 42%, is the upperlimit for skylight. Blues of real skies areless pure.

Another way of conveying the color of asource of light is by its color temperature,the temperature of a blackbody having thesame perceived color as the source. Sinceblackbodies do not span the entire gamutof colors, not all sources of light can beassigned color temperatures. But manynatural sources of light can. The colortemperature of light scattered accordingto Rayleigh’s law is infinite. This followsfrom Planck’s spectral emission functionebλ in the limit of high temperature,

ebλ ≈ 2πckT

λ4 ,hc

λ� kT, (3)

where h is Planck’s constant, k is Boltz-mann’s constant, c is the speed of lightin vacuo, and T is absolute temperature.Thus, the emission spectrum of a black-body with an infinite temperature has thesame functional form as Rayleigh’s scat-tering law.

2.4Variation of Sky Color and Brightness

Not only is skylight not pure blue,but its color and brightness vary acrossthe vault of the sky, with the bestblues at zenith. Near the astronomicalhorizon the sky is brighter than overheadbut of considerably lower purity. Thatthis variation can be observed from anairplane flying at 10 km, well abovemost particles, suggests that the skyis inherently nonuniform in color andbrightness (Fig. 2). To understand whyrequires invoking multiple scattering.

Multiple scattering gives rise to observ-able phenomena that cannot be explainedsolely by single-scattering arguments. Thisis easily demonstrated. Fill a blackened pan

Fig. 2 Even at an altitude of 10 km, well abovemost particles, the sky brightness increasesmarkedly from the zenith to theastronomical horizon


with clean water, then add a few drops ofmilk. The resulting dilute suspension il-luminated by sunlight has a bluish cast.But when more milk is added, the suspen-sion turns white. Yet the properties of thescatterers (fat globules) have not changed,only their optical thickness: the blue suspen-sion being optically thin, the white beingoptically thick.

Optical thickness is physical thicknessin units of scattering mean free path, andhence is dimensionless. The optical thick-ness τ between any two points connectedby an arbitrary path in a medium populatedby (incoherent) scatterers is an integralover the path:

τ =∫ 2

1β ds. (4)

The normal optical thickness τn of theatmosphere is that along a radial pathextending from the surface of the Earthto infinity. Figure 3 shows τn over thevisible spectrum for a purely molecularatmosphere. Because τn is generally smallcompared with unity, a photon fromthe sun traversing a radial path in theatmosphere is unlikely to be scatteredmore than once. But along a tangential

Fig. 3 Normal optical thickness of a puremolecular atmosphere

path, the optical thickness is about 35 timesgreater (Fig. 4), which leads to severalobservable phenomena.

Even an intrinsically black object isluminous to an observer because ofairlight, light scattered by all the moleculesand particles along the line of sightfrom observer to object. Provided thatthis is uniformly illuminated by sunlightand that ground reflection is negligi-ble, the airlight radiance L is approxi-mately

L = GL0(1 − e−τ ), (5)

where L0 is the radiance of incidentsunlight along the line of sight with opticalthickness τ . The term G accounts forgeometric reduction of radiance becauseof scattering of nearly monodirectionalsunlight in all directions. If the line ofsight is uniform in composition, τ = βd,where β is the scattering coefficient and dis the physical distance to the black object.

If τ is small (�1), L ≈ GL0τ . In apurely molecular atmosphere, τ varieswith wavelength according to Rayleigh’slaw; hence the distant black object insuch an atmosphere is perceived to be

Fig. 4 Optical thickness (relative to the normaloptical thickness) of a molecular atmospherealong various paths with zenith angles between0◦ (normal) and 90◦ (tangential)


bluish. As τ increases so does L but notproportionally. Its limit is GL0: The airlightradiance spectrum is that of the source ofillumination. Only in the limit d = 0 isL = 0 and the black object truly black.

Variation of the brightness and color ofdark objects with distance was called aerialperspective by Leonardo. By means of it weestimate distances to objects of unknownsize such as mountains.

Aerial perspective belongs to the samefamily as the variation of color andbrightness of the sky with zenith angle.Although the optical thickness along a pathtangent to the Earth is not infinite, it issufficiently large (Figs. 3 and 4) that GL0

is a good approximation for the radiance ofthe horizon sky. For isotropic scattering (acondition almost satisfied by molecules),G is around 10−5, the ratio of the solidangle subtended by the sun to the solidangle of all directions (4π ). Thus, thehorizon sky is not nearly so bright asdirect sunlight.

Unlike in the milk experiment, whatis observed when looking at the hori-zon sky is not multiply scattered light.Both have their origins in multiple scat-tering but manifested in different ways.Milk is white because it is weakly ab-sorbing and optically thick, and hence allcomponents of incident white light aremultiply scattered to the observer eventhough the blue component traverses ashorter average path in the suspensionthan the red component. White horizonlight has escaped being multiply scat-tered, although multiple scattering is whythis light is white (strictly, has the spec-trum of the source). More light at theshort-wavelength end of the spectrum isscattered toward the observer than at thelong-wavelength end. But long-wavelengthlight has the greater likelihood of being

transmitted to the observer without be-ing scattered out of the line of sight.For a long optical path, these two pro-cesses compensate, resulting in a hori-zon radiance spectrum which is that ofthe source.

Selective scattering by molecules is notsufficient for a blue sky. The atmospherealso must be optically thin, at least formost zenith angles (Fig. 4) (the black-ness of space as a backdrop is takenfor granted but also is necessary, asLeonardo recognized). A corollary of thisis that the blue sky is not inevitable: anatmosphere composed entirely of nonab-sorbing, selectively scattering moleculesoverlying a nonselectively reflecting earthneed not be blue. Figure 5 shows calcu-lated spectra of the zenith sky over blackground for a molecular atmosphere withthe present normal optical thickness aswell as for hypothetical atmospheres 10and 40 times thicker. What we take tobe inevitable is accidental: If our atmo-sphere were much thicker, but identicalin composition, the color of the skywould be quite different from what itis now.

Fig. 5 Spectrum of overhead skylight for thepresent molecular atmosphere (solid curve), aswell as for hypothetical atmospheres 10 (dashes)and 40 (dots) times thicker


2.5Sunrise and Sunset

If short-wavelength light is preferentiallyscattered out of direct sunlight, long-wavelength light is preferentially trans-mitted in the direction of sunlight.Transmission is described by an expo-nential law (if light multiply scatteredback into the direction of the sunlightis negligible):

L = L0e−τ , (6)

where L is the radiance at the observerin the direction of the sun, L0 is theradiance of sunlight outside the atmo-sphere, and τ is the optical thickness alongthis path.

If the wavelength dependence of τ isgiven by Rayleigh’s law, sunlight is red-dened upon transmission: The spectrumof the transmitted light is comparativelyricher than the incident spectrum inlight at the long-wavelength end of thevisible spectrum. But to say that trans-mitted sunlight is reddened is not thesame as saying it is red. The perceivedcolor can be yellow, orange, or red, de-pending on the magnitude of the opticalthickness. In a molecular atmosphere,the optical thickness along a path fromthe sun, even on or below the horizon,is not sufficient to give red light upontransmission. Although selective scatter-ing by molecules yields a blue sky, redsare not possible in a molecular atmo-sphere, only yellows and oranges. Thiscan be observed on clear days, when thehorizon sky at sunset becomes succes-sively tinged with yellow, then orange, butnot red.

Equation (6) applies to the radiance onlyin the direction of the sun. Oranges andreds can be seen in other directionsbecause reddened sunlight illuminates

scatterers not lying along the line ofsight to the sun. A striking exampleof this is a horizon sky tinged withoranges and pinks in the direction oppositethe sun.

The color and brightness of the sunchanges as it arcs across the sky becausethe optical thickness along the line of sightchanges with solar zenith angle �. If theEarth were flat (as some still aver), thetransmitted solar radiance would be

L = L0eτn/ cos �. (7)

This equation is a good approximationexcept near the horizon. On a flat earth,the optical thickness is infinite for horizonpaths. On a spherical earth, optical thick-nesses are finite although much larger forhorizon than for radial paths.

The normal optical thickness of anatmosphere in which the number densityof scatterers decreases exponentially withheight z above the surface, exp(−z/H), isthe same as that for a uniform atmosphereof finite thickness:

τn =∫ ∞

0β dz = β0H, (8)

where H is the scale height and β0 isthe scattering coefficient at sea level.This equivalence yields a good approx-imation even for the tangential opti-cal thickness. For any zenith angle,the optical thickness is given approxi-mately by

τ

τn=

√R2

e

H2 cos2 � + 2Re

H+ 1

− Re

Hcos �, (9)

where Re is the radius of the Earth.A flat earth is one for which Re isinfinite, in which instance Eq. (9) yields


the expected relation

limRe→∞

τ

τn= 1

cos �. (10)

For Earth’s atmosphere, the molecularscale height is about 8 km. According tothe approximate relation Eq. (9), therefore,the horizon optical thickness is about39 times greater than the normal opticalthickness. Taking the exponential decreaseof molecular number density into accountyields a value about 10% lower.

Variations on the theme of reds andoranges at sunrise and sunset can beseen even when the sun is overhead. Theradiance at an observer an optical distanceτ from a (horizon) cloud is the sumof cloudlight transmitted to the observerand airlight:

L = L0G(1 − e−τ ) + L0Gce−τ , (11)

where Gc is a geometrical factor thataccounts for scattering of nearly monodi-rectional sunlight into a hemisphere ofdirections by the cloud. If the cloudis approximated as an isotropic reflec-tor with reflectance R and illuminatedat an angle �, the geometrical factorGc is �sR cos�/π , where �s is thesolid angle subtended by the sun atthe Earth. If Gc > G, the observed ra-diance is redder (i.e., enriched in lightof longer wavelengths) than the incidentradiance. If Gc < G, the observed radi-ance is bluer than the incident radiance.Thus, distant horizon clouds can be red-dish if they are bright or bluish if theyare dark.

Underlying Eq. (11) is the implicit as-sumption that the line of sight is uniformlyilluminated by sunlight. The first termin this equation is airlight; the secondis transmitted cloudlight. Suppose, how-ever, that the line of sight is shadowed

from direct sunlight by clouds (that donot, of course, occlude the distant cloudof interest). This may reduce the firstterm in Eq. (11) so that the second termdominates. Thus, under a partly over-cast sky, distant horizon clouds may bereddish even when the sun is high inthe sky.

The zenith sky at sunset and twilightis the exception to the general rule thatmolecular scattering is sufficient to ac-count for the color of the sky. In theabsence of molecular absorption, the spec-trum of the zenith sky would be essentiallythat of the zenith sun (although greatlyreduced in radiance), hence would notbe the blue that is observed. This waspointed out by Hulburt [7], who showedthat absorption by ozone profoundly af-fects the color of the zenith sky when thesun is near the horizon. The Chappuisband of ozone extends from about 450 to700 nm and peaks at around 600 nm. Pref-erential absorption of sunlight by ozoneover long horizon paths gives the zenithsky its blueness when the sun is nearthe horizon. With the sun more thanabout 10◦ above the horizon, however,ozone has little effect on the color ofthe sky.

3Polarization of Light in a MolecularAtmosphere

3.1The Nature of Polarized Light

Unlike sound, light is a vector wave, anelectromagnetic field lying in a plane nor-mal to the propagation direction. Thepolarization state of such a wave is deter-mined by the degree of correlation of any


two orthogonal components into whichits electric (or magnetic) field is resolved.Completely polarized light corresponds tocomplete correlation; completely unpolar-ized light corresponds to no correlation;partially polarized light corresponds to par-tial correlation.

If an electromagnetic wave is completelypolarized, the tip of its oscillating electricfield traces out a definite elliptical curve,the vibration ellipse. Lines and circles arespecial ellipses, the light being said tobe linearly or circularly polarized, respec-tively. The general state of polarization iselliptical.

Any beam of light can be consid-ered an incoherent superposition of twocollinear beams, one unpolarized, theother completely polarized. The radianceof the polarized component relative tothe total is defined as the degree of po-larization (often multiplied by 100 andexpressed as a percentage). This can bemeasured for a source of light (e.g., lightfrom different sky directions) by rotat-ing a (linear) polarizing filter and notingthe minimum and maximum radiancestransmitted by it. The degree of (linear)polarization is defined as the differencebetween these two radiances divided bytheir sum.

3.2Polarization by Molecular Scattering

Unpolarized light can be transformed intopartially polarized light upon interactionwith matter because of different changes inamplitude of the two orthogonal field com-ponents. An example of this is the partialpolarization of sunlight upon scatteringby atmospheric molecules, which can bedetected by looking at the sky through a po-larizing filter (e.g., polarizing sunglasses)while rotating it. Waxing and waning of the

observed brightness indicates some degreeof partial polarization.

In the analysis of any scattering prob-lem, a plane of reference is required. Thisis usually the scattering plane, determinedby the directions of the incident and scat-tered waves, the angle between them beingthe scattering angle. Light polarized perpen-dicular (parallel) to the scattering plane issometimes said to be vertically (horizon-tally) polarized. Vertical and horizontal inthis context, however, are arbitrary termsindicating orthogonality and bear no rela-tion, except by accident, to the directionof gravity.

The degree of polarization P of lightscattered by a tiny sphere illuminated byunpolarized light is (Fig. 6)

P = 1 − cos2 θ

1 + cos2 θ, (12)

where the scattering angle θ ranges from0◦ (forward direction) to 180◦ (backwarddirection); the scattered light is partiallylinearly polarized perpendicular to thescattering plane. Although this equation

Fig. 6 Degree of polarization of the lightscattered by a small (compared with thewavelength) sphere for incident unpolarizedlight (solid curve). The dashed curve is for asmall spheroid chosen such that the degree ofpolarization at 90◦ is that for air


is a first step toward understandingpolarization of skylight, more often thannot it also has been a false step, having ledcountless authors to assert that skylight iscompletely polarized at 90◦ from the sun.Although P = 1 at θ = 90◦ according toEq. (12), skylight is never 100% polarizedat this or any other angle, and forseveral reasons.

Although air molecules are very smallcompared with the wavelengths of visiblelight, a requirement underlying Eq. (12),the dominant constituents of air are notspherically symmetric.

The simplest model of an asymmetricmolecule is a small spheroid. Althoughit is indeed possible to find a directionin which the light scattered by such aspheroid is 100% polarized, this directiondepends on the spheroid’s orientation.In an ensemble of randomly orientedspheroids, each contributes its mite to thetotal radiance in a given direction, buteach contribution is partially polarized tovarying degrees between 0 and 100%. Itis impossible for beams of light to beincoherently superposed in such a way thatthe degree of polarization of the resultantis greater than the degree of polarizationof the most highly polarized beam.

Because air is an ensemble of randomlyoriented asymmetric molecules, sunlightscattered by air never is 100% polarized.The intrinsic departure from perfection isabout 6%. Figure 6 also includes a curvefor light scattered by randomly orientedspheroids chosen to yield 94% polarizationat 90◦. This angle is so often singledout that it may deflect attention fromnearby scattering angles. Yet, the degree ofpolarization is greater than 50% for a rangeof scattering angles 70◦ wide centeredabout 90◦.

Equation (12) applies to air, not tothe atmosphere, the distinction being

that in the atmosphere, as opposed tothe laboratory, multiple scattering is notnegligible. Also, atmospheric air is almostnever free of particles and is illuminatedby light reflected by the ground. We musttake the atmosphere as it is, whereasin the laboratory we often can eliminateeverything we consider extraneous.

Because of both multiple scatteringand ground reflection, light from anydirection in the sky is not, in general,made up solely of light scattered in asingle direction relative to the incidentsunlight but is a superposition of beamswith different scattering histories, hencedifferent degrees of polarization. As aconsequence, even if air molecules wereperfect spheres and the atmosphere werecompletely free of particles, skylight wouldnot be 100% polarized at 90◦ to the sun orat any other angle.

Reduction of the maximum degree ofpolarization is not the only consequenceof multiple scattering. According to Fig. 6,there should be two neutral points in thesky, directions in which skylight is unpo-larized: directly toward and away from thesun. Because of multiple scattering, how-ever, there are three such points. Whenthe sun is higher than about 20◦ above thehorizon there are neutral points within 20◦of the sun, the Babinet point above it, theBrewster point below. They coincide whenthe sun is directly overhead and moveapart as the sun descends. When the sunis lower than 20◦, the Arago point is about20◦ above the antisolar point, the directionopposite the sun.

One consequence of the partial polar-ization of skylight is that the colors ofdistant objects may change when viewedthrough a rotated polarizing filter. If thesun is high in the sky, horizontal airlightwill have a fairly high degree of polariza-tion. According to the previous section,


airlight is bluish. But if it also is partiallypolarized, its radiance can be diminishedwith a polarizing filter. Transmitted cloud-light, however, is unpolarized. Because theradiance of airlight can be reduced morethan that of cloudlight, distant clouds maychange from white to yellow to orangewhen viewed through a rotated polariz-ing filter.

4Scattering by Particles

Up to this point we have considered only anatmosphere free of particles, an idealizedstate rarely achieved in nature. Particlesstill would inhabit the atmosphere evenif the human race were to vanish fromthe Earth. They are not simply by-products of the ‘‘dark satanic mills’’ ofcivilization.

All molecules of the same substance areessentially identical. This is not true ofparticles: They vary in shape and size,and may be composed of one or morehomogeneous regions.

4.1The Salient Differences between Particlesand Molecules: Magnitude of Scattering

The distinction between scattering bymolecules when widely separated andwhen packed together into a droplet isthat between scattering by incoherent andcoherent arrays. Isolated molecules areexcited primarily by incident (external)light, whereas the same molecules forminga droplet are excited by incident light andby each other’s scattered fields. The totalpower scattered by an incoherent array ofmolecules is the sum of their scatteredpowers. The total power scattered by acoherent array is the square of the total

scattered field, which in turn is the sumof all the fields scattered by the individualmolecules. For an incoherent array we mayignore the wave nature of light, whereasfor a coherent array we must take itinto account.

Water vapor is a good example to ponderbecause it is a constituent of air andcan condense to form cloud droplets. Thedifference between a sky containing watervapor and the same sky with the sameamount of water but in the form of a cloudof droplets is dramatic.

According to Rayleigh’s law, scatteringby a particle small compared with thewavelength increases as the sixth powerof its size (volume squared). A droplet ofdiameter 0.03 µm, for example, scattersabout 1012 times more light than does oneof its constituent molecules. Such a dropletcontains about 107 molecules. Thus,scattering per molecule as a consequenceof condensation of water vapor intoa coherent water droplet increases byabout 105.

Cloud droplets are much larger than0.03 µm, a typical diameter being about10 µm. Scattering per molecule in sucha droplet is much greater than scatter-ing by an isolated molecule, but not tothe extent given by Rayleigh’s law. Scat-tering increases as the sixth power ofdroplet diameter only when the moleculesscatter coherently in phase. If a dropletis sufficiently small compared with thewavelength, each of its molecules is ex-cited by essentially the same field andall the waves scattered by them inter-fere constructively. But when a dropletis comparable to or larger than the wave-length, interference can be constructive,destructive, and everything in between,and hence scattering does not increaseas rapidly with droplet size as predicted byRayleigh’s law.


The figure of merit for comparingscatterers of different size is their scatter-ing cross section per unit volume, which,except for a multiplicative factor, is the scat-tering cross section per molecule. A scat-tering cross section may be looked uponas an effective area for removing radiantenergy from a beam: the scattering crosssection times the beam irradiance is theradiant power scattered in all directions.

The scattering cross section per unitvolume for water droplets illuminatedby visible light and varying in sizefrom molecules (10−4 µm) to raindrops(103 µm) is shown in Fig. 7. Scattering bya molecule that belongs to a cloud droplet isabout 109 times greater than scattering byan isolated molecule, a striking exampleof the virtue of cooperation. Yet inmolecular as in human societies there arelimits beyond which cooperation becomesdysfunctional: Scattering by a moleculethat belongs to a raindrop is about 100times less than scattering by a moleculethat belongs to a cloud droplet. Thistremendous variation of scattering bywater molecules depending on their stateof aggregation has profound observationalconsequences. A cloud is optically so much

Fig. 7 Scattering (per molecule) of visible light(arbitrary units) by water droplets varying in sizefrom a single molecule to a raindrop

different from the water vapor out of whichit was born that the offspring bears noresemblance to its parents. We can seethrough tens of kilometers of air ladenwith water vapor, whereas a cloud a fewtens of meters thick is enough to occultthe sun. Yet a rainshaft born out of acloud is considerably more translucentthan its parent.

4.2The Salient Differences between Particlesand Molecules: Wavelength Dependence ofScattering

Regardless of their size and composi-tion, particles scatter approximately asthe inverse fourth power of wavelengthif they are small compared with the wave-length and absorption is negligible, twoimportant caveats. Failure to recognizethem has led to errors, such as that yel-low light penetrates fog better becauseit is not scattered as much as light ofshorter wavelengths. Although there maybe perfectly sound reasons for choosingyellow instead of blue or green as thecolor of fog lights, greater transmissionthrough fog is not one of them: Scat-tering by fog droplets is essentially in-dependent of wavelength over the visiblespectrum.

Small particles are selective scatterers;large particles are not. Particles nei-ther small nor large give the reverse ofwhat we have come to expect as nor-mal. Figure 8 shows scattering of visiblelight by oil droplets with diameters 0.1,0.8, and 10 µm. The smaller dropletsscatter according to Rayleigh’s law; thelarger droplets (typical cloud droplet size)are nonselective. Between these two ex-tremes are droplets (0.8 µm) that scatterlong-wavelength light more than short-wavelength. Sunlight or moonlight seen


Fig. 8 Scattering of visible light by oil dropletsof diameter 0.1 µm (solid curve), 0.8 µm(dashes), and 10 µm (dots)

through a thin cloud of these intermediatedroplets would be bluish or greenish. Thisrequires droplets of just the right size, andhence it is a rare event, so rare that it oc-curs once in a blue moon. Astronomers,for unfathomable reasons, refer to the sec-ond full moon in a month as a blue moon,but if such a moon were blue it would beonly by coincidence. The last reliably re-ported outbreak of blue and green sunsand moons occurred in 1950 and wasattributed to an oily smoke produced inCanadian forest fires.

4.3The Salient Differences between Particlesand Molecules: Angular Dependence ofScattering

The angular distribution of scattered lightchanges dramatically with the size ofthe scatterer. Molecules and particles thatare small compared with the wavelengthare nearly isotropic scatterers of unpo-larized light, the ratio of maximum (at0◦ and 180◦) to minimum (at 90◦) scat-tered radiance being only 2 for spheres,and slightly less for other spheroids. Al-though small particles scatter the same in

the forward and backward hemispheres,scattering becomes markedly asymmetricfor particles comparable to or larger thanthe wavelength. For example, forward scat-tering by a water droplet as small as 0.5 µmis about 100 times greater than backwardscattering, and the ratio of forward to back-ward scattering increases more or lessmonotonically with size (Fig. 9).

The reason for this asymmetry is foundin the singularity of the forward direc-tion. In this direction, waves scatteredby two or more scatterers excited solelyby incident light (ignoring mutual ex-citation) are always in phase regardlessof the wavelength and the separationof the scatterers. If we imagine a par-ticle to be made up of N small sub-units, scattering in the forward direc-tion increases as N2, the only direc-tion for which this is always true. Forother directions, the wavelets scattered bythe subunits will not necessarily all bein phase. As a consequence, scatteringin the forward direction increases withsize (i.e., N) more rapidly than in anyother direction.

Fig. 9 Angular dependence of scattering ofvisible light (0.55 µm) by water droplets smallcompared with the wavelength (dashes),diameter 0.5 µm (solid curve), and diameter10 µm (dots)


Many common observable phenom-ena depend on this forward-backwardasymmetry. Viewed toward the illumi-nating sun, glistening fog droplets on aspider’s web warn us of its presence. Butwhen we view the web with our backsto the sun, the web mysteriously disap-pears. A pattern of dew illuminated bythe rising sun on a cold morning seemsetched on a windowpane. But if we gooutside to look at the window, the patternvanishes. Thin clouds sometimes hoverover warm, moist heaps of dung, but maygo unnoticed unless they lie between usand the source of illumination. These arebut a few examples of the consequencesof strongly asymmetric scattering by sin-gle particles comparable to or larger thanthe wavelength.

4.4The Salient Differences between Particlesand Molecules: Degree of Polarization ofScattered Light

All the simple rules about polarizationupon scattering are broken when we turnfrom molecules and small particles toparticles comparable to the wavelength.For example, the degree of polarization oflight scattered by small particles is a simplefunction of scattering angle. But simplicitygives way to complexity as particles grow(Fig. 10), the scattered light being partiallypolarized parallel to the scattering planefor some scattering angles, perpendicularfor others.

The degree of polarization of lightscattered by molecules or by small particlesis essentially independent of wavelength.But this is not true for particles comparableto or larger than the wavelength. Scatteringby such particles exhibits dispersion ofpolarization: The degree of polarization at,

Fig. 10 Degree of polarization of light scatteredby water droplets illuminated by unpolarizedvisible light (0.55 µm). The dashed curve is for adroplet small compared with the wavelength; thesolid curve is for a droplet of diameter 0.5 µm;the dotted curve is for a droplet of diameter1.0 µm. Negative degrees of polarizationindicate that the scattered light is partiallypolarized parallel to the scattering plane

Fig. 11 Degree of polarization at a scatteringangle of 90◦ of light scattered by a water dropletof diameter 0.5 µm illuminated byunpolarized light

say, 90◦ may vary considerably over thevisible spectrum (Fig. 11).

In general, particles can act as polarizersor retarders or both. A polarizer transformsunpolarized light into partially polarizedlight. A retarder transforms polarized lightof one form into that of another (e.g.,


linear into elliptical). Molecules and smallparticles, however, are restricted to rolesas polarizers. If the atmosphere wereinhabited solely by such scatterers, skylightcould never be other than partially linearlypolarized. Yet particles comparable toor larger than the wavelength oftenare present; hence skylight can acquirea degree of ellipticity upon multiplescattering: Incident unpolarized light ispartially linearly polarized in the firstscattering event, then transformed intopartially elliptically polarized light insubsequent events.

Bees can navigate by polarized sky-light. This statement, intended to evokegreat awe for the photopolimetric pow-ers of bees, is rarely accompanied byan important caveat: The sky must beclear. Figures 10 and 11 show two rea-sons – there are others – why bees, re-markable though they may be, cannot dothe impossible. The simple wavelength-independent relation between the posi-tion of the sun and the direction inwhich skylight is most highly polarized,an underlying necessity for navigatingby means of polarized skylight, is oblit-erated when clouds cover the sky. Thiswas recognized by the decoder of beedances himself von Frisch, [8]: ‘‘Some-times a cloud would pass across the areaof sky visible through the tube; when thishappened the dances became disoriented,and the bees were unable to indicate thedirection to the feeding place. Whateverphenomenon in the blue sky served to ori-ent the dances, this experiment showedthat it was seriously disturbed if theblue sky was covered by a cloud.’’ Butvon Frisch’s words often have been for-gotten by disciples eager to spread thestory about bee magic to those just aseager to believe what is charming eventhough untrue.

4.5The Salient Differences between Particlesand Molecules: Vertical Distributions

Not only are the scattering properties ofparticles quite different, in general, fromthose of molecules; the different verticaldistributions of particles and molecules bythemselves affect what is observed. Thenumber density of molecules decreasesmore or less exponentially with heightz above the surface: exp(−z/Hm), wherethe molecular scale height Hm is around8 km. Although the decrease in numberdensity of particles with height is also ap-proximately exponential, the scale heightfor particles Hp is about 1–2 km. As aconsequence, particles contribute dispro-portionately to optical thicknesses alongnear-horizon paths. Subject to the approxi-mations underlying Eq. (9), the ratio of thetangential (horizon) optical thickness forparticles τtp to that for molecules τtm is

τtp

τtm= τnp

τnm

√Hm

Hp, (13)

where the subscript t indicates a tangentialpath and n indicates a normal (radial) path.Because of the incoherence of scatteringby atmospheric molecules and particles,scattering coefficients are additive, andhence so are optical thicknesses. For equalnormal optical thicknesses, the tangentialoptical thickness for particles is at leasttwice that for molecules. Molecules bythemselves cannot give red sunrises andsunsets; molecules need the help ofparticles. For a fixed τnp, the tangentialoptical thickness for particles is greaterthe more they are concentrated nearthe ground.

At the horizon the relative rate of changeof transmission T of sunlight with zenith


angle is1

T

dT

d�= τn

Re

H, (14)

where the scale height and normal opti-cal thickness may be those for moleculesor particles. Not only do particles, be-ing more concentrated near the surface,give disproportionate attenuation of sun-light on the horizon, but they magnifythe angular gradient of attenuation there.A perceptible change in color across thesun’s disk (which subtends about 0.5◦)on the horizon also requires the helpof particles.

5Atmospheric Visibility

On a clear day can we really see for-ever? If not, how far can we see? Toanswer this question requires qualifyingit by restricting viewing to more or lesshorizontal paths during daylight. Starsat staggering distances can be seen atnight, partly because there is no sky-light to reduce contrast, partly becausestars overhead are seen in directionsfor which attenuation by the atmosphereis least.

The radiance in the direction of a blackobject is not zero, because of light scatteredalong the line of sight (see Sec. 2.4). Atsufficiently large distances, this airlight isindistinguishable from the horizon sky.An example is a phalanx of parallel darkridges, each ridge less distinct than thosein front of it (Fig. 12). The farthest ridgesblend into the horizon sky. Beyond somedistance we cannot see ridges because ofinsufficient contrast.

Equation (5) gives the airlight radi-ance, a radiometric quantity that de-scribes radiant power without taking into

Fig. 12 Because of scattering by molecules andparticles along the line of sight, each successiveridge is brighter than the ones in front of it eventhough all of them are covered with the samedark vegetation

account the portion of it that stimu-lates the human eye or by what relativeamount it does so at each wavelength.Luminance (also sometimes called bright-ness) is the corresponding photometricquantity. Luminance and radiance arerelated by an integral over the visi-ble spectrum:

B =∫

K(λ)L(λ) dλ, (15)

where the luminous efficiency of the hu-man eye K peaks at about 550 nm andvanishes outside the range 385–760 nm.

The contrast C between any object andthe horizon sky is

C = B − B∞B∞

, (16)

where B∞ is the luminance for an infinitehorizon optical thickness. For a uniformlyilluminated line of sight of length d,uniform in its scattering properties, and


with a black backdrop, the contrast is

C = −

∫KGL0 exp(−βd) dλ∫

KGL0 dλ

. (17)

The ratio of integrals in this equationdefines an average optical thickness:

C = − exp(−〈τ 〉). (18)

This expression for contrast reductionwith (optical) distance is mathematically,but not physically, identical to Eq. (6),which perhaps has engendered the mis-conception that atmospheric visibility isreduced because of attenuation. Yet asthere is no light from a black object to beattenuated, its finite visual range cannotbe a consequence of attenuation.

The distance beyond which a darkobject cannot be distinguished from thehorizon sky is determined by the contrastthreshold: the smallest contrast detectableby the human observer. Although thisdepends on the particular observer, theangular size of the object observed,the presence of nearby objects, and theabsolute luminance, a contrast thresholdof 0.02 is often taken as an average. Thisvalue in Eq. (18) gives

− ln |C| = 3.9 = 〈τ 〉 = 〈βd〉. (19)

To convert an optical distance into aphysical distance requires the scatteringcoefficient. Because K is peaked at around550 nm, we can obtain an approximatevalue of d from the scattering coefficientat this wavelength in Eq. (19). At sealevel, the molecular scattering coefficientin the middle of the visible spectrumcorresponds to about 330 km for ‘‘forever’’:the greatest distance at which a blackobject can be seen against the horizon

sky assuming a contrast threshold of 0.02and ignoring the curvature of the earth.

We also observe contrast between ele-ments of the same scene, a hillside mottledwith stands of trees and forest clearings,for example. The extent to which we canresolve details in such a scene depends onsun angle as well as distance.

The airlight radiance for a nonreflectingobject is Eq. (5) with G = p(�)�s, wherep(�) is the probability (per unit solid angle)that light is scattered in a direction makingan angle � with the incident sunlight and�s is the solid angle subtended by the sun.When the sun is overhead, � = 90◦; withthe sun at the observer’s back, � = 180◦;for an observer looking directly into thesun, � = 0◦.

The radiance of an object with a finite re-flectance R and illuminated at an angle �

is given by Eq. (11). Equations (5) and (11)can be combined to obtain the contrast be-tween reflecting and nonreflecting objects:

C = Fe−τ

1 + (F − 1)e−τ,

F = R cos �

πp(�). (20)

All else being equal, therefore, contrastdecreases as p(�) increases. As shown inFig. 9, p(�) is more sharply peaked in theforward direction the larger the scatterer.Thus, we expect the details of a distantscene to be less distinct when lookingtoward the sun than away from it if theoptical thickness of the line of sight hasan appreciable component contributed byparticles comparable to or larger than thewavelength.

On humid, hazy days, visibility isoften depressingly poor. Haze, however,is not water vapor but rather waterthat has ceased to be vapor. At highrelative humidities, but still well below


100%, small soluble particles in theatmosphere accrete liquid water to becomesolution droplets (haze). Although thesedroplets are much smaller than clouddroplets, they markedly diminish visualrange because of the sharp increase inscattering with particle size (Fig. 7). Thesame number of water molecules whenaggregated in haze scatter vastly more thanwhen apart.

6Atmospheric Refraction

6.1Physical Origins of Refraction

Atmospheric refraction is a consequenceof molecular scattering, which is rarelystated given the historical accident thatbefore light and matter were well un-derstood refraction and scattering werelocked in separate compartments and sub-sequently have been sequestered morerigidly than monks and nuns in neigh-boring cloisters.

Consider a beam of light propagating inan optically homogeneous medium. Lightis scattered (weakly but observably) later-ally to this beam as well as in the directionof the beam (the forward direction). Theobserved beam is a coherent superposi-tion of incident light and forward-scatteredlight, which was excited by the incidentlight. Although refractive indices are of-ten defined by ratios of phase velocities,we may also look upon a refractive indexas a parameter that specifies the phaseshift between an incident beam and theforward-scattered beam that the incidentbeam excites. The connection between(incoherent) scattering and refraction (co-herent scattering) can be divined from theexpressions for the refractive index n of a

gas and the scattering cross section σs of agas molecule:

n = 1 + 12αN, (21)

σs = k4

6π|α|2, (22)

where N is the number density (not massdensity) of gas molecules, k = 2π/λ is thewave number of the incident light, and α

is the polarizability of a molecule (induceddipole moment per unit inducing electricfield). The appearance of the polarizabil-ity in Eq. (21) but its square in Eq. (22) isthe clue that refraction is associated withelectric fields whereas lateral scatteringis associated with electric fields squared(powers). Scattering, without qualification,often means incoherent scattering in alldirections. Refraction, in a nutshell, is co-herent scattering in a particular direction.

Readers whose appetites have beenwhetted by the preceding brief discussionof the physical origins of refraction aredirected to a beautiful paper by Doyle [9]in which he shows how the Fresnelequations can be dissected to reveal thescattering origins of (specular) reflectionand refraction.

6.2Terrestrial Mirages

Mirages are not illusions, any more sothan are reflections in a pond. Reflectionsof plants growing at its edge are notinterpreted as plants growing into thewater. If the water is ruffled by wind,the reflected images may be so distortedthat they are no longer recognizableas those of plants. Yet we still wouldnot call such distorted images illusions.And so is it with mirages. They areimages noticeably different from what theywould be in the absence of atmospheric


refraction, creations of the atmosphere,not of the mind.

Mirages are vastly more common thanis realized. Look and you shall see them.Contrary to popular opinion, they arenot unique to deserts. Mirages can beseen frequently even over ice-coveredlandscapes and highways flanked by deepsnowbanks. Temperature per se is notwhat gives mirages but rather temperaturegradients.

Because air is a mixture of gases, thepolarizability for air in Eq. (21) is anaverage over all its molecular constituents,although their individual polarizabilitiesare about the same (at visible wavelengths).The vertical refractive index gradient canbe written so as to show its dependence onpressure p and (absolute) temperature T :

d

dzln(n − 1) = 1

p

dp

dz− 1

T

dT

dz. (23)

Pressure decreases approximately ex-ponentially with height, where the scaleheight is around 8 km. Thus, the first termon the right-hand side of Eq. (23) is around0.1 km−1. Temperature usually decreaseswith height in the atmosphere. An averagelapse rate of temperature (i.e., its decreasewith height) is around 6 ◦C/km. The aver-age temperature in the troposphere (withinabout 15 km of the surface) is around280 K. Thus, the magnitude of the secondterm in Eq. (23) is around 0.02 km−1. Onaverage, therefore, the refractive-index gra-dient is dominated by the vertical pressuregradient. But within a few meters of thesurface, conditions are far from average.On a sun-baked highway your feet maybe touching asphalt at 50 ◦C while yournose is breathing air at 35 ◦C, which cor-responds to a lapse rate a thousand timesthe average. Moreover, near the surface,temperature can increase with height. In

shallow surface layers, in which the pres-sure is nearly constant, the temperaturegradient determines the refractive indexgradient. It is in such shallow layers thatmirages, which are caused by refractive-index gradients, are seen.

Cartoonists by their fertile imaginationsunfettered by science, and textbook writersby their carelessness, have engenderedthe notion that atmospheric refraction canwork wonders, lifting images of ships, forexample, from the sea high into the sky.A back-of-the-envelope calculation dispelssuch notions. The refractive index of air atsea level is about 1.0003 (Fig. 13). Lightfrom empty space incident at glancingincidence onto a uniform slab with thisrefractive index is displaced in angularposition from where it would have beenin the absence of refraction by

δ = √2(n − 1). (24)

This yields an angular displacement ofabout 1.4◦, which as we shall see is a roughupper limit.

Trajectories of light rays in nonuniformmedia can be expressed in different ways.According to Fermat’s principle of least

Fig. 13 Sea-level refractive index versuswavelength at −15 ◦C (dashes) and 15 ◦C (solidcurve). Data from Penndorf, R. (1957), J. Opt.Soc. Am. 47, 176–182 [2]


time (which ought to be extreme time),the actual path taken by a ray between twopoints is such that the path integral

∫ 2

1n ds (25)

is an extremum over all possible paths.This principle has inspired piffle about thealleged efficiency of nature, which directslight over routes that minimize travel time,presumably freeing it to tend to importantbusiness at its destination.

The scale of mirages is such that inanalyzing them we may pretend that theEarth is flat. On such an earth, withan atmosphere in which the refractiveindex varies only in the vertical, Fermat’sprinciple yields a generalization

n sin θ = constant (26)

of Snel’s law, where θ is the angle betweenthe ray and the vertical direction. Wecould, of course, have bypassed Fermat’sprinciple to obtain this result.

If we restrict ourselves to nearly hori-zontal rays, Eq. (26) yields the followingdifferential equation satisfied by a ray:

d2z

dy2 = dn

dz, (27)

where y and z are its horizontal and verticalcoordinates, respectively. For a constantrefractive-index gradient, which to goodapproximation occurs for a constant tem-perature gradient, Eq. (27) yields parabolasfor ray trajectories. One such parabola fora constant temperature gradient about 100times the average is shown in Fig. 14.Note the vastly different horizontal andvertical scales. The image is displaceddownward from what it would be in theabsence of atmospheric refraction; hencethe designation inferior mirage. This is the

Fig. 14 Parabolic ray paths in an atmospherewith a constant refractive-index gradient (inferiormirage). Note the vastly different horizontal andvertical scales

familiar highway mirage, seen over high-ways warmer than the air above them. Thedownward angular displacement is

δ = 1

2s

dn

dz, (28)

where s is the horizontal distance betweenobject and observer (image). Even fora temperature gradient 1000 times thetropospheric average, displacements ofmirages are less than a degree at distancesof a few kilometers.

If temperature increases with height,as it does, for example, in air over acold sea, the resulting mirage is calleda superior mirage. Inferior and superior arenot designations of lower and higher castebut rather of displacements downwardand upward.

For a constant temperature gradient,one and only one parabolic ray tra-jectory connects an object point to animage point. Multiple images thereforeare not possible. But temperature gra-dients close to the ground are rarelylinear. The upward transport of energyfrom a hot surface occurs by molecularconduction through a stagnant boundary


layer of air. Somewhat above the surface,however, energy is transported by air inmotion. As a consequence, the tempera-ture gradient steepens toward the groundif the energy flux is constant. This vari-able gradient can lead to two observableconsequences: magnification and multi-ple images.

According to Eq. (28), all image pointsat a given horizontal distance are dis-placed downward by an amount propor-tional to the (constant) refractive indexgradient. A corollary is that the closeran object point is to a surface, wherethe temperature gradient is greatest, thegreater the downward displacement of thecorresponding image point. Thus, non-linear vertical temperature profiles maymagnify images.

Multiple images are seen frequentlyon highways. What often appears tobe water on the highway ahead butevaporates before it is reached is theinverted secondary image of either thehorizon sky or horizon objects lighter thandark asphalt.

6.3Extraterrestrial Mirages

When we turn from mirages of terrestrialobjects to those of extraterrestrial bodies,most notably the sun and moon, wecan no longer pretend that the Earthis flat. But we can pretend that theatmosphere is uniform and bounded.The total phase shift of a vertical rayfrom the surface to infinity is the samein an atmosphere with an exponentiallydecreasing molecular number density as ina hypothetical atmosphere with a uniformnumber density equal to the surface valueup to height H.

A ray refracted along a horizon pathby this hypothetical atmosphere and

originating from outside it had to havebeen incident on it from an angle δ belowthe horizon:

δ =√

2H

R−

√2H

R− 2(n − 1), (29)

where R is the radius of the Earth. Thus,when the sun (or moon) is seen to be onthe horizon it is actually more than halfwaybelow it, δ being about 0.36◦, whereas theangular width of the sun (or moon) isabout 0.5◦.

Extraterrestrial bodies seen near thehorizon also are vertically compressed. Thesimplest way to estimate the amount ofcompression is from the rate of change ofangle of refraction θr with angle of inci-dence θi for a uniform slab

dθr

dθi= cos θi√

n2 − sin2 θi

, (30)

where the angle of incidence is that fora curved but uniform atmosphere suchthat the refracted ray is horizontal. Theresult is

dθr

dθi=

√1 − R

H(n − 1), (31)

according to which the sun near thehorizon is distorted into an ellipse withaspect ratio about 0.87. We are unlikelyto notice this distortion, however, be-cause we expect the sun and moon tobe circular, and hence we see themthat way.

The previous conclusions about thedownward displacement and distortion ofthe sun were based on a refractive-indexprofile determined mostly by the verti-cal pressure gradient. Near the ground,however, the temperature gradient is theprime determinant of the refractive-indexgradient, as a consequence of which thesun on the horizon can take on shapes


Fig. 15 A nearly triangular sun on the horizon.The serrations are a consequence of horizontalvariations in refractive index

more striking than a mere ellipse. Forexample, Fig. 15 shows a nearly triangu-lar sun with serrated edges. Assigninga cause to these serrations provides alesson in the perils of jumping to con-clusions. Obviously, the serrations are theresult of sharp changes in the temper-ature gradient – or so one might think.Setting aside how such changes could beproduced and maintained in a real at-mosphere, a theorem of Fraser [10] givespause for thought. According to this the-orem, ‘‘In a horizontally (spherically) ho-mogeneous atmosphere it is impossiblefor more than one image of an extrater-restrial object (sun) to be seen above theastronomical horizon.’’ The serrations onthe sun in Fig. 15 are multiple images.But if the refractive index varies onlyvertically (i.e., along a radius), no mat-ter how sharply, multiple images are notpossible. Thus, the serrations must owetheir existence to horizontal variations ofthe refractive index, a consequence ofgravity waves propagating along a tem-perature inversion.

6.4The Green Flash

Compared to the rainbow, the greenflash is not a rare phenomenon. Beforeyou dismiss this assertion as the ravingsof a lunatic, consider that rainbowsrequire raindrops as well as sunlight toilluminate them, whereas rainclouds oftencompletely obscure the sun. Moreover,the sun must be below about 42◦. As aconsequence of these conditions, rainbowsare not seen often, but often enough thatthey are taken as the paragon of colorvariation. Yet tinges of green on the upperrim of the sun can be seen every dayat sunrise and sunset given a sufficientlylow horizon and a cloudless sky. Thus,the conditions for seeing a green flashare more easily met than those for seeinga rainbow. Why then is the green flashconsidered to be so rare? The distinctionhere is that between a rarely observedphenomenon (the green flash) and a rarelyobservable one (the rainbow).

The sun may be considered to be acollection of disks, one for each visiblewavelength. When the sun is overhead,all the disks coincide and we see thesun as white. But as it descends in thesky, atmospheric refraction displaces thedisks by slightly different amounts, the redless than the violet (see Fig. 13). Most ofeach disk overlaps all the others exceptfor the disks at the extremes of the visiblespectrum. As a consequence, the upperrim of the sun is violet or blue, its lowerrim red, whereas its interior, the region inwhich all disks overlap, is still white.

This is what would happen in the ab-sence of lateral scattering of sunlight. Butrefraction and lateral scattering go hand inhand, even in an atmosphere free of par-ticles. Selective scattering by atmosphericmolecules and particles causes the color


of the sun to change. In particular, theviolet-bluish upper rim of the low sun canbe transformed to green.

According to Eq. (29) and Fig. 13, theangular width of the green upper rim ofthe low sun is about 0.01◦, too narrow tobe resolved with the naked eye or even tobe seen against its bright backdrop. Butdepending on the temperature profile, theatmosphere itself can magnify the upperrim and yield a second image of it, therebyenabling it to be seen without the aid of atelescope or binoculars. Green rims, whichrequire artificial magnification, can beseen more frequently than green flashes,which require natural magnification. Yetboth can be seen often by those who knowwhat to look for and are willing to look.

7Scattering by Single Water Droplets

All the colored atmospheric displays thatresult when water droplets (or ice crystals)are illuminated by sunlight have the sameunderlying cause: light is scattered indifferent amounts in different directionsby particles larger than the wavelength,and the directions in which scattering isgreatest depends on wavelength. Thus,when particles are illuminated by whitelight, the result can be angular separationof colors even if scattering integrated overall directions is independent of wavelength(as it essentially is for cloud droplets andice crystals). This description, althoughcorrect, is too general to be completelysatisfying. We need something morespecific, more quantitative, which requirestheories of scattering.

Because superficially different theorieshave been used to describe different op-tical phenomena, the notion has becomewidespread that they are caused by these

theories. For example, coronas are said tobe caused by diffraction and rainbows byrefraction. Yet both the corona and therainbow can be described quantitatively tohigh accuracy with a theory (the Mie the-ory for scattering by a sphere) in whichdiffraction and refraction do not explicitlyappear. No fundamentally impenetrablebarrier separates scattering from (specu-lar) reflection, refraction, and diffraction.Because these terms came into generaluse and were entombed in textbooks be-fore the nature of light and matter was wellunderstood, we are stuck with them. Butif we insist that diffraction, for example, issomehow different from scattering, we doso at the expense of shattering the unityof the seemingly disparate observable phe-nomena that result when light interactswith matter. What is observed dependson the composition and disposition of thematter, not on which approximate theoryin a hierarchy is used for quantitative de-scription.

Atmospheric optical phenomena arebest classified by the direction in whichthey are seen and by the agents respon-sible for them. Accordingly, the followingsections are arranged in order of scatteringdirection, from forward to backward.

When a single water droplet is illumi-nated by white light and the scatteredlight projected onto a screen, the resultis a set of colored rings. But in the atmo-sphere we see a mosaic to which individualdroplets contribute. The scattering patternof a single droplet is the same as themosaic provided that multiple scatteringis negligible.

7.1Coronas and Iridescent Clouds

A cloud of droplets narrowly distributed insize and thinly veiling the sun (or moon)


can yield a spectacular series of coloredconcentric rings around it. This coronais most easily described quantitatively bythe Fraunhofer diffraction theory, a sim-ple approximation valid for particles largecompared with the wavelength and forscattering angles near the forward direc-tion. According to this approximation, thedifferential scattering cross section (crosssection for scattering into a unit solidangle) of a spherical droplet of radius ailluminated by light of wave number k is

|S|2k2 , (32)

where the scattering amplitude is

S = x2 1 + cos θ

2

J1(x sin θ)

x sin θ. (33)

The term J1 is the Bessel function offirst order and the size parameter x =ka. The quantity (1 + cos θ )/2 is usuallyapproximated by 1 since only near-forwardscattering angles θ are of interest.

The differential scattering cross section,which determines the angular distributionof the scattered light, has maxima forx sin θ = 5.137, 8.417, 11.62, . . . Thus,the dispersion in the position of the firstmaximum is

dθ

dλ≈ 0.817

a(34)

and is greater for higher-order maxima.This dispersion determines the upper limiton drop size such that a corona can beobserved. For the total angular dispersionover the visible spectrum to be greaterthan the angular width of the sun (0.5◦),the droplets cannot be larger than about60 µm in diameter. Drops in rain, evenin drizzle, are appreciably larger thanthis, which is why coronas are not seenthrough rainshafts. Scattering by a dropletof diameter 10 µm (Fig. 16), a typical cloud

Fig. 16 Scattering of light near the forwarddirection (according to Fraunhofer theory) by asphere of diameter 10 µm illuminated by red andgreen light

droplet size, gives sufficient dispersion toyield colored coronas.

Suppose that the first angular maxi-mum for blue light (0.47 µm) occurs for adroplet of radius a. For red light (0.66 µm)a maximum is obtained at the same an-gle for a droplet of radius a + �a. Thatis, the two maxima, one for each wave-length, coincide. From this we concludethat coronas require narrow size distri-butions: if cloud droplets are distributedin radius with a relative variance �a/agreater than about 0.4, color separation isnot possible.

Because of the stringent requirementsfor the occurrence of coronas, they arenot observed often. Of greater occur-rence are the corona’s cousins, iridescentclouds, which display colors but usuallynot arranged in any obviously regular ge-ometrical pattern. Iridescent patches inclouds can be seen even at the edges ofthick clouds that occult the sun.

Coronas are not the unique signatures ofspherical scatterers. Randomly oriented icecolumns and plates give similar patternsaccording to Fraunhofer theory [11]. As apractical matter, however, most coronas


probably are caused by droplets. Manyclouds at temperatures well below freezingcontain subcooled water droplets. Onlyif a corona were seen in a cloud at atemperature lower than −40 ◦C could oneassert with confidence that it must be anice-crystal corona.

7.2Rainbows

In contrast with coronas, which are seenlooking toward the sun, rainbows areseen looking away from it, and arecaused by water drops much larger thanthose that give coronas. To treat therainbow quantitatively we may pretendthat light incident on a transparent sphereis composed of individual rays, each ofwhich suffers a different fate determinedonly by the laws of specular reflection andrefraction. Theoretical justification for thisis provided by van de Hulst’s ([12], p. 208)localization principle, according to whichterms in the exact solution for scattering bya transparent sphere correspond to moreor less localized rays.

Each incident ray splinters into an infi-nite number of scattered rays: externallyreflected, transmitted without internal re-flection, transmitted after one, two, and soon internal reflections. At any scatteringangle θ , each splinter contributes to thescattered light. Accordingly, the differen-tial scattering cross section is an infiniteseries with terms of the form

b(θ)

sin θ

db

dθ. (35)

The impact parameter b is a sin �i, where�i is the angle between an incidentray and the normal to the sphere. Eachterm in the series corresponds to oneof the splinters of an incident ray. Arainbow angle is a singularity (or caustic)

of the differential scattering cross sectionat which the conditions

dθ

db= 0,

b

sin θ= 0 (36)

are satisfied. Missing from Eq. (35) arevarious reflection and transmission coeffi-cients (Fresnel coefficients), which displayno singularities and hence do not deter-mine rainbow angles.

A rainbow is not associated with raysexternally reflected or transmitted withoutinternal reflection. The succession ofrainbow angles associated with one, two,three . . . internal reflections are calledprimary, secondary, tertiary . . . rainbows.Aristotle recognized that ‘‘Three or morerainbows are never seen, because eventhe second is dimmer than the first, andso the third reflection is altogether toofeeble to reach the sun (Aristotle’s viewwas that light streams outward from theeye)’’. Although he intuitively grasped thateach successive ray is associated withever-diminishing energy, his statementabout the nonexistence of tertiary rainbowsin nature is not quite true. Althoughreliable reports of such rainbows are rare(unreliable reports are as common as dirt),at least one observer who can be believedhas seen one [13].

An incident ray undergoes a totalangular deviation as a consequence oftransmission into the drop, one or moreinternal reflections, and transmission outof the drop. Rainbow angles are angles ofminimum deviation.

For a rainbow of any order to exist,

cos �i =√

n2 − 1

p(p + 1)(37)

must lie between 0 and 1, where �i isthe angle of incidence of a ray that givesa rainbow after p internal reflections and


n is the refractive index of the drop. Aprimary bow therefore requires drops withrefractive index less than 2; a secondarybow requires drops with refractive indexless than 3. If raindrops were composedof titanium dioxide (n ≈ 3), a commonlyused opacifier for paints, primary rainbowswould be absent from the sky and wewould have to be content with onlysecondary bows.

If we take the refractive index of water tobe 1.33, the scattering angle for the primaryrainbow is about 138◦. This is measuredfrom the forward direction (solar point).Measured from the antisolar point (thedirection toward which one must lookin order to see rainbows in nature), thisscattering angle corresponds to 42◦, thebasis for a previous assertion that rainbows(strictly, primary rainbows) cannot beseen when the sun is above 42◦. Thesecondary rainbow is seen at about 51◦from the antisolar point. Between thesetwo rainbows is Alexander’s dark band, aregion into which no light is scatteredaccording to geometrical optics.

The colors of rainbows are a conse-quence of sufficient dispersion of therefractive index over the visible spectrumto give a spread of rainbow angles thatappreciably exceeds the width of the sun.The width of the primary bow from violetto red is about 1.7◦; that of the secondarybow is about 3.1◦.

Because of its band of colors arcingacross the sky, the rainbow has becomethe paragon of color, the standard againstwhich all other colors are compared. Leeand Fraser [14, 15], however, challengedthis status of the rainbow, pointing outthat even the most vivid rainbows arecolorimetrically far from pure.

Rainbows are almost invariably dis-cussed as if they occurred literally in avacuum. But real rainbows, as opposed

to the pencil-and-paper variety, are nec-essarily observed in an atmosphere, themolecules and particles of which scat-ter sunlight that adds to the light fromthe rainbow but subtracts from its purityof color.

Although geometrical optics yields thepositions, widths, and color separationof rainbows, it yields little else. Forexample, geometrical optics is blind tosupernumerary bows, a series of narrowbands sometimes seen below the primarybow. These bows are a consequenceof interference, and hence fall outsidethe province of geometrical optics. Sincesupernumerary bows are an interferencephenomenon, they, unlike primary andsecondary bows (according to geometricaloptics), depend on drop size. This posesthe question of how supernumerary bowscan be seen in rain showers, the dropsin which are widely distributed in size. Ina nice piece of detective work, Fraser [16]answered this question.

Raindrops falling in a vacuum are spher-ical. Those falling in air are distorted byaerodynamic forces, not, despite the de-pictions of countless artists, into teardropsbut rather into nearly oblate spheroids withtheir axes more or less vertical. Fraser ar-gued that supernumerary bows are causedby drops with a diameter of about 0.5 mm,at which diameter the angular positionof the first (and second) supernumerarybow has a minimum: interference causesthe position of the supernumerary bowto increase with decreasing size whereasdrop distortion causes it to increase withincreasing size. Supernumerary patternscontributed by drops on either side of theminimum cancel, leaving only the contri-bution from drops at the minimum. Thiscancellation occurs only near the tops ofrainbows, where supernumerary bows areseen. In the vertical parts of a rainbow, a


horizontal slice through a distorted dropis more or less circular, and hence thesedrops do not exhibit a minimum supernu-merary angle.

According to geometrical optics, allspherical drops, regardless of size, yieldthe same rainbow. But it is not necessaryfor a drop to be spherical for it to yieldrainbows independent of its size. Thismerely requires that the plane defined bythe incident and scattered rays intersectthe drop in a circle. Even distorteddrops satisfy this condition in the verticalpart of a bow. As a consequence, theabsence of supernumerary bows there iscompensated for by more vivid colorsof the primary and secondary bows [17].Smaller drops are more likely to bespherical, but the smaller a drop, theless light it scatters. Thus, the dominantcontribution to the luminance of rainbowsis from the larger drops. At the top of abow, the plane defined by the incident andscattered rays intersects the large, distorteddrops in an ellipse, yielding a range ofrainbow angles varying with the amount ofdistortion, and hence a pastel rainbow. Tothe knowledgeable observer, rainbows areno more uniform in color and brightnessthan is the sky.

Although geometrical optics predictsthat all rainbows are equal (neglectingbackground light), real rainbows do notslavishly follow the dictates of this approx-imate theory. Rainbows in nature rangefrom nearly colorless fog bows (or cloudbows) to the vividly colorful vertical por-tions of rainbows likely to have inspiredmyths about pots of gold.

7.3The Glory

Continuing our sweep of scattering direc-tions, from forward to backward, we arrive

at the end of our journey: the glory. Becauseit is most easily seen from airplanes itsometimes is called the pilot’s bow. Anothername is anticorona, which signals that itis a corona around the antisolar point. Al-though glories and coronas share somecommon characteristics, there are differ-ences between them other than directionof observation. Unlike coronas, which maybe caused by nonspherical ice crystals, glo-ries require spherical cloud droplets. Anda greater number of colored rings may beseen in glories than in coronas becausethe decrease in luminance away from thebackward direction is not as steep as thataway from the forward direction. To seea glory from an airplane, look for coloredrings around its shadow cast on clouds be-low. This shadow is not an essential partof the glory, it merely directs you to theantisolar point.

Like the rainbow, the glory may belooked upon as a singularity in the dif-ferential scattering cross section Eq. (35).Equation (36) gives one set of conditionsfor a singularity; the second set is

sin θ = 0, b(θ) = 0. (38)

That is, the differential scattering crosssection is infinite for nonzero impactparameters (corresponding to incidentrays that do not intersect the center of thesphere) that give forward (0◦) or backward(180◦) scattering. The forward directionis excluded because this is the directionof intense scattering accounted for by theFraunhofer theory.

For one internal reflection, Eq. (38) leadsto the condition

sin �i = n

2

√4 − n2, (39)

which is satisfied only for refractive indicesbetween 1.414 and 2, the lower refractiveindex corresponding to a grazing-incidence


ray. The refractive index of water liesoutside this range. Although a conditionsimilar to Eq. (39) is satisfied for rays un-dergoing four or more internal reflections,insufficient energy is associated with suchrays. Thus, it seems that we have reachedan impasse: the theoretical condition fora glory cannot be met by water droplets.Not so, says van de Hulst [18] in a sem-inal paper. He argues that 1.414 is closeenough to 1.33 given that geometrical op-tics is, after all, an approximation. Clouddroplets are large compared with the wave-length, but not so large that geometricaloptics is an infallible guide to their opticalbehavior. Support for the van de Hulstianinterpretation of glories was provided byBryant and Cox [19], who showed that thedominant contribution to the glory is fromthe last terms in the exact series for scat-tering by a sphere. Each successive termin this series is associated with ever largerimpact parameters. Thus, the terms thatgive the glory are indeed those correspond-ing to grazing rays. Further unraveling ofthe glory and vindication of van de Hulst’sconjectures about the glory were providedby Nussenzveig [20].

It sometimes is asserted that geometricaloptics is incapable of treating the glory.Yet the same can be said for the rainbow.Geometrical optics explains rainbows onlyin the sense that it predicts singularities forscattering in certain directions (rainbowangles). But it can predict only the anglesof intense scattering, not the amount.Indeed, the error is infinite. Geometricaloptics also predicts a singularity in thebackward direction. Again, this simpletheory is powerless to predict more.Results from geometrical optics for bothrainbows and glories are not the endbut rather the beginning, an invitationto take a closer look with more powerfulmagnifying glasses.

8Scattering by Single Ice Crystals

Scattering by spherical water drops in theatmosphere gives rise to three distinct dis-plays in the sky: coronas, rainbows, andglories. Ice particles (crystals) also can in-habit the atmosphere, and they introducetwo new variables in addition to size: shapeand orientation, the second a consequenceof the first. Given this increase in thenumber of degrees of freedom, it is hardlycause for wonder that ice crystals are thesource of a greater variety of displays thanare water drops. As with rainbows, thegross features of ice-crystal phenomenacan be described simply with geometricaloptics, various phenomena arising fromthe various fates of rays incident on crys-tals. Colorless displays (e.g., sun pillars)are generally associated with reflected rays,colored displays (e.g., sun dogs and halos)with refracted rays. Because of the wealthof ice-crystal displays, it is not possible totreat all of them here, but one exampleshould point the way toward understand-ing many of them.

8.1Sun Dogs and Halos

Because of the hexagonal crystalline struc-ture of ice it can form as hexagonal platesin the atmosphere. The stable position ofa plate falling in air is with the normalto its face more or less vertical, which iseasy to demonstrate with an ordinary busi-ness card. When the card is dropped withits edge facing downward (the supposedlyaerodynamic position that many peopleinstinctively choose), the card somersaultsin a helter-skelter path to the ground. Butwhen the card is dropped with its face par-allel to the ground, it rocks back and forthgently in descent.


A hexagonal ice plate falling throughair and illuminated by a low sun islike a 60◦ prism illuminated normally toits sides (Fig. 17). Because there is nomechanism for orienting a plate withinthe horizontal plane, all plate orientationsin this plane are equally probable. Statedanother way, all angles of incidence fora fixed plate are equally probable. Yet allscattering angles (deviation angles) of raysrefracted into and out of the plate are notequally probable.

Figure 18 shows the range of scatteringangles corresponding to a range of raysincident on a 60◦ ice prism that is part of ahexagonal plate. For angles of incidenceless than about 13◦, the transmittedray is totally internally reflected in theprism. For angles of incidence greaterthan about 70◦, the transmittance plunges.Thus, the only rays of consequence arethose incident between about 13◦ and70◦.

All scattering angles are not equallyprobable. The (uniform) probabilitydistribution p(θi) of incidence angles θi

is related to the probability distribution

Fig. 17 Scattering by a hexagonal ice plateilluminated by light parallel to its basal plane.The particular scattering angle θ shown is anangle of minimum deviation. The scattered lightis that associated with two refractions bythe plate

Fig. 18 Scattering by a hexagonal ice plate (seeFig. 17) in various orientations (angles ofincidence). The solid curve is for red light, thedashed for blue light

P(θ) of scattering angles θ by

P(θ) = p(θi)

dθ/ dθi. (40)

At the incidence angle for whichdθ/ dθi = 0, P(θ) is infinite and scat-tered rays are intensely concentratednear the corresponding angle of mini-mum deviation.

The physical manifestation of this singu-larity (or caustic) at the angle of minimumdeviation for a 60◦ hexagonal ice plate isa bright spot about 22◦ from either orboth sides of a sun low in the sky. Thesebright spots are called sun dogs (becausethey accompany the sun) or parhelia ormock suns.

The angle of minimum deviation θm,hence the angular position of sun dogs,depends on the prism angle � (60◦for the plates considered) and refrac-tive index:

θm = 2 sin−1(

n sin�

2

)− �. (41)

Because ice is dispersive, the separationbetween the angles of minimum deviationfor red and blue light is about 0.7◦ (Fig. 18),


somewhat greater than the angular widthof the sun. As a consequence, sun dogsmay be tinged with color, most noticeablytoward the sun. Because the refractiveindex of ice is least at the red end of thespectrum, the red component of a sun dogis closest to the sun. Moreover, light of anytwo wavelengths has the same scatteringangle for different angles of incidenceif one of the wavelengths does notcorrespond to red. Thus, red is the purestcolor seen in a sun dog. Away from its redinner edge a sun dog fades into whiteness.

With increasing solar elevation, sundogs move away from the sun. A falling iceplate is roughly equivalent to a prism, theprism angle of which increases with solarelevation. From Eq. (41) it follows that theangle of minimum deviation, hence thesun dog position, also increases.

At this point you may be wondering whyonly the 60◦ prism portion of a hexagonalplate was singled out for attention. Asevident from Fig. 17, a hexagonal platecould be considered to be made up of 120◦prisms. For a ray to be refracted twice, itsangle of incidence at the second interfacemust be less than the critical angle. Thisimposes limitations on the prism angle.For a refractive index 1.31, all incident raysare totally internally reflected by prismswith angles greater than about 99.5◦.

A close relative of the sun dog is the22◦ halo, a ring of light approximately 22◦from the sun (Fig. 19). Lunar halos arealso possible and are observed frequently(although less frequently than solar halos);even moon dogs are possible. UntilFraser [21] analyzed halos in detail, theconventional wisdom had been that theyobviously were the result of randomlyoriented crystals, yet another example ofjumping to conclusions. By combiningoptics and aerodynamics, Fraser showedthat if ice crystals are small enough

to be randomly oriented by Brownianmotion, they are too small to yield sharpscattering patterns.

But completely randomly oriented platesare not necessary to give halos, especiallyones of nonuniform brightness. Each partof a halo is contributed to by plates witha different tip angle (angle between thenormal to the plate and the vertical).The transition from oriented plates (zerotip angle) to randomly oriented platesoccurs over a narrow range of sizes. Inthe transition region, plates can be smallenough to be partially oriented yet largeenough to give a distinct contribution tothe halo. Moreover, the mapping betweentip angles and azimuthal angles on thehalo depends on solar elevation. Whenthe sun is near the horizon, plates cangive a distinct halo over much of itsazimuth.

Fig. 19 A 22◦ solar halo. The hand is not forartistic effect but rather to occlude the bright sun


When the sun is high in the sky,hexagonal plates cannot give a sharp halobut hexagonal columns – another possibleform of atmospheric ice particles – can.The stable position of a falling column iswith its long axis horizontal. When thesun is directly overhead, such columnscan give a uniform halo even if they all liein the horizontal plane. When the sun isnot overhead but well above the horizon,columns also can give halos.

A corollary of Fraser’s analysis is thathalos are caused by crystals with a range ofsizes between about 12 and 40 µm. Largercrystals are oriented; smaller particlesare too small to yield distinct scatter-ing patterns.

More or less uniformly bright halos withthe sun neither high nor low in the skycould be caused by mixtures of hexagonalplates and columns or by clusters of bullets(rosettes). Fraser opines that the latter ismore likely.

One of the by-products of his analysis isan understanding of the relative rarity ofthe 46◦ halo. As we have seen, the angle ofminimum deviation depends on the prismangle. Light can be incident on a hexagonalcolumn such that the prism angle is 60◦for rays incident on its side or 90◦ forrays incident on its end. For n = 1.31,Eq. (41) yields a minimum deviation angleof about 46◦ for � = 90◦. Yet, although 46◦halos are possible, they are seen much lessfrequently than 22◦ halos. Plates cannotgive distinct 46◦ halos although columnscan. Yet they must be solid and mostcolumns have hollow ends. Moreover, therange of sun elevations is restricted.

Like the green flash, ice-crystal phenom-ena are not intrinsically rare. Halos andsun dogs can be seen frequently – onceyou know what to look for. Neuberger [22]reports that halos were observed in StateCollege, Pennsylvania, an average of 74

days a year over a 16-year period, withextremes of 29 and 152 halos a year. Al-though the 22◦ halo was by far the mostfrequently seen display, ice-crystal displaysof all kinds were seen, on average, moreoften than once every four days at a loca-tion not especially blessed with clear skies.Although thin clouds are necessary for ice-crystal displays, clouds thick enough toobscure the sun are their bane.

9Clouds

Although scattering by isolated particlescan be studied in the laboratory, parti-cles in the atmosphere occur in crowds(sometimes called clouds). Implicit in theprevious two sections is the assumptionthat each particle is illuminated solelyby incident sunlight; the particles do notilluminate each other to an appreciable de-gree. That is, clouds of water droplets orice grains were assumed to be opticallythin, and hence multiple scattering wasnegligible. Yet the term cloud evokes fluffywhite objects in the sky, or perhaps anovercast sky on a gloomy day. For suchclouds, multiple scattering is not negligi-ble, it is the major determinant of theirappearance. And the quantity that deter-mines the degree of multiple scattering isoptical thickness (see Sec. 2.4).

9.1Cloud Optical Thickness

Despite their sometimes solid appearance,clouds are so flimsy as to be almostnonexistent – except optically. The fractionof the total cloud volume occupied bywater substance (liquid or solid) is about10−6 or less. Yet although the massdensity of clouds is that of air to within


a small fraction of a percent, their opticalthickness (per unit physical thickness) ismuch greater. The number density of airmolecules is vastly greater than that ofwater droplets in clouds, but scattering permolecule of a cloud droplet is also muchgreater than scattering per air molecule(see Fig. 7).

Because a typical cloud droplet is muchlarger than the wavelengths of visible light,its scattering cross section is to goodapproximation proportional to the squareof its diameter. As a consequence, thescattering coefficient [see Eq. (2)] of a cloudhaving a volume fraction f of droplets isapproximately

β = 3f〈d2〉〈d3〉 , (42)

where the brackets indicate an averageover the distribution of droplet diametersd. Unlike molecules, cloud droplets aredistributed in size. Although cloud parti-cles can be ice particles as well as waterdroplets, none of the results in this and thefollowing section hinge on the assumptionof spherical particles.

The optical thickness along a cloudpath of physical thickness h is βh fora cloud with uniform properties. Theratio 〈d3〉/〈d2〉 defines a mean dropletdiameter, a typical value for which is10 µm. For this diameter and f = 10−6, theoptical thickness per unit meter of physicalthickness is about the same as the normaloptical thickness of the atmosphere in themiddle of the visible spectrum (see Fig. 3).Thus, a cloud only 1 m thick is equivalentoptically to the entire gaseous atmosphere.

A cloud with (normal) optical thicknessabout 10 (i.e., a physical thickness of about100 m) is sufficient to obscure the disk ofthe sun. But even the thickest cloud doesnot transform day into night. Clouds are

usually translucent, not transparent, yetnot completely opaque.

The scattering coefficient of clouddroplets, in contrast with that of airmolecules, is more or less independentof wavelength. This is often invoked asthe cause of the colorlessness of clouds.Yet wavelength independence of scatter-ing by a single particle is only sufficient,not necessary, for wavelength indepen-dence of scattering by a cloud of particles(see Sec. 2.4). Any cloud that is opticallythick and composed of particles for whichabsorption is negligible is white uponillumination by white light. Although ab-sorption by water (liquid and solid) is notidentically zero at visible wavelengths, andselective absorption by water can lead toobservable consequences (e.g., colors ofthe sea and glaciers), the appearance of allbut the thickest clouds is not determinedby this selective absorption.

Equation (42) is the key to the vastlydifferent optical characteristics of cloudsand of the rain for which they are theprogenitors. For a fixed amount of water(as specified by the quantity fh), opticalthickness is inversely proportional to meandiameter. Rain drops are about 100 timeslarger on average than cloud droplets, andhence optical thicknesses of rain shafts arecorrespondingly smaller. We often can seethrough many kilometers of intense rainwhereas a small patch of fog on a well-traveled highway can result in carnage.

9.2Givers and Takers of Light

Scattering of visible light by a singlewater droplet is vastly greater in theforward (θ < 90◦) hemisphere than in thebackward (θ > 90◦) hemisphere (Fig. 9).But water droplets in a thick cloudilluminated by sunlight collectively scatter


much more in the backward hemisphere(reflected light) than in the forwardhemisphere (transmitted light). In eachscattering event, incident photons aredeviated, on average, only slightly, butin many scattering events most photonsare deviated enough to escape from theupper boundary of the cloud. Here is anexample in which the properties of anensemble are different from those of itsindividual members.

Clouds seen by passengers in an airplanecan be dazzling, but if the airplane wereto descend through the cloud these samepassengers might describe the cloudy skyoverhead as gloomy. Clouds are both giversand takers of light. This dual role isexemplified in Fig. 20, which shows thecalculated diffuse downward irradiancebelow clouds of varying optical thickness.On an airless planet the sky would be blackin all directions (except directly towardthe sun). But if the sky were to be filledfrom horizon to horizon with a thin cloud,the brightness overhead would markedlyincrease. This can be observed in a partlyovercast sky, where gaps between clouds(blue sky) often are noticeably darker than

Fig. 20 Computed diffuse downward irradiancebelow a cloud relative to the incident solarirradiance as a function of cloudoptical thickness

their surroundings. As so often happens,more is not always better. Beyond acertain cloud optical thickness, the diffuseirradiance decreases. For a sufficientlythick cloud, the sky overhead can be darkerthan the clear sky.

Why are clouds bright? Why are theydark? No inclusive one-line answers can begiven to these questions. Better to ask, Whyis that particular cloud bright? Why is thatparticular cloud dark? Each observationmust be treated individually; generaliza-tions are risky. Moreover, we must keepin mind the difference between bright-ness and radiance when addressing thequeries of human observers. Brightness isa sensation that is a property not only ofthe object observed but of its surround-ings as well. If the luminance of an objectis appreciably greater than that of its sur-roundings, we call the object bright. If theluminance is appreciably less, we call theobject dark. But these are relative ratherthan absolute terms.

Two clouds, identical in all respects,including illumination, may still appeardifferent because they are seen againstdifferent backgrounds, a cloud against thehorizon sky appearing darker than whenseen against the zenith sky.

Of two clouds under identical illumi-nation, the smaller (optically) will be lessbright. If an even larger cloud were to ap-pear, the cloud that formerly had been de-scribed as white might be demoted to gray.

With the sun below the horizon, twoidentical clouds at markedly differentelevations might appear quite differentin brightness, the lower cloud beingshadowed from direct illumination bysunlight.

A striking example of dark clouds cansometimes be seen well after the sunhas set. Low-lying clouds that are notilluminated by direct sunlight but are


seen against the faint twilight sky maybe relatively so dark as to seem likeink blotches.

Because dark objects of our everydaylives usually owe their darkness to absorp-tion, nonsense about dark clouds is rife:they are caused by pollution or soot. Yet ofall the reasons that clouds are sometimesseen to be dark or even black, absorptionis not among them.

Glossary

Airlight: Light resulting from scattering byall atmospheric molecules and particlesalong a line of sight.

Antisolar Point: Direction opposite thesun.

Astronomical Horizon: Horizontal direc-tion determined by a bubble level.

Brightness: The attribute of sensation bywhich an observer is aware of differencesof luminance (definition recommendedby the 1922 Optical Society of AmericaCommittee on Colorimetry).

Contrast Threshold: The minimum rela-tive luminance difference that can beperceived by the human observer.

Inferior Mirage: A mirage in which imagesare displaced downward.

Irradiance: Radiant power crossing unitarea in a hemisphere of directions.

Lapse Rate: The rate at which a physicalproperty of the atmosphere (usually tem-perature) decreases with height.

Luminance: Radiance integrated over thevisible spectrum and weighted by the

spectral response of the human ob-server. Also sometimes called photometricbrightness..5.5

Mirage: An image appreciably differentfrom what it would be in the absence ofatmospheric refraction.

Neutral Point: A direction in the sky forwhich the light is unpolarized.

Normal Optical Thickness: Optical thick-ness along a radial path from the surfaceof the earth to infinity.

Optical Thickness: The thickness of a scat-tering medium measured in units ofphoton mean free paths. Optical thick-nesses are dimensionless.

Radiance: Radiant power crossing a unitarea and confined to a unit solid angleabout a particular direction.

Scale Height: The vertical distance overwhich a physical property of the at-mosphere is reduced to 1/e of itsvalue.

Scattering Angle: Angle between incidentand scattered waves.

Scattering Coefficient: The product of scat-tering cross section and number density ofscatterers.

Scattering Cross Section: Effective area of ascatterer for removal of light from a beamby scattering.

Scattering Plane: Plane determined byincident and scattered waves.

Solar Point: The direction toward thesun.


Superior Mirage: A mirage in which im-ages are displaced upward.

Tangential Optical Thickness: Opticalthickness through the atmosphere alonga horizon path.

References

Many of the seminal papers in atmosphericoptics, including those by Lord Rayleigh, arebound together in Bohren, C. F. (Ed.) (1989),Selected Papers on Scattering in the Atmosphere,Bellingham, WA: SPIE Optical EngineeringPress. Papers marked with an asterisk are inthis collection.

[1] Moller, F. (1972), Radiation in the at-mosphere, in D. P. McIntyre (Ed.), Me-teorological Challenges: A History. Ottawa:Information Canada, pp. 43–71.

[2]∗ Penndorf, R. (1957), J. Opt. Soc. Am. 47,176–182.

[3]∗ Young, A. T. (1982), Phys. Today 35(1),2–8.

[4] Einstein, A. (1910), Ann. Phys. (Leipzig)33, 175; English translation in Alexan-der, J. (Ed.) (1926), Colloid Chemistry,Vol. I. New York: The Chemical CatalogCompany, pp. 323–339.

[5] Zimm, B. H. (1945), J. Chem. Phys. 13,141–145.

[6] Thekaekara, M. P., Drummond, A. J.(1971), Nat. Phys. Sci. 229, 6–9.

[7]∗ Hulburt, E. O. (1953), J. Opt. Soc. Am. 43,113–118.

[8] von Frisch, K. (1971), Bees: Their Vision,Chemical Senses, and Language, revisededition, Ithaca, NY: Cornell UniversityPress, p. 116.

[9] Doyle, W. T. (1985), Am. J. Phys. 53,463–468.

[10]∗ Fraser, A. B. (1975), Atmosphere 13, 1–10.[11] Takano, Y., Asano, S. (1983), J. Meteor.

Soc. Jpn. 61, 289–300.[12] van de Hulst, H. C. (1957), Light Scattering

by Small Particles. New York: Wiley-Interscience.

[13] Pledgley, E. (1986), Weather 41, 401.[14] Lee, R., Fraser, A. (1990), New Scientist

127(September), 40–42.

[15] Lee, R. (1991), Appl. Opt. 30, 3401–3407.[16]∗ Fraser, A. B. (1983), J. Opt. Soc. Am. 73,

1626–1628.[17] Fraser, A. B. (1972), J. Atmos. Sci. 29, 211,

212.[18]∗ van de Hulst, H. C. (1947), J. Opt. Soc.

Am. 37, 16–22.[19]∗ Bryant, H. C., Cox, A. J. (1966), J. Opt. Soc.

Am. 56, 1529–1532.[20]∗ Nussenzveig, H. M. (1979), J. Opt. Soc.

Am. 69, 1068–1079.[21]∗ Fraser, A. B. (1979), J. Opt. Soc. Am. 69,

1112–1118.[22] Neuberger, H. (1951), Introduction to Phys-

ical Meteorology. University Park, PA: Col-lege of Mineral Industries, PennsylvaniaState University.

Further Reading

Minnaert, M. (1954), The Nature of Light andColour in the Open Air. New York: DoverPublications, is the bible for those interestedin atmospheric optics. Like accounts of naturalphenomena in the Bible, those in Minnaert’sbook are not always correct, despite which,again like the Bible, it has been and willcontinue to be a source of inspiration.

A book in the spirit of Minnaert’s but with awealth of color plates is by Lynch, D. K., Liv-ingston, W. (1995), Color and Light in Nature.Cambridge, UK: Cambridge University Press.

A history of light scattering, From Leonardo tothe Graser: Light Scattering in Historical Per-spective, was published serially by Hey, J. D.(1983), S. Afr. J. Sci. 79, 11–27, 310–324;Hey, J. D. (1985), S. Afr. J. Sci. 81, 77–91,601–613; Hey, J. D., (1986), S. Afr. J. Sci. 82,356–360. The history of the rainbow is re-counted by Boyer, C. B. (1987), The Rainbow.Princeton, NJ: Princeton University Press.

A unique, beautifully written and illustratedtreatise on rainbows in science and art,both sacred and profane, is by Lee, R. L.,Fraser, A. B. (2001), The Rainbow Bridge,University Park, PA: Penn State UniversityPress.

Special issues of Journal of the Optical Societyof America (August 1979 and December 1983)and Applied Optics (20 August 1991 and 20 July1994) are devoted to atmospheric optics.


Several monographs on light scattering byparticles are relevant to and contain exam-ples drawn from atmospheric optics: vande Hulst, H. C. (1957), Light Scattering bySmall Particles. New York: Wiley-Interscience;reprint (1981), New York: Dover Publica-tions; Deirmendjian, D. (1969), Electromag-netic Scattering on Polydispersions. New York:Elsevier; Kerker, M. (1969), The Scatteringof Light and Other Electromagnetic Radiation.New York: Academic Press; Bohren, C. F.,Huffman, D. R. (1983), Light Scattering bySmall Particles. New York: Wiley-Interscience;Nussenzveig, H. M. (1992), Diffraction Effectsin Semiclassical Scattering. Cambridge, UK:Cambridge University Press.

The following books are devoted to a widerange of topics in atmospheric optics:Tricker, R. A. R. (1970), Introduction to Mete-orological Optics. New York: Elsevier; McCart-ney, E. J. (1976), Optics of the Atmosphere. NewYork: Wiley; Greenler, R. (1980), Rainbows,Halos, and Glories. Cambridge, UK: CambridgeUniversity Press. Monographs of more limitedscope are those by Middleton, W. E. K. (1952),Vision Through the Atmosphere. Toronto: Uni-versity of Toronto Press; O’Connell, D. J. K.(1958), The Green Flash and Other LowSun Phenomena. Amsterdam: North Holland;Rozenberg, G. V. (1966), Twilight: A Study inAtmospheric Optics. New York: Plenum; Hen-derson, S. T. (1977), Daylight and its Spectrum,(2nd ed.), New York: Wiley; Tricker, R. A. R.(1979), Ice Crystal Haloes. Washington, DC:Optical Society of America; Konnen, G. P.(1985), Polarized Light in Nature. Cambridge,UK: Cambridge University Press; Tape, W.(1994), Atmosphere Halos. Washington, DC:American Geophysical Union.

Although not devoted exclusively to atmosphericoptics, Humphreys, W. J. (1964), Physics of theAir. New York: Dover Publications, contains

a few relevant chapters. Two popular sciencebooks on simple experiments in atmosphericphysics are heavily weighted toward atmo-spheric optics: Bohren, C. F. (1987), Clouds ina Glass of Beer. New York: Wiley; Bohren, C. F.(1991), What Light Through Yonder WindowBreaks? New York: Wiley.

For an expository article on colors of the skysee Bohren, C. F., Fraser, A. B. (1985), Phys.Teacher 23, 267–272.

An elementary treatment of the coherenceproperties of light waves was given by For-rester, A. T. (1956), Am. J. Phys. 24, 192–196.This journal also published an expository arti-cle on the observable consequences of multiplescattering of light: Bohren, C. F. (1987), Am. J.Phys. 55, 524–533.

Although a book devoted exclusively to atmo-spheric refraction has yet to be published,an elementary yet thorough treatment of mi-rages was given by Fraser, A. B., Mach, W. H.(1976), Sci. Am. 234(1), 102–111.

Colorimetry, the often (and unjustly) neglectedcomponent of atmospheric optics, is treatedin, for example, Optical Society of Amer-ica Committee on Colorimetry (1963), TheScience of Color. Washington, DC: OpticalSociety of America. Billmeyer, F. W., Saltz-man, M. (1981), Principles of Color Technol-ogy, (2nd ed.), New York: Wiley-Interscience.MacAdam, D. L. (1985), Color Measurement,(2nd ed.), Berlin: Springer.

Understanding atmospheric optical phenom-ena is not possible without acquiring atleast some knowledge of the propertiesof the particles responsible for them. Tothis end, the following are recommended:Pruppacher, H. R., Klett, J. D. (1980), Micro-physics of Clouds and Precipitation. Dor-drecht, Holland: D. Reidel. Twomey, S. A.(1977), Atmospheric Aerosols. New York:Elsevier.

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