Blue moons and Martian sunsets

Blue moons and Martian sunsets

Kurt Ehlers,1,2,* Rajan Chakrabarty,2 and Hans Moosmüller2

1Mathematics Department Truckee Meadows Community College, Reno, Nevada 89512, USA2Desert Research Institute, Nevada System of Higher Education, Reno, Nevada 89512, USA

*Corresponding author: [email protected]

Received 21 October 2013; revised 9 January 2014; accepted 10 February 2014;posted 12 February 2014 (Doc. ID 199822); published 18 March 2014

The familiar yellow or orange disks of the moon and sun, especially when they are low in the sky, andbrilliant red sunsets are a result of the selective extinction (scattering plus absorption) of blue light byatmospheric gas molecules and small aerosols, a phenomenon explainable using the Rayleigh scatteringapproximation. On rare occasions, dust or smoke aerosols can cause the extinction of red light to exceedthat for blue, resulting in the disks of the sun and moon to appear as blue. Unlike Earth, the atmosphereof Mars is dominated bymicron-size dust aerosols, and the sky during sunset takes on a bluish glow. Herewe investigate the role of dust aerosols in the blue Martian sunsets and the occasional blue moons andsuns on Earth.We use theMie theory and the Debye series to calculate the wavelength-dependent opticalproperties of dust aerosols most commonly found on Mars. Our findings show that while wavelengthselective extinction can cause the sun’s disk to appear blue, the color of the glow surrounding thesun as observed from Mars is due to the dominance of near-forward scattering of blue light by dustparticles and cannot be explained by a simple, Rayleigh-like selective extinction explanation. © 2014Optical Society of AmericaOCIS codes: (010.0010) Atmospheric and oceanic optics; (330.0330) Vision, color, and visual optics;

(290.0290) Scattering.http://dx.doi.org/10.1364/AO.53.001808

1. Introduction

Optical phenomena in the Earth’s atmosphere suchas blue skies, red sunsets, halos, rainbows, and coro-nas are enjoyed by most humans and raise curiosityabout the optics causing them. Therefore, thesephenomena are frequently discussed in physicsand optics education. For example, the color of thesun is mostly observed during sunsets and sunrisesor when part of the sunlight is obscured by thinclouds or large aerosol concentrations, making it pos-sible to safely view the sun with the naked eye. Thecolor of the moon can easily be observed at all times.While the reddish appearance of the sun and themoon, especially when low in the sky, has beennoticed by most, the much rarer appearance ofblue moons and suns requires a specific and narrow

particle size distribution [1] that has only beenknown to occur after large forest fires and volcaniceruptions. (Note that when we speak of a blue moon,we mean it in the literal sense, rather than in theliterary or calendar sense.)

Recent images taken on the surface of Mars re-mind us that interesting optical phenomena arenot limited to the Earth’s atmosphere but can alsobe found on other planets. Of particular interest tous is the unfamiliar blue color of Martian sunsetsand sunrises (Fig. 1). Here we make a theoreticalstudy using a model for Martian dust based on theoptical properties and size distribution of Martiandust measured during recent Mars Rover missions.Because the size of Martian dust particles is compa-rable to the wavelength of visible light [2], the opticsare complicated and solutions toMaxwell’s equationsare required to model the scattered electric and mag-netic fields. Our theoretical analysis is based on Mietheory [3], the Debye series [4], and the anomalous

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1808 APPLIED OPTICS / Vol. 53, No. 9 / 20 March 2014

http://dx.doi.org/10.1364/AO.53.001808

diffraction theory of van de Hulst [5]. The goals of ouranalysis are to study the optical properties of aero-sols with a wavelength-dependent refractive index,especially as they influence the color of the sunand moon, and to uncover the basic physics behindthe blue Martian sunset.

The blue sky and red sunset on Earth are a resultof scattering of blue light by atmospheric gases andsmall particles. Mars, the red planet, has an atmos-phere that is dominated by dust [2]. In analogy withwhat happens on Earth, extinction of red lightthrough scattering by Martian dust is a likely candi-date for causing blue sunsets and sunrises on Mars.On the other hand, Martian dust absorbs blue lightmore strongly than red light [6]. We show that whilestronger absorption in the blue can diminish the ex-tent to which extinction of red light exceeds that forblue light, it can also increase the range of particlesizes for which extinction in red dominates. Basedon our understanding of the optical properties andsize distribution of Martian dust, the extinction ofred light by the Martian atmosphere is slightlygreater than the extinction of blue light. The diskof the sun would appear slightly blue, especially atsunset when the optical path is greatest. However,wavelength-dependent extinction of light is notsufficient to explain the intriguing blue glow sur-rounding the sun in Fig. 1.

WhileMartian dust removes only slightly more redlight than blue from the sun’s beam, the pattern inwhich it scatters red and blue light is very different.To investigate this, we use the scattering diagram: aplot of scattering intensity versus scattering angle.For a model Martian dust, the scattering intensityof blue light within a cone of about 10° about the axisof sunlight propagation, exceeds that for red by a fac-tor of more than 6. Outside this cone, blue light’sdominance decreases until about 28° where the in-tensity of red light surpasses that for blue. The blueglow surrounding the sun at sunset is created bylight scattered at small angles by dust particles.The physical processes leading to the dominance ofblue in near-forward scattering from Martian dustare external reflection and diffraction. While the diskof the sun would only appear blue at sunset, whenthe optical path is longest, or during a dust storm,the blue glow around the sun should be visiblethroughout the Martian day. On Earth, the reddishcolor of the sun at sunset and the red sunset createdby scattered light are both products of wavelength-selective extinction, while onMars, only the blue diskof the sun is. The blue glow surrounding the sun is aproduct of the angular pattern of scattering fromdust particles.

We begin by summarizing explanations for the redand rare blue appearances of the sun andmoon in the

Fig. 1. Martian sunset over the Gusev crater overlaid with scattering angles. The image of the sunset was taken by NASA’s Mars Ex-ploration Rover Spirit on May 19, 2005. Image credit: NASA/JPL/Texas A&M/Cornell.

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Earth’s atmosphere. This is followed by a discussionof the wavelength-dependent optical properties ofaerosols as they relate to the bluing of light. Herewe expand on earlier studies of the bluing andreddening of light by aerosols by considering a wave-length-dependent refractive index, which is particu-larly relevant to Martian dust. We conclude with aninvestigation of possible reasons for the blue sunsetsand sunrises, which have been observed on Mars.

2. Color of the Moon and Sun as Seen on Earth

The textbook explanation of why the sky is blue isbased on Rayleigh’s law of scattering. Without anatmosphere, the sky would simply appear black.The light we see in the sky is light scattered byatmospheric gases and aerosols. An aerosol is a sus-pension of small particles such as smoke particles,dust, or water droplets within atmospheric gases.Earth’s atmosphere is generally dominated by gasesand particles that are much smaller that the wave-length of visible light. John William Strutt (alsoknown as the third Baron Rayleigh or Lord Rayleighfor short) showed that scattering from small particlesis proportional to 1∕λ4 where λ is the wavelength oflight [7]. As a consequence, blue light with a wave-length of 425 nm is scattered more than seven timesas strongly as red light with a wavelength of 700 nm.The scattered sunlight we see in the sky during thedaytime is therefore dominated by blue unless largeparticles are present. Conversely, when looking atthe disk of the moon or sun at sunset, we are seeingthe direct beam of light from these objects. Since theblue light is scattered away, the remaining light isreddened, making the disks of the moon and sunappear yellow to red. This phenomenon is most pro-nounced when the sun or moon is low in the sky andthe optical depth is greatest. When the reddenedlight scatters off larger atmospheric aerosols suchas water droplets in clouds, which scatter all wave-lengths without prejudice, spectacular red sunsetsoccur. A key observation here is that the red diskof the sun and the redness of the surrounding skyat sunset are both caused by wavelength-dependentextinction.

On rare occasions, atmospheric aerosols can causethe disks of the sun and moon to appear blue. In1853, the volcano Krakatoa erupted, spewing dustinto the atmosphere. For nearly a month afterward,the moon appeared blue over tropical regions ofEarth [8]. Soon after the eruption, Reverend SerenoBishop reported from Hawai’i the appearance of ananomalous corona encircling the sun consisting ofa bluish inner region surrounded by a brownish ring[9]. This optical phenomenon, known as a Bishop’sring, is generally associated with dust from volcaniceruptions. Blue moons and suns also occur indust-prone areas of Northern Africa and the ArabianPeninsula. In [10], six of 14 AERONET sites reporteda negative Ångström coefficient, which would indi-cate a blue sun at these locations, at some time. Inthe early 19th century, a blue glow of area one and

a half diameters surrounding the sun was observedin Cairo during a dust storm that turned the sky ayellowish white color [11]. In this instance, the sunremained a pale yellow, and it was noted that therewas no brownish ring present. Smoke from forestfires can also cause blue moons and suns. Duringa particularly intense fire season in Alberta andBritish Columbia in 1950, smoke caused the sunand moon to appear blue from locations in easternCanada and America, then later in Europe [12].

While on Earth, blue suns, moons, and sunsets area rare occurrence, blue sunsets appear to be the normon Mars. The photograph of the sunset on Marstaken from the Gusev crater by theMars ExplorationRover Spirit (Fig. 1) shows a blue Martian sunset.Phenomenal movies created using the Mars Explora-tion Rover Opportunity’s Pancam by Mark Lemmonof Texas A&M and Jim Bell of Cornell Universityshowing theMartian sunset are available on a NASAwebsite [13].

On Earth, for the moon or sun to appear blue, ex-tinction (scattering plus absorption) of red light mustexceed that for blue light. For an aerosol whose re-fractive index is constant over the visible spectrum,this only occurs when the aerosol has a very narrowsize distribution [1]. For example, for an aerosol ofspherical water droplets with index of refractionn � 1.35, the radius needs to be about 0.75 μm.Waterdroplets in clouds or fog are typically 10–20 μm, sofog and clouds do not cause blue suns or moons.For an aerosols to cause the moon or sun to appearblue, the optical depth must be sufficient to overcomethe normal extinction of blue light by atmosphericgases and small aerosols. On rare occasions, a suffi-cient number of particles of just the right size havebeen produced by volcanic eruptions and large forestfires to cause the moon and sun to appear blue.

The situation on Mars is complicated by the factthat the Martian atmosphere is dominated by dustparticles whose size is close to the wavelength ofvisible light (400–700 nm). For particles of this sizethere is no simple law analogous to Rayleigh’s law.Scattering for these particles is sensitive to size,shape, and composition. The density of atmosphericgases on Mars, which are composed of approximately95% carbon dioxide, 3% nitrogen, and 1.6% argon, ismuch lower than it is on Earth. OnMars the density ofatmospheric gases is 0.016 kg∕m3 while on Earth it isnearly 80 times greater at 1.2 kg∕m3. Rayleigh scat-tering is much less important on Mars than on Earth.In addition, the refractive index n � m� ik ofMartian dust depends significantly on the wavelengthof light over the visible spectrum. Here, the realpart m is the ratio of the speed of light in avacuum to that through the substance, and the imagi-nary part k is a measure of the absorption of lightby the bulk substance. For Martian dust, the imaginary part increases strongly toward shorter wave-lengths [6], so Martian dust strongly absorbs bluelight, which has an effect on the bluing and reddeningof light as it passes through the Martian atmosphere.


A. Scattering, Absorption, and Extinction of Light byAerosols

The color of the disks of the moon and sun and thecolor of the sunset on Earth are a product of extinc-tion of certain wavelengths of light by atmosphericaerosols. Light becomes extinct from a beam if it isscattered (its course is deviated) or it is absorbed(changed to another form of energy such as heat).To quantify extinction, we use the absorption, scat-tering, and extinction cross sections and efficiencies.These parameters can be found experimentally ortheoretically using solutions to Maxwell’s equations.The following is a brief review of scattering and ab-sorption by an individual particle and can be skippedby experts. Additional details can be found in [14].

For an individual particle, absorption and scatter-ing are characterized by the absorption cross sectionσabs and scattering cross section σsca, which haveunits ofm2. For a suspension of n particles containedin a volume V, absorption and scattering are charac-terized by the absorption and scattering coefficientsdefined as

βabs �Pn

i�1 σiabs

Vand βsca �

Pni�1 σ

isca

V; (1)

which have units of m−1. The extinction cross sectionand extinction coefficient are then defined as

σext � σsca � σabs and βext � βsca � βabs: (2)

Physically, the extinction coefficient βext representsthe fractional loss of the beam’s power per unit pathlength as expressed through the Beer–Lambertlaw [15]:

PPo

� e−βextz: (3)

Here Po is the radiant power of the incident beam,and P is the power after the beam travels a distancez through the aerosol. Bluing of light occurs whenextinction of red light exceeds that of blue lightand reddening of light occurs when extinction of bluelight exceeds that of red light.

Most of our analysis will be based on the dimen-sionless absorption, scattering, and extinctionefficiency factors, Qabs, Qsca, and Qext, respectively,which are defined to be the ratio of the absorption,scattering, and extinction cross sections and thegeometric cross section. For a spherical particlewith radius r, the extinction efficiency factor isQext � σext∕�πr2�. In terms of extinction efficiencies,bluing occurs when Qext is greater for red light thanit is for blue.

The scattering and absorption efficiencies dependon varying degrees of size, shape, and mineralogicaland chemical composition of the particle. In thepresent work, we focus on the effect of the particlesize and refractive index on the appearance of themoon, sun, and sunset. Our analysis is based on

the scattering and absorption of light by a homo-geneous spherical particles of radius r with complexrefractive index n � m� ik.

Traditionally, scattering and absorption are ex-pressed as a function of the dimensionless sizeparameter x, defined to be the circumference of theparticle divided by the wavelength λ of the light:x � 2πr∕λ. For particles much smaller than the wave-length (x ≪ 1), absorption is proportional to λ−1 andscattering is proportional to λ−4 [16–18].

When the size of the particle and the wavelength ofthe light are about equal (x ≈ 1), as happens withMartian dust, the situation becomes much moredifficult, and solutions to Maxwell’s equations arerequired to determine scattering and absorption.Gustav Mie solved the problem of scattering and ab-sorption of a plane electromagnetic wave from ahomogeneous isotropic sphere in 1908 [3]. It is wellknown that dust particles are nonspherical andtherefore have substantially different optical proper-ties from volume equivalent spheres. While thisleads to substantial errors in retrieving aerosoloptical thickness from satellite reflectance measure-ments [19], phase functions in the forward hemi-sphere, especially in the near-forward direction,are nearly identical for spherical and nonsphericalparticles [19,20]. Therefore nonsphericity is largelyirrelevant to the optical phenomena discussed here,and Mie theory can be used.

Mie’s solution expresses the scattering amplitudefrom a sphere as the infinite series:

S1 �Xn

2n� 1n�n� 1� �anπn � bnτn�

S2 �Xn

2n� 1n�n� 1� �anτn � bnπn�: (4)

The πn and τn are spherical Bessel functions, whichare referred to as partial waves in the present con-text. The functions S1 and S2 give the amplitudeof the scattered light at the azimuthal angle θ whenmultiplied by the amplitude of the incident beam.Once the scattering amplitudes are obtained, the ab-sorption, scattering, and extinction efficiencies arereadily computed. A full derivation and explanationof Mie’s solution can be found in [14]. The number ofterms of the series that must be kept for a goodapproximation is on the order of the size parameterx. For our problem, this is easily accomplished usinga computer algebra system such as Mathematica, orwith dedicated Mie theory software such as MiePlot[21], both of which have been employed in thepresent work.

While Mie’s partial wave expansion provides accu-rate solutions, the complicated series of partialwaves provides little physical intuition about themechanism behind the scattering and absorption.Also in 1908, Debye [4] solved for the scatteringfrom an infinite cylinder. Subsequently, Debye’ssolution was adapted to the case of a spherical


scatterer [22–24]. The advantage of Debye’s solutionis that it lends itself to physical interpretation. TheDebye decomposition expresses the partial wave co-efficients of the Mie solution in terms of a set of am-plitudes corresponding to transmitted and reflectedwaves at the interface of two media. For a homo-geneous sphere, the Debye series expansion of thepartial wave scattering amplitudes an and bn areof the form:

12

�1 − R22

n −

X∞p�1

T21n �R11

n �p−1T12n

�:

[25] Each term has a typical physical interpretationwhen substituted for Mie scattered partial waveamplitudes in expressions for electromagnetic scat-tering by a homogeneous sphere. The first term,1∕2, is taken to describe the component of the scat-tered electromagnetic field due to diffraction. Thesecond term, −�1∕2�R22, is taken to describe thecomponent of the scattered field due to external re-flection from the sphere. The third term is an infinitesum, each individual term of which is taken torepresent the component of the field, which haspenetrated the sphere, undergone p − 1 internalreflections and then subsequently emerged againinto the surrounding medium. If all the terms ofthe Debye series are summed and then substitutedfor the Mie scattered partial wave amplitudes, thenthe results are identical to ordinary Mie scattering.Thus the Debye series is taken to interpret Miescattering as the result of many scattering processes,each with a unique number of internal/externalreflections.

The MiePlot software package provides the optionof computing the Debye series for scattering from ahomogeneous sphere, term by term. In the presentwork we employ the Mie series, the Debye series,as well as the anomalous diffraction approximation[5] to compute the optical properties of spherical par-ticles as they relate to the occurrence of blue moonsand suns, especially when the refractive index isallowed to depend on the wavelength and to isolate

the physical processes behind the blue Martiansunset.

Bluing and the extinction curve. Viewed fromspace, the disks of the sun and moon appear white.Their colors, as observed from Earth, are a result ofthe extinction (removal) light of certain wavelengthsas the direct beam of light passes through the atmos-phere. The terms bluing and reddening are usedto describe these processes. Bluing refers to theremoval of red light, and reddening refers to the re-moval of blue light. Under normal circumstances,reddening of light by atmospheric gases and othersmall particles causes the disks of the moon andsun to appear yellow or red, especially when theyare low in the sky. In the evening, the reddened lightilluminates water droplets in clouds and other largeparticles (which effectively scatter all wavelengths oflight), and they appear red. Blue moons and suns areobserved under the rare circumstances when atmos-pheric aerosols cause bluing.

For a particular aerosol, the bluing or reddening oflight can be inferred using its extinction curve: theplot of Qext versus the size parameter x, the wave-length of light, or the phase shift parameter(described below). If Qext decreases with wavelength,reddening occurs, and if Qext increases with wave-length, bluing occurs.

Figure 2(a) shows a plot of Qext versus the sizeparameter for a spherical particle with purely realrefractive index n � 1.4� 0i computed using Mietheory. By fixing a radius ro, the visible spectrum0.4μm<λ<0.7μm defines an interval 2πro∕0.7 μm <x < 2πro∕0.4 μm. Note that higher wavelengths areto the left, so that positive slopes correspond to red-dening and negative slopes correspond to bluing.Note also that the intervals with positive slopesget larger as the particle size increases. For verysmall particles, the interval is on the far left sideof the graph, which has a positive slope. As predictedby Lord Rayleigh, an aerosol of small particles red-dens light. For particles of radius 0.65 μm, the inter-val corresponding to visible light is 5.8 < x < 10.2.On this interval the slope is negative and bluingoccurs. A monodisperse aerosol of particles with

(a) (b)

0 10 20 30 400

1

2

3

4

5

Size parameter x

Qex

t

0 10 20 30 400

1

2

3

4

Phase shift parameter

Qex

t

r = 0.65 µm

Fig. 2. Extinction curve. (a) The extinction curve for a spherical particle with refractive index n � 1.4 computed using Mie’s solution(upper curve) and the anomalous diffraction approximation (lower curve). The interval corresponding to the visible spectrum for a particleof radius 0.65 μm is indicated. (b) Anomalous diffraction approximation of the extinction efficiency versus the size parameter: Qext versusphase shift parameter.


refractive index n � 1.4� 0i with radius 0.65 μmwould lead to a blue moon or sun. For larger par-ticles, the interval gets larger and contains manysmaller oscillations of the curve. Neither bluingnor reddening occurs. For a particle with refractiveindex n � 1.4� 0i, bluing occurs only when the ra-dius is very nearly 0.65 μm. The small range of par-ticle sizes leading to bluing of light together withstrong scattering of blue light by atmospheric gasesexplains the rarity of blue moons and suns on Earth.

Understanding the bluing and reddening of lightboils down to understanding the shape of the extinc-tion curve and how the shape is changed when therefractive index is varied. The extinction curve ischaracterized by large low-frequency oscillationsabout, and asymptotic to, the geometric optics valueof Qext � 2, superimposed with smaller high-frequency ripples. Physically, the ripples are associ-ated with resonances of internally reflected light[26]. Bluing and reddening of light are associatedwith the large oscillations known as the interferencestructure. The interference structure is so namedbecause the large oscillations are a result of the con-structive and destructive interference between lightthat is diffracted around the particle and light that istransmitted through the particle.

Insight into how the refractive index is related tothe interference structure can be gained using theanomalous diffraction approximation derived byvan de Hulst [5]. This approximation is valid foroptically soft particles, i.e., particles for which therefractive index is very close to unity. Here the ex-tinction efficiency Qext is plotted against the phaseshift parameter ρ � 2x�m − 1�. Physically, the phaseshift parameter represents the phase lag suffered bya centrally transmitted ray. The anomalous diffrac-tion approximation qualitatively captures the inter-ference structure thus allowing the intervals ofbluing and reddening to be approximated for a par-ticular refractive index.

For optically soft particles, light changes directionvery little as it passes through a boundary of the par-ticle, but, since it travels slower within the particle, itsuffers a phase lag compared to light diffractedaround the particle. Applying Huygen’s principle tocombine the diffracted and transmitted light atpoints beyond the particle, van de Hulst derivedthe formula:

~Qext � 2 −

4ρ

sin ρ� 4

ρ2�1 − cos ρ�; (5)

[Fig. 2(b)]. The peaks and valleys of the interferencestructure are a result of constructive and destructiveinterference of the combined light.

By rescaling the horizontal axis, we can obtain anapproximation to the extinction curve (Qext versus x)for any purely real refractive index. To compare withthe extinction curve obtained using Mie’s solution,we have approximated the extinction curve for a par-ticle with a refractive index of n � 1.4 using Eq. (5)

[Fig. 2(a)]. Note that, because the internally reflectedlight is ignored in the anomalous diffraction approxi-mation, the ripples are absent and that the peaksand valleys have smaller amplitude than those ob-tained using Mie’s solution. On the other hand, thequalitative features of the interference structureare preserved. This remains true when the refractiveindex is as high as n � 2 [5]. For instance, the peaksin the interference structure occur with nearly thesame period, which is 2π in ρ or π∕�m − 1� in x. Fora particle with refractive index n � 1.4, the periodis about 7.9 in x. An immediate observation is thatchanges in the real part of the refractive indexstretches or compresses the extinction curve in thehorizontal direction.

The anomalous diffraction approximation allowsus to directly estimate the particle size that wouldlead to bluing for a particular refractive index.Bluing is associated with the region between the firstmaximum and the following minimum in the extinc-tion curve [Fig. 2(b)]. The center of this intervaloccurs at about ρ � 5.85. Using the definition ofthe phase shift parameter, we can estimate theradius leading to bluing for a given refractive indexn � m� 0i. Indeed, we set 5.85 � 2x�m − 1� �2��2πr�∕λ��m − 1� and set λ � 0.55 μm (the center ofthe visible spectrum) then

r � 0.26 μm∕�m − 1�; (6)

[5] p. 423, and [27]. For a mono-disperse aerosol, theradius leading to a bluing is driven by the real part ofthe index of refraction. For water, m equals approx-imately 1.35 so the radius should be approximately0.74 μm, which is in agreement with the findingsof [1].

3. Bluing by Absorbing Particles witha Wavelength-Dependent Refractive Index

The theory described so far is well known and satis-factorily describes many of the known appearances ofblue moons and suns on Earth. Blue moons and sunsare associated with smoke or dust from volcaniceruptions with nearly identical particle size andwhose refractive index is nearly constant over thevisible spectrum. The situation on Mars (as well asfor dust storms with iron oxide containing dust onEarth [10]) is more complicated. The atmosphereon Mars is dominated by dust particles that havea fairly wide distribution of sizes and whose refrac-tive index varies significantly over the visible spec-trum [6]. Specifically, Martian dust preferentiallyabsorbs blue light. To understand the bluing and red-dening of light on Mars, we must investigate theeffect of the imaginary part of the refractive indexon the shape of the extinction curve, especially whenit is allowed to vary with wavelength.

A nonzero imaginary part of the index of refractionleads to absorption of the internally transmittedlight. A small imaginary part dampens the high-frequency ripples and has only a small effect on


the interference structure [see the dashed curve inFig. 3(a) where n � 1.4� 0.05i]. Intuitively, this isthe case because the transmitted light travels onlya short distance through the sphere thus sufferingnegligible absorption and interference with dif-fracted light is little affected. The high-frequencyripples are more affected since they are associatedwith light that is internally reflected and takes alonger path through the particle, thereby sufferinggreater extinction. As the imaginary part increases,the interference structure is lost [see the dottedcurve in Fig. 3(a)]. The effect of increasing the imagi-nary part of the refractive index is to dampen peaksin the interference structure toward the valueQext � 2. Note, however, that there is no change inthe period of the peaks. In the limit k → ∞, the par-ticle becomes opaque. All of the light incident on theparticle is removed from the beam and an equalamount of light is removed through diffraction(Babinet’s principle) and Qext → 2 [14].

For an absorbing sphere, the anomalous diffractionapproximation is given by

~Qext � f �ρ�

� 2 − 4e−ρ tan β cos β

ρsin�ρ − β�

− 4e−ρ tan β

�cos β

ρ

�2cos�ρ − 2β�

� 4�cos β

ρ

�2cos 2β (7)

where

tan β � km − 1

;

[5]. This formula simplifies to Eq. (5) for a nonabsorb-ing sphere (k � 0). Figure 3(b) gives the plot of theresulting extinction curve normalized for a refractiveindex n � 1.4� ik for several values of k. As with thenonabsorbing case, this formula captures the salientfeatures of the extinction curve as they relate to blu-ing, namely, the interference structure and the effect

of absorption on it. By absorbing light, the interfer-ence is reduced, and extinction can thereby beincreased or decreased.

A refractive index that depends on a wavelengthchanges the shape of the extinction curve in differentways. Increasing the imaginary part of the refractiveindex bends the curve in the vertical direction towardthe value Qext � 2 while increasing the real part ofthe refractive index compresses the plot horizontally.For iron oxide containing dust such as found onMars,the dependence is most pronounced over the visiblespectrum for the imaginary part of the refractiveindex, so we focus on this case.

As an example of the effect on bluing of a wave-length-dependent refractive index see Fig. 4. Com-pare the extinction efficiency, computed usingMie’s solution, for a particle of radius 0.475 μm withrefractive index n � 1.35 (upper curve) to that for aparticle with refractive index n � 1.35� ik, where kdecreases linearly from 0.5 to 0 (lower curve). Thereis no bluing for the nonabsorbing particle; accordingto Eq. (6) the radius should be 0.74 μm for bluing tooccur. On the other hand, for the same-sized particlethat absorbs more strongly in blue, the extinction

Fig. 3. Extinction curve for an absorbing particle. (a) The extinction curves for a spherical particle with refractive index n � 1.4� ikwhere k � 0 (solid), 0.05 (dashed), and 0.15 (dotted). (b) The extinction curve Qext estimated using formula 7 versus size parameter x for aspherical particle with refractive index n � 1.4� ik where k � 0 (solid), 0.05 (dashed), and 0.15 (dotted).

Fig. 4. Effect of wavelength-dependent absorption on the shape ofthe extinction curve for a particle with radius r � 0.475 μm. Theindex of refraction is n � 1.35� ik where k � 0 for the uppercurve, and k decreases linearly from 0.5 to 0 as the wavelengthincreases from 0.4 to 0.7 μm in the lower curve.


efficiency increases with wavelength and bluingdoes occur. The destructive interference in bluewas partially removed by absorption of blue lighttransmitted through the particle. A substance witha wavelength-dependent refractive index can havea wider range of particle sizes leading to bluing.

The widened range of particle sizes can be visual-ized by plotting ~Qext versus both wavelength andradius using the anomalous diffraction model inEq. (7) expressed as

~Qext � q�r; λ� � f �4πr�m − 1�∕λ�: (8)

Using this to approximate the extinction in a particu-lar case, Fig. 5 shows how damping the transmittedlight in the blue (thus lowering the destructive inter-ference) widens the r interval where extinction in redis dominant. The graph is bent toward the geometriclimit of 2 for wavelengths near λ � 0.4 μm, while itremains close to the nonabsorbing value for wave-lengths near 0.7 μm. Because Martian dust is astrong absorber of blue light, the range of particlesizes over which bluing occurs is widened.

4. Blue Sunset of Mars

The thin yet dusty atmosphere of Mars makes its skyappear much different than Earth’s sky. While it isdifficult to reconstruct exactly the color one wouldsee while standing on the surface of Mars, data fromthe Mars Exploration Rover Pancam instruments in-dicate that the sky typically is “yellowish brown”[28]. Both the sun’s disk and the sky surroundingthe sun at sunset on Mars appear blue. What proper-ties of the Martian atmosphere are responsible forthese phenomena?

A natural question to ask is whether the presenceof hematite, which is a strong absorber of blue lightand is responsible for the reddish appearance ofMars, is also responsible for the blue sunset. Basedon a simplified model of Martian dust, we find thatthere is slight bluing associated with the observeddust size distribution. This is responsible for themildly bluish appearance of the sun’s disk at sunset.On the other hand, the blue glow surrounding thesun’s disk is not caused by wavelength-selective

extinction, but rather is caused by the dominanceof blue in near-forward scattered light from Martiandust. Near-forward scattered light is largely a prod-uct of diffraction and external reflection and is there-fore mostly invariant under changes in lightabsorption by the particle.

Martian dust contains hematite (α − Fe2O3) [29], aniron oxide whose refractive index depends stronglyon wavelength over the visible spectrum (Fig. 6). Itsimaginary part ranges from the hundredths in thered to more than one in the blue. Fine grained hem-atite with a particle radius of less than 2.5–5 μm scat-ters strongly in the longer visible wavelengths andappears red, while course grained particles withradius greater than 2.5–5 μm appears gray [31]. Bothforms of Hematite exist on Mars. The existence ofgray hematite is significant since its formation onEarth is a water-driven process [29], and this is takenas evidence that water may have existed on Mars.The fine grained red hematite, which becomes sus-pended in the Martian atmosphere during seasonaldust storms, gives Mars its reddish color.

Thermal infrared spectra acquired by the Mariner9 Infrared Interferometer Spectrometer and theMars Global Surveyor Thermal Emissions Spectrom-eter suggest that Martian dust is dominated byfeldspar and/or zeolite with lesser amounts of olivine,pyroxene, amorphous material, hematite, and mag-netite [32]. Using images from the Imager for theMars Pathfinder, Tomasko et al. determined thatthe geometric cross-section-weighted mean particle

Fig. 5. Effect of wavelength dependence on bluing using the anomalous diffraction approximation. Left: PlotQext versus wavelength (λ inμm) and radius (r in μm) computed using Eq. (7) with refractive index n � 1.4. Right: Plot Qext versus wavelength and radius computedusing Eq. (7), where n � 1.4� ki with k decreasing linearly from 0.1 at λ � 0.4 μm to 0 at 0.7 μm.

Fig. 6. Imaginary part (solid) and real part (dashed) of the refrac-tive index of hematite from [30].


radius (effective radius) of the atmospheric dust is1.6� 0.15 μm [6]. (See also [33] and [29].)

As a simplified model of Martian dust, we take anonabsorbing substrate with index of refraction n �1.5� 0i (representing feldspar or zeolite) containinga small percentage of hematite. To determine therefractive index of the resulting dust, we use theMaxwell–Garnett mixing rule for a nonabsorbingsubstrate with refractive index no and absorbinginclusion nA. The refractive index of the mixture isgiven by

n2 � n20

n2A � 2n2

0 � 2vA�n2A − n2

0�n2A � 2n2

0 − vA�n2A − n2

0�; (9)

where vA is the volume fraction of the absorbing in-clusion. This rule was chosen because its resultsshow good agreement with experiments [34], andit is simple to implement in our algorithm. By choos-ing a 3% hematite mixture, we obtain a wavelength-dependent index of refraction in good agreementwith that found by Tomasko et al. [6] [see Fig. 7(a)].

Our analysis is based on a modified gamma distri-bution of particle sizes [35]. The number of particles

with radius between r and r� dr in a unit volume isgiven by N�r�dr, where

N�r� � Kr�1−3b�∕b exp�−r∕�ab��: (10)

Here a is the effective radius and b is the effectivevariance. The effective radius is the ratio of the thirdand second moments of the radius distributions:

a �R∞0 r3N�r�drR∞0 r2N�r�dr :

The dimensionless effective variance is

b �R∞0 �r − a�2r2N�r�dra2

R∞0 r2N�r�dr :

Using an effective radius of 1.6 μm and an effectivevariance of 0.2 [6], we obtain the size distribution de-picted in Fig. 7(b).

Figure 8(a) shows a contour plot of Qext versusradius and wavelength with lighter regions corre-sponding to higher values of Qext. As predicted usingEq. (6), radii near r � 0.26 μm∕�1.53 − 1� � 0.5 μm

(a) (b)

Fig. 7. Dust model parameters. (a) The real (dashed) and imaginary (solid) parts of the index of refraction of the simulated Martian dust.(b) Size distribution of the simulated Martian dust.

400 450 500 550 600 650 7000.0

0.5

1.0

1.5

2.0

Wavelength nm

Part

icle

Rad

ius

m

400 450 500 550 600 650 7000.0

0.5

1.0

1.5

2.0

2.5

3.0

Wavelength nm

Qab

s,Q

sca,

Qex

t

(a) (b)

Fig. 8. Extinction efficiencies for the simulated Martian dust. (a) Contour plot of Qext versus radius and wavelength. (b) Plot of Qabs(dashed), Qsca (dotted), and Qext (solid) computed using Mie theory for a 3% hematite mixture (black), a 5% hematite mixture (blue),and a 20% mixture (red).


lead to the strongest bluing. By averaging over theparticle size distribution over a unit volume, wecan determine the normalized single particle valueof Qext. For a 3% hematite mixture with effective ra-dius 1.6 μm and effective variance of 0.2, the absorp-tion, scattering, and extinction efficiencies are shownin Fig. 8(b). The slope of the curve is positive over thevisible spectrum indicating mild bluing. Todetermine the role played by hematite in bluing,we repeat the computations for 5% and 20%mixtures[Fig. 8(b)]. While the contributions of absorption andscattering vary greatly, the extinction curve remainsnearly invariant: for our simulatedMars dust, bluingis largely independent of the percentage of hematite.

While there is bluing of sunlight as it passesthrough the Martian atmosphere, it is mild and isnot likely responsible for the blue glow surroundingthe sun. The secret to this blue glow is revealed in thescattering diagram (Fig. 9). The scattering angle ismeasured relative to the direction of light propaga-tion so that 0° corresponds to forward scatteringand 180° corresponds to back scattering. In this dia-gram it is seen that the intensity of blue light is about

6.5 times that for red at a scattering angle of 0° andremains above that for red until an angle of about28°. The domination by blue is most pronouncedfor angles less than about 10°. An observer on Marssees a blue glow around the sun caused by the domi-nance of blue light scattered in the near-forwarddirection (Fig. 10).

Figure 9 shows that, while the presence of hema-tite plays a significant role for large scattering an-gles, it plays almost no role in blue’s dominancefor angles less than about 10°. Insight into why bluelight dominates in the near-forward direction can begained using the Debye series for the scattering in-tensity. LikeMie’s solution, the Debye series

Ppi is a

rigorous solution to Maxwell’s equations for lightincident on a sphere. It too contains an infinitenumber of terms. However, each term of the Debyeseries has a physical interpretation: the p0 termcorresponds to light that is diffracted and externallyreflected from the surface of the particle, the p1 termcorresponds to twice-refracted light, the p2 corre-sponds to light that is internally reflected once,the p3 term corresponds to light internally reflectedtwice, and so forth.

In Fig. 11, the scattering intensity is computed us-ing Mie’s solution (the solid curves), the p0 term ofthe Debye series (the dotted curves), and diffractiontheory for blue and red light. These plots show thatthe blue sunset of Mars is primarily an effect of dif-fraction and external reflection. For both blue andred light, the p0 terms are in good agreement withMie’s solution for scattering angles of less than10°. This is the region where the dominance in theintensity of blue light is most significant, so wecan conclude that diffraction and external reflectionare the primary mechanisms behind the appearanceof the blue Martian sunset.

The calculation of intensity for diffraction uses theformula:

I�θ� � Io

�x2

1� cos2 θ2

J1�x sin θ�x sin θ

�2; (11)

where J1 is a Bessel function of the first kind [5].While diffraction does not completely capture theintensity in the near-forward direction (the graphis on a logarithmic scale), it does drive the significant

0 50 100 150

10

100

1000

104

Scattering angle degrees

Rel

ativ

ein

tens

ity

S 12S 22

Fig. 9. Relative intensity (S21 � S2

2) of scattered light versus scat-tering angle for a 3% hematite dust mixture (solid curves) and thepure substrate with no hematite (dashed curves). The blue curvescorrespond to blue light (425 nm), and the red curves correspond tored light (694 nm). Note that in the near-forward directions, thecurves are insensitive to hematite, while in the region approachinga scattering angle of 180°, hematite concentration plays a signifi-cant role. Without hematite, red and blue light would be backscat-tered in equal amounts, and Mars would not be known as the redplanet.

1 2

θ φ

Fig. 10. Observation of scattered light at sunset. Light seen by the observer scattered from a dust particle at position 1 appears blue sinceθ < 28°, so the intensity of near-forward blue light is much greater than that for red. The dominance by blue is lost at position 2 whereϕ > 28°.


difference between the intensity of blue and red lightin these directions. For a fixed radius, most of thediffracted light is concentrated in the region betweenθ � 0° and the first minimum of I (the Airydisk), which occurs at approximately x sin θ ��2πr∕λ� sin θ � 3.83. The smaller the wavelength,the narrower and longer this region becomes.Roughly speaking, destructive interference by dif-fracted light out of phase with the undisturbed fieldoccurs closer to the axis through the particle withθ � 0° as either the radius becomes larger or thewavelength becomes smaller.

5. Conclusion

The typical color of the disks of the sun and moon onEarth, and the color of sunsets on Earth are amongthe simplest optical phenomena to explain. Gasesand very small particles, which generally dominateEarth’s atmosphere, scatter blue light much moreeffectively than red light. The reddening of lightcaused by the enhanced scattering of blue lightcauses the disks of the sun and moon to appear red-dish (especially when they are low in the sky) andcreates a red sunset: both a reddish sun and reddishlight scattered by large particles in the surroundingsky. Blue suns and moons are observed on Earthwhen large aerosol particles cause bluing of lightsufficient to overcome the reddening of light byatmospheric gases and small particles. These occur-rences are rare because the size range of particlesthat cause bluing of light is limited, especially whenthe refractive index is nearly constant over the vis-ible spectrum.

The situation on Mars is complicated by the factthat its atmosphere is dominated by dust particleswhose radius is close to the wavelength of visiblelight. Scattering and absorption by large aerosol par-ticles is sensitive to size and composition. To furthercomplicate things, the refractive index of Martiandust depends strongly on wavelengths over the vis-ible spectrum. Simple tautologies such as “Mars ap-pears red therefore scattering of red light causesbluing of light, hence the blue sunset” are false.One such subtlety relevant to the Martian atmos-phere is that selective absorption of blue light by aparticle can enhance bluing by reducing destructiveinterference of transmitted and diffracted light. This

may also be relevant to bluing of sunlight duringdust storms on Earth, where the imaginary part ofthe refractive index is driven by the presence of ironoxides [10,36].

Bluing is only responsible for one aspect of theMartian sunset: the slightly bluish disk of the sun.The blue glow surrounding sun as viewed from Marsis most likely a product of the pattern of scatteredlight from dust particles. As such, the blue glowshould not be a feature unique to sunrise or sunset,but should follow the sun as it traverses the Martiansky from sunrise to sunset. However, it will be mostintense during sunrise and sunset due to the in-creased optical path length through the atmosphere.At small scattering angles, the intensity of blue lightdominates over that for red light. The primary sourceof this near-forward scattered light is diffraction andexternal reflection (the p0 term of the Debye series).For our simplified model (using spherical particles),blue light’s intensity dominates for scattering anglesup to about 28° with the greatest dominance for an-gles up to about 10°. This is in good agreement withthe photograph of the Martian sunset (Fig. 1). To addscattering angles to this figure, we used the image ofthe sun as scale. The sun has a diameter of 0.0093 au(astronomical unit � 149.6 × 106 km), and the dis-tance betweenMars and the sun is 1.52 au. Thereforethe angular diameter of the sun seen from Mars is0.35°, and its angular radius 0.175°. In Fig. 1, wehave drawn a circle around the image of the sunand used the linear scaling of its angular radius(i.e., 0.175°) to draw additional circles with angularradii of 5°, 10°,15°, 20°, and 25°. These angular radiiequal the respective scattering angles and helprelate the results of scattering calculations to theimage of the Martian sunset.

The authors are grateful to Jair Koiller and theanonymous reviewers for their insightful and helpfulcomments. This material is based upon work sup-ported by NASA EPSCoR under cooperative agree-ment no. NNX10AR89A and by NASA ROSESunder grant no. NNX11AB79G.

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5 10 20 50 1001

10

100

1000


Rel

ativ

ein

tens

ity

5 10 20 50 100

10

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104


Rel

ativ

ein

tens

ity

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