The familiar yellow or orange disks of the moon and sun, especially when they are low in the sky, and brilliant red sunsets are a result of the selective extinction (scattering plus absorption) of blue light by atmospheric gas molecules and small aerosols, a phenomenon explainable using the Rayleigh scattering approximation. On rare occasions, dust or smoke aerosols can cause the extinction of red light to exceed that for blue, resulting in the disks of the sun and moon to appear as blue. Unlike Earth, the atmosphere of Mars is dominated by micron-size dust aerosols, and the sky during sunset takes on a bluish glow. Here we investigate the role of dust aerosols in the blue Martian sunsets and the occasional blue moons and suns on Earth. We use the Mie theory and the Debye series to calculate the wavelength-dependent optical properties of dust aerosols most commonly found on Mars. Our findings show that while wavelength selective extinction can cause the sun’s disk to appear blue, the color of the glow surrounding the sun as observed from Mars is due to the dominance of near-forward scattering of blue light by dust particles and cannot be explained by a simple, Rayleigh-like selective extinction explanation.
© 2014 Optical Society of America
Optical phenomena in the Earth’s atmosphere such as blue skies, red sunsets, halos, rainbows, and coronas are enjoyed by most humans and raise curiosity about the optics causing them. Therefore, these phenomena are frequently discussed in physics and optics education. For example, the color of the sun is mostly observed during sunsets and sunrises or when part of the sunlight is obscured by thin clouds or large aerosol concentrations, making it possible to safely view the sun with the naked eye. The color of the moon can easily be observed at all times. While the reddish appearance of the sun and the moon, especially when low in the sky, has been noticed by most, the much rarer appearance of blue moons and suns requires a specific and narrow particle size distribution  that has only been known to occur after large forest fires and volcanic eruptions. (Note that when we speak of a blue moon, we mean it in the literal sense, rather than in the literary or calendar sense.)
Recent images taken on the surface of Mars remind us that interesting optical phenomena are not limited to the Earth’s atmosphere but can also be found on other planets. Of particular interest to us is the unfamiliar blue color of Martian sunsets and sunrises (Fig. 1). Here we make a theoretical study using a model for Martian dust based on the optical properties and size distribution of Martian dust measured during recent Mars Rover missions. Because the size of Martian dust particles is comparable to the wavelength of visible light , the optics are complicated and solutions to Maxwell’s equations are required to model the scattered electric and magnetic fields. Our theoretical analysis is based on Mie theory , the Debye series , and the anomalous diffraction theory of van de Hulst . The goals of our analysis are to study the optical properties of aerosols with a wavelength-dependent refractive index, especially as they influence the color of the sun and moon, and to uncover the basic physics behind the blue Martian sunset.
The blue sky and red sunset on Earth are a result of scattering of blue light by atmospheric gases and small particles. Mars, the red planet, has an atmosphere that is dominated by dust . In analogy with what happens on Earth, extinction of red light through scattering by Martian dust is a likely candidate for causing blue sunsets and sunrises on Mars. On the other hand, Martian dust absorbs blue light more strongly than red light . We show that while stronger absorption in the blue can diminish the extent to which extinction of red light exceeds that for blue light, it can also increase the range of particle sizes for which extinction in red dominates. Based on our understanding of the optical properties and size distribution of Martian dust, the extinction of red light by the Martian atmosphere is slightly greater than the extinction of blue light. The disk of the sun would appear slightly blue, especially at sunset when the optical path is greatest. However, wavelength-dependent extinction of light is not sufficient to explain the intriguing blue glow surrounding the sun in Fig. 1.
While Martian dust removes only slightly more red light than blue from the sun’s beam, the pattern in which it scatters red and blue light is very different. To investigate this, we use the scattering diagram: a plot of scattering intensity versus scattering angle. For a model Martian dust, the scattering intensity of blue light within a cone of about 10° about the axis of sunlight propagation, exceeds that for red by a factor of more than 6. Outside this cone, blue light’s dominance decreases until about 28° where the intensity of red light surpasses that for blue. The blue glow surrounding the sun at sunset is created by light scattered at small angles by dust particles. The physical processes leading to the dominance of blue in near-forward scattering from Martian dust are external reflection and diffraction. While the disk of the sun would only appear blue at sunset, when the optical path is longest, or during a dust storm, the blue glow around the sun should be visible throughout the Martian day. On Earth, the reddish color of the sun at sunset and the red sunset created by scattered light are both products of wavelength-selective extinction, while on Mars, only the blue disk of the sun is. The blue glow surrounding the sun is a product of the angular pattern of scattering from dust particles.
We begin by summarizing explanations for the red and rare blue appearances of the sun and moon in the Earth’s atmosphere. This is followed by a discussion of the wavelength-dependent optical properties of aerosols as they relate to the bluing of light. Here we expand on earlier studies of the bluing and reddening of light by aerosols by considering a wavelength-dependent refractive index, which is particularly relevant to Martian dust. We conclude with an investigation of possible reasons for the blue sunsets and 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 is based on Rayleigh’s law of scattering. Without an atmosphere, the sky would simply appear black. The light we see in the sky is light scattered by atmospheric gases and aerosols. An aerosol is a suspension of small particles such as smoke particles, dust, or water droplets within atmospheric gases. Earth’s atmosphere is generally dominated by gases and particles that are much smaller that the wavelength of visible light. John William Strutt (also known as the third Baron Rayleigh or Lord Rayleigh for short) showed that scattering from small particles is proportional to where is the wavelength of light . As a consequence, blue light with a wavelength of 425 nm is scattered more than seven times as strongly as red light with a wavelength of 700 nm. The scattered sunlight we see in the sky during the daytime is therefore dominated by blue unless large particles are present. Conversely, when looking at the disk of the moon or sun at sunset, we are seeing the direct beam of light from these objects. Since the blue light is scattered away, the remaining light is reddened, making the disks of the moon and sun appear yellow to red. This phenomenon is most pronounced when the sun or moon is low in the sky and the optical depth is greatest. When the reddened light scatters off larger atmospheric aerosols such as water droplets in clouds, which scatter all wavelengths without prejudice, spectacular red sunsets occur. A key observation here is that the red disk of the sun and the redness of the surrounding sky at sunset are both caused by wavelength-dependent extinction.
On rare occasions, atmospheric aerosols can cause the disks of the sun and moon to appear blue. In 1853, the volcano Krakatoa erupted, spewing dust into the atmosphere. For nearly a month afterward, the moon appeared blue over tropical regions of Earth . Soon after the eruption, Reverend Sereno Bishop reported from Hawai’i the appearance of an anomalous corona encircling the sun consisting of a bluish inner region surrounded by a brownish ring . This optical phenomenon, known as a Bishop’s ring, is generally associated with dust from volcanic eruptions. Blue moons and suns also occur in dust-prone areas of Northern Africa and the Arabian Peninsula. In , six of 14 AERONET sites reported a negative Ångström coefficient, which would indicate a blue sun at these locations, at some time. In the early 19th century, a blue glow of area one and a half diameters surrounding the sun was observed in Cairo during a dust storm that turned the sky a yellowish white color . In this instance, the sun remained a pale yellow, and it was noted that there was no brownish ring present. Smoke from forest fires can also cause blue moons and suns. During a particularly intense fire season in Alberta and British Columbia in 1950, smoke caused the sun and moon to appear blue from locations in eastern Canada and America, then later in Europe .
While on Earth, blue suns, moons, and sunsets are a rare occurrence, blue sunsets appear to be the norm on Mars. The photograph of the sunset on Mars taken from the Gusev crater by the Mars Exploration Rover Spirit (Fig. 1) shows a blue Martian sunset. Phenomenal movies created using the Mars Exploration Rover Opportunity’s Pancam by Mark Lemmon of Texas A&M and Jim Bell of Cornell University showing the Martian sunset are available on a NASA website .
On Earth, for the moon or sun to appear blue, extinction (scattering plus absorption) of red light must exceed that for blue light. For an aerosol whose refractive index is constant over the visible spectrum, this only occurs when the aerosol has a very narrow size distribution . For example, for an aerosol of spherical water droplets with index of refraction , the radius needs to be about 0.75 μm. Water droplets in clouds or fog are typically 10–20 μm, so fog and clouds do not cause blue suns or moons. For an aerosols to cause the moon or sun to appear blue, the optical depth must be sufficient to overcome the normal extinction of blue light by atmospheric gases and small aerosols. On rare occasions, a sufficient number of particles of just the right size have been produced by volcanic eruptions and large forest fires to cause the moon and sun to appear blue.
The situation on Mars is complicated by the fact that the Martian atmosphere is dominated by dust particles whose size is close to the wavelength of visible light (400–700 nm). For particles of this size there is no simple law analogous to Rayleigh’s law. Scattering for these particles is sensitive to size, shape, and composition. The density of atmospheric gases on Mars, which are composed of approximately 95% carbon dioxide, 3% nitrogen, and 1.6% argon, is much lower than it is on Earth. On Mars the density of atmospheric gases is while on Earth it is nearly 80 times greater at . Rayleigh scattering is much less important on Mars than on Earth. In addition, the refractive index of Martian dust depends significantly on the wavelength of light over the visible spectrum. Here, the real part is the ratio of the speed of light in a vacuum to that through the substance, and the imaginary part is a measure of the absorption of light by the bulk substance. For Martian dust, the imag inary part increases strongly toward shorter wavelengths , so Martian dust strongly absorbs blue light, which has an effect on the bluing and reddening of light as it passes through the Martian atmosphere.
A. Scattering, Absorption, and Extinction of Light by Aerosols
The color of the disks of the moon and sun and the color of the sunset on Earth are a product of extinction of certain wavelengths of light by atmospheric aerosols. Light becomes extinct from a beam if it is scattered (its course is deviated) or it is absorbed (changed to another form of energy such as heat). To quantify extinction, we use the absorption, scattering, and extinction cross sections and efficiencies. These parameters can be found experimentally or theoretically using solutions to Maxwell’s equations. The following is a brief review of scattering and absorption by an individual particle and can be skipped by experts. Additional details can be found in .
For an individual particle, absorption and scattering are characterized by the absorption cross section and scattering cross section , which have units of . For a suspension of particles contained in a volume , absorption and scattering are characterized by the absorption and scattering coefficients defined as15]:
Most of our analysis will be based on the dimensionless absorption, scattering, and extinction efficiency factors, , , and , respectively, which are defined to be the ratio of the absorption, scattering, and extinction cross sections and the geometric cross section. For a spherical particle with radius , the extinction efficiency factor is . In terms of extinction efficiencies, bluing occurs when is greater for red light than it is for blue.
The scattering and absorption efficiencies depend on varying degrees of size, shape, and mineralogical and chemical composition of the particle. In the present work, we focus on the effect of the particle size and refractive index on the appearance of the moon, sun, and sunset. Our analysis is based on the scattering and absorption of light by a homogeneous spherical particles of radius with complex refractive index .
Traditionally, scattering and absorption are expressed as a function of the dimensionless size parameter , defined to be the circumference of the particle divided by the wavelength of the light: . For particles much smaller than the wavelength (), absorption is proportional to and scattering is proportional to [16–18].
When the size of the particle and the wavelength of the light are about equal (), as happens with Martian dust, the situation becomes much more difficult, and solutions to Maxwell’s equations are required to determine scattering and absorption. Gustav Mie solved the problem of scattering and absorption of a plane electromagnetic wave from a homogeneous isotropic sphere in 1908 . It is well known that dust particles are nonspherical and therefore have substantially different optical properties from volume equivalent spheres. While this leads to substantial errors in retrieving aerosol optical thickness from satellite reflectance measurements , phase functions in the forward hemisphere, especially in the near-forward direction, are nearly identical for spherical and nonspherical particles [19,20]. Therefore nonsphericity is largely irrelevant to the optical phenomena discussed here, and Mie theory can be used.
Mie’s solution expresses the scattering amplitude from a sphere as the infinite series:14]. The number of terms of the series that must be kept for a good approximation is on the order of the size parameter . For our problem, this is easily accomplished using a computer algebra system such as Mathematica, or with dedicated Mie theory software such as MiePlot , both of which have been employed in the present work.
While Mie’s partial wave expansion provides accurate solutions, the complicated series of partial waves provides little physical intuition about the mechanism behind the scattering and absorption. Also in 1908, Debye  solved for the scattering from an infinite cylinder. Subsequently, Debye’s solution was adapted to the case of a spherical scatterer [22–24]. The advantage of Debye’s solution is that it lends itself to physical interpretation. The Debye decomposition expresses the partial wave coefficients of the Mie solution in terms of a set of amplitudes corresponding to transmitted and reflected waves at the interface of two media. For a homogeneous sphere, the Debye series expansion of the partial wave scattering amplitudes and are of the form:25] Each term has a typical physical interpretation when substituted for Mie scattered partial wave amplitudes in expressions for electromagnetic scattering by a homogeneous sphere. The first term, , is taken to describe the component of the scattered electromagnetic field due to diffraction. The second term, , is taken to describe the component of the scattered field due to external reflection from the sphere. The third term is an infinite sum, each individual term of which is taken to represent the component of the field, which has penetrated the sphere, undergone internal reflections and then subsequently emerged again into the surrounding medium. If all the terms of the Debye series are summed and then substituted for the Mie scattered partial wave amplitudes, then the results are identical to ordinary Mie scattering. Thus the Debye series is taken to interpret Mie scattering as the result of many scattering processes, each with a unique number of internal/external reflections.
The MiePlot software package provides the option of computing the Debye series for scattering from a homogeneous sphere, term by term. In the present work we employ the Mie series, the Debye series, as well as the anomalous diffraction approximation  to compute the optical properties of spherical particles as they relate to the occurrence of blue moons and suns, especially when the refractive index is allowed to depend on the wavelength and to isolate the physical processes behind the blue Martian sunset.
Bluing and the extinction curve. Viewed from space, the disks of the sun and moon appear white. Their colors, as observed from Earth, are a result of the extinction (removal) light of certain wavelengths as the direct beam of light passes through the atmosphere. The terms bluing and reddening are used to describe these processes. Bluing refers to the removal of red light, and reddening refers to the removal of blue light. Under normal circumstances, reddening of light by atmospheric gases and other small particles causes the disks of the moon and sun to appear yellow or red, especially when they are low in the sky. In the evening, the reddened light illuminates water droplets in clouds and other large particles (which effectively scatter all wavelengths of light), and they appear red. Blue moons and suns are observed under the rare circumstances when atmospheric aerosols cause bluing.
For a particular aerosol, the bluing or reddening of light can be inferred using its extinction curve: the plot of versus the size parameter , the wavelength of light, or the phase shift parameter (described below). If decreases with wavelength, reddening occurs, and if increases with wavelength, bluing occurs.
Figure 2(a) shows a plot of versus the size parameter for a spherical particle with purely real refractive index computed using Mie theory. By fixing a radius , the visible spectrum defines an interval . Note that higher wavelengths are to the left, so that positive slopes correspond to reddening and negative slopes correspond to bluing. Note also that the intervals with positive slopes get larger as the particle size increases. For very small particles, the interval is on the far left side of the graph, which has a positive slope. As predicted by Lord Rayleigh, an aerosol of small particles reddens light. For particles of radius 0.65 μm, the interval corresponding to visible light is . On this interval the slope is negative and bluing occurs. A monodisperse aerosol of particles with refractive index with radius 0.65 μm would lead to a blue moon or sun. For larger particles, the interval gets larger and contains many smaller oscillations of the curve. Neither bluing nor reddening occurs. For a particle with refractive index , bluing occurs only when the radius is very nearly 0.65 μm. The small range of particle sizes leading to bluing of light together with strong scattering of blue light by atmospheric gases explains the rarity of blue moons and suns on Earth.
Understanding the bluing and reddening of light boils down to understanding the shape of the extinction curve and how the shape is changed when the refractive index is varied. The extinction curve is characterized by large low-frequency oscillations about, and asymptotic to, the geometric optics value of , superimposed with smaller high-frequency ripples. Physically, the ripples are associated with resonances of internally reflected light . Bluing and reddening of light are associated with the large oscillations known as the interference structure. The interference structure is so named because the large oscillations are a result of the constructive and destructive interference between light that is diffracted around the particle and light that is transmitted through the particle.
Insight into how the refractive index is related to the interference structure can be gained using the anomalous diffraction approximation derived by van de Hulst . This approximation is valid for optically soft particles, i.e., particles for which the refractive index is very close to unity. Here the extinction efficiency is plotted against the phase shift parameter . Physically, the phase shift parameter represents the phase lag suffered by a centrally transmitted ray. The anomalous diffraction approximation qualitatively captures the interference structure thus allowing the intervals of bluing and reddening to be approximated for a particular refractive index.
For optically soft particles, light changes direction very little as it passes through a boundary of the particle, but, since it travels slower within the particle, it suffers a phase lag compared to light diffracted around the particle. Applying Huygen’s principle to combine the diffracted and transmitted light at points beyond the particle, van de Hulst derived the formula:2(b)]. The peaks and valleys of the interference structure are a result of constructive and destructive interference of the combined light.
By rescaling the horizontal axis, we can obtain an approximation to the extinction curve ( versus ) for any purely real refractive index. To compare with the extinction curve obtained using Mie’s solution, we have approximated the extinction curve for a particle with a refractive index of using Eq. (5) [Fig. 2(a)]. Note that, because the internally reflected light is ignored in the anomalous diffraction approximation, the ripples are absent and that the peaks and valleys have smaller amplitude than those obtained using Mie’s solution. On the other hand, the qualitative features of the interference structure are preserved. This remains true when the refractive index is as high as . For instance, the peaks in the interference structure occur with nearly the same period, which is in or in . For a particle with refractive index , the period is about 7.9 in . An immediate observation is that changes in the real part of the refractive index stretches or compresses the extinction curve in the horizontal direction.
The anomalous diffraction approximation allows us to directly estimate the particle size that would lead to bluing for a particular refractive index. Bluing is associated with the region between the first maximum and the following minimum in the extinction curve [Fig. 2(b)]. The center of this interval occurs at about . Using the definition of the phase shift parameter, we can estimate the radius leading to bluing for a given refractive index . Indeed, we set and set (the center of the visible spectrum) then5] p. 423, and . For a mono-disperse aerosol, the radius leading to a bluing is driven by the real part of the index of refraction. For water, equals approximately 1.35 so the radius should be approximately 0.74 μm, which is in agreement with the findings of .
3. Bluing by Absorbing Particles with a Wavelength-Dependent Refractive Index
The theory described so far is well known and satisfactorily describes many of the known appearances of blue moons and suns on Earth. Blue moons and suns are associated with smoke or dust from volcanic eruptions with nearly identical particle size and whose refractive index is nearly constant over the visible spectrum. The situation on Mars (as well as for dust storms with iron oxide containing dust on Earth ) is more complicated. The atmosphere on Mars is dominated by dust particles that have a fairly wide distribution of sizes and whose refractive index varies significantly over the visible spectrum . Specifically, Martian dust preferentially absorbs blue light. To understand the bluing and reddening of light on Mars, we must investigate the effect of the imaginary part of the refractive index on the shape of the extinction curve, especially when it is allowed to vary with wavelength.
A nonzero imaginary part of the index of refraction leads to absorption of the internally transmitted light. A small imaginary part dampens the high-frequency ripples and has only a small effect on the interference structure [see the dashed curve in Fig. 3(a) where ]. Intuitively, this is the case because the transmitted light travels only a short distance through the sphere thus suffering negligible absorption and interference with diffracted light is little affected. The high-frequency ripples are more affected since they are associated with light that is internally reflected and takes a longer path through the particle, thereby suffering greater extinction. As the imaginary part increases, the interference structure is lost [see the dotted curve in Fig. 3(a)]. The effect of increasing the imaginary part of the refractive index is to dampen peaks in the interference structure toward the value . Note, however, that there is no change in the period of the peaks. In the limit , the particle becomes opaque. All of the light incident on the particle is removed from the beam and an equal amount of light is removed through diffraction (Babinet’s principle) and .
For an absorbing sphere, the anomalous diffraction approximation is given by5]. This formula simplifies to Eq. (5) for a nonabsorbing sphere (). Figure 3(b) gives the plot of the resulting extinction curve normalized for a refractive index for several values of . As with the nonabsorbing case, this formula captures the salient features of the extinction curve as they relate to bluing, namely, the interference structure and the effect of absorption on it. By absorbing light, the interference is reduced, and extinction can thereby be increased or decreased.
A refractive index that depends on a wavelength changes the shape of the extinction curve in different ways. Increasing the imaginary part of the refractive index bends the curve in the vertical direction toward the value while increasing the real part of the refractive index compresses the plot horizontally. For iron oxide containing dust such as found on Mars, the dependence is most pronounced over the visible spectrum for the imaginary part of the refractive index, so we focus on this case.
As an example of the effect on bluing of a wavelength-dependent refractive index see Fig. 4. Compare the extinction efficiency, computed using Mie’s solution, for a particle of radius 0.475 μm with refractive index (upper curve) to that for a particle with refractive index , where decreases linearly from 0.5 to 0 (lower curve). There is no bluing for the nonabsorbing particle; according to Eq. (6) the radius should be 0.74 μm for bluing to occur. On the other hand, for the same-sized particle that absorbs more strongly in blue, the extinction efficiency increases with wavelength and bluing does occur. The destructive interference in blue was partially removed by absorption of blue light transmitted through the particle. A substance with a wavelength-dependent refractive index can have a wider range of particle sizes leading to bluing.
The widened range of particle sizes can be visualized by plotting versus both wavelength and radius using the anomalous diffraction model in Eq. (7) expressed as5 shows how damping the transmitted light in the blue (thus lowering the destructive interference) widens the interval where extinction in red is dominant. The graph is bent toward the geometric limit of 2 for wavelengths near , while it remains close to the nonabsorbing value for wavelengths near 0.7 μm. Because Martian dust is a strong absorber of blue light, the range of particle sizes over which bluing occurs is widened.
4. Blue Sunset of Mars
The thin yet dusty atmosphere of Mars makes its sky appear much different than Earth’s sky. While it is difficult to reconstruct exactly the color one would see while standing on the surface of Mars, data from the Mars Exploration Rover Pancam instruments indicate that the sky typically is “yellowish brown” . Both the sun’s disk and the sky surrounding the sun at sunset on Mars appear blue. What properties of the Martian atmosphere are responsible for these phenomena?
A natural question to ask is whether the presence of hematite, which is a strong absorber of blue light and is responsible for the reddish appearance of Mars, is also responsible for the blue sunset. Based on a simplified model of Martian dust, we find that there is slight bluing associated with the observed dust size distribution. This is responsible for the mildly bluish appearance of the sun’s disk at sunset. On the other hand, the blue glow surrounding the sun’s disk is not caused by wavelength-selective extinction, but rather is caused by the dominance of blue in near-forward scattered light from Martian dust. Near-forward scattered light is largely a product of diffraction and external reflection and is therefore mostly invariant under changes in light absorption by the particle.
Martian dust contains hematite () , an iron oxide whose refractive index depends strongly on wavelength over the visible spectrum (Fig. 6). Its imaginary part ranges from the hundredths in the red to more than one in the blue. Fine grained hematite with a particle radius of less than 2.5–5 μm scatters strongly in the longer visible wavelengths and appears red, while course grained particles with radius greater than 2.5–5 μm appears gray . Both forms of Hematite exist on Mars. The existence of gray hematite is significant since its formation on Earth is a water-driven process , and this is taken as evidence that water may have existed on Mars. The fine grained red hematite, which becomes suspended in the Martian atmosphere during seasonal dust storms, gives Mars its reddish color.
Thermal infrared spectra acquired by the Mariner 9 Infrared Interferometer Spectrometer and the Mars Global Surveyor Thermal Emissions Spectrometer suggest that Martian dust is dominated by feldspar and/or zeolite with lesser amounts of olivine, pyroxene, amorphous material, hematite, and magnetite . Using images from the Imager for the Mars Pathfinder, Tomasko et al. determined that the geometric cross-section-weighted mean particle radius (effective radius) of the atmospheric dust is . (See also  and .)
As a simplified model of Martian dust, we take a nonabsorbing substrate with index of refraction (representing feldspar or zeolite) containing a small percentage of hematite. To determine the refractive index of the resulting dust, we use the Maxwell–Garnett mixing rule for a nonabsorbing substrate with refractive index and absorbing inclusion . The refractive index of the mixture is given by34], and it is simple to implement in our algorithm. By choosing a 3% hematite mixture, we obtain a wavelength-dependent index of refraction in good agreement with that found by Tomasko et al.  [see Fig. 7(a)].
Our analysis is based on a modified gamma distribution of particle sizes . The number of particles with radius between and in a unit volume is given by , where6], we obtain the size distribution depicted in Fig. 7(b).
Figure 8(a) shows a contour plot of versus radius and wavelength with lighter regions corresponding to higher values of . As predicted using Eq. (6), radii near lead to the strongest bluing. By averaging over the particle size distribution over a unit volume, we can determine the normalized single particle value of . For a 3% hematite mixture with effective radius 1.6 μm and effective variance of 0.2, the absorption, scattering, and extinction efficiencies are shown in Fig. 8(b). The slope of the curve is positive over the visible spectrum indicating mild bluing. To determine the role played by hematite in bluing, we repeat the computations for 5% and 20% mixtures [Fig. 8(b)]. While the contributions of absorption and scattering vary greatly, the extinction curve remains nearly invariant: for our simulated Mars dust, bluing is largely independent of the percentage of hematite.
While there is bluing of sunlight as it passes through the Martian atmosphere, it is mild and is not likely responsible for the blue glow surrounding the sun. The secret to this blue glow is revealed in the scattering diagram (Fig. 9). The scattering angle is measured relative to the direction of light propagation so that 0° corresponds to forward scattering and 180° corresponds to back scattering. In this diagram it is seen that the intensity of blue light is about 6.5 times that for red at a scattering angle of 0° and remains above that for red until an angle of about 28°. The domination by blue is most pronounced for angles less than about 10°. An observer on Mars sees a blue glow around the sun caused by the dominance of blue light scattered in the near-forward direction (Fig. 10).
Figure 9 shows that, while the presence of hematite plays a significant role for large scattering angles, it plays almost no role in blue’s dominance for angles less than about 10°. Insight into why blue light dominates in the near-forward direction can be gained using the Debye series for the scattering intensity. Like Mie’s solution, the Debye series is a rigorous solution to Maxwell’s equations for light incident on a sphere. It too contains an infinite number of terms. However, each term of the Debye series has a physical interpretation: the term corresponds to light that is diffracted and externally reflected from the surface of the particle, the term corresponds to twice-refracted light, the corresponds to light that is internally reflected once, the term corresponds to light internally reflected twice, and so forth.
In Fig. 11, the scattering intensity is computed using Mie’s solution (the solid curves), the term of the Debye series (the dotted curves), and diffraction theory for blue and red light. These plots show that the blue sunset of Mars is primarily an effect of diffraction and external reflection. For both blue and red light, the terms are in good agreement with Mie’s solution for scattering angles of less than 10°. This is the region where the dominance in the intensity of blue light is most significant, so we can conclude that diffraction and external reflection are the primary mechanisms behind the appearance of the blue Martian sunset.
The calculation of intensity for diffraction uses the formula:5]. While diffraction does not completely capture the intensity in the near-forward direction (the graph is on a logarithmic scale), it does drive the significant difference between the intensity of blue and red light in these directions. For a fixed radius, most of the diffracted light is concentrated in the region between and the first minimum of (the Airy disk), which occurs at approximately . The smaller the wavelength, the narrower and longer this region becomes. Roughly speaking, destructive interference by diffracted light out of phase with the undisturbed field occurs closer to the axis through the particle with as either the radius becomes larger or the wavelength becomes smaller.
The typical color of the disks of the sun and moon on Earth, and the color of sunsets on Earth are among the simplest optical phenomena to explain. Gases and very small particles, which generally dominate Earth’s atmosphere, scatter blue light much more effectively than red light. The reddening of light caused by the enhanced scattering of blue light causes the disks of the sun and moon to appear reddish (especially when they are low in the sky) and creates a red sunset: both a reddish sun and reddish light scattered by large particles in the surrounding sky. Blue suns and moons are observed on Earth when large aerosol particles cause bluing of light sufficient to overcome the reddening of light by atmospheric gases and small particles. These occurrences are rare because the size range of particles that cause bluing of light is limited, especially when the refractive index is nearly constant over the visible spectrum.
The situation on Mars is complicated by the fact that its atmosphere is dominated by dust particles whose radius is close to the wavelength of visible light. Scattering and absorption by large aerosol particles is sensitive to size and composition. To further complicate things, the refractive index of Martian dust depends strongly on wavelengths over the visible spectrum. Simple tautologies such as “Mars appears red therefore scattering of red light causes bluing of light, hence the blue sunset” are false. One such subtlety relevant to the Martian atmosphere is that selective absorption of blue light by a particle can enhance bluing by reducing destructive interference of transmitted and diffracted light. This may also be relevant to bluing of sunlight during dust storms on Earth, where the imaginary part of the refractive index is driven by the presence of iron oxides [10,36].
Bluing is only responsible for one aspect of the Martian sunset: the slightly bluish disk of the sun. The blue glow surrounding sun as viewed from Mars is most likely a product of the pattern of scattered light from dust particles. As such, the blue glow should not be a feature unique to sunrise or sunset, but should follow the sun as it traverses the Martian sky from sunrise to sunset. However, it will be most intense during sunrise and sunset due to the increased optical path length through the atmosphere. At small scattering angles, the intensity of blue light dominates over that for red light. The primary source of this near-forward scattered light is diffraction and external reflection (the term of the Debye series). For our simplified model (using spherical particles), blue light’s intensity dominates for scattering angles up to about 28° with the greatest dominance for angles up to about 10°. This is in good agreement with the photograph of the Martian sunset (Fig. 1). To add scattering angles to this figure, we used the image of the sun as scale. The sun has a diameter of 0.0093 au (astronomical ), and the distance between Mars and the sun is 1.52 au. Therefore the angular diameter of the sun seen from Mars is 0.35°, and its angular radius 0.175°. In Fig. 1, we have drawn a circle around the image of the sun and used the linear scaling of its angular radius (i.e., 0.175°) to draw additional circles with angular radii of 5°, 10°,15°, 20°, and 25°. These angular radii equal the respective scattering angles and help relate the results of scattering calculations to the image of the Martian sunset.
The authors are grateful to Jair Koiller and the anonymous reviewers for their insightful and helpful comments. This material is based upon work supported by NASA EPSCoR under cooperative agreement no. NNX10AR89A and by NASA ROSES under grant no. NNX11AB79G.
1. H. Horvath, G. Metzig, O. Preining, and R. Pueschel, “Observation of a blue sun over New Mexico, U.S.A.,” Atmos. Environ. 28, 621–630 (1994). [CrossRef]
2. M. Lemmon, M. Wolff, M. Smith, R. Clancy, D. Banfield, G. A. Landis, A. Ghosh, P. H. Smith, N. Spanovich, B. Whitney, P. Whelley, R. Greeley, S. Thompson, J. F. Bell III, and S. W. Squyres, “Atmospheric imaging results from the Mars exploration rovers: spirit and opportunity,” Science 306, 1753–1756 (2004). [CrossRef]
3. G. Mie, “Beiträge Zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Annalen der Physik, Vierte Folge 25, 377–445 (1908).
4. P. Debye, “Das Elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908).
5. H. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
6. M. Tomasko, L. Doose, M. Lemmon, P. Smith, and E. Wegryn, “Properties of dust in the Martian atmosphere from the Imager on Mars Pathfinder,” J. Geophys. Res. 104, 8987–9007 (1999). [CrossRef]
7. L. Rayleigh, “On the light from the sky, its polarization and color,” Philos. Mag. 41, 107–120, 274–279 (1929).
8. J. Aitken, “On some phenomena connected with cloudy condensation,” in Collected Scientific Papers of John Aitken, LL.D. F.R.S. (Cambridge, 1923), pp. 279–280.
9. S. Bishop, “The remarkable sunsets,” Nature 29, 259–260 (1884). [CrossRef]
10. D. Kim, M. Chin, H. Yu, T. F. Eck, A. Sinyuk, A. Smirnov, and B. N. Holben, “Dust optical properties over North Africa and Arabian Peninsula derived from the AERONET Dataset,” Atmos. Chem. Phys. 11, 10733–10741 (2011). [CrossRef]
11. J. Durward, “Blue colour of the sun and surrounding sky in a dust storm,” Q. J. R. Meteorol. Soc. 63, 54–64 (1937). [CrossRef]
12. H. Hogg, “Blue Sun,” JR Astron. Soc. Can. 44, 241–245 (1950).
14. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
15. P. Bouguer, Essai d’Optique, sur la gradation de la lumiere (Chez Claude Jombert, 1729).
16. H. Moosmüller and P. Arnott, “Particle optics in the Rayleigh regime,” J. Air Waste Manage. Assoc. 59, 1028–1031 (2009). [CrossRef]
17. H. Moosmüller, R. Chakrabarty, and W. Arnott, “Aerosol light absorption and its measurement: a review,” J. Quant. Spectrosc. Radiat. Transfer 110, 844–878 (2009). [CrossRef]
18. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Patterns in Mie scattering: evolution when normalized by the Rayleigh cross section,” Appl. Opt. 44, 7487–7493 (2005). [CrossRef]
19. M. I. Mishchenko, A. A. Lacis, B. E. Carlson, and L. D. Travis, “Nonsphericity of dust-tropospheric aerosols: implications for aerosol remote-sensing and climate modeling,” Geophys. Res. Lett. 22, 1077–1080 (1995). [CrossRef]
20. M. I. Mishchenko, L. D. Travis, R. A. Kahn, and R. A. West, “Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids,” J. Geophys. Res. 102, 16831–16847 (1997). [CrossRef]
21. L. P. Mie, plot simulator, www.philiplaven.com/mieplot.htm.
22. B. van der Pol and H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finite conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos Mag. 24, 141–176, 825–864 (1937).
23. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere I. Direct reflection and direct transmission,” J. Math. Phys. 10, 82–124 (1969). [CrossRef]
24. J. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988). [CrossRef]
25. E. Hovenac and J. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A , 9781–795 (1992). [CrossRef]
26. S. Hill and R. Brenner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. Barber and R. Chang, eds. (World Scientific, 1988), pp. 3–61.
27. G. Lothian, “Blue sun and moon,” Nature 168, 1086–1087 (1951). [CrossRef]
28. J. Bell III, D. Savransky, and M. Wolff, “Chromaticity of the Martian sky as observed by the Mars Exploration Rover Pancam instruments,” J. Geophys. Res. 111, E12S05 (2006).
29. P. Christensen, R. Morris, M. Lane, J. Bandfield, and M. Malin, “Global mapping of Martian hematite mineral deposits: remnants of water-driven processes on early Mars,” J. Geophys. Res. 106, 873–885 (2001).
30. M. Querry, “Optical constants,” Rep. CRDEC-CR-88009 (U.S. Army Chemical Research and Development Center, 1985).
31. M. Lane, R. Morris, and P. Christensen, “Spectral behavior of hematite at visible/near infrared and mid-infrared wavelengths,” in The Fifth International Conference on Mars, Pasadena, California, 1999.
32. V. Hamilton, H. McSween, and B. Hapke, “Mineralogy of Martian atmospheric dust inferred from thermal infrared spectra of aerosols,” J. Geophys. Res. 110, E12006 (2005). [CrossRef]
33. J. Pollack, D. Colburn, F. Flaser, R. Kahn, C. Carlston, and D. Pidek, “Properties and effects of dust particles suspended in the Martian atmosphere,” J. Geophys. Res. , 842929–2945 (1979). [CrossRef]
34. P. Chýlek, V. Srivastava, R. Pinnick, and R. Wang, “Scattering of electromagnetic waves by composite spherical particles: experiment and effective medium approximations,” Appl. Opt. 27, 2396–2404 (1988). [CrossRef]
35. J. Hansen and L. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974). [CrossRef]
36. H. Moosmüller, J. P. Engelbrecht, M. Skiba, G. Frey, R. K. Chakrabarty, and W. P. Arnott, “Single scattering Albedo of fine mineral dust aerosols controlled by iron concentration,” J. Geophys. Res.117, doi:10.1029/2011JD016909 (2012). [CrossRef]