Icarus 239 (2014) 85–95

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Icarus

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Spectral bluing induced by small particles under the Mie and Rayleighregimes

http://dx.doi.org/10.1016/j.icarus.2014.05.0420019-1035/� 2014 Elsevier Inc. All rights reserved.

E-mail address: abrown@seti.org

Adrian J. BrownSETI Institute, 189 Bernardo Ave, Mountain View, CA 94043, USA

a r t i c l e i n f o

Article history:Received 8 November 2013Revised 30 April 2014Accepted 29 May 2014Available online 6 June 2014

Keywords:Radiative transferMarsCometsIcesSatellites, surfaces

a b s t r a c t

Scattering by particles significantly smaller than the wavelength is an important physical process in theicy and rocky bodies in our Solar System and beyond. A number of observations of spectral bluing(referred to in those papers as ‘Rayleigh scattering’) on planetary surfaces and cometary comas have beenrecently reported, however, the necessary mathematical modeling of this phenomenon has not yetachieved maturity. This paper is a first step to this effect, by examining the effect of grain size and opticalindex on the albedo of small conservative and absorbing particles as a function of wavelength. The con-ditions necessary for maximization of spectral bluing effects in real-world situations are identified. Wefind that any sufficiently narrow size distribution of scattering particles will cause spectral bluing in somepart of the EM spectrum regardless of its optical index. We also investigate the effect of including adistribution of particle sizes.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Clark et al. (2008, 2012) recently used data from the Visual andInfrared Mapping Spectrometer (VIMS) instrument on the Cassinispacecraft currently orbiting Saturn to observe spectral bluing inthe visible part of the spectrum. They used the term ‘Rayleigh scat-tering’ to describe the observed increase in albedo with decreasingwavelength. In addition, Clark et al. (2010) used data from theM-cubed instrument on the Chandrayaan spacecraft to report‘Rayleigh scattering’ on the lunar surface. Their abstract suggesteda simple model for small particles based on Eqs. (5.13) and (5.14)of Hapke (1993) which is developed further herein. Brown et al.(2010) reported spectral bluing of spectra from the area aroundthe retreating martian north polar ice cap. Yang et al. (2009)reported observations of Comet 17P/Holmes with a strong negative(bluing) spectral slope that distinguished that comet from manyother types of small bodies that show neutral or red spectral curves(Jewitt and Meech, 1986). These observations of spectral bluing areall candidates for explanations using the models presented in thispaper.

The aim of this paper is to examine the mathematical develop-ment behind spectral bluing caused by small particles using Mieand Rayleigh scattering models and examine a range of hypothet-ical, but physically realizable situations that could be the cause ofthis phenomenon.

In this study, we explicitly model the single scattering albedo ofabsorbing particles small compared with the wavelength. Thisshould be distinguished from transmission models of atmosphericRayleigh scattering of the sky (Rayleigh, 1871b; Chandrasekhar,1960; Hannay, 2007; Pesic, 2008) and observations of non-absorb-ing molecular Rayleigh scattering (Young, 1982). We ignore theeffects of polarization in this paper and intend to return to thewavelength dependence of polarization in future work. Polariza-tion measurements of Solar System bodies displaying spectral blu-ing phenomena are important and significant (Chandrasekhar,1960; Hovenier, 1969; Coulson, 1989; Mishchenko et al., 2010),particularly when measurements of the polarization hemisphericalphase function through symmetry maps (Brown and Xie, 2012)and through a range of phase angles are available (Painter andDozier, 2004; Sun and Zhao, 2011).

In this study, in order to remain focused and to draw succinctconclusions, we adopt the following simplifications:

(Restriction a) EM spectrum range limitation. We consider primar-ily the visible to near infrared (VNIR: 0.25–2.5 lm) frequency range– this is not strictly necessary but enables us to address a reducedsearch space and remain relevant for a large range of astronomical,terrestrial (Goetz et al., 1985; Green et al., 1998; Brown et al., 2006),laboratory (Clark and Roush, 1984; Brown et al., 2008b) and plane-tary science reflectance instruments (Brown et al., 2004; Murchieet al., 2007; Green et al., 2011). It also allows us to apply already-developed spectral analysis techniques to the measurements(Clark et al., 1987; Brown, 2006) and this spectral range possesses

86 A.J. Brown / Icarus 239 (2014) 85–95

unique absorption bands critical for determining optical pathlengths that can be used to infer grain size (independent of the tech-nique discussed in this paper) – for water ice and CO2 ice, for exam-ple (Clark and Lucey, 1984; Brown et al., 2008a, 2010, 2012). Wepresent our predictive results for the VNIR range, but the extensionsto other parts of the EM spectrum should be rather direct.

(Restriction b) independent scattering. We consider only Mie andRayleigh single scattering (no multiple scattering), and we chooseto overlook the fact that these are only ideal mathematical descrip-tions of single scattering particles and do not address the issue ofclose packing of particles in regoliths, which will inevitably com-plicate the situation (Petrova et al., 2001; Mishchenko et al.,2006, 2007). We justify our approach as an exploratory one – ouraim here is to examine the question: ‘‘at what point does a particlebecome scattering, and at what point does if become absorbing,due to its size?’’. In order to answer this question, we are focusingour attention on the Rayleigh approximation of the scattering pro-cess, and ignoring close-packing effects.

(Restriction c) constant optical index. We assume in this studythat the optical index m = n + ij remains constant across thewavelength range we consider here. This is an admitted majorsimplification of real world conditions, and may introduce com-plications when comparing the model to physical situations,however, in order to isolate the effects of grain size on singlescattering albedo, we found it eminently desirable to vary ‘onlyone parameter at a time’. We justify the approach in the follow-ing manner: if the optical index of the target varies significantlyover the spectral bluing region (e.g. there is a vibration absorp-tion band) then the spectral bluing will be superimposed uponthis feature. In the worst case, the spectrum of the material willhave a broad absorption band that replicates the spectral slopewe consider here (e.g. the water ice absorption band at 3 lm).In this event, interpretations using the approach advocated herewould have to be conducted with great caution. When using theresults of this study to quantitatively estimate the bluing powerof a particular material, one might take the approach of usingthe most pessimistic estimates of m = n + ij in the region ofinterest (and as shown below, this may not equate to the largestvalues of n and j).

Dispersion effects. As a result of restriction (c) we are also ignor-ing the effects of dispersion that results when waves propagatethrough a medium according to Maxwell’s equations (Stratton,1941). Our neglect of dispersion can also be justified on thegrounds that normal dispersion in dielectrics and metals has theeffect of increasing the attenuation of a wave with increasing fre-quency. Dispersion therefore causes spectral reddening, which isnot the focus of this paper.

(Restriction d) particle shape considerations. Van de Hulst (1957)and Kerker (1969) in particular have presented models of ellipsoi-dal particles smaller than the wavelength. We address below onlythe effect on homogeneous spheres in this paper, and the effect ofparticle shape on albedo as a function of wavelength will be thetopic of future work.

Restrictions (a–d) are significant simplifications, but they areimportant to state as clearly as possible in order to understandthe restrictions on the conclusions drawn in this paper.

As shown below all isolated particles sufficiently small com-pared with the wavelength are good absorbers. If we decreasethe wavelength (holding all other factors constant) the absorptioneffects will decrease, causing spectral bluing.

The purpose of this paper is to identify the region where theeffects of spectral bluing process are maximized and investigatethe conditions under which the spectral bluing phenomenon mightbe observed. As we show below, a sufficiently narrow size distribu-tion of particles will show spectral bluing at some point in the EMspectrum, regardless of the optical index.

A small aside on terminology: Hapke notes on p. 73 (Hapke,1993) that ‘‘particles small compared with the wavelength areknown as Rayleigh scatterers’’ and on the following page that ‘‘smallabsorbing particles are called ‘Rayleigh absorbers’’’ and Clark et al.(2010) has followed this terminology. Bohren and Huffman(1983) acknowledge on p. 132 that Rayleigh did not addressabsorbing particles, however they ‘‘attach the name ‘Rayleigh’ tosmall particle scattering for convenience’’.

2. Theory

2.1. Rayleigh scattering approximation

When an incident plane electromagnetic wave is scattered by anobject smaller than the wavelength of the incident light, the objectscatterers the light approximately as an electric dipole. Two condi-tions must be met for the dipole approximation to be a good one:

(1) scatterer size �k, so the scatterer is in a homogenous EMfield, and

(2) scatterer size�k/m where m is the complex index of refrac-tion, so that incident light penetrates so quickly that the par-ticle’s field is set up in a short time compared to the incidentradiation (Hansen and Travis, 1974).

We start with the auxiliary field, D, of the Maxwell equations,which for a simple dielectric gas with permittivity e and relativedielectric constant er is defined as:

D ¼ eE ¼ ere0E ð1Þ

When an electric field is applied or present in the dielectric gas,we can imagine a spherical cell with radial coordinate r within thedielectric which experiences the direct field plus the inducedpolarization, P, which originates from other cells within the vol-ume. Using Gaussian (cgs) units,

DðrÞ ¼ EðrÞ þ 4pPðrÞ ð2Þ

Assuming the scatterer is a homogenous dielectric, the incidentlight sets up a polarization within the particle proportional to the elec-tric field, where the proportionality constant is called the polarizabil-ity, a of the dielectric, N is the density of molecules in the dielectric.PðrÞ ¼ NaEðrÞ ð3Þ

therefore using (1)–(3), e = 1 + Na. However, the polarization feltby imaginary cell within a dielectric is best approximated when wesubtract the ‘‘Lorentz average’’ of the ‘‘self-field’’ (4pP(r)/3) producedby the material within the imaginary spherical cell (Zangwill, 2013).

PðrÞ ¼ Na EðrÞ � 4pPðrÞ3

� �¼ 4pNa

1� 4pNa3

EðrÞ ð4Þ

The second equality comes from using (3).Substituting (4) into (2) and comparing to (1) we obtain the

Clausius–Mossotti relation (Mossotti, 1850; Clausius, 1867;Feynman et al., 1963)

Na ¼ 34p

er � 1er þ 2

ð5Þ

where N is the number density – the number of particles per unitvolume, for one particle, N ¼ 1

V, er is the relative dielectric constant.Taking the convention of representing the index of refraction asm = n � ij, and under the assumption of a relative magnetic perme-ability lr = 1, so that m ¼ ffiffiffiffiffiffielp �

ffiffiffiep

we get this form of the Lor-entz–Lorenz relation (Lorentz, 1880; Lorenz, 1880):

a ¼ 34p

m2 � 1m2 þ 2

� �V ¼ m2 � 1

m2 þ 2

� �a3 ð6Þ

A.J. Brown / Icarus 239 (2014) 85–95 87

where V is the volume and a is the radius of the sphere. We can usethe following formulas for the scattering and absorption cross sec-tion (van de Hulst, 1957):

Csca ¼8p3

k4jaj2 ð7Þ

Cabs ¼ 4pk � ReðiaÞ ð8Þ

plugging in (6) into (7) and (8) we get:

Csca ¼8p3

k4 m2 � 1m2 þ 2

��������

2

a6 ¼ 83

X4 m2 � 1m2 þ 2

��������2

a2p ð9Þ

Cabs ¼ 4pkImm2 � 1m2 þ 2

� �a3 ¼ �4XIm

m2 � 1m2 þ 2

� �pa2 ð10Þ

where a is the radius of the spheres, k = 2p/k, and X = ka is the sizefactor. We can convert the scattering cross sections into scatteringefficiencies, Qsca and Qabs, by dividing (9) and (10) by pa2.

We then compute the single scattering albedo of a Mie orRayleigh particle using:

w ¼ Q sca

Q ext¼ Qsca

Q sca þ Q absð11Þ

2.2. Mie scattering

Fig. 1 is an adaptation of Fig. 22 from Hapke’s treatise on spaceweathering and the effect of nanophase particles on the albedo ofrocky objects (Hapke, 2001). Fig. 1 plots the absorption and scatter-ing coefficients of a Mie target (calculated using the Mie scatteringcode of Wiscombe, 1980) with m = 3.0 + 3.0i. We have noted in theleft of Fig. 1, where X < 1 and the absorption coefficient is decreas-ing linearly with X, the scattering coefficient is decreasing accord-ing to X4. This results in very low albedo for experimentsreplicating this physical situation.

In the central part of Fig. 1, we have marked an area of bluingwhere the single scattering albedo is increasing most rapidly with

Fig. 1. Adapted from Hapke (2001). We have plotted the Mie scattering and absorptiongeneral regions of most effective spectral bluing, Rayleigh absorbing and ‘Large Size Scatcan still occur in the ‘Rayleigh absorbing’ region of the plot.

increasing size factor – it is this part of the graph that is the subjectof this paper. As we show below, this region is present for all par-ticles, no matter what their optical index, m.

2.3. Spectral bluing effects with wavelength for small particles

Spectral bluing associated with small particles can be dividedaccording to the conservative or absorbing nature of the scatteringparticle.

(a) Conservative scattering. One way in which small particles canscatter efficiently is if they are truly conservative scatterers, i.e.j = 0. As discussed by Bohren and Huffman on p. 132 (Bohren andHuffman, 1983), this is the situation studied by Rayleigh (1871a)and results in a scattering cross section varying with k�4 as in Eq.(7). Thus, a particulate medium of small (X� 1) particles withj = 0 will be blue. However, it should be noted that an optically thickmedium of particles with j = 0 is white, not blue, because photons ofall wavelengths eventually escape from such a medium. But if themedium is thin enough that it is not optically thick, it becomes blue,even though j = 0 for all wavelengths, because the optical thicknessis different for each wavelength. This is the situation in the Earth’satmosphere (Bohren and Fraser, 1985).

(b) Absorbing scatterers. In Fig. 1, we noted a region where it waspossible for the scattering coefficient to be higher than the absorp-tion coefficient, where the size factor X < 1, this region was marked‘spectral bluing’. We now show that particles where Qabs < Qsca andX < 1 have the capacity to produce higher albedo signatures whichmay appear as bluing of a continuum spectrum.

3. Numerical results

3.1. Mathematical approach to determining and optimizing spectralbluing and absorbing conditions

(a) Rayleigh approximation to single scattering albedo. Substitut-ing Eqs. (9) and (10) into Eq. (11), we arrive at:

coefficients for a particle with constant optical index m = 3.0 + 3.0i. We note thetering’ based on the results presented in this paper. Note that weak spectral bluing

Fig. 2. The Rayleigh approximation of w, the single scattering albedo (see Eq. (6)) plotted for varying size factor X on the range axis and absorption coefficient, j, on thedomain axis. Note both axes are logarithmic. The left of each plot (small grains) is black, showing regions where small particles are absorbing. Regions of increased value of won the center of the plot track the regions of the search space of X and j where bluing is likely to occur. In the right of the plot (the ‘large grain size’ scattering region) isconsistently high in the Rayleigh approximation (which does not hold in this part of the plot, see discussion in Figs. 4 and 5). Four plots are presented, showing how theseresults change for variations in the real index of refraction, n = 1.1, 3, 5 and 10. See text for further discussion.

1 For interpretation of color in Figs. 3, 4 and 8, the reader is referred to the webversion of this article.

88 A.J. Brown / Icarus 239 (2014) 85–95

wðX;n;jÞ ¼ X3ððn� 1Þ2 þ j2Þððnþ 1Þ2 þ j2ÞX3ððn2 þ j2Þ2 � 2ðn2 � j2Þ þ 1Þ þ 9jn

ð12Þ

expanding this expression using a partial fraction decomposition,we get:

wðX;n;jÞ ¼ 1� 9njX3ððn2 þ j2Þ2 � 2ðn2 � j2Þ þ 1Þ þ 9jn

ð13Þ

Fig. 2 shows the Rayleigh approximation for the singlescattering albedo w, according to Eqs. (12) and (13). The singlescattering albedo is plotted against size factor X and absorptioncoefficient, j. The left of each plot (smaller grains) is black,showing regions where small particles are absorbing. Regions ofincreasing value of w on the center of the plot track the regionsof the search space of X and j where bluing is likely to occur.In the right of the plot, the Rayleigh approximation is notexpected to hold (e.g. for X > 0.4, see below), and so the constantvalues of w(=1) in the plot should be disregarded. Plots for n = 1.1,3, 5 and 10 are presented to show how variations in the realindex of refraction affect the Rayleigh approximation of w. Forincreasing n, the region of bluing moves further to the left, andthe convex bend (at X = 1, j = 0.1 for n = 1.1) moves ‘upwards’(to higher j) with increasing n.

The most important conclusion from Fig. 2 is that spectral blu-ing occurs for particles of any optical index m, even for largeabsorption index j. The real index n and absorption index j weaklyinfluence the critical size of the particles where they act as Ray-leigh absorbers or Rayleigh scatterers.

We can obtain the third order Taylor’s series expansion at X = 0for Eq. (13) as follows:

wðX ! 0;n;jÞ � X3ððn� 1Þ2 þ j2Þððnþ 1Þ2 þ j2Þ9jn

þ Oðx4Þ ð14Þ

(b) Rayleigh approximation to dw/dX. Differentiating Eq. (12)once (using the quotient rule) with respect to X, we arrive at thefollowing relation:

dwdXðX;n;jÞ ¼ X2ð27jnððn� 1Þ2 þ j2Þððnþ 1Þ2 þ j2ÞÞ

ðX3ððn2 þ j2Þ2 � 2ðn2 � j2Þ þ 1Þ þ 9jnÞ2 ð15Þ

Fig. 3 shows the first derivative of the Rayleigh approximation(dw/dX) according to Eq. (15) plotted for varying size factor X onthe range axis and absorption coefficient, j, on the domain axis.Plots for n = 1.1, 3, 5 and 10 are presented, showing how theseresults change for variations in the real index of refraction.

We can interpret the differential graphs as showing us regionsof the n, j space that will provide the best opportunities for max-imal bluing effects. Those areas where the derivative values aregreatest will give the strongest bluing effect.

(c) Small size factor approximations. When X� 1, an approxima-tion to Eq. (15) is obtained by setting the X6 term in the denomina-tor to zero:

dwdXðX;n;jÞ� X2ð27jn5þð54j3�54jÞn3þð27j5þ54j3þ27jÞnÞ

ð18jn5þð36j3�36jÞn3þð18j5þ36j3þ18jÞnÞX3þ81j2n2

ð16Þ

we can also derive a Taylor’s series for dw/dX at X = 0, obtaining

dwdXðX ! 0;n;jÞ � X2ððn� 1Þ2 þ j2Þððnþ 1Þ2 þ j2Þ

3jnþ OðX5Þ ð17Þ

Eqs. (16) and (17) approximate the behavior of the dw/dX at X = 0.The strength of dw/dX grows with increasing n from n = 1.1 to

n = 10 (note the scale of each plot in Fig. 3 is different from theother color1 scales). dw/dX increases in each plot with decreasing

Fig. 3. The first derivative of the Rayleigh approximation to dw/dX (Eq. (11)) plotted for varying size factor X on the range axis and absorption coefficient, j, on the domainaxis. Note both axes are logarithmic. The regions of increased value of the first derivative track the regions of the search space of X and j where bluing is likely to occur. Fourplots are presented, showing how these results change for variations in the real index of refraction, n = 1.1, 3, 5 and 10. Note the scale of each image is different. A cheat sheetfor converting size factor to grain size in VNIR is provided on the lop left. See text for further discussion.

A.J. Brown / Icarus 239 (2014) 85–95 89

j, for j < 0.11. However, dw/dX increases above a certain value of j,dependent on the value of n. For n = 1.1, this is around j = 0.1.

(d) Mie single scattering albedo calculations. It should be remem-bered that Rayleigh approximation is only an approximation to thelight scattering process, and thus the above results are not beapplicable outside a restricted range of size factors. Penndorf(1962) reminds us that the Rayleigh approximation is onlyexpected to hold for nX < 1, and Cox et al. (2002) report goodmatches of the Rayleigh approximation to experiment fornX � 1.21, but not beyond that point. Kim et al. (1996) showed thatRayleigh scattering is a good approximation for Mie scattering fork < 0.4X. Kerker et al. (1978) and Ku and Felske (1984) have alsopointed out the limitations of the Rayleigh scattering approxima-tion. For this reason we also calculate the Mie absorption and scat-tering coefficients in this paper, with the understanding thatspherical scatterers are also likely to be an over-simplification ofthe real world physical situation (e.g. on planetary surfaces).

Fig. 4 presents the calculated Mie single scattering albedo usingEq. (11) and the Mie scattering code of Wiscombe (1980) forabsorption coefficient j versus size factor X. In Fig. 5 we presenta comparison with the Rayleigh approximation by calculating theabsolute difference between the Rayleigh (Fig. 2) and Mie (Fig. 4)solutions. This primarily highlights the difference between thetwo solutions due to Mie resonances at X � 1.

It is of particular interest to this study that there is very littledifference in the most effective bluing region (the colored regionof Fig. 3, which corresponds the blue region in Fig. 4, which isthe slope up from 0 from left to right). Small particles that areabsorbing due to their size in the Rayleigh approximation are alsoabsorbing in the Mie theory, and although there is a decrease in themagnitude of the albedos around 0.1 < k < 1 (particularly apparentin the top two images for n = 1.1 and n = 3, and also in the bottomleft for n = 5) the region of increasing albedo (the spectral bluingregion) is almost identical. For this reason, we feel confident ininterpreting our Rayleigh derivative results in Fig. 3 as the regionsof most effective spectral bluing, which is the key finding of thispaper.

(e) Effects of a size distribution. To examine the effects of a vari-ation in the size of the Mie particles, we have used a modifiedGamma function (Deirmendjian, 1969) to specify the dispersionof particle radii. The following formula (we have set c = 1 andb = a/rm in Deirmendijian’s original formula) allows us to specifya mode radius (most common radius in the polydispersion) andspecify the spread of the modified gamma distribution using justtwo active parameters: rm the mode radius and a the spreadparameter.

nðrÞ ¼ Crae�arrm ð18Þ

Fig. 4. Mie theory calculations of single scattering albedo for varying index of absorption j and size factor, X. Four instances of real index n = 1.1, 3, 5 and 10 are presented. Seetext for further discussion.

Fig. 5. The calculated absolute difference between the Rayleigh approximation and Mie single scattering albedos for varying index of absorption j and size factor X. Note thescales of each plot are different. Most of the observed difference is caused by the Mie resonances around X � 1. See text for further discussion.

90 A.J. Brown / Icarus 239 (2014) 85–95

where C ¼ 1Cðaþ1Þ ð a

rmÞaþ1.

It should be pointed out that this size distribution is symmetri-cal, unlike the size distributions advocated by Hansen (1971),which makes more them suitable for the testing carried out here.

By discretizing the size distribution into 100 Riemannian binsand applying Gaussian quadrature (Stroud and Secrest, 1966) at10 points within each bin, we numerically integrated across the

polydispersion, between maxima and minima specified as beingwith a tolerance level (1�10) of zero. The scattering and extinc-tion coefficients within each Gaussian division were normalizedby the Gaussian weighting coefficients and the normalizationdue to the size distribution, and summed, to arrive at Csca andCext. We then used Eq. (11) to calculate the single scatteringalbedo, w.

20.0

17.5

15.0

12.5

10.0

7.5

2.5

5.0

0.00.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

a b

Fig. 6. The effect of a polydispersion on the spectral bluing of small particles. (a) The Mie single scattering albedo for a polydispersion with optical properties n = 1.1, k = 0.01,mode radius rm = 0.2. (b) The effect of the a (spread) parameter on the modified gamma parameter. The critical size dispersion at a = 5 is shown in blue. See text for furtherdiscussion. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. An estimate of the most effective grain diameter (‘sweet spot’) for bluing atparticular wavelengths in the VNIR wavelength range. Note: this relationship onlyholds for optical indexes in a restricted range 1.1 < n < 3 and 0.03 < j < 3. Becausethe size factor range 0.5 < X < 1.2 is most effective for bluing (see Fig. 3, particularlythe top two panels) in this range of optical indexes, we have highlighted this rangeof X in red and yellow. To estimate the grain diameter range that is most effective atany wavelength in the VNIR, one may take the limits of this band for any particulark in lm. See text for further discussion. (For interpretation of the references to colorin this figure legend, the reader is referred to the web version of this article.)

A.J. Brown / Icarus 239 (2014) 85–95 91

Fig. 6 shows the effect of the polydispersion on the spectral blu-ing. Fig. 6a shows the Mie single scattering albedo for the polydi-spersion as a whole. For easy comparison with Figs. 3–5, Fig. 6ashows the size factor X on the domain (x-axis) in the same manneras the earlier figures. The a (spread) parameter is shown on therange (y-axis). The effect of the a (spread) parameter on the sizedistribution is demonstrated in Fig. 6b. Large values of the param-eter lead to tighter spread of the size distribution. Fig. 6a uses amode radius of 0.2 lm, and optical properties of n = 1.1, k = 0.01.This means that the calculations can be directly compared withthe monomodal results shown in the top left diagram in Fig. 4,by drawing a horizontal line out from k = 0.01.

The size distribution data show that there is very little differ-ence in the spectral bluing effect for size distributions wherea > 5. For more disperse distributions (bottom half of Fig. 6a) thereis a peak and spectral reddening begins to appear for large size fac-tors X (corresponding to short wavelengths, e.g. in the UV). For thisreason, we conclude that if one makes the assumption that apolydispersion behaves similarly to the modified gamma distribu-tion we have used here, then (approximating a normal distributionunapologetically so we can infer an acceptable, and more widelyunderstood, standard distribution) for the mode radius we haveexamined, a grain radius of 0.2 ± 0.1 lm (standard distribution�0.1) still gives spectral bluing effects in the VNIR range similarto the monomodal sizes we used earlier. Beyond this range, forbroader size distributions, reddening effects will occur in the UV.

3.2. Predictions and conditions for achieving spectral bluing in theVNIR

With the preceding discussion, we are now in a position tomake some predictions regarding the appearance of bluing in theVNIR as reported by Clark et al. (2008, 2010, 2012), Brown andCalvin (2010) and Yang et al. (2009). We interpret the observedbluing reported in these three cases by trying to approximatelyfit the bluing range in each case. We do not attempt to numericallyfit the data, but give a qualitative prediction for the grain size rangethat would best fit the observations given the simplifications wehave already discussed.

In order to make a reasonable prediction of the most likely grainsize responsible for spectral bluing, we make a further simplifyingassumption that the visible optical index of the small particles is inthe range 1.1 < n < 3 and 0.03 < j < 3, because in this region thebluing is limited to a relatively restricted band of size factors (from0.5 < X < 1.2, see Fig. 3). This covers a large portion of naturallyoccurring materials.

We have created a plot showing the span of the band of0.5 < X < 1.2 for a range of grain diameters, 0.1 < D < 1 (lm) andwavelengths 0.2 < k < 2.5 (lm), this is shown as Fig. 7. For thisrestricted range of optical indexes, Fig. 7 provides an estimate atthe most range of most effective grain diameters to produce bluing.For example, for particle diameters of 0.5 lm, the most effectivebluing will occur in the 1–2.5 lm range. In this region, the size fac-tor X of 0.5 lm particles varies from 1.2 to 0.5.

It bears repeating that according to Restriction (a) of this paper,we have chosen to show the most effective bluing regions in theVNIR (0.25–2.5 lm) range. The spectral bluing effect naturally con-tinues beyond this range, and our earlier analysis (e.g. Figs. 2–6)was wavelength-range agnostic because we used the size parame-ter X.

We will now use the results in Fig. 7 to interpret three fourspectral bluing measurements reported to date in the literature.

The six examples of spectral bluing we shall examine appear inFig. 8.

(a) Epimetheus. As seen in Fig. 8a, Clark et al. (2008) reported arestricted blue peak for Epimetheus (from 0.4 to 0.7 lm) for datacollected from the Cassini VIMS instrument. Based on these obser-vations, and with the assumptions stated above, the grain diameterobserved in this region of Epimetheus would best fit 0.2 lm, andmust lie in the 0.05–0.4 lm range.

(b) Dione. As seen in Fig. 8b, Clark et al. (2008) observed a widerincrease in reflectance in the visible range at Dione (from 0.4 to1.0 lm). Based on these observations, and with the assumptions

a b

c d

e f

Fig. 8. A Rogues’ Gallery of spectral bluing examples that are used to estimate grain sizes of small particles in this paper. (a) VIMS/Cassini spectrum of Epimetheus, adaptedfrom Clark et al. (2008, Fig. 10). (b) VIMS/Cassini spectrum of the dark regions of Dione, adapted from Fig. 11 of Clark et al. (2008). (c) VIMS/Cassini spectra of Iapetus and theCassini Division of Saturn’s Rings (Clark et al., 2012). (d) A set of continuum removed spectra taken by M-cubed/Chandrayaan spectrum of the lunar far side showing varyingdegrees of spectral bluing, adapted from Fig. 2 of Clark et al. (2010). (e) CRISM/MRO spectra from a mosaic of Utopia Planitia in the north polar region of Mars adapted fromFig. 2 of Brown and Calvin (2010). (f) SpeX/IRTF telescopic observation of the coma of Comet 17P/Holmes adapted from Fig. 2 of Yang et al. (2009). See text for interpretationsof spectral bluing implications in each case.

92 A.J. Brown / Icarus 239 (2014) 85–95

stated above, the grain diameter of the small scatterers on Dionewould best fit 0.3 lm, and must lie within the 0.1–0.5 lm range.

(c) Lunar regolith. As seen in Fig. 8c, Clark et al. (2010) reported asteep increase in reflectance in the visible range (starting at 1 lmand increasing most steeply at 0.5 lm) using spectra from theMoon Mineralogy Mapper (M-cubed) on Chandrayaan data. Basedon the range of the observed spectral bluing effect, and making theassumptions discussed above, we can bracket the most likely graindiameter of the scatterers between 0.1 and 0.5 lm, with best fitsaround 0.3 lm.

(d) Iapetus and Saturn’s Rings (Cassini Division). As seen in Fig. 8d,Clark et al. (2012) observed similar ‘bumps’ in the visible part ofthe VIMS spectra of Iapetus and the Cassini Division portion of Sat-urn’s Ring. The bluing extends from 0.4 to 0.7 lm, and thereforeour best estimate is that this feature is caused by scatterers witha mean size of 0.2 lm, and their sizes must lie within the 0.05–0.4 lm envelope. The Ring spectrum was sourced from a study ofRing particles by Cuzzi et al. (2009). That paper explores the grainsizes of ring particles in detail and points out that there is a largerange of scattering particles sizes present. In the blue spectrumof the Cassini Division, in Fig. 8c, the Rayleigh scattering peakallows us to surmise that when the spectrum was taken, the lightdetected into the instrument was scattered by �0.2 lm size parti-cles. They may have been in close proximity, or may have coated,

larger Ring particles. Further investigation of the occurrence rateof this spectral bluing within Saturn’s Rings will be of great valueand may lead to a better understanding of their formation, perma-nence and modern aggregation processes.

(e) Martian regolith. Brown and Calvin (2010) reported a numberof VNIR spectra from the Compact Reconnaissance Imaging Spec-trometer for Mars (CRISM) instrument on the Mars ReconnaissanceOrbiter (MRO) of martian soil/regolith (observed after the winterpolar CO2 ice cap had recently disappeared) that showed a gradualincrease in reflectance from 1 to 2 lm. Based on the assumptionslisted above, a grain diameter of 0.5 lm would best fit the data,and the scattering particles must lie within the range of 0.3–0.8 lm in order to produce the observed effect.

(f) Comet 17P/Holmes. Yang et al. (2009) reported observationsof Comet 17P/Holmes with a strong negative (bluing) spectralslope from 1 to 2.4 lm. As in case (e) above, a grain diameter of0.5 lm would best fit the data, and the scattering particles mustlie within the range of 0.3–0.8 lm in order to produce the observedeffect.

4. Discussion

Most effective ‘bluing’ regions. One might be tempted ask –what is the best use of Figs. 2 and 3 showing w and dw/dX for k

A.J. Brown / Icarus 239 (2014) 85–95 93

versus X? The answer is that they show regions where bluing willbe most effective. It should be noted that weak regions of bluingmight be possible in regions where the dw/dX is small, howeverthe most effective spectral bluing regions are shown as maximain Fig. 3. It should be remembered that these particles are likelyobserved with other ‘non-bluing’ agents (within the field of view,or intimately associated with the bluing grains) so the effect ofbluing will be diminished, thus making it important that the scat-tering agents possess strong bluing capabilities.

Non-uniqueness of solution. Dellago and Horvath (1993)showed that when a size distribution is introduced, the scatteringproblem is an ill posed Fredholm equation and we encounter non-unique solutions of the problem. This is an inevitable part of theinverse problem of remote sensing, and it should always beremembered that the solutions attained using the methodadvocated are not unique. In fact, all we can say for sure that obser-vations Rayleigh scattering and spectral bluing does imply thepresence of small particles (X < 1), and this alone can yield crtiticalinformation.

‘‘There is an unavoidable indeterminacy in the calculation of obser-vable results, the theory enabling us to calculate in general only theprobability of our obtaining a particular result when we make anobservation’’ (Dirac, 1947).

Although Dirac was speaking of quantum mechanics, this quoteholds true for the approach outlined here regarding particle sizedetermination from spectral bluing.

The approach to the spectral bluing problem advocated here,while not unique by mathematical standards, is an addition tothe available arsenal of attacks on the problem of small particlescattering. We have presented what is effectively a ‘likelihood’argument when predicting the size of scattering particles, by iso-lating the regions of ‘most effective bluing’. Future laboratory test-ing and further theoretical development along these lines isneeded to test the true usefulness of this approach.

Use as initial conditions in regularization techniques. Not-withstanding the non-uniqueness problem, many studies existwhere the inverse problem of size distribution determination usingspectroscopy is examined using regularization techniques (Mülleret al., 1998; Liu et al., 1999; Doicu et al., 2010; Mroczka andSzczuczynski, 2013; Osterloh et al., 2013). These inversion tech-niques rely on reasonable initial conditions, and the techniqueadvocated here could be used in such a manner.

Rayleigh scattering in regolith. It should also be borne in mindthat the spectral bluing caused by small particles apparently incontact with icy bodies (on saturnian satellites and ring particles)calls for a different approach to the inverse method normally usedin atmospheric sciences (with widely separated and independentscatterers). The spectral bluing observations by Cassini andM-cubed in particular (Fig. 8a–d) are constrained to a certainwavelength range and the effect is not apparent across the wholespectrum, and thus is possibly subdued by multiple scattering fromlarger particles. Thus a different approach is called for whenaddressing the size distribution of these particles. Future labora-tory studies, already started by Clark et al. (2012), will be of greatuse for better understanding of the differences of closely packedparticles on Rayleigh scattering.

Related (but different) studies. We wish to mention a coupleof related studies of small particle scattering and mention howthey differ from this study.

(a) Blue suns and moons. As mentioned above and discussed byWilson (1951), Pesic (2008) and Adler and Lock, 2002, the phenom-enon of blue suns and blue moons is explained well as a transmis-sion effect that is related to variations in the Mie scattering cross

section of a small particle due to Mie resonances, and the particlessizes we have studied here (with X < 1) are too small to achieve thiseffect. A simple diffraction-based model for blue moons and bluesuns has been suggested by Pesic (2008), however he consideredthe effects of transmission through a cloudy atmosphere and theeffects of decreasing Qext over various ranges of size factor X.Instead, in this paper we have considered the spectral bluing effectby concentrating on the variation of the single scattering albedo ofthe small particles.

(b) Rayleigh scattering near terrestrial clouds. ‘Rayleigh scattering’of reflected sunlight near clouds has also been of some concern inthe terrestrial environments. Most suggested explanations for thiseffect invoke molecular Rayleigh scattering above cloud tops ingaps between clouds might be responsible for this effect(Marshak et al., 2008; Wen et al., 2008).

A similar, but angular-dependent, coloration of the spectrumoccurs when light is reflected in the specular direction from a ran-domly rough surface, as discussed in Dashtdar and Tavassoly (2009).

(c) Interstellar extinction. Light traveling to us from distant starsis reddened because of the increased absorption of blue light scat-tered by interstellar dust grains (Cardelli et al., 1989). Becauseinterstellar extinction causes spectral reddening, it is due to a pre-ponderance of larger grains than discussed in this paper.

(d) Rayleigh scattering in the Cosmic Microwave Background(CMB). The effect of Rayleigh scattering of photons from neutralhydrogen in the Cosmic Microwave Background may contribute2–3% of the Cosmic Microwave Background, in addition to Thomp-son scattering from free electrons (Takahara and Sasaki, 1991; Yuet al., 2001). Analysis of the effect of Rayleigh scattering by futureCMB missions suggests that it may be reveal more information onthe era immediately following recombination (Lewis, 2013) andthus may lead to greater understanding of the Sunyaev–Zel’dovicheffect (Sunyaev and Zeldovich, 1970; Sunyaev and Zeldovich,1980). Safari et al. (2012a, 2012b) have recently calculated the rel-ativistic scattering cross sections and linear and circular polariza-tion cross sections using a multipole expansion of the scatteringterms in the Dirac equation.

(e) Scattering by small absorbing particles. Many studies havebeen made of the variable absorption properties of small particlesusing Mie theory. Deirmendjian et al. (1961), Plass (1966) andKattawar and Plass (1967) provided the first extensive investiga-tions into scattering by highly absorbing spheres using Mie scatter-ing codes, and emphasized the variations in Qsca and Qext as afunction of size factor X, with particular attention on the patternsof Mie resonances. Faxvog and Roessler (1978) conducted a similarstudy of absorbing Mie spheres and found the grain diameter forgreatest reduction in visibility for aerosols of carbon(m = 1.96 + 0.66i), iron (m = 3.51 + 3.95i), silica (m = 1.55) andwater (m = 1.33).

Mishchenko et al. (2011) studied the effect of a ‘dusting’ of sub-microscopic particles on wavelength size particles using a numer-ically exact T-matrix formulation. They calculated scattering effi-ciencies as a function of scattering angle and found that thesmall particles are less significant than a major asphericity of thescattering object. None of these studies touched on the potentialfor spectral bluing for particles small with the wavelength.

5. Conclusions

We have investigated the theoretical dependence of Rayleighand Mie single scattering albedo of small particles as a functionof grain size and size distribution for a wide range of the opticalindex m. Our findings can be summarized thus:

94 A.J. Brown / Icarus 239 (2014) 85–95

� Spectral bluing effects are predicted by Mie and Rayleigh theoryand that the single scattering albedo of any isolated particlewith X < 1 is strongly determined by its size and weakly by itsoptical index, m.� All sufficiently narrow size distribution of spherical scatterers

will cause spectral bluing at some point in the EM spectrumregardless of their optical index m.� Spectral bluing effects occur for particles of any optical index m,

even for large absorption index j. The real index n and absorp-tion index j weakly influence the critical size of the particleswhere they act as bluing agents.� The region of spectral bluing (the region where the Rayleigh

approximation to w (Eq. (13)) is increasing rapidly with increas-ing size factor) varies weakly with optical index m.� We have presented a mathematical relation to describe the con-

ditions of maximal effectiveness of spectral bluing (Eq. (16)). Inorder to achieve a bluing effect in the VNIR part of the spectrum,for a large range of optical index found in natural settings(1.1 < n < 3 and 0.03 < j < 3) size factors from 0.5 < X < 1.2 pro-vide the most effective bluing capability (Fig. 3).� We have applied the results of this investigation to several cases

of spectral bluing known to us at this time – on Dione and Epi-methius, the Moon, Mars, and Comet 17P/Holmes. Using thedata presented in Fig. 7, future observations of spectral bluingcan be associated with a size range of scattering particles withgreater confidence.

Acknowledgments

This work was partly supported by two grants (NNX11AP23Gand NNX13AN21G) from the NASA Planetary Geology and Geo-physics program run by Dr. Mike Kelley. We thank Roger Clarkfor bringing this fascinating phenomenon to our attention.

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