Physical Properties of Cometary Dust from Light Scattering ... · 578 Comets II 2. OBSERVATIONAL...

Kolokolova et al.: Physical Properties of Cometary Dust 577

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Physical Properties of Cometary Dust fromLight Scattering and Thermal Emission

Ludmilla KolokolovaUniversity of Florida

Martha S. HannerJet Propulsion Laboratory/California Institute of Technology

Anny-Chantal Levasseur-RegourdAéronomie CNRS-IPSL, Université Paris

Bo Å. S. GustafsonUniversity of Florida

This chapter explores how physical properties of cometary dust (size distribution, compo-sition, and grain structure) can be obtained from characteristics of the electromagnetic radia-tion that the dust scatters and emits. We summarize results of angular and spectral observationsof brightness and polarization in continuum as well as thermal emission studies. We reviewmethods to calculate light scattering starting with solutions to Maxwell’s equations as well asapproximations and specific techniques used for the interpretation of cometary data. Laboratoryexperiments on light scattering and their results are also reviewed. We discuss constraints onphysical properties of cometary dust based on the results of theoretical and experimental simula-tions. At the present, optical and thermal infrared observations equally support two models ofcometary dust: (1) irregular polydisperse particles with a predominance of submicrometer parti-cles, or (2) porous aggregates of submicrometer particles. In both models the dust should containsilicates and some absorbing material. Comparison with the results obtained by other than light-scattering methods can provide further constraints.

1. INTRODUCTION

The main technique to reveal properties of cometary dustfrom groundbased observations is to study characteristicsof the light it scatters and emits. Such a study can includedynamical properties of the dust particles, presuming thatthe spatial distribution of the dust resulted from the dustinteraction with gravitation and radiation. However, the pri-mary way is to study the scattered light or emitted thermalinfrared radiation to search for signatures typical of spe-cific particles. The main focus of this chapter is a reviewof methods in the interpretation and the extraction of thedata on the size distribution, composition, and structure ofcometary dust (hereafter referred to as physical properties)from the observed angular and spectral characteristics ofthe brightness and polarization of scattered and emitted light(hereafter referred to as observational characteristics). Wewill not detail here observational techniques (for this, see,e.g., the review by Jockers, 1999), but will mainly concen-trate on the results of the observations and methods of theirinterpretation.

We review how observational characteristics such as thedust color, albedo, polarization, etc., can be estimated fromvisual, near-infrared, and mid-infrared observations and

which regularities have been found in the variation of theseparameters with phase angle, heliocentric distance andwithin the coma (section 2). Then we describe theoreticallight-scattering techniques developed to solve the inverseproblem, whereby dust physical characteristics are extractedfrom the characteristics of the light it scatters and emits(section 3). Since the concentration of the dust particles incometary coma at cometocentric distances correspondingto the resolvable scale in the observations is sufficiently low,the particles can be considered to be independent scatter-ers. This means that the intensity of the light (and its otherStokes parameters, see below) scattered by a collection ofdust particles is equal to the sums of the intensity (or re-spective Stokes parameters) of the light scattered by all theparticles individually. We therefore consider the methodsdeveloped to describe the light scattering by single particles,indicating how light-scattering characteristics depend on theparticle shape, size, composition, and structure. In section 3we also show how the application of these techniques tothe observational data leads to the current views on thephysical properties of cometary dust. Section 4 comparesour knowledge about cometary dust obtained using light-scattering methods with results obtained using other tech-niques and outlines future research.

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2. OBSERVATIONAL DATA

2.1. Brightness Characteristics ofScattered Light

The most straightforward way to characterize cometarydust is to study how much light it scatters and absorbs andhow this depends on the geometry of observations, mainlydetermined by the phase angle, α (the Sun–comet–Earthangle). The efficiency of scattering and absorption is usu-ally characterized by the albedo and geometrical character-ization of the scattering by the angular scattering function.These characteristics, as well as spectral dependence of thescattered light, expressed through the dust colors, are con-sidered below.

2.1.1. Albedo. Single-scattering albedo is defined inthe most general way as the ratio of the energy scattered inall directions to the total energy removed from the incidentbeam by an isolated particle (van de Hulst, 1957; Hanneret al., 1981). This definition includes all components (dif-fracted, refracted, reflected) of the scattered radiation andcan be applied to small particles where the diffracted ra-diation is spread over a wide angular range and cannotreadily be separated. In practice, this definition is not veryuseful when analyzing cometary dust, because it requiresknowledge of the radiation scattered at all directions. Gehrzand Ney (1992) compared the scattered energy to the ab-sorbed energy reradiated in the infrared to derive an albedoat the phase angle of observation. This method was usedby a number of observers, and the results are summarizedin Fig. 1. The albedo estimates thus obtained for dust indifferent comets should be compared at the same phaseangle since the radiation scattered in the direction of obser-vation depends on the relative positions of the Sun, Earth,and comet. Using such an approach, Mason et al. (2001)found the albedo of the dust in Comet C/1995 O1 Hale-Boppto be roughly 50% higher than that of P/Halley at α ~ 40°.A few maps of albedo obtained so far by combining visi-ble light and thermal infrared images demonstrate increas-ing albedo with the distance from the nucleus (Hammel etal., 1987; Hayward et al., 1988).

The geometric albedo of a particle, Ap, is defined as theratio of the energy scattered at α = 0° to that scattered by awhite Lambert disk of the same geometric cross section(Hanner et al., 1981). Since comets are rarely observed atα = 0°, it is convenient to define Ap(α) as the product ofthe geometric albedo and the normalized scattering func-tion at the angle α. Hanner and Newburn (1989) presenteda plot of Ap(α) in the J bandpass (1.2 µm) for 10 comets.The total dust cross section within the field of view wasdetermined by fitting a dust emission model to the thermalspectral energy distribution, and then applying the totalcross section to the scattered intensity to derive an averagealbedo. The resulting Ap are typically very low, close to0.025 at α = 35°–80° and about 0.05 at α near zero. Thereis some indication that Ap is higher for comets beyond 3 AU(see Fig. 3 in Hanner and Newburn, 1989). The albedo was

observed to increase by 50% in Comet P/Halley’s coma dur-ing episodes of strong jet activity (Tokunaga et al., 1986).

Difference in albedo may result from different composi-tion, particle size, shape, or internal structure. For example,Mason et al. (2001) found that high albedo of the dust inComet Hale-Bopp is consistent with the domination of smallparticles in the grain population (see also section 3.4.1).There may be several components of the dust, with differingalbedos and temperatures, and the average albedo may notrepresent the actual albedo of any of the components. How-ever, the low average albedo rules out a large population ofcold, bright grains that contribute to the scattered light butnot to the thermal emission.

2.1.2. Angular scattering function. The angular scat-tering function indicates how the intensity of the scatteredlight is changing with phase angle. Since a comet can beobserved at only one phase angle on a given date, the an-gular scattering function has to be acquired by observing acomet over time as the Sun–Earth–comet geometry changes.Thus this function of individual cometary dust particles isdifficult to determine, because the observed brightness de-pends not only on the physical properties of the dust, butalso on its amount, and the total cross section of dust inthe coma contributing to the scattered light intensity doesnot remain constant over time. Two methods have been usedto normalize the observed intensity.

One method is to assume that the ratio of the dust to gasproduction rates remains constant over time, and to normal-ize the scattered light intensity to the gas production rate.Millis et al. (1982) derived the angular scattering functionfor the dusty Comet P/Stephan-Oterma by normalizing thescattered intensity to the C2 production rate. The scattering

Fig. 1. Dependence of albedo on the phase angle. The data arefrom (0) Mason et al. (2001); (1) Mason et al. (1998); (2)–(7) Gehrz and Ney (1992); (8) Grynko and Jockers (2003). Thesolid line represents the least-squares fit to the data for CometsC/1975 V1 West and C/1980 Y1 Bradfield and is interpolated tosmoothly connect to the backscattering data for Comet P/Stephan-Oterma (Millis et al., 1982) normalized at the angle α = 30°.


function was a factor of 2 higher at 3°–4° than at 30°, cor-responding to a slope of ~0.02 mag/°. Meech and Jewitt(1987) determined a linear slope of 0.02–0.035 mag/° forfour comets observed at α = 0°–25°. They saw no evidenceof an opposition surge larger than 20% in P/Halley within1.4°–9°. More detailed data by Schleicher et al. (1998) showan evident curvature of the scattering function for P/Halleythat can be represented by a quadratic fit.

Another method is to compare the measured scatteredlight to the measured thermal emission from the same vol-ume in the coma, with the assumption that the emittingproperties of the dust remain constant. This method wasdescribed above as the means for extracting an albedo forthe dust. The method has been applied to a number of com-ets (see Fig. 1) and yields a relatively flat scattering func-tion from 35° to 80° (Tokunaga et al., 1986; Hanner andNewburn, 1989; Gehrz and Ney, 1992). Two comets havebeen observed at large phase angles 120°< α < 150° usingthis method, and they display strong forward scattering (Neyand Merrill, 1976; Ney, 1982; Gehrz and Ney, 1992). Thuswe can characterize a typical angular scattering function forcometary dust as possessing a distinct forward-scatteringsurge, a rather gentle backscattering peak, and a flat shapeat medium phase angles (Fig. 1).

2.1.3. Color. The dust color indicates trends in thewavelength dependence of the light scattered by the dust.Traditionally the color of cometary dust was determinedthrough measurements of the comet magnitude m in twodifferent continuum filters, e.g., blue (B, ~0.4–0.45 µm) andred (R, ~0.64–0.68 µm), and was expressed as CB–R = mblue–mred. This color was a unitless characteristic expressed asthe logarithm of the ratio of intensities in two filters. Al-though this definition is still used, spectrophotometry ofcomets resulted in the definition of color as the spectral gra-dient of reflectivity, usually measured in % per 0.1 µm withan indication of the range of wavelength it was measured in.

The determination of cometary dust colors requires useof continuum bands that are truly free of gas contamina-tion [see discussion on gas-contamination influence on thecolors in A’Hearn et al. (1995)]. This tends to make colorsobtained using filters less trustworthy than those obtainedusing spectrophotometry to target gas-free spectral regions.The exception is near-infrared colors that are considered tobe free from gas contamination and thus more reliable, al-though thermal emission from the warm dust will contributeto the K (2.2 µm) bandpass for comets within 1 AU of theSun. We summarize the data for B–R color in the visibleand J–H and H–K color in the near-infrared in Fig. 2.

The scattered light is generally redder than the Sun; thereflectivity gradient decreases with wavelength from 5–18%per 0.1 µm at wavelengths 0.35–0.65 µm to 0–2% per0.1 µm at 1.6–2.2 µm (Jewitt and Meech, 1986). Hartmannet al. (1982) and Hartmann and Cruikshank (1984) foundthat the near-infrared dust colors depend on heliocentricdistance. However, data by Jewitt and Meech (1986) andTokunaga et al. (1986) demonstrate no heliocentric depen-dence. Hanner and Newburn (1989) noted that only the H–

K color was less red in comets observed at R > 3 AU, whilethe J–H color showed no trend with heliocentric distance.The B–R color shows no dependence on heliocentric dis-tance although Schleicher and Osip (2002) found that thecolor m0.4845–m0.3650 for Comet Hyakutake (1996 B2) gotredder with increasing heliocentric distance within 0.6–1.9 AU. No color dependence on phase angle was found inthe visible or near-infrared.

Decrease in the near-infrared colors was recorded at peri-ods of enhanced comet activity in Comet P/Halley (Toku-naga et al., 1986; Morris and Hanner, 1993) A more redcolor in the visible was associated with strong jet activityin Comet P/Halley (Hoban et al., 1989), but the spiral struc-tures in Comet Hale-Bopp had less red color (Jockers et al.,1999). Less red and even blue color was associated withoutburst in Comet C/1999 S4 LINEAR during its disruption(Bonev et al., 2002). Although no regular dependence onthe field of view was found, a smooth change in color wasobserved in the central part of the coma of some comets,including Hale-Bopp (Jockers et al., 1999; Laffont et al.,1999; Kolokolova et al., 2001a).

2.2. Polarization from Cometary Dust

The light scattered by particles is usually polarized, i.e.,its electromagnetic wave has a preferential plane of oscil-lation. The degree of linear polarization (P, hereafter calledpolarization) is defined as

P = (I⊥ – I||)/(I⊥ + I||)

where I⊥ and I|| are the intensity components perpendicularand parallel to the scattering plane. For randomly orientedparticles the electromagnetic wave predominantly oscillates

Fig. 2. Color as a function of heliocentric distance: J–H (+), H–K ( ), and B–R (×) colors are by Jewitt and Meech (1986); B–Rcolors ( ) are by Kolokolova et al. (1997) ( for Comet P/Halley).Color is measured in ∆m/∆λ (mag/µm) and the solar color is sub-tracted. The data cover the range of the phase angles 0°–110°.

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either perpendicular (by convention positive polarization) orparallel (by convention negative polarization) to the scatter-ing plane.

The convenience of polarization is that it is already anormalized characteristic of the scattered light, whereas thebrightness, I = I⊥ + I||, depends upon the distances to theSun and observer, and the spatial distribution of the dustparticles. Polarization variations within a coma relate tochanges in the dust physical properties, and different po-larization observed for different comets or in features (jets,shells, etc.) indicates a diversity of dust particles. For a giventype of cometary dust particles, the polarization mainly de-pends on the phase angle α and wavelength λ.

2.2.1. Angular dependence. The changing geometry ofthe comet and observer with respect to the Sun defines apolarization phase curve. Small phase angles can be ex-plored for distant comets, while α > 90° can only be reachedat R < 1 AU. Most of the available polarization observationshave been performed within the heliocentric distances 0.5–2 AU using groundbased instruments, although large phaseangles were studied using the Solar and Heliospheric Ob-servatory (SOHO) C3 coronagraph (Jockers et al., 2002).

The data, which represent an average value of the polar-ization on the projected coma, are obtained either by aper-ture polarimetry (e.g., Dollfus et al., 1988; Kiselev andVelichko, 1999) or deduced from the integrated flux of thepolarized brightness images (e.g., Renard et al., 1996;Kiselev et al., 2000; Hadamcik and Levasseur-Regourd,2003). The (real or virtual) diaphragm is centered on thecenter of brightness, which corresponds to the nucleus. Itsaperture, in terms of projected distance on the cometarycoma, needs to be large enough to include the coma fea-tures that may alter the global polarization. It is usuallyfound that, once the aperture is sufficiently large, the re-sulting polarization of the comet does not change with in-creasing aperture.

The phase dependence of polarization, which has beendocumented so far in the range 0.3°–122°, is smooth andsimilar to that of atmosphereless solar system bodies. Allcomets show a shallow branch of negative polarization atthe backscattering region, first observed by Kieselev andChernova (1978), that inverts to positive polarization at α0 ~21° with a slope at inversion h ~ 0.2–0.4%/°, and a posi-tive branch with a broad maximum near 90°–100°. The datamay be fitted by a typically fifth-order polynomial or trigo-nometric function, e.g., as suggested by Lumme and Muino-nen (1993), P(α) ~ (sinα)a(cosα/2)bsin(α – α0). These func-tions cannot be used for extrapolation but only within thephase angle range where well-distributed data points areavailable.

While the minimum polarization is typically –2% (Mukaiet al., 1991; Chernova et al., 1993; Levasseur-Regourd et al.,1996), a significant dispersion is noticed for α > 30°–40°.Once the data are separated in different wavelength ranges,the dispersion on the positive branch is reduced, and Pλ(α)for a variety of comets can be compared (see Levasseur-Regourd et al., 1996, and references therein). Comets tend to

divide into three classes corresponding to different maximain polarization (Fig. 3): (1) comets with a low maximum,about 10–15% depending on the wavelength; (2) cometswith a higher maximum, about 25–30% depending on thewavelength; and (3) Comet C/1995 O1 Hale-Bopp, whosepolarization is distinctively higher, although it was not ob-served for α > 48°.

Numerous observations obtained by Dollfus et al. (1988)for P/Halley showed that the polarization at a given phaseangle and wavelength does not vary with heliocentric dis-tance. Some transient increases in polarization were foundcorrelated with cometary outbursts. Similarly, the polariza-tion of Comet Hale-Bopp obtained at small phase angles(large heliocentric distances) was consistent with that ob-tained closer to the Sun, but increased after outbursts events(Ganesh et al., 1998; Manset and Bastien, 2000). Observa-tions of Comet C/1999 S4 LINEAR during its near-periheliondisruption (Kiselev et al., 2002; Hadamcik and Levasseur-Regourd, 2003) indicated an increase in polarization of about4%, a value comparable with what had been noticed dur-ing the P/Halley outbursts.

2.2.2. Spectral dependence. The data have been re-trieved in the ultraviolet, visible, and near-infrared domainsto avoid any contribution from the thermal emission. Nar-row-band cometary filters are now available at 0.3449,0.4453, 0.5259, and 0.7133 µm (Farnham et al., 2000).However, observers often need to make a trade-off betweennarrow filters that remove gaseous emissions and wider fil-ters that improve the signal-to-noise ratio. It is highly advis-able to analyze spectra of the observed comet and estimatethe contribution of some faint molecular lines to allow forthe depolarizing effect of molecular emissions when cal-culating the dust polarization (Le Borgne and Crovisier,

Fig. 3. Polarization in the narrow-band red filter vs. phase angle:(×) comets with low maximum in polarization, (+) comets withhigher maximum, ( ) Comet C/1995 O1 Hale-Bopp, ( ) CometC/1999 S4 LINEAR at disruption.


1987; Dollfus and Suchail, 1987; Kiselev et al., 2001).Spectropolarimetric cometary data remain rare, and so fartheir resolution is not better than the resolution providedby narrowband cometary filters (Myers and Nordsieck,1984).

It was noticed for P/Halley (Dollfus et al., 1988) andlater confirmed for other comets (Chernova et al., 1993;also Kolokolova and Jockers, 1997, and references therein)that the polarization usually increases with increasing wave-length, at least in the visible domain, and the increase growsfor greater values of the polarization.

If polarization data are available for two wavelengths,the spectral gradient of polarization is defined as polarimet-ric color, ∆P/∆λ = [P(λ2) – P(λ1)]/[λ2 – λ1].

The polarimetric color changes with the phase angle(Chernova et al., 1993; Kolokolova and Jockers, 1997) fromzero and even negative values to gradually increasing posi-tive values at α > 50°. Figure 4 summarizes systematicmultiwavelength observations of polarization, performed forComets Halley and Hale-Bopp. Within α = 25°–60° the po-larization increases with the wavelength in the visibledomain; the gradient seems to decrease in the near-infra-red [Fig. 4; see also Hadamcik and Levasseur-Regourd(2003)]. For α ~ 50°, the polarimetric color is about 9%/µm for Halley and 14%/µm for Hale-Bopp (Levasseur-Regourd and Hadamcik, 2003), which agrees with the value11%/µm at 45° obtained by Kiselev and Velichko (1999).The polarimetric gradient for Comet Hale-Bopp is higherthan for other comets but its sign and phase-angle trend aretypical for comets, whereas asteroids and other atmosphere-less bodies usually have negative polarimetric color (e.g.,Mukai et al., 1997). So far a negative polarimetric color at

large phase angles was observed only for Comet P/Giacobini-Zinner (Kiselev et al., 2000). Polarimetric color of variablesign was observed for Comet C/1999 S4 LINEAR duringits disruption (Kiselev et al., 2002).

2.2.3. Variations within the coma. A coma whose op-tical thickness exceeds 0.1 was found so far only for CometHale-Bopp and only at distances closer than 1000 km fromthe nucleus (Fernández, 2002). Thus, except for possiblythe very innermost coma where multiple scattering mighttake place, a change in the polarization with the distancefrom the nucleus indicates an evolution in the physical prop-erties of the particles ejected from the nucleus. Polarizationimages of P/Halley (Eaton et al., 1988; Sen et al., 1990),C/1990 K1 Levy (Renard et al., 1992), 109P/Swift-Tuttle(Eaton et al., 1995), 47P/Ashbrook-Jackson (Renard et al.,1996), and C/1995 O1 Hale-Bopp (Hadamcik et al., 1999;Jockers et al., 1999; Kolokolova et al., 2001a); variable-aperture polarimetry of P/Halley (Dollfus et al., 1988); andhigh-resolution in situ observations with the Optical ProbeExperiment (OPE) onboard the Giotto spacecraft (Renardet al., 1996; Levasseur-Regourd et al., 1999) revealed a re-gion in the innermost coma characterized by a lower polari-zation. Comet Hale-Bopp could be observed at phase anglesranging from about 7° to 48°. The low polarization regionin Comet Hale-Bopp has been monitored within this phase-angle range (Hadamcik and Levasseur-Regourd, 2003). Itshowed highly negative values of polarization at small phaseangles, e.g., –5% at α = 8°, whereas a typical value shouldbe –0.5%. The low negative values of polarization, noticedat the cometocentric distances less than 2000 km, could notbe explained by multiple scattering but is most likely dueto the physical properties of the local dust grains.

Fig. 4. Wavelength dependence of polarization for (a) Comet Hale-Bopp [data from Furusho et al. (1999), Ganesh et al. (1998),Hadamcik et al. (1999), Jockers et al. (1999), Jones and Gehrz (2000), Kiselev and Velichko (1999), Manset and Bastien (2000)] and(b) Comet Halley [data from Bastien et al. (1986), Brooke et al. (1987), Dollfus and Suchail (1987), Le Borgne et al. (1987), Mukaiet al. (1987), Sen et al. (1990)] for phase angles 40° (bottom curve) and 50°.

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A further proof of different physical properties of thedust near the nucleus is its negative polarimetric color ob-served in situ (OPE/Giotto) (Levasseur-Regourd et al., 1999)and a smaller polarimetric color in the innermost coma sus-pected for Comet Hale-Bopp (Jockers et al., 1999). Somecomets do not show lower polarization [e.g., C/1989 X1(Eaton et al., 1992), C/2001 A2 (Rosenbush et al., 2002)]in the innermost coma, possibly hidden by highly polarizeddust jets or, as Kiselev et al. (2001) suggest, resulted fromstrong gas contamination.

Aperture polarimetry with an offset of the center, as wellas cuts through polarization images, may be used to studythe variation of the polarization with the cometocentric dis-tance. The increase within the jets and decrease in the inner-most coma are clearly visible on such graphs, and somedifferences may be noticed between the sunward and anti-sunward side. Although the data are line-of-sight integrated,they emphasize the temporal evolution in the physical prop-erties of the dust particles ejected from the nucleus. Exceptfor the above-mentioned features, the polarization decreasedgradually within 5000–8000 km from the nucleus for C/1996 B2 Hyakutake and C/1996 Q1 Tabur (Kolokolova et al.,2001a) and increased for P/Halley (Levasseur-Regourd et al.,1999) and Hale-Bopp (Kolokolova et al., 2001a), accom-panied by similar trends in the B–R color.

The polarization images often reveal elongated fan-shaped structures with polarization higher than in the sur-rounding coma. They generally correspond to bright jet-likefeatures on the brightness images. This effect has beennoticed for quite a few comets, e.g., 1P/Halley (Eaton et al.,1988), C/1990 K1 Levy (Renard et al., 1992), 109P/Swift-Tuttle (Eaton et al., 1995), 47P/Ashbrook-Jackson (Renardet al., 1996), or C/1996 B2 Hyakutake (Tozzi et al., 1997).The case of the bright and active Comet Hale-Bopp is mostdocumented; straight jets at large heliocentric distances areseen to evolve in arcs or shells around perihelion passage(see, e.g., Hadamcik et al., 1999; Jockers et al., 1999;Furusho et al., 1999). Such conspicuous features could beproduced by an alignment of elongated particles or byfreshly ejected dust particles different from the particles inthe regular coma.

2.2.4. Plane of polarization and circular polarization.The plane of polarization is in almost all cases perpendicularor parallel to the scattering plane, indicating randomly ori-ented particles. However, some variations by a few degreesmay be noticed with changing aperture size (Manset andBastien, 2000). Deviations in the polarization plane were no-ticed for Comet P/Halley near the inversion angle (Beskrov-naya et al., 1987); they were mapped by Dollfus and Suchail(1987). Changes in the polarization plane often accompanythe polarization variations at the outburst activity and wereregistered in Comets 29P/Schwassmann-Wachmann 1(Kiselev and Chernova, 1979), P/Halley (Dollfus et al.,1988), C/1990 K1 Levy (Rosenbush et al., 1994), and C/2001 A2 LINEAR (Rosenbush et al., 2002). For CometsHalley, Hale-Bopp, and S4 LINEAR, the circular polariza-tion was estimated and found to be nonzero but below 2%

(Dollfus and Suchail, 1987; Metz and Haefner, 1987; Rosen-bush et al., 1999; Manset and Bastien, 2000; Rosenbush andShakhovskoj, 2002).

The existence of the faint circular polarization, variationsin the plane of polarization, and nonzero polarization ob-served at forward-scattering direction during stellar occul-tations (Rosenbush et al., 1994) may be indications ofaligned elongated or optically anisotropic particles in comet-ary atmospheres.

2.3. Dust Thermal Emission

Thermal data provide two types of results that are sen-sitive to the physical properties of the particles: the ther-mal spectral energy distribution (SED) and the detection ofspectral features (e.g., the silicate feature at 10 µm and 16–30 µm). In addition, the ratio of the thermal energy to thescattered energy provides a measure of albedo, as describedpreviously.

Thermal emission from the dust in the coma results fromabsorption of the solar radiation followed by its reemission.The thermal radiation from a single grain depends upon itstemperature, T, and wavelength-dependent emissivity, ε

F(λ) = (r/∆)2ε(r,λ)πB(λ,T) (1)

where r is some specific dimension of the particle, e.g., itsradius, ∆ the geocentric distance, and B(λ,T) the Planckfunction for grain of temperature T. The observed SED fromthe ensemble of grains in a coma having differing tempera-tures and wavelength-dependent emissivities generally issimilar to (but broader than) a blackbody curve (Fig. 5). TheSED can be characterized by a single temperature (the colortemperature, Tc) over a defined wavelength interval, but itis important to remember that the color temperature is notthe physical temperature of the grains.

The physical temperature of a particle in the solar radia-tion field depends strongly on the latent heat of sublima-tion and sublimation rate of any material that may be sub-limating, but once the sublimation has ceased and thermalequilibrium is established, the temperature depends on thebalance between the solar energy absorbed at visual wave-lengths and the energy radiated in the infrared. For a particleof arbitrary shape we may write the energy balance equation

R–2 ∫ Cabs(r,m,λ)S(λ)dλ = ζ ∫ πB(λ,T)Cabs(r,m,λ)dλ (2)

where S(λ) is the solar flux at 1 AU, R is the heliocentricdistance in AU, and the factor ζ is the ratio of the total ra-diating area to the exposed area. This ratio is equal to 4 forspheres and any convex particle averaged over random ori-entations (van de Hulst, 1957, section 8.41). The crucialparameter in equation (2) is Cabs(r,m,λ), the absorption crosssection, which depends on grain size, morphology, and opti-cal constants. By Kirchoff’s law, Cabs(r,m,λ) is equal to theemissivity, ε(r,λ), times the emitting area. We assume thatthe coma is optically thin (no attenuation of sunlight and no


heating by other grains) and that sublimation is negligible.The observed color temperature can be compared with theSED predicted from equations (1)–(2) for grains of varioussizes and optical properties in order to infer the physicalcharacteristics of the cometary dust.

The 3–20-µm thermal SED has been observed for manycomets over the past three decades, with improving pre-cision. P/Halley was monitored regularly at R < 2.8 AU(Tokunaga et al., 1986, 1988; Gehrz and Ney, 1992) andHale-Bopp was observed from the ground and from Earthorbit at 0.9–4.9 AU (Mason et al., 2001; Grün et al., 2001;Hanner et al., 1999; Wooden et al., 1999; Hayward et al.,2000). The 5–18-µm color temperature is typically 5–25%higher than a perfect blackbody at the same heliocentricdistance that can be calculated as T = 280 * R–0.5.

Comets display a heliocentric dependence, T ~ R–0.5; inthe case of P/Halley the 8–20-µm color temperature variedas T = 315.5 * R–0.5 (Tokunaga et al., 1988). The cometswith the strongest dust emission, such as P/Halley and Hale-Bopp, display a higher color temperature at 3.5–8 µm thanat 8–20 µm (Tokunaga et al., 1988; Williams et al., 1997).

Thermal emission observations in the far-infrared andsubmillimeter spectral regions can potentially provide in-formation about the larger particles in the coma (see sec-tion 3.1). Far-infrared observations at λ < 200 µm have beenacquired by the InfraRed Astronomical Satellite (IRAS),Cosmic Background Explorer (COBE), and Infrared SpaceObservatory (ISO) satellites. The Diffuse InfraRed Back-ground Experiment (DIRBE) instrument on COBE meas-ured the SED from 3.5 to 100 µm for three comets (Lisseet al., 1994, 1998). Among these, only Comet Levy (C/1990K1) had a color temperature 10% higher than the black-

body temperature and a drop in the average grain emissiv-ity at 60 and 100 µm, indicating that the flux contributionat 60 and 100 µm was primarily from particle sizes smallerthan the wavelength (see section 3.1). Continuum emissionat submillimeter wavelengths has been measured for sev-eral comets including Hale-Bopp (Jewitt and Luu, 1990,1992; Jewitt and Matthews, 1997, 1999), indicating thepresence of millimeter-sized particles.

In addition to the smoothly varying infrared continuumemission, broad spectral features attributed to small silicategrains are seen in some comets near 10 µm (Fig. 5) and18 µm. The strength of the features depends upon grain sizeand temperature (Hanner et al., 1987). The 16–45-µm spec-tra of Hale-Bopp recorded by the Short Wavelength Spec-trometer (SWS) onboard the ISO satellite (Crovisier et al.,1997) displayed five distinct spectral peaks that correspondto laboratory features of crystalline Mg-olivine (forsterite)(see Hanner and Bradley, 2004). Not all comets display astrong 10-µm silicate feature (Hanner et al., 1994; Hannerand Bradley, 2004). In particular, the feature tends to beweak (~20% above the continuum) or absent entirely inJupiter-family comets (e.g., Hanner et al., 1996; Li andGreenberg, 1998a).

2.4. Correlations Among Scattering andEmitting Observational Characteristics

It has been well established that the observable scatter-ing parameters of cometary dust tend to correlate. Thesecorrelations can provide insight into the relative importanceof particle size, composition, and structure on the observ-able scattering properties.

We have already noted that comets divide into someclasses, based upon their maximum polarization (Fig. 3).The high Pmax comets are mostly comets with a strong dustcontinuum, while the low Pmax comets are comets with aweak dust continuum (Chernova et al., 1993). Comets ex-hibiting higher polarization generally display a strongersilicate feature (Levasseur-Regourd et al., 1996), indicatingthat small silicate particles have a major influence on Pmax(see section 3.1). The strength of the silicate feature alsocorrelates with higher color temperature and stronger fluxat 3–5 µm (Gehrz and Ney, 1992), both indicators of hot,submicrometer-sized absorbing grains (see section 3.1.1).

The fact that the B–R, J–H, and H–K colors do not cor-relate, e.g., show different tendencies with the heliocentricdistance or with the distance from the nucleus, supports theidea that the changes in the colors cannot be explained bysimple change of particle size or smooth change in theoptical constants. A smooth simultaneous change in B–Rcolor and polarization was observed in the innermost comaof some comets (Kolokolova et al., 2001a), indicating asmooth change in some dust properties that occurs as dustis moving out of the nucleus.

Hanner (2002, 2003) summarized the correlations andcompared the scattering properties in Comets P/Halley andHale-Bopp: The dust in Hale-Bopp displayed higher po-

Fig. 5. Spectral energy distribution for Comet Hale-Bopp (Wil-liams et al., 1997). The left curve is the solar spectrum, the rightcurve is the continuum fit to the sum of a 5800 K blackbody com-ponent due to scattered solar radiation and a 475 K blackbody dueto thermal emission. The heavy solid line shows the silicate fea-ture observed with the HIFOGS (High-efficiency Infrared FaintObject Grating Spectrometer). Although Comet Hale-Bopp wasnot a typical comet, the figure provides an illustration of the maincharacteristics of cometary dust thermal emission.

584 Comets II

larization at comparable α, redder polarimetric color, higheralbedo, stronger silicate feature, higher infrared color tem-perature compared to the blackbody temperature, and higher3–5-µm thermal emission.

2.5. Summary

The observational characteristics of cometary dust areextremely variable and depend on phase angle, heliocen-tric distance, and position within the coma, especially as-sociated with comet activity (jets, fans, etc.). The mostcomplex characteristic is the brightness that depends on thespatial distribution of dust particles, and thus to a largeextent is affected by temporary variations in dust produc-tion rate. Parameters, formed as ratios of brightness suchas color, polarization, and albedo, are more suitable forstudying physical properties of the dust grains. However,measured using aperture instruments, they demonstrate av-erage characteristics related to a mixture of particles of avariety of properties and can vary due to comet activity.Coma images obtained in narrowband continuum filters pro-vide the most adequate source of information about prop-erties of cometary dust.

The general regularities found in observational charac-teristics of cometary dust that may be used as a basis forstudying physical properties of the dust particles are listedlater in this chapter (see Table 2). Variability of these char-acteristics within the coma, correlations between them, andexceptions to the listed rules are also an interesting subjectfor light-scattering modeling.

3. LIGHT-SCATTERING THEORETICALAND EXPERIMENTAL METHODS

Study of cometary dust using its interaction with solarradiation is a typical example of remote sensing that usu-ally is associated with the scattering inverse problem. Thesought quantities define a set of particle parameters suchas size, shape, and composition (refractive index) from thescattering/emission data; the set can include the parametersof more than one type of particle. The rigorous way to solvethe inverse problem would be:

• to measure the characteristics of the scattered light (in-tensity, I; degree of linear polarization, P; position ofthe plane of polarization, defined by the angle θ; de-gree of circular polarization, V) at a given wavelength,λ, and given phase angle, α;

• to calculate the Stokes vector I = (I, Q, U, V) of thescattered light whose components are related to theobserved characteristics as I = I, I/)UQ(P 22 += ,tan(2θ) = Q/U, V = V/I;

• from the Stokes vectors of the scattered and incidentlight, I and I0, determine elements of the scatteringmatrix F through (I, Q, U, V) = F/(kR)2 * (I0, Q0, U0,V0), where R is the distance between the detector andthe scatterer and k is the wave number in empty spaceequal to k = 2π/λ.

The scattering matrix F is defined by the particle prop-erties, thus the inverse problem can be solved if the ele-ments of the scattering matrix, obtained from observations,can be also found from a theory that describes scatteringby particles, i.e., most rigorously, from Maxwell’s equa-tions. To determine all 16 elements of the matrix F one mustobtain four independent combinations of vectors I0 and I,but since sunlight is incoherent and essentially unpolarized,this is usually not possible from astronomical observations.In practice, one therefore fits only the two upper-left ele-ments, or equivalently the intensity and polarization. Whilethe intensity or brightness is plagued by sensitivity to varia-tions in the particle number density, the polarization is a ratioof intensities and is therefore independent of the amountof dust. These observational constraints make it even moreimportant to fit all available observations and to extend theobservations to as broad a range of phase angles as possibleand wavelengths as well as thermal emission spectra. Thisstill cannot guarantee the uniqueness of the solution, espe-cially as the dominating grains in a mixture of grain typescan vary strongly both with wavelength and with scatter-ing angle. Therefore some authors emphasize the impor-tance in finding a solution that also adheres to limitationsset by, for example, cosmic abundances (Greenberg andHage, 1990) and constraints from the dynamics of the par-ticles (Gustafson, 1994). However, in this chapter we focuson constraints from optical and infrared observations.

For electromagnetic waves propagating in a homoge-neous medium, characterized by its complex refractive in-dex m = n + iκ, Maxwell’s equations can be reduced to

curl H = ikm2E(3)

curl E = –ikH

Since the magnetic vector H can be expressed throughthe electrical vector E, equation (3) can be further reducedto one so-called Helmholtz equation

∇2E + k2E = 0 (4)

The electromagnetic vector E is related to the Stokes pa-rameters through the values of its two complex perpendicu-lar amplitudes E|| and E⊥ and their conjugates as (van deHulst, 1957; Bohren and Huffman, 1983)

I = E|| |E || * + E⊥ E⊥*Q = E|| E || * – E⊥ E⊥*U = E|| E⊥* + E⊥ E|| *V = i (E|| E⊥* – E⊥ E|| *)

In principal, a complete solution to the problem can beachieved if we consider equation (4) outside the particle,where the electromagnetic field is a superposition of inci-dent and scattered fields, and inside the particle (internalfield), and apply the boundary conditions. The boundaryconditions (van de Hulst, 1957, section 9.13) mean that anytangential and normal components of E are continuous


across the two materials at the particle boundary. Thus tosolve the problem we need to know the shape of the bound-ary, i.e., the particle shape. The particle shape means amathematically sharp boundary between a particle interiorand empty space. We therefore tend to distinguish betweenparticle shape and structure (surface and internal). Avail-able solutions are specific to classes of particle shapes, e.g.,spheres, cylinders, and spheroids. These are usually forhomogeneous and isotropic internal structures, althoughsolutions for some anisotropic or inhomogeneous structureshave been derived for some specific shapes.

3.1. Light Scattering by HomogeneousSpherical Particles (Mie Theory)

Mie theory [attributed to Mie (1908); credit is also usu-ally given to earlier works by Clebsch, Lorentz, Debye, andothers (see Kerker, 1969)] accurately predicts the scatteringby any homogeneous and isotropic sphere. It solves equa-tion (4) in spherical coordinates (�, θ, ϕ) using the separa-tion of variables technique, i.e., presenting the electromag-netic wave E as

E = E(�,θ,ϕ)e = Φ(ϕ)Θ(θ)R(�)e

where vector e indicates the direction of oscillations. Thisgives E in the form of a sum of products of trigonometricand spherical Bessel functions and associated Legendrepolynomials with the size parameter x = kr (where r is theparticle radius) and refractive index of the particle in theargument. In practice, one has to take care with the numeri-cal code in order to not degrade the accuracy for very smallor very large particles. Detailed discussion on Mie solutioncan be found in van de Hulst (1957) and Bohren and Huff-man (1983). The latter book also contains a simple but effi-cient Mie code. A modern code is available from M. Mish-chenko at ftp://ftp.giss.nasa.gov/pub/crmim/spher.f.

Even when the shape and structure of the particle is assimple as that of a homogeneous sphere, the complexity ofthe solution does not allow the scattering properties of aparticle to be expressed through a simple analytical func-tion of the size and refractive index. The numerically de-rived solutions are also generally found to not be uniqueto one combination of particle size and refractive index. Theproblem of uniqueness is more acute if the observationscover only a limited range of phase angles. Thus the inverseproblem is often solved indirectly using analysis of thequalitative light-scattering behavior and incorporating otherempirical constraints on particle properties.

Use of Mie theory has helped exclude spherical andhomogeneous particles as the main component of cometarydust. Even if some specific size distribution and refractiveindex could fit some observations (e.g., polarization) atsome wavelengths (see examples in Mukai et al., 1987), thesame characteristics of spherical particles could not providea reasonable fit over a broad range of wavelengths and toother observational parameters, e.g., scattering function,

color, or temperature. However, the calculations for spherescan be used in a qualitative manner to narrow the range ofcometary grain parameters, as illustrated by the use of three-dimensional or color-contour graphics of the type shown inFig. 6 (see also numerous examples in Hansen and Travis,1974; Mishchenko and Travis, 1994; Mishchenko et al.,2002). From Fig. 6 one can see that light scattering fromcometary dust is not dominated by particles much smallerthan the optical wavelengths, since for such particles polari-zation is always positive with very high maximum polariza-tion at α = 90°. Also, cometary dust cannot be representedby an ensemble of monodisperse spherical or quasisphericalparticles of medium size (1 < x < 20) since their light-scat-tering characteristics experience the oscillating behaviorseen even in Fig. 6 (although the oscillations are smoothedby the size distribution). Note also that for spherical par-ticles, strong color dependence on the phase angle is seenfor all sizes and the polarimetric color is mainly negative.

The contour graphics and a more quantitative approachbased on statistical multifactor analysis (Kolokolova et al.,1997) can be used to estimate influence of dust propertieson observational characteristics. As a result of such ananalysis, the reason for the trends observed within the comaand correlations between light-scattering parameters can befound. For example, such an approach (Kolokolova et al.,2001a) showed that the correlation between color and po-larization observed in the innermost coma of Comets Tabur,Hyakutake, and Hale-Bopp cannot be a result of changingparticle size, but is more likely an indication of changingcomposition.

For certain applications, we might expect that arbitrarilyshaped, randomly oriented particles can be approximatedby equal-volume, equal-projected-area, or, in the case ofconvex particles, equal-surface-area spheres. They appearappropriate when some specific term dominates the scat-tering. For example, we might expect scattering in the for-ward domain (large phase angles) to be dominated by dif-fraction, which is strongly dependent on the particle sizesince only the particle cross section enters calculations ofthe diffraction pattern (van de Hulst, 1957, section 3.3). De-pending on the absorption of the particle material at someparticle sizes, the transmitted light starts reducing the mag-nitude of the forward-scattering peak but does not stronglyaffect the angular distribution of the intensity. In particu-lar, the angle of the first minimum in intensity counted fromthe direction of backscattering seems to be a good size in-dicator even for aggregated particles (Zerull et al., 1993).

Scattering by particles of size x << 1 is proportional tothe square of their polarizability (see section 3.2) and there-fore to their volume squared (van de Hulst, 1957, section6.11) so that equal volume particles in random orientationscatter identically. For the other extreme of large convexparticles in the geometric optics regime van de Hulst (1957,section 8.42) showed that the scattering caused by reflec-tion from randomly oriented particles is identical with thescattering by a sphere of the same material and surface area.He also showed that such particles have the same geometri-

586 Comets II

Fig. 6. Top panel: Intensity and polarization of scattered light vs. phase angle and effective size parameter xeff for spheres of therefractive index m = 1.55 + i0.005. The particle size distribution has the form n(x) = x–3 and is determined by its cross-sectional-area-weighted mean size parameter xeff, and the width of the size distribution defined by the effective variance veff = 0.05 (for details seeMishchenko and Travis, 1994). Bottom panel: Same for color and polarimetric color but instead of size parameter the effective size ofparticle varies at the fixed wavelengths of 0.45 and 0.65 µm. Darker colors indicate smaller values.


cal cross section as the sphere. The approximations workbetter the closer the particle shape is to spherical. For ex-ample, an error in the absorption cross section is less than5% for a spheroid of the axes ratio a : b = 1.4 regardless ofthe particle size, but can exceed 15% for a spheroid withthe axes ratio a : b = 2 at x = 1 (Mishchenko et al., 1996).Important limitations can be high porosity of the particles(Henning and Stognienko, 1993) or the presence of sharpedges (Yanamandra-Fisher and Hanner, 1998).

3.1.1. Use of Mie theory to estimate grain emissivity atthermal infrared wavelengths. As we will see below, thethermal emission by cometary dust comes mainly fromsubmicrometer particles. Therefore the equal-volume-sphererepresentation provides a good estimate for the absorptioncross section as it works for particles that are small in alldimensions, i.e., represent quasi-equidimensional (not nee-dles or disks) compact particles of x << 1. It is also a goodapproximation for submicrometer particles in a porous ag-gregate so that the interaction between particles is weak.Using the thus estimated absorption cross section Cabs one canuse equation (2) to find the temperature for grains of a vari-ety of sizes and composition (Mukai, 1977; Hanner, 1983).This revealed a set of general regularities in thermal emis-sion from small grains shown in Fig. 7.

Particles of absorptive materials, e.g., glassy carbon ormagnetite, showed temperatures higher than for a black-body; the smaller the particles, the higher the temperature.Particles made of transparent materials were found to becooler than a blackbody and their temperature did not de-pend much on particle size. Small grains of an absorbing(e.g., carbonaceous) material absorb strongly at visual wave-lengths, but cannot radiate efficiently in the infrared at

wavelengths greater than about 10 times their size. Theyheat up until the energy radiated at the shorter infraredwavelengths (roughly 3–8 µm for comets at 0.5–3 AU fromthe Sun) balances the absorbed energy. In fact, for smallabsorbing grains, their size controls their temperature, re-gardless of their specific composition (Hanner, 1983).

In contrast to carbon grains, silicate grains radiate effi-ciently in the infrared because of the resonance in theiroptical constants near 10 and 20 µm; the amount of absorp-tion at visual wavelengths controls their temperature. Theabsorption at visual wavelengths depends strongly on theFe content of the silicates (Dorschner et al., 1995). PureMg-rich silicates have very low absorption; the imaginarypart of the refractive index κ ~ 0.0003 at 0.5 µm for a glasswith Fe/Mg κ ~ 0.05 at 0.5 µm for a pyroxene glass withFe/Mg = 1 and k ~ 0.1 for an olivine glass with Fe/Mg = 1.0.Consequently, pure Mg silicates in a comet coma near 1 AUwould be much colder than a blackbody while Fe-rich sili-cate grains would be warmer than a blackbody. Composi-tional measurements indicate that the silicates in comets areMg-rich (Hanner and Bradley, 2004), so one might expectthem to be cold. However, even a small admixture of ab-sorbing material can increase the temperature significantly.

The computed temperature for a submicrometer absorb-ing grain varies approximately as R–0.35 instead of observedR–0.5 (Fig. 7) that would be also expected for a blackbodyin equilibrium. Whether this indicates a change in size dis-tribution with R, dominance by larger particles or the short-coming of the computations for small spheres is not clear.Chances are that this is a result of combination of opticalcharacteristics of both small and large particles exhibited byaggregates of small grains (see section 3.3.3).

The observed 3–20-µm SED for comets cannot gener-ally be fit by the F(λ) computed for a single grain of anysize or composition. However, models using a size distri-bution of absorbing particles dominated by micrometer- tosubmicrometer-sized grains provide a match to the observedSED (Li and Greenberg, 1998b; Hanner et al., 1999). Com-ets with the strongest dust emission (such as P/Halley andHale-Bopp) that also display a higher color temperature at3–8 µm than at 8–20 µm indicate an enhanced abundanceof hot, submicrometer-sized grains.

By fitting the observed SED with a model of thermalemission from a size distribution of grains, the total dustmass within the coma can be estimated. When combinedwith a size-dependent velocity distribution for the dust, therate of dust production from the nucleus can also be esti-mated (Hanner and Hayward, 2003).

Extensive observations of Comet Hale-Bopp from theEuropean Space Agency’s ISO (the photometer PHOTmeasured the thermal flux through filters at 7–160 µm,while the two spectrometers recorded the spectrum from 5to 160 µm) enabled the SED to be obtained at long wave-lengths and fit by an emission model having a size distri-bution of the form n(r) ~ r–a, with a ≤ 3.5 (Grün et al.,2001). For an outflow velocity v(r) ~ r–0.5, this result impliesthat the dust production size distribution from the nucleushad a slope a ≤ 4 and the mass was concentrated in large

Fig. 7. Temperature of a grain vs. heliocentric distance. Dottedlines show results for absorptive particles (glassy carbon) of ra-dius 0.1, 0.5, and 10 µm. Dashed lines are for olivine particles ofradius 0.1 and 10 µm. The thin solid line is the blackbody tem-perature. The thick solid line shows the results for Comet Halley(Tokunaga et al., 1988). The comet dust temperature is higher thana blackbody, indicating the presence of submicrometer absorptiveparticles.

588 Comets II

particles. Yet Hale-Bopp displayed the highest 3–13-µmcolor temperature and strongest silicate feature ever seenin a comet, clearly indicating that the thermal emission atshorter wavelengths arose from a high abundance of sub-micrometer grains compared to other comets.

While Mie theory can be used to compute the absorp-tion (or emission) cross section Cabs for a compact particlethat is small in comparison to the wavelength in all its di-mensions, in cases where Cabs varies slowly with wave-length, it cannot be applied to calculations for an emissionfeature, where the optical constants are rapidly varying.Analysis of the 10-µm silicate emission feature in cometsindicates the presence of both glassy and crystalline sili-cate components (Hanner et al., 1999; Wooden et al., 1999;Hanner and Bradley, 2004). The optical constants of theglassy silicates vary slowly across the 8–13-µm range andMie theory will not be too inaccurate for computing Cabs.However, the crystalline components, especially crystallineolivine, which has a strong resonance at 11.2 µm in natu-rally occurring samples, cannot be treated using Mie theorysince a resonance is very sensitive to particle shape (Bohrenand Huffman, 1983). Mie theory does not even predict thecorrect wavelength of the peak, let alone the correct peakheight (Yanamandra-Fisher and Hanner, 1998). Thus, mostresearchers make use of measured reflectivity or mass ab-sorption coefficients to obtain Cabs within the spectral feature(Wooden et al., 1999, 2000; Hayward et al., 2000; Harkeret al., 2002).

Mie theory influenced our understanding of cometarydust, constraining it as an ensemble of dark, polydisperse,and nonspherical particles whose main contribution to thethermal emission comes from particles of size parameterclose to one unit. It still remains an important tool whenestimating the thermal radiation from cometary dust, sincethe particle shape does not affect the absorption cross sec-tions as long as the particles can be considered as compactand roughly equidimensional or sufficiently porous ag-gregates of such particles so that particle interactions canbe neglected. However, angular and spectral dependenciesof intensity and polarization of cometary dust indicate thatthese properties cannot be calculated using the model ofspherical homogeneous particles.

3.2. Approximations to Treat NonsphericalInhomogeneous Particles

Along with techniques that consider light scattering inrigorous terms through the solution of Maxwell’s equations,there are some useful approximations that allow fast com-putations subject to the size parameter and/or refractiveindex of the particle.

When x << 1 and |mx| << 1 the Rayleigh (1897) approxi-mation applies. Such a particle can be considered as a di-pole with an inherent polarizability that defines the dipolemoment induced in the particle by the incident electromag-netic field. In the case of a very small particle, the field nearit can be treated as an electrostatic field. This allows thederivation of a simple equation for the particle polarizabil-

ity, which in many cases [spheres, spheroids, ellipsoids,disks, cylinders, see, e.g., Bohren and Huffman (1983)]admits analytical solution.

When x|m – 1| << 1 and |m – 1| << 1, the Rayleigh-Gansapproximation applies. This approximation is often used notby itself but as a first approximation for the successive it-erations to obtain more general approximations (see, e.g.,Haracz et al., 1984). Muinonen (1996) applied this approxi-mation to stochastic irregular particles. Closely related tothe Rayleigh-Gans approximation is the coherent Mie scat-tering approximation developed to treat porous aggregatesof spheres (Zerull et al., 1993). Only phase interrelationsbetween waves scattered by constituent particles in an ag-gregate of spheres are taken into account. This approxima-tion works better with increasing porosity of the aggregate;usually the porosity should be more than 90% (Xu andWang, 1998).

When x >> 1, |m – 1| << 1, we may apply the anomalousdiffraction approximation (van de Hulst, 1957). This ap-proximation is based on the assumption that the rays arenegligibly deviated inside the particle, thus only absorptioninside the particle and interference of light passed throughthe particle and around it can be taken into account. Thisallows easy calculation of the absorption and extinctioncross sections, which sometimes can be expressed throughanalytical formulas. This method was widely applied toestimate cross sections of particles very different fromspheres, such as prismatic and hexagonal columns, cubes,and finite cylinders. The accuracy of the method increaseswith decreasing absorption.

For the sake of completeness, we mention an approxima-tion called perturbation theory, which treats the radius r of anirregular particle as a function of coordinates ϑ and ϕ andr = r0(1 + ξf(ϑ,ϕ)), where r0 is the radius of “unperturbed”sphere and |ξf(ϑ,ϕ)| < 1. The solution is obtained as an ex-pansion of the boundary condition in series in ξ (Oguchi,1960). This method provides accurate results only for parti-cles of x ≤ 7 and only if the deviations from spherical shapeare smaller than the wavelength. It is a convenient methodwhen effects of the surface roughness are of interest.

x >> 1 is the geometric optics case, now often referredto as the “ray-tracing approximation.” It considers the in-cident field as a set of independent rays, each of whichreflects according to Snell’s law and Fresnel’s equations onthe surface of the particle. The size restriction is in partrelaxed when geometric optics is combined with Fraunhoferdiffraction. For particles of complicated shape, the ray trac-ing is usually performed using the Monte Carlo approach.On this basis results have been obtained for stochastic parti-cles (Muinonen, 2000) and particles with inclusions (Macke,2000). The Web site by A. Macke (http://www.ifm.uni-kiel.de/) contains a collection of ray-tracing codes. The com-parison of ray-tracing calculations with exact solutions, e.g.,for spheres or cylinders, shows that the discrepancies be-tween the approximation and the exact solution start fortransparent particles of size parameter x < 100 (Wielaardet al., 1997); for larger absorption the ray-tracing techniquecan be applied to smaller size parameters. Waldemarsson


and Gustafson (2003) developed a combination of ray-trac-ing technique with diffraction on the edge for light scatter-ing by thin plates (flakes) that provides a reasonable accu-racy down to particle sizes of x ~ 1.

Cometary dust is not dominated by very small or verylarge particles, but the approximations described above canbe applied to calculations for wide size distributions ofnonspherical particles at the edges of the size distribution.For example, Bonev et al. (2002) calculated the color ofcometary grains assuming them to be polydisperse sphe-roids and using the T-matrix method (see section 3.3.2) forsize parameters x < 25 and the ray-tracing technique forlarger particles. A similar approach can be used to calcu-late light scattering when a broad range of wavelengths needto be covered, as was done by Min et al. (2003), who alsofound a criterion for selecting the ray-tracing or anomalousdiffraction approximations to calculate cross sections.

Kolokolova et al. (1997) showed that in the color, pre-sented as the difference of logarithms of intensity at twowavelengths, the contribution of intermediate particles iscanceled and small and large particles dominate. If the sizedistribution is smooth (e.g., of power-law type) and wide[i.e., includes small (Rayleigh) and large (ray-tracing) par-ticles], then the color can be expressed analytically throughthe radii of the smallest and largest particles, the power ofthe size distribution, and two parameters that describe theparticle composition. These analytical expressions have fiveunknowns that can be found if five colors are measured ashas been done for Comet C/1996 Q1 Tabur (Kolokolova etal., 2001b).

The approximations listed above were developed to treatnonspherical but homogeneous particles. To describe lightscattering by heterogeneous particles one can use approxi-mations called mixing rules or effective medium theories.The main idea behind these approximations is to substitutea composite material with some “effective” material whoserefractive index provides the correct light-scattering char-acteristics of the heterogeneous medium. It is a reasonableapproximation when inhomogeneities (inclusions) are muchsmaller than the wavelength (Rayleigh-type) and the sizeof the macroscopic particle is much larger than the size ofthe inclusions. In large scale, such a medium acts as anisotropic homogeneous medium for which the displacementD is related to the electrostatic field E by a linear relation-ship D = εE where ε is the complex dielectric constant ofthe material related to the refractive index as εm = . For amacroscopically homogeneous isotropic mixture, D and Ecan be averaged within the medium as

∫ D(x,y,z)dxdydz = ∫ ε(x,y,z)E(x,y,z)dxdydz =εeff ∫ E(x,y,z)dzdydz

(5)

where E, D, and ε are determined for a point inside the ma-terial with coordinates (x,y,z). Using the Maxwell equationdivD = 0 and assuming some size, shape, and distributionof the inclusions, one can obtain expressions, in some caseanalytical, for the effective dielectric constant εeff. Dozens

of mixing rules have been developed for a variety of inclu-sion types (non-Rayleigh, nonspherical, layered, anisotro-pic, chiral) and topology of their distribution within themedium, including aligned inclusions and fractal structures(see an extensive review by Sihvola, 1999). However, stillthe most popular remain the simplest Maxwell Garnett(Garnett, 1904)

effeff

mii

effeff

meff

2f

2 ε+εε−ε=

ε+εε−ε

(6)

and Bruggeman (Bruggeman, 1935)

effm

effmm

effi

ieffi 2

f2

fε+ε

ε−ε=

ε+εε−ε

(7)

mixing rules. In equations (6) and (7) εi and εm are dielec-tric constants of inclusions and matrix respectively, and fiis the volume fraction of the inclusions in the mixture. TheMaxwell Garnett rule represents the medium as inclusionsembedded into the matrix material with εm, the result ofwhich depends on the material chosen as the matrix. TheBruggeman rule was obtained for particles with εi and εmembedded into the material with ε = εeff, so the formula issymmetric with respect to the interchange of materials. Thismakes it easy to be generalized for the n-component me-dium

∑=

=ε+εε−εn

1ieffieffii 0)2/()(f (8)

Since the derivation of the mixing rules was based onassuming the external field as electrostatic, the inclusionswere assumed to be much smaller than the wavelength ofelectromagnetic waves. More exactly, the criterion of theirvalidity is xRe(m) << 1 (Chylek et al., 2000), where x is thesize parameter of inclusions and Re(m) is the real part ofthe refractive index for the matrix material. Comparison ofeffective medium theories with DDA (see section 3.3.3) cal-culations (Lumme and Rahola, 1994; Wolff et al., 1998) andexperiments (Kolokolova and Gustafson, 2001) show thateven for xRe(m) ~ 1 effective medium theories provide rea-sonable results. The best accuracy can be obtained for crosssections and the worst for polarization, especially at phaseangles α < 50° and α > 120°.

There were a number of attempts to consider heteroge-neous cometary grains using effective-medium theories, par-ticularly to treat aggregates as a mixture of constituent par-ticles (inclusions) and voids (matrix material) (e.g., Green-berg and Hage, 1990; Mukai et al., 1992; Li and Greenberg,1998b). In the visual region the cometary aggregates withthe size parameter of constituent particles x > 1 are, mostlikely, out of the range of the validity of the effective me-dium theories. For the thermal infrared region, cometaryaggregates can be treated with effective medium theoriesif they are sufficiently large (remember that the number of

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inclusions must allow the macroscopic particle to be con-sidered as a medium). Using the mixing rules, one shouldbe careful to calculate the refractive index for core-mantleparticles. Even for particles of size parameter x ≥ 0.3, sig-nificant differences in cross sections from exact core mantlecalculations appeared (Gustafson et al., 2001).

3.3. Numerical Solutions for InhomogeneousNonspherical Particles

Further characterization of cometary dust requires moreadvanced light scattering theories. We consider them in thissection, mainly outlining the strengths and drawbacks of themethods that have been applied to study cometary dust. Aswell as Mie theory, solutions of the Maxwell equations fornonspherical particles cannot be used directly to solve theinverse problem for cometary dust, but instead provide someconstraints through fitting procedures and investigation ofgeneral trends in the dependence of light-scattering char-acteristics on particle properties.

3.3.1. Separation of variables method. The main ideabehind Mie theory, expansion of the electromagnetic wavesin the coordinate system that corresponds to the shape ofthe body (e.g., cylindrical, spheroidal) and then separatesthe variables, has provided a number of solutions for non-spherical but regular particles: multilayered, optically an-isotropic, and radially inhomogeneous spheres, cylinders,slabs, and spheroids (see the review by Ciric and Cooray,2000). Among solutions obtained with this method, the mostpopular are solutions for cylinders (Bohren and Huffman,1983), core-mantle spheres (Toon and Ackerman, 1981),spheroids (Asano and Yamamoto, 1975; Voshchinnikov andFarafonov, 1993), and core-mantle spheroids (Farafonov etal., 1996). Although this method can provide exact solu-tions to Maxwell’s equations, the problem requires numeri-cal treatment to solve the boundary condition even forsimple particle shapes. This limits the applicability of themethod to rather small size parameters (x < 40 for sphe-roids) or to nonabsorbing materials. However, in the rangeof their applicability, these theories provide highly accurateresults, which makes them a good tool for testing otherapproaches.

The solution of the separation of variables for a singlesphere can be extended to aggregates of spheres representingthe total scattering field as a superposition of spherical wavefunctions, which expand the field scattered by each sphereand use the translation addition theorem for spherical func-tions (Bruning and Lo, 1971). The method was extendedfor aggregates of spheroids, cylinders, multilayered spheres,and spheres with inclusions (see Xu, 1997; Fuller and Mac-kowski, 2000, and references therein). The computationalcomplexity of the method (the number of numerical opera-tions in the algorithm) depends on the number of particlesand their size parameter, shape, and configuration.

3.3.2. T-matrix approach. A popular T-matrix ap-proach was initially introduced by Waterman (1965) underthe name “extended boundary condition.” This method ex-

pands the incident and scattered fields into vector spheri-cal functions with the scattered field expanded outside asphere circumscribing a nonspherical particle. The main ad-vantage of the resulting formulation is that the T-matrix isdefined only by physical characteristics of the particle it-self, i.e., it is computed only once and then can be used forany incidence/scattering direction and polarization of theincident light. A significant development of the method wasan analytical orientation-averaging procedure (Mishchenko,1991) that made calculations for randomly oriented particlesas efficient as for a fixed orientation of the same particle.T-matrix computations are especially efficient for axisym-metric, including multilayered, particles; however, the meth-od can be extended to any shape of the scatterer. Althoughcomputational complexity greatly increases for particleswithout rotational symmetry, recently solutions were ob-tained for ellipsoids, cubes, and aggregates of sphericalparticles. The latter case is of special interest in the field ofastrophysics. It presents the T-matrix development of theseparate-variable method for the aggregates of spheres con-sidered above. Since the scattered field for the individualsphere does not depend on the incident light direction andpolarization, the expansions for individual spheres can betransformed into a single expansion centered inside the ag-gregate. Such an expansion is equivalent to the T-matrix ofthe aggregate and thus can be used in analytical averaging ofscattering characteristics over aggregate orientations (Mish-chenko et al., 1996).

A variety of computational T-matrix codes are availableon line at Mishchenko’s Web site (http://www.giss.nasa.gov/~crmim), among them codes for polydisperse axial-sym-metric particles and aggregates of spheres. The computa-tional complexity of the T-matrix method increases as x3–x4,i.e., the method is a fast and convenient way to test resultsobtained with other codes (see, e.g., Lumme and Rahola,1998) or for survey-type studies. Kolokolova et al. (1997)used it to check the sensitivity of color and polarization tothe size and composition of nonspherical particles. Theresults were found to be similar to the results for polydis-perse spheres. However, details in polarization for non-spherical particles differ significantly from those calculatedfor equal-volume or equal-area spheres (Mishchenko andTravis, 1994). For example, randomly oriented spheroidsshow no oscillations in angular dependencies of intensityand polarization at size distributions 10–20× narrower thanis necessary to wash out the oscillations for spheres. Polar-ization for polydisperse silicate spheroids (Fig. 8) at inter-mediate phase angles changes from negative, typical forspheres (Fig. 6), to positive, leaving a negative branch atα < 30°. For elongated particles at α ~ 90°, polarimetriccolor is positive and shows the angular trend similar to theobserved one (it gets larger with increasing phase angle)for a broad range of particle size (Fig. 8). The angular de-pendence of the intensity is also reminiscent of the cometaryscattering function (Fig. 1) except at phase angles 10°–60°,where it reaches a deep minimum instead of being flat. Thespheroids demonstrate color that changes rather rapidly its


Fig. 8. Same as Fig. 6 but for prolate spheroids (ratio of axis is 1 : 3) of the refractive index m = 1.55 + i0.005; the effective sizeparameter xeff corresponds to this for equal-volume sphere.

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value and even sign with phase angle. Randomly orientedcylinders exhibit similar tendencies (Mishchenko et al.,2002). Using higher absorption one can get higher valuesof color and polarimetric color, but the negative polariza-tion and backscattering peak disappear, eliminating the re-semblance of the calculated scattering function and polari-zation curve to the observed ones.

Mishchenko et al. (1997) showed that appending themodel by a shape distribution (a range of aspect ratios ofspheroids) can provide a good fit to the cometary scatter-ing function for polydisperse silicate spheroids at 6 < xeff <24. For the size distribution with xeff > 8 and a constantrefractive index m = 1.53 + i0.008, the color becomes neu-tral and then red with a very weak angular change. How-ever, polyshaped silicate spheroids fail to reproduce the cor-rect polarization (Mishchenko, 1993).

Petrova et al. (2000) used the T-matrix code by Mac-kowski and Mishchenko (1996) for a systematic compari-son of cometary observations with the calculations for ag-gregated particles. They considered aggregates built fromparticles of size parameters 1 < x < 3.5 for a set of refrac-tive indexes with the real part equal to 1.65 and the imagi-nary part varying within the range 0.002–0.1. The maxi-mum number of particles in each aggregate was limited to43 by the code convergence. “Cometary” type polarizationwith low maximum at α ~ 90° and negative branch at smallphase angles was obtained for a power-law size distributionof aggregates built of 8–43 constituent particles of x = 1.3,1.5, and 1.65, although a smooth shape of the curve andthe quantitative fit to the observational polarization curvewas not achieved. The characteristics of the negative po-larization depend on the composition (it becomes less pro-nounced with increasing absorption from 0.01 to 0.05) andporosity (more pronounced for more compact aggregates).A rather good qualitative fit to the cometary scattering func-tion was achieved for the polydisperse aggregates with x =1.3 and κ = 0.01. In a later paper, Petrova and Jockers(2002) studied the spectral dependence of polarization fora variety of size distributions and a range of refractive in-dexes within n = 1.65, 0 < κ < 0.1, assuming the refrac-tive index independent of wavelength. They showed that theaggregate model successfully reproduced the observed posi-tive polarimetric color. However, the color of such aggre-gates was found to be blue, unlike the observed red color.The extinction calculations by Petrova et al. (2000) aremainly sensitive to the overall size of the aggregate and areclose to the results for equal-volume spheres if the aggre-gates have size parameters x < 3, but becomes larger forlarger aggregates. This is an indication that the equal-vol-ume sphere calculations can be used to estimate cross sec-tions of aggregates in the thermal infrared but not in thevisual.

3.3.3. Coupled dipole approximation and similar tech-niques. The methods described above are based on thesolution to the boundary conditions on the particle surfaces,while a set of more flexible methods rely on solutions of theinternal field from which the scattered field is calculated.

The most popular among such methods is probably thecoupled dipole approximation, often called the discrete di-pole approximation (DDA) since it represents a particle asconsisting of elementary cells, polarizable units called di-poles (Purcell and Pennypacker, 1973). Each dipole is ex-cited by the external field and the fields scattered by allother dipoles. This representation allows one for a N-dipoleparticle to derive N linear equations describing the N fieldsthat excite each dipole. The numerical solution of the sys-tem of these N equations provides the partial field scatteredby each dipole, and finally the total scattered field is cal-culated. Draine and Flatau (1994) and Lumme and Rahola(1994) improved the computational efficiency of the DDA,which allowed them to study larger particles, in particularaggregates of larger size. Lumme et al. (1997) consideredaggregates up to 200 particles of x = 1.2–1.9. The mostpopular has been the DDSCAT code by Draine and Flatau(see http://www.astro.princeton.edu/~draine/DDSCAT.html).The code includes geometry implementations for ellipsoids,finite circular cylinders, rectangular parallelepipeds, hexag-onal prisms, and tetrahedra, uniaxial and coated particlesas well as for systems of particles. The great advantage ofthe coupled dipole approximation is that it can be appliedto particles of arbitrary shape, structure, and composition.However, it has limited numerical accuracy, requires a sepa-rate calculation for each orientation of particle, and slowsdown dramatically with increasing N, so that its computa-tional complexity increases as the ninth order of the parti-cle size parameter.

Using DDA, West (1991) undertook the first systematictheoretical study of aggregates. He considered the constantrefractive index m = 1.7 + i0.029. West demonstrated thatfor compact aggregates both intensity and polarization be-have as is more typical for the spherical equivalent of theaggregate. For loose aggregates the projected area of thewhole aggregate mostly defines the intensity in the forward-scattering domain, whereas the polarization is defined bythe size of the constituent particles. Kozasa et al. (1993)confirmed that even aggregates of thousand constituentparticles of x < 1 behave like Rayleigh particles in polariza-tion regardless their composition.

Xing and Hanner (1997) accomplished a detailed studyof aggregates with a variety of sizes, shapes, and composi-tions of constituent particles and their packing within theaggregate. They confirmed that polarization is more sen-sitive to the properties of constituent particles whose sizeand refractive indexes determine most of the polarization,whereas shape, number, and packing generate scatteringeffects of the second order. A reasonable fit to the observedcometary polarization and intensity was achieved for themixture of carbon and silicate aggregates of an intermedi-ate (touching-particle) porosity. The calculations of crosssections showed that in the case of short wavelengths (sizeparameter x > 1) the values of extinction cross section perunit volume is close to those typical for constituent particles,but at long wavelengths (x << 1) constituent particles andaggregates show significant differences that are more pro-


nounced the more compact the aggregate. When the poros-ity is less than 60%, the temperature of the particle is closeto that of an equal-volume sphere. Loose aggregates havea temperature typical for the constituent particle, i.e., ther-mal emission from a large but aggregated particle looks thesame as its small constituent particle regardless of its shape(see also Greenberg and Hage, 1990; Li and Greenberg,1998b). Davidsson and Skorov (2002) obtained similarDDA results for single-scattering albedo. This means thatthermal infrared data do not allow distinguishing loose ag-gregates from single particles of the size of their constitu-ent particles.

The DDA was used by Yanamandra-Fisher and Hanner(1998) to study the light scattering and emission by non-spherical but regular particle shapes of size parameter 1 <x < 5, as well as heterogeneous and porous particles. Theyshowed that compact particles of different shapes producevery different angular dependence of polarization for atransparent material (silicate), but the dependence on par-ticle shape is reduced for absorbing particles (carbon). Theyconcluded that negative polarization at small phase anglescould be produced by nonspherical silicate particles of x ≥2. Introducing porosity for a transparent (silicate) particlecaused the polarization to resemble that of a smaller par-ticle, while porosity had less of an effect on an absorbingparticle provided that the particle remained opaque. Al-though the use of regular shapes caused oscillations thatwould be absent in the scattering by an irregularly shapedparticle, the authors showed that a macroscopic mixture ofsilicate and absorbing particles with a distribution of sizeparameters 1 < x < 5 could produce curves for polarizationand angular scattering qualitatively similar to the curves ob-served for cometary dust. A similar conclusion was reachedby Xing and Hanner (1997) based on the scattering by smallaggregates.

Yanamandra-Fisher and Hanner (1998) also applied theDDA code to examine particle shape effects in the mid-infrared emission features produced by crystalline olivine.Both the shape and the peak wavelength of the spectralfeatures depend on the particle shape; moreover, olivine isan anisotropic crystal, and the differing optical constantsgenerate peaks at different wavelengths. To reproduce theobserved silicate feature the authors had to rule out extremeshapes (disks, needles) as well as ideal spheres. Polariza-tion within the 10-µm silicate feature was studied by Hen-ning and Stognienko (1993), who demonstrated that shapeand porosity of particles affect the spectral dependence ofpolarization, both the strength and shape of the feature, evenstronger than the spectral dependence of the intensity.

Combining the T-matrix technique with the DDA, Ki-mura (2001) considered aggregated particles of fractal struc-ture, ballistic particle-cluster aggregates (BPCA) and ballis-tic cluster-cluster aggregates (BCCA) (see Mukai et al.,1992), of hundreds and thousands of constituent particles.He showed that aggregates of silicate particles of x = 1.57reproduce the cometary polarization with a very good fit.They also provide rather good fit to the “cometary” shape

of the scattering function. The number of particles in theaggregates, at least if it is within 64 ≤ N ≤ 256, as well astheir porosity, was found not to be important. But a changein composition (carbon instead of silicate) could dramati-cally affect the value of maximum polarization, eliminatethe negative branch, and make the scattering function moreflat at the medium phase angles. Kimura’s (2001) resultsdemonstrate the best fit to the observational data achievedso far, although the spectral characteristics of intensity andpolarization have not been checked.

Similar to DDA, the method of moments (MOM) ap-proach also divides a particle into small elements, but con-siders them not as dipoles but as small cells with a constantrefractive index. The price of this straightforwardness ofMOM in comparison with DDA is the necessity of calcu-lating a self-interaction term; different approaches in MOMusually diverge in the way they treat this term. The MOMhas strengths and drawbacks similar to those for DDA.Using MOM, Lumme and Rahola (1998) considered comet-ary particles as stochastically shaped, i.e., particles whoseshape can be described by a mean radius and the covari-ance function of the radius given as a series of Legendrepolynomials (for details, see Muinonen, 2000). Lumme andRahola (1998) made computations for a variety of particleshapes and size parameter (x = 1–6) using the refractive in-dex m = 1.5 + i0.005. Like many previous authors, Lummeand Rahola found that the particles should be of x > 1 toprovide negative polarization at small phase angles and lowmaximum polarization. Also, they showed that monodis-perse particles, even of very complicated, irregular shape,demonstrate oscillating behavior of light-scattering charac-teristics that can be eliminated by considering a polydis-persion of particles. The same oscillating behavior is shownby color and polarimetric color that can be seen from thecomparison of the intensity and polarization for x = 2, 4,6. Particles with the size distribution of the form n(r) ~ r–3

show angular functions of intensity and polarization that arerather similar to cometary ones. However, one also needsto fit to the spectral characteristics to judge how close themodel is to real cometary dust.

3.3.4. Other numerical methods. Above we considerednumerical techniques that have been used in cometary phys-ics. However, there are other light-scattering techniques thatmay be worth applying, among them the finite differencetime domain method (FDTD) and the finite element method(FEM). Both these methods are based on a straightforwardsolution to equation (4). The FDTD discretizes space andtime, making a grid in the time-space domain and thensolves equation (4) for each space-time cell, whereas theFEM discretizes the scattering particle itself into small vol-ume cells. Then boundary conditions (continuity of the elec-trical vector across the surface) are applied on the boundaryof each cell. Both methods are applicable to particles ofarbitrary shape, structure, and composition. The main dis-advantage of these methods is time-consuming computa-tions, aggravated by the fact that the computations shouldbe repeated for each orientation of the particle with respect

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to the incident wave. The computational complexity of theFDTD increases as the fourth power of the particle sizeparameter and varies between x4 and x7 for FEM depend-ing on the computational technique.

More details of the methods discussed above and othernumerical methods and approximations can be found inrecent reviews (Mishchenko et al., 2000, 2002; Kahnert,2003). Some codes are available via online libraries suchas http://atol.ucsd.edu/~pflatau/scatlib/ (P. Flatau), http://www.t-matrix.de/ (T. Wriedt), http://www.emclab.umr.edu/codes.html, http://urania.astro.spbu.ru/DOP/, and http://www.astro.uni-jena.de/Users/database/. The latter Web sitealso contains a huge collection of refractive indexes for ma-terials of astrophysical interest (Jäger et al., 2003).

3.4. Experimental Simulations

The complexity of cometary and other cosmic dust andplanetary aerosols and the ensuing difficulty treating themusing theoretical means generated several approaches forsolving the scattering problem through experimental simu-lation. As before, we will not discuss all experiments con-ducted so far to understand light scattering by particles, butonly those representative examples that provided significantresults for cometary dust. We will not discuss here the ex-perimental setups detailed in the papers cited below.

One experimental technique to study light scattering byparticles uses streams or suspensions of small particles,often powdered terrestrial rocks, that are illuminated by asource of light, recently mainly by a laser. The light scat-tered by the particles is measured to obtain intensity, polari-zation, Stokes parameters, and even the complete scatteringmatrix for a range of phase angles. The other technique,called the microwave analog method, introduced by Green-berg et al. (1961), simulates light scattering using micro-wave radiation. It actually takes advantage of the samescaling used in all theoretical approaches where the particledimensions are given through its size parameter, i.e., as aratio to the wavelength. In such a simulation the light scat-tering problem is scaled to longer wavelengths that preciselypreserve the characteristics of the scattering problem as longas the size parameter of the scatterer and its refractive in-dex are preserved. This allows light scattering by a singlesubmicrometer or micrometer particle to be simulated us-ing a manageable millimeter or centimeter analog model.

3.4.1. Microwave analog experiments. The early micro-wave simulations were measurements of cross sections andscattering functions for spheroids, finite cylinders, rods, anda variety of compact irregular particles (Greenberg et al.,1961; Zerull et al., 1977) directed to determine the appli-cability of Mie theory to nonspherical particles. The studywas then extended to polarization and color of nonsphericalparticles (Zerull and Giese, 1983). A systematic study ofparticle shape effects was carried out by Schuerman et al.(1981). Significant discrepancy between the data for non-spherical particles and Mie theory was obtained for both theshape of measured curves and values of light-scattering

parameters. Giese et al. (1978) showed that, unlike com-pact and rather regular particles, irregular, porous particlesin the size range of a few micrometers exhibit positive andnegative polarization typical for cometary dust. Thus in the1970s the microwave method provided the results obtainedonly in theory in the late 1990s. Greenberg and Gustafson(1981) showed that models consisting of a tangle of rods,so-called “Bird’s Nests,” could provide the observed angu-lar dependencies of intensity and polarization. This experi-mental test of the aggregated nature of cometary dust thenwas extended to the study of a variety of aggregates ofspherical, including core-mantle, particles by Zerull et al.(1993).

The systematic study of light scattering by complexparticles has been performed using a new generation ofmicrowave facilities designed and built at the University ofFlorida (Gustafson, 1996, 2000). This facility works acrossa 2.7–4-mm waveband to simulate a region 0.4–0.65 µmin the visual, and thus can study not only angular but alsospectral dependencies of light scattering (colors). Gustafsonand Kolokolova (1999) reported results from a systematicstudy of light scattering by aggregates. Among these resultsis that the polarization is mainly determined by the size andcomposition of constituent particles and the size parameterof cometary constituents must be 1 < x < 10 to produce alow maximum polarization and negative polarization atsmall phase angles that confirms the theoretical resultsconsidered above (e.g., West, 1991; Xing and Hanner, 1997;Lumme et al., 1997; etc.). The color and polarimetric colorof aggregates also primarily depend on the size and com-position of the constituent particles. For aggregates madeof particles whose size parameters covered the range 0.5–20, an increase in the size of constituent particles results inlarger (more red) color and smaller polarimetric color (therefractive index was m = 1.74 + i0.005, constant through-out the wavelength range). The same tendencies result fromincreasing the imaginary part of the refractive index thatmay provide a clue to the negative polarimetric color ob-served for Comet P/Giacobini-Zinner (Kiselev et al., 2000).A combination of positive polarimetric color and blue coloris typical of aggregates of nonabsorbing particles of 0.5 <x < 10, whereas red color and positive polarimetric color areindicative of dark, absorbing aggregates. From the above,the aggregates, which similarly scatter light to the cometarydust, consist of absorptive constituent particles of size pa-rameter x > 1.The shape of constituent particles could notbe seen to influence the aggregates’ color and only slightlychanges the position and amount of maximum polarization.Notice that the results were obtained for the refractive in-dex that does not change with wavelength. The intensity andpolarization data for a variety of particles, including aggre-gates, are given at http://www.astro.ufl.edu/~aplab.

Even though implementation of some theoretical meth-ods, e.g., DDA, on modern computers allows a study ofsimilar accuracy for aggregates of similar characteristics asat microwave measurements, the microwave method is stillmuch faster and can provide a large scope of the results in


a reasonable time. For example, single-wavelength DDAcalculations for a silicate aggregate of 1024 constituentparticles of size parameter x = 0.733 (Kimura, 2001) tookmore than a month on the Alpha station, whereas scatter-ing from an aggregate of any number of particles of x > 1can be measured in one week providing data at 50 discretewavelengths. This allows systematic studies to obtain quan-titative characteristics of the light scattering, e.g., multifactoranalysis (similar to that described in Kolokolova et al.,1997) of sensitivity of light-scattering characteristics tophysical properties of particles. Some results of such a studyof aggregates are presented in Table 1, which summarizesa study of large (500–5000 constituent particles) aggregates.The aggregates were built to satisfy the most plausiblemodel of cometary dust aggregates resulted from theoreti-cal simulations: The size parameter of the constituent par-ticles was in the range 1.5–5, the constant refractive index,m = 1.74 + i0.005, simulated a silicate-organic mixture, andthe porosity varied within 50–90%. The larger numbers inTable 1 indicate a stronger correlation between the dustcharacteristic and the light-scattering parameter; negativevalues show that they anticorrelate. Table 1 shows quanti-tatively that the size of constituent particles is the main char-acteristic that determines all light-scattering parameters. Theinfluence of the porosity is much smaller, and within therange investigated the number of particles in the aggregatehas almost no influence on the light-scattering parameters.

The data from Table 1 show that the correlations betweencolor and polarization (their simultaneous decrease or in-crease) in the near-nucleus coma observed by K. Jockers andcolleagues (Kolokolova et al., 2001a) could not be explainedby the changing size of the aggregates (overall size influ-ences neither color nor polarization) or the changing sizeof the constituent particles (color and polarization shouldthen anticorrelate). Change in the porosity could producea correlation, but then the color should anticorrelate withthe polarimetric color that has not been observed. More-over, the simultaneous increase in color and polarization ob-served for Comet Hale-Bopp could be provided only byunrealistic increasing compactness of the aggregate on itsway out from the nucleus. Microwave measurements formaterials of a variety of refractive indexes show that sucha correlation may be a consequence of evaporation/destruc-tion of some dark material contained in the aggregates. Sucha slowly evaporating dark material can be an organic com-

pound, e.g., kerogen or tholin (Wallis et al., 1987) or anorganic refractory (Greenberg and Li, 1998). This assump-tion is consistent with the observed increase in albedo withthe distance from the nucleus and might also explain thegradient in the near-infrared colors.

3.4.2. Light-scattering experiments in the visual. Amongearly experimental works that contributed to our under-standing of light scattering by cosmic grains, Weiss-Wrana(1983) showed a significant difference between light scatter-ing by terrestrial/meteoritic particles and Mie calculations.The work also showed that transparent compact irregularparticles ~30 µm in size had the polarization curve with themaximum located at α ≈ 45° and negative polarization atlarge phase angles; large (28–80 µm) absorbing (κ = 0.66)compact particles had a bell-shaped polarization curve withhigh (about 50%) polarization maximum; and fly ash,slightly absorbing (κ = 0.01) loose aggregates with an aver-age size of 41 µm had an angular dependence of polariza-tion similar to the cometary one.

Extensive measurements of all 16 elements of the scatter-ing matrix F were performed by the group from Amsterdam(Hovenier, 2000; Hovenier et al., 2003; see also http://www.astro.uva.nl/scatter/). Muñoz et al. (2000) studied the scat-tering matrix of polydisperse, heterogeneous, irregular par-ticles presented by terrestrial (light-colored olivine) andmeteoritic (dark carbonaceous chondrite) powders at wave-lengths of 0.442 and 0.633 µm. The effective radius of theparticles was approximately a few micrometers so that thesize parameter of the particles peaked within the range x =10–20. Both terrestrial and meteoritic samples showed po-larization curves of shape similar to the cometary ones andpositive polarimetric color. However, the values and angleof the minimum polarization almost twice exceeded the ob-served ones, whereas the maximum polarization was almosthalf the observed values. Also, the color was blue, i.e., theintensity at 0.442 µm exceeded the intensity at 0.633 µm.The study shows that polydisperse heterogeneous irregularparticles cannot be ruled out as candidates for cometarydust, although, at least for size distributions that peak at x =10–20, they are not able to correctly reproduce all the ob-served characteristics.

Light-scattering experiments under reduced gravity (mi-crogravity) have been proposed to avoid sedimentation ofthe studied dust as well as particle sorting or orientationthat can occur for particles dropped or suspended in air-flow. The microgravity PROGRA2 experiment was concen-trated on light-scattering measurements of levitated particlesduring parabolic flights (Worms et al., 2000). The angulardependences of polarization were retrieved for various typesof compact particles (Worms et al., 1999) and aggregates(Hadamcik et al., 2002). This experiment later providedpolarization images in two wavelengths (Hadamcik et al.,2003).

Hadamcik et al. (2002) performed a systematic micro-gravity study of aggregated particles. The majority of thesamples were made of 12–14-nm particles of silica, alumina,titanium dioxide, or carbon. The particles formed complex

TABLE 1. The strength of correlation (normalized tothe color/radius correlation) between light-scattering

characteristics and properties of the aggregates.

Radius of Number ofConstituent Constituent

Particles Particles Porosity

Color 1.00 0.212 –0.923Polarization –3.23 0.030 –0.262Polarimetric color –2.36 –0.316 1.34

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structures of three-level hierarchy: submicrometer-sizedchains of particles were combined into several-micrometer-sized aggregates, and the aggregates in turn formed milli-meter-sized particles, “agglomerates,” that were levitatedand measured. The most striking result of this study wasthat the particles, even though composed of nanometer con-stituents, did not show Rayleigh-type scattering propertiesunlike other aggregates of small particles considered above(Zerull et al., 1993; Gustafson and Kolokolova, 1999; West,1991; Kozasa et al., 1993). The microscopic photographsshow that although “agglomerates” are rather porous struc-tures, “aggregates” look as if they were made of solid ma-terial, demonstrating rather high packing of “chains” in theaggregates. Since the size of constituents in “aggregates”is of the order of some micrometers, their size can be thefactor that determines the light scattering. The polarimet-ric curves have a low maximum at α ~ 100° and demon-strate positive polarimetric color at large phase angles. Atsmall phase angles the samples made of a single type ofconstituent demonstrate positive polarization, which maybecome negative for mixtures of small and large or dark andlight particles that can be evidence of a heterogeneous struc-ture of cometary grains.

The most recent microgravity study of the wavelengthdependence of polarization (Hadamcik et al., 2003) showedthat a mixture of fluffy aggregates of submicrometer silicaand carbon grains demonstrates a positive polarimetriccolor, whereas gray compact particles with size greater thanthe wavelength exhibit a negative polarimetric color. Thiscan explain the difference between the polarimetric colorof comets and asteroids as well as changes in polarimetriccolor within cometary comae.

The COsmic Dust Aggregation (CODAG) experimentintends to study dust aggregation during rocket flights (Blumet al., 2000; Levasseur-Regourd et al., 2001). These experi-ments (1) study the formation and evolution of dust grainsand aggregates under a variety of physical conditions rep-resentative of the protoplanetary nebula, (2) monitor thedust particles in three colors at α = 5°–175° to study thechanges in their light scattering that accompany the aggre-gation process, and (3) provide validation of the light-scat-tering codes. Low-temperature studies to reveal the dustlight-scattering properties during condensation or sublima-tion of ices on the grains will hopefully be performed on-board the International Space Station (Levasseur-Regourdet al., 2001).

3.5. Summary

Table 2 summarizes the observational data and resultsof the theoretical and laboratory simulations of light scat-tering by cometary dust. One can see that the theoreticaland laboratory simulations of light scattering and thermalemission by cometary dust show that a plausible model ofcometary dust can be heterogeneous (silicates and someabsorbing material) particles that are either irregular, poly-

disperse grains with the predominance of submicrometerparticles or are aggregates of submicrometer particles. Themajor constraints on the dust properties came from polari-metric and thermal infrared data. Angular dependence ofpolarization puts constraints on the size of particles andlimits the refractive index. The SED of the thermal emis-sion implies a broad size distribution of the dust; the ex-cess color temperature above a blackbody at λ < 20 µmindicates grains of r < 1 µm, while the slow decrease in fluxat longer wavelengths indicates larger particles with a sizeparameter comparable to the wavelength. Aggregates ofsubmicrometer grains with a range of porosities would becompatible with thermal SED. Spectral midinfrared featuresidentify silicates as a component of the dust material. Thecomposition could be refined using the spectral changes inthe scattered light (color and polarimetric color); it is likelythat they indicate spectral dependence of the average re-fractive index of the material. A change in the characteris-tics of the scattered and emitted radiation with distance fromthe nucleus is a helpful tool to further determine the sizedistribution and composition of cometary dust as it reflectsthe dynamic sorting of particles and composition changes,e.g., sublimation of volatiles. Although observational dataaccumulated so far, as well as their interpretation, are notsufficient to provide reliable information about cometarydust from the correlations between the light-scattering andthermal-emission characteristics (section 2.4), such corre-lations can be valuable to further constrain the propertiesof cometary dust and to compare the dust in different partsof the coma and in different comets.

4. CONCLUSIONS AND CONSISTENCYWITH THE RESULTS OBTAINED

USING OTHER TECHNIQUES

Below we discuss the physical properties of cometarydust obtained using light-scattering techniques, comparingthem with the results of studying cometary dust using itsdynamical properties, in situ measurements, and study ofinterplanetary dust particles (IDPs).

4.1. Shape

The shape of cometary particles is significantly non-spherical. Evidence of this is the complete absence of reso-nant oscillations in intensity, polarization, and color. Suchoscillations are typical for non-Rayleigh spherical particleseven with a rather wide size distribution. None of the at-tempts to fit the scope of observations with a model basedon spherical particles has been successful. Additional sup-port for this conclusion are observations of star occultationsthat show nonzero polarization at α = 180° and the presenceof circular polarization and polarization whose plane is in-clined to the scattering plane. Such observations require thatthe symmetry inherent in spherical particles be broken. Sim-ulated mid-infrared spectral features demonstrate noticeable


Observational Fact and Section Where It is Discussed

Brightness characteristics:1. Low geometric albedo of the particles (2.1.1)

2. Prominent forward-scattering and gentle backscatteringpeaks in the angular dependence of intensity, “flat”behavior at medium phase angles (α) (2.1.2)

Polarization characteristics:1. For a broad range of wavelengths angular dependence

of linear polarization (2.2.1) demonstrates• negative branch of polarization for α < 20° with the

minimum P ≈ –2%• bell-shaped positive branch with low maximum of

value P ≈ 10–30% at α ≈ 90°–100°.

2. Small circular polarization V < 2% (2.2.4).

Spectral optical characteristics:1. Usually red or neutral color for a broad range of

wavelength λ (2.1.3).

2. Polarization at a given α above ≈30° usually increaseswith λ (positive polarimetric color) (2.2.2).

Spatial distribution of optical characteristics within the coma:1. Increase in albedo with the distance from the nucleus

(2.1.1).

2. Variations in the value of color and polarization(2.1.1, 2.2.3)

3. Deviations in the polarization plane throughout thecoma (2.2.4)

Thermal infrared characteristics (2.3):1. 3- to 20-µm color temperature higher than the blackbody

temperature.

2. Midinfrared emission features typical of silicates insome comets.

3. Thermal emission does not decrease more steeply thana blackbody at λ > 20 µm

Cometary Dust Model that can Reproduce the ObservationalFact and Section Where It is Discussed

1. Absorbing particles (κ > 0.02) (3.4.1). Decrease in the particlesize increases albedo for slightly absorbing particles.

2. Aggregates of particles of x = 1–5 (3.3.2, 3.3.3, 3.4.1), silicatepolydisperse (power law) elongated particles (e.g., spheroids)with a distribution of the aspect ratios (3.3.2), silicate polydis-perse (power law) irregular particles (3.3.3).

1. Polydisperse (power law) irregular submicrometer particles(3.3.2, 3.3.3; 3.4.2); aggregates of particles of x = 1–5 (3.3.2,3.3.3, 3.4.2). The best fit is provided by aggregates of largenumber of constituent particles (3.3.3) or polydisperse aggregates(3.3.2). Absorption seems to reduce the negative polarization butincreases the maximum polarization (3.3.2, 3.3.3).

2. Nonspherical particles or particles containing optically anisotro-pic materials.

1. Slightly absorbing particles of x > 6 (3.3.2); absorbing particlesof x >1 (3.4.1); particles, containing material with a spectrallydependent refractive index (3.3.3).

2. Aggregates of particles of x = 1–5 (3.3.2, 3.3.3, 3.4.1, 3.4.2) orpolydisperse nonspherical particles of x > 1 (but not x >> 1)(3.4.2) for a broad range of refractive indexes independent ofwavelength; particles made of materials with spectrally depen-dent refractive index.

1. Change in the particle composition (evaporation of some darkmaterial) (3.4.1) or decreasing size of silicate particles

2. Variations in size distribution of cometary dust or in theircomposition (if the changes in color and polarization correlate)(3.1, 3.4.1)

3. Nonspherical particles or particles containing optically anisotro-pic materials

1. Absorbing grains of r < 1 µm (3.1.1); porous aggregates ofsubmicrometer absorbing grains (3.3.2).

2. Silicate grains of size r < 1 µm or porous aggregates ofsubmicrometer silicate grains (Hanner and Bradley, 2004).

3. Evidence of a broad size distribution that includes larger thanmicrometer-sized particles.

TABLE 2. Observational facts and their interpretations.

598 Comets II

dependence of the feature’s position and shape on the shapeof dust particles, and thus mid-infrared spectral data canbe used to further constrain the particle shape.

4.2. Size Distribution

As was shown in Table 2, observations in the visibleand their interpretation based on light-scattering methodsequally support two models of cometary dust: (1) ratherlarge or polydisperse aggregates of submicrometer particles,and (2) irregular particles with a power-law size distributiondominated by submicrometer particles. The study of thedust thermal emission allows us to constrain characteristicsof the size distribution. This can be done by fitting the ob-served spectral energy distribution (SED) with a calculatedone using energy balance equation (2). The increase in colortemperature indicates the contribution of submicrometergrains, while the slow decrease in flux at longer wave-lengths indicates larger particles. Grün et al. (2001) fit thebroad thermal flux from 7 to 160 µm from Comet Hale-Bopp with the size distribution of form n(r) ~ r–a with a ≤3.5. Observations in the submillimeter spectral range indi-cate the presence of millimeter-sized particles. Results forother studies confirm that the dust emitted from cometsspans a broad size range, from submicrometer to millime-ter or centimeter and larger. This is known from analysisof the trajectories of dust particles in comet tails (Sekanina,1996) and measurements of the mass of particles impacting

the Giotto spacecraft during the flyby of Comet P/Halley(McDonnell et al., 1991). If the measured Giotto mass dis-tribution is typical of comets, then most of the particulatemass shed from the nucleus is concentrated in large par-ticles, whereas the cross section is broadly distributed orconcentrated in the small particles (Fig. 9), thus makingsubmicrometer- and micrometer-sized particles dominant inthe visual and thermal infrared. This most likely is true forall comets, although the detailed size distribution may varyfrom comet to comet as well as within a coma. For example,size distribution is different for jets and regular coma withpredominance of smaller particles in the jets. The size dis-tribution varies with the distance from the nucleus as a re-sult of both dynamical sorting of particles by radiation pres-sure and sublimation of volatiles.

4.3. Composition

Cometary dust consists of heterogeneous particles thatinclude a dark material, which determines the low albedo,as well as silicates responsible for the spectral features inthe mid-infrared. A distributed source of CO near the cometnucleus, trends in color, polarization, and albedo supportthe notion of a slowly sublimating organic material (Green-berg and Li, 1998; Kolokolova et al., 2001a). Thus, not onlythe size distribution but also the composition of cometarygrains changes as they move out from the nucleus. A sig-nificant specific of the cometary dust is its red color, whichdoes not vary with the phase angle. This cannot be easilysimulated even for polydisperse nonspherical particles. Itmay be the result not of a specific size or structure of parti-cles but of a spectral change in the refractive index. Compo-sitional analysis of dust particles during the Halley flybys aswell as analysis of IDPs and infrared spectra indicate thatthe dust grains are composed largely of silicates and carbo-naceous (CHON) material (Hanner and Bradley, 2004)mixed together on a very fine scale. The composition ofthe cometary organics may be inferred from its sublimationconstants gauged by studying the trends in colors, polariza-tion, and thermal-emission characteristics with the cometo-centric distance.

4.4. Structure

As mentioned above, polydisperse irregular particles canreproduce most of the light-scattering properties of comet-ary dust. The same can be achieved with a model of ag-gregated particles. Porous aggregates of submicrometerparticles are better candidates to explain the data on ther-mal emission. An aggregated structure of cometary particlesis also supported by other studies. Thus, evolutionary ideasabout growing cometesimals in protoplanetary disks sup-port the idea of an aggregated nature of cometary refracto-ries (see, e.g., Greenberg, 1982). The most developed modelby Greenberg and Li (1999) describes cometary grains asfluffy (porous) aggregates of particles that are 0.1-µm sili-cate cores coated by an organic refractory mantle and outer

Fig. 9. Relative cross section (\\\), mass (/// ), and thermalemission (– – –) vs. particle radius for the size distribution meas-ured by Giotto, as adapted to fit the SED of Comet 81P/Wild 2(Hanner and Hayward, 2003). For power-law size distributions witha ≥ 3, submicrometer particles dominate in the light scattering.


mantle of predominantly water ice, which contains embed-ded carbonaceous and polycyclic aromatic hydrocarbon(PAH)-type particles with sizes in the 1–10-nm range. Whenlimited by relative cosmic abundances, the water is con-strained to be close to 30% by mass and the refractory tovolatile ratio is close to 1 : 1.

Captured IDPs of probable cometary origin can give usinsight into the morphology of cometary dust. These IDPsconsist primarily of submicrometer-sized compact, nonspheri-cal grains and angular micrometer-sized crystals clumpedinto aggregates having a range in porosity.

4.5. Future Work

Further progress in remotely determining physical prop-erties of dust from their light-scattering and emission re-quires new observations and new theoretical and laboratorysimulations.

The most useful observations are likely to be multiwave-length monitoring of comets within a broad range of he-liocentric distances and phase angles. The observations canprovide new insight into the classification of comets accord-ing to their maximum polarization; heliocentric tendenciesin colors (including near-infrared); correlated or anti-cor-related change in such characteristics as color, polarization,and albedo; and thermal emission characteristics (tempera-ture, shape, and strength of a silicate feature) with heliocen-tric distance and within the coma. The correlation betweenlight-scattering and emission properties, e.g., between thevalue of maximum polarization and the strength of the sili-cate feature, might be indicative of the diversity of dustfrom one comet to another. In the framework of light-scat-tering and emission methods, more information about thecomet dust size distribution might be obtained from detailedstudies of forward-scattering characteristics of the scatteredlight (as with, e.g., the Sun-grazing comets detected bySOHO), heliocentric dependence of the dust temperature,and measurements of the SED at far-infrared and submilli-meter wavelengths (or shorter wavelengths at larger helio-centric distances). Spectrophotometric and spectropolari-metric data of improved spectral resolution will be impor-tant to better eliminate the influence of gas contaminationas well as to get more precise values of spectral gradientsof intensity and polarization and their change with wave-length. Good-quality images of comets or high-spatial-reso-lution scans along the coma are necessary to see the divers-ity of cometary dust within the coma (e.g., in jets, shells,etc.) and the dust changes with the distance from the nu-cleus. The latter can result from both dynamic sorting ofparticles by radiation pressure (i.e., can be used to refinemeasurements of the dust size distribution) and sublimationof volatiles (i.e., can be used to gauge sublimation constantsof cometary organics). Detailed data for the innermost coma(within a few thousand kilometers from the nucleus) canprovide properties of the fresh, recently lifted dust and mayallow extrapolation to characterize the nucleus material.More data on the spatial variations of the elliptical polar-

ization and the polarization plane are necessary, not onlyto study any alignment of the dust particles within the coma,but also to obtain information about the shape of the par-ticles and possible optical anisotropy and optical activityof the cometary dust material.

Theoretical and laboratory surveys are necessary to chartthe differences in observable characteristics for aggregatesand irregular polydisperse particles. The surveys should bedirected to simulate not individual observational facts butthe whole scope of the observational data, i.e., angular andspectral characteristics of both scattered and emitted radia-tion. Contribution from submicrometer- and micrometer-sized particles is required by the realistic size distributions,and thus light scattering by aggregates of hundreds andthousands of constituent particles or large (but with size stillbelow geometric-optics validity) irregular particles shouldbe simulated. The influence of the structure of the aggre-gates, the morphology and shape of the constituent parti-cles, as well as the morphology and specific shapes of irreg-ular large particles should be studied. Such studies mostlikely require development of new theoretical techniques orsystematic use of controlled laboratory measurements. Morecomprehensive modeling of cometary dust composition isnecessary: The refractive indexes used in calculationsshould satisfy the confirmed presence of both silicates andabsorbing constituents in the dust and must be consistentwith the Kramers-Kronig relations (Bohren and Huffman,1983, section 2.3.2) across all wavelengths. More realisticmixtures of silicates and carbonaceous materials or core-mantle particles should be a point of special interest. Theaverage, effective refractive index can be considered whenthe silicate and dark materials are mixed on a very smallscale compared to the wavelength, e.g., when an admixturesof FeNi or FeS typical for IDPs (Hanner and Bradley, 2004)in submicrometer silicate grains are simulated. The need forstudies of spectral characteristics of the light scattered byparticles whose material has a realistic wavelength-depen-dent refractive index is apparent. This also refers to labora-tory light-scattering simulations that should address spec-tral dependencies, and we therefore need to concentrate onparticles with realistic and controlled refractive indexesacross the studied spectral range. Simulations should alsoprovide an explanation of stability or diversity of the ob-servational characteristics, as well as correlations betweenthem, and should be consistent with the data obtained fromother studies of cometary dust, e.g., dynamical, evolutional,and cosmic abundance studies, studies of IDPs, and in situmeasurements.

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Physical Properties of Cometary Dust from Light Scattering ... · 578 Comets II 2. OBSERVATIONAL...

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