Physical Properties of Cometary Dust from Light Scattering and Thermal Emission

Kolokolova et al.: Physical Properties of Cometary Dust 577

Ludmilla Kolokolova University of Florida

Martha S. Hanner Jet Propulsion Laboratory/California Institute of Technology

Anny-Chantal Levasseur-Regourd Aéronomie CNRS-IPSL, Université Paris

Bo Å. S. Gustafson University of Florida

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

1. INTRODUCTION which regularities have been found in the variation of these parameters with phase angle, heliocentric distance and The main technique to reveal properties of cometary dust within the coma (section 2). Then we describe theoretical from groundbased observations is to study characteristics light-scattering techniques developed to solve the inverse of the light it scatters and emits. Such a study can include problem, whereby dust physical characteristics are extracted dynamical properties of the dust particles, presuming that from the characteristics of the light it scatters and emits the spatial distribution of the dust resulted from the dust (section 3). Since the concentration of the dust particles in interaction with gravitation and radiation. However, the pri- cometary coma at cometocentric distances corresponding mary way is to study the scattered light or emitted thermal to the resolvable scale in the observations is sufficiently low, infrared radiation to search for signatures typical of spe- the particles can be considered to be independent scatter- cific particles. The main focus of this chapter is a review ers. This means that the intensity of the light (and its other of methods in the interpretation and the extraction of the Stokes parameters, see below) scattered by a collection of data on the size distribution, composition, and structure of dust particles is equal to the sums of the intensity (or re- cometary dust (hereafter referred to as physical properties) spective Stokes parameters) of the light scattered by all the from the observed angular and spectral characteristics of particles individually. We therefore consider the methods the brightness and polarization of scattered and emitted light developed to describe the light scattering by single particles, (hereafter referred to as observational characteristics). We indicating how light-scattering characteristics depend on the will not detail here observational techniques (for this, see, particle shape, size, composition, and structure. In section 3 e.g., the review by Jockers, 1999), but will mainly concen- we also show how the application of these techniques to trate on the results of the observations and methods of their the observational data leads to the current views on the interpretation. physical properties of cometary dust. Section 4 compares We review how observational characteristics such as the our knowledge about cometary dust obtained using light- dust color, albedo, polarization, etc., can be estimated from scattering methods with results obtained using other tech- visual, near-infrared, and mid-infrared observations and niques and outlines future research.

577 578 Comets II

2. OBSERVATIONAL DATA

2.1. Brightness Characteristics of Scattered Light

The most straightforward way to characterize cometary dust is to study how much light it scatters and absorbs and how this depends on the geometry of observations, mainly determined by the phase angle, α (the Sun–comet–Earth angle). The efficiency of scattering and absorption is usually characterized by the albedo and geometrical characterization of the scattering by the angular scattering function. These characteristics, as well as spectral dependence of the scattered light, expressed through the dust colors, are considered below. 2.1.1. Albedo. Single-scattering albedo is defined in the most general way as the ratio of the energy scattered in Fig. 1. Dependence of albedo on the phase angle. The data are all directions to the total energy removed from the incident from (0) Mason et al. (2001); (1) Mason et al. (1998); (2)– beam by an isolated particle (van de Hulst, 1957; Hanner (7) Gehrz and Ney (1992); (8) Grynko and Jockers (2003). The et al., 1981). This definition includes all components (dif- solid line represents the least-squares fit to the data for Comets fracted, refracted, reflected) of the scattered radiation and C/1975 V1 West and C/1980 Y1 Bradfield and is interpolated to can be applied to small particles where the diffracted ra- smoothly connect to the backscattering data for Comet P/Stephan- Oterma (Millis et al., 1982) normalized at the angle α = 30°. diation is spread over a wide angular range and cannot readily be separated. In practice, this definition is not very useful when analyzing cometary dust, because it requires knowledge of the radiation scattered at all directions. Gehrz observed to increase by 50% in Comet P/Halley’s coma dur- and Ney (1992) compared the scattered energy to the ab- ing episodes of strong jet activity (Tokunaga et al., 1986). sorbed energy reradiated in the infrared to derive an albedo Difference in albedo may result from different composi- at the phase angle of observation. This method was used tion, particle size, shape, or internal structure. For example, by a number of observers, and the results are summarized Mason et al. (2001) found that high albedo of the dust in in Fig. 1. The albedo estimates thus obtained for dust in Comet Hale-Bopp is consistent with the domination of small different comets should be compared at the same phase particles in the grain population (see also section 3.4.1). angle since the radiation scattered in the direction of obser- There may be several components of the dust, with differing vation depends on the relative positions of the Sun, Earth, albedos and temperatures, and the average albedo may not and comet. Using such an approach, Mason et al. (2001) represent the actual albedo of any of the components. How- found the albedo of the dust in Comet C/1995 O1 Hale-Bopp ever, the low average albedo rules out a large population of to be roughly 50% higher than that of P/Halley at α ~ 40°. cold, bright grains that contribute to the scattered light but A few maps of albedo obtained so far by combining visi- not to the thermal emission. ble light and thermal infrared images demonstrate increas- 2.1.2. Angular scattering function. The angular scat- ing albedo with the distance from the nucleus (Hammel et tering function indicates how the intensity of the scattered al., 1987; Hayward et al., 1988). light is changing with phase angle. Since a comet can be The geometric albedo of a particle, Ap, is defined as the observed at only one phase angle on a given date, the an- ratio of the energy scattered at α = 0° to that scattered by a gular scattering function has to be acquired by observing a white Lambert disk of the same geometric cross section comet over time as the Sun–Earth–comet geometry changes. (Hanner et al., 1981). Since comets are rarely observed at Thus this function of individual cometary dust particles is α α = 0°, it is convenient to define Ap( ) as the product of difficult to determine, because the observed brightness de- the geometric albedo and the normalized scattering func- pends not only on the physical properties of the dust, but tion at the angle α. Hanner and Newburn (1989) presented also on its amount, and the total cross section of dust in α a plot of Ap( ) in the J bandpass (1.2 µm) for 10 comets. the coma contributing to the scattered light intensity does The total dust cross section within the field of view was not remain constant over time. Two methods have been used determined by fitting a dust emission model to the thermal to normalize the observed intensity. spectral energy distribution, and then applying the total One method is to assume that the ratio of the dust to gas cross section to the scattered intensity to derive an average production rates remains constant over time, and to normal- albedo. The resulting Ap are typically very low, close to ize the scattered light intensity to the gas production rate. 0.025 at α = 35°–80° and about 0.05 at α near zero. There Millis et al. (1982) derived the angular scattering function is some indication that Ap is higher for comets beyond 3 AU for the dusty Comet P/Stephan-Oterma by normalizing the (see Fig. 3 in Hanner and Newburn, 1989). The albedo was scattered intensity to the C2 production rate. The scattering Kolokolova et al.: Physical Properties of Cometary Dust 579

function was a factor of 2 higher at 3°–4° than at 30°, corresponding to a slope of ~0.02 mag/°. Meech and Jewitt (1987) determined a linear slope of 0.02–0.035 mag/° for four comets observed at α = 0°–25°. They saw no evidence of an opposition surge larger than 20% in P/Halley within 1.4°–9°. More detailed data by Schleicher et al. (1998) show an evident curvature of the scattering function for P/Halley that can be represented by a quadratic fit. Another method is to compare the measured scattered light to the measured thermal emission from the same volume in the coma, with the assumption that the emitting properties of the dust remain constant. This method was described above as the means for extracting an albedo for the dust. The method has been applied to a number of comets (see Fig. 1) and yields a relatively flat scattering function from 35° to 80° (Tokunaga et al., 1986; Hanner and Newburn, 1989; Gehrz and Ney, 1992). Two comets have Fig. 2. Color as a function of heliocentric distance: J–H (+), H– been observed at large phase angles 120°< α < 150° using K ( ), and B–R (×) colors are by Jewitt and Meech (1986); B–R this method, and they display strong forward scattering (Ney colors ( ) are by Kolokolova et al. (1997) ( for Comet P/Halley). and Merrill, 1976; Ney, 1982; Gehrz and Ney, 1992). Thus Color is measured in ∆m/∆λ (mag/µm) and the solar color is sub- we can characterize a typical angular scattering function for tracted. The data cover the range of the phase angles 0°–110°. cometary dust as possessing a distinct forward-scattering surge, a rather gentle backscattering peak, and a flat shape at medium phase angles (Fig. 1). K color was less red in comets observed at R > 3 AU, while 2.1.3. Color. The dust color indicates trends in the the J–H color showed no trend with heliocentric distance. wavelength dependence of the light scattered by the dust. The B–R color shows no dependence on heliocentric dis- Traditionally the color of cometary dust was determined tance although Schleicher and Osip (2002) found that the through measurements of the comet magnitude m in two color m0.4845–m0.3650 for Comet Hyakutake (1996 B2) got different continuum filters, e.g., blue (B, ~0.4–0.45 µm) and redder with increasing heliocentric distance within 0.6– µ red (R, ~0.64–0.68 m), and was expressed as CB–R = mblue– 1.9 AU. No color dependence on phase angle was found in mred. This color was a unitless characteristic expressed as the visible or near-infrared. the logarithm of the ratio of intensities in two filters. Al- Decrease in the near-infrared colors was recorded at peri- though this definition is still used, spectrophotometry of ods of enhanced comet activity in Comet P/Halley (Toku- comets resulted in the definition of color as the spectral gra- naga et al., 1986; Morris and Hanner, 1993) A more red dient of reflectivity, usually measured in % per 0.1 µm with color in the visible was associated with strong jet activity an indication of the range of wavelength it was measured in. in Comet P/Halley (Hoban et al., 1989), but the spiral struc- The determination of cometary dust colors requires use tures in Comet Hale-Bopp had less red color (Jockers et al., of continuum bands that are truly free of gas contamina- 1999). Less red and even blue color was associated with tion [see discussion on gas-contamination influence on the outburst in Comet C/1999 S4 LINEAR during its disruption colors in A’Hearn et al. (1995)]. This tends to make colors (Bonev et al., 2002). Although no regular dependence on obtained using filters less trustworthy than those obtained the field of view was found, a smooth change in color was using spectrophotometry to target gas-free spectral regions. observed in the central part of the coma of some comets, The exception is near-infrared colors that are considered to including Hale-Bopp (Jockers et al., 1999; Laffont et al., be free from gas contamination and thus more reliable, al- 1999; Kolokolova et al., 2001a). though thermal emission from the warm dust will contribute to the K (2.2 µm) bandpass for comets within 1 AU of the 2.2. Polarization from Cometary Dust Sun. We summarize the data for B–R color in the visible and J–H and H–K color in the near-infrared in Fig. 2. The light scattered by particles is usually polarized, i.e., The scattered light is generally redder than the Sun; the its electromagnetic wave has a preferential plane of oscil- reflectivity gradient decreases with wavelength from 5–18% lation. The degree of linear polarization (P, hereafter called per 0.1 µm at wavelengths 0.35–0.65 µm to 0–2% per polarization) is defined as 0.1 µm at 1.6–2.2 µm (Jewitt and Meech, 1986). Hartmann et al. (1982) and Hartmann and Cruikshank (1984) found P = (I⊥ – I||)/(I⊥ + I||) that the near-infrared dust colors depend on heliocentric distance. However, data by Jewitt and Meech (1986) and where I⊥ and I|| are the intensity components perpendicular Tokunaga et al. (1986) demonstrate no heliocentric depen- and parallel to the scattering plane. For randomly oriented dence. Hanner and Newburn (1989) noted that only the H– particles the electromagnetic wave predominantly oscillates 580 Comets II either perpendicular (by convention positive polarization) or parallel (by convention negative polarization) to the scattering plane. The convenience of polarization is that it is already a normalized characteristic of the scattered light, whereas the brightness, I = I⊥ + I||, depends upon the distances to the Sun and observer, and the spatial distribution of the dust particles. Polarization variations within a coma relate to changes in the dust physical properties, and different polarization observed for different comets or in features (jets, shells, etc.) indicates a diversity of dust particles. For a given type of cometary dust particles, the polarization mainly depends on the phase angle α and wavelength λ. 2.2.1. Angular dependence. The changing geometry of the comet and observer with respect to the Sun defines a polarization phase curve. Small phase angles can be ex- plored for distant comets, while α > 90° can only be reached at R < 1 AU. Most of the available polarization observations Fig. 3. Polarization in the narrow-band red filter vs. phase angle: have been performed within the heliocentric distances 0.5– (×) comets with low maximum in polarization, (+) comets with 2 AU using groundbased instruments, although large phase higher maximum, ( ) Comet C/1995 O1 Hale-Bopp, ( ) Comet angles were studied using the Solar and Heliospheric Ob- C/1999 S4 LINEAR at disruption. servatory (SOHO) C3 coronagraph (Jockers et al., 2002). The data, which represent an average value of the polarization on the projected coma, are obtained either by aperture polarimetry (e.g., Dollfus et al., 1988; Kiselev and divide into three classes corresponding to different maxima Velichko, 1999) or deduced from the integrated flux of the in polarization (Fig. 3): (1) comets with a low maximum, polarized brightness images (e.g., Renard et al., 1996; about 10–15% depending on the wavelength; (2) comets Kiselev et al., 2000; Hadamcik and Levasseur-Regourd, with a higher maximum, about 25–30% depending on the 2003). The (real or virtual) diaphragm is centered on the wavelength; and (3) Comet C/1995 O1 Hale-Bopp, whose center of brightness, which corresponds to the nucleus. Its polarization is distinctively higher, although it was not ob- aperture, in terms of projected distance on the cometary served for α > 48°. coma, needs to be large enough to include the coma fea- Numerous observations obtained by Dollfus et al. (1988) tures that may alter the global polarization. It is usually for P/Halley showed that the polarization at a given phase found that, once the aperture is sufficiently large, the re- angle and wavelength does not vary with heliocentric dis- sulting polarization of the comet does not change with in- tance. Some transient increases in polarization were found creasing aperture. correlated with cometary outbursts. Similarly, the polariza- The phase dependence of polarization, which has been tion of Comet Hale-Bopp obtained at small phase angles documented so far in the range 0.3°–122°, is smooth and (large heliocentric distances) was consistent with that ob- similar to that of atmosphereless solar system bodies. All tained closer to the Sun, but increased after outbursts events comets show a shallow branch of negative polarization at (Ganesh et al., 1998; Manset and Bastien, 2000). Observa- the backscattering region, first observed by Kieselev and tions of Comet C/1999 S4 LINEAR during its near-perihelion α Chernova (1978), that inverts to positive polarization at 0 ~ disruption (Kiselev et al., 2002; Hadamcik and Levasseur- 21° with a slope at inversion h ~ 0.2–0.4%/°, and a posi- Regourd, 2003) indicated an increase in polarization of about tive branch with a broad maximum near 90°–100°. The data 4%, a value comparable with what had been noticed dur- may be fitted by a typically fifth-order polynomial or trigo- ing the P/Halley outbursts. nometric function, e.g., as suggested by Lumme and Muino- 2.2.2. Spectral dependence. The data have been re- α α a α b α α nen (1993), P( ) ~ (sin ) (cos /2) sin( – 0). These func- trieved in the ultraviolet, visible, and near-infrared domains tions cannot be used for extrapolation but only within the to avoid any contribution from the thermal emission. Nar- phase angle range where well-distributed data points are row-band cometary filters are now available at 0.3449, available. 0.4453, 0.5259, and 0.7133 µm (Farnham et al., 2000). While the minimum polarization is typically –2% (Mukai However, observers often need to make a trade-off between et al., 1991; Chernova et al., 1993; Levasseur-Regourd et al., narrow filters that remove gaseous emissions and wider fil- 1996), a significant dispersion is noticed for α > 30°–40°. ters that improve the signal-to-noise ratio. It is highly advis- Once the data are separated in different wavelength ranges, able to analyze spectra of the observed comet and estimate the dispersion on the positive branch is reduced, and Pλ(α) the contribution of some faint molecular lines to allow for for a variety of comets can be compared (see Levasseur- the depolarizing effect of molecular emissions when cal- Regourd et al., 1996, and references therein). Comets tend to culating the dust polarization (Le Borgne and Crovisier, Kolokolova et al.: Physical Properties of Cometary Dust 581

1987; Dollfus and Suchail, 1987; Kiselev et al., 2001). large phase angles was observed only for Comet P/Giacobini- Spectropolarimetric cometary data remain rare, and so far Zinner (Kiselev et al., 2000). Polarimetric color of variable their resolution is not better than the resolution provided sign was observed for Comet C/1999 S4 LINEAR during by narrowband cometary filters (Myers and Nordsieck, its disruption (Kiselev et al., 2002). 1984). 2.2.3. Variations within the coma. A coma whose op- It was noticed for P/Halley (Dollfus et al., 1988) and tical thickness exceeds 0.1 was found so far only for Comet later confirmed for other comets (Chernova et al., 1993; Hale-Bopp and only at distances closer than 1000 km from also Kolokolova and Jockers, 1997, and references therein) the nucleus (Fernández, 2002). Thus, except for possibly that the polarization usually increases with increasing wave- the very innermost coma where multiple scattering might length, at least in the visible domain, and the increase grows take place, a change in the polarization with the distance for greater values of the polarization. from the nucleus indicates an evolution in the physical prop- If polarization data are available for two wavelengths, erties of the particles ejected from the nucleus. Polarization the spectral gradient of polarization is defined as polarimet- images of P/Halley (Eaton et al., 1988; Sen et al., 1990), ∆ ∆λ λ λ λ λ ric color, P/ = [P( 2) – P( 1)]/[ 2 – 1]. C/1990 K1 Levy (Renard et al., 1992), 109P/Swift-Tuttle The polarimetric color changes with the phase angle (Eaton et al., 1995), 47P/Ashbrook-Jackson (Renard et al., (Chernova et al., 1993; Kolokolova and Jockers, 1997) from 1996), and C/1995 O1 Hale-Bopp (Hadamcik et al., 1999; zero and even negative values to gradually increasing posi- Jockers et al., 1999; Kolokolova et al., 2001a); variable- tive values at α > 50°. Figure 4 summarizes systematic aperture polarimetry of P/Halley (Dollfus et al., 1988); and multiwavelength observations of polarization, performed for high-resolution in situ observations with the Optical Probe Comets Halley and Hale-Bopp. Within α = 25°–60° the po- Experiment (OPE) onboard the Giotto spacecraft (Renard larization increases with the wavelength in the visible et al., 1996; Levasseur-Regourd et al., 1999) revealed a re- domain; the gradient seems to decrease in the near-infra- gion in the innermost coma characterized by a lower polari- red [Fig. 4; see also Hadamcik and Levasseur-Regourd zation. Comet Hale-Bopp could be observed at phase angles (2003)]. For α ~ 50°, the polarimetric color is about 9%/ ranging from about 7° to 48°. The low polarization region µm for Halley and 14%/µm for Hale-Bopp (Levasseur- in Comet Hale-Bopp has been monitored within this phase- Regourd and Hadamcik, 2003), which agrees with the value angle range (Hadamcik and Levasseur-Regourd, 2003). It 11%/µm at 45° obtained by Kiselev and Velichko (1999). showed highly negative values of polarization at small phase The polarimetric gradient for Comet Hale-Bopp is higher angles, e.g., –5% at α = 8°, whereas a typical value should than for other comets but its sign and phase-angle trend are be –0.5%. The low negative values of polarization, noticed typical for comets, whereas asteroids and other atmosphere- at the cometocentric distances less than 2000 km, could not less bodies usually have negative polarimetric color (e.g., be explained by multiple scattering but is most likely due Mukai et al., 1997). So far a negative polarimetric color at to 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), Mukai et al. (1987), Sen et al. (1990)] for phase angles 40° (bottom curve) and 50°. 582 Comets II

A further proof of different physical properties of the (Dollfus and Suchail, 1987; Metz and Haefner, 1987; Rosen- dust near the nucleus is its negative polarimetric color ob- bush et al., 1999; Manset and Bastien, 2000; Rosenbush and served in situ (OPE/Giotto) (Levasseur-Regourd et al., 1999) Shakhovskoj, 2002). and a smaller polarimetric color in the innermost coma sus- The existence of the faint circular polarization, variations pected for Comet Hale-Bopp (Jockers et al., 1999). Some in the plane of polarization, and nonzero polarization ob- comets do not show lower polarization [e.g., C/1989 X1 served at forward-scattering direction during stellar occul- (Eaton et al., 1992), C/2001 A2 (Rosenbush et al., 2002)] tations (Rosenbush et al., 1994) may be indications of in the innermost coma, possibly hidden by highly polarized aligned elongated or optically anisotropic particles in comet- dust jets or, as Kiselev et al. (2001) suggest, resulted from ary atmospheres. strong gas contamination. Aperture polarimetry with an offset of the center, as well 2.3. Dust Thermal Emission as cuts through polarization images, may be used to study the variation of the polarization with the cometocentric dis- Thermal data provide two types of results that are sen- tance. The increase within the jets and decrease in the inner- sitive to the physical properties of the particles: the ther- most coma are clearly visible on such graphs, and some mal spectral energy distribution (SED) and the detection of differences may be noticed between the sunward and anti- spectral features (e.g., the silicate feature at 10 µm and 16– sunward side. Although the data are line-of-sight integrated, 30 µm). In addition, the ratio of the thermal energy to the they emphasize the temporal evolution in the physical prop- scattered energy provides a measure of albedo, as described erties of the dust particles ejected from the nucleus. Except previously. for the above-mentioned features, the polarization decreased Thermal emission from the dust in the coma results from gradually within 5000–8000 km from the nucleus for C/ absorption of the solar radiation followed by its reemission. 1996 B2 Hyakutake and C/1996 Q1 Tabur (Kolokolova et al., The thermal radiation from a single grain depends upon its 2001a) and increased for P/Halley (Levasseur-Regourd et al., temperature, T, and wavelength-dependent emissivity, ε 1999) and Hale-Bopp (Kolokolova et al., 2001a), accom- panied by similar trends in the B–R color. F(λ) = (r/∆)2ε(r,λ)πB(λ,T) (1) The polarization images often reveal elongated fan- shaped structures with polarization higher than in the sur- where r is some specific dimension of the particle, e.g., its rounding coma. They generally correspond to bright jet-like radius, ∆ the geocentric distance, and B(λ,T) the Planck features on the brightness images. This effect has been function for grain of temperature T. The observed SED from noticed for quite a few comets, e.g., 1P/Halley (Eaton et al., the ensemble of grains in a coma having differing tempera- 1988), C/1990 K1 Levy (Renard et al., 1992), 109P/Swift- tures and wavelength-dependent emissivities generally is Tuttle (Eaton et al., 1995), 47P/Ashbrook-Jackson (Renard similar to (but broader than) a blackbody curve (Fig. 5). The et al., 1996), or C/1996 B2 Hyakutake (Tozzi et al., 1997). SED can be characterized by a single temperature (the color The case of the bright and active Comet Hale-Bopp is most temperature, Tc) over a defined wavelength interval, but it documented; straight jets at large heliocentric distances are is important to remember that the color temperature is not seen to evolve in arcs or shells around perihelion passage the physical temperature of the grains. (see, e.g., Hadamcik et al., 1999; Jockers et al., 1999; The physical temperature of a particle in the solar radia- Furusho et al., 1999). Such conspicuous features could be tion field depends strongly on the latent heat of sublima- produced by an alignment of elongated particles or by tion and sublimation rate of any material that may be sub- freshly ejected dust particles different from the particles in limating, but once the sublimation has ceased and thermal the regular coma. equilibrium is established, the temperature depends on the 2.2.4. Plane of polarization and circular polarization. balance between the solar energy absorbed at visual wave- The plane of polarization is in almost all cases perpendicular lengths and the energy radiated in the infrared. For a particle or parallel to the scattering plane, indicating randomly ori- of arbitrary shape we may write the energy balance equation ented particles. However, some variations by a few degrees –2 ∫ λ λ λ ζ ∫ π λ λ λ may be noticed with changing aperture size (Manset and R Cabs(r,m, )S( )d = B( ,T)Cabs(r,m, )d (2) Bastien, 2000). Deviations in the polarization plane were noticed for Comet P/Halley near the inversion angle (Beskrov- where S(λ) is the solar flux at 1 AU, R is the heliocentric naya et al., 1987); they were mapped by Dollfus and Suchail distance in AU, and the factor ζ is the ratio of the total ra- (1987). Changes in the polarization plane often accompany diating area to the exposed area. This ratio is equal to 4 for the polarization variations at the outburst activity and were spheres and any convex particle averaged over random ori- registered in Comets 29P/Schwassmann-Wachmann 1 entations (van de Hulst, 1957, section 8.41). The crucial λ (Kiselev and Chernova, 1979), P/Halley (Dollfus et al., parameter in equation (2) is Cabs(r,m, ), the absorption cross 1988), C/1990 K1 Levy (Rosenbush et al., 1994), and C/ section, which depends on grain size, morphology, and opti- λ 2001 A2 LINEAR (Rosenbush et al., 2002). For Comets cal constants. By Kirchoff’s law, Cabs(r,m, ) is equal to the Halley, Hale-Bopp, and S4 LINEAR, the circular polariza- emissivity, ε(r,λ), times the emitting area. We assume that tion was estimated and found to be nonzero but below 2% the coma is optically thin (no attenuation of sunlight and no Kolokolova et al.: Physical Properties of Cometary Dust 583

body temperature and a drop in the average grain emissivity at 60 and 100 µm, indicating that the flux contribution at 60 and 100 µm was primarily from particle sizes smaller than the wavelength (see section 3.1). Continuum emission at submillimeter wavelengths has been measured for several comets including Hale-Bopp (Jewitt and Luu, 1990, 1992; Jewitt and Matthews, 1997, 1999), indicating the presence of millimeter-sized particles. In addition to the smoothly varying infrared continuum emission, broad spectral features attributed to small silicate grains are seen in some comets near 10 µm (Fig. 5) and 18 µm. The strength of the features depends upon grain size and temperature (Hanner et al., 1987). The 16–45-µm spectra of Hale-Bopp recorded by the Short Wavelength Spec- trometer (SWS) onboard the ISO satellite (Crovisier et al., Fig. 5. Spectral energy distribution for Comet Hale-Bopp (Wil- 1997) displayed five distinct spectral peaks that correspond liams et al., 1997). The left curve is the solar spectrum, the right to laboratory features of crystalline Mg-olivine (forsterite) curve is the continuum fit to the sum of a 5800 K blackbody com- (see Hanner and Bradley, 2004). Not all comets display a ponent due to scattered solar radiation and a 475 K blackbody due strong 10-µm silicate feature (Hanner et al., 1994; Hanner to thermal emission. The heavy solid line shows the silicate fea- and Bradley, 2004). In particular, the feature tends to be ture observed with the HIFOGS (High-efficiency Infrared Faint weak (~20% above the continuum) or absent entirely in Object Grating Spectrometer). Although Comet Hale-Bopp was Jupiter-family comets (e.g., Hanner et al., 1996; Li and not a typical comet, the figure provides an illustration of the main characteristics of cometary dust thermal emission. Greenberg, 1998a).

2.4. Correlations Among Scattering and heating by other grains) and that sublimation is negligible. Emitting Observational Characteristics The observed color temperature can be compared with the SED predicted from equations (1)–(2) for grains of various It has been well established that the observable scatter- sizes and optical properties in order to infer the physical ing parameters of cometary dust tend to correlate. These characteristics of the cometary dust. correlations can provide insight into the relative importance The 3–20-µm thermal SED has been observed for many of particle size, composition, and structure on the observ- comets over the past three decades, with improving pre- able scattering properties. cision. P/Halley was monitored regularly at R < 2.8 AU We have already noted that comets divide into some (Tokunaga et al., 1986, 1988; Gehrz and Ney, 1992) and classes, based upon their maximum polarization (Fig. 3). Hale-Bopp was observed from the ground and from Earth The high Pmax comets are mostly comets with a strong dust orbit at 0.9–4.9 AU (Mason et al., 2001; Grün et al., 2001; continuum, while the low Pmax comets are comets with a Hanner et al., 1999; Wooden et al., 1999; Hayward et al., weak dust continuum (Chernova et al., 1993). Comets ex- 2000). The 5–18-µm color temperature is typically 5–25% hibiting higher polarization generally display a stronger higher than a perfect blackbody at the same heliocentric silicate feature (Levasseur-Regourd et al., 1996), indicating –0.5 distance that can be calculated as T = 280 * R . that small silicate particles have a major influence on Pmax Comets display a heliocentric dependence, T ~ R–0.5; in (see section 3.1). The strength of the silicate feature also the case of P/Halley the 8–20-µm color temperature varied correlates with higher color temperature and stronger flux as T = 315.5 * R–0.5 (Tokunaga et al., 1988). The comets at 3–5 µm (Gehrz and Ney, 1992), both indicators of hot, with the strongest dust emission, such as P/Halley and Hale- submicrometer-sized absorbing grains (see section 3.1.1). Bopp, display a higher color temperature at 3.5–8 µm than The fact that the B–R, J–H, and H–K colors do not cor- at 8–20 µm (Tokunaga et al., 1988; Williams et al., 1997). relate, e.g., show different tendencies with the heliocentric Thermal emission observations in the far-infrared and distance or with the distance from the nucleus, supports the submillimeter spectral regions can potentially provide in- idea that the changes in the colors cannot be explained by formation about the larger particles in the coma (see sec- simple change of particle size or smooth change in the tion 3.1). Far-infrared observations at λ < 200 µm have been optical constants. A smooth simultaneous change in B–R acquired by the InfraRed Astronomical Satellite (IRAS), color and polarization was observed in the innermost coma Cosmic Background Explorer (COBE), and Infrared Space of some comets (Kolokolova et al., 2001a), indicating a Observatory (ISO) satellites. The Diffuse InfraRed Back- smooth change in some dust properties that occurs as dust ground Experiment (DIRBE) instrument on COBE meas- is moving out of the nucleus. ured the SED from 3.5 to 100 µm for three comets (Lisse Hanner (2002, 2003) summarized the correlations and et al., 1994, 1998). Among these, only Comet Levy (C/1990 compared the scattering properties in Comets P/Halley and K1) had a color temperature 10% higher than the black- Hale-Bopp: The dust in Hale-Bopp displayed higher po- 584 Comets II larization at comparable α, redder polarimetric color, higher The scattering matrix F is defined by the particle prop- albedo, stronger silicate feature, higher infrared color tem- erties, thus the inverse problem can be solved if the ele- perature compared to the blackbody temperature, and higher ments of the scattering matrix, obtained from observations, 3–5-µm thermal emission. can be also found from a theory that describes scattering by particles, i.e., most rigorously, from Maxwell’s equa- 2.5. Summary tions. To determine all 16 elements of the matrix F one must obtain four independent combinations of vectors I0 and I, The observational characteristics of cometary dust are but since sunlight is incoherent and essentially unpolarized, extremely variable and depend on phase angle, heliocen- this is usually not possible from astronomical observations. tric distance, and position within the coma, especially as- In practice, one therefore fits only the two upper-left ele- sociated with comet activity (jets, fans, etc.). The most ments, or equivalently the intensity and polarization. While complex characteristic is the brightness that depends on the the intensity or brightness is plagued by sensitivity to varia- spatial distribution of dust particles, and thus to a large tions in the particle number density, the polarization is a ratio extent is affected by temporary variations in dust produc- of intensities and is therefore independent of the amount tion rate. Parameters, formed as ratios of brightness such of dust. These observational constraints make it even more as color, polarization, and albedo, are more suitable for important to fit all available observations and to extend the studying physical properties of the dust grains. However, observations to as broad a range of phase angles as possible measured using aperture instruments, they demonstrate av- and wavelengths as well as thermal emission spectra. This erage characteristics related to a mixture of particles of a still cannot guarantee the uniqueness of the solution, espe- variety of properties and can vary due to comet activity. cially as the dominating grains in a mixture of grain types Coma images obtained in narrowband continuum filters pro- can vary strongly both with wavelength and with scatter- vide the most adequate source of information about prop- ing angle. Therefore some authors emphasize the impor- erties of cometary dust. tance in finding a solution that also adheres to limitations The general regularities found in observational charac- set by, for example, cosmic abundances (Greenberg and teristics of cometary dust that may be used as a basis for Hage, 1990) and constraints from the dynamics of the par- studying physical properties of the dust particles are listed ticles (Gustafson, 1994). However, in this chapter we focus later in this chapter (see Table 2). Variability of these char- on constraints from optical and infrared observations. acteristics within the coma, correlations between them, and For electromagnetic waves propagating in a homoge- exceptions to the listed rules are also an interesting subject neous medium, characterized by its complex refractive in- for light-scattering modeling. dex m = n + iκ, Maxwell’s equations can be reduced to

3. LIGHT-SCATTERING THEORETICAL curl H = ikm2E (3) AND EXPERIMENTAL METHODS curl E = –ikH

Study of cometary dust using its interaction with solar Since the magnetic vector H can be expressed through radiation is a typical example of remote sensing that usu- the electrical vector E, equation (3) can be further reduced ally is associated with the scattering inverse problem. The to one so-called Helmholtz equation sought quantities define a set of particle parameters such as size, shape, and composition (refractive index) from the ∇2E + k2E = 0 (4) scattering/emission data; the set can include the parameters of more than one type of particle. The rigorous way to solve The electromagnetic vector E is related to the Stokes pa- the inverse problem would be: rameters through the values of its two complex perpendicu- • to measure the characteristics of the scattered light (in- lar amplitudes E|| and E⊥ and their conjugates as (van de tensity, I; degree of linear polarization, P; position of Hulst, 1957; Bohren and Huffman, 1983) the plane of polarization, defined by the angle θ; degree of circular polarization, V) at a given wavelength, I=E|| |E*|| + E⊥ E*⊥ λ α , and given phase angle, ; Q= E|| E*|| – E⊥ E*⊥ • to calculate the Stokes vector I = (I, Q, U, V) of the U= E|| E*⊥ + E⊥ E*|| scattered light whose components are related to the V= i (E|| E*⊥ – E⊥ E*)|| observed characteristics as I = I, P = (Q2 + U2)/I , tan(2θ) = Q/U, V = V/I; In principal, a complete solution to the problem can be • from the Stokes vectors of the scattered and incident achieved if we consider equation (4) outside the particle, light, I and I0, determine elements of the scattering where the electromagnetic field is a superposition of inci- 2 matrix F through (I, Q, U, V) = F/(kR) * (I0, Q0, U0, dent and scattered fields, and inside the particle (internal V0), where R is the distance between the detector and field), and apply the boundary conditions. The boundary the scatterer and k is the wave number in empty space conditions (van de Hulst, 1957, section 9.13) mean that any equal to k = 2π/λ. tangential and normal components of E are continuous Kolokolova et al.: Physical Properties of Cometary Dust 585

across the two materials at the particle boundary. Thus to color, or temperature. However, the calculations for spheres solve the problem we need to know the shape of the bound- can be used in a qualitative manner to narrow the range of ary, i.e., the particle shape. The particle shape means a cometary grain parameters, as illustrated by the use of three- mathematically sharp boundary between a particle interior dimensional or color-contour graphics of the type shown in and empty space. We therefore tend to distinguish between Fig. 6 (see also numerous examples in Hansen and Travis, particle shape and structure (surface and internal). Avail- 1974; Mishchenko and Travis, 1994; Mishchenko et al., able solutions are specific to classes of particle shapes, e.g., 2002). From Fig. 6 one can see that light scattering from spheres, cylinders, and spheroids. These are usually for cometary dust is not dominated by particles much smaller homogeneous and isotropic internal structures, although than the optical wavelengths, since for such particles polari- solutions for some anisotropic or inhomogeneous structures zation is always positive with very high maximum polariza- have been derived for some specific shapes. tion at α = 90°. Also, cometary dust cannot be represented by an ensemble of monodisperse spherical or quasispherical 3.1. Light Scattering by Homogeneous particles of medium size (1 < x < 20) since their light-scat- Spherical Particles (Mie Theory) tering characteristics experience the oscillating behavior seen even in Fig. 6 (although the oscillations are smoothed Mie theory [attributed to Mie (1908); credit is also usu- by the size distribution). Note also that for spherical par- ally given to earlier works by Clebsch, Lorentz, Debye, and ticles, strong color dependence on the phase angle is seen others (see Kerker, 1969)] accurately predicts the scattering for all sizes and the polarimetric color is mainly negative. by any homogeneous and isotropic sphere. It solves equa- The contour graphics and a more quantitative approach tion (4) in spherical coordinates (r, θ, ϕ) using the separa- based on statistical multifactor analysis (Kolokolova et al., tion of variables technique, i.e., presenting the electromag- 1997) can be used to estimate influence of dust properties netic wave E as on observational characteristics. As a result of such an analysis, the reason for the trends observed within the coma E = E(r,θ,ϕ)e = Φ(ϕ)Θ(θ)R(r)e and correlations between light-scattering parameters can be found. For example, such an approach (Kolokolova et al., where vector e indicates the direction of oscillations. This 2001a) showed that the correlation between color and po- gives E in the form of a sum of products of trigonometric larization observed in the innermost coma of Comets Tabur, and spherical Bessel functions and associated Legendre Hyakutake, and Hale-Bopp cannot be a result of changing polynomials with the size parameter x = kr (where r is the particle size, but is more likely an indication of changing particle radius) and refractive index of the particle in the composition. argument. In practice, one has to take care with the numeri- For certain applications, we might expect that arbitrarily cal code in order to not degrade the accuracy for very small shaped, randomly oriented particles can be approximated or very large particles. Detailed discussion on Mie solution by equal-volume, equal-projected-area, or, in the case of can be found in van de Hulst (1957) and Bohren and Huff- convex particles, equal-surface-area spheres. They appear man (1983). The latter book also contains a simple but effi- appropriate when some specific term dominates the scat- cient Mie code. A modern code is available from M. Mish- tering. For example, we might expect scattering in the for- chenko at ftp://ftp.giss.nasa.gov/pub/crmim/spher.f. ward domain (large phase angles) to be dominated by dif- Even when the shape and structure of the particle is as fraction, which is strongly dependent on the particle size simple as that of a homogeneous sphere, the complexity of since only the particle cross section enters calculations of the solution does not allow the scattering properties of a the diffraction pattern (van de Hulst, 1957, section 3.3). De- particle to be expressed through a simple analytical func- pending on the absorption of the particle material at some tion of the size and refractive index. The numerically de- particle sizes, the transmitted light starts reducing the mag- rived solutions are also generally found to not be unique nitude of the forward-scattering peak but does not strongly to one combination of particle size and refractive index. The affect the angular distribution of the intensity. In particu- problem of uniqueness is more acute if the observations lar, the angle of the first minimum in intensity counted from cover only a limited range of phase angles. Thus the inverse the direction of backscattering seems to be a good size in- problem is often solved indirectly using analysis of the dicator even for aggregated particles (Zerull et al., 1993). qualitative light-scattering behavior and incorporating other Scattering by particles of size x << 1 is proportional to empirical constraints on particle properties. the square of their polarizability (see section 3.2) and there- Use of Mie theory has helped exclude spherical and fore to their volume squared (van de Hulst, 1957, section homogeneous particles as the main component of cometary 6.11) so that equal volume particles in random orientation dust. Even if some specific size distribution and refractive scatter identically. For the other extreme of large convex index could fit some observations (e.g., polarization) at particles in the geometric optics regime van de Hulst (1957, some wavelengths (see examples in Mukai et al., 1987), the section 8.42) showed that the scattering caused by reflec- same characteristics of spherical particles could not provide tion from randomly oriented particles is identical with the a reasonable fit over a broad range of wavelengths and to scattering by a sphere of the same material and surface area. other observational parameters, e.g., scattering function, 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 the refractive 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 see Mishchenko and Travis, 1994). Bottom panel: Same for color and polarimetric color but instead of size parameter the effective size of particle varies at the fixed wavelengths of 0.45 and 0.65 µm. Darker colors indicate smaller values. Kolokolova et al.: Physical Properties of Cometary Dust 587

cal cross section as the sphere. The approximations work wavelengths greater than about 10 times their size. They better the closer the particle shape is to spherical. For ex- heat up until the energy radiated at the shorter infrared ample, an error in the absorption cross section is less than wavelengths (roughly 3–8 µm for comets at 0.5–3 AU from 5% for a spheroid of the axes ratio a : b = 1.4 regardless of the Sun) balances the absorbed energy. In fact, for small the particle size, but can exceed 15% for a spheroid with absorbing grains, their size controls their temperature, re- the axes ratio a : b = 2 at x = 1 (Mishchenko et al., 1996). gardless of their specific composition (Hanner, 1983). Important limitations can be high porosity of the particles In contrast to carbon grains, silicate grains radiate effi- (Henning and Stognienko, 1993) or the presence of sharp ciently in the infrared because of the resonance in their edges (Yanamandra-Fisher and Hanner, 1998). optical constants near 10 and 20 µm; the amount of absorp- 3.1.1. Use of Mie theory to estimate grain emissivity at tion at visual wavelengths controls their temperature. The thermal infrared wavelengths. As we will see below, the absorption at visual wavelengths depends strongly on the thermal emission by cometary dust comes mainly from Fe content of the silicates (Dorschner et al., 1995). Pure submicrometer particles. Therefore the equal-volume-sphere Mg-rich silicates have very low absorption; the imaginary representation provides a good estimate for the absorption part of the refractive index κ ~ 0.0003 at 0.5 µm for a glass cross section as it works for particles that are small in all with Fe/Mg κ ~ 0.05 at 0.5 µm for a pyroxene glass with dimensions, i.e., represent quasi-equidimensional (not nee- Fe/Mg = 1 and k ~ 0.1 for an olivine glass with Fe/Mg = 1.0. dles or disks) compact particles of x << 1. It is also a good Consequently, pure Mg silicates in a comet coma near 1 AU approximation for submicrometer particles in a porous ag- would be much colder than a blackbody while Fe-rich sili- gregate so that the interaction between particles is weak. cate grains would be warmer than a blackbody. Composi- Using the thus estimated absorption cross section Cabs one can tional measurements indicate that the silicates in comets are use equation (2) to find the temperature for grains of a vari- Mg-rich (Hanner and Bradley, 2004), so one might expect ety of sizes and composition (Mukai, 1977; Hanner, 1983). them to be cold. However, even a small admixture of ab- This revealed a set of general regularities in thermal emis- sorbing material can increase the temperature significantly. sion from small grains shown in Fig. 7. The computed temperature for a submicrometer absorb- Particles of absorptive materials, e.g., glassy carbon or ing grain varies approximately as R–0.35 instead of observed magnetite, showed temperatures higher than for a black- R–0.5 (Fig. 7) that would be also expected for a blackbody body; the smaller the particles, the higher the temperature. in equilibrium. Whether this indicates a change in size dis- Particles made of transparent materials were found to be tribution with R, dominance by larger particles or the short- cooler than a blackbody and their temperature did not de- coming of the computations for small spheres is not clear. pend much on particle size. Small grains of an absorbing Chances are that this is a result of combination of optical (e.g., carbonaceous) material absorb strongly at visual wave- characteristics of both small and large particles exhibited by lengths, but cannot radiate efficiently in the infrared at aggregates of small grains (see section 3.3.3). The observed 3–20-µm SED for comets cannot generally be fit by the F(λ) computed for a single grain of any size or composition. However, models using a size distribution of absorbing particles dominated by micrometer- to submicrometer-sized grains provide a match to the observed SED (Li and Greenberg, 1998b; Hanner et al., 1999). Com- ets with the strongest dust emission (such as P/Halley and Hale-Bopp) that also display a higher color temperature at 3–8 µm than at 8–20 µm indicate an enhanced abundance of hot, submicrometer-sized grains. By fitting the observed SED with a model of thermal emission from a size distribution of grains, the total dust mass within the coma can be estimated. When combined with a size-dependent velocity distribution for the dust, the rate of dust production from the nucleus can also be estimated (Hanner and Hayward, 2003). Extensive observations of Comet Hale-Bopp from the European Space Agency’s ISO (the photometer PHOT measured the thermal flux through filters at 7–160 µm, Fig. 7. Temperature of a grain vs. heliocentric distance. Dotted while the two spectrometers recorded the spectrum from 5 lines show results for absorptive particles (glassy carbon) of ra- to 160 µm) enabled the SED to be obtained at long wave- dius 0.1, 0.5, and 10 µm. Dashed lines are for olivine particles of radius 0.1 and 10 µm. The thin solid line is the blackbody tem- lengths and fit by an emission model having a size distri- –a ≤ perature. The thick solid line shows the results for Comet Halley bution of the form n(r) ~ r , with a 3.5 (Grün et al., (Tokunaga et al., 1988). The comet dust temperature is higher than 2001). For an outflow velocity v(r) ~ r–0.5, this result implies a blackbody, indicating the presence of submicrometer absorptive that the dust production size distribution from the nucleus particles. had a slope a ≤ 4 and the mass was concentrated in large 588 Comets II particles. Yet Hale-Bopp displayed the highest 3–13-µm ity, which in many cases [spheres, spheroids, ellipsoids, color temperature and strongest silicate feature ever seen disks, cylinders, see, e.g., Bohren and Huffman (1983)] in a comet, clearly indicating that the thermal emission at admits analytical solution. shorter wavelengths arose from a high abundance of sub- When x|m – 1| << 1 and |m – 1| << 1, the Rayleigh-Gans micrometer grains compared to other comets. approximation applies. This approximation is often used not While Mie theory can be used to compute the absorp- by itself but as a first approximation for the successive it- tion (or emission) cross section Cabs for a compact particle erations to obtain more general approximations (see, e.g., that is small in comparison to the wavelength in all its di- Haracz et al., 1984). Muinonen (1996) applied this approxi- mensions, in cases where Cabs varies slowly with wave- mation to stochastic irregular particles. Closely related to length, it cannot be applied to calculations for an emission the Rayleigh-Gans approximation is the coherent Mie scat- feature, where the optical constants are rapidly varying. tering approximation developed to treat porous aggregates Analysis of the 10-µm silicate emission feature in comets of spheres (Zerull et al., 1993). Only phase interrelations indicates the presence of both glassy and crystalline sili- between waves scattered by constituent particles in an ag- cate components (Hanner et al., 1999; Wooden et al., 1999; gregate of spheres are taken into account. This approxima- Hanner and Bradley, 2004). The optical constants of the tion works better with increasing porosity of the aggregate; glassy silicates vary slowly across the 8–13-µm range and usually the porosity should be more than 90% (Xu and Mie theory will not be too inaccurate for computing Cabs. Wang, 1998). However, the crystalline components, especially crystalline When x >> 1, |m – 1| << 1, we may apply the anomalous olivine, which has a strong resonance at 11.2 µm in natu- diffraction approximation (van de Hulst, 1957). This ap- rally occurring samples, cannot be treated using Mie theory proximation is based on the assumption that the rays are since a resonance is very sensitive to particle shape (Bohren negligibly deviated inside the particle, thus only absorption and Huffman, 1983). Mie theory does not even predict the inside the particle and interference of light passed through correct wavelength of the peak, let alone the correct peak the particle and around it can be taken into account. This height (Yanamandra-Fisher and Hanner, 1998). Thus, most allows easy calculation of the absorption and extinction researchers make use of measured reflectivity or mass ab- cross sections, which sometimes can be expressed through sorption coefficients to obtain Cabs within the spectral feature analytical formulas. This method was widely applied to (Wooden et al., 1999, 2000; Hayward et al., 2000; Harker estimate cross sections of particles very different from et al., 2002). spheres, such as prismatic and hexagonal columns, cubes, Mie theory influenced our understanding of cometary and finite cylinders. The accuracy of the method increases dust, constraining it as an ensemble of dark, polydisperse, with decreasing absorption. and nonspherical particles whose main contribution to the For the sake of completeness, we mention an approxima- thermal emission comes from particles of size parameter tion called perturbation theory, which treats the radius r of an close to one unit. It still remains an important tool when irregular particle as a function of coordinates ϑ and ϕ and ξ ϑ ϕ estimating the thermal radiation from cometary dust, since r = r0(1 + f( , )), where r0 is the radius of “unperturbed” the particle shape does not affect the absorption cross sec- sphere and |ξf(ϑ,ϕ)| < 1. The solution is obtained as an ex- tions as long as the particles can be considered as compact pansion of the boundary condition in series in ξ (Oguchi, and roughly equidimensional or sufficiently porous ag- 1960). This method provides accurate results only for parti- gregates of such particles so that particle interactions can cles of x ≤ 7 and only if the deviations from spherical shape be neglected. However, angular and spectral dependencies are smaller than the wavelength. It is a convenient method of intensity and polarization of cometary dust indicate that when effects of the surface roughness are of interest. these properties cannot be calculated using the model of x >> 1 is the geometric optics case, now often referred spherical homogeneous particles. to as the “ray-tracing approximation.” It considers the incident field as a set of independent rays, each of which 3.2. Approximations to Treat Nonspherical reflects according to Snell’s law and Fresnel’s equations on Inhomogeneous Particles the surface of the particle. The size restriction is in part relaxed when geometric optics is combined with Fraunhofer Along with techniques that consider light scattering in diffraction. For particles of complicated shape, the ray trac- rigorous terms through the solution of Maxwell’s equations, ing is usually performed using the Monte Carlo approach. there are some useful approximations that allow fast com- On this basis results have been obtained for stochastic parti- putations subject to the size parameter and/or refractive cles (Muinonen, 2000) and particles with inclusions (Macke, index of the particle. 2000). The Web site by A. Macke (http://www.ifm.uni-kiel. When x << 1 and |mx| << 1 the Rayleigh (1897) approxi- de/) contains a collection of ray-tracing codes. The com- mation applies. Such a particle can be considered as a di- parison of ray-tracing calculations with exact solutions, e.g., pole with an inherent polarizability that defines the dipole for spheres or cylinders, shows that the discrepancies be- moment induced in the particle by the incident electromag- tween the approximation and the exact solution start for netic field. In the case of a very small particle, the field near transparent particles of size parameter x < 100 (Wielaard it can be treated as an electrostatic field. This allows the et al., 1997); for larger absorption the ray-tracing technique derivation of a simple equation for the particle polarizabil- can be applied to smaller size parameters. Waldemarsson Kolokolova et al.: Physical Properties of Cometary Dust 589

and Gustafson (2003) developed a combination of ray-trac- of mixing rules have been developed for a variety of inclu- ing technique with diffraction on the edge for light scatter- sion types (non-Rayleigh, nonspherical, layered, anisotro- ing by thin plates (flakes) that provides a reasonable accu- pic, chiral) and topology of their distribution within the racy down to particle sizes of x ~ 1. medium, including aligned inclusions and fractal structures Cometary dust is not dominated by very small or very (see an extensive review by Sihvola, 1999). However, still large particles, but the approximations described above can the most popular remain the simplest Maxwell Garnett be applied to calculations for wide size distributions of (Garnett, 1904) nonspherical particles at the edges of the size distribution. For example, Bonev et al. (2002) calculated the color of ε − ε ε − ε cometary grains assuming them to be polydisperse sphe- eff m = f i m (6) ε + ε i ε + ε roids and using the T-matrix method (see section 3.3.2) for eff 2 eff eff 2 eff size parameters x < 25 and the ray-tracing technique for larger particles. A similar approach can be used to calcu- and Bruggeman (Bruggeman, 1935) late light scattering when a broad range of wavelengths need to be covered, as was done by Min et al. (2003), who also ε − ε ε − ε found a criterion for selecting the ray-tracing or anomalous f eff i = f m eff (7) i ε + ε m ε + ε diffraction approximations to calculate cross sections. i 2 eff m 2 eff Kolokolova et al. (1997) showed that in the color, pre- ε ε sented as the difference of logarithms of intensity at two mixing rules. In equations (6) and (7) i and m are dielec- wavelengths, the contribution of intermediate particles is tric constants of inclusions and matrix respectively, and fi canceled and small and large particles dominate. If the size is the volume fraction of the inclusions in the mixture. The distribution is smooth (e.g., of power-law type) and wide Maxwell Garnett rule represents the medium as inclusions ε [i.e., includes small (Rayleigh) and large (ray-tracing) par- embedded into the matrix material with m, the result of ticles], then the color can be expressed analytically through which depends on the material chosen as the matrix. The ε ε the radii of the smallest and largest particles, the power of Bruggeman rule was obtained for particles with i and m ε ε the size distribution, and two parameters that describe the embedded into the material with = eff, so the formula is particle composition. These analytical expressions have five symmetric with respect to the interchange of materials. This unknowns that can be found if five colors are measured as makes it easy to be generalized for the n-component me- has been done for Comet C/1996 Q1 Tabur (Kolokolova et dium al., 2001b). The approximations listed above were developed to treat n f ε − ε ε + ε = (8) nonspherical but homogeneous particles. To describe light ∑ i ( i eff )/( i 2 eff) 0 = scattering by heterogeneous particles one can use approxi- i 1 mations called mixing rules or effective medium theories. Since the derivation of the mixing rules was based on The main idea behind these approximations is to substitute assuming the external field as electrostatic, the inclusions a composite material with some “effective” material whose were assumed to be much smaller than the wavelength of refractive index provides the correct light-scattering char- electromagnetic waves. More exactly, the criterion of their acteristics of the heterogeneous medium. It is a reasonable validity is xRe(m) << 1 (Chylek et al., 2000), where x is the approximation when inhomogeneities (inclusions) are much size parameter of inclusions and Re(m) is the real part of smaller than the wavelength (Rayleigh-type) and the size the refractive index for the matrix material. Comparison of of the macroscopic particle is much larger than the size of effective medium theories with DDA (see section 3.3.3) cal- the inclusions. In large scale, such a medium acts as an culations (Lumme and Rahola, 1994; Wolff et al., 1998) and isotropic homogeneous medium for which the displacement experiments (Kolokolova and Gustafson, 2001) show that D is related to the electrostatic field E by a linear relation- even for xRe(m) ~ 1 effective medium theories provide rea- ship D = εE where ε is the complex dielectric constant of sonable results. The best accuracy can be obtained for cross the material related to the refractive index as m = ε . For a sections and the worst for polarization, especially at phase macroscopically homogeneous isotropic mixture, D and E angles α < 50° and α > 120°. can be averaged within the medium as There were a number of attempts to consider heterogeneous cometary grains using effective-medium theories, par- ∫ D(x,y,z)dxdydz = ∫ ε(x,y,z)E(x,y,z)dxdydz = ticularly to treat aggregates as a mixture of constituent par- ε ∫ (5) eff E(x,y,z)dzdydz 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 with where E, D, and ε are determined for a point inside the ma- the size parameter of constituent particles x > 1 are, most terial with coordinates (x,y,z). Using the Maxwell equation likely, out of the range of the validity of the effective me- divD = 0 and assuming some size, shape, and distribution dium theories. For the thermal infrared region, cometary of the inclusions, one can obtain expressions, in some case aggregates can be treated with effective medium theories ε analytical, for the effective dielectric constant eff. Dozens if they are sufficiently large (remember that the number of 590 Comets II inclusions must allow the macroscopic particle to be con- pands the incident and scattered fields into vector spheri- sidered as a medium). Using the mixing rules, one should cal functions with the scattered field expanded outside a be careful to calculate the refractive index for core-mantle sphere circumscribing a nonspherical particle. The main ad- particles. Even for particles of size parameter x ≥ 0.3, sig- vantage of the resulting formulation is that the T-matrix is nificant differences in cross sections from exact core mantle defined only by physical characteristics of the particle it- calculations appeared (Gustafson et al., 2001). self, i.e., it is computed only once and then can be used for any incidence/scattering direction and polarization of the 3.3. Numerical Solutions for Inhomogeneous incident light. A significant development of the method was Nonspherical Particles an analytical orientation-averaging procedure (Mishchenko, 1991) that made calculations for randomly oriented particles Further characterization of cometary dust requires more as efficient as for a fixed orientation of the same particle. advanced light scattering theories. We consider them in this T-matrix computations are especially efficient for axisym- section, mainly outlining the strengths and drawbacks of the metric, including multilayered, particles; however, the meth- methods that have been applied to study cometary dust. As od can be extended to any shape of the scatterer. Although well as Mie theory, solutions of the Maxwell equations for computational complexity greatly increases for particles nonspherical particles cannot be used directly to solve the without rotational symmetry, recently solutions were ob- inverse problem for cometary dust, but instead provide some tained for ellipsoids, cubes, and aggregates of spherical constraints through fitting procedures and investigation of particles. The latter case is of special interest in the field of general trends in the dependence of light-scattering char- astrophysics. It presents the T-matrix development of the acteristics on particle properties. separate-variable method for the aggregates of spheres con- 3.3.1. Separation of variables method. The main idea sidered above. Since the scattered field for the individual behind Mie theory, expansion of the electromagnetic waves sphere does not depend on the incident light direction and in the coordinate system that corresponds to the shape of polarization, the expansions for individual spheres can be the body (e.g., cylindrical, spheroidal) and then separates transformed into a single expansion centered inside the ag- the variables, has provided a number of solutions for non- gregate. Such an expansion is equivalent to the T-matrix of spherical but regular particles: multilayered, optically an- the aggregate and thus can be used in analytical averaging of isotropic, and radially inhomogeneous spheres, cylinders, scattering characteristics over aggregate orientations (Mish- slabs, and spheroids (see the review by Ciric and Cooray, chenko et al., 1996). 2000). Among solutions obtained with this method, the most A variety of computational T-matrix codes are available popular are solutions for cylinders (Bohren and Huffman, on line at Mishchenko’s Web site (http://www.giss.nasa.gov/ 1983), core-mantle spheres (Toon and Ackerman, 1981), ~crmim), among them codes for polydisperse axial-sym- spheroids (Asano and Yamamoto, 1975; Voshchinnikov and metric particles and aggregates of spheres. The computa- Farafonov, 1993), and core-mantle spheroids (Farafonov et tional complexity of the T-matrix method increases as x3–x4, al., 1996). Although this method can provide exact solu- i.e., the method is a fast and convenient way to test results tions to Maxwell’s equations, the problem requires numeri- obtained with other codes (see, e.g., Lumme and Rahola, cal treatment to solve the boundary condition even for 1998) or for survey-type studies. Kolokolova et al. (1997) simple particle shapes. This limits the applicability of the used it to check the sensitivity of color and polarization to method to rather small size parameters (x < 40 for sphe- the size and composition of nonspherical particles. The roids) or to nonabsorbing materials. However, in the range results were found to be similar to the results for polydis- of their applicability, these theories provide highly accurate perse spheres. However, details in polarization for non- results, which makes them a good tool for testing other spherical particles differ significantly from those calculated approaches. for equal-volume or equal-area spheres (Mishchenko and The solution of the separation of variables for a single Travis, 1994). For example, randomly oriented spheroids sphere can be extended to aggregates of spheres representing show no oscillations in angular dependencies of intensity the total scattering field as a superposition of spherical wave and polarization at size distributions 10–20× narrower than functions, which expand the field scattered by each sphere is necessary to wash out the oscillations for spheres. Polar- and use the translation addition theorem for spherical func- ization for polydisperse silicate spheroids (Fig. 8) at inter- tions (Bruning and Lo, 1971). The method was extended mediate phase angles changes from negative, typical for for aggregates of spheroids, cylinders, multilayered spheres, spheres (Fig. 6), to positive, leaving a negative branch at and spheres with inclusions (see Xu, 1997; Fuller and Mac- α < 30°. For elongated particles at α ~ 90°, polarimetric kowski, 2000, and references therein). The computational color is positive and shows the angular trend similar to the complexity of the method (the number of numerical opera- observed one (it gets larger with increasing phase angle) tions in the algorithm) depends on the number of particles for a broad range of particle size (Fig. 8). The angular de- and their size parameter, shape, and configuration. pendence of the intensity is also reminiscent of the cometary 3.3.2. T-matrix approach. A popular T-matrix ap- scattering function (Fig. 1) except at phase angles 10°–60°, proach was initially introduced by Waterman (1965) under where it reaches a deep minimum instead of being flat. The the name “extended boundary condition.” This method ex- spheroids demonstrate color that changes rather rapidly its Kolokolova et al.: Physical Properties of Cometary Dust 591

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 size parameter xeff corresponds to this for equal-volume sphere. 592 Comets II value and even sign with phase angle. Randomly oriented The most popular among such methods is probably the cylinders exhibit similar tendencies (Mishchenko et al., coupled dipole approximation, often called the discrete di- 2002). Using higher absorption one can get higher values pole approximation (DDA) since it represents a particle as of color and polarimetric color, but the negative polariza- consisting of elementary cells, polarizable units called di- tion and backscattering peak disappear, eliminating the re- poles (Purcell and Pennypacker, 1973). Each dipole is ex- semblance of the calculated scattering function and polari- cited by the external field and the fields scattered by all zation curve to the observed ones. other dipoles. This representation allows one for a N-dipole Mishchenko et al. (1997) showed that appending the particle to derive N linear equations describing the N fields model by a shape distribution (a range of aspect ratios of that excite each dipole. The numerical solution of the sys- spheroids) can provide a good fit to the cometary scatter- tem of these N equations provides the partial field scattered ing function for polydisperse silicate spheroids at 6 < xeff < by each dipole, and finally the total scattered field is cal- 24. For the size distribution with xeff > 8 and a constant culated. Draine and Flatau (1994) and Lumme and Rahola refractive index m = 1.53 + i0.008, the color becomes neu- (1994) improved the computational efficiency of the DDA, tral and then red with a very weak angular change. How- which allowed them to study larger particles, in particular ever, polyshaped silicate spheroids fail to reproduce the cor- aggregates of larger size. Lumme et al. (1997) considered rect polarization (Mishchenko, 1993). aggregates up to 200 particles of x = 1.2–1.9. The most Petrova et al. (2000) used the T-matrix code by Mac- popular has been the DDSCAT code by Draine and Flatau kowski and Mishchenko (1996) for a systematic compari- (see http://www.astro.princeton.edu/~draine/DDSCAT.html). son of cometary observations with the calculations for ag- The code includes geometry implementations for ellipsoids, gregated particles. They considered aggregates built from finite circular cylinders, rectangular parallelepipeds, hexag- particles of size parameters 1 < x < 3.5 for a set of refrac- onal prisms, and tetrahedra, uniaxial and coated particles tive indexes with the real part equal to 1.65 and the imagi- as well as for systems of particles. The great advantage of nary part varying within the range 0.002–0.1. The maxi- the coupled dipole approximation is that it can be applied mum number of particles in each aggregate was limited to to particles of arbitrary shape, structure, and composition. 43 by the code convergence. “Cometary” type polarization However, it has limited numerical accuracy, requires a sepa- with low maximum at α ~ 90° and negative branch at small rate calculation for each orientation of particle, and slows phase angles was obtained for a power-law size distribution down dramatically with increasing N, so that its computa- of aggregates built of 8–43 constituent particles of x = 1.3, tional complexity increases as the ninth order of the parti- 1.5, and 1.65, although a smooth shape of the curve and cle size parameter. the quantitative fit to the observational polarization curve Using DDA, West (1991) undertook the first systematic was not achieved. The characteristics of the negative po- theoretical study of aggregates. He considered the constant larization depend on the composition (it becomes less pro- refractive index m = 1.7 + i0.029. West demonstrated that nounced with increasing absorption from 0.01 to 0.05) and for compact aggregates both intensity and polarization be- porosity (more pronounced for more compact aggregates). have as is more typical for the spherical equivalent of the A rather good qualitative fit to the cometary scattering func- aggregate. For loose aggregates the projected area of the tion was achieved for the polydisperse aggregates with x = whole aggregate mostly defines the intensity in the forward- 1.3 and κ = 0.01. In a later paper, Petrova and Jockers scattering domain, whereas the polarization is defined by (2002) studied the spectral dependence of polarization for the size of the constituent particles. Kozasa et al. (1993) a variety of size distributions and a range of refractive in- confirmed that even aggregates of thousand constituent dexes within n = 1.65, 0 < κ < 0.1, assuming the refrac- particles of x < 1 behave like Rayleigh particles in polariza- tive index independent of wavelength. They showed that the tion regardless their composition. aggregate model successfully reproduced the observed posi- Xing and Hanner (1997) accomplished a detailed study tive polarimetric color. However, the color of such aggre- of aggregates with a variety of sizes, shapes, and composi- gates was found to be blue, unlike the observed red color. tions of constituent particles and their packing within the The extinction calculations by Petrova et al. (2000) are aggregate. They confirmed that polarization is more sen- mainly sensitive to the overall size of the aggregate and are sitive to the properties of constituent particles whose size close to the results for equal-volume spheres if the aggre- and refractive indexes determine most of the polarization, gates have size parameters x < 3, but becomes larger for whereas shape, number, and packing generate scattering larger aggregates. This is an indication that the equal-vol- effects of the second order. A reasonable fit to the observed ume sphere calculations can be used to estimate cross sec- cometary polarization and intensity was achieved for the tions of aggregates in the thermal infrared but not in the mixture of carbon and silicate aggregates of an intermedi- visual. ate (touching-particle) porosity. The calculations of cross 3.3.3. Coupled dipole approximation and similar tech- sections showed that in the case of short wavelengths (size niques. The methods described above are based on the parameter x > 1) the values of extinction cross section per solution to the boundary conditions on the particle surfaces, unit volume is close to those typical for constituent particles, while a set of more flexible methods rely on solutions of the but at long wavelengths (x << 1) constituent particles and internal field from which the scattered field is calculated. aggregates show significant differences that are more pro- Kolokolova et al.: Physical Properties of Cometary Dust 593 nounced the more compact the aggregate. When the poros- of the scattering function. The number of particles in the ity is less than 60%, the temperature of the particle is close aggregates, at least if it is within 64 ≤ N ≤ 256, as well as to that of an equal-volume sphere. Loose aggregates have their porosity, was found not to be important. But a change a temperature typical for the constituent particle, i.e., ther- in composition (carbon instead of silicate) could dramati- mal emission from a large but aggregated particle looks the cally affect the value of maximum polarization, eliminate same as its small constituent particle regardless of its shape the negative branch, and make the scattering function more (see also Greenberg and Hage, 1990; Li and Greenberg, flat at the medium phase angles. Kimura’s (2001) results 1998b). Davidsson and Skorov (2002) obtained similar demonstrate the best fit to the observational data achieved DDA results for single-scattering albedo. This means that so far, although the spectral characteristics of intensity and thermal infrared data do not allow distinguishing loose ag- polarization have not been checked. gregates from single particles of the size of their constitu- Similar to DDA, the method of moments (MOM) ap- ent particles. proach also divides a particle into small elements, but con- The DDA was used by Yanamandra-Fisher and Hanner siders them not as dipoles but as small cells with a constant (1998) to study the light scattering and emission by non- refractive index. The price of this straightforwardness of spherical but regular particle shapes of size parameter 1 < MOM in comparison with DDA is the necessity of calcu- x < 5, as well as heterogeneous and porous particles. They lating a self-interaction term; different approaches in MOM showed that compact particles of different shapes produce usually diverge in the way they treat this term. The MOM very different angular dependence of polarization for a has strengths and drawbacks similar to those for DDA. transparent material (silicate), but the dependence on par- Using MOM, Lumme and Rahola (1998) considered comet- ticle shape is reduced for absorbing particles (carbon). They ary particles as stochastically shaped, i.e., particles whose concluded that negative polarization at small phase angles shape can be described by a mean radius and the covari- could be produced by nonspherical silicate particles of x ≥ ance function of the radius given as a series of Legendre 2. Introducing porosity for a transparent (silicate) particle polynomials (for details, see Muinonen, 2000). Lumme and caused the polarization to resemble that of a smaller par- Rahola (1998) made computations for a variety of particle ticle, while porosity had less of an effect on an absorbing shapes and size parameter (x = 1–6) using the refractive in- particle provided that the particle remained opaque. Al- dex m = 1.5 + i0.005. Like many previous authors, Lumme though the use of regular shapes caused oscillations that and Rahola found that the particles should be of x > 1 to would be absent in the scattering by an irregularly shaped provide negative polarization at small phase angles and low particle, the authors showed that a macroscopic mixture of maximum polarization. Also, they showed that monodis- silicate and absorbing particles with a distribution of size perse particles, even of very complicated, irregular shape, parameters 1 < x < 5 could produce curves for polarization demonstrate oscillating behavior of light-scattering charac- and angular scattering qualitatively similar to the curves ob- teristics that can be eliminated by considering a polydis- served for cometary dust. A similar conclusion was reached persion of particles. The same oscillating behavior is shown by Xing and Hanner (1997) based on the scattering by small by color and polarimetric color that can be seen from the aggregates. comparison of the intensity and polarization for x = 2, 4, Yanamandra-Fisher and Hanner (1998) also applied the 6. Particles with the size distribution of the form n(r) ~ r –3 DDA code to examine particle shape effects in the mid- show angular functions of intensity and polarization that are infrared emission features produced by crystalline olivine. rather similar to cometary ones. However, one also needs Both the shape and the peak wavelength of the spectral to fit to the spectral characteristics to judge how close the features depend on the particle shape; moreover, olivine is model is to real cometary dust. an anisotropic crystal, and the differing optical constants 3.3.4. Other numerical methods. Above we considered generate peaks at different wavelengths. To reproduce the numerical techniques that have been used in cometary phys- observed silicate feature the authors had to rule out extreme ics. However, there are other light-scattering techniques that shapes (disks, needles) as well as ideal spheres. Polariza- may be worth applying, among them the finite difference tion within the 10-µm silicate feature was studied by Hen- time domain method (FDTD) and the finite element method ning and Stognienko (1993), who demonstrated that shape (FEM). Both these methods are based on a straightforward and porosity of particles affect the spectral dependence of solution to equation (4). The FDTD discretizes space and polarization, both the strength and shape of the feature, even time, making a grid in the time-space domain and then stronger than the spectral dependence of the intensity. solves equation (4) for each space-time cell, whereas the Combining the T-matrix technique with the DDA, Ki- FEM discretizes the scattering particle itself into small vol- mura (2001) considered aggregated particles of fractal struc- ume cells. Then boundary conditions (continuity of the elec- ture, ballistic particle-cluster aggregates (BPCA) and ballis- trical vector across the surface) are applied on the boundary tic cluster-cluster aggregates (BCCA) (see Mukai et al., of each cell. Both methods are applicable to particles of 1992), of hundreds and thousands of constituent particles. arbitrary shape, structure, and composition. The main dis- He showed that aggregates of silicate particles of x = 1.57 advantage of these methods is time-consuming computa- reproduce the cometary polarization with a very good fit. tions, aggravated by the fact that the computations should They also provide rather good fit to the “cometary” shape be repeated for each orientation of the particle with respect 594 Comets II to the incident wave. The computational complexity of the parameters. Giese et al. (1978) showed that, unlike com- FDTD increases as the fourth power of the particle size pact and rather regular particles, irregular, porous particles parameter and varies between x4 and x7 for FEM depend- in the size range of a few micrometers exhibit positive and ing on the computational technique. negative polarization typical for cometary dust. Thus in the More details of the methods discussed above and other 1970s the microwave method provided the results obtained numerical methods and approximations can be found in only in theory in the late 1990s. Greenberg and Gustafson recent reviews (Mishchenko et al., 2000, 2002; Kahnert, (1981) showed that models consisting of a tangle of rods, 2003). Some codes are available via online libraries such so-called “Bird’s Nests,” could provide the observed angu- as http://atol.ucsd.edu/~pflatau/scatlib/ (P. Flatau), http:// lar dependencies of intensity and polarization. This experi- www.t-matrix.de/ (T. Wriedt), http://www.emclab.umr.edu/ mental test of the aggregated nature of cometary dust then codes.html, http://urania.astro.spbu.ru/DOP/, and http:// was extended to the study of a variety of aggregates of www.astro.uni-jena.de/Users/database/. The latter Web site spherical, including core-mantle, particles by Zerull et al. also contains a huge collection of refractive indexes for ma- (1993). terials of astrophysical interest (Jäger et al., 2003). The systematic study of light scattering by complex particles has been performed using a new generation of 3.4. Experimental Simulations microwave facilities designed and built at the University of Florida (Gustafson, 1996, 2000). This facility works across The complexity of cometary and other cosmic dust and a 2.7–4-mm waveband to simulate a region 0.4–0.65 µm planetary aerosols and the ensuing difficulty treating them in the visual, and thus can study not only angular but also using theoretical means generated several approaches for spectral dependencies of light scattering (colors). Gustafson solving the scattering problem through experimental simu- and Kolokolova (1999) reported results from a systematic lation. As before, we will not discuss all experiments con- study of light scattering by aggregates. Among these results ducted so far to understand light scattering by particles, but is that the polarization is mainly determined by the size and only those representative examples that provided significant composition of constituent particles and the size parameter results for cometary dust. We will not discuss here the ex- of cometary constituents must be 1 < x < 10 to produce a perimental setups detailed in the papers cited below. low maximum polarization and negative polarization at One experimental technique to study light scattering by small phase angles that confirms the theoretical results particles uses streams or suspensions of small particles, considered above (e.g., West, 1991; Xing and Hanner, 1997; often powdered terrestrial rocks, that are illuminated by a Lumme et al., 1997; etc.). The color and polarimetric color source of light, recently mainly by a laser. The light scat- of aggregates also primarily depend on the size and com- tered by the particles is measured to obtain intensity, polari- position of the constituent particles. For aggregates made zation, Stokes parameters, and even the complete scattering of particles whose size parameters covered the range 0.5– matrix for a range of phase angles. The other technique, 20, an increase in the size of constituent particles results in called the microwave analog method, introduced by Green- larger (more red) color and smaller polarimetric color (the berg et al. (1961), simulates light scattering using micro- refractive index was m = 1.74 + i0.005, constant through- wave radiation. It actually takes advantage of the same out the wavelength range). The same tendencies result from scaling used in all theoretical approaches where the particle increasing the imaginary part of the refractive index that dimensions are given through its size parameter, i.e., as a may provide a clue to the negative polarimetric color ob- ratio to the wavelength. In such a simulation the light scat- served for Comet P/Giacobini-Zinner (Kiselev et al., 2000). tering problem is scaled to longer wavelengths that precisely A combination of positive polarimetric color and blue color preserve the characteristics of the scattering problem as long is typical of aggregates of nonabsorbing particles of 0.5 < as the size parameter of the scatterer and its refractive in- x < 10, whereas red color and positive polarimetric color are dex are preserved. This allows light scattering by a single indicative of dark, absorbing aggregates. From the above, submicrometer or micrometer particle to be simulated us- the aggregates, which similarly scatter light to the cometary ing a manageable millimeter or centimeter analog model. dust, consist of absorptive constituent particles of size pa- 3.4.1. Microwave analog experiments. The early micro- rameter x > 1.The shape of constituent particles could not wave simulations were measurements of cross sections and be seen to influence the aggregates’ color and only slightly scattering functions for spheroids, finite cylinders, rods, and changes the position and amount of maximum polarization. a variety of compact irregular particles (Greenberg et al., Notice that the results were obtained for the refractive in- 1961; Zerull et al., 1977) directed to determine the appli- dex that does not change with wavelength. The intensity and cability of Mie theory to nonspherical particles. The study polarization data for a variety of particles, including aggre- was then extended to polarization and color of nonspherical gates, are given at http://www.astro.ufl.edu/~aplab. particles (Zerull and Giese, 1983). A systematic study of Even though implementation of some theoretical meth- particle shape effects was carried out by Schuerman et al. ods, e.g., DDA, on modern computers allows a study of (1981). Significant discrepancy between the data for non- similar accuracy for aggregates of similar characteristics as spherical particles and Mie theory was obtained for both the at microwave measurements, the microwave method is still shape of measured curves and values of light-scattering much faster and can provide a large scope of the results in Kolokolova et al.: Physical Properties of Cometary Dust 595

a reasonable time. For example, single-wavelength DDA pound, e.g., kerogen or tholin (Wallis et al., 1987) or an calculations for a silicate aggregate of 1024 constituent organic refractory (Greenberg and Li, 1998). This assump- particles of size parameter x = 0.733 (Kimura, 2001) took tion is consistent with the observed increase in albedo with more than a month on the Alpha station, whereas scatter- the distance from the nucleus and might also explain the ing from an aggregate of any number of particles of x > 1 gradient in the near-infrared colors. can be measured in one week providing data at 50 discrete 3.4.2. Light-scattering experiments in the visual. Among wavelengths. This allows systematic studies to obtain quan- early experimental works that contributed to our under- titative characteristics of the light scattering, e.g., multifactor standing of light scattering by cosmic grains, Weiss-Wrana analysis (similar to that described in Kolokolova et al., (1983) showed a significant difference between light scatter- 1997) of sensitivity of light-scattering characteristics to ing by terrestrial/meteoritic particles and Mie calculations. physical properties of particles. Some results of such a study The work also showed that transparent compact irregular of aggregates are presented in Table 1, which summarizes particles ~30 µm in size had the polarization curve with the a study of large (500–5000 constituent particles) aggregates. maximum located at α ≈ 45° and negative polarization at The aggregates were built to satisfy the most plausible large phase angles; large (28–80 µm) absorbing (κ = 0.66) model of cometary dust aggregates resulted from theoreti- compact particles had a bell-shaped polarization curve with cal simulations: The size parameter of the constituent par- high (about 50%) polarization maximum; and fly ash, ticles was in the range 1.5–5, the constant refractive index, slightly absorbing (κ = 0.01) loose aggregates with an aver- m = 1.74 + i0.005, simulated a silicate-organic mixture, and age size of 41 µm had an angular dependence of polariza- the porosity varied within 50–90%. The larger numbers in tion similar to the cometary one. Table 1 indicate a stronger correlation between the dust Extensive measurements of all 16 elements of the scatter- characteristic and the light-scattering parameter; negative ing matrix F were performed by the group from Amsterdam values show that they anticorrelate. Table 1 shows quanti- (Hovenier, 2000; Hovenier et al., 2003; see also http://www. tatively that the size of constituent particles is the main char- astro.uva.nl/scatter/). Muñoz et al. (2000) studied the scat- acteristic that determines all light-scattering parameters. The tering matrix of polydisperse, heterogeneous, irregular par- influence of the porosity is much smaller, and within the ticles presented by terrestrial (light-colored olivine) and range investigated the number of particles in the aggregate meteoritic (dark carbonaceous chondrite) powders at wave- has almost no influence on the light-scattering parameters. lengths of 0.442 and 0.633 µm. The effective radius of the The data from Table 1 show that the correlations between particles was approximately a few micrometers so that the color and polarization (their simultaneous decrease or in- size parameter of the particles peaked within the range x = crease) in the near-nucleus coma observed by K. Jockers and 10–20. Both terrestrial and meteoritic samples showed po- colleagues (Kolokolova et al., 2001a) could not be explained larization curves of shape similar to the cometary ones and by the changing size of the aggregates (overall size influ- positive polarimetric color. However, the values and angle ences neither color nor polarization) or the changing size of the minimum polarization almost twice exceeded the ob- of the constituent particles (color and polarization should served ones, whereas the maximum polarization was almost then anticorrelate). Change in the porosity could produce half the observed values. Also, the color was blue, i.e., the a correlation, but then the color should anticorrelate with intensity at 0.442 µm exceeded the intensity at 0.633 µm. the polarimetric color that has not been observed. More- The study shows that polydisperse heterogeneous irregular over, the simultaneous increase in color and polarization ob- particles cannot be ruled out as candidates for cometary served for Comet Hale-Bopp could be provided only by dust, although, at least for size distributions that peak at x = unrealistic increasing compactness of the aggregate on its 10–20, they are not able to correctly reproduce all the ob- way out from the nucleus. Microwave measurements for served characteristics. materials of a variety of refractive indexes show that such Light-scattering experiments under reduced gravity (mi- a correlation may be a consequence of evaporation/destruc- crogravity) have been proposed to avoid sedimentation of tion of some dark material contained in the aggregates. Such the studied dust as well as particle sorting or orientation a slowly evaporating dark material can be an organic com- that can occur for particles dropped or suspended in air- flow. The microgravity PROGRA2 experiment was concentrated on light-scattering measurements of levitated particles TABLE 1. The strength of correlation (normalized to during parabolic flights (Worms et al., 2000). The angular the color/radius correlation) between light-scattering dependences of polarization were retrieved for various types characteristics and properties of the aggregates. of compact particles (Worms et al., 1999) and aggregates (Hadamcik et al., 2002). This experiment later provided Radius of Number of polarization images in two wavelengths (Hadamcik et al., Constituent Constituent 2003). Particles Particles Porosity Hadamcik et al. (2002) performed a systematic micro- Color 1.00 0.212 –0.923 gravity study of aggregated particles. The majority of the Polarization –3.23 0.030 –0.262 samples were made of 12–14-nm particles of silica, alumina, Polarimetric color –2.36 –0.316 1.34 titanium dioxide, or carbon. The particles formed complex 596 Comets II structures of three-level hierarchy: submicrometer-sized disperse grains with the predominance of submicrometer chains of particles were combined into several-micrometer- particles or are aggregates of submicrometer particles. The sized aggregates, and the aggregates in turn formed milli- major constraints on the dust properties came from polari- meter-sized particles, “agglomerates,” that were levitated metric and thermal infrared data. Angular dependence of and measured. The most striking result of this study was polarization puts constraints on the size of particles and that the particles, even though composed of nanometer con- limits the refractive index. The SED of the thermal emis- stituents, did not show Rayleigh-type scattering properties sion implies a broad size distribution of the dust; the ex- unlike other aggregates of small particles considered above cess color temperature above a blackbody at λ < 20 µm (Zerull et al., 1993; Gustafson and Kolokolova, 1999; West, indicates grains of r < 1 µm, while the slow decrease in flux 1991; Kozasa et al., 1993). The microscopic photographs at longer wavelengths indicates larger particles with a size show that although “agglomerates” are rather porous struc- parameter comparable to the wavelength. Aggregates of tures, “aggregates” look as if they were made of solid ma- submicrometer grains with a range of porosities would be terial, demonstrating rather high packing of “chains” in the compatible with thermal SED. Spectral midinfrared features aggregates. Since the size of constituents in “aggregates” identify silicates as a component of the dust material. The is of the order of some micrometers, their size can be the composition could be refined using the spectral changes in factor that determines the light scattering. The polarimet- the scattered light (color and polarimetric color); it is likely ric curves have a low maximum at α ~ 100° and demon- that they indicate spectral dependence of the average re- strate positive polarimetric color at large phase angles. At fractive index of the material. A change in the characteris- small phase angles the samples made of a single type of tics of the scattered and emitted radiation with distance from constituent demonstrate positive polarization, which may the nucleus is a helpful tool to further determine the size become negative for mixtures of small and large or dark and distribution and composition of cometary dust as it reflects light particles that can be evidence of a heterogeneous struc- the dynamic sorting of particles and composition changes, ture of cometary grains. e.g., sublimation of volatiles. Although observational data The most recent microgravity study of the wavelength accumulated so far, as well as their interpretation, are not dependence of polarization (Hadamcik et al., 2003) showed sufficient to provide reliable information about cometary that a mixture of fluffy aggregates of submicrometer silica dust from the correlations between the light-scattering and and carbon grains demonstrates a positive polarimetric thermal-emission characteristics (section 2.4), such corre- color, whereas gray compact particles with size greater than lations can be valuable to further constrain the properties the wavelength exhibit a negative polarimetric color. This of cometary dust and to compare the dust in different parts can explain the difference between the polarimetric color of the coma and in different comets. of comets and asteroids as well as changes in polarimetric color within cometary comae. 4. CONCLUSIONS AND CONSISTENCY The COsmic Dust Aggregation (CODAG) experiment WITH THE RESULTS OBTAINED intends to study dust aggregation during rocket flights (Blum USING OTHER TECHNIQUES et al., 2000; Levasseur-Regourd et al., 2001). These experiments (1) study the formation and evolution of dust grains Below we discuss the physical properties of cometary and aggregates under a variety of physical conditions rep- dust obtained using light-scattering techniques, comparing resentative of the protoplanetary nebula, (2) monitor the them with the results of studying cometary dust using its dust particles in three colors at α = 5°–175° to study the dynamical properties, in situ measurements, and study of changes in their light scattering that accompany the aggre- interplanetary dust particles (IDPs). gation process, and (3) provide validation of the light-scattering codes. Low-temperature studies to reveal the dust 4.1. Shape light-scattering properties during condensation or sublimation of ices on the grains will hopefully be performed on- The shape of cometary particles is significantly non- board the International Space Station (Levasseur-Regourd spherical. Evidence of this is the complete absence of reso- et al., 2001). nant oscillations in intensity, polarization, and color. Such oscillations are typical for non-Rayleigh spherical particles 3.5. Summary even with a rather wide size distribution. None of the attempts to fit the scope of observations with a model based Table 2 summarizes the observational data and results on spherical particles has been successful. Additional sup- of the theoretical and laboratory simulations of light scat- port for this conclusion are observations of star occultations tering by cometary dust. One can see that the theoretical that show nonzero polarization at α = 180° and the presence and laboratory simulations of light scattering and thermal of circular polarization and polarization whose plane is in- emission by cometary dust show that a plausible model of clined to the scattering plane. Such observations require that cometary dust can be heterogeneous (silicates and some the symmetry inherent in spherical particles be broken. Sim- absorbing material) particles that are either irregular, poly- ulated mid-infrared spectral features demonstrate noticeable Kolokolova et al.: Physical Properties of Cometary Dust 597

TABLE 2. Observational facts and their interpretations.

Cometary Dust Model that can Reproduce the Observational Observational Fact and Section Where It is Discussed Fact and Section Where It is Discussed Brightness characteristics: 1. Low geometric albedo of the particles (2.1.1) 1. Absorbing particles (κ > 0.02) (3.4.1). Decrease in the particle size increases albedo for slightly absorbing particles.

2. Prominent forward-scattering and gentle backscattering 2. Aggregates of particles of x = 1–5 (3.3.2, 3.3.3, 3.4.1), silicate peaks in the angular dependence of intensity, “flat” polydisperse (power law) elongated particles (e.g., spheroids) behavior at medium phase angles (α) (2.1.2) with a distribution of the aspect ratios (3.3.2), silicate polydisperse (power law) irregular particles (3.3.3).

Polarization characteristics: 1. For a broad range of wavelengths angular dependence 1. Polydisperse (power law) irregular submicrometer particles of linear polarization (2.2.1) demonstrates (3.3.2, 3.3.3; 3.4.2); aggregates of particles of x = 1–5 (3.3.2, • negative branch of polarization for α < 20° with the 3.3.3, 3.4.2). The best fit is provided by aggregates of large minimum P ≈ –2% number of constituent particles (3.3.3) or polydisperse aggregates • bell-shaped positive branch with low maximum of (3.3.2). Absorption seems to reduce the negative polarization but value P ≈ 10–30% at α ≈ 90°–100°. increases the maximum polarization (3.3.2, 3.3.3).

2. Small circular polarization V < 2% (2.2.4). 2. Nonspherical particles or particles containing optically anisotropic materials.

Spectral optical characteristics: 1. Usually red or neutral color for a broad range of 1. Slightly absorbing particles of x > 6 (3.3.2); absorbing particles wavelength λ (2.1.3). of x >1 (3.4.1); particles, containing material with a spectrally dependent refractive index (3.3.3).

2. Polarization at a given α above ≈30° usually increases 2. Aggregates of particles of x = 1–5 (3.3.2, 3.3.3, 3.4.1, 3.4.2) or with λ (positive polarimetric color) (2.2.2). polydisperse nonspherical particles of x > 1 (but not x >> 1) (3.4.2) for a broad range of refractive indexes independent of wavelength; particles made of materials with spectrally dependent refractive index.

Spatial distribution of optical characteristics within the coma: 1. Increase in albedo with the distance from the nucleus 1. Change in the particle composition (evaporation of some dark (2.1.1). material) (3.4.1) or decreasing size of silicate particles

2. Variations in the value of color and polarization 2. Variations in size distribution of cometary dust or in their (2.1.1, 2.2.3) composition (if the changes in color and polarization correlate) (3.1, 3.4.1)

3. Deviations in the polarization plane throughout the 3. Nonspherical particles or particles containing optically anisotro- coma (2.2.4) pic materials

Thermal infrared characteristics (2.3): 1. 3- to 20-µm color temperature higher than the blackbody 1. Absorbing grains of r < 1 µm (3.1.1); porous aggregates of temperature. submicrometer absorbing grains (3.3.2).

2. Midinfrared emission features typical of silicates in 2. Silicate grains of size r < 1 µm or porous aggregates of some comets. submicrometer silicate grains (Hanner and Bradley, 2004).

3. Thermal emission does not decrease more steeply than 3. Evidence of a broad size distribution that includes larger than a blackbody at λ > 20 µm micrometer-sized particles. 598 Comets II

the Giotto spacecraft during the flyby of Comet P/Halley (McDonnell et al., 1991). If the measured Giotto mass distribution is typical of comets, then most of the particulate mass shed from the nucleus is concentrated in large particles, whereas the cross section is broadly distributed or concentrated in the small particles (Fig. 9), thus making submicrometer- and micrometer-sized particles dominant in the visual and thermal infrared. This most likely is true for all comets, although the detailed size distribution may vary from comet to comet as well as within a coma. For example, size distribution is different for jets and regular coma with predominance of smaller particles in the jets. The size distribution varies with the distance from the nucleus as a result of both dynamical sorting of particles by radiation pressure and sublimation of volatiles.

4.3. Composition

Cometary dust consists of heterogeneous particles that include a dark material, which determines the low albedo, as well as silicates responsible for the spectral features in the mid-infrared. A distributed source of CO near the comet Fig. 9. Relative cross section (\\\), mass (/// ), and thermal nucleus, trends in color, polarization, and albedo support emission (– – –) vs. particle radius for the size distribution meas- the notion of a slowly sublimating organic material (Green- ured by Giotto, as adapted to fit the SED of Comet 81P/Wild 2 berg and Li, 1998; Kolokolova et al., 2001a). Thus, not only (Hanner and Hayward, 2003). For power-law size distributions with the size distribution but also the composition of cometary ≥ a 3, submicrometer particles dominate in the light scattering. grains changes as they move out from the nucleus. A significant specific of the cometary dust is its red color, which does not vary with the phase angle. This cannot be easily dependence of the feature’s position and shape on the shape simulated even for polydisperse nonspherical particles. It of dust particles, and thus mid-infrared spectral data can may be the result not of a specific size or structure of parti- be used to further constrain the particle shape. cles but of a spectral change in the refractive index. Compo- sitional analysis of dust particles during the Halley flybys as 4.2. Size Distribution well as analysis of IDPs and infrared spectra indicate that the dust grains are composed largely of silicates and carbo- As was shown in Table 2, observations in the visible naceous (CHON) material (Hanner and Bradley, 2004) and their interpretation based on light-scattering methods mixed together on a very fine scale. The composition of equally support two models of cometary dust: (1) rather the cometary organics may be inferred from its sublimation large or polydisperse aggregates of submicrometer particles, constants gauged by studying the trends in colors, polariza- and (2) irregular particles with a power-law size distribution tion, and thermal-emission characteristics with the cometo- dominated by submicrometer particles. The study of the centric distance. dust thermal emission allows us to constrain characteristics of the size distribution. This can be done by fitting the ob- 4.4. Structure served spectral energy distribution (SED) with a calculated one using energy balance equation (2). The increase in color As mentioned above, polydisperse irregular particles can temperature indicates the contribution of submicrometer reproduce most of the light-scattering properties of comet- grains, while the slow decrease in flux at longer wave- ary dust. The same can be achieved with a model of ag- lengths indicates larger particles. Grün et al. (2001) fit the gregated particles. Porous aggregates of submicrometer broad thermal flux from 7 to 160 µm from Comet Hale- particles are better candidates to explain the data on ther- Bopp with the size distribution of form n(r) ~ r–a with a ≤ mal emission. An aggregated structure of cometary particles 3.5. Observations in the submillimeter spectral range indi- is also supported by other studies. Thus, evolutionary ideas cate the presence of millimeter-sized particles. Results for about growing cometesimals in protoplanetary disks sup- other studies confirm that the dust emitted from comets port the idea of an aggregated nature of cometary refracto- spans a broad size range, from submicrometer to millime- ries (see, e.g., Greenberg, 1982). The most developed model ter or centimeter and larger. This is known from analysis by Greenberg and Li (1999) describes cometary grains as of the trajectories of dust particles in comet tails (Sekanina, fluffy (porous) aggregates of particles that are 0.1-µm sili- 1996) and measurements of the mass of particles impacting cate cores coated by an organic refractory mantle and outer Kolokolova et al.: Physical Properties of Cometary Dust 599 mantle of predominantly water ice, which contains embed- ization and the polarization plane are necessary, not only ded carbonaceous and polycyclic aromatic hydrocarbon to study any alignment of the dust particles within the coma, (PAH)-type particles with sizes in the 1–10-nm range. When but also to obtain information about the shape of the par- limited by relative cosmic abundances, the water is con- ticles and possible optical anisotropy and optical activity strained to be close to 30% by mass and the refractory to of the cometary dust material. volatile ratio is close to 1 : 1. Theoretical and laboratory surveys are necessary to chart Captured IDPs of probable cometary origin can give us the differences in observable characteristics for aggregates insight into the morphology of cometary dust. These IDPs and irregular polydisperse particles. The surveys should be consist primarily of submicrometer-sized compact, nonspheri- directed to simulate not individual observational facts but cal grains and angular micrometer-sized crystals clumped the whole scope of the observational data, i.e., angular and into aggregates having a range in porosity. spectral characteristics of both scattered and emitted radiation. Contribution from submicrometer- and micrometer- 4.5. Future Work sized particles is required by the realistic size distributions, and thus light scattering by aggregates of hundreds and Further progress in remotely determining physical prop- thousands of constituent particles or large (but with size still erties of dust from their light-scattering and emission re- below geometric-optics validity) irregular particles should quires new observations and new theoretical and laboratory be simulated. The influence of the structure of the aggre- simulations. gates, the morphology and shape of the constituent parti- The most useful observations are likely to be multiwave- cles, as well as the morphology and specific shapes of irreg- length monitoring of comets within a broad range of he- ular large particles should be studied. Such studies most liocentric distances and phase angles. The observations can likely require development of new theoretical techniques or provide new insight into the classification of comets accord- systematic use of controlled laboratory measurements. More ing to their maximum polarization; heliocentric tendencies comprehensive modeling of cometary dust composition is in colors (including near-infrared); correlated or anti-cor- necessary: The refractive indexes used in calculations related change in such characteristics as color, polarization, should satisfy the confirmed presence of both silicates and and albedo; and thermal emission characteristics (tempera- absorbing constituents in the dust and must be consistent ture, shape, and strength of a silicate feature) with heliocen- with the Kramers-Kronig relations (Bohren and Huffman, tric distance and within the coma. The correlation between 1983, section 2.3.2) across all wavelengths. More realistic light-scattering and emission properties, e.g., between the mixtures of silicates and carbonaceous materials or core- value of maximum polarization and the strength of the sili- mantle particles should be a point of special interest. The cate feature, might be indicative of the diversity of dust average, effective refractive index can be considered when from one comet to another. In the framework of light-scat- the silicate and dark materials are mixed on a very small tering and emission methods, more information about the scale compared to the wavelength, e.g., when an admixtures comet dust size distribution might be obtained from detailed of FeNi or FeS typical for IDPs (Hanner and Bradley, 2004) studies of forward-scattering characteristics of the scattered in submicrometer silicate grains are simulated. The need for light (as with, e.g., the Sun-grazing comets detected by studies of spectral characteristics of the light scattered by SOHO), heliocentric dependence of the dust temperature, particles whose material has a realistic wavelength-depen- and measurements of the SED at far-infrared and submilli- dent refractive index is apparent. This also refers to labora- meter wavelengths (or shorter wavelengths at larger helio- tory light-scattering simulations that should address spec- centric distances). Spectrophotometric and spectropolari- tral dependencies, and we therefore need to concentrate on metric data of improved spectral resolution will be impor- particles with realistic and controlled refractive indexes tant to better eliminate the influence of gas contamination across the studied spectral range. 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