Effect of absorption on light scattering by agglomerated debris
particles
Evgenij Zubko,1,2 Hiroshi Kimura,3 Yuriy Shkuratov,2 Karri Muinonen,4Tetsuo Yamamoto,3 Hajime Okamoto,1 and Gorden Videen5
1Graduate School of Science, Tohoku University, Japan; 2 Institute of Astronomy, Kharkov NationalUniversity, Ukraine; 3 Institute of Low Temperature Science, Hokkaido University, Japan;
4Observatory, University of Helsinki, Finland; 5 Space Science Institute, USA
Abstract. We study the influence of material absorption on light scattering by agglomerated debris particles
whose sizes are comparable with the wavelength. We find that the angular profile of linear polarization isextremely sensitive to the imaginary part of refractive index, and there are some unique features that may assist
in the retrieval of physical properties of particles using remote-sensing techniques. Most notably, the position of
the positive polarization maximum αmax changes monotonically with the imaginary part of refractive index,
allowing it to be used to characterize this property. In addition, the amplitude of the negative polarization branch(NPB) is significantly greater for dielectric particles than for non-dielectric particles. It disappears in thetransition region between dielectric and conducting particles before reappearing as the imaginary part of therefractive index is increased further. Further increasing the imaginary part of the refractive index may see theNPB disappearing and reappearing in quasi-periodic fashion. This recurrent NPB has a much smaller amplitude
than that of dielectric particles. This suggests that the cometary circumnuclear haloes, which have significantNPBs, cannot contain significant quantities of absorbing particles. In addition, combined observations suggestthat the polarization maximum of circumnuclear haloes are relatively small Pmax ~ 12%, and occur at relatively
small phase angles αmax ~ 60°, which is also consistent with dielectric particles.

Model of agglomerated debris particles
We simulate the light scattered by irregularly shaped particles using our
implementation of the discrete dipole approximation (DDA) (see, e.g., [1–4]).
We generate irregularly shaped model particles by damaging a perfect spherewith the following algorithm (see Fig. 1). A spherical volume is filled with aregular cubic lattice that is considered as the initial matrix of the irregularparticles. All cubic cells forming this initial matrix are divided into twogroups: cells belonging to the surface layer and cells internal to the surface
layer. Among surface dipoles, we choose randomly 100 cells that areconsidered as seed cells of empty space. In the set of internal dipoles, werandomly choose 21 seed cells of material and 20 seed cells of empty space.
Then, step-by-step, each cell distinct from the seed cells is marked with thesame optical properties as that of the nearest seed cell. Images of six sample
particles generated in this way are shown in the bottom of Fig 1. As can beseen, the particles appear to be agglomerates of very irregular shape. We
identify such particles as agglomerated debris particles.

Figure 1

The total numbers of cells forming the initial matrices are 17,256, 137,376 and1,099,136; the choice depends on size parameter x. Material in these
agglomerated debris particles occupies on average approximately 26% of thesevolumes; thus, these agglomerated debris particles have a rather sparsestructure.

Previously, we have studied the light scattering of some manifestations of this kind of particles [2–4].

Results and Discussion
We consider a set of four size parameters x = 5, 10, 20, and 30 (x = 2πrcs/λ, where rcs is radius of circumscribing
sphere, λ – wavelength). In the simulations, the real part of the refractive index has been fixed at 1.5. Excluding thecase of x = 30, the imaginary part is varied from 0 to 1.3. For the largest particles convergence cannot be reached atIm(m) = 0. For averaging, we consider a minimumof 500 particle realizations for each size parameter.
Scattering properties of each particle have been
averaged additionally over 100 scattering planes,
evenly distributed around the wavevector of theincoming light.
Fig. 2 shows the degree of linear polarization P(corresponding to the ratio of Mueller matrix
elements -S21/S11) as a function of phase angle α andthe imaginary part of refractive index Im(m). Fourpanels are shown for size parameters ranging from x= 5 to x = 30. One can see that the smallest particles
show only positive linear polarization at all phaseangles. When Im(m) exceeds 0.5, the angularprofiles of linear polarization of the small particles
are bimodal and further increasing Im(m) makes thisFigure 2
effect more explicit. This feature is mostpronounced in the case of the smallest particles (x = 5) studied as increasing particle size tends to wash it away as
does decreasing particle size. At x = 30, the second maximum is already inexpressive. It is interesting to note that asimilar bimodal profile of the polarization curve can be seen in laboratory measurements of hematite particles [5].

In general, the polarization maximum Pmax depends nonmonotonically on Im(m) (see also Fig. 3). The largest polarizations
tend to occur in the transition zone between dielectric and conducting
particles. Increasing the particle size tends to make it more
pronounced. Unlike Pmax, the phase angle of maximum polarization
αmax increases monotonically with Im(m). The range of variation
increases with particle size. Potentially, this may be an indicator ofparticle absorption.
Both single-scattering and geometrical albedos (see bottom panels inFig. 3) reveal an inverse correlation with the positive polarization
maximum Pmax. The later parameter is defined as the ratio of theenergy scattered at phase angle 0° to that scattered by a white Lambertdisk of the same geometric cross-section (albedoG=(S11π)/(k2G), where
S11 is the element of Mueller matrix, k – wavenumber and G –geometrical cross-section of the particle). Note that, the term
“geometrical albedo” is more appropriate to describe light-scattering
properties of surfaces; whereas, in the case of independent particles, it
is just intensity at backscattering which is normalized. Nevertheless,the geometrical albedo is being actively used in the cometary physics,
Figure 3
and, therefore, we consider this parameter here. As one can see in Fig.
3, single-scattering albedo is visibly better correlated with the inverse of Pmax rather than geometrical albedo.
Nevertheless, for both types of albedo, there is a range of Im(m) where the inverse correlation with the maximum of
positive polarization fails. For both albedos, this occurs for transient materials, but in the case of single-scatteringalbedo, the range at which the inverse correlation fails is smaller in Im(m).

Our simulations demonstrate a negative polarization branch (NPB) at small phase angles that depends strongly on
size parameter and material absorption. This feature is a region of negative polarization at very small yet non-zerophase angles, extending out to 10°-20°. It is not visible for the smallest particles (x = 5), but appears for larger
particles and becomes more persistent as size parameter increases. Interestingly, at Re(m) = 1.5 there is a strong
NPB for very low absorption that completely disappears for all considered particles as Im(m) > 0.1. When the size
parameter of the particles is larger than x = 20, the negative polarization reappears again at larger values of Im(m).
However, in all cases, this feature is no longer as deep as for particles having very low absorption. The position ofthe NPB minimum does not reveal a systematic behavior when the particles are in the dielectric domain, but in thetransient and conductive domains, i.e. Im(m) > 0.1, it tends to shift to larger phase angles. This resurgent NPB may
also disappear and reappear quasi-periodically with increasing absorption(e.g., x = 20). The depth of the NPB remains approximately the same forthese larger absorptions and is much smaller than for the very low
absorption case, suggesting that it may have a different physicalmechanism than that of the near-dielectric particles. Notice also that theNPB for the more absorbing particles often are periodically accompanied
by a positive polarization branch (PPB) near α = 0 (e.g., x = 20, 30).
Considering the specific properties of conductors, we can guess that, in
this case, the negative polarization may appear due to surface waves.

Figure 4

The dependence of the NPB on Im(m) leads us to one very important
conclusion concerning the properties of dust in cometary circumnuclear
haloes. Those haloes are relatively small but bright clouds surrounding
cometary nuclei. Polarimetry of comets demonstrates that the haloes havea very deep NPB at small phase angles [6–8] (see Fig. 4). Different comets
show similar behavior: the minimum polarization Pmin is typically -6%.

Using results of our simulations, we can immediately derive threshold value for Im(m) that is approximately 0.05: if
the imaginary part of the refractive index is greater than this value, then it is impossible to achieve such a deep
negative polarization. Additionally, we must consider that our results concern only particles monodispersed in size;
whereas, cometary particles are in all likelihood polydispersed in size, which would further reduce the depth of theminimum, especially since very small particles do not display any NPB. This suggests that the threshold of Im(m) ison the order of 0.02, corresponding to the relatively deep initial NPB. The results of the imaging polarimetry ofcomets coupled with our phase curves allow us to place restrictions on the materials present in the circumnuclear
haloes.
In situ measurement of dust of comet 1P/Halley has shown that two the most abundant components are so-called
ROCK and CHON [9]. ROCK corresponds to magnesia-rich silicates; whereas, CHON is a mixture of fourchemical elements: carbon, hydrogen, oxygen, and nitrogen. The origin of CHON remains unclear, and there is anobjective reason for this, since these four basic elements form a huge number of combinations. Even primitivemeteorites reveal over 650 individual organic molecules [10]; therefore, the determination of optical constants oforganic material could be quite helpful, at least, in order to reduce the number of variables concerning the organicmaterial. From the restriction on the imaginary part of refractive index discussed above, we can conclude that theorganic material based on that of the Greenberg model [11] cannot exist in considerable quantities in thecircumnuclear haloes; neither can hydrogenated carbon [12]. The reason is that their absorption properties are too
great and their significant presence would dampen the NPBs beyond what has been measured and recorded in thepolarimetric images. We cannot rule out the presence of ice tholin [13] and kerogen [14] in significant quantities,
since these have significantly lower absorption. We would like to stress that we do not claim that the organic
material proposed by the Greenberg model and/or hydrogenated carbon are totally absent in comets, but only that it
does not make up a significant proportion of their circumnuclear haloes. Indeed, the circumnuclear haloes are notthe only features in polarimetric images of comets; there are also jets, which remain positively polarized over theentire range of phase angles [6].

The conclusion that dust particles in circumnuclear haloes are primarily weakly absorbing allows us to estimate thelocation and amplitude of the maximum of the positive polarization Pmax. At large phase angles, ground-based
observations of circumnuclear haloes are met with a fundamental difficulty, since large phase angles can only beachieved when the distance from the comet to the Sun is comparatively small. When the comet is close to the sun it
becomes very active and its circumnuclear halo is partially or completely hidden by jets [6]. Because of thisdifficulty, there are only two polarimetric measurements of circumnuclear haloes at large phase angles; both havebeen made with the Optical Probe Experiment (OPE) on board of Giotto space probe. The first measurement
corresponds to comet 1P/Halley at a phase angle of 73 degrees [7]; whereas, the second corresponds to comet
26P/Grigg-Skjellerup at a phase angle of 90.4 degrees [8]. In Figure 4 we compile these data with those obtainedusing ground-based polarimetry of circumnuclear haloes [6]. The data suggests the location of the maximum of
positive polarization from circumnuclear haloes to be located in the vicinity of αmax = 60°, with Pmax being
approximately 12%. Note that both the position and amplitude of the positive polarization for the circumnuclear
haloes are consistent with the dielectric particles shown in Figure 3. When aperture-averaged polarimetry of theentire cometary comae is performed [6], the estimation of αmax approaches 100–110°, which is well beyond what
can be expected from the data of Figure 4, suggesting that the particles in the circumnuclear halo are significantlydifferent from the particles in the rest of the comae.
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