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1995 O1 (Hale–Bopp)
Icarus 149, 351–356 (2001) doi:10.1006/icar.2000.6565, available online at http://www.idealibrary.com on
Analysis of Coma Dust Optical Properties in Comet C/1995 O1 (Hale–Bopp) II. Effects of Polarization Dana Xavier Kerola and Stephen M. Larson Lunar and Planetary Laboratory, University of Arizona, Tucson, Arizona 85721 E-mail: [email protected] Received February 23, 2000; revised June 30, 2000
Efforts to apply a single-scattering polarized radiative transfer code to interpret photopolarimetric measurements of coma dust optical properties in Comet Hale–Bopp corroborate previous photometrically derived conclusions concerning the predominance of small sized particles in Hale–Bopp’s coma. Calculations of the degree of linear polarization (DP) as a function of observation phase angle (α) produced by prolate spheroidal crystalline olivine particles with effective radii (a = 0.216 µm) are compatible with the comet’s measured polarization in standard filters at λ = 0.4845 and 0.684 µm. Our rudimentary “trade-off” studies highlight the extreme sensitivity of DP to dust particle size and shape. A combination of viewing geometry effects in association with enhanced multiple scattering might provide a quantitative explanation of the negative polarization for 0◦ ≤ α ≤ 20◦ seen in Hale–Bopp and other comets. °c 2001 Academic Press Key Words: dust optical properties; polarization of light.
nevertheless have been testing the G–S code in our continuing cometary and related planetary atmosphere simulations. Most certainly for the innermost portions of the coma, multiple scattering is operative. Inquiries one of us (D.X.K.) is currently making into reasons for the location (as a function of viewing geometry) and magnitude of the slightly negative branch of polarization exhibited in all observations of solar system dust might require invocation of select aspects of multiple scattering theory. A single-scattering polarized code for use in analyzing the scattered light from cometary comae has been created as well. The use of either one of these codes should help significantly in any attempts to quantitatively model particle size distributions of dust particles found in tenuous atmospheres of primitive solar system bodies. Polarization studies have the unique capability, which sheer photometry usually does not, to extract particle shapes from remotely sensed data. 1.2. Recent Related Work
1. INTRODUCTION
1.1. Objectives Recently planned and currently ongoing spacecraft rendevous missions to conduct in situ examination of comets and asteroids provide a real impetus for performing baseline analytical studies. Such numerically oriented investigations could serve as a preliminary backdrop for eventual sample returns of primordial solar system materials. Along with the various observations and experiments to be carried out (e.g., by the recently launched Stardust spacecraft), what is needed now is some reliable analytic framework within which to interpret the host of possible spectrophotometric and polarimetric measurements of cometary dust and gas molecular properties. A computer program developed by Kerola (1996) relies on a Gauss–Seidel (G–S) multiple-scattering approach to treat both the intensity and the polarization of light. Even though use of a single-scattering treatment should suffice in most situations when dealing with mainly very thin comet atmospheres, we
For ascertaining the basic chemical composition of Comet Hale–Bopp’s particles, the authors have been able to adapt the single-scattering and multiple-scattering algorithms for the express purpose of computing the spectral radiances from Hale– Bopp’s coma. That initial modeling effort aimed at deducing dust particle properties strictly from comparisons to calibrated measurements of scattered intensities has been completed (cf. Kerola and Larson, 1999; hereafter referred to as Part I). Our results suggested there was an abundance of tiny olivine particles streaming from the comet during the end of 1996. Part I established a fairly firm baseline upon which to further constrain the physical properties of cometary dust grains. It is now recognized in the cometary astrophysics community that there may be a separation, taxonomically, between comets (Hale–Bopp, e.g.) that appear to have a high preponderance of very small particles composing their often rather quite dusty comae and another class of comets, perhaps distinct from the first group in terms of their place and mode of origin, that evince
351 0019-1035/01 $35.00 c 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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a greater proportion of much larger particles. Precisely in this context lies the significance of the polarization of the light scattered in cometary comae. It is from a portrayal of curves of partial linear polarization vs phase angle that A. C. Levasseur-Regourd (1999) argues in behalf of such a taxonomical distinction between comets. Those that have high polarizations have been found to have a structured silicate feature composed of small, nonspherical grains (cf. Hanner et al. 1994). Furthermore, as Li and Greenberg (1998) argue, if the particles are porous as well, they might not necessarily have to be exceptionally small. Additional corroboration of the belief that comet dust must be composed of nonspherical particles comes most recently for Hale–Bopp by way of measurements of circular polarization (cf. Rosenbush et al. 1997). These measurements revealed a small (less than 0.3% in the coma) circular polarization, which could be indicative of aligned nonspherical grains. Moreover, the overall shape of the polarization phase curve, along with the belief that particle formation processes in the early Solar System were triggered by inelastic collisions, favors the formation of fractal aggregates. The analyses we undertook in our present work have not been directed toward fractal particle properties. Only now are accurate enough theoretical and computational techniques becoming available for treatment of those kinds of noncompact particles. 2. COMPUTATION OF LINEAR POLARIZATION
As seen in Part I, Eq. (6), we have the ability of computing not only the intensity of the scattered light but also the degree of polarization (DP). We can explicitly construct the full angular scattering properties of the coma dust particles. In general, the phase matrix for randomly oriented aerosol particles, each of which has a plane of symmetry, can be written in the form
P11 P 21 P˜ (2) = 0 0
P12 P22 0 0
0 0 P33 P43
0 0 , P34 P44
(1)
where 2 is the scattering angle, i.e., the angle formed by the incident solar ray’s direction and the scattered ray’s direction. The observation phase angle is given by α = 180.0◦ − 2. In general, P12 = P21 and P34 = −P43 . Furthermore, for Mie particles, P11 = P22 and P33 = P44 . We can use the elements P11 , P21 , and P22 to give us the two relevant Stokes intensity components, Il and Ir , parallel and perpendicular to the scattering plane, respectively. Then the degree of linear polarization will be given by DP =
Ir − Il . Ir + Il
(2)
Our computer codes have been structured so that the phase matrix elements can be found for Mie and nonspherical particles
alike. The modeling we did in Part I relied solely on calculations for Mie particles. Now we expand our inquiry so that the degree of polarization produced by ensembles of irregularly shaped particles can be found. The public domain T-matrix computer codes of Michael Mishchenko (the underlying algorithms of which are reported in Mishchenko 1991) were acquired and adapted for our present purposes. 3. POLARIMETRIC MEASUREMENTS OF HALE–BOPP’S COMA
The results of an extensive campaign to perform optical photopolarimetry on the coma of Comet C/1995 O1 have quite recently been reported by Manset and Bastien (2000; hereafter referred to as MB). We make use of their observational datasets as a primary means of comparison with our theoretical radiative transfer modeling. They made a host of polarization measurements over the course of many nights from mid-1996 to Spring, 1997. A variety of apertures and several different spectral filters were employed. There were no systematic trends in the measured polarizations as a function of aperture or wavelength. They also provide formulas for calculating DP in concentric rings about the comet’s opto-center. But for us to be consistent with our Part I philosophy, which treated Hale–Bopp’s inner coma in a full-disk averaged sense, we will limit our modeling here to their full disk measurements. All of their measurements with which we make direct comparisons to our modeled results are for an aperture of 15.5”. Also, in order to mimimize unneeded interferences and ambiguities in our modeling efforts (arising at wavelengths where gas emission bands reside), we will restrict the analyses to the regions centered on λ = 0.4845 and 0.684 µm, the standard International Halley Watch (IHW) wavelengths. Two other observational agendas comparable in scope to the MB work were carried out during the same time period by Ganesh et al. (1998) and Kiselev and Velichko (1997). Their results with the IHW filters parallel closely the MB work, instilling confidence in the reliability of the three groups’ measurements. We include selected results of all of these studies for intercomparison (cf. especially Figs. 3 and 4). Hadamcik and Levasseur–Regourd (1997) also obtained good phase angle coverage in red light, though their measurements were in a broaderband (width 1λ = 0.10 µm) filter centered at λ = 0.650 µm. Additional summaries of Hale–Bopp optical polarization studies have been given by Jockers et al. (1997), Rosenbush et al. (1997), Tanga et al. (1997), and Furusho et al. (1999). In our ensuing best-fit plots given in Fig. 3 and Fig. 5, we are able to overlay the results of the former two investigations (cf. Fig. 3 for Rosenbush et al. and Fig. 5 for Jockers et al.), but cannot directly make a quantitative comparison with the latter two studies, since both the Furusho et al. and Tanga et al. findings were presented mainly as colorized maps of polarization in various shell structures throughout the inner coma. Hasegawa et al. (1997) have extended the Hale–Bopp polarimetry measurements to include
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the near-IR region. Though the thrust of our present work is concentrated at visible wavelengths, a cursory post-facto application of our best-fit model to the J, K, and H bands (at 1.24, 1.65, and 2.2 µm) appears not to be inconsistent with our retrieved dust optical characteristics. To match their infrared measurements, we seem to require slightly larger particles with a wider size spread than we need to match the observations in blue and red light. Certainly a separate, comprehensive interpretation of the photopolarimetry of Hale–Bopp at near-IR wavelengths would be needed before any definitive claims of consistency with the visible model can be made. 4. DUST PARTICLE SIZE AND SHAPE EFFECTS
A favorable circumstance in the present study, absent in Part I, is that now there is no need to perform any kind of calibration of the observations or of the computed scattering quantities. In the case of the single-scattering regime, the degree of polarization is simply DP =
−P21 . P11
FIG. 1. Comparison of the calculated variation of DP vs α (dotted curve) using optimum Mie particle parameters (a = 0.145 µm, b = 0.10) against the observationally derived values (diamonds) in the red continuum (λ = 0.684 µm) taken from Manset and Bastien. Their actual measurements were within the phase angle range 3.0◦ ≤ α ≤ 47.0◦ . Their trigonometric extrapolation is shown outside that range. Also plotted is the linear fit (solid line with slope 0.0064/◦ ) to the individually measured points of Kiselev and Velichko.
(3)
The main goal here, as it was in Part I, is the application of mathematically rigorous, self-consistent procedures for interpreting photopolarimetric observations of dust in cometary comae. Once again, we conducted a limited “sensitivity study,” i.e., a systematic variation of the key physical parameters, to see if a self-consistent fit to the observations can be achieved or, failing attainment of any unique agreement of the model and the measurements, if some combinations of parameters might possibly be ruled out. Along with the results reported by us in Part I, the strong indications from Hale–Bopp thermal infrared data (cf. Hanner et al. 1997, Williams et al. 1997) point to the presence of crystalline olivine as an abundant constituent in its coma. In continuing our exploration of dust optical characteristics, we can therefore remove one degree of freedom by presupposing olivine as the main mineralogic component of Hale–Bopp’s coma. Concentrating initially on analysis of the reflected light at λ = 0.684 µm, let us begin by depicting in Fig. 1 the degree of polarization versus α for an ensemble of Mie particles that come closest to matching the full curve of the Manset and Bastien observations. It should be made clear that their observationally derived curve consists of the actual measured DP values for 3.0◦ ≤ α ≤ 47.0◦ , plus their best-fit trigonometric extrapolations down to α = 0.0◦ and out to α = 120.0◦ . A Standard-0 (cf. Hansen and Travis 1974) particle size distribution with particle effective radius (a = 0.145 µm) and variance (b = 0.10) for olivine spheres with index of refraction n = 1.63nr + 0.00003n i , when input to the single-scattering code, yields the calculated results given in Fig. 1. Interestingly enough, we inferred nearly the same combination of parameters, from photometry alone, in Part I. In the present study, numerous other combinations involving small sizes (0.12 µm ≤ a ≤ 0.15 µm) and a range of variances were
tried. None of these permutations permitted an overall fitting to the observational curve. So, not surprisingly, we confirm what myriad other studies have found; namely, the particles in Hale– Bopp’s coma are not perfectly spherical. To calculate next the linear polarization for irregularly shaped particles, we invoke Mishchenko’s T-matrix routines. A number of runs must be made at the outset to become comfortable with the combination of inputs that his code can tolerate. The best way to execute the T-matrix program in this and all other cases is in a mode whereby a power law aerosol size distribution is used, so that the minimum and the maximum particle radius are automatically set for each and every run merely by specifying the desired particle effective radius and variance (identically equal to the Standard-0 values of a and b). The interested reader is urged to refer to Mishchenko and Travis (1998) for a concise summary of the methodology, applicability, and great virtue of the T-matrix approach. A few early iterations using contrasting values of one of the prime inputs, the ratio of the horizontal to the rotational axis of the particle (designated as EPS, are instructive. Figure 2 highlights how dramatically the curves change for an assortment of EPS values for small prolate spheroids (for which EPS < 1.0, by definition). As we try to retrieve some set of self-consistent particle properties, it is important to remind ourselves that because of the ultra-complicated interplay of wavelength-dependent particle size and shape effects occurring amid the changing hydrodynamic activity of the coma as a function of time and distance from the sun, a single least-squares fit is ludicrous to expect. Instead, we “freeze” as many of the nagging, unpredictably changing parameters as possible, in order to work our way toward establishing limiting values for size and shape
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FIG. 2. Comparisons of T-matrix calculations against the Hale–Bopp observed red continuum polarization using prolate spheroid particles having effective radius a = 0.216 µm and variance b = 0.0105 for differing values of the ratio of horizontal to rotational axis (EPS) of the aerosol; EPS = 0.280 (dashed), EPS = 0.415 (dot-dash), and EPS = 0.580 (dotted). A curve using the same size distribution for Mie particles (i.e., EPS = 1.0) is also included (triple-dot, dash).
combinations. The MB measurements for both the blue and red continua were also plotted for several different dates near the time of perihelion (01 April 1997) and for one month after that. Figures 3 and 4 show that reasonably good agreement can be achieved with one set of “spherical volume-equivalent” values of a, b, and EPS (0.216 µm, 0.0105, and 0.415, respectively) for prolate spheroids at λ = 0.4845 and 0.684 µm, across a range of observation phase angles lying within the linear region of DP
FIG. 3. Good fit of T-matrix polarization calculations at λ = 0.4845 µm (dot-dash curve) using prolate spheroids (a = 0.216 µm, b = 0.0105, and EPS = 0.415). The measured DP values (diamonds) of Manset and Bastien (2000) correspond to 1997 observations on 3 April (α = 47.0◦ ), 16 April (α = 40.0◦ ), and 3 May (α = 28.9◦ ). Also shown are the measurements of Ganesh et al. (asterisks) and approximate values transcribed from Kiselev and Velichko (crosses). One additional point is the measurement of Rosenbush et al. (triangle).
FIG. 4. Same as Fig. 3 except at λ = 0.684 µm.
vs α. The measured points correspond to 1997 observations on 3 April (α = 47.0◦ ), 16 April (40◦ ), and 3 May (28.9◦ ). Clearly the closest fit to the observations occurs for 16 April. In fact, it appears that adjustments could be made to fit the 3 April and 3 May times as well, by pivoting the calculated curve, using 16 April as the “fulcrum.” These adjustments ought to depend primarily upon a slight change in the particle effective radius. We once again execute the T-matrix program, now for particles slightly smaller than were used to match the mid-April measurements (this time with a = 0.2135 µm). A very slightly smaller variance had to be used, too (b = 0.0085). With that combination, we indeed can match simultaneously the blue and the red continuum observations for 3 April. Likewise, by adopting slightly larger particles (now a = 0.2245 µm, b = 0.0105), we can achieve close agreement with the measurements made 1 month after perihelion. In Fig. 5 we show how well our model agrees with the MB measurements of DP vs λ on the three chosen dates. We include two additional measurements made by Jockers et al. (1997) on the day of perihelion as confirmation of the consistency of the results obtained by the various observing groups. Given the measurement uncertainties, along with a lack of precise knowledge of the true index of refraction of the particles, the best we can hope to do is “bracket” the sizes of the particles between 0.21 µm ≤ a ≤ 0.23 µm, as we have done. Simulations such as those we have just now reported for prolate spheroids, done by triggering the selection of oblate spheroids (with EPS > 1.0), lead likewise to good agreement in red light, but not nearly so good in blue light (for the same particle size and shape parameters). Figure 6 is a distillation of the modeling for oblate spheroids, where we show the results at both wavelengths. Other particles shapes can be invoked too. Many trials were made using right circular cylinders (essentially plates, pillboxes, and needles). Once again, as in Fig. 6, the results agree well in red light but not simultaneously in blue. Of course, it can be argued that in actuality cometary dust particles would not be configured in such smooth, deterministic
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We are not claiming the exact shapes of the particles with certitude. A completely proper and comprehensive study would require the use of linear programming methods (involving the use of systems of equations with algebraic constraints) for the determination of the best combinations of particle shapes and sizes. Such a nontrivial analysis is beyond the scope of the rudimentary study done here. More work will also be needed to build new procedures for handling any optical effects arising from the chemical compositional heterogeneities inside the particles, as well as from their fluffiness or fractally aggregated shapes. Our modeling though is a first step in the right direction toward setting stringent constraints on cometary dust physical properties. 5. TOWARD A QUANTITATIVE ANALYSIS OF THE NEGATIVE POLARIZATION BRANCH FIG. 5. Variation of the polarization in the coma of Hale–Bopp as a function of continuum wavelength based on the Manset and Bastien (MB) measurements (diamonds) and the T-matrix modeling for prolate spheroids (asterisks) for the three indicated dates in 1997. The solid lines connect the MB observations, while the dot-dashed lines connect the calculated points. The MB observation phase angles were 47.0◦ , 40.0◦ , and 28.9◦ , from top to bottom. The particle ensemble parameters (i.e., effective radius and variance) were (top) a = 0.2135 µm, b = 0.0085; (middle) a = 0.216 µm, b = 0.0105; and (bottom) a = 0.2245 µm, b = 0.0105. In all cases, the ratio of horizontal to rotational axis length of the particles was EPS = 0.415. Also indicated are two measurements from Jockers et al. (open squares) made at λ = 0.443 and 0.642 µm on 01 April 1997.
shapes as prolate or oblate spheroids. It might be interesting to remark that Greenberg and Li (1996) found prolate spheroids to be more satisfactory than other shapes they explored in the course of the modeling they did of interstellar dust polarization. They remind us that prolate spheroids are a natural result of the process of “clumping” in the proto-solar nebulae.
FIG. 6. Similar to Figs. 3 and 4 with the combined blue and red continuum results using oblate spheroids. Using particle ensemble parameters a = 0.214 µm, b = 0.0275, and EPS = 2.18 for both colors, a good fit in red light (dashed curve) is found, but not good in blue light (dot-dashed) with the same particle size distribution. The observations of Manset and Bastien at λ = 0.4845 µm are shown as diamonds, and those at λ = 0.684 µm as asterisks.
The myriad records of polarimetric measurements of comets, past and present, all document the existence of a region of slight negative linear polarization within about 20–30◦ of the exact backscattering direction. Refer, for example, to the review by Levasseur-Regourd et al. (1998) for a collective digest of this phenomenon, not just for comets but for other kinds of solar system dusty bodies as well. They note that because lower polarization is found in the near-nucleus region of comets where dusty jets are most pronounced, perhaps multiple scattering is operative. In the work we alluded to early on, Tanga et al. (1997) declare their similar kind of predilection for a multiple-scattering explanation. Much has been written to explain in great theoretical detail the causation of the negative polarization branch. Most prominently, within the past decade or so, the mechanism of coherent backscattering has been promoted as a possible explanation. Muinonen (1993) gives an in-depth review of the subject, presenting all of its ramifications and uncertainties. There are also strong advocates of the so-called “fluffy aggregate” model originally proposed by Greenberg and Hage (1990), and recently adopted by Xing and Hanner (1997), as the preferred way in which cometary dust can exhibit negative polarization. In concluding our present study of coma dust in Hale–Bopp, perhaps at least a short commentary regarding the negative branch is in order. Nowhere in Figs. 1 through 5, nor in all of the other single-scattering simulations that were performed, did we match the observed DP vs α curve for a phase angle of about 20◦ or less. What could be happening here? There might be (1) particles (such as fluffy aggregates) having properties different from those of Figs. 3 and 4 that are more abundant in certain spatial regions of Hale–Bopp’s coma, (2) enhanced multiple scattering, which might be producing an effect like coherent backscattering, occurring preferentially at select phase angles, or (3) a combination of (1) and (2), obviously. Current work is now under way to find a few “benchmark” cases for which we can compare our G–S code results against those of the scant number of other amenable vector radiative transfer programs in existence. When we execute our plane parallel program for particles similar in size to those of
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Figs. 3 and 4, with a line-of-sight optical depth of a few tenths, we do see the slightly negative polarizations for small phase angles (where deviations caused by curvature of the quasi-parabolically shaped coma should not be sizable). So, from our preliminary simulations, it appears that for viewing geometries for which the line-of-sight is straight down-range through the main parts of the comet’s tail, on one side or the other of the direction to the nucleus, where some quasi-symmetrical enhanced accumulation of dust can be expected when the comet is very active, multiple scattering quite plausibly could be the mechanism that would explain the negative polarization. We will require longrange careful review and continued verification of our multiplescattering code (along with our exploration of alternative codes to treat fractal aggregates) to become confident of our ability to disentangle the simultaneous geometrical, optical, chemical, and hydrodynamic effects occurring in cometary comae. ACKNOWLEDGMENTS This research was supported by NASA Planetary Atmospheres Grant NAG53751 and Comet Hale–Bopp Grant NAG5-4350.
REFERENCES Furusho, R., B. Suzuki, N. Yamamoto, H. Kawakita, T. Sasaki, Y. Shimizu, and T. Kurakami 1999. Imaging polarimetry and color of the inner coma of Comet Hale–Bopp (C/1995 O1). Pub. Astron. Soc. Japan 51, 367–373. Ganesh, S., U. C. Joshi, K. S. Baliyan, and M. R. Deshpande 1998. Polarimetric observations of the Comet Hale–Bopp. Astron. Astrophys. Suppl. 129, 489– 493. Greenberg, J. M., and J. I. Hage 1990. From interstellar dust to comets: A unification of observational constraints. Astrophys. J. 361, 260–274. Greenberg, J. M., and A. Li 1996. What are the true astronomical silicates? Astron. Astrophys. 309, 258–266. Hadamcik, E., A. C. Levasseur–Regourd, and J. B. Renard 1997. CCD polarimetric imaging of Comet Hale–Bopp (C/1995 O1). Earth, Moon, & Planets 78, 365–371. Hanner, M. S., D. K. Lynch, and R. W. Russell 1994. The 8–13 µm spectra of comets and the composition of the silicate grains. Astrophys. J. 425, 275– 285. Hanner, M. S., R. D. Gehrz, D. E. Harker, T. L. Hayward, D. K. Lynch, C. G. Mason, R. W. Russell, D. M. Williams, D. H. Wooden, and C. E. Woodward 1997. Thermal emission and dust properties of Comet Hale–Bopp. Earth, Moon, & Planets 79, 247–264. Hansen, J. E., and L. D. Travis 1974. Light scattering in planetary atmospheres. Space Sci. Rev. 16, 527–610.
Hasegawa, H., T. Ichikawa, A. Shinsuke, S. Hamamura, L. Ohnishi, and J. Watanabe 1997. Near-infrared photometric and polarimetric observations of Comet Hale–Bopp. Earth, Moon, & Planets 78, 353–358. Jockers, K., V. K. Rosenbush, T. Bonev, and T. Credner 1997. Images of polarization and colour in the inner coma of Comet Hale–Bopp. Earth, Moon, & Planets 78, 373–379. Jones, T. J., and R. D. Gehrz 2000. Infrared imaging and polarimetry of Comet C/1995 O1 (Hale–Bopp). Icarus 143 338–346. Kerola, D. X. 1996. Polarization and intensity of sunlight in the Earth’s atmosphere: Revisiting the Gauss–Seidel Modelling Approach. Bull. Am. Astron. Soc. 28, 1158. Kerola, D. X., and S. M. Larson 1999. Adaptation of planetary atmosphere modeling methods for analysis of coma dust optical properties in Comet C/1995 O1 (Hale–Bopp). I. Icarus 140, 439–445. Kiselev N. N., and F. P. Velichko 1997. Aperture polarimetry and photometry of Comet Hale–Bopp. Earth, Moon, & Planets 78, 347–352. Levasseur-Regourd, A.-C. 1999. Polarization of light scattered by cometary dust particles: Observations and tentative interpretations. Space Sci. Rev. 90, 163–168. Levasseur-Regourd, A.-C., M. Cabane, and V. Haudebourg 1998. Observational evidence for the scattering properties of dust clouds in the Solar System: An update. In Conference on Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, 29 September–1 October 1998, New York, pp. 223–226. Li, A., and J. M. Greenberg 1998. From interstellar dust to comets: Infrared emission from Comet Hale–Bopp (C/1995 O1). Astrophys. J. 498, L83. Manset, N., and P. Bastien 2000. Polarimetric observations of Comets C/1995 O1 Hale–Bopp and C/1996 B2 Hyakutake. Icarus 145, 203–219. Mishchenko, M. I. 1991. Light scattering by randomly oriented axially symmetric particles. J. Opt. Soc. Am. A 8, 871–882. Mishchenko, M. I., and L. D. Travis 1998. T-matrix method and its application. In Conference on Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, 29 September–1 October 1998, New York, pp. 81–84. Muinonen, K. 1993. Coherent backscattering by Solar System dust particles. In IAU Symposium 160. Asteroids, Meteors, Comets 1993, pp. 271–296. Rosenbush, V. K., N. M. Shakhovskoj, and A. E. Rosenbush 1997. Polarimetry of Comet Hale–Bopp: Linear and circular polarization, stellar occultation. Earth, Moon, & Planets 78, 381–386. Tanga, P., A. Cellino, and M. Di Martino 1997. Imaging polarimetry of Comet Hale–Bopp (C/1995 O1) around perihelion. Earth, Moon, & Planets 78, 359– 364. Williams, D. M., C. G. Mason, R. D. Gehrz, T. J. Jones, C. H. Woodward, D. E. Harker, M. S. Hanner, D. H. Wooden, F. C. Witteborn, and H. M. Butner 1997. Measurement of submicron grains in the coma of Comet Hale–Bopp C/1995 O1 during 1997 February 15–20 UT. Astrophys. J. 489, L91–L94. Xing, Z., and M. S. Hanner 1997. Light scattering by aggregate particles. Astron. Astrophys. 324, 805–820.
Analysis of Coma Dust Optical Properties in Comet C/1995 O1 (Hale–Bopp) II. Effects of Polarization Dana Xavier Kerola and Stephen M. Larson Lunar and Planetary Laboratory, University of Arizona, Tucson, Arizona 85721 E-mail: [email protected] Received February 23, 2000; revised June 30, 2000
Efforts to apply a single-scattering polarized radiative transfer code to interpret photopolarimetric measurements of coma dust optical properties in Comet Hale–Bopp corroborate previous photometrically derived conclusions concerning the predominance of small sized particles in Hale–Bopp’s coma. Calculations of the degree of linear polarization (DP) as a function of observation phase angle (α) produced by prolate spheroidal crystalline olivine particles with effective radii (a = 0.216 µm) are compatible with the comet’s measured polarization in standard filters at λ = 0.4845 and 0.684 µm. Our rudimentary “trade-off” studies highlight the extreme sensitivity of DP to dust particle size and shape. A combination of viewing geometry effects in association with enhanced multiple scattering might provide a quantitative explanation of the negative polarization for 0◦ ≤ α ≤ 20◦ seen in Hale–Bopp and other comets. °c 2001 Academic Press Key Words: dust optical properties; polarization of light.
nevertheless have been testing the G–S code in our continuing cometary and related planetary atmosphere simulations. Most certainly for the innermost portions of the coma, multiple scattering is operative. Inquiries one of us (D.X.K.) is currently making into reasons for the location (as a function of viewing geometry) and magnitude of the slightly negative branch of polarization exhibited in all observations of solar system dust might require invocation of select aspects of multiple scattering theory. A single-scattering polarized code for use in analyzing the scattered light from cometary comae has been created as well. The use of either one of these codes should help significantly in any attempts to quantitatively model particle size distributions of dust particles found in tenuous atmospheres of primitive solar system bodies. Polarization studies have the unique capability, which sheer photometry usually does not, to extract particle shapes from remotely sensed data. 1.2. Recent Related Work
1. INTRODUCTION
1.1. Objectives Recently planned and currently ongoing spacecraft rendevous missions to conduct in situ examination of comets and asteroids provide a real impetus for performing baseline analytical studies. Such numerically oriented investigations could serve as a preliminary backdrop for eventual sample returns of primordial solar system materials. Along with the various observations and experiments to be carried out (e.g., by the recently launched Stardust spacecraft), what is needed now is some reliable analytic framework within which to interpret the host of possible spectrophotometric and polarimetric measurements of cometary dust and gas molecular properties. A computer program developed by Kerola (1996) relies on a Gauss–Seidel (G–S) multiple-scattering approach to treat both the intensity and the polarization of light. Even though use of a single-scattering treatment should suffice in most situations when dealing with mainly very thin comet atmospheres, we
For ascertaining the basic chemical composition of Comet Hale–Bopp’s particles, the authors have been able to adapt the single-scattering and multiple-scattering algorithms for the express purpose of computing the spectral radiances from Hale– Bopp’s coma. That initial modeling effort aimed at deducing dust particle properties strictly from comparisons to calibrated measurements of scattered intensities has been completed (cf. Kerola and Larson, 1999; hereafter referred to as Part I). Our results suggested there was an abundance of tiny olivine particles streaming from the comet during the end of 1996. Part I established a fairly firm baseline upon which to further constrain the physical properties of cometary dust grains. It is now recognized in the cometary astrophysics community that there may be a separation, taxonomically, between comets (Hale–Bopp, e.g.) that appear to have a high preponderance of very small particles composing their often rather quite dusty comae and another class of comets, perhaps distinct from the first group in terms of their place and mode of origin, that evince
351 0019-1035/01 $35.00 c 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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a greater proportion of much larger particles. Precisely in this context lies the significance of the polarization of the light scattered in cometary comae. It is from a portrayal of curves of partial linear polarization vs phase angle that A. C. Levasseur-Regourd (1999) argues in behalf of such a taxonomical distinction between comets. Those that have high polarizations have been found to have a structured silicate feature composed of small, nonspherical grains (cf. Hanner et al. 1994). Furthermore, as Li and Greenberg (1998) argue, if the particles are porous as well, they might not necessarily have to be exceptionally small. Additional corroboration of the belief that comet dust must be composed of nonspherical particles comes most recently for Hale–Bopp by way of measurements of circular polarization (cf. Rosenbush et al. 1997). These measurements revealed a small (less than 0.3% in the coma) circular polarization, which could be indicative of aligned nonspherical grains. Moreover, the overall shape of the polarization phase curve, along with the belief that particle formation processes in the early Solar System were triggered by inelastic collisions, favors the formation of fractal aggregates. The analyses we undertook in our present work have not been directed toward fractal particle properties. Only now are accurate enough theoretical and computational techniques becoming available for treatment of those kinds of noncompact particles. 2. COMPUTATION OF LINEAR POLARIZATION
As seen in Part I, Eq. (6), we have the ability of computing not only the intensity of the scattered light but also the degree of polarization (DP). We can explicitly construct the full angular scattering properties of the coma dust particles. In general, the phase matrix for randomly oriented aerosol particles, each of which has a plane of symmetry, can be written in the form
P11 P 21 P˜ (2) = 0 0
P12 P22 0 0
0 0 P33 P43
0 0 , P34 P44
(1)
where 2 is the scattering angle, i.e., the angle formed by the incident solar ray’s direction and the scattered ray’s direction. The observation phase angle is given by α = 180.0◦ − 2. In general, P12 = P21 and P34 = −P43 . Furthermore, for Mie particles, P11 = P22 and P33 = P44 . We can use the elements P11 , P21 , and P22 to give us the two relevant Stokes intensity components, Il and Ir , parallel and perpendicular to the scattering plane, respectively. Then the degree of linear polarization will be given by DP =
Ir − Il . Ir + Il
(2)
Our computer codes have been structured so that the phase matrix elements can be found for Mie and nonspherical particles
alike. The modeling we did in Part I relied solely on calculations for Mie particles. Now we expand our inquiry so that the degree of polarization produced by ensembles of irregularly shaped particles can be found. The public domain T-matrix computer codes of Michael Mishchenko (the underlying algorithms of which are reported in Mishchenko 1991) were acquired and adapted for our present purposes. 3. POLARIMETRIC MEASUREMENTS OF HALE–BOPP’S COMA
The results of an extensive campaign to perform optical photopolarimetry on the coma of Comet C/1995 O1 have quite recently been reported by Manset and Bastien (2000; hereafter referred to as MB). We make use of their observational datasets as a primary means of comparison with our theoretical radiative transfer modeling. They made a host of polarization measurements over the course of many nights from mid-1996 to Spring, 1997. A variety of apertures and several different spectral filters were employed. There were no systematic trends in the measured polarizations as a function of aperture or wavelength. They also provide formulas for calculating DP in concentric rings about the comet’s opto-center. But for us to be consistent with our Part I philosophy, which treated Hale–Bopp’s inner coma in a full-disk averaged sense, we will limit our modeling here to their full disk measurements. All of their measurements with which we make direct comparisons to our modeled results are for an aperture of 15.5”. Also, in order to mimimize unneeded interferences and ambiguities in our modeling efforts (arising at wavelengths where gas emission bands reside), we will restrict the analyses to the regions centered on λ = 0.4845 and 0.684 µm, the standard International Halley Watch (IHW) wavelengths. Two other observational agendas comparable in scope to the MB work were carried out during the same time period by Ganesh et al. (1998) and Kiselev and Velichko (1997). Their results with the IHW filters parallel closely the MB work, instilling confidence in the reliability of the three groups’ measurements. We include selected results of all of these studies for intercomparison (cf. especially Figs. 3 and 4). Hadamcik and Levasseur–Regourd (1997) also obtained good phase angle coverage in red light, though their measurements were in a broaderband (width 1λ = 0.10 µm) filter centered at λ = 0.650 µm. Additional summaries of Hale–Bopp optical polarization studies have been given by Jockers et al. (1997), Rosenbush et al. (1997), Tanga et al. (1997), and Furusho et al. (1999). In our ensuing best-fit plots given in Fig. 3 and Fig. 5, we are able to overlay the results of the former two investigations (cf. Fig. 3 for Rosenbush et al. and Fig. 5 for Jockers et al.), but cannot directly make a quantitative comparison with the latter two studies, since both the Furusho et al. and Tanga et al. findings were presented mainly as colorized maps of polarization in various shell structures throughout the inner coma. Hasegawa et al. (1997) have extended the Hale–Bopp polarimetry measurements to include
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the near-IR region. Though the thrust of our present work is concentrated at visible wavelengths, a cursory post-facto application of our best-fit model to the J, K, and H bands (at 1.24, 1.65, and 2.2 µm) appears not to be inconsistent with our retrieved dust optical characteristics. To match their infrared measurements, we seem to require slightly larger particles with a wider size spread than we need to match the observations in blue and red light. Certainly a separate, comprehensive interpretation of the photopolarimetry of Hale–Bopp at near-IR wavelengths would be needed before any definitive claims of consistency with the visible model can be made. 4. DUST PARTICLE SIZE AND SHAPE EFFECTS
A favorable circumstance in the present study, absent in Part I, is that now there is no need to perform any kind of calibration of the observations or of the computed scattering quantities. In the case of the single-scattering regime, the degree of polarization is simply DP =
−P21 . P11
FIG. 1. Comparison of the calculated variation of DP vs α (dotted curve) using optimum Mie particle parameters (a = 0.145 µm, b = 0.10) against the observationally derived values (diamonds) in the red continuum (λ = 0.684 µm) taken from Manset and Bastien. Their actual measurements were within the phase angle range 3.0◦ ≤ α ≤ 47.0◦ . Their trigonometric extrapolation is shown outside that range. Also plotted is the linear fit (solid line with slope 0.0064/◦ ) to the individually measured points of Kiselev and Velichko.
(3)
The main goal here, as it was in Part I, is the application of mathematically rigorous, self-consistent procedures for interpreting photopolarimetric observations of dust in cometary comae. Once again, we conducted a limited “sensitivity study,” i.e., a systematic variation of the key physical parameters, to see if a self-consistent fit to the observations can be achieved or, failing attainment of any unique agreement of the model and the measurements, if some combinations of parameters might possibly be ruled out. Along with the results reported by us in Part I, the strong indications from Hale–Bopp thermal infrared data (cf. Hanner et al. 1997, Williams et al. 1997) point to the presence of crystalline olivine as an abundant constituent in its coma. In continuing our exploration of dust optical characteristics, we can therefore remove one degree of freedom by presupposing olivine as the main mineralogic component of Hale–Bopp’s coma. Concentrating initially on analysis of the reflected light at λ = 0.684 µm, let us begin by depicting in Fig. 1 the degree of polarization versus α for an ensemble of Mie particles that come closest to matching the full curve of the Manset and Bastien observations. It should be made clear that their observationally derived curve consists of the actual measured DP values for 3.0◦ ≤ α ≤ 47.0◦ , plus their best-fit trigonometric extrapolations down to α = 0.0◦ and out to α = 120.0◦ . A Standard-0 (cf. Hansen and Travis 1974) particle size distribution with particle effective radius (a = 0.145 µm) and variance (b = 0.10) for olivine spheres with index of refraction n = 1.63nr + 0.00003n i , when input to the single-scattering code, yields the calculated results given in Fig. 1. Interestingly enough, we inferred nearly the same combination of parameters, from photometry alone, in Part I. In the present study, numerous other combinations involving small sizes (0.12 µm ≤ a ≤ 0.15 µm) and a range of variances were
tried. None of these permutations permitted an overall fitting to the observational curve. So, not surprisingly, we confirm what myriad other studies have found; namely, the particles in Hale– Bopp’s coma are not perfectly spherical. To calculate next the linear polarization for irregularly shaped particles, we invoke Mishchenko’s T-matrix routines. A number of runs must be made at the outset to become comfortable with the combination of inputs that his code can tolerate. The best way to execute the T-matrix program in this and all other cases is in a mode whereby a power law aerosol size distribution is used, so that the minimum and the maximum particle radius are automatically set for each and every run merely by specifying the desired particle effective radius and variance (identically equal to the Standard-0 values of a and b). The interested reader is urged to refer to Mishchenko and Travis (1998) for a concise summary of the methodology, applicability, and great virtue of the T-matrix approach. A few early iterations using contrasting values of one of the prime inputs, the ratio of the horizontal to the rotational axis of the particle (designated as EPS, are instructive. Figure 2 highlights how dramatically the curves change for an assortment of EPS values for small prolate spheroids (for which EPS < 1.0, by definition). As we try to retrieve some set of self-consistent particle properties, it is important to remind ourselves that because of the ultra-complicated interplay of wavelength-dependent particle size and shape effects occurring amid the changing hydrodynamic activity of the coma as a function of time and distance from the sun, a single least-squares fit is ludicrous to expect. Instead, we “freeze” as many of the nagging, unpredictably changing parameters as possible, in order to work our way toward establishing limiting values for size and shape
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FIG. 2. Comparisons of T-matrix calculations against the Hale–Bopp observed red continuum polarization using prolate spheroid particles having effective radius a = 0.216 µm and variance b = 0.0105 for differing values of the ratio of horizontal to rotational axis (EPS) of the aerosol; EPS = 0.280 (dashed), EPS = 0.415 (dot-dash), and EPS = 0.580 (dotted). A curve using the same size distribution for Mie particles (i.e., EPS = 1.0) is also included (triple-dot, dash).
combinations. The MB measurements for both the blue and red continua were also plotted for several different dates near the time of perihelion (01 April 1997) and for one month after that. Figures 3 and 4 show that reasonably good agreement can be achieved with one set of “spherical volume-equivalent” values of a, b, and EPS (0.216 µm, 0.0105, and 0.415, respectively) for prolate spheroids at λ = 0.4845 and 0.684 µm, across a range of observation phase angles lying within the linear region of DP
FIG. 3. Good fit of T-matrix polarization calculations at λ = 0.4845 µm (dot-dash curve) using prolate spheroids (a = 0.216 µm, b = 0.0105, and EPS = 0.415). The measured DP values (diamonds) of Manset and Bastien (2000) correspond to 1997 observations on 3 April (α = 47.0◦ ), 16 April (α = 40.0◦ ), and 3 May (α = 28.9◦ ). Also shown are the measurements of Ganesh et al. (asterisks) and approximate values transcribed from Kiselev and Velichko (crosses). One additional point is the measurement of Rosenbush et al. (triangle).
FIG. 4. Same as Fig. 3 except at λ = 0.684 µm.
vs α. The measured points correspond to 1997 observations on 3 April (α = 47.0◦ ), 16 April (40◦ ), and 3 May (28.9◦ ). Clearly the closest fit to the observations occurs for 16 April. In fact, it appears that adjustments could be made to fit the 3 April and 3 May times as well, by pivoting the calculated curve, using 16 April as the “fulcrum.” These adjustments ought to depend primarily upon a slight change in the particle effective radius. We once again execute the T-matrix program, now for particles slightly smaller than were used to match the mid-April measurements (this time with a = 0.2135 µm). A very slightly smaller variance had to be used, too (b = 0.0085). With that combination, we indeed can match simultaneously the blue and the red continuum observations for 3 April. Likewise, by adopting slightly larger particles (now a = 0.2245 µm, b = 0.0105), we can achieve close agreement with the measurements made 1 month after perihelion. In Fig. 5 we show how well our model agrees with the MB measurements of DP vs λ on the three chosen dates. We include two additional measurements made by Jockers et al. (1997) on the day of perihelion as confirmation of the consistency of the results obtained by the various observing groups. Given the measurement uncertainties, along with a lack of precise knowledge of the true index of refraction of the particles, the best we can hope to do is “bracket” the sizes of the particles between 0.21 µm ≤ a ≤ 0.23 µm, as we have done. Simulations such as those we have just now reported for prolate spheroids, done by triggering the selection of oblate spheroids (with EPS > 1.0), lead likewise to good agreement in red light, but not nearly so good in blue light (for the same particle size and shape parameters). Figure 6 is a distillation of the modeling for oblate spheroids, where we show the results at both wavelengths. Other particles shapes can be invoked too. Many trials were made using right circular cylinders (essentially plates, pillboxes, and needles). Once again, as in Fig. 6, the results agree well in red light but not simultaneously in blue. Of course, it can be argued that in actuality cometary dust particles would not be configured in such smooth, deterministic
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We are not claiming the exact shapes of the particles with certitude. A completely proper and comprehensive study would require the use of linear programming methods (involving the use of systems of equations with algebraic constraints) for the determination of the best combinations of particle shapes and sizes. Such a nontrivial analysis is beyond the scope of the rudimentary study done here. More work will also be needed to build new procedures for handling any optical effects arising from the chemical compositional heterogeneities inside the particles, as well as from their fluffiness or fractally aggregated shapes. Our modeling though is a first step in the right direction toward setting stringent constraints on cometary dust physical properties. 5. TOWARD A QUANTITATIVE ANALYSIS OF THE NEGATIVE POLARIZATION BRANCH FIG. 5. Variation of the polarization in the coma of Hale–Bopp as a function of continuum wavelength based on the Manset and Bastien (MB) measurements (diamonds) and the T-matrix modeling for prolate spheroids (asterisks) for the three indicated dates in 1997. The solid lines connect the MB observations, while the dot-dashed lines connect the calculated points. The MB observation phase angles were 47.0◦ , 40.0◦ , and 28.9◦ , from top to bottom. The particle ensemble parameters (i.e., effective radius and variance) were (top) a = 0.2135 µm, b = 0.0085; (middle) a = 0.216 µm, b = 0.0105; and (bottom) a = 0.2245 µm, b = 0.0105. In all cases, the ratio of horizontal to rotational axis length of the particles was EPS = 0.415. Also indicated are two measurements from Jockers et al. (open squares) made at λ = 0.443 and 0.642 µm on 01 April 1997.
shapes as prolate or oblate spheroids. It might be interesting to remark that Greenberg and Li (1996) found prolate spheroids to be more satisfactory than other shapes they explored in the course of the modeling they did of interstellar dust polarization. They remind us that prolate spheroids are a natural result of the process of “clumping” in the proto-solar nebulae.
FIG. 6. Similar to Figs. 3 and 4 with the combined blue and red continuum results using oblate spheroids. Using particle ensemble parameters a = 0.214 µm, b = 0.0275, and EPS = 2.18 for both colors, a good fit in red light (dashed curve) is found, but not good in blue light (dot-dashed) with the same particle size distribution. The observations of Manset and Bastien at λ = 0.4845 µm are shown as diamonds, and those at λ = 0.684 µm as asterisks.
The myriad records of polarimetric measurements of comets, past and present, all document the existence of a region of slight negative linear polarization within about 20–30◦ of the exact backscattering direction. Refer, for example, to the review by Levasseur-Regourd et al. (1998) for a collective digest of this phenomenon, not just for comets but for other kinds of solar system dusty bodies as well. They note that because lower polarization is found in the near-nucleus region of comets where dusty jets are most pronounced, perhaps multiple scattering is operative. In the work we alluded to early on, Tanga et al. (1997) declare their similar kind of predilection for a multiple-scattering explanation. Much has been written to explain in great theoretical detail the causation of the negative polarization branch. Most prominently, within the past decade or so, the mechanism of coherent backscattering has been promoted as a possible explanation. Muinonen (1993) gives an in-depth review of the subject, presenting all of its ramifications and uncertainties. There are also strong advocates of the so-called “fluffy aggregate” model originally proposed by Greenberg and Hage (1990), and recently adopted by Xing and Hanner (1997), as the preferred way in which cometary dust can exhibit negative polarization. In concluding our present study of coma dust in Hale–Bopp, perhaps at least a short commentary regarding the negative branch is in order. Nowhere in Figs. 1 through 5, nor in all of the other single-scattering simulations that were performed, did we match the observed DP vs α curve for a phase angle of about 20◦ or less. What could be happening here? There might be (1) particles (such as fluffy aggregates) having properties different from those of Figs. 3 and 4 that are more abundant in certain spatial regions of Hale–Bopp’s coma, (2) enhanced multiple scattering, which might be producing an effect like coherent backscattering, occurring preferentially at select phase angles, or (3) a combination of (1) and (2), obviously. Current work is now under way to find a few “benchmark” cases for which we can compare our G–S code results against those of the scant number of other amenable vector radiative transfer programs in existence. When we execute our plane parallel program for particles similar in size to those of
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Figs. 3 and 4, with a line-of-sight optical depth of a few tenths, we do see the slightly negative polarizations for small phase angles (where deviations caused by curvature of the quasi-parabolically shaped coma should not be sizable). So, from our preliminary simulations, it appears that for viewing geometries for which the line-of-sight is straight down-range through the main parts of the comet’s tail, on one side or the other of the direction to the nucleus, where some quasi-symmetrical enhanced accumulation of dust can be expected when the comet is very active, multiple scattering quite plausibly could be the mechanism that would explain the negative polarization. We will require longrange careful review and continued verification of our multiplescattering code (along with our exploration of alternative codes to treat fractal aggregates) to become confident of our ability to disentangle the simultaneous geometrical, optical, chemical, and hydrodynamic effects occurring in cometary comae. ACKNOWLEDGMENTS This research was supported by NASA Planetary Atmospheres Grant NAG53751 and Comet Hale–Bopp Grant NAG5-4350.
REFERENCES Furusho, R., B. Suzuki, N. Yamamoto, H. Kawakita, T. Sasaki, Y. Shimizu, and T. Kurakami 1999. Imaging polarimetry and color of the inner coma of Comet Hale–Bopp (C/1995 O1). Pub. Astron. Soc. Japan 51, 367–373. Ganesh, S., U. C. Joshi, K. S. Baliyan, and M. R. Deshpande 1998. Polarimetric observations of the Comet Hale–Bopp. Astron. Astrophys. Suppl. 129, 489– 493. Greenberg, J. M., and J. I. Hage 1990. From interstellar dust to comets: A unification of observational constraints. Astrophys. J. 361, 260–274. Greenberg, J. M., and A. Li 1996. What are the true astronomical silicates? Astron. Astrophys. 309, 258–266. Hadamcik, E., A. C. Levasseur–Regourd, and J. B. Renard 1997. CCD polarimetric imaging of Comet Hale–Bopp (C/1995 O1). Earth, Moon, & Planets 78, 365–371. Hanner, M. S., D. K. Lynch, and R. W. Russell 1994. The 8–13 µm spectra of comets and the composition of the silicate grains. Astrophys. J. 425, 275– 285. Hanner, M. S., R. D. Gehrz, D. E. Harker, T. L. Hayward, D. K. Lynch, C. G. Mason, R. W. Russell, D. M. Williams, D. H. Wooden, and C. E. Woodward 1997. Thermal emission and dust properties of Comet Hale–Bopp. Earth, Moon, & Planets 79, 247–264. Hansen, J. E., and L. D. Travis 1974. Light scattering in planetary atmospheres. Space Sci. Rev. 16, 527–610.
Hasegawa, H., T. Ichikawa, A. Shinsuke, S. Hamamura, L. Ohnishi, and J. Watanabe 1997. Near-infrared photometric and polarimetric observations of Comet Hale–Bopp. Earth, Moon, & Planets 78, 353–358. Jockers, K., V. K. Rosenbush, T. Bonev, and T. Credner 1997. Images of polarization and colour in the inner coma of Comet Hale–Bopp. Earth, Moon, & Planets 78, 373–379. Jones, T. J., and R. D. Gehrz 2000. Infrared imaging and polarimetry of Comet C/1995 O1 (Hale–Bopp). Icarus 143 338–346. Kerola, D. X. 1996. Polarization and intensity of sunlight in the Earth’s atmosphere: Revisiting the Gauss–Seidel Modelling Approach. Bull. Am. Astron. Soc. 28, 1158. Kerola, D. X., and S. M. Larson 1999. Adaptation of planetary atmosphere modeling methods for analysis of coma dust optical properties in Comet C/1995 O1 (Hale–Bopp). I. Icarus 140, 439–445. Kiselev N. N., and F. P. Velichko 1997. Aperture polarimetry and photometry of Comet Hale–Bopp. Earth, Moon, & Planets 78, 347–352. Levasseur-Regourd, A.-C. 1999. Polarization of light scattered by cometary dust particles: Observations and tentative interpretations. Space Sci. Rev. 90, 163–168. Levasseur-Regourd, A.-C., M. Cabane, and V. Haudebourg 1998. Observational evidence for the scattering properties of dust clouds in the Solar System: An update. In Conference on Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, 29 September–1 October 1998, New York, pp. 223–226. Li, A., and J. M. Greenberg 1998. From interstellar dust to comets: Infrared emission from Comet Hale–Bopp (C/1995 O1). Astrophys. J. 498, L83. Manset, N., and P. Bastien 2000. Polarimetric observations of Comets C/1995 O1 Hale–Bopp and C/1996 B2 Hyakutake. Icarus 145, 203–219. Mishchenko, M. I. 1991. Light scattering by randomly oriented axially symmetric particles. J. Opt. Soc. Am. A 8, 871–882. Mishchenko, M. I., and L. D. Travis 1998. T-matrix method and its application. In Conference on Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, 29 September–1 October 1998, New York, pp. 81–84. Muinonen, K. 1993. Coherent backscattering by Solar System dust particles. In IAU Symposium 160. Asteroids, Meteors, Comets 1993, pp. 271–296. Rosenbush, V. K., N. M. Shakhovskoj, and A. E. Rosenbush 1997. Polarimetry of Comet Hale–Bopp: Linear and circular polarization, stellar occultation. Earth, Moon, & Planets 78, 381–386. Tanga, P., A. Cellino, and M. Di Martino 1997. Imaging polarimetry of Comet Hale–Bopp (C/1995 O1) around perihelion. Earth, Moon, & Planets 78, 359– 364. Williams, D. M., C. G. Mason, R. D. Gehrz, T. J. Jones, C. H. Woodward, D. E. Harker, M. S. Hanner, D. H. Wooden, F. C. Witteborn, and H. M. Butner 1997. Measurement of submicron grains in the coma of Comet Hale–Bopp C/1995 O1 during 1997 February 15–20 UT. Astrophys. J. 489, L91–L94. Xing, Z., and M. S. Hanner 1997. Light scattering by aggregate particles. Astron. Astrophys. 324, 805–820.