, , , q
i i I 1 i i i 1 i i
1 i i i
0 20 40 60 80 100
Phase Angle, 8
0 20 40 60 80 100
Phase Angle, 8
Figure 1. Hale-Bopp polarization (red filter A, - 0.67 |Am). Filled circles: [9,15], Squares: . Filled triangles:  A. - 0.76 |i,m. Dashed curve: polarization for comets in high Pmax group ,
In single-aperture measurements, negative polarization of order -0.5% to -1% was recorded at 0 < 20 similar to that in other comets at similar phase angles [8,13,12]. However, spatially resolved observations revealed differences in the negative polarization branch with position in the coma, with values as low as -5% near the photometric center at 0 = 7° , while integrated measurements over the polarization image showed little or no negative polarization at all [15,9,16],
Widespread use of CCD arrays to construct polarization images for Hale-Bopp has facilitated the study of polarization variations within the coma [7,15,9,11,16], Structure in the polarization images generally correlates with the structure seen in images of total intensity I x p, where p is the projected distance from the photometric center. The visible jets and arcs correspond to regions of higher polarization.
There is an inner region, extending a few thousand km in diameter, with lower polarization than the surrounding coma. This circumnuclear zone is evident in both 1996 and 1997, although the morphology of the inner coma changed considerably during that time. At small phase angles, this zone exhibits negative polarization down to -5% at 0.67 nm . A similar circumnuclear region of lower polarization was detected in several other comets, including lP/Halley , C/1990 K1 Levy , 47P/Ashbrook-Jackson  and 81PAVild 2 , However, C/1996 Q1 Tabur and C/1996 Hyakutake displayed higher polarization in the inner 2000 km ,
Interpreting the polarization in terms of particle properties is complex. The degree of polarization and its angular dependence, P(0), depend on particle size, shape, composition, surface roughness, and aggregation. Trends, for example with composition, may differ in different particle size regimes. Predictions based on Mie theory will be inaccurate, even misleading in the general trends, for any particle shape except a smooth sphere.
Yanamandra-Fisher and Hanner  studied shape effects for particles with size parameter X = 2n a/k in the range 1-5. While silicate spheres with X = 2.5 exhibit negative polarization at all 0, other shapes show P^ ~ +25% and negative polarization at 0 < 0 c, where 0 c ranges from 20 0 to 70°. In contrast, carbon particles with X = 2.5 display mainly positive polarization, Pmax ~ 50%; the phase angle of maximum polarization varies with particle shape. Mishchenko  computed the phase matrix for shape distributions of dirty silicate spheroids, X = 3.5. As the axial ratio of the spheroids increased, the polarization at intermediate phase angles changed from negative (characteristic of spheres) to positive, leaving a negative branch at 0°-30° phase angles, similar to the cometary P(0).
In recent years, there has been encouraging progress in understanding the polarization by irregular particles, especially aggregates. Numerical methods have allowed exploration of particle shape and aggregate structure while direct scattering measurements have contributed data on particle structures too large to be handled easily by the numerical codes. West  and Zerull et al.  demonstrated via computation and laboratory measurements, respectively, that the polarization of a fluffy aggregate is determined primarily by the polarization properties of the constituent grains. Calculations by Kozasa et al.  confirmed that, indeed, fluffy aggregates composed of spheres much smaller than the wavelength produce Rayleigh polarization. Xing and Hanner  found that aggregates of intermediate porosity (~ 60%) generated polarization intermediate between that of the constituent grains and that of a compact particle with size parameter equal to that of the aggregate. They concluded that a mixture of silicate and carbonaceous aggregates of porosity ~ 60% and constituent grains of X, ~ 2.5 could produce a polarization phase curve similar to that of comet dust.
Petrova et al.  computed P(9) for aggregates of 15-33 constitutent grains having size parameter X, = 0.7-2.5 and refractive index 1.65-ik, £=0.01-0.05. They find that aggregates of constituent grains with X,- = 1.5-1.65 can reproduce the cometary P(0), including the small negative polarization branch at 0 < 25°. However, this small negative branch quickly disappears for other Xt.
Recently, two groups have published results of experiments designed to measure the polarization by fluffy aggregate structures:
The PROGRA2 experiments make use of microgravity during parabolic airplane flights to levitate particles in the path of a laser and measure P(0) [29,30,15,31], In the recent experiments, the particles consist of submicron grains, radius < 0.1 nm (X < 1) forming fluffy aggregates. The aggregates can agglomerate into very loose structures up to millimeter size. Although the P^ is generally high, because of the small grain size, mixtures of silica and carbon aggregates show promise for reproducing the cometary P(B), including negative polarization at small phase angles.
The microwave scattering laboratory at the University of Florida permits study of cometary analogue particles by scaling both the particle dimensions and the wavelength to the microwave domain . Measurements can be made over a range of frequencies, to determine color and polarimetric color. Gustafson and Kolokolova  have presented color and polarization measurements for aggregates of silicate and absorbing grains, 0.5 < X < 20 and have analyzed the trends of color and polarimetric color with grain size and composition. These results have led Kolokolova et al.  to interpret the change in polarization with distance from the nucleus in terms of sublimating organic refractory mantles on silicate grains.
Thus, both modeling and laboratory measurements show that irregular aggregates having constituent grains with X/ in a relatively narrow range of roughly 1.5 to 2 can exhibit P(0) similar to that observed, including a small negative branch. However, scattering by comet dust generates a similar phase function, including the small negative branch, over a range of a factor 4 or more in wavelength, with only a modest increase in Pmax.. Reproducing this behavior in the modeled particles remains a challenge. The cometary aggregate particles apparently have a range in structure and porosity such that they appear similar when viewed at different spatial scales.
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