7 Conclusions and scope for further improvements

A model has been described to obtain intensity and polarization maps of coma and shell (jet) of comets. The intensity profile across the simulated coma and shells for the geometry on 10 April, 1997 of comet Hale-Bopp is compared with the observed profile. Published values of polarization and colour in the coma and shells of comet Hale-Bopp on a close by date are used to investigate the nature of the grains, their organic content and porosity. Published aperture photometry at $0.6840~\mu {\rm m}$, $0.4845~\mu {\rm m}$ and $0.3650~\mu {\rm m}$ are initially used to short-list the possible range in porosities and the silicate to organic mass ratio. The relative weights of grain types in the coma and shell are further constrained such that the combined coma and shell population 1) reproduces the polarization vs. phase curve, 2) explains the high polarization on the shells and 3) their bluer colour (compared to coma). The results indicate that although it is impossible to determine the nature of the silicate (pyroxenes/olivines or amorphous/crystalline) using the present technique, the porosity, organic fraction and size distribution of the grains can be constrained. The scattered light from coma has polarization and colour characteristics which point to presence of at least $60 \%\, \pm\, 5 \%$ of grains with $1 \le m_{\rm sil}/m_{\rm org} \le 2$ and of medium porosity in the range $0 {-} 50 \%$. A narrow size distribution between $0.08~\mu {\rm m} {-} 0.6~\mu {\rm m}$ is inferred for $70 \%$ of the grains in the coma and shell. These grains could be the building blocks of larger fluffy grains of porosity larger than $90 \%$. At these porosities the individual units will interact independently with the incoming light and will act as isolated scatterers (Xing & Hanner 1997). The silicate in these grains could be Astronomical silicate (SiA) or a combination of Mg poor and Mg rich olivines or pyroxenes in crystalline or amorphous state. The shells of comet Hale-Bopp which produce larger polarization and are bluer compared to the coma appear to contain grains of silicate with lower organic fraction ( $r_{\rm m} \ge 8$) and of moderate porosity ( $30{-}50 \%$). In order to explain the presence of as many as 8-10 shells in well processed images, the fit using the present dynamical model imposes the condition that the shells must contain grains of $\beta <1$. This is independent of detailed assumptions on the optical properties of the grains or the scattering theory. Our Mie scattering calculations using Effective Median Theory to calculate the overall optical properties suggest that highly porous grains with $p \ge 90 \%$ made up of sub units in the form of Rayleigh inclusions (uniform density) have this property. This inference can be tested or further refined by using more elegant light scattering theories or techniques as mentioned below. Alternatively, the fluffy grains made of sub micron grains as sub units may also contain larger grains which may control the dynamics by reducing the effective $\beta$. The change in slope of the polarization vs phase curves at high phase angles which occurred at closer heliocentric distances and when the northern region of the comet was sunlit, could be better explained by increasing the fraction of the Mg poor olivine grains with $r_{\rm m} \ge 12$ at these epochs. This may be due to the limitations of the Mie theory in constructing the phase vs. polarization of natural grains. Considering the geometry of the apparition of the comet, a dichotomy in the surface composition of the comet may also be responsible to some extent for such an enhancement of these grain types required to fit the data set near perihelion.

In spite of the limitations of EMT theory with Mie solutions, the present results which indicate predominance of grains in the size range $0.08~\mu {\rm m} {-} 0.6~\mu {\rm m}$, and porosity in the range $0 {-} 45 \%$are similar to that suggested by Xing & Hanner (1997) using DDSCAT. They find that a mixture of aggregates of carbonaceous and silicate monomers of a tenth of micron in size and of intermediate porosity of $60 \%$ resembles the observed phase dependence of polarization of comets. For this intermediate porosity, it therefore appears that, at least for the polydisperse grains, EMT solutions may approximate the real scattering solutions. The DDA technique and assuming the grains to be made up of organic coated silicate monomers (Greenberg & Hage 1990) will be attempted as the next step to match the polarizations in the range from $0.3650~\mu {\rm m}$ to $2.2~\mu {\rm m}$. The use of computational techniques can be dispensed with and one could directly use (1) the microwave and laser facilities to determine scattering and colour signatures related to the physical properties of dust grains (Weiss-Wrana et al. 1985a, 1985b), (2) the database created by Gustafson et al. (1999) using microwave analogues of complex grain systems or (3) the data from the scattering experiments (PROGRA) conducted under micro gravity conditions on fluffy grains (Levasseur-Regourd et al. 1998; Worms et al. 1999) with optical constants similar to Astronomical silicate. In any case, by adapting a superior light scattering theory or using results of laboratory simulations, the present model can be used to fully extract the information available from polarization and intensity maps of cometary shells and coma.

Acknowledgements

I am grateful to Dr. P. Chakraborty of the Indian Institute of Astrophysics for the processed image of 10 April, 1997 and access to his software to convert the simulated images to IRAF images. I am also grateful to Dr. A. Li for the optical constants of the silicates. I am indebted to the referee for the valuable comments.