4 Application of the model

4.1 Observational inputs

1.

Image of comet Hale-Bopp on 10 April, 1997 in R filter obtained from the Vainu Bappu observatory was used to investigate the intensity variations in the shells and the coma.

2.

Published polarizations on 10 June, 1996, 30 September, 1996, and 9th April, 1997 separately in the coma and bright regions by Hadamcik et al. (1997) were used to investigate the nature of the grains in the coma.

3.

As the colour of the grains do not change drastically with phase angle, we set a conservative limit of $5 {-} 10 \%$ per 1000 Å on the shells and $10 {-} 20 \%$ per 1000  Å on the regions between the shells representing the coma based on the colour reported by Bellucci (1998), Furusho et al. (1999), Kiselev & Velichko (1997) and Kolokolova et al. (2001).

4.

Published aperture polarimetric observations at a range of phase angles by several groups (Sect. 1 and references therein).

4.2 Selection of grain material

Cometary grains are known to be porous and contain an organic component (Greenberg & Hage 1990; Hage & Greenberg 1990; Krishnaswamy et al. 1988; Xing & Hanner 1997). The dynamics as well as the scattering properties of the grains are affected by porosity and organic content, as both these factors modify the complex refractive indices of the grains significantly. The silicaceous components in the grain have been identified from their emission signatures in the IR spectra in the $10~\mu {\rm m}$region by Hayward et al. (2000) and Wooden et al. (1999). In the visible and near IR regions, it is futile to identify the Mg rich silicates either in glassy or crystalline form in the composite grains using their scattered radiation, as presence of even a small amount of organic content of 1/12 by weight will be sufficient to significantly alter the refractive index of the composite grain. Hence silicates were sub grouped into three classes depending on the absorption index k:

1.

Mg rich silicates with 0.0001<k<0.001, we refer to this group as SiX. Forsterite, with the optical constants by Scott & Duley (1996) was taken as the sample representative of this class (class 1).

2.

Astronomical silicate (Draine 1985; Draine & Lee 1984) with $k \approx 0.03$ (SiA). The present study uses the constants of crystalline olivine constructed by Li & Greenberg (1997) which uses the absorptive index of SiA in the visible region. Since polarization mainly depends on the absorptive index, we refer to this sample also as SiA (class 2).

3.

Mg poor silicates characterized by an absorption at wavelengths shortward of 0.5 $~\mu {\rm m}$ and with k > 0.03. Amorphous olivine (Aol) and pyroxene (Apy) with Mg number of 0.5 (Dorschner et al. 1995) were taken as representatives of this class (class 3).

At $r_{\rm m} \leq 12$, distinction between SiA and SiX almost vanishes. Following Li & Greenberg (1998), the organic refractory component was assumed to be a composite of interstellar refractory (Li & Greenberg 1997) and amorphous carbon (Rouleau & Martin 1991) of equal mass. Effective refractive indices of the composite grains were determined using Bruggmann's mixing rule. Collections of individual grains of these silicates with the organic refractory component in the mass ratios $r_{\rm m} = 0$, 1, 2, 8, 12 and $\infty$ and the porosities $p= 0\%$, $30 \%$, $35 \%$, $40 \%$, $45 \%$, $50 \%$, $60 \%$, $70 \%$, $80 \%$ and $95 \%$ were considered.

4.3 Short-listing of grain types - preliminary fit to three colour phase vs. polarization data

Phase dependence of polarization carries valuable information on the interaction between the reflected and refracted components from the grain along different scattering directions. The fraction of the refracted component depends on the optical properties of the grain, its size and shape. Polarization of light scattered by an ensemble of spherical grains at the three continuum wavelengths 0.6840 $\mu$m, 0.4845 $\mu$m and 0.3650 $\mu$m corresponding to aperture polarimetry observations by Ganesh et al. (1998) and Manset & Bastien (1998) were computed. For a proper match with these observations over the entire range of phase angles, combinations of grain types and proper choice of their size distribution are essential. Hanner size distribution (Hanner 1985)

$$n(s){\rm d}s=(1-s_0/s)^{M^{\prime}}(s_0/s)^{N^{\prime}}$$ (20)

was used. The constants s₀, $M^{\prime}$ and $N^{\prime}$ determine the slopes at the lower and upper size domains and the radius $s_{\rm p}$ at which n(s) is maximum. As a first guess, four component equal weight inter-combinations of the three classes of silicates, each with 30 different combinations of p and $r_{\rm m}$ were considered. A total of 810000 combinations were compared in the least square sense with the observed phase curves of Ganesh et al. (1998). The size distribution parameters were varied to improve the $\chi^2$ values. The constants in set 'a' in Table 1 yielded best fit.

 

 

Table 1: Size distributions investigated in the present study.

Distribution	$s_{\rm o}$	$N^{\prime}$	$M^{\prime}$	Grain sizes
				$s_{\min}$	$s_{\max}$
	$\mu {\rm m}$			$\mu {\rm m}$	$\mu {\rm m}$
a1	0.001	3.85	92	0.05	30
b2	0.05	3.70	8.47	0.05	30
c1	0.1	3.60	12.9	0.10	30
1. Derived in the present work.
2. Adopted from Hayward et al. (2000).

The intensity contribution I(s)n(s)ds for this distribution at a mean wavelength of $0.5~\mu {\rm m}$is shown as the solid line in Fig. 2.

Figure 2: Intensity contribution of grains of sizes between s and $s+{\rm d}s$, along the scattering angle of $136^\circ$ and at a wavelength of $0.5~\mu {\rm m}$ for the size distribution constants in Table 1. All the three curves have been normalized to enable a direct comparison. The solid vertical line indicates the lower limit of 0.05 $~\mu {\rm m}$ on the grain size used in the present investigation.

Table 2 gives the grain types which were common in the first 100 best fitting combinations.

 

 

Table 2: Short-listed grain types.

	Mg rich			Mg poor
	SiA1/	SiA	SiX	Aol3/	SiA/SiX/
	SiX2			Apy4	Aol/Apy5
$r_{\rm m}$	1,2,8,12	$\infty$	$\infty$	$\infty$	$1 {-} \infty$
$p (\%)$	0-50	30-50	30-50	30-50	95
1. Close to Astronomical silicate.
2. Olivine or Pyroxene in crystalline or amorphous state.
3. Mg2yFe2-2ySiO4 with y=.5.
4. MgxFe1-xSiO3 with x=.5.
5. At $p \ge 90 \%$, $r_{\rm m}$ is poorly constrained. These grains may
represent porous grains with Rayleigh type inclusions.

It is striking to note that except for the highly porous grains, all grain types have porosities in the range between $0 \%$ (compact) and $50 \%$.

The size distribution 'a' represents the combined grain population in the coma and shell which are sampled in the aperture of $26\hbox{$.\!\!^{\prime\prime}$ }5$. In the present work it is attempted to investigate the relative fraction of the grain type and their size distributions separately in the coma and the shells.

4.4 Dust emission geometry

The simulations of the intensity, colour and polarization maps of comet Hale-Bopp were carried out for the geometry on 10 April, 1997, in the wavelength band corresponding to the cometary continuum wavelengths at $0.4450~\mu {\rm m}$ and $0.6840~\mu {\rm m}$ for the grain types in Table 2. The pole positions and latitude of the sources are adapted from Paper II. The grain velocities computed using Eq. (2) with v₀ = 0.63 kms^-1 were found to be within $\pm 5 \%$-$\pm 10 \%$ to the values computed using Eq. (1). The latter was used in the simulations. The coma was simulated by considering emission from a grid on the comet at intervals of $10^\circ$ in latitude and $10^\circ \cos (\phi)$ in longitude, where $\phi$ is the latitude of the point on the grid. The sources are assumed to be active when sunlit. The trajectories of the dust grains are computed assuming collimated emissions. In reality, the emission may be spread over a cone. Further, the velocity of the grains will have a spread due to deviation from their assumed sphericity, finite size of the active regions and possible irregular local terrain. Their cumulative effect will produce a broadening of the shells. As it is difficult to model each of these effects, the sky plane position of the grains were convolved with a Gaussian-random profile with a sigma of 1 $.\!\!^{\prime\prime}$5. To include the effect of spreading with time, the sigma was allowed to increase by 0 $.\!\!^{\prime\prime}$25 per day. Due to rotation of the nucleus, grains in the expanding jets are swept along the azimuth of the comet giving the appearance of a set of shells, one shell per rotation. The simulated images contain 8 shells and coma dust ejected between 0 - 150 hr.

4.5 Selection of size distribution of the grains in the coma and the shells

Larger grains in the coma compared to the shells or a different grain composition for the two regions have been inferred by Hadamcik et al. (1997), Jockers et al. (1997) and Tanga et al. (1997). High polarization and high silicate IR emission (Sect. 1 and references therein) in the shells suggest predominance of smaller grains in these regions. The size distribution 'a' obtained by overall fit to coma and the shells may represent a distribution which is the average of a larger grain population for the coma and finer grains in the shells. The simulations were carried out for the distribution 'b' (Table 1), used by Hayward et al. (2000) to include larger grains. The dot dashed line in Fig. 2 shows this distribution. The size distribution of the smaller set of grains should be such that the total ensemble should produce the observed trend of increase of polarization with wavelength. As shown in the next section, the distribution 'c' with the coefficients given in Table 1 and the intensity distribution curve shown as the "dashed'' line in Fig. 2 produces this trend.

4.5.1 Effect of grain size distribution on the spatial variation of intensity, colour and polarization

Effects of variation of porosity, organic fraction and size distributions are demonstrated in the left, middle and right columns of Fig. 3 for distributions 'c',

Figure 3: Effects of grain size distribution ('c': left, 'a': middle, 'b': right) and organic content on the spatial variation of intensity (lower panel, superposed on the observations), colour (middle panel), polarizations $P_{\rm r}$ and $P_{\rm b}$ ( solid and dashed lines in top panel), across the coma along the sunward - anti sunward direction for the geometry on April 10, 1997. The horizontal line at $10.0 \%$ polarization represents $P_{\rm r}$in the dark regions (Hadamcik et al. 1997). The lower horizontal line at $7.2 \%$ represents the expected value of $P_{\rm b}$.

'a' and 'b' respectively for a few sample grain types in Table 2. Variation of intensity, colour and polarizations along the solar - anti-solar direction across the simulated comae are shown in the lower, middle and top sections respectively. For distribution 'c', at porosities less than $45 \%$, the polarization in $0.6840 ~\mu$, $P_{\rm r}$ (solid line) is larger compared to that in $0.4550 ~\mu$, $P_{\rm b}$ (dashed line). The horizontal line at $10.0 \%$ polarization level indicates the reported value in dark regions by Hadamcik et al. (1997) in red. The lower horizontal line at $7.2 \%$is the expected polarization in $0.4450~\mu {\rm m}$ based on the $P_{\rm r}{-}P_{\rm b}$ values from published aperture photometry. The error bars in the observed intensity cuts represent the uncertainty in the determination of sky background which is the largest source of error in the image. The upper and lower limits correspond to sky fluxes which were $0.7\%$ and $2.4\%$ of the central flux. The solid curve in the middle section of this panel shows the simulated colour variation. The steep fall in intensity and polarization beyond $25\,000$ km on the sunward side in the coma of grains with $r_{\rm m} < 2$ and distributions 'a' and 'c' occurs because of their large $\beta$ (Eqs. (4) and (5)) values. The increase in polarization beyond $35\,000$ km in the sunward direction has opposite polarization colour as this is caused by the larger grains in the tail end of distribution.

Figure 3 illustrates clearly the effect of size distribution on the colour and polarization. For grains of medium porosity ($p <45 \%$), $P_{\rm r}>P_{\rm b}$ for distribution 'c', $P_{\rm r} \leq P_{\rm b}$ for 'a' and $P_{\rm r}<P_{\rm b}$ for 'b'. The anti-correlation in colour and polarization as pointed out by Kolokolova et al. (2001) is clearly noticeable for 'a' and 'b': the collection 'b' (right) which has larger grains has lower polarization but larger colour than the distribution 'a' (middle). Contrary to scattering behavior of sub-micron sized grains, the absorptive grains in population 'c' are redder compared to that in 'a'which contains additional larger grains. The reddening is found to increase with the refractive index of the grain either due to increase of organic fraction or reduction in porosity. It is marginally higher for Aol compared to the less absorptive SiA. As pointed out by Kolokolova et al. (2001), a grain may be redder if its size parameter x is just beyond the scattering peak. The scattering peak occurs at $x \approx 2$ for absorptive grains with an effective refractive index of $\approx$ $1.6 {-}i\,0.2$ corresponding to $1 < r_{\rm m} < 2$ and $p \leq 45 \%$. For less absorptive grains of pure silicates of effective refractive index $\approx$ $1.4 {-}i\,0.01$ corresponding to $45 \%$ porosity, the peak occurs at $x \approx 6$ (Wickramasinghe 1973). For the average grain size of $0.24~\mu {\rm m}$ at the broad peak in Fig. 2 for distribution 'c', the size parameters at $0.6840~\mu {\rm m}$ and $0.3650~\mu {\rm m}$are 2.2 and 4.4 respectively. This range in size parameter is beyond the scattering peak for absorptive grains and before for the pure silicates. This explains the red colour of the absorptive grains and the blue colour of the less absorptive ones. Further, for this distribution $P_{\rm r} \geq P_{\rm b}$, the reason for this large polarization colour is that in this domain of $0.08~\mu {\rm m} < s < 0.6~\mu {\rm m}$, the grain becomes increasingly forward scattering with increase in its size parameter. Blue light is therefore more forward scattered than the red. With increase in the transmitted part of blue light which is parallelly polarized (negative polarization) the resultant polarization is lower compared to that in red. On the other hand, for the large grain population 'b', $P_{\rm r}<P_{\rm b}$, which is opposite to the observed trend for comets. Therefore, the difference in the grain populations in the coma and shells may in most part be due to difference in the silicate to organic ratio rather than difference in grain sizes.

4.5.2 Effect of grain size distribution on the variation of polarization and polarization colour with solar phase angle

The selected grain types should also explain the phase dependent polarization in various colours. Figure 4 shows the phase dependence of the polarization produced by the same grain types as in Fig. 3 at seven continuum wavelengths for distributions c (left),

Figure 4: Effects of grain size distribution and organic content on the phase dependence of polarization for the same sample grain types as in Fig. 3. Details of the computed curves and the data are same as for Fig. 6.

a (middle) and b (right) respectively. The polarizations at $0.3650~\mu {\rm m}$ (short dashed line), $0.4845~\mu {\rm m}$ (dotted line), $0.6840~\mu {\rm m}$ (solid line), $1.24~\mu {\rm m}$ (short dash - dot), $1.65~\mu {\rm m}$ (short dash - dot, thick line) and $2.2~\mu {\rm m}$ (short dash - long dash) show different trends for the three distributions. The observed polarizations in the optical by Ganesh et al. (1998), Hadamcik et al. (1997), Kiselev & Velichko (1997) and Manset & Bastien (1998) and the infrared observations by Hasegawa (1999) are over plotted. The phase dependence and colour of polarization again critically depend on the size distribution. The relative polarization colour represented by ( $(P_{\rm r} -P_{\rm b})/(P_{\rm b}-P_{\rm u}))$ matches with the observed ratio for grains of distribution 'c' with $1 < r_{\rm m} < 2$and $p < 30 \%$, where $P_{\rm u}$ is polarization in $0.3650~\mu {\rm m}$. Organic refractory component has increased absorption in blue and UV. Comparing SiA grains of $45 \%$ porosity of $r_{\rm m}=2$(SiA0245) and $r_{\rm m}=\infty$ (SiA$\infty 45$), in this grain size domain, polarizations in $0.4845~\mu {\rm m}$ and 0.3650 are therefore lower in the latter compared to the former because of lesser attenuation of the refracted component. Grains of Aol$\infty 45$ show their characteristic signature. The absorption in blue causes higher positive polarization in all size domains. The fluctuations in the large pure silicate grains are caused by the interference between the refracted and reflected components of light (Kolokolova et al. 1997).

Figures 3 and 4 clearly demonstrate that porosity, organic fraction and the size distribution of grains in the coma and shells play an important role in controlling the phase dependence of polarization, colour of polarization, photometric colour and spatial variation of intensity in a comet. We exploit this to investigate the composition of grains in the coma and shell separately.

4.6 Investigation of grain materials and dust size distribution in the coma

Synthetic comae were constructed by combining the grain types in Table 2. The relative weights were adjusted to fulfill the following four observational constrains:

1.

The published values of red polarization between the shells (dark regions) of $-3.5 \%$, $-0.8 \%$ and $10 \%$ at phase angles of $7 \hbox{$.\!\!^\circ$ }5$, $19 \hbox{$.\!\!^\circ$ }6$ and $44 \hbox{$.\!\!^\circ$ }2$ respectively by Hadamcik et al. (1997), and blue polarizations deduced from the published aperture photometry;

2.

Colour of the coma in the range $10 \% {-} 20 \%$;

3.

Higher polarization on the anti-sunward direction.

All the three size distributions were considered for each grain type. The weights and the size distributions which met this test are given in Table 3.

 

 

Table 3: Possible grain types in the coma and shells.

Contribution:	Coma				Shell
$m_{\rm sil}/m_{\rm org}$:	$1 \le r_{\rm m} \le 2$		$8 \le r_{\rm m} \le \infty$		$8 \le r_{\rm m} \le \infty$	$1 \le r_{\rm m} \le \infty$
Porosity:	$0 \% \le p \le 50 \%$	$40 \%\le p \le 50 \%$	$30 \%\le p \le 50 \%$		$40 \%\le p \le 50 \%$	$p \ge 90 \%$
Silicate Type:	SiA/SiX	SiA/SiX	SiA/SiX	Aol1	SiA	SiA/SiX
Fractional weight2:	$0.47 \pm 0.05$	$0.14 \pm 0.05$	$0.05 \pm 0.05$	$0.14 \pm 0.05$	$0.12 \pm 0.02$	$0.06 \pm 0.02$
Size Distribution:	c	b	c	b	c	a
1. Weight near perihelion.
2. Relative contribution to intensity in red.

These weights correspond to the relative contribution to the red intensity rather than the number of grains. Thus the coma may contain a large fraction of grains with distribution 'c'. However such grains will produce a very flattened coma. Hence these grains may be part of larger fluffy grains some of which may also be physically associated with grains of distribution 'b'. The dynamics will then be controlled by an average value of $\beta$ and the coma profile will follow the observed coma (regions between the shells). Alternatively, some of the grains may have distributed sources and hence have a synthetic radial profile mimicking the grains of distribution 'b'.

4.7 Investigation of grain material in the shells

The weights of the grain types in the shells were adjusted so that the simulated shells have the following attributes reported by Hadamcik et al. (1997), Jockers et al. (1997) and Tanga et al. (1997):

1.

High polarization on the shells;

2.

Bluer colour of the shells by $5 \% {-} 8 \%$;

3.

Rapid decline of polarization of the shells on the sunward side;

4.

Intensity contrast between the shells and the underlying coma.

Figure 5 shows the simulations of shells made up of grain types which produce polarizations larger than the published aperture polarization measurements.

Figure 5: Column 1: observed processed image from VBO $\times$ (projected cometocentric distance)-1. North is top and East is left. The sets of shells expanding from the nucleus in the direction of N-W, S-W and S-E are from the sources at $-5 ^\circ$, $65 ^\circ$ and $+5 ^\circ$ respectively. Columns 2 and 3: simulated intensity (shell) and polarization (shell + coma) maps. Column 4: intensity scans of the shells superposed on the coma (lower panel), colour in % (middle panel), $P_{\rm r}$ (solid line in top panel), $P_{\rm b}$ (short dashed line in top panel) and $P_{\rm r}{-}P_{\rm b}$ (long dashed line in top panel).

The first column shows the observed processed image from VBO. The shell structures are enhanced by dividing by a synthetic image of a coma (Chakraborty 2001) in which the intensity declines as the inverse of projected cometocentric distance. The coma image was scaled so that the resultant normalized flux in the N-E edge of the image which is devoid of shells, was unity. The sets of shells expanding from the nucleus in the direction of N-W, S-W and S-E are from the sources at $-5 ^\circ$, $65 ^\circ$ and $+5 ^\circ$ respectively (Papers I and II). The simulated intensity map of the shells and polarization map of the shells superposed on the coma are shown in second and third columns respectively. Each simulated image is divided into $195 \times 195$ square pixels of size 0 $.\!\!^{\prime\prime}$5 on a side. The number density of the grains in the jets were adjusted to match the intensity variation in the shells. The fourth column shows the intensity scans of the shells superposed on the coma (lower panel), colour in % (middle panel), $P_{\rm r}$ (solid line in top panel), $P_{\rm b}$ (short dashed line in top panel) and $P_{\rm r}{-}P_{\rm b}$ (long dashed line in top panel). Each grain type has its characteristic signature in the intensity, colour and polarization map.

1.

Row 1 (starting from below): grains of the type "Si $\infty$45'' with distribution 'a'produce shells that resemble the observed ones. The shells are bluer but the polarization is only marginally higher than the ambient coma.

2.

Row 2: grains of the type "Si $\infty$45'' but with distribution 'c'appear to be likely candidate as $P_{\rm r}$ is larger on the shell compared to coma. However neither is there a rapid fall in polarization in the sun ward direction nor an increase in polarization in the anti sunward direction.

3.

Row 3 & 4: increasing the organic fraction to $r_{\rm m}=8\ \&\ 12$ causes the older shells to be depleted of finer grains as these are blown away towards the anti-sunward direction. The shells are compressed and a piling up of shells older than four rotations causes a "shell pause'' like structure in the polarization map and an abrupt decline in polarization. In the intensity map, the pile up of matter merges with the coma. This region appears bluer in the colour map because of the shell material. For a detailed comparison with observed colour and polarization maps, shells ejected over a larger numbers of the nuclear rotations (>8) and coma ejections over an extended duration (t > 150 hr) are required.

4.

Row 5 & 6: with increase in the organic fraction to $r_{\rm m} \leq 2$, the radiation pressure blows out grains from the shells older than two rotations and produces a very steep increase in polarization in the anti-sunward direction and an abrupt fall in the sunward direction. The contrast in colour between the coma and the shells disappears The effect is pronounced for $r_{\rm m}=1$.

5.

Row 7: porous grains are known to produce large polarizations. Hence we considered a collection of highly porous grains with distribution 'a' which are assumed to be homogeneous in the Rayleigh limit. This set produces well constrained shells which survive several rotation periods. This is because of lower $\beta$ values which vary slowly with the grain size. Although the present approach of EMT + Mie theory is known to have serious limitations for porous grains, these simulations help to visualize the signature of highly porous homogeneous grains. The streaks in the polarization map streaming away in the anti-sun direction represent the small grains in the population.

From Fig. 5, it is apparent that larger polarization and bluer colour on the shells are produced by grains of SiA with $r_{\rm m} >8$. However, not more than 3 shells survive the radiation pressure forces while the observed image after careful evaluation and removal of the underlying coma shows up to 8 shells (Chakraborty 2001). Therefore a population of grains with high polarization but low $\beta$, similar to the over simplified example of a uniform porous grain in Row 7 in Fig. 5 also appear to be needed to explain the survival of older shells. A reliable estimate of the fractional weights of these grain types can only be made by a quantitative comparison of the polarization maps. In the absence of availability of such a data set, we make use of the aperture polarimetry observations. As the aperture photometry represents combined contribution of coma and shells, the grains which may explain the polarization on the shell, when added in the same proportion to the selected combination representing the coma should explain the aperture polarimetric data. The best fitting weights for the shells which fitted polarization vs. phase curves are given in Table 3. Figure 6 shows the resultant simulated polarization vs. phase curves.

Figure 6: Fitted polarization vs. phase angle curves for comet Hale-Bopp: computed curves: $0.6840~\mu {\rm m}$ (solid line), $0.4845~\mu {\rm m}$ (dotted line), $0.3650~\mu {\rm m}$ (dashed line), $0.7200~\mu {\rm m}$ (long dashed line), $1.24~\mu {\rm m}$ (short dash - dot, thin line), $1.65~\mu {\rm m}$ (short dash - dot, thick line) and $2.20~\mu {\rm m}$ (long dash - short dash).

At phase angles larger than $30 ^\circ$, the fits were better if the relative fraction of Mg poor silicate grains with distribution b were increased gradually with heliocentric distance. A possible explanation for this is given in Sect. 6.

The resultant simulated intensity maps of the shells and polarization maps of shells and coma using weights in Table 3 are shown in the top, middle and right panels of Fig. 7.

Figure 7: Top left: the processed observed image as in Fig. 5. Top middle and right: simulated intensity maps of the shells and polarization maps of shells+coma using weights in Table 3. The lower panel shows the intensity, colour and polarization scans in the sun-ward - anti-sun-ward direction.

The first panel is the processed observed image as in Fig. 5. Separation of the high $\beta$ ($p=45 \%$) grains and low $\beta$ grains ($p=95 \%$) components produces the splitting of the shells in the N-W and S-E region of the simulated image (Table 3).

The lower panel of Fig. 7 shows the intensity, colour and polarization scans in the sunward - anti-sunward direction. In order to reduce the fluctuations arising due to discretization of cometary longitudes and latitudes, the scans have been averaged over 9 pixels ( $4 \hbox{$.\!\!^{\prime\prime}$ }5$). The polarization contrast between the coma and the shells are therefore reduced in these plots. The maximum and minimum value of polarization in the map however correspond to $16 \%$ in the bright regions and $2 \%$ beyond the 4th shell in the sunward direction.

4.8 Sources of errors and limitations of the present technique

The width and separation of the shells depend on the pole position, velocity $v_{\rm gr}$ and the acceleration $\alpha$ of the grains. The later two quantities depend on the size distribution and the nature of the grains. For the geometry on 10 April, 1997, the uncertainty in the computed separation of the shells from the source at $65 ^\circ$, is found to be $\pm 10 \%$ along the projected sunward direction for a change in $\alpha_{\rm p}$by $\pm 15 ^\circ$ and $\delta_{\rm p}$ by $\pm 10 ^\circ$. The model thus reproduces the shells within an accuracy of $\pm 5 \%$ in the shell separation corresponding to the uncertainty of $\pm 5^\circ$ in the pole position used in the present study following Paper II. The grain velocities cannot therefore be constrained better than $\pm 5 \%$. For a more rigorous check on the derived pole positions and the grain velocities, intensity scans across the observed image along the directions $184 \hbox{$.\!\!^\circ$ }5$ (b), $194 \hbox{$.\!\!^\circ$ }5$ (c), $214 \hbox{$.\!\!^\circ$ }5$ (d), $224 \hbox{$.\!\!^\circ$ }5$ (e, the sunward direction), $234 \hbox{$.\!\!^\circ$ }5$ (f), $244 \hbox{$.\!\!^\circ$ }5$ (g) and $254 \hbox{$.\!\!^\circ$ }5$ (h), divided by the scans along the same directions across the simulated coma of composition in Table 3 are compared with the simulated scans in Fig. 8 (lower panel).

Figure 8: Top: the processed observed image as in Fig. 5 showing the directions of scans. Lower panel: intensity scans across the observed (dashed line) and simulated (solid line) images along directions $145 \hbox{$.\!\!^\circ$ }5$ a), $184 \hbox{$.\!\!^\circ$ }5$ b), $194 \hbox{$.\!\!^\circ$ }5$ c), $214 \hbox{$.\!\!^\circ$ }5$ d), $224 \hbox{$.\!\!^\circ$ }5$  e), $234 \hbox{$.\!\!^\circ$ }5$ f), $244 \hbox{$.\!\!^\circ$ }5$ g), $254 \hbox{$.\!\!^\circ$ }5$ h) and $285 \hbox {$^\circ $ }$ i), divided by the scans along the same directions across the simulated coma of composition in Table 3.

The directions are marked on the observed image in the top panel. The nucleus rotated by $154 ^\circ$ during the interval of time between the ejections of the two extreme sections of the shells. The cuts along the directions $145 \hbox{$.\!\!^\circ$ }5$(a) and $285 \hbox{$.\!\!^\circ$ }5$ (i) are across the equatorial shells. The location and width of the shells match reasonably well over a range of hour angle of the Sun spanning $154 ^\circ$ for the high latitude shells as well as the two cuts across the equatorial shells. The cuts along the directions 'b', 'c' and 'h' are contaminated by the equatorial shells. For all the cuts, at distances larger than $30 \hbox{$^{\prime\prime}$ }$- $35 \hbox{$^{\prime\prime}$ }$, deviations between the observed and simulated cuts become noticeable. This may be attributed to low intensity level of the observed image at these distances as a result of which the errors associated with improper sky subtraction increase rapidly, as indicated by the large error bars in Figs. 5 and 7. The simulated image also suffers from limitations due to under-sampling of the shells of only 8 rotations and coma of 13 rotations. The deviations may also be due to the simplicity of the model which does not take into account the grain evolution processes like evaporation or splitting. These processes alter $\beta$ of the grains and hence their dynamics. There is also a gradual increase in the intensity of the first shell from morning till evening (a-i). This may be due to the assumption of uniform dust emission from dawn till dusk made in the simulation of the coma, which is used to normalize the intensity. The simulated "shell+coma'' profile may be unaffected because of similar assumptions for simulating the shells. The local E-W asymmetry appears to smoothen out in the outer regions. In any case, inferring the grain composition by matching the observed separation and width of the first few shells alone will be questionable because a variety of factors such as grain size distribution, spread in porosity and organic content of the grains determine the average value and spread in $\beta$ of the ensemble. Fortunately, these factors influence the colour and the phase dependence of polarization at different wavelengths in different ways (Figs. 3 and 4). Hence as attempted in the present investigation, the spatial match must be accompanied by comparisons of (1) colour and (2) polarization of the observed and simulated shells (Figs. 3 and 5). The derived range in porosity and organic fraction must explain the published phase curves in three colours (Fig. 6). This places further constraints on the derived values of p and $r_{\rm m}$. However, there are limitations in the present model which should not be overlooked. As explained in the following sections, Mie theory does not accurately predict the polarization vs. phase curves of porous grains which are not homogeneous. Lack of our understanding of the nature of the organic component limits the reliability of the derived compositions. If the composition of the organic material assumed in Sect. 4.2 is changed by including the extremely reddish material inferred by Tegler and Romanishin (2000) in the Kuiper-belt objects, the fitted fraction of Aol will be modified. This may also change the best fitting size distribution as this parameter was constrained to explain the colour of the grains. Further, despite the availability of an extensive and wide range of data set on this comet, the inclusion of a large parameter space will hinder in getting a unique solution due to cross talk between the effects of different parameters.

For reasons explained in Sect. 4.2, the present technique cannot be used to identify the mineralogy of the silicates in the grains although distinction can be made between the low and high Mg content silicates based on the photometric and polarization colours (Figs. 3 and 4). Such investigations are best made using IR spectroscopy (Hayward et al. 2000; Wooden et al. 1999). However, the derived porosities and organic contents in the present work can help in constraining the IR continuum, and hence the two techniques are complementary to one another.