3 Brightness estimates of the shells and the coma

The intensity of light $I_{\rm s}(\theta, \lambda)$ and polarization $P_{\rm s}(\theta, \lambda)$ scattered by the grain towards the observer are given by (Krishnaswamy & Shah 1987):

$$I_{\rm s}(\theta,\lambda) \frac{I_{0}(\lambda) \lambda^2}{8 \pi r^2 \Delta^2}\left( i_{\perp} + i_{\parallel}\right)$$ = (18)

$$P_{\rm s}(\theta,\lambda) \left( i_{\perp} - i_{\parallel}\right)/\left( i_{\perp} + i_{\parallel}\right)$$ = (19)

where $\theta$ is the scattering angle and $I_0(\lambda)$ is the solar intensity at 1 AU. The intensity scattering functions $i_{\perp}$ & $i_{\parallel}$ polarized perpendicular and parallel to the scattering plane, and the scattering efficiency for radiation pressure $Q_{\rm pr}(s, \lambda)$ of the grain of radius s at the wavelength of radiation $\lambda$ (Eq. (6)) were estimated using Mie scattering using the code BHMIE by Bohren & Huffman (1983).

The position of the grain in the shell projected on the sky plane at a given time was thus calculated using Eqs. (13) and (14) and the intensity and polarization of light scattered by it was computed using the Eqs. (18) and (19). The colour is calculated following the definition by Jewitt & Meech (1986) as the normalized reflectivity gradient expressed in $\%$, $S^{\prime}(\lambda_1, \lambda_2) = ({\rm d}S/{\rm d}\lambda)\times 1000/S_{\rm mean}$ between wavelengths $\lambda_1$ and $\lambda_2$, where dS/d$\lambda$ is the rate of change of reflectivity with wavelength and $S_{\rm mean}$ is the mean reflectivity.