2 The model

The computer simulations are based on the basic concepts introduced by Sekanina (1981, 1991) and Sekanina & Larson (1984). Except for the outburst events during 1995, which were sporadic activities, the dust grains are assumed to be ejected continuously from the sources radially outwards, from local sunrise to sunset. The ejected grains are subjected to the gravitational attraction of the Sun and the Solar radiation pressure force in the opposite direction. In the present work, since we study the shells which are close to the nucleus within a distance of $\approx$$100\,000$ km, it is assumed that dust grains follow the same Keplerian motion as that of the comet due to solar gravity. Thus, relative to the comet, the position of the dust grain can be calculated from its initial velocity, the radiation pressure force and the ejection geometry.

2.1 Velocity and acceleration of the grains

The velocity $v_{\rm gr}$ attained by the grains due to gas drag (Probstein 1969) can be calculated using the relation by Sekanina (1981):

$$1/v_{\rm gr} = a + b/\sqrt{\beta} \\$$ (1)

and also the following equation by Fulle (1987)

$$v_{\rm gr}=v_0 \beta^{1/6}$$ (2)

where $\beta$ is the ratio of the force due to solar radiation pressure on the grain to the gravitational force. Sekanina & Larson (1984) have used Eq. (1) successfully for emission from discrete sources, and pointed out that it is valid for slightly absorbing grains with $\beta \leq 0.6$. The acceleration $\alpha$ due to solar radiation pressure depends on the nature of the grain and the heliocentric distance. Using the definition of $\beta$,

$$\alpha = (\beta g_{\rm sun(1)})/r^2 ,$$ (3)

where $g_{\rm sun(1)}$ is the acceleration due to solar gravity at one AU ( $0.6 \times 10^{-5}$ kms^-2) and r the heliocentric distance of the comet. The value of $\beta$ can be calculated for a grain of given size and composition using the relation by Finson & Probstein (1968):

$$\beta F_{\rm rad}/F_{\rm grav}$$ = (4)

$$\frac{3 Q_{\rm pr} E_{\odot}}{4 \pi c G M_{\odot}}\frac{1}{\rho s }$$ = (5)

where $Q_{\rm pr}$ is the scattering efficiency for radiation pressure of the grain, s its radius and $\rho$ its density. $E_{\odot}$ is the total solar radiation per second, c the velocity of light, $M_{\odot}$ the mass of the Sun and G the universal gravitational constant. In the present investigation, the grain size is allowed to vary from $s_{\min} = 0.05~\mu {\rm m}$ to $s_{\max} = 30.0~\mu {\rm m}$ with Hanner (1985) size distribution law. Since the radiation pressure parameter $Q_{\rm pr}$ of a grain material depends both on its size and the wavelength of radiation, the value of $\beta$ for a grain of radius s was calculated using the more precise relation

$$\beta= \frac { 3}{4 \pi c G M_{\odot}} \frac {1}{\rho s }\int... ...a 1} Q_{\rm pr}(s,\lambda) F_{\odot}(\lambda) {\rm d}\lambda ,$$ (6)

where, $F_{\odot}(\lambda)$ is the mean total solar radiation from the sun per second per Angstrom at the wavelength $\lambda$. The above integration was carried out from $\lambda 1 = 0.1~\mu {\rm m}$to $\lambda 2 = 5.0~\mu {\rm m}$.

2.2 Comet-o-centric spherical coordinates of the grains referred to comet's equator

In the simulation, the path of the dust grain ejected radially from an active region at a comet-o-centric longitude u, latitude $\phi$ and the nuclear radius R is calculated as a function of time. Figure 1 shows the ideal case of a spherically symmetric nucleus centered at C.

Figure 1: A model to compute the sky plane coordinates of the ejected dust grains. The comet centered at C is assumed to be spherically symmetric. The grain from the source at G is ejected with velocity $v_{\rm gr}$. The north pole of the comet is at $N_{\rm C}$, CPE is the direction of Earth's celestial pole, E and S are the sub-Earth and sub-Sun points respectively. See text for more details.

The sub-Sun and sub-Earth points are S and E respectively and $N_{\rm C}$ is the North pole of the comet. The direction of the Earth's north pole is along $CP_{\rm E}$. The ascending node of the comet's equator on the Earth's equator is $N_{\rm equ}$ and that on the ecliptic is indicated by $N_{\rm ecl}$. Components of $\alpha$ parallel to the plane of the comet's equator and perpendicular to it are $\alpha_{\parallel}=\alpha \cos B^{\prime}$ parallel to SF and $\alpha_{\perp}=\alpha \sin B^{\prime}$ parallel to ST respectively where $B^{\prime }$ is the comet-o-centric latitude of the Sun. The longitude of the source and the dust grains are measured from $N_{\rm equ}$along the direction of rotation of the comet. The longitude of the sub-Earth point E is $U-\pi$ and the angle $B = \widehat{E_{\rm E}CE}$ is its latitude. Dust grains are ejected radially outwards with the velocity $v_{\rm gr}$ from an active source $G(u, \phi, R)$, where $u = N_{\rm equ}E_{\rm D}$ and $\phi = E_{\rm D}G$. Strictly speaking, the grain attains this velocity after it is dragged by the subliming gas to a few nuclear radii. However, considering the vast spatial extent of the shell and jet structures of several rotations which are being modeled, this distance can be ignored. Relative to the comet, during time t, the dust traverses a distance $v_{\rm gr}t$ in the radial direction and a distance $1/2 \alpha t^2$ along the Sun - comet direction due to solar radiation pressure.

The position ( $u^{\prime}, \phi^{\prime}, r^{\prime}, t$) of the grain at time t reckoned after ejection is given by:

$$r^{\prime} \cos \phi ^{\prime} \cos u^{\prime} = v_{\rm gr} t \cos \phi \cos u - (1/2) \alpha t^2 \cos B^{\prime} \cos U^{\prime\prime}$$

$$r^{\prime} \cos \phi^{\prime} \sin u^{\prime} = v_{\rm gr} t \cos \phi \sin u - (1/2) \alpha t^2 \cos B^{\prime} \sin U^{\prime\prime}$$

$$r^{\prime} \sin \phi^{\prime} = v_{\rm gr} t \sin \phi - (1/2) \alpha t^2 \sin B^{\prime}$$ (7)

where $U^{\prime \prime}$, the longitude of the sub-Sun point is given by

$$U^{\prime \prime} = N_{\rm equ}E_{\rm S} = \omega + U^{\prime} - \pi.$$

The expressions in Eq. (7) are valid provided the direction and magnitude of $\alpha$ is constant. For the observations during February-May 1997, since we model only up to a maximum of 8 shells, a constant value for $\alpha$ during the period of 8 rotations may be a valid assumption for comet Hale-Bopp with a period of 11.34 hr (Jorda et al. 1997; Licandro et al. 1998). For the 1995-96 images, slow apparent motion of the comet again justifies such an assumption.

The angles $\omega=N_{\rm equ} N_{\rm ecl}$, $U^{\prime} - \pi =N_{\rm ecl} E_{\rm S}$can be calculated from the following relations:

$$\sin \omega \sin i = \sin \epsilon \sin N$$

$$\cos \omega \sin i = \cos \epsilon \sin J - \sin \epsilon \cos J \cos N$$

$$\cos i = \cos \epsilon \cos J + \sin \epsilon \sin J \cos N,$$ (8)

where i is the inclination of the comet's equator to the ecliptic. The angles $N = \gamma N_{\rm equ}$ and J, the inclination of the comet's equator to the Earth's equator are related to the right ascension and declination of the comet's pole:

$$N = \alpha_p + \pi/2$$

$$J = \pi/2 - \delta_{\rm p}.$$ (9)

The angles U, $U^{\prime}$, B, $B^{\prime }$ and the position angle Pof the north pole of the comet projected on the sky plane were calculated using the equations used for calculating the planet-o-centric positions of the satellite with respect to the planets (Rhode & Sinclair 1992):

$$\cos B \sin U = \cos J \cos \delta \sin (\alpha -N) + \sin J \sin \delta$$

$$\cos B \cos U = \cos \delta \cos (\alpha - N)$$

$$\sin B = \sin J \cos \delta \sin (\alpha - N) - \cos J \sin \delta$$

$$\cos B \sin P = -\sin J \cos ( \alpha - N)$$

$$\cos B \cos P = \sin J \sin \delta \sin (\alpha -N) +\cos J \cos \delta$$

$$\cos B^{\prime} \sin U^{\prime} = \cos i \cos b \sin (l-\Omega) + \sin i \sin b$$

$$\cos B^{\prime} \cos U^{\prime} = \cos b \cos (l - \Omega)$$

$$\sin B^{\prime} = \sin i \cos b \sin (l - \Omega) - \cos i \sin b ,$$ (10)

where $\alpha$, $\delta$, l and b are the geocentric right ascension, declination, heliocentric ecliptic longitude and latitude of the comet respectively. The angle $\Omega = \gamma N_{\rm ecl}$ can be calculated using the following relations:

$$\cos \Omega \sin i -\cos J \sin \epsilon + \sin J \cos \epsilon \cos N$$ =

$$\sin \Omega \sin i \sin N \sin J.$$ = (11)

The angles U, P and B are not directly needed for calculating the trajectory of the dust grain. These are nevertheless useful to visualize the defect of solar illumination on the comet, the orientation and aspect angle of the dust jet and shell patterns.

2.3 Comet-o-centric spherical coordinates of the dust grain referred to the Earth's equator

The comet-o-centric coordinates $(u^{\prime}, \phi^{\prime}, r^{\prime}, t)$ referred to the comet's equator were calculated using Eq. (7). These were transformed to the comet-o-centric Earth's equatorial coordinates $(A^{\prime}, D^{\prime}, r^{\prime}, t)$ of the grain using the following equations:

$$\cos (A^{\prime}-N) \cos D^{\prime} = \cos \phi^{\prime} \cos u^{\prime}$$

$$\sin (A^{\prime}-N) \cos D^{\prime} = \cos \phi^{\prime} \sin u^{\prime} \cos J - \sin \phi^{\prime} \sin J$$

$$\sin D^{\prime} = \cos \phi^{\prime} \sin u^{\prime} \sin J + \sin \phi ^{\prime} \cos J .$$ (12)

The comet-o-centric equatorial coordinate of the source at $G(A,\ D,\ R,\ t=0)$ is shown in Fig. 1.

2.4 Differential sky plane coordinates of the dust grain with respect to the comet's center

The projected location of the grain on the sky plane with respect to the comet center is obtained by computing its differential Earth's equatorial co-ordinates using the following equations (Gurnette & Woolley 1960):

$$\tan (\alpha_{\rm D} - \alpha ) = \frac{\xi}{( 1 + \zeta )\cos \delta - \eta \sin \delta}$$ (13)

$$\tan (\delta_{\rm D} - \delta) = \frac {\eta - \xi \tan \frac... ...) + \xi \tan\frac{1}{2}(\alpha_{\rm D} -\alpha ) \cos \delta}$$ (14)

where

$$\xi r^{\prime} \cos D^{\prime} \sin \left( A^{\prime} - \alpha \right)/ \Delta$$ = (15)

$$\eta r^{\prime} \left(\sin D^{\prime} \cos \delta - \cos D^{\prime} \sin \delta \cos \left( A^{\prime} - \alpha \right) \right ) /\Delta$$ = (16)

$$\zeta r^{\prime} \left(\sin D^{\prime} \sin \delta + \cos D^{\prime} \cos \delta \cos \left( A^{\prime} - \alpha \right ) \right)/\Delta ,$$ = (17)

where $\alpha_{\rm D}$ and $\delta_{\rm D}$ are the geocentric right ascension and declination of the grain and $\Delta$ its geocentric distance. These rigorous expressions in Eqs. (13)-(17) are actually used for calculating the differential coordinates of satellites with respect to the primary without any assumptions regarding their latitudes. These are therefore applicable in the present case of the comet-dust geometry.