Astron. Astrophys. 354, 1086-1090 (2000)

3. Theoretical nucleus model

The nucleus models presented in this paper have been computed with a numerical code, solving the unidimensional heat conduction and gas diffusion equations through an idealized spherical comet nucleus. This model was applied many times and is accurately described in various papers (Capria et al. 1996; Coradini et al. 1997a,b; De Sanctis et al. 1999), to whom the reader should refer to have more details: here only the main characteristics will be briefly described.

The comet nucleus is assumed to be initially homogeneous and uniformly porous, composed by different ices (, and CO) and dust particles embedded in the ice matrix. The initial dust grain size distribution is given. In our simulations the dynamical history is taken into account: the evolution starts when the cometary nucleus is in the Kuiper belt, far from the Sun. We then compute a number of revolutions large enough to reach a quasi stationary state. The resulting nucleus conditions are then used as initial parameters for the next stage, with the aim to simulate the so-called multi-stage capture process (Fernandez 1985) from the Kuiper belt to the final Jupiter family orbit of P/Gehrels 3 (Table 1). When the nucleus reaches the present orbit of P/Gehrels 3, the internal structure is affected by the previous thermal history and the upper layers are already differentiated.

The thermal evolution of the nucleus leads to its differentiation due to sublimation-recondensation processes taking place in its interior. The dust grains are released by the sublimation of the ices and undergo the drag exerted by the escaping gas, so that they can be blown off by the flux and be lost in space, or they can accumulate on the nucleus surface to form a dust mantle.

Thermal evolution and chemical differentiation models of P/Gehrels 3 have been computed in order to study the characteristics of this body during the epoch of the observation. Some of the initial parameters assumed in the models are derived by the observations, while some others are commonly accepted in cometary nuclei modelling.

The estimated nucleus radius is about 3 km, obtained from the comet absolute magnitude assuming an albedo of 0.04. At the beginning of the evolution, in the Kuiper belt, an initial temperature of 30 K is assumed. The initial mean pore radius has been fixed to 10^-5 m. Due to the lack of information on the rotation period, we assumed a period of 10 hours, a value typical for the rotation periods of minor bodies. The models assume a dust density value of 1000 , according to the fact that grains are considered the result of an accumulation process and are therefore highly porous, like Brownlee particles. The initial bulk porosity is 0.8. The models are computed in the fast rotator approximation.

In model A we have assumed a set of parameters considered "standard" for a comet nucleus (Rickman & Hübner 1990; Hübner et al. 1999), while in model B we have tried to use the characteristics that can favor the activity.

In all the models dust on ice mass ratio was assumed to be 1, but the dust distributions are different. The models are developed with two different initial dust distributions, the "primordial" (a) (Coradini et al. 1977) and the "small grains" (b), to see the effects of a large amount of very small dust particles on the comet evolution. In Fig. 2 are shown the two dust distributions.

Fig. 2a and b. Initial dust distributions embedded in comet matrix: a Primordial, b Small grains.

The accumulation of dust on the nucleus surface depends on the forces exerted on the single particles: as long as they are embedded in the icy matrix they are considered trapped and cannot be dragged away from the nucleus. When the ices (water, CO, CO₂) begin to sublimate the dust particles that are embedded in them, are considered free and not interacting among themselves.

To determine quantitatively how many particles can be blown off by the gas flow, and how many can be accumulated on the nucleus surface, we compare the different forces that act on the single dust grains. The balance between the drag of the outflowing gas plus the centrifugal force and the gravitational attraction force exerted by the nucleus has to be determined.

For each model, for each time step, we compute the amount of free dust particles and how this amount is redistributed between the grains that are ejected from the surface, forming the dust flux, and those that are deposited on the nucleus surface. The dust crust is formed by particles of different sizes when the surface layer is completely depleted of ice. This layer of crust is very porous and the gas can flow through the dust crust. In these models we consider two different surface regimes: the free sublimation regime and the gas diffusion regime. In the free sublimation regime, the grains can reside on the nucleus but these are considered isolated from each other and do not interact with the escaping gas. In the second case, a coherent dust layer obstructs the free passage of gas molecules: the gas can diffuse through the dust layers with different regimes (viscous, Knudsen and an average between the two). This scheme is applied to models A and B while for the model C we have introduced a "trapping" mechanism favoring crust formation (De Sanctis at al. 1999).

Model A . The dust distribution assumed in model A, a , is derived from studies of grains accretion in the primordial Solar System (Coradini et al. 1977) and has been largely used and discussed in previous papers (Coradini et al. 1997a,b; Capria et al. 1996). The initial value of CO/H₂O and CO₂/H₂O was 0.01. The orbit of P/Gehrels 3 has a very low eccentricity and a semiaxis of 4.04 AU: the comet never arrives close to the Sun and the maximum surface temperature, reached at perihielion, is quite low ( 157 K) and not very different from that at aphelion ( 136 K). Accordingly, the activity level is low: the peak of water flux is molec s^-1 (Fig. 3), while those of CO and CO₂ are respectively molec s^-1 and molec s^-1. At the perihelion the gas flux is dominated by the water, while the CO flux is constant along the orbit because it is coming from deep layers at a quasi-constant temperature. The CO₂ emission tends to diminish due to the sink of the CO₂ sublimation front. The low gas flux is coupled with low dust flux: only the smaller particles can be ejected from the nucleus, the larger ones tend to accumulate slowly on the comet surface. Revolution after revolution a thin, porous layer of dust tends to cover the comet surface.

Fig. 3. Model A. Gas flux versus time on the current orbit of P/Gehrels 3.

Model B . In model B we have tried to assume values of parameters that can favor the comet activity, such as the amount of volatile ices (CO/H₂O=0.1 and CO₂/H₂O=0.05), stressing out the influence of the dust distribution, using very small dust particles (dust distribution b ). In this case the smaller grains, with very low mass, are blown off from the nucleus contributing to the dust flux. The dust flux resulting is very low: only few kilograms for second (Fig. 4). The gas flux, from the beginning of the evolution (in the Kuiper Belt) to the current orbit is shown in Fig. 5a. The water flux (Fig. 5b) is still low but similar to that of the previous model, while the CO₂ and CO flux are much higher than in model A. After few revolution the water is the dominant species at the perihelion. Sublimation fronts and the crystallization front drop down slowly in the nucleus. It is possible to foresee a quite constant flux of volatiles along the whole orbit of P/Gehrles 3.

Fig. 4. Model B. Dust flux versus heliocentric distance along one orbit.

Fig. 5a and b. Model B. a Water, CO and CO2 flux evolution from the Kuiper belt to the current orbit of P/Gehrels 3. b Enlarged view of the final part of Fig. 5a: gas flux versus time on the current orbit of P/Gehrels 3

Model C . In the model C, that is the analogue of model A, we introduced a mechanism that favors the crust formation (De Sanctis et al. 1999). The idea of trapping was introduced by Shul'man (1972): as the large dust grains accumulate on the surface, the interstices between the particles become too small to allow the escaping of the smaller grains, even if these particles are smaller than the critical radius. Here we try to simulate the behaviour of cometary nuclei covered by a dust crust. This crust should be cohesive (Komle et al. 1996), but still porous and permitting the gas flux passage. The resulting thermal conductivity of the dust layers is higher than that of the ice/dust mixture. In this model the time needed for a stable crust formation is less than in the previous cases and the crust thickness is higher. The presence of the crust strongly influences the dust and gas flux: the dust flux is quenched and the water activity is reduced (Fig. 6). The water flux is related to the crust thickness and decreases proportionally to the distance of the sublimation front from the surface. CO and CO₂ fluxes are not so strongly influenced by the presence of a stable crust: the depth of the sublimation fronts tends to increase with time and, consequently, flux tends to decrease.

Fig. 6. Model C. Gas flux versus time on the current orbit of P/Gehrels 3.

The distance from the surface of the CO sublimation front remains quite stable, increasing very slowly: due to the near presence of crystallization front and to the high volatility, the CO emission tends to increase very slowly. It appears that model C reaches earlier a kind of "extinction", due to the presence of a stable and thick crust that strongly reduces the gas and dust emissions.

© European Southern Observatory (ESO) 2000

Online publication: February 25, 2000
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