Numerical Models

We have computed detailed models of comet thermal evolution. The nucleus was assumed to be spherically symmetric, and heated uniformly from all directions. The orbit was taken to be circular, with a semimajor axis (radius) of 100 AU. The actual details of the orbit are not important, other than that they ensure a low surface temperature. The nucleus itself was assumed to be composed of a mixture of ice and dust, with the mass ratio of these two components being 1:1. It was also assumed that 1% CO and 1% CO2 were originally present, either as ices or as gas occluded in the amorphous ice. The amorphous ice was allowed to crystallize according to Eq. (2) with the resulting release of latent heat. The ice was assumed to be porous.

The baseline model had an initial porosity of 0.1, and an initial pore size of 100 |m. Upon heating, the CO and CO2 flow through the porous nucleus, along with the water vapor, and contribute to the heat transport. In addition, if their pressure is high enough, these gases can cause the pores to expand, and change the permeability of the medium.

Heating is caused by the radioactive materials 40K, 235U, 238U, 232Th, and 26Al. The first four isotopes are assumed to be present in the dust in their solar ratio to silicon, and the initial 26 Al mass fraction in the dust is taken to be 5 x 10~8 in the baseline model. In addition, account is taken of the latent heat released by the amorphous to crystalline phase transition, as well as the phase changes between gases, liquids, and solids. Details of how the various processes are modelled can be found in Prialnik (1992), Prialnik et al. (1987), Mekler et al. (1990), Prialnik et al. (1993), Podolak and Prialnik (1996), and Prialnik et al. (2005).

The actual melting temperature of the ice in the nucleus is somewhat problematic, since it depends on the details of the impurities present. The presence in the ice of 1% ammonia, by mass, for example, will lower the melting temperature of the ice by roughly 1 K. The observed abundance of ammonia in comets is less than this, however. Other materials, such as methanol, though present with a greater abundance, are less effective in lowering the melting point. Finally, the abundance of nonvolatile materials is not known in any detail. In view of this we feel that the ice in the nucleus will almost certainly not melt if the temperature is below 260 K. Above this temperature we will consider the ice liquid, provided there is sufficient pressure to stabilize the liquid state.

Figure 10.3 shows the results of these computations for a series of models starting with the baseline model, and varying only the radius and porosity. As can be seen from the figure, a 1-km comet will never reach melting. The main cooling mechanism is gas diffusion. For such a small radius the gas can

Porosity

Fig. 10.3. Maximum central temperature for a series of comets of different radii as a function of porosity. Other parameters are like those in the baseline model.

escape easily for any reasonable porosity, and even for very small porosities the heat can escape through thermal diffusion. As the porosity increases, the efficiency of gas diffusion does not change substantially, but the thermal diffusion coefficient decreases somewhat, so that the maximum temperature increases slightly.

As the radius of the comet increases, the time for thermal diffusion quickly exceeds the time for radioactive heat production, and the temperature goes up. However, if the porosity increases sufficiently so that gas cooling becomes effective, the maximum central temperature drops quickly, as can be seen in the figure. For increasing radii, this critical porosity increases until a radius of around 30 km is reached. At this point the gas diffusion timescale is roughly equal to the radioactive heating timescale so that for larger bodies the maximum central temperature is fairly independent of porosity. It does, however, drop as the porosity increases, even for a 100-km body. The maximum central temperature will be in the vicinity of melting only for low porosities 0.1).

Figure 10.4 shows the temperature inside a 100-km comet (baseline model) at two particular times: 6.3 x 104 years after formation and 2.3 x 105 years after formation. In both cases the inner 90% of the body is at temperatures above 260 K. High temperature is not sufficient for guaranteeing liquid water, however. An additional requirement is that the ambient pressure be high enough to ensure that the water is in the liquid and not in the gaseous state. Figure 10.5 shows the pressure throughout the comet for the two times shown in Fig. 10.4. As can be seen, the pressures at 6.3 x 104 years are high enough to ensure that water will be present in the liquid phase. By 2.3 x 105 years, however, the pressures have fallen to about 2 mbar throughout most of the volume. At this pressure water at 260 K will be a vapor. Thus, although the

Fig. 10.4. Temperature inside baseline model at 6.3 x 104 years and 2.3 x 105 years after formation as a function of radius.

Depth (km)

Fig. 10.5. Pressure inside baseline model at 6.3 x 104 years and 2.3 x 105 years after formation as a function of radius.

Depth (km)

Fig. 10.5. Pressure inside baseline model at 6.3 x 104 years and 2.3 x 105 years after formation as a function of radius.

body of the comet may remain warm for a considerable time, the pressures will be too low to maintain liquid water for much of that time.

The models show that several criteria are needed for the formation of liquid water in a comet interior: The body must have a radius of at least 10 km. It must have a low porosity, of the order of 0.1. The 26 Al mass fraction must be around 5 x 10~8, and occluded gases must be present in amounts less than a few percent. Finally, if the pores themselves are too large, they will allow the gas to cool the comet efficiently. A maximum pore radius of around a millimeter is indicated (Podolak and Prialnik, 2000).

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