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Astron. Astrophys. 319, 995-1006 (1997)

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4. Physical and numerical parameters

The model nucleus has a uniform temperature of [FORMULA] at the beginning of the simulation so that the condensed CO remains in solid state as its net sublimation rate at this temperature is small in a porous matter. The initial composition has a dust to ice mass ratio of [FORMULA] which corresponds to the lower value evaluated for comet 1P/Halley (McDonnell et al. 1991). The ice contains condensed CO with a molar fraction of (free) [FORMULA]. An additional amount of [FORMULA] is trapped by the amorphous water ice. The compact dust density is assumed with [FORMULA]. One single density is chosen for the ice [FORMULA]. The initial total mass density is [FORMULA] which is a mean value of Rickman's density estimation for comet 1P/Halley (Rickman 1986). The porosity is [FORMULA]. The material is supposed to be built by accretion of micrometre-sized aggregates. Accordingly, we have chosen a characteristic pore size [FORMULA].

The thermal conductivity of compact water ice has been evaluated by Klinger (1980). For crystalline ice he found a [FORMULA] dependency by means of laboratory experiments.

[EQUATION]

For amorphous ice Klinger suggested a very low conductivity derived from the classical phonon model for solids.

[EQUATION]

Recently, Kouchi et al. (1992) have measured values several orders of magnitude lower. An explanation of this huge difference might be a weakly connected structure of the condensed sample. Nevertheless, already Klinger's value is so small that the main contribution to the heat transport should come from the gas phase and is not due to heat conduction. In our model we use Klinger's expression as an upper limit of the contribution of the amorphous ice. The conductivity of condensed CO is expected to be small if the molecules are in the amorphous phase. Therefore we use for CO the same conductivity as for amorphous [FORMULA]. The bulk conductivity of the dust is adopted from the mean conductivity of terrestrial minerals given by Drury et al. (1984).

[EQUATION]

The specific heat of crystalline water ice has been fitted by Klinger (1980) from measured data given by Giauque & Stout (1936).

[EQUATION]

The same expression is adopted for amorphous ice. For CO we use the upper limit suggested by Tancredi et al. (1994) although small deviation occurs below [FORMULA] K: [FORMULA]. The specific heat of the dust is supposed to be:

[EQUATION]

This formula fits the specific heat of terrestrial minerals measured at room temperature (Drury et al. 1984). The specific heat of water vapour and CO gas is inferred from the ideal gas law approximation: [FORMULA] and [FORMULA].

The latent heat of sublimation of pure [FORMULA] is given by Cowan & A'Hearn (1979) from a fit of older data (Washburn 1928).

[EQUATION]

The lower limit of latent heat of sublimation of volatile molecules condensed in a water ice matrix is estimated from the adsorption energy (Schmitt 1991): [FORMULA].

The energy balance of the crystallisation process is calculated using the latent heat of crystallisation of pure amorphous water ice of [FORMULA] (Ghormley 1968). The energy of evaporation of trapped CO is approximated by its latent heat of sublimation. This description is consistent with the endothermic process observed during the crystallisation of water ice being strongly enriched with [FORMULA] (Kouchi 1995). For the chosen parameter the threshold amount of trapped CO for an endothermic process is [FORMULA] 23 % (molar). Other works (Prialnik et al. 1992; Tancredi et al. 1994) consider that the trapped CO behaves as a gas and the energy reduction during the gas release is therefore much lower.

The sticking (or condensation) coefficient [FORMULA] (Eq. 38) for [FORMULA] on ice surfaces has been measured by Haynes et al. (1992). [FORMULA] is temperature dependent. After fitting the measured data by a linear law we get

[EQUATION]

The saturated vapour pressures are approximated by

[EQUATION]

where [FORMULA], [FORMULA] K (Fanale & Salvail 1984) and [FORMULA], [FORMULA] K (Fanale & Salvail 1990). More precise formula of the vapour pressure exist (e.g. Washburn 1928, Kouchi 1987). However, taking into account the uncertainties in the description of the sublimation process we have chosen Eq. (50) as it allows more efficient computation.

The nucleus surface is supposed to be dark which is the actual image influenced by the analysis of the optical properties of comet 1P/Halley. It is also consistent with the presence of complex hydrocarbons suggested by Greenberg (1982). We use an albedo of 0.05 and an infrared emissivity of 0.9.

The model nucleus has a spherical shape with a radius of [FORMULA] km (see Meech et al. 1993). The nucleus radius is discretised by [FORMULA] grid points. The first layers under the surface have a thickness of 0.1 m. The deepest layers have a thickness of a few hundred metres. Each meridian contains [FORMULA] grid points. The nucleus rotation around its spin axis is cut into [FORMULA] steps leading to a data matrix with the size of [FORMULA] elements per rotation.

To resolve the diurnal heat wave the numerical integration time step [FORMULA] has to be sufficiently small. But the smaller the time step the larger the computation time for a given problem size. In order to know how good the diurnal heat wave is resolved we compare a characteristic propagation time [FORMULA] with [FORMULA]. The characteristic time can be defined as the ratio of the space resolution to the propagation velocity: [FORMULA]. The propagation velocity of a diffusion wave is

[EQUATION]

where P is the nucleus spin period. Using a heat conductivity of [FORMULA], a mass density of [FORMULA], a specific heat of [FORMULA] and [FORMULA] days we get with [FORMULA] m a characteristic time scale of [FORMULA] seconds. This value is compared to the chosen time step.

[EQUATION]

The integration time step is small compared to the diffusion time scale. A further necessary condition is on the size of the space resolution. [FORMULA] should be smaller than a characteristic depth of penetration of the diurnal heat wave. We consider the damping of a harmonic diffusion wave by a factor x (with [FORMULA]) at the depth r

[EQUATION]

For [FORMULA] the penetration depth is equal to about [FORMULA] m corresponding to 5 layers. One can therefore expect that the diurnal heat wave can be resolved with the chosen time and space resolution.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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