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

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2. Model description

2.1. Composition

The model nucleus is composed of fluffy ice-dust aggregates. These aggregates are supposed to have a similar fractal dimension (see Eq. 55). The dust component stands for a non-volatile phase with a low visible albedo. The icy constituent is composed of carbon monoxide enriched water ice which is initially in the amorphous state. Some amount of the condensed CO may evaporate by temperature controlled sublimation. However, as shown by Schmitt et al. (1989), a small fraction is trapped by the amorphous ice and can only escape during the crystallisation of the amorphous matrix. We consider that amorphous and crystalline water ice can coexist before complete crystallisation. The composition of the solid ice-dust matrix is described by the mass concentration of each constituent. We use [FORMULA] for the mass of dust per unit volume, [FORMULA] for the water ice concentration and [FORMULA] for the concentration of solid carbon monoxide. We will later also use [FORMULA] and [FORMULA] ([FORMULA]) to distinguish between amorphous and crystalline ice. Additionally we denote [FORMULA] and [FORMULA] ([FORMULA]) for the concentrations of trapped and free CO. The mass density of the matrix can then be written as

[EQUATION]

The porosity, which is here defined as the ratio of the pore volume to total volume of the matrix, is given by comparing the mass concentration of each constituent with the density of a compact solid ([FORMULA], [FORMULA], [FORMULA]).

[EQUATION]

The amount of trapped CO is supposed in the model to be directly related to the amorphous ice aggregate. The CO mass concentration is proportional to the mass concentration of [FORMULA].

[EQUATION]

The dust to ice mass ratio of the solid matrix is given by

[EQUATION]

At the beginning of the simulation the model nucleus has a uniform composition.

2.2. Physico-chemical reactions

Below the surface the material is initially in thermodynamic equilibrium. A small temperature rise induces sublimation. The statistical sublimation (or condensation) rate of a species i ([FORMULA]) can be estimated using the kinetic gas theory.

[EQUATION]

([FORMULA]). This expression holds for the sublimation rate within a unit volume. T stands for the temperature which is supposed to be identical in the gas and in the solid phase. [FORMULA] is the Boltzmann constant and [FORMULA] represents the molecular mass of species i. The saturated vapour pressure [FORMULA] and the partial pressure [FORMULA] are corrected by the sticking (or condensation) coefficient [FORMULA]. [FORMULA] is the volume specific surface. It can be shown that the characteristic time scale of sublimation is much shorter than those of heat and gas diffusion (e.g. Tancredi et al. 1994). Hence, the gas should be close to saturation if sufficient ice is present. Significant deviation from saturation can exist within the first few centimetres beneath surface. This effect has been confirmed by means of laboratory experiments (Kömle et al. 1992). Saturation is reached if [FORMULA]. We introduce a phenomenological description of the sublimated gas per unit volume during a given time interval

[EQUATION]

This leads to an enthalpy change due to sublimation of

[EQUATION]

where [FORMULA] is the latent heat of sublimation.

The crystallisation rate is determined using the time scale for complete crystallisation found by Schmitt et al. (1989).

[EQUATION]

This time scale has been found by analysing the time evolution of the shape and position of both the 3.1 and 6.0 [FORMULA] infrared absorption bands of [FORMULA] in pure and gas enriched water ice. The crystallisation of amorphous water ice is an irreversible exothermic process. This process starts by forming a cubic lattice. In a following stage cubic ice transforms to hexagonal ice. The enthalpy change during the last phase transition is small compared to the first one. Amorphous water ice has a particularly large specific surface area which plays an important role for the adsorption of very volatile molecules like CO. When the amorphous ice matrix is heated, molecules being adsorbed in deep sites can be trapped due to surface rearrangements. The evaporation rate of CO during crystallisation is given for a constant release by

[EQUATION]

The net enthalpy change of this process has to account for the latent heat of crystallisation and the evaporation energy of trapped CO.

[EQUATION]

2.3. Diffusion equations

The transport of heat and gas is defined by means of a system of three coupled diffusion equations.

[EQUATION]

K stands for the effective thermal conductivity of the solid matrix. Additional heat sources (sublimation, crystallisation and advection) are represented by a source term [FORMULA]. The gas diffusion equations apply to viscous, free molecular (or Knudsen) and mutual (or Fick) diffusion. [FORMULA] is the partial pressure of species i and [FORMULA] is the total gas pressure of a binary gas mixture. [FORMULA] and [FORMULA] are the diffusion coefficients for diffusive and viscous flow. Additional gas sources are denoted with [FORMULA].

The thermal conductivity of cometary material is unknown so far. One can expect that its value is very sensitive to structure parameters of the porous matrix which may change during the comet's thermodynamic evolution. We use a parametrised expression for the effective thermal conductivity. As an upper limit of the conduction we use a parallel network.

[EQUATION]

The parameter h is generally called hertz factor. Throughout one simulation we use a constant value for h. The last term of Eq. (13) accounts for the heat transport by means of infrared radiation, where [FORMULA] is a characteristic pore size, [FORMULA] is the infrared emissivity and [FORMULA] is the Stefan-Boltzmann constant.

The gas diffusion coefficients are adopted from the Chapman-Enskog method (Chapman & Cowling 1960). For a binary gas mixture the coefficients are

[EQUATION]

where [FORMULA] is the molar fraction of species i. [FORMULA] is a geometrical parameter describing the ratio of the gas flux across a porous material to the gas flux in a capillary of radius [FORMULA]. This parameter will be discussed more in detail at the end of this section. [FORMULA] is given by

[EQUATION]

[FORMULA] and [FORMULA] are the molecular masses of species i and j. [FORMULA] expresses the ratio of the unreduced diffusive diffusion coefficient [FORMULA] to the mutual diffusion coefficient [FORMULA].

[EQUATION]

[FORMULA] is defined by

[EQUATION]

[FORMULA] is the free molecular diffusion coefficient. [FORMULA] is given by

[EQUATION]

The viscosity [FORMULA] for a binary gas mixture is given by

[EQUATION]

where [FORMULA]. [FORMULA] and [FORMULA] are the diameters of the molecules treated as rigid elastic spheres. The mutual diffusion coefficient is (e.g. Present 1958)

[EQUATION]

where [FORMULA] is the reduced mass. The free molecular diffusion coefficient inside a disordered porous medium is strongly dependent on the interfacial geometry and the connectivity of the pore network. Following Levitz (1993) (see also Dullien 1992) we introduce

[EQUATION]

where [FORMULA] is the mean thermal gas velocity.

[EQUATION]

The random walk of gas molecules is considered by a succession of chords belonging to the pore network. One chord begins and ends at a solid surface. For a random packing of hard spheres the mean free path is given by

[EQUATION]

The tortuosity of such systems is

[EQUATION]

We relate a similar geometrical correction to the viscous diffusion coefficient. The parameter [FORMULA] of Eq. (15) then becomes

[EQUATION]

An additional heat source is taken into account for advection. The gas transports enthalpy and is supposed to thermalise immediately with the solid phase.

[EQUATION]

with the specific heat [FORMULA] of species i.

[FIGURE] Fig. 1. Diffusion flux ([FORMULA]) of a binary gas mixture across a capillary (diameter 1 [FORMULA]) at 150 K with constant pressure gradient. The gas is composed of [FORMULA] (50 %) and CO (50 %). The 3 curves are obtained with the Poiseuille (viscous, dashed line) and Knudsen (free molecular, dotted line) formula as well as with the Chapman-Enskog method (solid line). For comparison the Knudsen number is plotted below.

2.4. Boundary conditions and singularities

Each diffusion equation has two boundary conditions per dimension. In our model these equations are solved for two spatial dimensions (r, radius and [FORMULA], colatitude) using spherical coordinates. After writing out all terms of these equations there appear singularities at [FORMULA], [FORMULA] and [FORMULA]. If the problem is symmetrical with respect to the origin it follows by doing Maclaurin's expansion that the diffusion equation at [FORMULA] becomes

[EQUATION]

(with [FORMULA]). Fortunately the singularities at [FORMULA] and [FORMULA] can be approximated with the de l'Hopital rule

[EQUATION]

Each segment of the sphere is treated independently and no flux can pass the inner borders (von Neumann conditions).

[EQUATION]

[EQUATION]

The last boundary condition has to be determined for each point of the spherical surface (Dirichlet conditions). The received solar flux per unit surface at a given time, corresponding to a declination [FORMULA] of the Sun and to a heliocentric distance [FORMULA] (in AU), is given by

[EQUATION]

[FORMULA] is the solar constant for 1 AU. The zenith distance of the Sun z is found by using the theorem of spherical triangles.

[EQUATION]

where [FORMULA] is the latitude and [FORMULA] is the hour angle of the considered point. During the night phase ([FORMULA]) the direct solar flux is equal to zero and only a small arbitrary background radiation of 20 K due to scattered light in the coma is considered. The declination [FORMULA] is related to the true longitude of the Sun ([FORMULA]) by

[EQUATION]

where [FORMULA] is the obliquity or the tilt of the nucleus rotation axis relative to the orbital plane. [FORMULA] is connected to the true anomaly [FORMULA] by

[EQUATION]

[FORMULA] is the true anomaly of the vernal equinox. Let us note that [FORMULA] and [FORMULA] define the orientation of the nucleus rotation axis with respect to the Sun. The comet - Sun distance is given by the ellipse equation.

[EQUATION]

The ellipse is defined by its semi-major axis a and its eccentricity e. In the case of P/SW1 [FORMULA] AU and [FORMULA]. The angular velocity of the orbital motion is

[EQUATION]

where G is the gravitational constant and M is the mass of the Sun.

At surface, the temperature is inferred from the energy balance of a sublimating, conducting layer of finite thickness.

[EQUATION]

where Al is the albedo. [FORMULA] is the latent heat of sublimating water ice and [FORMULA] is the mass loss rate per unit surface.

[EQUATION]

Any temperature change will be lessened by the enthalpy change term [FORMULA]. In the model we suppose that in any case there is no condensed CO or amorphous ice present within the surface layer.

The pressure change at surface is

[EQUATION]

and [FORMULA]. The CO partial pressure at surface is supposed to be zero in the model. This hypothesis holds if no condensed CO is present in the surface layer and the potential effusion is larger than the diffusion flux just below the surface.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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