Astron. Astrophys. 319, 995-1006 (1997)
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
for the mass of dust per unit volume,
for the water ice concentration and
for the concentration of solid carbon monoxide.
We will later also use and
( ) to distinguish between amorphous and
crystalline ice. Additionally we denote and
( ) for the concentrations
of trapped and free CO. The mass density of the matrix can then be
written as
![[EQUATION]](img14.gif)
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
( , ,
).
![[EQUATION]](img18.gif)
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 .
![[EQUATION]](img20.gif)
The dust to ice mass ratio of the solid matrix is given by
![[EQUATION]](img21.gif)
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
( ) can be estimated using the kinetic gas
theory.
![[EQUATION]](img23.gif)
( ). 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.
is the Boltzmann constant and
represents the molecular mass of species
i. The saturated vapour pressure and the
partial pressure are corrected by the sticking
(or condensation) coefficient .
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 . We introduce a phenomenological
description of the sublimated gas per unit volume during a given time
interval
![[EQUATION]](img32.gif)
This leads to an enthalpy change due to sublimation of
![[EQUATION]](img33.gif)
where 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]](img35.gif)
This time scale has been found by analysing the time evolution of
the shape and position of both the 3.1 and 6.0
infrared absorption bands of 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]](img38.gif)
The net enthalpy change of this process has to account for the
latent heat of crystallisation and the evaporation energy of trapped
CO.
![[EQUATION]](img39.gif)
2.3. Diffusion equations
The transport of heat and gas is defined by means of a system of
three coupled diffusion equations.
![[EQUATION]](img40.gif)
K stands for the effective thermal conductivity of the solid
matrix. Additional heat sources (sublimation, crystallisation and
advection) are represented by a source term .
The gas diffusion equations apply to viscous, free molecular (or
Knudsen) and mutual (or Fick) diffusion. is the
partial pressure of species i and is the
total gas pressure of a binary gas mixture. and
are the diffusion coefficients for diffusive
and viscous flow. Additional gas sources are denoted with
.
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]](img46.gif)
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 is a characteristic pore size,
is the infrared emissivity and
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]](img50.gif)
where is the molar fraction of species
i. is a geometrical parameter describing
the ratio of the gas flux across a porous material to the gas flux in
a capillary of radius . This parameter will be
discussed more in detail at the end of this section.
is given by
![[EQUATION]](img54.gif)
and are the molecular
masses of species i and j.
expresses the ratio of the unreduced diffusive diffusion coefficient
to the mutual diffusion coefficient
.
![[EQUATION]](img59.gif)
is defined by
![[EQUATION]](img60.gif)
is the free molecular diffusion coefficient.
is given by
![[EQUATION]](img63.gif)
The viscosity for a binary gas mixture is
given by
![[EQUATION]](img65.gif)
where . and
are the diameters of the molecules treated as
rigid elastic spheres. The mutual diffusion coefficient is (e.g.
Present 1958)
![[EQUATION]](img69.gif)
where 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]](img71.gif)
where is the mean thermal gas velocity.
![[EQUATION]](img73.gif)
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]](img74.gif)
The tortuosity of such systems is
![[EQUATION]](img75.gif)
We relate a similar geometrical correction to the viscous diffusion
coefficient. The parameter of Eq. (15) then
becomes
![[EQUATION]](img76.gif)
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]](img77.gif)
with the specific heat of species
i.
![[FIGURE]](img79.gif) |
Fig. 1. Diffusion flux ( ) of a binary gas mixture across a capillary (diameter 1 ) at 150 K with constant pressure gradient. The gas is composed of (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.
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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 , colatitude) using
spherical coordinates. After writing out all terms of these equations
there appear singularities at ,
and . If the problem is
symmetrical with respect to the origin it follows by doing Maclaurin's
expansion that the diffusion equation at
becomes
![[EQUATION]](img85.gif)
(with ). Fortunately the singularities at
and can be approximated
with the de l'Hopital rule
![[EQUATION]](img87.gif)
Each segment of the sphere is treated independently and no flux can
pass the inner borders (von Neumann conditions).
![[EQUATION]](img88.gif)
![[EQUATION]](img89.gif)
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
of the Sun and to a heliocentric distance
(in AU), is given by
![[EQUATION]](img92.gif)
is the solar constant for 1
AU. The zenith distance of the Sun z is found by using
the theorem of spherical triangles.
![[EQUATION]](img94.gif)
where is the latitude and
is the hour angle of the considered point.
During the night phase ( ) 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
is related to the true longitude of the Sun
( ) by
![[EQUATION]](img99.gif)
where is the obliquity or the tilt of the
nucleus rotation axis relative to the orbital plane.
is connected to the true anomaly
by
![[EQUATION]](img102.gif)
is the true anomaly of the vernal equinox.
Let us note that and
define the orientation of the nucleus rotation axis with respect to
the Sun. The comet - Sun distance is given by the ellipse
equation.
![[EQUATION]](img104.gif)
The ellipse is defined by its semi-major axis a and its
eccentricity e. In the case of P/SW1
AU and . The angular velocity of
the orbital motion is
![[EQUATION]](img106.gif)
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]](img107.gif)
where Al is the albedo. is the
latent heat of sublimating water ice and is
the mass loss rate per unit surface.
![[EQUATION]](img110.gif)
Any temperature change will be lessened by the enthalpy change term
. 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]](img112.gif)
and . 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.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998
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