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heat of crystallization
Assuming a spherical cometary nucleus, temperature T at time
t at distance r from a center of the nucleus with density
and specific heat
per unit mass is described by the following equation of heat conduction:
where
and
are the heating rates by the two sources discussed in the previous section.
The heating rate
due to radiogenic activity is given by
where x is the mass fraction of silicate in a cometary nucleus.
The heating rate
due to latent heat deposition by crystallization of amorphous ice is expressed
by
where the factor (1-x) indicates the mass fraction of ice (both
amorphous and crystalline), and
is the rate of increase of the crystalline fraction
.
The rate of crystallization is expressed with the use of eq. (10) by
These are the equations describing the time variation of temperature
in a cometary nucleus. Note that
in eq.. (11)
depends on
, which varies according to eq. (14),
and temperature T (see §8.2).
Also the specific heat
is a function of T, and is approximately expressed (Haruyama et
al., 1993) by
To solve these equations, we need initial and boundary conditions. For
the initial conditions, we assume that at t=0 the nucleus is of
uniform temperature of
K and that the ice is initially amorphous, i.e.
. The initial temperature of
K corresponds to the solar nebula temperature at the presumed distance
of
AU from the sun (see §5.2;
Yamamoto and Kozasa, 1988). The temperature variation during ejection from
AU to the Oort cloud distance can be ignored (Kouchi et al., 1992b).
With regard to the boundary conditions, the temperature gradient should
vanish at the center of the nucleus because of spherical symmetry:
For the surface boundary condition, we take a so-called flux boundary condition. Namely the energy flux of absorption of solar and interstellar radiations, thermal emission from the nucleus surface, and conductive heat flow should balance at the nucleus surface of r=a:
Here
is the surface temperature of the nucleus,
emissivity of the nucleus surface, and F is the energy flux of solar
and interstellar radiations given by
where
is solar luminosity,
is albedo of a nucleus surface, R is the distance from the sun,
and
is the flux of interstellar radiation field, which gives a black-body temperature
of
K (Greenberg, 1971). The first term in the right-hand side of eq. (18)
assumes that the nucleus is a rapid rotator. If the nucleus does not rotate,
the numerical factor 16 in the denominator is replaced by 4. In the numerical
results shown in the next section, we take
AU, a typical Oort cloud distance,
and
in vew of the very low albedo of comet Halley.