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Equations Describing the
One of the typical results is shown in Fig. 8,
which shows time variation of the temperature at distances r/a=0
to 0.95 and at the surface r/a=1 for the nucleus radius
km, the silicate mass fraction x=0.5, and the reduction factor
.
The thermal history is divided into three stages.
At the first stage, the internal temperature increases gradually with
time. The temperature increase at this stage is due to slow decay of radiogenic
nuclides;
K is the major source (see Table 5). Note that the internal temperature
is rather uniform except in the immediate vicinity of the surface, indicating
that each region of the nucleus is heated locally by decay of the radioactive
nuclides with little heat conduction owing to the low thermal conductivity
of amorphous ice.
The second stage begins when the temperature reaches around
K. Then crystallization becomes substantial, since the transition rate
given by eq. (10)
increases very rapidly with temperature. The transition time scale from
amorphous to crystalline ice
is shorter than
yr at
K. As the temperature becomes higher, the transition proceeds more rapidly,
and release of the latent heat accelerates the temperature increase. This
leads to further latent heat release due to further crystallization, and
so on. Namely, the crystallization is a sort of a positive feedback process.
The temperature attains a maximum of about
K within a short period of
yr. The temperature increase is due to the fact that the heating rate by
the latent heat release is greater than the cooling rate by heat conduction
towards the surface. At this stage the ice is completely crystallized in
the entire nucleus except just beneath the surface.
At the third stage when the crystallization completes in almost whole
interior of the nucleus, the temperature drops very rapidly to the ambient
temperature of
K within
yr, and remains this temperature after that. The rapid temperature decrease
is due to high thermal conductivity of crystalline ice compared with that
of amorphous ice.
For a larger thermal conductivity, the situation becomes quite different.
Figure 9 shows a temperature history for
; other parameters are the same as in Fig. 8.
It is seen that the temperature history is similar to the case of
, but the maximum temperature of about
K is much lower than that for
. This is because even a small fraction of crystallization increases the
thermal conductivity, and thus the temperature increase is suppressed.
As a result the degree of crystallization is less than a few percent throughout
the interior of the nucleus.
To see the dependence of degree of crystallization
on the reduction factor
, Figure 10 shows
at the center of the nucleus after
yr as a function of
for
km and x=0.5.
It is clearly seen that there is a critical value
. The crytical value is
for
km and x=0.5. For
the crystallization degree is less than a few percent even at the center
of the nucleus, whereas for
complete crystallization over the entire nucleus except the surface and
its immediate vicinity.