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Under what conditions does amorphous ice in a cometary nucleus crystallize in general? We saw in the previous section that the crystallization is a competitive process between the heating by latent heat deposition due to crystallization triggered by radiogenic heating and the cooling by heat conduction. We shall derive a general condition for the crystallization by analyzing the heating and cooling processes.
Amorphous ice will crystallize eventually if the heating rate is always
higher than the cooling rate. From eqs. (12)
and (13),
the heating rate
is written as
with
where we have ignored the abundance decrease of the radiogenic nuclides
due to decay since
at the time when the temperature becomes maximum according to the results
given in §10.
On the other hand, the cooling rate by heat conduction
may roughly be evaluated to be
with the use of eq. (9)
for
. Thus the ratio of the heating and cooling rates is given by
where
is given by eq. (14).
At low temperatures, the radiogenic heating is dominant compared with
latent heat deposition by crystallization (i.e.
), and
may be approximated as
Since the crystallization degree
increases with increasing temperature,
decreases with increasing temperature. At high temperatures where
but still
, on the other hand, the latent heat deposition by crystallization becomes
dominant compared with the radiogenic heating. Then the ratio
given by eq. (22)
may be approximated as
which indicates that
increases with increasing temperature and thus
.
Combining behaviors of
given by eqs. (23)
and (24),
we see that
takes a minimum at a certain temperature. The bahavior of
is illustrated schematically in Fig. 11.
Whether the minimum of
is larger than unity or not is detrmined by the coefficients of the right-hand-sides
of eqs. (23)
and (24).
Namely, when the constant C defined by
is large, complete crystallization occurs, whereas amorphous ice is preserved when C is small. According to more detailed analysis (Haruyama et al., 1993), the criterion for crystallization of amorphous ice is given by:
where
is a proportional coefficient of the specific heat defined by eq. (15).
For large a and
or small
, E,
, and
, the ice tends to crystallize as is expected qualitatively. Furthermore
note that the criterion is independent of the thermal conductivity of amorphous
ice, indicating that whether the ice crystallize or not does not depend
on the value of
so long as
.