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Physics of the Thermal History

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 tex2html_wrap_inline1306 is written as

  equation410

with

  equation415

where we have ignored the abundance decrease of the radiogenic nuclides due to decay since tex2html_wrap_inline1734 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

displaymath1736

may roughly be evaluated to be

  equation424

with the use of eq. (9) for tex2html_wrap_inline1470 . Thus the ratio of the heating and cooling rates is given by

  equation430

where tex2html_wrap_inline1614 is given by eq. (14).

At low temperatures, the radiogenic heating is dominant compared with latent heat deposition by crystallization (i.e. tex2html_wrap_inline1742 ), and tex2html_wrap_inline1744 may be approximated as

  equation441

Since the crystallization degree tex2html_wrap_inline1464 increases with increasing temperature, tex2html_wrap_inline1744 decreases with increasing temperature. At high temperatures where tex2html_wrap_inline1750 but still tex2html_wrap_inline1470 , on the other hand, the latent heat deposition by crystallization becomes dominant compared with the radiogenic heating. Then the ratio tex2html_wrap_inline1744 given by eq. (22) may be approximated as

  equation450

which indicates that tex2html_wrap_inline1744 increases with increasing temperature and thus tex2html_wrap_inline1464 .

Combining behaviors of tex2html_wrap_inline1744 given by eqs. (23) and (24), we see that tex2html_wrap_inline1744 takes a minimum at a certain temperature. The bahavior of tex2html_wrap_inline1744 is illustrated schematically in Fig. 11.

Whether the minimum of tex2html_wrap_inline1744 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

displaymath1770

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:

  equation464

where tex2html_wrap_inline1774 is a proportional coefficient of the specific heat defined by eq. (15). For large a and tex2html_wrap_inline1778 or small tex2html_wrap_inline1490 , E, tex2html_wrap_inline1784 , and tex2html_wrap_inline1774 , 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 tex2html_wrap_inline1404 so long as tex2html_wrap_inline1790 .


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Next: Acknowledgments Up: No Title Previous: Numerical Results

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Mon Sep 16 16:23:29 JST 1996