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Numerical Results

  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 tex2html_wrap_inline1668 km, the silicate mass fraction x=0.5, and the reduction factor tex2html_wrap_inline1506 .

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; tex2html_wrap_inline1508 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 tex2html_wrap_inline1676 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 tex2html_wrap_inline1678 is shorter than tex2html_wrap_inline1680 yr at tex2html_wrap_inline1682 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 tex2html_wrap_inline1684 K within a short period of tex2html_wrap_inline1686 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 tex2html_wrap_inline1688 K within tex2html_wrap_inline1686 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 tex2html_wrap_inline1692 ; other parameters are the same as in Fig. 8.

It is seen that the temperature history is similar to the case of tex2html_wrap_inline1506 , but the maximum temperature of about tex2html_wrap_inline1676 K is much lower than that for tex2html_wrap_inline1506 . 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 tex2html_wrap_inline1464 on the reduction factor tex2html_wrap_inline1490 , Figure 10 shows tex2html_wrap_inline1464 at the center of the nucleus after tex2html_wrap_inline1706 yr as a function of tex2html_wrap_inline1490 for tex2html_wrap_inline1668 km and x=0.5.

It is clearly seen that there is a critical value tex2html_wrap_inline1714 . The crytical value is tex2html_wrap_inline1716 for tex2html_wrap_inline1668 km and x=0.5. For tex2html_wrap_inline1722 the crystallization degree is less than a few percent even at the center of the nucleus, whereas for tex2html_wrap_inline1724 complete crystallization over the entire nucleus except the surface and its immediate vicinity.


next up previous
Next: Physics of the Thermal Up: No Title Previous: Basic Equations Describing the

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Mon Sep 16 16:23:29 JST 1996