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Basic Equations Describing the Thermal History

Assuming a spherical cometary nucleus, temperature T at time t at distance r from a center of the nucleus with density tex2html_wrap_inline1598 and specific heat tex2html_wrap_inline1532 per unit mass is described by the following equation of heat conduction:

  equation329

where tex2html_wrap_inline1602 and tex2html_wrap_inline1604 are the heating rates by the two sources discussed in the previous section. The heating rate tex2html_wrap_inline1602 due to radiogenic activity is given by

  equation342

where x is the mass fraction of silicate in a cometary nucleus. The heating rate tex2html_wrap_inline1604 due to latent heat deposition by crystallization of amorphous ice is expressed by

  equation349

where the factor (1-x) indicates the mass fraction of ice (both amorphous and crystalline), and tex2html_wrap_inline1614 is the rate of increase of the crystalline fraction tex2html_wrap_inline1464 .

The rate of crystallization is expressed with the use of eq. (10) by

  equation355

These are the equations describing the time variation of temperature in a cometary nucleus. Note that tex2html_wrap_inline1486 in eq.. (11) depends on tex2html_wrap_inline1464 , which varies according to eq. (14), and temperature T (see §8.2). Also the specific heat tex2html_wrap_inline1532 is a function of T, and is approximately expressed (Haruyama et al., 1993) by

  equation366

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 tex2html_wrap_inline1630 K and that the ice is initially amorphous, i.e. tex2html_wrap_inline1632 . The initial temperature of tex2html_wrap_inline1634 K corresponds to the solar nebula temperature at the presumed distance of tex2html_wrap_inline1172 AU from the sun (see §5.2; Yamamoto and Kozasa, 1988). The temperature variation during ejection from tex2html_wrap_inline1172 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:

  equation376

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:

  equation381

Here tex2html_wrap_inline1642 is the surface temperature of the nucleus, tex2html_wrap_inline1324 emissivity of the nucleus surface, and F is the energy flux of solar and interstellar radiations given by

  equation388

where tex2html_wrap_inline1648 is solar luminosity, tex2html_wrap_inline1650 is albedo of a nucleus surface, R is the distance from the sun, and tex2html_wrap_inline1654 is the flux of interstellar radiation field, which gives a black-body temperature of tex2html_wrap_inline1656 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 tex2html_wrap_inline1658 AU, a typical Oort cloud distance, tex2html_wrap_inline1660 and tex2html_wrap_inline1662 in vew of the very low albedo of comet Halley.


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Next: Numerical Results Up: No Title Previous: Latent heat of crystallization

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Mon Sep 16 16:23:29 JST 1996