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This model assumes that cometary ice was formed in the course of the formation of the solar system from the interstellar-cloud stage to the planetesimal-formation stage as described in §2. For simplicity the formation of cometary ice is divided roughly into two stages.
The first stage is the parent interstellar cloud stage. At this stage,
gaseous molecules in the cloud condensed onto grain surfaces to form icy
mantles on them. The grain temperature in dense regions of the cloud is
as low as
K, so that even very volatile gaseous species such as N
and CO could condense onto the grain surfaces. The mantle composition reflects
the interstellar molecule composition, apart from positive ions. The grains
in dense molecular clouds have been characterized by Greenberg (1982).
Namely, the volatile mantle is composed of a mixture of oxidized and reduced
species. It should be pointed out that the chemical composition of the
gas and mantles are far from the composition expected in chemical equilibrium.
This is because the chemistry prevailing in interstellar clouds is not
thermal chemistry as is in the primordial solar nebula, but is rather based
on ion-molecule reactions in the gas phase because of the low density and
temperature of the cloud as well as because of the irradiation by UV and
cosmic rays (see papers by Strazzulla and Pirronello in this volume).
The second stage occurs when and after the solar nebula formed from the interstellar cloud. The important physical quantities are the temperature distribution of the solar nebula and its time variation, which, however, are not well clarified at present. In the inner region, most of the grains would have sublimed completely, and as the gas cooled down subsequently, grains would have recondensed. In the outer solar nebula where the temperature rise was not much, on the other hand, there must have been a region where the the grains coated with ice mantles survived. Since the solar nebula was warmer than the interstellar cloud because of heating by solar radiation, however, very volatile species would have been lost by sublimation from the ice mantles. The degree of sublimation would depend upon the distance from the sun. The formation region of comets is the region where the observed volatile abundances are realized. Comets are regarded as planetesimals accreted from grains in this region.
On the basis of this scenario, Yamamoto et al. (1983) and Yamamoto (1985) deduced the following results:
The sublimation temperature for each molecular species is defined by
the gas temperature at which the sublimation rate from a grain surface
equals the sticking rate of the molecules from a gas phase. The sublimation
rate
of molecular species X of molecular weight
per unit surface area of a grain of temperature
is expressed by
where
is the vapor pressure of the molecular species X,
mass of an hydrogen atom, and k the Boltzmann constant. On the other
hand, the sticking rate
of the molecules X in the gas (mainly of H
in the solar nebula conditions) of temperature T and density n
onto grain's unit surface area is given by
where
is abundance of molecular species X relative to H
, and
is the mean thermal velocity of molecules X. At
, both sticking and sublimation balance, i.e.
, and we have
The relation between the gas temperature T and the grain temperature
is determined from energy balance of a grain in the solar nebula. It is
assumed that grains settle toward the midplane in the solar ne bula just
before formation of a cometary nucleus by gravitational fragmentation of
the dust layer (see §2).
The grains are heated by collisions of gaseous molecules in the solar nebula;
heating by solar radiation may be ignored since the dust layer is optically
thick for the solar visible and UV radiation. Gaseous molecules contributing
most to the grain heating are H
molecules in the solar nebula. Ignoring other molecular species, the heating
rate
for a grain of radius a is given by
where
is mean thermal velocity of H
molecules, and
is the accommodation coefficient of H
, which is an efficiency of energy transfer at collisions. On the other
hand, grains cool by thermal emission with temperature
. The cooling rate
of a grain of radius a and emissivity
is given by
where
is the Stefan-Boltzmann constant, and
is a background radiation temperature. The grain temperature
is obtained from the energy balance
The sublimation temperature
and the grain temperature
corresponding to
obtained by solving simultaneously eqs. (3)
and (6)
with putting
in eq. (6).
Figure 6 shows
and corresponding
thus calculated for molecular species relevant to cometary ice, together
with the temperature-gas density profiles of the radiative equilibrium
(Hayashi, 1981) and adiabatic (Cameron, 1978) models of the solar nebula.
The formation distance of cometary nuclei corresponding to the formation
temperature of
K is estimated to be 14 to 15 < r < 80 to 110 AU, depending
upon the temperature distribution of the solar nebula (these values assume
a radiative equilibrium (Hayashi, 1981), or adiabatic distribution, (Cameron,
1978). It must be mentioned that there remain ambiguities in translating
the formation temperature into the formation distance, since the latter
depends upon the actual temperature distribution in the solar nebula and
its time variation, and upon the degree of radial transport of the nebular
materials.