A&A 384, 1107-1118 (2002)
DOI: 10.1051/0004-6361:20020086
D. Bockelée-Morvan1 - D. Gautier1 - F. Hersant 1 - J.-M. Huré1,2 - F. Robert 3
1 - Observatoire de Paris, 5 place Jules Janssen, 92195 Meudon,
France
2 -
Université de Paris 7, 2 place Jussieu, 75251,
Paris Cedex 05, France
3 -
Laboratoire de Minéralogie, Muséum d'Histoire Naturelle,
61 rue Buffon, 75005 Paris, France
Received 3 October 2001 / Accepted 9 January 2002
Abstract
There is much debate about the origin of crystalline silicates in
comets. Silicates in the protosolar cloud were likely amorphous,
however the
temperature of the outer solar nebula was too cold
to allow their formation in this region by thermal annealing or
direct condensation. This paper investigates the formation of crystalline
silicates
in the inner hot regions of the solar nebula, and their diffusive
transport out to the comet formation zone, using a turbulent
evolutionary model of the solar nebula. The model
uses time-dependent temperature and surface density
profiles generated from the 2-D -disk model
of Hersant et al. (2001). It is shown that
turbulent diffusion is an efficient process to carry crystalline silicates
from inner to outer disk regions within timescales of a few
104 yr.
The warmest solar nebula models which reproduce the D/H ratios measured
in meteorites, comets, Uranus and Neptune (Hersant et al. 2001)
provide a mass fraction of crystalline silicates in
the Jupiter-Neptune region in agreement with that measured in comet Hale-Bopp.
Key words: solar system: formation - comets: general - comets: individual: C/1995 O1 (Hale-Bopp) - planetary systems: fonction - planetary systems: protoplanetary disks
The origin of cometary silicates is controversial. Infrared spectra obtained from ground-based telescopes and the Infrared Space Observatory (ISO) have shown that Mg-rich olivines and pyroxenes are present in cometary grains in both amorphous and crystalline form (Hanner et al. 1994; Hayward et al. 2000; Crovisier et al. 1997, 2000; Wooden et al. 1999). Since the formation of crystalline silicates requires high temperatures, in contrast to the amorphous variety, the puzzling question is how comets could have incorporated both high- and low temperature materials, including ices.
Crystalline silicates are encountered in various environments.
They are found in interplanetary dust particles (IDPs), sometimes
coexisting with the amorphous form in the aggregate IDPs thought to be
of cometary origin (Bradley et al. 1992, 1999). Silicates in
chondritic meteorites,
formed in the inner hot solar nebula, are entirely crystalline. Crystalline
silicates have been detected in disks around pre-main-sequence
Herbig Ae/Be stars (e.g., Malfait et al. 1998;
Bouwman et al. 2001). These disks, surrounding intermediate-mass
stars (2-20 ), are believed to be similar to the
primitive solar nebula from which the Solar System was formed.
Crystalline silicates are also found in the debris disks present around
young main sequence stars as, e.g.,
-Pic (Knacke et al. 1993).
On the other hand, there is no spectral evidence for their presence
in the diffuse interstellar medium or molecular clouds, nor in young
stellar objects (Demyk et al. 1999; Hanner et al. 1998). It
is thus likely that the crystalline
silicates present in cometary grains were produced during or after the
collapse of the presolar cloud. Crystalline silicates can form at high
temperature by direct condensation or thermal annealing of amorphous
silicates. This latter process requires temperatures of at least 800 K
(e.g., Gail 1998). The detection of silica SiO2 in Herbig Ae/Be systems
and comets is consistent with compositional changes associated to
thermal annealing of amorphous grains (Bouwman et al. 2001).
We follow the current opinion that silicates infalling from the presolar cloud onto the nebula discoid were all amorphous. Chick & Cassen (1997) studied the thermal processing of silicates during the collapse of the presolar cloud in order to assess their survivability. Combining results of envelope radiative heating with heating in the accretion shock above the disk (Neufeld & Hollenback 1994), they show that grains of silicates infalling at distances larger than 1 AU from the Sun escaped vaporization. From the temperature radial profiles they compute, we can deduce that, during this stage, the formation of crystalline silicates by annealing of amorphous silicates was inefficient at distances greater than 2-3 AU.
Gail (1998) improved the analysis of the problem.
Assuming that the nebula was
already formed, he studied the fate of amorphous silicates radially infalling
onto the Sun. Using a stationary model of the nebula, he showed that
the annealing of amorphous silicates was
inefficient at heliocentric distances greater than 1.5 AU. In other words,
the temperature was too cold in the outer nebula where comets are
currently supposed to have been formed, for the
interstellar silicate grains to be thermally processed prior to
their incorporation to comets. Therefore, either crystallization was
initiated by a low temperature process, as proposed for disks around evolved
stars (Molster et al. 1999),
or there was a
significant radial mixing between the warm and cold parts of the solar
nebula. However, Gail (1998) did not take into account in his
calculations any mixing process. More recently, Gail (2001)
studied mixing by turbulent diffusion in a stationary Keplerian
-disk.
We investigate here the thermal annealing of
amorphous silicates in the hot regions of the solar nebula and their
diffusive transport out to the comet formation zone by using a turbulent
evolutionary model of the nebula. In fact, large-scale
radial mixing in the nebula has previously been shown
to be a requisite in order to fit measurements of the D/H ratio in water
in LL3 meteorites and Oort cloud comets (Drouart et al. 1999;
Mousis et al. 2000). While these authors used a 1-D turbulent model,
we take advantage here of
the recent 2-D evolutionary model of Hersant et al. (2001)
which provides an improved description of the nebula. The model
generates time-dependent temperature, density, and pressures profiles
throughout the nebula. Hersant et al. further studied
the radial distribution and
time evolution of the deuterium enrichment in water
in the solar nebula due to isotopic
exchange with H2 (the main reservoir of deuterium)
and diffusive turbulent transport. Using the D/H values measured
in LL3 meteorites and Oort cloud comets, they constrained the parameters
of the nebula, namely its initial accretion rate ,
initial
outer radius
,
and the coefficient of turbulent viscosity
(Shakura & Sunyaev 1973).
The goal of the present paper is to investigate whether the solar nebula models selected by Hersant et al. (2001), which fit the D/H ratio in comets, are also able to explain the amount of crystalline silicates detected in these objects. Section 2 summarizes observational constraints obtained on the nature of silicates in comets. The model is presented in Sect. 3 and the results are discussed in Sect. 4.
Date | ![]() |
Cry Ol | Cry Pyr | Am Ol | Am Pyr | Cry Tota | Reference |
7 Oct. 1996 | 2.8 | 0.22 | 0.08 | -b | 0.72 | 0.30 | Crovisier et al. (2000) |
0.50 | -b | 0.10 | 0.40 | 0.50 | Colangeli et al. (1999)c | ||
11 Apr. 1997 | 0.93 | 0.33 | 0.20 | 0.31 | 0.13 | 0.54 | Wooden et al. (1999) |
0.08 | 0.80d | 0.08 | 0.03 | 0.88 | Wooden et al. (1999)e | ||
28 Dec. 1997 | 3.9 | 0.22 | 0.14 | -b | 0.64 | 0.56 | Crovisier et al. (2000) |
a Relative mass fraction of silicates in crystalline form, to be
directly compared to model calculations of ![]() b Silicate component not considered in the fit of the spectrum. c Spectral fit includes amorphous carbon. Silicate mass fractions have been here renormalized. d Sum of ortho- and clino-pyroxene contributions. Ortho-pyroxene contribution is dominant. e In contrast to other spectral analyses, crystalline pyroxenes are assumed to be cooler than other silicates. |
Measurements concerning cometary silicates are reviewed in detail by Hanner et al. (1997) and Hanner (1999). A summary is given here.
The nature of cometary silicates was mainly investigated from
their Si-O stretching bands falling around 10 m. Low resolution
8-13
m spectra have been acquired for a dozen
comets (see Hanner et al. 1994; Hanner 1999 and
references therein). Strong structured
emission features were observed for half of them. The detection of a peak
at 11.2-11.3
m in 1P/Halley and some of the later observed comets
pointed out the presence of Mg-rich crystalline olivines
mixed with Fe-bearing amorphous olivines and pyroxenes
(Hanner et al. 1994; Wooden et al. 1999).
Observations of comet C/1995 O1 (Hale-Bopp) provided new insights on silicate mineralogy
(Hanner et al. 1997; Hayward et al. 2000;
Wooden et al. 1999; Crovisier et al. 1997, 2000).
The 10-m spectra exhibit several discrete peaks, which have
been attributed to the presence of at least five different Mg-rich silicates
in cometary dust: amorphous
pyroxene, amorphous olivine, crystalline ortho-pyroxene, crystalline
clino-pyroxene, and crystalline olivine (Wooden et al. 1999).
ISO observations of the full 2-45
m spectra of comet Hale-Bopp
showed several broad emission features longward 16
m, all attributed
to bending mode vibrations of Mg-rich crystalline olivine
(Crovisier et al. 1997, 2000; Brucato et al. 1999a;
Colangeli et al. 1999).
All interpretations of Hale-Bopp spectra agree on the presence of
olivines
and pyroxenes in both amorphous and crystalline form.
However, they differ quantitatively. A summary of inferred relative
abundances is given in
Table 1. The relative mass fraction of silicates in crystalline
form inferred from Hale-Bopp spectra varies from 30 to 90%. Not included
in Table 1 is the estimation made by Hayward et al.
(2000) of 20% of
crystalline silicates . As explained by Hanner (1999), it is
difficult
to determine accurately the relative abundances of minerals in cometary dust from their
infrared spectra. With respect to other results, Wooden et al. (1999)
suggest a much higher mass fraction of crystalline
pyroxenes (Table 1). In contrast to other studies, they
attribute the 9.3 m peak to crystalline pyroxene. In order to
reproduce the
relative strength of this feature observed at various heliocentric
distances with a similar relative mass fraction of crystalline pyroxenes
with respect to the other dust components, they argue
that crystalline pyroxenes must be cooler than other silicate components,
hence the different result.
As discussed by Wooden et al., this requires that pyroxene crystals are
separate particles in the Hale-Bopp's coma. This doesn't fit with
current views of cometary dust as porous aggregate particles mixing
silicates and carbonaceous material, and might deserve further studies
(see the discussion of Hayward et al. 2000).
10-m spectra obtained in other long-period comets indicate a silicate
mineralogy in these comets similar to that in comet Hale-Bopp (Wooden et
al. 1999). However, significant spectral differences are seen between comets.
Colangeli et al. (1995, 1996) attribute these differences to
variations in the relative amounts of the various silicates or
in the Mg/Fe ratio.
Hayward et al. (2000) argue that they might rather
reflect variations in the dust size distribution and temperature.
Data on
short-period comets, presumably formed in the Kuiper Belt, are very sparse.
They usually exhibit weak and broad featureless 10 m emission,
presumably due to their deficiency in small particles
(Hanner et al. 1994). However, crystalline olivines might be
present in short-period comets as well, as suggested by the ISO spectrum
of comet 103P/Hartley 2 which shows the presence of a peak at 11.3
m
(Crovisier et al. 1999, 2000).
The goal of our modelling is to investigate whether crystalline silicates formed in the inner warm regions of the solar nebula can be efficiently transported by turbulent diffusion to the outermost cold regions where comets presumably formed. This model treats both the amorphous-to-crystalline phase transition of small silicate particles and their diffusive transport throughout the nebula as a function of time.
We use a simplistic approach in which only two dust components are considered, regardless of their mineralogy: amorphous and crystalline silicates. The amorphous-to-crystalline transition by thermal heating can be treated thanks to recent laboratory measurements (Sect. 3.3).
The model solves time-dependent equations which describe the diffusive
transport of small dust particles
in a turbulent accretion disk (Morfill & Völk 1984). Defining
as the mass density of the disk (dominated by H2 and He),
and
as the mass density of dust particles of chemical index k,
we have:
Using Eqs. (1) and (2), it can be shown that, in the midplane of
the disk,
the relative concentration
follows the diffusion
equation:
The annealing process of amorphous
silicates and the diffusive transport of crystalline silicates in the
outer regions of the solar nebula can be investigated by solving two
diffusion equations which are coupled by source and sink terms:
We sum up here the solar nebula model that we use, and which is
described in length by Hersant et al. (2001). The temporal
evolution of the solar nebula is
modelled by a sequence of
stationary solutions provided by the 2-D accretion disk model of
Huré (2000), where the
vertical stratification of the physical quantities is explicitly calculated.
In a 2-D model, the midplane temperature,
pressure, and volumetric density are generally slightly higher than in a
1-D model where
a vertically homogeneous disk is assumed (Huré & Galliano 2001).
This model uses the well known
prescription of Shakura &
Sunyaev (1973), which permits modellers to parameterize
the effects of small scale turbulence. The prescription
consists of writing the turbulent viscosity as
,
where
is the sound velocity and
is the
Keplerian angular velocity.
is a non-dimensional parameter whose value depends on the nature
of the turbulence and is usually taken, for simplicity, as constant with
respect to the heliocentric distance in the disk and time.
The 2-D model is calculated for a given
accretion rate
which governs the inflow of matter from the disk
onto the proto-Sun. As the disk evolves and
cools down, the turbulence is less and less efficient, and this accretion
rate
diminishes. The evolutionary model
begins when the formation of the Sun
is almost complete. The subsequent evolution of the nebula is obtained
by calculating
2-D models for a temporal sequence of accretion rates following the
prescription of Makalkin & Dorofeyeva (1991). These authors
have shown that the
evolution of
as a function of time can be described
by a simple power law which depends upon the accretion rate and
the outer radius of the nebula
at t = 0 according to:
The model takes also into account that, under the action of turbulence,
the nebula spreads out viscously with time. It is assumed that
the evolution of its outer radius
follows:
![]() |
(10) |
With this solar nebula model, Hersant et al. studied the radial
distribution and time evolution of the
deuterium enrichment in water in the solar nebula due to isotopic
exchange with H2 and diffusive turbulent transport. Assuming that the
highly D-enriched component measured in LL3 meteorites
(D/H = (7312
)
is representative of
the grains in the presolar cloud (Drouart et al. 1999;
Mousis et al. 2000), Hersant et al. were able to explain
the D/H value of
measured in cometary water
as due to some mixing between water vapor reprocessed in the inner regions
of the solar nebula by isotopic exchange with H2, and the unprocessed
component of its outer regions.
Hersant et al. (2001) showed that the three
parameters defining the nebula can be constrained by the deuterium enrichments
measured in LL3 meteorites, comets, proto-Uranian and proto-Neptunian
ices, taking into account a few other
physical constraints, namely: 1) the mass of the nebula cannot exceed
0.3
in order to maintain the stability
of the disk (Shu et al. 1990); 2) the angular momentum must have
been transported outward to Neptune by turbulence in
yr;
3) the
initial temperature of the nebula was higher than 1000 K inside 3 AU to
secure the crystallization of silicates in the inner regions.
Hersant et al. concluded that
was in between
and 10-5
yr-1,
between
12.8 and
39 AU, and
between 0.006 and 0.02. For the purpose
of this paper in which the temperature of the
nebula is crucial for determining where and when the conversion of
amorphous silicates into the crystalline variety occurred, they selected
among all nebulae fitting observational constraints one of the coldest and one
of the hottest nebulae, named cold nebula and warm nebula, respectively.
Their parameters are given in Table 2.
They also selected a nebula intermediate between the cold and
warm nebulae, called nominal nebula (Table 2).
Radial profiles of
and T as
a function of time are shown in Fig. 1 for these three nebulae.
Adiabats (P,T) of the nominal solar nebula at various epochs are
shown in Fig. 2.
Type | ![]() |
![]() |
![]() |
[![]() |
[AU] | ||
cold | 2 ![]() |
4.0 ![]() |
27.0 |
nominal | 9 ![]() |
5.0 ![]() |
17.0 |
warm | 8 ![]() |
9.8 ![]() |
12.8 |
![]() |
Figure 1:
Surface density (![]() ![]() ![]() ![]() |
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Solving Eqs. (6) and (7) requires, in addition to a solar nebula model, data on silicates crystallization and sublimation.
The thermally induced amorphous-to-crystalline transition has been studied
both theoretically (see Gail 1998 and references therein) and in the
laboratory (e.g. Ashworth et al. 1984; Nuth & Donn
1982; Hallenbeck et al. 1998;
Brucato et al. 1999b; Fabian et al. 2000).
The crystallization process corresponds, on
a microscopic scale, to the movement of oxygen and silicon atoms to new
locations characterized by deeper local energy minima. The characteristic
timescale for the amorphous-crystalline transition can be expressed as:
Experiments revealed a significant distinction in the
amorphous-to-crystalline transition between micrometre-sized glass and smoke
particles (Fabian et al. 2000). This is something interesting
to consider, especially given the large size
of interstellar dust (typically 0.1
m) compared to that of
the laboratory smokes.
First, the timescale at 1000 K for
crystallization of MgSiO3 glass particles exceeds that of MgSiO3
smoke particles. In addition, the activation temperature determined for
these micrometre particles with
= 2
1013 s-1
decreases with increasing annealing temperature
(Fabian et al. 2000). We performed a least-square fit to the
annealing times measured for temperatures
between 1030 K and 1121 K (median values of Table 3 in Fabian et al.
2000), which shows that the measurements can be fitted by an
exponential law (Eq. (11)) with
s-1 and
K (note that the uncertainties on
and
are
large: respectively at least
2 orders of magnitude and
5000 K,
taking the range of estimated annealing times as the uncertainty on the
measurements). Although this law may not apply over a large temperature
range, it permits us to investigate larger timescales for
crystallization with respect to those derived for nanometre smoke particles.
Table 3 gives the laws for
considered in this paper and
values at T = 1000 K.
The laws (1) to (3) come from the
mentioned experiments on smoke particles. Law numbered (4) is that
derived in the paragraph above, from measurements on micrometre grains.
The law numbered (5) (
s-1 and
K) is possibly more appropriate for 0.1
m sized particles. It was derived from law (4), assuming the linear
grain size dependence suggested
by annealing experiments made on silicate glassy sheets
(Fabian et al. 2000).
The annealing timescales we use are those determined for nearly pure magnesium silicates. It can be objected that these measurements are not fully appropriate for our purpose, since amorphous silicates in the protosolar cloud contained some iron. Demyk et al. (1999) studied the infrared signatures of the silicates around two massive protostellar objects and showed that they can be well reproduced with Mg-Fe pyroxenes amorphous grains with a Fe/Mg ratio of about 0.5. It would be thus interesting to use annealing timescales characteristic of such compounds. Annealing experiments have been made for pure iron-silicate and magnesium-iron silicates (Hallenbeck et al. 1998), but the activation temperatures required for our model are not available. Hallenbeck et al. (1998) found that annealing timescales are much larger for iron silicates than for magnesium silicates. On the other hand, the evolution of mixed magnesium-iron silicates resembles that of magnesium silicate samples (Hallenbeck et al. 1998).
Lawa | Compound | ![]() |
![]() |
![]() |
[s-1] | [K] | 1000 K [s] | ||
(1) | Mg2SiO4 smoke | 2 ![]() |
39100 | 4.8 ![]() |
(2) | MgSiO3 smoke | 2 ![]() |
42040 | 9.1 ![]() |
(3) | MgSiO3 smoke | 2.5 ![]() |
47500 | 1.7 ![]() |
(4) | MgSiO3 glass | 5 ![]() |
70000 | 5.0 ![]() |
(5) | MgSiO3 glass | 2.5 ![]() |
70000 | 1.0 ![]() |
The stability of Mg-rich silicates against vaporization has been studied
by Duschl et al. (1996). It is shown that silicates decompose in a near
equilibrium process with the gas phase and gradually disappear over
a finite temperature interval of 300 K. We have considered the
(
,
T) equilibrium curve with f = 0.9 given in Fig. 4 of
this paper,
which corresponds to the situation where a fraction f = 0.9 of the Si
is still bound into grains. This curve is plotted in Fig. 2 over
the (P, T) adiabat of the solar nebula. The last grains
disappear at a
temperature
100 K higher (Duschl et al. 1996). Taking the
(
,
T) equilibrium curve with f = 0 that corresponds to the full
disappearance of silicate grains would not change significantly our
results, keeping in mind that sublimation curves depend somewhat on the
silicate composition.
![]() |
Figure 2:
Adiabat (P, T) of the nominal nebula as a function
of time. Curves from top to bottom correspond to increasing times.
Plain lines are for times 103, 104, 105,
and 106 years. Dotted lines are for times
![]() ![]() ![]() ![]() ![]() ![]() |
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Numerical integration of the coupled partial differential Eqs. (6)
and (7) requires initial
conditions and boundary conditions at the inner r0 and outer
radii at
any time. This is a classical
problem which can be
solved by finite differencing, given a r-grid and time step.
Equations (6) and (7) mix advective and diffusion terms, each
of them
requiring a specific differencing scheme to avoid numerical dissipation
and stability problems. Therefore, we used the "operator splitting'' method
and computed separately, and successively within one time step,
the time evolution of
and
due to the advective and
diffusion terms. The time evolutions resulting from the chemical and
diffusion terms were treated together.
For the transport term, we used the explicit spatial differencing scheme suggested by Hawley et al. (1984) (Sect. II.c.ii) which mixes the second-order (in space) averaging scheme of Wilson (1978) and the first-order upwind scheme. For treating the diffusion, we used the Crank-Nicholson differencing scheme which is second-order in both space and time (see Drouart et al. 1999).
The r-grid has been defined between
r0 = 0.1 AU and
with
logarithmic steps such that
.
The time step
has been set
to
yr
.
For minimizing the
computer time,
values less than 50 yr (5 yr for the cold nebula)
were set to 50 yr (5 yr for the cold nebula).
As initial condition, we took
and
at any r, where
Ct is a constant setting the relative mass concentration of silicates with
respect to H2. Since Eqs. (6) and (7) are linear in Ct and
we are interested in the relative fraction of crystalline versus amorphous
silicates, we took Ct = 1.
and
then provides directly the
relative mass fractions of amorphous and
crystalline silicates, respectively. With this initial condition,
we assume that the silicates are
only in amorphous form throughout the nebula at t = 0. Due to hot
temperatures
at small r, the initial concentrations are in fact determined by the thermal
radial profile of the nebula at t = 0.
Figure 3 shows that silicates
are essentially amorphous at distances larger than 10 AU for the
considered nebulae. These initial conditions are
valid, providing that the grains outside 10 AU were not thermally processed
in the preceding stages of solar nebula formation and evolution. As
discussed in Sects. 1 and 4.1, silicates suffered little processing
during the collapse phase (Chick & Cassen 1997).
The boundary spatial condition
was taken at
.
This implies that we neglect radial influx
of material from the outside.
When the sublimation boundary
extends
farther than r0 = 0.1 AU, the inner spatial boundary condition is
at
.
When this is not the case
(for the considered
nebulae, this happens only at
yr, Fig. 3),
we assumed, for
simplicity,
.
The temporal and radial evolution of the relative mass fraction of crystalline
silicates
is shown in Fig. 3, for the three considered nebulae.
In Fig. 4 is plotted the mass fraction of crystalline silicates
integrated over the whole nebula
as a function of time:
![]() |
(12) |
At time slightly above 0 yr, silicate grains are vaporized or
are in crystalline form within some distance which depends
on the initial thermal structure of the nebula. This distance, that
we call the crystalline front, is 5 to 7 AU for the considered
nebulae. As time goes on, amorphous silicates, present in the outer regions,
are transported towards the inner regions by advection and turbulent
diffusion, and
then thermally
annealed. This radial inward flux of amorphous material inside the
region where crystallization takes place balances the loss
of crystalline silicates by advective transport within the evaporation
zone. As shown in Fig. 4, the integrated mass fraction of crystalline
silicates is maintained
to a value that slightly exceeds the value acquired at
up to times
yr due to on-going crystallization.
When mass input from crystallization no longer balances
loss by evaporation, the integrated mass fraction of crystalline silicates
then decreases to reach a plateau value. The ratio between
this long term plateau value and the crystalline mass fraction at
t
0 is
more than 25% for the nebulae considered here (Fig. 4). This shows
that
turbulent diffusion of both amorphous and crystalline silicates is an
efficient process for maintaining a high proportion
of crystalline silicates in the solar nebula, as it is cooling down.
The effects of turbulent diffusion are seen on the radial profiles of
shown in Figs. 3A-C. Crystalline silicates are
progressively mixed with the
amorphous grains
present in the outer regions, so that
continuously increases in the
outer regions to finally reach a plateau equal to the
plateau value mentioned above. In turn, the crystalline front
moves towards the Sun, as the nebula cools with time.
The characteristic timescale for radial mixing is approximately given by the
accretion rate timescale t0 defined by Eq. (9) and equal
to
,
and
yr for the cold, nominal and warm nebulae,
respectively (Hersant et al. 2001).
Figure 3 shows that, indeed, the plateau is reached faster for the
warm
nebula than for the nominal nebula, and faster for the nominal nebula than for
the cold nebula. For example, at t = 104 yr,
the value of
at r = 15 AU is
80% the plateau value for the warm nebula, while,
at the outermost
distances of the cold nebula, the silicates are still essentially amorphous.
At times larger than
5-
yr, the mass
fraction of crystalline silicates at r > 10 AU no longer evolves for the
three nebulae considered here. The timescale for radial mixing is
governed by the diffusion timescale
,
which
increases rapidly with increasing distance, and also increases
significantly with
increasing time, due to decreasing temperature and disk thickness with time
(Fig. 5).
Lawa | Compound | ![]() |
||
Solar nebula | ||||
cold | nominal | warm | ||
(1) | Mg2SiO4 smoke | 0.022 | 0.123 | 0.58 |
(2) | MgSiO3 smoke | 0.018 | 0.094 | 0.44 |
(3) | MgSiO3 smoke | 0.013 | 0.064 | 0.29 |
(4) | MgSiO3 glass | 0.013 | 0.066 | 0.30 |
(5) | MgSiO3, 0.1 ![]() |
0.015 | 0.076 | 0.35 |
![]() |
Figure 3:
Mass fraction of crystalline silicates as a function of
radius and time in the solar nebula. Thermal annealing is modelled
using law (1) of Table 3 (
![]() ![]() |
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The mass fraction of crystalline silicates in the plateau is, as expected,
strongly dependent on the thermal structure of the early nebula.
The cold and warm nebulae considered here are extreme, so that the
crystalline fraction varies from
0.02 to 0.6 for the annealing timescales considered in the calculations
shown in Fig. 3. Table 4 shows that, in contrast, the
degree of
crystallinity reached in the plateau is not very sensitive to the
law. Comparing results obtained for laws (1) and
(2), the
model predicts similar (within 30%) degrees of crystallinity for
olivine and pyroxene grains. The degree of crystallinity is lower for
Mg-rich pyroxene grains than for olivine, due to their slightly higher
activation temperature
.
The
laws used for modelling the
annealing of MgSiO3 glass particles (laws (4) and (5))
provide results
not very different (within 40%) from those derived using experimental data
on smoke MgSiO3 particles (law (2)).
![]() |
Figure 4:
Mass fraction of crystalline silicates integrated over the
whole nebula
![]() |
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The present work is based on the assumption that crystalline
silicates were formed from the annealing of amorphous silicates
moving radially in the disk equatorial plane towards the inner hot regions.
Another mechanism for the formation of crystalline silicates may result from
the re-condensation of silicates. Silicates infalling onto the
disk underwent heating in the collapsing presolar cloud or during
the shock when arriving onto the disk. Chick & Cassen (1997) have
investigated the temperatures experienced by
infalling interstellar grains during this early stage of Solar System
formation and conclude that silicates evaporate within 1 AU. Both
theoretical considerations and compositional constraints from meteorites
argue that the thermal regime experienced in the midplane of the
accretion disk
was more severe and controlled the vaporization distances of refractory
grains (Chick & Cassen 1997). In the solar nebula models
investigated
here, the vaporization distance of silicates is 1 AU, a value
less than the value
AU required to explain the relative
abundances of moderately volatile elements of carbonaceous meteorites
(Cassen 1996, 2001).
We do not take into account in our model the condensation of silicates in the vapor phase which takes place near the Sun when the nebula cools down. This would require a kinetic treatment of the chemical surface reactions during grain growth for which data are lacking (Gail 1998; Gail & Sedlmayr 1999), and would demand a coupling of the dust and gas diffusion equations. Should the condensation of silicates actually produce the crystalline form, as is expected, neglecting this process leads to an underestimation of the relative fraction of crystalline silicates with respect to the amorphous variety. Since the mass of material located within the sublimation region is much smaller than the mass located outside (<10% for the selected nebulae; see Fig. 14 of Hersant et al. 2001), we believe that neglecting the contribution of crystalline silicates condensed from the vapor phase does not significantly affect our results. There are however presumptions that vapour phase condensates are present in cometary grains. A class of very porous interplanetary dust particles called CP, and suspected to be of cometary origin, shows Mg-rich pyroxene crystals with whisker and platelet morphologies (Bradley et al. 1983). These micro-structures are expected to grow by condensation from a relatively low-pressure vapour phase.
Silicates in meteorites are all in crystalline form. However, crystalline silicates in meteorites were not necessarily produced by the annealing process investigated in the present work. Therefore, we cannot discuss the crystallinity of meteorites only in the context of our model, although there is no apparent conflict, since Fig. 3 shows that silicates within 3 AU remain essentially crystalline at times less than 105 yr.
Most meteorites - stony, stony-iron and iron - record melting events related to the metamorphism and magmatism of at least, seventy small parent bodies (Scott 1991). In addition, a large number of asteroids seem differentiated. Although the mechanism that yielded this increase in temperature of the asteroidal parent body is still largely debated, no amorphous interstellar silicates present in the protosolar nebula would have survived such thermal stress linked to the planetesimal formation and evolution. Therefore, in the inner zone of the solar system, all solid bodies appear crystalline.
Among the classes of meteorites that could have preserved a partial record of the crystalline state of their amorphous silicate precursors, are the carbonaceous and the unequilibrated type 3 chondrites (especially the type LL3.0). These meteorites consists of chondrules and matrix.
Chondrules are 100 to 1000 m size spheroidal silicates, mainly
made of olivine and pyroxene, which constitute the dominant fraction
(up to 70%) of chondrites. They are the products of partial melting of
precursor silicates - possibly the amorphous interstellar dust - with
negligible contribution of sulphides, oxides and iron. Although the
conditions of formation of chondrules are not totally understood,
experimental simulations have allowed a precise reconstruction of their
thermal history. (1) Their heating time during which sub-melting was
taking place varied from 100 s (Tsuchiyama & Nagahara 1981) to 20
h (Lofgren & Russell 1988). (2) Essentially based on the
distribution of the chemical elements between the different
mineralogical phases coexisting in chondrules (as for example the
Fe/Mg partitioning coefficients) cooling rates were estimated to lie
between 100 to 2000
C/hr (Radomsky & Hewins 1990). (3)
The range of
heating temperature depends almost exclusively on the Mg-Al-Si chemical
proportions of the chondrules and ranges from 1200 to 1700
C. The
occurrence of the three common textures (granular in Mg-rich olivine
chondrules, pophyritic in Fe-rich olivine chondrules, radiating in
pyroxene chondrules) are experimentally reproduced by heating the
precursor silicates to near-liquidus temperature (1300-1800
C) but not
higher (Hewins & Radomsky 1990).
These short heating and cooling times associated with these high silicate melting temperatures are consistent with the idea that chondrules formed by transient heating phenomena (as those involved in electric discharges) in the dust-rich regions of the protosolar nebula. These conditions are not accounted for in the present model. It is clear that amorphous precursors of chondrules, if any, would not have survived through such a violent thermal history.
As far as the matrix is concerned, its origin is poorly documented mainly because of its very fine grain structure. The matrix is essentially composed of broken fragments of chondrules, of clay minerals and sub-micronic silicates and sulfides which are also found as relic grains in chondrules. Matrix lumps are sometimes enclosed in chondrule rims and therefore the formation of the matrix post-date the chondrule formation. The fact that amorphous mineral are extremely rare in the matrix cannot be taken as an evidence that the dust was entirely crystalline at the time and location of the matrix formation. Indeed, an intense circulation of water in the parent body meteorites is attested by the presence of low temperature phyllosilicates (such as smectite) essentially located in the matrix. For a given chemical composition, the formation rate of clay minerals is enhanced for amorphous compared to crystalline silicates. Although the alteration rates of amorphous silicate has not been experimentally measured, it seems possible that all the pre-existing interstellar amorphous silicates - if any - would have been transformed in clays during mild hydrothermal events.
To summarize this discussion on silicates in meteorites, the post-formation geological processes have likely erased the amorphous structures of interstellar precursors.
We discuss now the model results in the context of the amount of crystalline
silicates present in the dust of long-period comets such as Hale-Bopp.
Dynamical considerations suggest that these comets formed mainly in the
Uranus-Neptune zone. Our model shows that microscopic grains made of
crystalline silicates present in the inner regions of the solar nebula are
efficiently transported outwards to the comet formation zone by
turbulence in timescales of a few 104 yr.
The mass fraction of crystalline silicates in the dust of comet Hale-Bopp
is estimated between 0.3-0.5 to 0.9 (Table 1). Comparing with model
results obtained in the outer regions (Table 4), we see that our
model well explains the low range of measured values in the case of the warm
nebula, regardless the crystallization law
adopted. On the other
hand, the interpretation of the high degree of crystallinity
%
inferred by Wooden et al. (1999) would require a still warmer nebula.
Radial mixing by turbulence is efficient as long as the grains are coupled to
the gas. The decoupling occurs when particles agglomerate and grow up to
sufficiently large sizes. There is no general consensus
on the critical size for decoupling: it is about 1 m according to
Weidenschilling (1997) and Supulver & Lin (2000), while
Stepinski & Valageas (1996) advocate for centimetre, or even smaller,
sizes. To be valid, our proposed scenario requires that timescales for
grain growth from micrometre sizes to centimetre or metre sizes are not
too short with respect to the diffusion timescale for radial mixing.
Dust coagulation in protoplanetary disks is a complex problem, which is
not yet well understood (see the review of Beckwith et al. 2000).
The sticking process depends both on the chemical and physical properties of
the grains, and on their relative velocity induced by thermal and
turbulent motions, gas drag and sedimentation. Theoretical calculations
predict that particle growth from sub-micron to metre sizes is very rapid
and occurs in timescales of typically a thousand orbital revolutions, that is
10 3 to
10 4 yr from 1 to 10 AU (Weidenschilling &
Cuzzi 1993; Stepinski & Valageas 1996; see also
Weidenschilling 1997,
for results obtained at 30 AU). Such short coagulation timescales
are in conflict
with observational evidence that disks around T Tauri and Herbig Ae/Be
stars of ages
105-107 years are still dominated in mass by
millimetre grains (see the review of Natta et al. 2000).
On the other hand, fast coagulation seems a requisite to explain the
presence of planets in the Solar System (Stepinski & Valageas
1996).
The coagulation timescales given above can be compared to our diffusion
timescales. These latter are plotted
as a function of radial distance and time in Fig. 5, for the warm nebula
parameters
which fit cometary silicate measurements.
Coagulation timescales are somewhat lower, but by a factor of 5 at most,
than the diffusion
timescales computed at t < 105 yr. Therefore, given
actual uncertainties on grain growth, destruction processes by
collisions (Benz 2000), and transport through turbulent eddies,
we conclude that turbulent radial mixing of dust grains in the solar
nebula is a plausible mechanism for explaining the large amount
of crystalline silicates in comets.
![]() |
Figure 5:
Diffusion timescale
![]() ![]() ![]() ![]() |
Open with DEXTER |
In order to explain the D/H ratios in cometary H2O and HCN,
Hersant et al. (2001) concluded that microscopic grains of H2O
and HCN ices, which subsequently accreted into cometesimals, condensed
and were mixed together in the 10-20 AU range at times
of typically 105 yr. Interestingly enough, this is also
the time when the crystalline mass fraction in this heliocentric
range has reached the plateau value (Fig. 3). In other words, at times
of
105 yr, silicates and icy particles with observed cometary
properties were available in the comet formation zone. Their sticking
and coagulation into kilometre-sized bodies could have then proceeded rapidly.
With a vertical 1-D model run at r = 30 AU which does not include
turbulence, Weidenschilling (1997) shows that the formation of comets
could have been completed in about
yr. To conclude,
our nebula model permits us to consistently explain both siliceous
and deuterium cometary composition by large scale turbulent radial
mixing in the solar nebula. In turn, the amount of
crystalline silicates in comets
provides us with additional constraints on the parameters of the
2-D model of Hersant et al. (2001).
One of our primary goals was to investigate whether solar
nebulae models which fit deuterium enrichments in primitive Solar System
bodies (Hersant et al. 2001), would be able to explain the
amount of crystalline silicates measured in cometary dust. In fact, the
present
analysis based on three selected nebulae shows
that producing the amount of cometary crystalline silicates
requires the warm nebula.
Since a number of other warm nebulae fitting deuterium enrichments can be
generated by the 2-D model, the relative proportion of
crystalline silicates in comets can be used as an additional constraint to the
selection of nebulae. Based on the observational data (Table 1),
we adopted the value of
for this constraint.
A number of
nebulae were then calculated, as in Hersant et al., each one
defined
by the initial accretion rate
and radius
of the nebula,
and the coefficient
of turbulent viscosity
.
The equations of diffusion 6
and 7
were then integrated as a function of radial distance and time, using the
surface
density and temperature radial profiles of these nebulae. Nebulae which do
not provide adopted constraints were rejected. The ranges of
acceptable nebula parameters fitting both D/H and
measurements are
substantially reduced with respect to those obtained by Hersant et al. (2001).
They are:
,
yr-1, and
AU.
The derived ranges of acceptable ,
and
values are very small and should not be taken too seriously.
First, the adopted physical and chemical constraints might not be
fully appropriate. Indeed, there are still large uncertainties on D/H values
and silicate composition in comets. More important, among
physical constraints adopted by Hersant et al. (2001), the
initial mass
of the solar nebula was limited to masses smaller than 0.3
,
in
order to satisfy the criterion of gravitational stability of the disk, and
the radius of the solar nebula was forced to reach Neptune distance in the
first
yr. These assumptions, which could be
certainly somewhat relaxed, result in the rejection
of nebulae with warm temperatures in their early life and constrain the
initial radius of the nebula to a small range. The second point to consider
is that the results are model dependent and obtained under several assumptions.
Among these, the value of the Prandtl number, which is not well known, has
a strong influence on the derived
,
and
ranges
(Hersant et al. 2001). The uncertainties on the timescales for
isotopic exchanges and thermal annealing are significant.
Finally, the validity of the disk model may be
also questioned. Standard
-disks are known to be not extended and
warm enough
to explain observations of circumstellar disks. Many alternative
and controversial solutions have been proposed and are beyond
the scope of the present paper. The initial high accretion
rate of
10-5
yr-1 that we derived
would correspond to that of Class 0 protostars when mass infall from
the envelope is feeding the disk (André et al. 2000).
Admittedly, this stage of the early
nebula is not well treated in our model and might affect the initial
and boundary conditions
of our transport equations. However, we must point out that
time-dependent models describing the early stages of the solar nebula
when most of its chemical evolution occurs, are not yet available.
This paper presents a model to explain the presence of a high amount of crystalline silicates in comets. The proposed scenario is that crystalline silicates formed in the inner warm regions of the solar nebula by the thermal annealing of amorphous silicates. They were then transported out to the comet formation zone by turbulent diffusion. This paper follows those of Drouart et al. (1999), Mousis et al. (2000) and Hersant et al. (2001), which interpret the variety of deuterium enrichments observed in primitive objects of the Solar System by isotopic exchange in the solar nebula and turbulent radial mixing.
Our study makes use of the 2-D evolutionary solar nebula model developed by Hersant et al. (2001). It takes into account the rapid evolution of temperatures and pressures during disk evolution. Such an approach is necessary to investigate quantitatively whether the proposed scenario is plausible, since both thermal annealing and turbulent mixing processes are strongly temperature dependent.
Using solar nebula parameters which fit D/H ratios in comets, we show that
turbulent diffusion is an efficient mechanism for
mixing amorphous and crystalline grains in the comet formation
zone. The timescale required for radial mixing is typically
yr, depending on the solar nebula model. These
values are comparable, within a factor of a few, to
theoretical expectations concerning the characteristic timescales for
particle growth in the solar nebula. It is also well below the
time needed to form kilometer-sized comets from a population of
microscopic grains (Weidenschilling 1997). Therefore, given actual
uncertainties on grain growth in the solar nebula, it seems likely that
fast dust coagulation did not preclude large-scale radial mixing,
allowing the transport of crystalline silicates from inner to outer regions.
One of the aims of our modelling was to investigate whether the solar nebulae
selected by Hersant et al. (2001) to fit cometary D/H ratios would
also explain quantitatively
the high amount of crystalline silicates. Our calculations show
that fitting both observational constraints is possible, provided that we
use model parameters which generate warm nebulae. Using the mass fraction
of crystalline silicates estimated in comet Hale-Bopp as an additional
constraint to the model of Hersant et al. (2001), we derived the
ranges of
nebula parameters ,
and
which fit both
cometary and silicate composition. The ranges of acceptable parameters
are small and may not represent the number of uncertainties and
assumptions in our model. However, this study qualitatively
demonstrates the need of
high temperatures in an extended zone of the solar nebula, to explain
both D/H and the properties of cometary silicates. In addition, it
shows that, in essence, the amount of crystalline silicates in comets
should permit us to check the validity of any model of the solar
nebula in which the temperature distribution is calculated.
More conclusive studies will have to await for the availability of
more realistic time-dependent models for solar nebula early evolution.
Our model predicts that, in a few 104 yr, the crystalline/amorphous silicates ratio reaches the same value in almost the whole nebula, except in the innermost regions. This might imply that all comets formed in the turbulent part of the solar nebula could have the same crystalline/amorphous silicates ratio, a result not in conflict with available data on long-period comets (Sect. 2). However, firm conclusions should await for more detailed models which include grain coagulation, since this process affects the efficiency of radial mixing by turbulence. In addition, measurements of the crystalline/amorphous silicates ratio in comets are still too sparse to allow detailed comparison with model predictions. Data on Jupiter family comets coming from the Kuiper Belt are specifically required. They may reveal a lower content of crystalline silicates, if they were formed in a non-turbulent part of the outer nebula.
Acknowledgements
We thank D. Fabian, for providing us with laboratory results before publication, and P. Michel and J. R. Brucato for enlightening discussions. This work was supported by the Programme national de planétologie de l'Institut national des sciences de l'univers (INSU) and the Centre de la recherche scientifique (CNRS).