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Formation of Planetesimals
STUART J. WEIDENSCHILLING
Planetary Science Institute
ABSTItACT
A widely accepted model for the formation of planetesimals is by
gravitational instability of a dust layer in the central plane of the solar
nebula. This mechanism does not eliminate the need for physical sticking
of particles, despite published claims to that effect. Such a dust layer
is extremely sensitive to turbulence, which would prevent gravitational
instability unless coagulation forms bodies large enough to decouple from
the gas (> meter-sized). Collisional accretion driven by differential motions
due to gas drag may bypass gravitational instability completely. Previous
models of coagulation assumed that aggregates were compact bodies with
uniform density, but it is likely that early stages of grain coagulation
produced fractal aggregates having densities that decreased with increasing
size. Fractal structure, even if present only at sub-millimeter size, greatly
slows the rate of coagulation due to differential settling and delays the
concentration of solid matter to the central plane. Low-density aggregates
also maintain higher opacity in the nebula than would result from compact
particles. The size distribution of planetesimals and the time scale of
their formation depend on poorly understood parameters, such as sticking
mechanisms for individual grains, mechanical properties of aggregates, and
the structure of the solar nebula.
INTRODUCTION
It is now generally accepted that the terrestrial planets and the cores
of the giant planets were formed by accretion of smaller solid bodies
82
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(planetesimals). The principal alternative, production of giant protoplanets
by large-scale gravitational instabilities of the gaseous component of the
solar nebula, has been abandoned (Cameron 1988~. Planetesimals, with
initial sizes of the order of kilometers, must have formed from much
smaller particles, perhaps consisting of a mixture of surviving presolar
grains and condensates from the nebular gas. Such grains were small,
probably sub-pm in size, and their motions were controlled by the drag of
the surrounding gas, rather than by gravitational forces. It is necessary to
understand the aerodynamic processes that affected these small bodies in
order to understand how planetesimals formed.
The most common assumption is that planetesimals resulted from
localized gravitational instabilities within a dust layer in the central plane
of the nebular disL Such a layer is assumed to form by settling of grains
through the quiescent gas due to the vertical component of solar gravity.
If the dust layer becomes sufficiently thin, and thereby sufficiently dense, it
is unstable with respect to density perturbations. This instability causes the
layer to fragment into self-gravitating clumps. These eventually collapse into
solid bodies, i.e., planetesimals. This process was described qualitatively
as early as 1949 (Wetherill 1980~. Quantitative expressions for the critical
density and wavelength were derived by Safronov (1969) and independently
by Goldreich and Ward (1973~. The critical density is approximately the
Roche density at the heliocentric distance a,
{c ~ 3M(3/27ra3,
(1)
where Me is the solar mass. Perturbations grow in amplitude if they are
smaller in size than a critical wavelength
Ac ~ 47r2Ga5/Q2,
(2)
where as is the surface density of the dust layer, G the gravitational
constant, and Q = (GM<~/a3~12 is the Kepler frequency. The characteristic
mass of a condensation is me ~ ~sAc2, and depends only on the two
parameters ~ and a. The assumption that a, equals the heavy element
content of a planet spread over a zone surrounding its orbit (Weidenschilling
1977a) implies ITS ~ fog cm-2 and me ~ lOi~g in the Earth's zone. Density
perturbations grow rapidly, on the time scale of the orbital period, but
condensations cannot collapse directly to solid bodies without first losing
angular momentum. Goldreich and Ward (1973) estimated the contraction
time to be ~ 103 years in Earth's zone, while Pechernikova and Vityazev
(1988) estimate ~ 105 - 106years.
The assumption that planetes~mals formed In this manner has in-
fluenced the choice of starting conditions for numerical simulations of
planetary accretion. Many workers (Greenberg et al. 1978; Nakagawa et al.
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1983; Horedt 1985; Spaute et al. 1985; Wetherill and Stewart 1989) have
assumed an initial swarm of roughly kilometer-sized bodies of uniform size
or with a narrow size distribution. This assumption was also due to the
lack of alternative models.
Another reason for popularity of the gravitational instability model
was the belief that it requires no mechanism for physical sticking of grains,
as was explicitly stated by Goldreich and Ward. Because the physical and
chemical properties of grains in the solar nebula are poorly characterized,
sticking mechanisms seem ad hoc. Still, there are reasons to believe that
grains in the solar nebula did experience sticking. It is common experi-
ence in the laboratory (and Earth's atmosphere) that microscopic particles
adhere on contact due to electrostatic or surface forces. Weidenschilling
(1980) argued that for typical relative velocities due to settling in the so-
lar nebula, van der Waals forces alone would allow aggregates to reach
centimeter sizes. There are additional arguments that coagulation must
have produced larger bodies, of meter size or larger, before gravitational
instability could produce planetesimals. In order to understand these argu-
ments, it is necessary to review the nature of the aerodynamic interactions
between solid bodies and gas in the solar nebula.
NEBULAdR STRUCTURE
We assume that the solar nebula is disk-shaped, has approximately
Keplerian rotation, and is in hydrostatic equilibrium" If the mass of the
disk is much less than the solar mass (< O.1M~), then its self-gravity can
be neglected compared with the vertical component of the Sun's attraction,
gz = GM`>z/a3 = Q2z, where z is the distance from the central plane. The
condition of hydrostatic equilibrium implies that the pressure at z = 0 is
Pc = Q~c/4,
(3)
where ~ is the surface density of the disk and c is the mean thermal velocity
of the gas molecules. Equation (3) is strictly true only if the temperature
is independent of z, but it is a good approximation even if the vertical
structure is adiabatic. It can be shown that
P(z) = Pcexp (-z2/H2),
where H = ~rc/2Q is the characteristic half-thickness or scale height.
(4)
There is also a radial pressure gradient in the disk. It is plausible
to assume that the temperature and density decrease with increasing he-
liocentric distance. Some accretion disk models of the nebula have ~
approximately constant, but equation (3) shows that even these will have a
strong pressure gradient, because Q is proportional to a-3/2 for Keplerian
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motion. The decrease in pressure is due primarily to the weakening of
solar gravity at larger distances. Because the gas is partially supported by
the pressure gradient in addition to the rotation of the disk, hydrostatic
equilibrium requires that its velocity be less than Keplerian;
v2 _ v2 + a UP
(5)
where p is the gas density. We define AV = Vk—Vg as the difference
between the Kepler velocity and the gas velocity. One can show (Weiden-
schilling, 1977b) that if one assumes a power law for the pressure gradient,
so that P or a-n, then the fractional deviation of the gas from Vk is
V _ nRT//1
Vk 2GMc'/a,
(6)
where R is the gas constant and ~ the molecular weight. We see that
AV/Vk is approximately the ratio of thermal and gravitational potential
energies of the gas. For most nebular models, this quantity is only a few
times 10-3, but this is enough to have a significant effect on the dynamics
of solid bodies embedded in the gas.
AERODYNAMICS OF THE SOLID BODIES IN THE NEBULA
The effects of gas on the motions of solid bodies in the solar nebula
have been described in detail by Adachi et al. (1976) and Weidenschilling
(1977b); here we summarize the most important points. The fundamental
parameter that characterizes a particle is its response time to drag,
te=mV/FD,
(7)
where m is the particle mass, V its velocity relative to the gas, and FD
is the drag force. The functional form of ED depends on the Knudsen
number (ratio of mean free path of gas molecules to particle radius) and
- Reynolds number (ratio of inertial to viscous forces). In the solar nebula,
the mean free path is typically a few centimeters, so dust particles are in
the free-molecular regime. In that case, a spherical particle of radius s,
bunk density p ,, has
te = Sp5/pC.
(~8'
The dynamical behavior of a particle depends on the ratio of te to its
orbital period, or, more precisely, to the inverse of the Kepler frequency.
A "small" particle, for which Ate << 1, is coupled to the gas, i.e., the drag
force dominates over solar gravity. It tends to move at the angular velocity
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of the gas. The residual radial component of solar gravity causes inward
radial drift at a terminal velocity given by
Vr = —2Q~ Ate -
(9)
Similarly, there is a drift velocity due to the vertical component of solar
gravity,
Vz =—Q2Zte.
(10)
For a "large" body with Ate ~ 1 gas drag is small compared with solar
gravity. Such a body pursues a Kepler orbit. Because the gas moves more
slowly, the body experiences a "headwind" of velocity /\V. The drag force
causes a gradual decay of its orbit at a rate
Vr = da/dt = - 2AV/Qte
(11)
For plausible nebular parameters, and particle densities of a few g cm-3,
the transition between '~small" and "large" regimes occurs at sizes of the
order of one meter. The peak radial velocity is equal to AV when Ate=
1. The dependence of a radial and transverse velocities on particle size are
shown for a typical case in Figure 1.
PROBLEMS WITH GRAVITATIONAL INSTABILITY
Could the gravitational instability mechanism completely eliminate the
need for particle coagulation? We first consider the case in which there is
no turbulence in the gas. The time scale for a particle to settle toward the
central plane of the nebula is in = z/Vz. From equations (8) and (10),
hi = pc/sp5Q2.
(12)
If we take p ~ 10-~°g cm-3, TX ~ (1O2JS) years, where s is in cm, so
if s = 1 ~m, ~: ~ 106 years. This is merely the e-folding time for z to
decrease. For an overall solids/gas mass ratio f = 3 x 10-3, corresponding
to the cosmic abundance of metal plus silicates, and an initial scale height
H ~ c/Q, the dust layer requires ~ 10 hi ~ 107 years to become thin
enough to be gravitationally unstable. This exceeds the probable lifetime
of a circumstellar disk. Undifferentiated meteorites and interplanetary dust
particles include sub-pm sized components. These presumably had to be
incorporated into their parent bodies not as separate grains, but as larger
aggregates, which settled more rapidly.
In addition to the problem of settling time scale, there is another
argument for growth of particles by sticking. A very slight amount of
turbulence in the gas would suffice to prevent gravitational instability.
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104
_
4, 1~ _
In
- AV
_
._
0
10-2
lo-2
Transverse
//\
/
/
/
/
_
/ —
/
me ,
/
/
X
/
-
\
1~1,//1 1 1 1 1
lo2
Particle Radius, cm
104
lot
FIGURE 1 Radial and transveme velocities relative to the surrounding gas for a spherical
body with density 1 g/cm3. Also shown is the escape velocity from the body's surface, Ve.
Numencal values are for the asteroid zone (Weidenschilling 1988) but behavior is similar
for other parts of the nebula.
Particles respond to turbulent eddies that have lifetimes longer than ~ te.
The largest eddies in a rotating system generally have timescales ~ 1/Q, so
bodies that are "small" in the dynamical sense of Ate < 1, or less than about
a meter in size, are coupled to turbulence. A particle would tend to settle
toward the central plane until systematic settling velocity is of the same
order as the turbulent velocity, Vie. From this condition, we can estimate
the turbulent velocity that allows the dust layer to reach a particular density
(Weidenschilling 1988). In order to reach a dust/gas ratio of unity
Vt ~ fQsp5/p.
(13)
Typical parameters give V' ~ (Sp5) cm see-1 when s and p5 are in cgs units,
i.e., if particles have density of order Unix, then the turbulent velocity of the
gas must be no greater than about one particle diameter per second in order
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for the space density of solids to exceed that of the gas. In order to reach
the critical density for gravitational instability, the dust density must exceed
that of the gas by about two orders of magnitude, with correspondingly
smaller Vie, ~ 10-2 particle diameters per second. For Capsized grains,
this would imply turbulent velocities of the order of a few meters per year,
which seems implausible for any nebular model.
Even if the nebula as a whole was perfectly laminar, formation of
a dense layer of particles would create turbulence. If the solids/gas ratio
exceeds unity, and the particles are strongly coupled to the gas by drag
forces, then the layer behaves as a unit, with gas and dust tending to
move at the local Kepler velocity. There is then a velocity difference of
magnitude /`V between the dust layer and the gas on either side. Goldreich
and Ward (1973) showed that density stratification in the region of shear
would not suffice to stabilize it, and this boundary layer would be turbulent.
Weidenschilling (1980) applied a similar analysis to the dust layer itself,
and showed that the shear would make it turbulent as well. Empirical
data on turbulence within boundary layers suggest that the eddy velocities
within the dust layer would be a few percent of the shear velocity Ale or
several meters per second. The preceding analysis argues that gravitational
instability would be possible only if the effective particle size were of the
order of a meter or larger. Bodies of this size must form by coagulation of
the initial population of small dust grains.
PARTICLE GROWTH BY COAGULATION
If it is assumed that particles stick upon contact, then their rate
of growth can be calculated. The rate of mass gain is proportional to the
product of the number of particles per unit volume, their masses and relative
velocities, and some collisional cross-section. We assume that the latter is
simply the geometric cross-section, frost + says (although electrostatic or
aerodynamic effects may alter this in some cases). For small particles (less
than a few tens of ~m), thermal motion dominates their relative velocities.
The mean thermal velocity is v = (3kT/m)~/2, where T is the temperature
and k is Boltzmann's constant. If we assume that all particles have the
same radius s (a reasonable approximation, as thermal coagulation tends
to produce a narrow size distribution), the number of particles per unit
volume is N = 3fp/41rp~s3. The mean particle size increases with time as
s(`t) = sO + ~15fp (kT/87rp53~12t]
(~14)
As a particle grows its thermal motion decreases, while its settling rate
increases. When settling dominates, a particle may grow by sweeping up
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smaller ones (particles of the same size, or te? have the same settling rate,
and hence do not collide). If it is much larger than its neighbors, then the
relative velocity is approximately the settling rate of the larger particle. It
grows at the rate
CR = fPVz/4ps = fQ2zs/4c,
giving
(15)
s(t) = sOexp (fQ2zt/4c).
(16)
A particle growing by this mechanism increases in size exponentially on a
time scale If = 4c/fQ2z, or ~ 4/fQ at z ~ c/Q. This time scale is a few
hundred years at a = 1 AU, and is independent of the particle density or
the gas density. The lack of dependence on particle density is due to the
fact that a denser particle settles faster, but has a smaller cross-section,
and can sweep up fewer grains; the two effects exactly cancel one another.
Likewise, if the gas density is increased, the settling velocity decreases,
but the number of accretable particles increases, provided f is constant.
The time scale increases with heliocentric distance; Tg OC a3/2. A particle
settling vertically from an initial height zO and sweeping up all grains that
it encounters can grow to a size climax) = fpzO/47r~, typically a few cm for zO
~ c/Q. Actually, vertical settling is accompanied by radial drift, so growth
to larger sizes is possible.
The growth rates and settling rates mentioned above have been used
(with considerable elaboration) to construct numerical models of particle
evolution in a laminar nebula (Weidenschilling 1980; Nakagawa et al. 1981~.
The disk is divided into a series of discrete levels. In each level the rate of
collisions between particles of different sizes is evaluated, and the changes
in the size distribution during a timestep At is computed. Then particles
are distributed to the next lower level at rates proportional to their settling
velocities. A typical result of those simulations shows particle growth
dominated by differential settling. Because the growth rate increases with
z, large particles form in the higher levels first, and "rain out" toward
the central plane through the lower levels. The size distribution remains
broad, with the largest particles much larger than the mean size, justifying
the assumptions used in deriving equation (16~. After ~ 10 fig, typically a
few thousand orbital periods, the largest bodies exceed one meter in size
and the solids/gas ratio in the central plane exceeds unity. The settling
is non-homologous, with a thin dense layer of large bodies containing ~
1-10% of the total surface density of solids, and the rest in the form of
small aggregates distributed through the thickness of the disk.
The further evolution of such a model population has not yet been
calculated. The main difficulty is the change in the nature of the interaction
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between particles and gas when the solids/gas ratio exceeds unity. As
mentioned previously, the particle layer begins to drag the gas with it. The
relative velocities due to drag no longer depend only on te, as in equations
(7-11), but on the local concentration of solids: there is also shear between
different levels. Work is presently under way to account for these effects,
at least approximately. A complete treatment may require the use of large
computers using computational fluid dynamics codes.
PROPERTIES OF FRACTAL AGGREGATES
The earlier numerical simulations mentioned above assumed that all
particles are spherical and have the same density, regardless of size. This is a
good assumption for coagulation of the liquid drops, but it does not apply to
solid particles. When two grains stick together, they retain their identities,
and the combined particle is not spherical. Aggregates containing many
grains have porous, fluffy structures. It has been shown (Mandelbrot 1982;
Meakin 1984) that such aggregates have fractal structures. A characteristic
of fractal aggregates is that the average number of particles found within a
distance s of any arbitrary point inside the aggregate varies as nisi or sD,
where D is the fractal dimension. The density varies with size according
to the relation p oc sD-3. "Normal" objects have D = 3 and uniform
density. Aggregates of particles generally have D ~ 2, so that their density
varies approximately inversely with size, i.e., they become more porous as
they grow larger. For the most simple aggregation processes (Jullien and
Botet 1986; Meakin 1988a,b) the fractal dimension lies in the range 1.7
< D < 2.2, but more complex mechanisms can lead to values of D lying
outside of this range. A hierarchical ballistic accretion in which clusters
of similar size stick at their point of contact yields D < 2. Building up a
cluster of successive accretion of single grains or small groups, or allowing
compaction after contact, leads to D > 2. An example of this type is shown
in Figure 2. This is a computer-generated model, but it corresponds closely
to soot particles observed in the laboratory (Meakin and Donn 1988~.
The structure of aggregates, which are very unlike uniform-density
spheres, affects their aerodynamic behavior in the solar nebula. For a
compact sphere, the response time is proportional to the size (equation
8~. For a fractal aggregate, te increases much more slowly with size.
Meakin has developed computer modeling procedures to determine the
mean projected area of an aggregate, as viewed from a randomly selected
direction (Meakin and Donn 1988; Meakin, unpublished). The variation of
the projected area with the number of grains in the aggregate depends on
the fractal dimension. If D < 2, aggregates become more open in structure
at larger sizes, and are asymptotically "transparent," i.e., the ratio of mass
to projected area never exceeds a certain limit. For D > 2, large aggregates
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e
BALLISTIC Cl-CI-3d ~
ONE STAGE RESTRUCTURING-
S =11186
~q
94 DIAMETERS
-
FIGURE 2 View of a computer-generated aggregate of fractal dimension ~ 2.11 and
containing ~ 104 individual grains.
are opaque; the mass/area ratio increases without limit, although more
slowly than for a uniform density object.
We assume that in the free molecular regime, when aggregates are
smaller than the mean free path of a gas molecule (> cm in typical nebular
models), te is proportional to the mass per unit projected area (m/A). From
the case of a spherical particle in this regime, as in equation (8), we infer
te = 3(m/A)/4pc.
(~17)
We can express te for aggregates conveniently in terms of the value
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-
a)
. _
a'
{
Q
Q
a)
rid
. _
a)
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-
-
-
D= 1.95
10 lo2 103 104
n ~ Number of grains per aggregate
. :T
105 lo6
FIGURE 3 Mass/area ratio vs. number of grains for aggregates of different dimensions.
D = 3 assumes coagulation produces a spherical body with the density of the separate
components (liquid drop coalescence). For D ~ 2, m/A or m1 2/D, or or ml/3 for D =
3, and oc m0 052 for D = Z11. For D < 2, m/A approaches an asymptotic value, shown
by the arrow.
for an individual grain, of size so and density pO, teo = sOpO/pc. E;its to
Meakin's data give
te = teO(0~343n 0 052 + 0 684n—0.262
(18)
for D = 2.11, where n is the number of grains in the aggregate. This relation
is plotted in Figure 3. For large aggregates of 106 grains, the mass/area
ratio and te are ~ 20-30 times smaller than for a compact particle of equal
mass.
Fractal properties of aggregates also have implications for the opacity
of the solar nebula. The dominant source of opacity is solid grains (Pollack
et al. 1985~. The usual assumption in computing opacity is that particles are
spherical (Weidenschilling 1984~. Their shapes are unimportant if they are
smaller than the wavelength considered (Rayleigh limit). However, when
sizes are comparable to the wavelength, the use of Mie scattering theory
for spherical particles is inappropriate. In the limit of geometrical optics
when particles are large compared with the wavelength, the opacity varies
inversely with the mass/area ratio. From Figure 3 we see that if aggregates
of 106 grains are in the geometrical optics regime, then the opacity is ~
20-30 times greater than for compact spherical particles of equal mass.
Actually, if individual grains are below the Rayleigh limit, aggregates may
not be in the geometrical optics regime, even if they are larger than the
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wavelength. The optical properties of fractal aggregates need further study,
but it is apparent that coagulation of grains is less effective for lowering
the nebula's opacity than has been generally assumed.
COAGULATION AND SETTLING OF FRACTAL AGGREGATES
We have modeled numerically the evolution of a population of particles
in the solar nebula with fractal dimension of 2.11, using the response time
of equation (18~. The modeling program is based on that of Weidenschilling
(1980~. The calculations assumed a laminar nebula with surface density of
gas 3 x 103g cm~2, surface density of solids 10 g cm-2, and temperature
of 500K at a heliocentric distance of 1 AU. At t = 0 the dust was in the form
of individual grains of diameter lam, uniformly mixed with the gas. With
the assumption that coagulation produced spherical particles of dimension
D = 3, or constant density (the actual value is unimportant; compare the
discussion of equation (16~), `'raining out" with growth of approximately
10-meter bodies in the central plane occurs in a few times 103 years.
For the case of fractal aggregates with D = 2.11, we assumed that
an individual M-sized grain (so = 0.5 Em) has a density p0 = 3 g cm-3.
Aggregates of such grains have densities that decrease with size according
to
pa = Po(s/SO)
(19)
until s = 0.5 mm, at which size p' ~ 0.01 g cm-3 (densities of this order
are achieved by some aggregates under terrestrial conditions; Donn and
Meakin 1988~. The density is assumed constant at this value until s = 1 cm,
and then increases approximately as s2 to a final density of 2 g cm-3 for s
~ 10 cm. This variation is arbitrary, but reflects the plausible assumption
that fractal structure eventually gives way to uniform density for sufficiently
large bodies due to collisional compaction. In a laminar nebula, relative
velocities may be low enough to allow fractal structure at larger sizes than
assumed here, so this assumption may be conservative. Experimental data
on the mechanical behavior of fractal aggregates are sorely needed.
Even the limited range of fractal behavior assumed here has a strong
effect on the evolution of the particles in the nebula. Growth and settling are
slowed greatly. After a model time of 2 x 104 years, the largest aggregates
are ~ one millimeter in size. The highest levels of the disk, above one scale
height, are slightly depleted in solids due to the assumption that the gas
is laminar. The largest aggregates at this time have settling velocities ~ 1
cm see-i, so there would be no concentration toward the central plane if
turbulence in the gas exceeded this value. Continuation of this calculation
results in more rapid growth by differential settling beginning at about 2.5
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1
lol2
H
1~
1
1
At =0
a = I A U
t= 3 x 104y
D= 2.11
10-4 10-3 lo-2
Solids / Gas
10-1 1 10
FIGURE 4 Outcome of a numerical simulation of coagulation and settling in the solar
nebula at a = 1 AU. Aggregates are assumed to have fractal dimension 2.11 at sizes <
0.1 cm. AT t = 0, dust/gas ratio is uniform at 0.0034, corresponding to cosmic abundance
of metal and silicates. At t = 3 X 104y, dust/gas exceeds unity in a narrow region near
the central plane (expanded scale at right).
X 104 years. By 3 x 104 years there is a high concentration of solids in
a narrow zone at the central plane of the disk (see Figure 4~. This layer
contains approximately 1% of the total mass of solids, in the form of bodies
several meters in diameter.
Evidently, the fractal nature of small aggregates greatly prolongs the
stage of well-mixed gas and dust, before "rainout" to the central plane.
The reason for this behavior is subtle. If we assume that the density varies
as equation (19), then a generalization of the thermal coagulation growth
rate of equation (14) gives
site = sO + (,(3D/2 - 2~3fp~kT/2~p3~il2s3(D-3)l2t~ i/(3D/2-2)
(20)
For D = 2.11, this gives s or t0 86, vs. s or t0 4 for D = 3. Thus, particle
sizes increase more rapidly for thermal coagulation of fractal aggregates,
due to their larger collisional cross-sections. For aggregates with D > 2,
the behavior of te at large sizes is to ~ teo(S/So)D-2. Using this relation
~09
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in equation (15), D appears in both the numerator and denominator,
leaving the growth rate unchanged. If thermal coagulation is faster for
fractal aggregates and the growth rate due to settling is independent of D,
then why is the evolution of the particle population slower? The answer
is that the derivation of equation (15) assumes that the mass available
to the larger aggregate is in much smaller particles, so that the relative
velocity is essentially equal to the larger body's settling rate. Thermal
coagulation tends to deplete the smallest particles most rapidly, creating
a narrowly peaked size distribution, and rendering differential settling
ineffective. Inserting the parameters used in our simulation into equation
(20) predicts s ~ 0.1 cm at t = 2 x 1()4 years, in good agreement with the
numerical results.
The qualitative behavior of this simulation, a long period of slow
coagulation followed by rapid growth and "rainout," is to some degree an
artifact of our assumption for the variation of particle density with size. A
transition from fractal behavior to compact bodies without abrupt changes
in slope would probably yield a more gradual onset of settling. It is likely
that fractal structure could persist to sizes larger than the one millimeter
we have assumed here, with correspondingly longer evolution time scales.
We have not yet modeled the evolution of the particle population after
the solids/gas ratio exceeds unity in the central plane and can only speculate
about possible outcomes. At the end of our simulations most of the mass in
this level is in bodies approximately 10 meters in size. These should undergo
further collisional growth, and in the absence of turbulence will settle into
an extremely thin layer. Because these bodies are large enough to be
nearly decoupled from the gas, it is conceivable that gravitational instability
could occur in this layer. However, the surface density represented by these
bodies is small, approximately 1% of the total surface density of solids.
The critical wavelength, as in equation (2), is correspondingly smaller. If
they grow to sizes >0.1 km before the critical density is reached, then their
mean spacing is ~ Ac, and gravitational instability is bypassed completely.
In any case, the first-formed planetesimals would continue to accrete the
smaller bodies that rain down to the central plane over a much longer time.
CONCLUSIONS
The settling of small particles to the central plane of the solar nebula is
very sensitive to the presence of turbulence in the gaseous disk. It appears
that a particle layer sufficiently dense to become gravitationally unstable
cannot form unless the particles are large enough to decouple from the gas,
i.e. ~ meter-sized. Thus, the simple model of formation of planetesimals
directly from dust grains is not realistic; there must be an intermediate
stage of particle coagulation into macroscopic aggregate bodies.
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96
PLANETARY SCIENCES
Coagulation of grains during settling alters the nature of the settling
process. The central layer of particles forms by the "raining out" of larger
aggregates that contain only a fraction of the total mass of solids. The
surface density of this layer varies with time in a manner that depends
on the nebular structure and particle properties. Thus, if planetesimals
form by gravitational instability of the particle layer, the scale of insta-
bilities and masses of planetes~mals are not simply related to the total
surface density of the nebula. It is possible that coll~sional coagulation due
to drag-induced differential motions may be sufficiently rapid to prevent
gravitational ~nstabili~ from occurring.
It is probable that dust aggregates in the solar nebula were low-densi~
fractal structures. The time scale for settling to the central plane may have
been one or two orders of magnitude greater than estimates which assumed
compact particles. The inefficiency of settling by "raining out" suggest that
a significant fraction of solids remained suspended in the form of small
particles until the gas was dispersed; the solar nebula probably remained
highly opaque. The mass of the nebula may have been greater than the
value represented by the present masses of the planets.
ACKNOWLEDGEMENTS
Research by S.J. Weidenschilling was supported by NASA Contract
NASW-4305. The Planetary Science Institute Is a division of Science
Applications International Corporation.
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Representative terms from entire chapter:
central plane
