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Planet Formation: The Early Years
Basic stages in nearly all models for planet formation. We are concerned
with whether we can get to the second phase (planetesimal scattering
collisional growth) from the first (small particle growth by coalescence)
Last time we saw that equation of motion for particle in the “Epstein
regime” will be:
dv/dt = mp K2 z - (4/3)  a2  vp
= mp K2 z - vp / fr
where fr = as/vth = “friction” or “stopping” time. But notice that the
friction term depends on mass/area for object, could be strange,
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Particle velocities
Even if the gas is in hydrostatic equilibrium and non-turbulent, it is
still necessary to treat the particle motions in detail, because they
always have some velocity relative to the gas (they feel no pressure
gradient, yet they have inertia and feel drag force when trying to
move relative to the gas), leading to a complex series of vertical and
radial motions that can lead to eventual loss of all the particles to the
central star. More optimistically, the relative motion of the grains
leads to collisions and, if the relative velocities are not so large that
they shatter, the collisions lead to coalescence, producing larger and
larger particles. We postpone the details for later, and just write the
equation for the motion of the grains that must be solved numerically
except in a few cases. The equation is:
dvp/dt = - (vp - u)/taufr - GMz/r2(x2 + r 2)1/2
For our thin disk approximation, which is good until you get out to ~
100 AU, recall that we can write the gravitational force term as K2 z.
Here taufr is a drag or friction timescale which we’ll explain
shortly. This innocent-looking equation can lead to a surprising variety
of collective motions, so generally must be treated carefully. Even
without the gravitational term, complex gas motions (e.g. a turbulent
u) makes this an interesting and difficult problem. We ignore such
complex gas motions for now and assume a laminar gas disk.
Drag forces:
The equation of motion for a particle in a fluid or gas is:
dvp/dt = - (vp - u)/taufr + external forces
where I write u for the fluid or gas velocity and it is always convenient
to write a coefficient like the drag coefficient in terms of a timescale,
taufr here, so it is transparent how the left and right sides have the
same units. I write “external forces” to be general, but in disks it is
mainly gravity that is pulling the grains toward the midplane in the
vertical direction. We’ll discuss the radial direction later.
Here taufr (fr means “friction” and we call this the friction time)
can be easily derived by solving the above equation with u=0, since it
just gives exponential decay of vp with e-folding time
taufr = as/  vth
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and vth is the rms thermal velocity of the gas at temperature T, say vth
= (kT/ mH)1/2 and the mass density  is given in terms of the betterto-memorize number denstiy n by  =  mH n.
Why does vth enter into this? Because we are assuming that the particles’
motion through the gas is slow compared to the thermal speed of the gas
molecules, so most of the momentum transfer is from atoms and molecules
moving at thermal speeds. You can imagine a grain that is moving
supersonically through a gas where this would not be satisfied.
The formula above for taufr assumes “Epstein drag”, which requires that
the mean free path of the gas molecules (1/(n)) is much larger than the size of
the particle. This condition is the same as
n ≪ a.
Recalling the useful fact that except in a fully ionized gas atomic-molecular-ionic
cross sections are all going to be around 10-16 to 10-15 (larger for heavier
molecules, like Earth’s atmosphere), gives the condition
n ≪ 1011 a cm-3
where a means a in units of 1 micron.
If this condition doesn’t hold, you have to use “Stokes drag”, which occurs
in very dense environments like planetary atmosphere, but also in the inner parts
of disks. The Stokes drag friction timescale depends on the square of the particle
size: (will fill in next time)
One can easily calculate terminal settling velocity (asuming that gravity is the
force that is pulling them down) and find that the heaviest particles settle fastest:
This “catching up” leads to particle collisions at the differential settling speed
between two grains (depends on their size, but usually settling speed of the
larger), so particles can grow on their “way down” if the collisions aren’t too
speedy, leading to fragmentation.
If most of the particles have a size a, how long will it take to get most of
the particles to the midplane?
Assuming Epstein drag (not true for a planetary atmsophere, and in the
inner parts of preplanetary disks), take the MMSN model (or some flatter model)
and calculate the friction time at the midplane as a function of r and z, including
the size dependence as a.
Fool around with what you have so far: What is the friction time as a
function of r in the midplane? If it is << an orbital period, then you expect the
particles to be very well-coupled to the gas, and so to have no “drifts”. Think
about how this condition starts to break down as you get to larger sizes.
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The ratio of taufr to some dynamical timescale in the fluid is called the
Stokes number of the particles, and is the fundamental thing to understand if you
are to understand planet formation. We can always write a fluid timescale as
some length L divided by some velocity u, so taufluid is L/u, but we have to be
careful how to pick L and u. For the disk we might pick L=2r and u=Keplerian
speed (do you know what that is? Find out) so that the characteristic timescale is
the orbital period P which astrophysicists like to write as a frequency K where K
stands for Keplerian. Using this as our comparison timescale gives a St number
that looks something lke:
Now think about this one: How long does it take for a grain starting at some
altitude z to fall to the midplane? (Assume constant density for starters.)
Calculate the difference in settling velocity between particles of two sizes. If they
collide, what will their relative velocity be? Are they likely to fragment during this
collision?
Vertical motion: results
The grains reach terminal velocity (d/dt = 0) rapidly, on timescale of friction
time:
taufr = as/vth ≈ 10 sec (a/1m) (n/1014)-1 rAU
In that case we can solve for the vertical and radial components of the drift
between the dust and gas:
In z-direction:
Vsettle = taufr K2 z
This is correct for either Epstein or Stokes drag once the appropriate expression
for taufr is used. With the gas in vertical hydrostatic equilibrium this shows several
things:
 The particles slow down as they approach the midplane
 The largest particles fall the fastest (because taufr  a s / vth,  particle size a
divided by the gas density )
 With the Keplerian frequency equal to (GM/r3)1/2, and a friction time that
decreases with distance from the star (because  vth increases with increasing r),
the dependence on distance from the star is weak. In fact if taufr = a s / vth, the
vertical settling goes like
Vz = fr K2 z  1/(vthr3) z
Our flatter disks had  ~ r-2, so there is some decrease to larger distances.
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Evaluating the expression we find
Vsettle = 0.1 cm/sec (a/1m)(z/H)
The associated timescale is
z/v(settle) = (K2 fr)-1 ~ 2x105 yr (a/1m)-1
A real integration of the equations of motion from 10H to 10-3H gives a number
about 10 times larger.
2. For disks, we haven’t been distinguishing between the particles
(what I will sometimes call grains, dust, aerosols, anything not in the
gas phase) and the gas. The sketch below emphasizes the thermal
structures of the gas and dust disks. Note that the grains are
assumed to have settled or sedimented to a particle subdisk.
Remember the equation of motion for a particle in a fluid or gas:
dvp/dt = - (vp - u)/taufr + external forces
where I write u for the fluid or gas velocity and it is always convenient
to write a coefficient like the drag coefficient in terms of a timescale,
so it is transparent how the left and right sides have the same units. I
write “external forces” to be general, but in disks it is mainly gravity
that is pulling the grains toward the midplane in the vertical direction.
We’ll discuss the radial direction later.
Here taufr (fr means “friction” and we call this the friction time)
can be easily derived by solving the above equation with u=0, since it
just gives exponential decay of vp with e-folding time
taufr = as/  vth
and vth is the rms thermal velocity of the gas at temperature T, say vth
= (kT/ mH)1/2 and the mass density  is given in terms of the betterto-memorize number denstiy n by  =  mH n.
Why does vth enter into this? Because we are assuming that the
particles’ motion through the gas is slow compared to the thermal
speed of the gas molecules, so most of the momentum transfer is from
atoms and molecules moving at thermal speeds. You can imagine a
grain that is moving supersonically through a gas where this would not
be satisfied.
The formula above for taufr assumes “Epstein drag”, which
requires that the mean free path of the gas molecules (1/(n)) is much
larger than the size of the particle. This condition is the same as
n ≪ a.
Recalling the useful fact that except in a fully ionized gas atomicmolecular-ionic cross sections are all going to be around 10-16 to 10-15
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(larger for heavier molecules, like Earth’s atmosphere), gives the
condition
n ≪ 1011 a
where a means a in units of 1 micron. [? If n ~ 1014 cm-3 at 1/AU,
this says Stokes rules until a ~ 1000 micron = 0.1m! Some mistake.]
If this condition doesn’t hold, you have to use “Stokes drag”,
which occurs in very dense environments like planetary atmosphere,
but also in the innermost parts of disks. The Stokes drag friction
timescale depends on the square of the particle size (see below).
. For now the important thing to know (because it is of more
general importance than the disks or brown dwarfs or planetary
atmospheres we are discussing), is that he friction timescale turns out
to be independent of the particle velocity, and is given by
taufrE = as/gas vth
for a < g (“Epstein regime”)
or
taufrS = 2a2s/(3g vth ) for a > g (“Stokes regime”).
In the last expression I have substituted the mfp version of the
expression for the kinematic viscosity ~ g vth (good to a factor of
better than two, an example of how a booklength worth of badelooking derivation [Chapman and Cowling 1933] does not always gain
you much), and  is the mass density of fluids, and I will omit the “g”
subscript from now on and write Zp  for p, i.e. the mass density of
particles with particle-fluid mass fraction Zp.
If you plug in some numerical values (in the Epstein result--see
below), you can convince yourself that the timescale for any initial
particle-fluid drift to decay by drag force is much less than either the
Keplerian time or the timescale of collisions between particles,
taucoll,p = (np a2 vp)
if you assume Brownian motion for the grains’ relative velocity
vB = vth (mH/mp)1/2.
Plugging in g (which comes from the fluid viscosity) as 1/(nmol) gives
taufrS = (2mol/3mH)(a2 s/vth) for a > g (“Stokes regime”)
Take note of the size-squared dependence on size and the lack of
dependence on fluid density.
The ratio of the Stokes to Epstein drag expressions is
(2/3)molna, which is larger than unity (of course this is approximately
the same as the condition that the mfp be < a) when
n > 7.5 x 1018 (mol/2x10-15)-1 am cm-3.
Using the n(r) given above for this specific MMSN model, with
normalization factor A, shows that Stokes friction only becomes more
important than Epstein for
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r < 0.03A4/11 am-4./11 AU.
It is hard to imagine that most disks are massive enough (A ~ 100
say) that this limit would suggest the importance of Stokes friction for
any reasonable planet formation scenario. Notice that 0.03AU is closer
than the silicate sublimation radius, so there wouldn’t be any solids
there anyway (except for some Fe maybe). Although there are giant
exoplanets this close to their parent stars, they presumably weren’t
born there (scattering from eccentric orbits, migration).
For your interest, most people who simulate planet formation take
their inner boundary condition (no more building blocks inside this
radius) at 0.4 AU, partly for computational reasons (the orbital time
varies with a3/2), but also (I would say mostly--I know that this was
only a computational problem 10 years ago) because they know they
would produce an embarasslingly massive Mercury due to the very
large column density at small distances from the star.
Ballistic Particle-Cluster Agglomeration and
Ballistic Cluster-Cluster Agglomeration
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simulations using monodisperse (equal-size) spheres.
An independent example of the two kinds of growth processes.
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We shouldn’t forget that other kinds of structures are entirely possible. This
is a close-up of a porous hydrated muco-gel.
This is a fibrous protein superhelix structure in collagen, but similar
nonbiological growth structures are found.
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Carbon Soot Superaggregate
(Dhoub...2006)
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Brownian motion. This represents what paths of smallest particles should be
like in earliest growth of grains to planetesimals.
Sample aggregates from a simulation of cluster-cluster eoalescence in which
velocities are due only to Brownian motion. Particles are labelled by fractal
dimension Dm~ 1.8 to 2.0 (mass ~ size Dm; not m~a3 for a compact object of
any shape)
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Experimentally-produced cluster-cluster aggregates (CCAs) produced by
collisions caused by Brownian motion (a), differential sedimentation (b), and
turbulence (c). Structures are similar, with fractal dimension Dm~ 1.8 to 2.0
(mass ~ size Dm).
Another reminder that the structures that grow in “real” particles could be
different. This is called a “polyampholyte chain necklace region.”
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Above some velocity, some of the collision KE should be able to go into
“compaction” of the fluffy fractal particle, after which its mass/area (drag)
will start increasing. This begins the vertical settling phase, as particles have
large enough m/A to fall rapidly through the disk gas.
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Experiment showing collisional breakup of two aggregates when collision
speed is too large (3.8 m/s here). Obviously there has to be some upper limit
to collision speed above which coalescence will not occur: fragmentation
limit.
Table below shows critical energies and velocities for (1) first restructuring,
(2) maximum compaction, (3) onset of loss of monomers, (4) onset of
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fragmentation, according to a model (Domink and Tielens 1997) and
experiments (CCAs shot into a wall).
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2. How to Follow the Evolving Size Distribution of Particles: The
Coalescence Equation
Let’s denote the relative velocity between particles of mass m1 and m2
as
Δvpp(m’, m; ug,...)
with the understanding that Δvpp could depend on more than the
relative masses (I indicated the local turbulent or other gas velocity ug
for example.) All the physics and crucial considerations are in the
calculation of this relative velocity, using the eqn. we discussed
above for the motion of each individual particle. For now let’s
assume this has been done and see how we would use them: The
goal is to calculate the size or mass distribution of particles, which
gives the relative number of particles in the size (mass) interval (a,
a+da) [(m, m+dm)] per unit size (mass). Technically it is a smoothed
probability density function (pdf), but it is easiest to just consider it a
histogram, with the height of each bar proportional to the fraction of
particles at that size (mass). You might get comfortable with pdfs and
their evolution by thinking about the pdf of human heights (weights).
The relative velocities, and the relative number of particles at
different masses, determine the collision rate and what we will define
below as the collision kernal. These can, and must, be used to solve
for the evolution of the mass distribution. The equation for this
evolution looks difficult, but is easy to understand, and comes up in
every field of physics and astronomy. Here it is called the
“coalescence equation”, and has to be solved at each (r, , z) in the
disk as a function of size (mass). This equation was first used to
calculate the rate at which corn crops will grow, and is also the
equation that planetary scientists and meteorologists solve for the
distribution of sizes of raindrops. Various equations encountered in
stellar dynamics are simplified versions of such an equation (known
more generally as the Chapmann-Kolmogorov equation); for example
the Fokker-Planck equation is what you get if the changes during
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encounters between entities is very small, so you can Taylor series
expand the awful-looking integrand. Before this step, the equation
would be called the Boltzmann equation. The best, and clearest,
treatment of this equation, for objects whose mass distribution is the
target, in the astrophysical literature is Silk and Takahashi 1979 ApJ).
∂ f (m)/∂t |coal =
∫ 0m f (m’ ) f (m − m’ )σcoll(m’, m − m’ ) Δvpp(m’, m-m’)dm’
− f (m ) ∫0∞ f (m’) σcoll(m’, m) Δvpp(m’, m)dm ,
(2)
This is the continuous form of a famous equation called the
Smoluchowski equation (Smoluchowski 1916), who wrote it with
summations instead of integrals. (There is endless argument about
whether the discrete version with sums is more accurate or
introduces artifacts.)
The first term on the right hand side represents the gain of
particles in the mass bin m by coagulation of two grains of mass m’
and m − m’ :
m’ + (m − m’ )
Notice the 2nd particle has to have a mass m-m’ so that the two
colliding masses add up to m’ + (m-m’) = m.
The quantity σcoll(m’, m − m’ ) is the collision cross section for
coalescence between the two particles. The geometric cross section
is (a’+ a)2 where a  m1/3 is the size of a particle. Δvpp (m’, m-m’) is
the average relative velocity between two particles of mass (m’, mm’). For example Δv could be due to Brownian motion, in which case
Δvpp(Brownian) = [8kT(m1 + m2) / (m1m2)]1/2
So the smallest grains move the fastest, and will undergo the most
collisions, if they mostly collide with larger particles. (In this case the
collision rate σΔv is proportional to m2/3m-1/2). Unfortunately there are
analytic solutions to this equation only for a few cases and this isn’t
one of them, so it has to be solved numerically or by simulation.
Experimentally, and in simulations of particle growth, it is known that
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Brownian motion-driven coagulation leads to a fairly monodisperse
size distribution of fluffy fractal-like particles with dimension~2. This
means m/A stays roughly constant while they grow, instead of
increasing (like a for a compact grain), so taufr stays constant.
Notice that in general the probability or rate of coalescence
must depend on the velocities of the two particles, since at large
enough relative speeds the collision could lead to fragmentation or a
bounce-off with some energy dissipation but no mass change, or
many other outcomes. So really we should be solving for f(x, v, m; t),
and include the velocity distribution of the particles as part of the
problem, including another integral over the velocity distribution of the
colliding particles--this makes the problem too difficult, worse than the
Boltzmann equation, so it is customary to neglect it, or to include it
only in Monte Carlo simulations.
The second term represents the loss of particles in the mass
bin m by coagulation of a particle of mass m with a particle of mass
m’. Notice that every collision of an “m” particle with a particle of any
other mass takes m out of mass bin m, if you assume all collisions
lead to coalescence. That’s why we can take f(m) out of the integral
and integrate over all masses.
The relative velocity Δv(m1,m2) denotes the average relative
velocity between two particles. This average velocity may consist of
random motions
but also of systematic drift between particles of different mass. That
is what we will spend our time estimating, because often you can tell
how fast the system will evolve, and in what way, just by knowing the
relative velocities that control the collision rates.
The combination K(m1, m2) ≡ σcoll(m1, m2)Δv(m1, m2) is called
the kernel of the coagulation equation. You might be interested in
thinking about the situation in which the collision kernal increases
with increasing mass, which leads to a runaway growth (formally, the
growth of a single huge particle in a finite time). This property is
popular in recent yers as models for black holes.
It is usual to try to include three processes leading to a Δv(m`,
m2): Brownian motion, differential settling or radial drift, and
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turbulence.
The relative velocities induced by turbulence is very difficult to
calculate, but the other relative velocities are straightforward. And in
most work it is assumed that the cross section is the geometric cross
section with all collisions leading to coalescence (even though papers
treating details of non-coalescing equations were studied in the
1970s, and have been studied in other problems, like the
coalescence model for interstellar cloud growth.
Because the relative velocity for Brownian motion is largest for
collisions in which at least one collision partner has a low mass, at
first the lowest-mass grains grow by Brownian coalescence. Once
these lowest mass particles get depleted, the next higher mass
particles (which are aggregates
of the smallest particles) will coagulate, etc. Brownian motion
will therefore lead to a narrow size distribution that slowly
moves to larger sizes. This hierarchical growth procedure leads
to aggregates with a fractal structure: so-called cluster-cluster
aggregates (CCA). If you visualize these growing clusters of particles
colliding randomly with each other, you might imagine how the
resulting “aggregates” will have a very irregular shape. The figure
below shows samples of simulations of random cluster-cluster
collisional growth. The average fractal dimension of these clusters is
about 1.95, meaning that the mass of an aggregate scales with its
size to the ~ 2 power (not 3 as would be the case for a “compact”
particle, since in that case mass  a3).
We have left out some important effects from the equation,
having to do with the spatial distribution of the grains of different
sizes, and how they could be transported either through some
relatively simple vertical (or azimuthal, or radial) drifts relative to the
gas, or by disk turbulence.
The usual way of including the settling of particles of different
sizes under vertical gravity is to use an advection term
∂[n(m)vZ(m)]/∂z  losing m through sedimentation of particles
=
where now we have to consider n(m) as a function of altitude above
the midplane z. Noting that n(m)vZ(m) is the flux of particles of mass
m in the z direction, we see that the derivative is the equivalent of the
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divergence of a flux, so it gives the rate at which particles are leaving
the (m,z) part of phase space by moving to another altitude.
We can use this kind of reasoning to include other processes,
in other kinds of coordinates. For example, the particles could be
accreting gas that they pass through at a rate dm/dt ≡ mdot, which
will be a function of the particle area, the flux of incoming gas that can
be accreted, and the density of the gas through which it is moving:
mdot ≈  a2 vth where  is usually called an “accomodation
coefficient,” which means a fudge factor to account for our lack of
knowledge about the probability that a molecule that hits the particle
will stick, usually taken about unity. [You should check the units in
that equation and be able to derive it yourself.] In this case there are
particles moving out of the (m, m+dm) interval continuously at a rate
given by the “divergence” of the rate at which particles are being
accreted into and out of this bin. The particle could be an iceball
wandering on an eccentric orbit inside the snow line (think of a
comet), in which case instead of accretion, sublimation might be
taking place at some rate mdot (which would depend on the vapor
pressure and other things--we haven’t discussed this process yet).
Any such continous process involving mass could be represented by
a contribution to ∂n(m)/∂t that is
∂n(m)/∂t|acc = ∂[n(m)mdot(m)]∂m.
This is completely analogous to the vertical settling term: consider
that z is the coordinate (like m) and the settling velocity vz is the
derivative of this coordinate, “zdot.”
We could do the same for any property of the particles for
which we can write an expression for its time derivative. For practice,
write down the equation for the distribution of heights of humans,
given a reasonalbe function for the growth rate as a function of size
(actually age). You should find it instructive to find that this kind of
growth rate gives a pileup at large sizes (because they grow slowest),
i.e. most humans are about the same height.
Returning to the gas velocity field advecting the particles as a
function of their mass, the other main gas velocity field is turbulence,
if the disk is turbulence. We can write this term down exactly:
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∂n(m)/∂t|turb = [n(m)ug(x,t)].
Unfortunately the velocity field ug(x,t) is a wildly complicated
stochastic field of strains (∂ui/∂xj) that cannot be specified easily, and
if it could, we would not be able to solve the coalescence equation
including it (the “turbulence problem” in the transport of any tracer
particle, e.g. in photochemistry of planetary atmospheres.
The traditional but definitely inadequate way of dealing with
turbulence is to write it as if it were a diffusion process with a certain
mean free path, and hence “diffusion coeffient” which is some
characteristic velocity times some characteristic length. You may
have encountered this in the theory of stellar interiors, where we hide
our ignorance of turbulent convection with the “mixing length” theory.
Yet for some questions, like the “puffing up” of the dust subdisk as it
tries to settle to the midplane, it gives us at least a rough way of
including the process. The turbulent diffusion term is included as the
first term on the right hand side below, with the second term being the
vertical settling term:
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
(3)
where K(m1, m2) ≡ σcoll(m1, m2)Δv(m1, m2) is again the collision kernal.
Numerical solutions to this equation are extremely difficult.
Consider that we want to cover a mass range from ~ 10-12 g (1
micron) to (wishful thinking) 1015 g (1 km). A good discussion of the
tricky aspects are in Ormel et al. 2007 A&A and Dullemond and
Dominik 2005 ApJ, who also show solutions.
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=
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Blum’s (2004) approximate but instructive solution.
In a 2004 review paper, Blum showed the kind of evolution to expect
by simplifying the size distribution equation in various ways, and then
solving for each “phase” of evolution that he imagined. Even though
he leaves out the veritcal fall to the midplane and the accompanying
growth of the subdisk, his results are similar to more modern work.
Blum’s result;
Results of Ormel, Spaans, and Tielens (2007 Astr.& Astrophys.)
Porosity of particle is evolved along with the mass of the aggregate,
starting at a size of 0.1 m. Particles grow with m ~ A with  = 0.95
until the velocities get so large that the collisions lead to
“compaction.”
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Critical velocities for compaction and fragmentation according to
Ormel et al. (2007). As coalescence proceeds, particles move from
right to left, with Brownian coagulation dominating at first (because
smallest).
After ~ 100 yr the particles are large enough for turbulent velocities to
take over, leading to compation as they grow toward the upper right.
Eventually they become so heavy that they rain out and fall to the
midplane, and are taken out of the computation box.
The evolution of the size distribution for several cases is shown
below.
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Evolution of the size distribution.
Top: compact particles; Bottom: porous particles.
Dotted line  first compaction event
Dashed line  rainout begins
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Effect of decreasing turbulence on the size distribution
Note that with very weak turbulence (right), the particles rain out
without compacting, and the “rain” consists of very small particles all
of about the same size. With strong turbulence (left), the particles
undergo growth to 1-10 cm before rainout.
Problems:
1.Micron-size grains disappear in 100-1000 yr. But there is strong
evidence from shape of IR solid state spectral features (mostly
silicate feature at 10 microns) that there are small (~ micron) grains in
most disks.
2. Largest rainout grains are about 0.1-10 cm, depending on level of
turbulence. Once in the midplane, however, their inward radial drifts
occur faster than they can grow through the “meter barrier” and the
particles will all fall into the the star, unless some other process
intervenes.
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An apparently more optimistic result.
Ciesla (2007 ApJL) has solved the same equation, but following
the vertical drift term in detail, although he doesn’t treat the change in
the shape of grains in as much detail. Again, only the vertical
direction is treated. He finds a runaway growth of the particles at the
midplane, producing 1-100 km-size planetesimals (the peak in the
size distribution below), a planet-maker’s dream come true! He claims
to have solved the most difficult problem in planet formation: the
growth of particles past the “1 meter barrier” that we’ll discuss below.
How is this possible??
What Ciesla doesn’t tell you is that he left out the radial drift of
particles, which we know will remove them as they approach a size of
1 meter. Here is a plot of his size distribution, as he allows the grains
to sit at the midplane, growing but not drifting.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
!!
!
Plotted are the mass density of particles in a layered disk when the
MRI-active region grows massive enough to "stir up" turbulence in the
dead zone. The lines represent the distribution of particles at the
midplane (solid line) at one scale height (1H; dashed line), 2H (dashdotted line), and 3H (dash–dot-dotted line). [From Ciesla 2007]
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