Download Formation of Planets

Formation of Planets
Revised and extended, 2011
Hans Rickman
Dept. of Astronomy & Space Physics, Uppsala Univ.
PAN Space Research Center, Warsaw
In this compendium we will deal with the question how the planets of the Solar System
may have formed out of the solar nebula. All the basic principles to be described are likely
applicable to exoplanets as well, but we will not discuss these explicitly, either as individual
objects or as dynamical systems. In terms of system properties like the distributions of orbital
semi-major axes or eccentricities, the discussion will focus entirely on the properties of our
Solar System. The problem of planet formation in general will be revisited in the compendium
about exoplanets.
Main ideas and arguments
We start from the basic structural properties of our planetary system. The four terrestrial
planets mark the innermost region, and they consist mainly of rocks and metals. The outer
region is inhabited by the giant planets, which can be separated into gas giants (Jupiter and
Saturn) and ice giants (Uranus and Neptune). In turn, the solar nebula can be described
compositionally as a rotating disk of gas and solids, where the composition of the solids varied
with distance from the center. In the innermost part they were metallic or rocky, and further
out they were icy.
There are two basic mechanisms to form planets out of the disk: planetesimal accretion (PA)
and gas disk instability (GI). In the first case there is a gradual transition from the tiny grains
of the presolar cloud into larger and larger structures, which grow by collision and sticking
until km-sized objects called planetesimals are formed, and then a gravitation-aided further
accretional growth of these into planets. In the second case a gravitational instability arises in
the whole gas disk practically independent of its dust content, leading to local concentrations
with masses similar to those of giant planets. In the first case only the solid component of
the disk is concerned, while the second mechanism involves all parts of the disk including its
dominating gas component. The following chart summarizes how terrestrial planets (T P) and
giant planets (GP) are generally believed to have formed:
Noble Gas Argument
According to current thinking the planetesimal accretion mechanism is the most likely one
for both kinds of planets. The terrestrial planets present the most straightforward case, since
they are in almost complete lack of the nebular gas component. An explanation of this lack
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based on atmospheric escape, whereby initially captured gases have been gradually lost from the
exospheres, might hold for hydrogen and helium, and possibly also neon, due to their low masses.
But the heavy noble gases (Ar, Kr and Xe) are also severely depleted with respect to common
elements like Si and Fe. Moreover, as shown by the diagram below, their relative depletion
pattern is not a simple function of atomic mass – as one would expect from atmospheric escape.
It rather appears that these elements were brought to the planets by a kind of meteoritic
bombardment and that the impactors in question had incorporated the noble gases with a
non-solar abundance pattern.
Capture of Volatiles
The planetesimal accretion scenario does not suffer from a problem to get rid of gases, since
the planetesimals (planetary building blocks) were made of solids. To some extent there is
instead the opposite problem, namely, how to incorporate any gases or volatiles, since from the
models of the solar nebula and the meteoritic evidence it is not evident that such materials
would be present at all in the solids of the innermost region. For instance, we have to explain
the presence of a certain amount of water as well as carbonaceous (C-bearing) and nitrous (Nbearing) compounds. We will return to this problem later and show that relevant explanations
may exist.
Time Scale Argument
The giant planets might present a different story, since they are so different in size and composition. One may think a priori that, with their relatively large content of hydrogen and helium,
the gas disk instability may be a viable idea. In addition, there is a time scale argument
against the planetesimal accretion scenario that goes as follows. The solar nebula only lasted
for a few Myr, and afterwards the hydrogen/helium gas was gone. If the giant planets grew
by planetesimal accretion, there had to be a two-stage process: first the growth of solid cores
with significant gravity, and then the gravitational capture of nebular gas. Hence we require
that massive cores (it is estimated that a few Earth masses are necessary) were formed within
a ∼ 1 Myr. This is difficult to realize, at least if the planets formed in the regions where they
are found at the present time. While it seems possible for Jupiter and Saturn to form this way,
the planetesimal accretion time scale at Neptune’s orbit is far too long.
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Compositional Argument
The gas disk instability, being essentially instantaneous, does not suffer from any time scale
problem. But it has problems of other kinds that make it less attractive than planetesimal
accretion to most scientists. One is that it cannot readily explain the difference in bulk composition between the different giant planets. None of them is solar, but Jupiter is much closer
to solar than Saturn, and Uranus and Neptune are much further still. The figure below illustrates the chemical compositions of the Sun (solar atmosphere) and the giant planets (bulk
compositions) in terms of mass, where in each case the shaded sector corresponds to all elements heavier than hydrogen and helium. The results for the planets are based on somewhat
preliminary models of the interior structures and are not very accurate, but the general, rough
picture is reliable.
Sun
Jup
Sat
Ura
Nep
Some enrichment of refractories might have occurred during planetary formation by gas disk
instability, in case the gravitational collapse did not involve all layers of the disk equally but
prioritized the layers near the midplane. These were probably enriched in rocks and ices at the
expense of those near the disk surface on either side. However, the very gas-poor compositions
of Uranus and Neptune cannot be explained in this way, and in addition the masses of these
two ice giants appear too small for the gas disk instability mechanism. One may then consider
planetesimal accretion as a viable alternative for Uranus and Neptune after all, but these are the
two planets for which the time scale argument appeared the hardest. There are only two escape
routes from this dilemma. Either one formulates a modified gas disk instability mechanism,
where much of the initial gas components is lost from Uranus and Neptune (we will describe
such a model below), or one argues that they formed by planetesimal accretion much closer to
the Sun and then migrated outward (also to be described below).
Disk Mass Argument
There are other arguments too. The gravitational instability of a disk requires a large mass –
> 0.1 M . Since protoplanetary disks observed in star-forming regions do not
typically, Mdisk ∼
appear to be this massive, it is believed that the solar nebula was less massive as well, and
hence it should not have been susceptible to the planet-forming instability. In addition, giant
planets have been found to exist around many other stars, and if these planets had been born
out of very massive disks, it would seem strange that the observations of forming stars do not
reveal such massive disks but instead disks of significantly lower masses.
Metallicity Argument
There is also evidence that the occurrence of extrasolar planets – including the giant ones –
is strongly correlated with the so-called metallicity of the host stars, i.e., their abundance of
heavy elements like the rock-forming ones (Si, Mg, Fe). Such a correlation is easy to understand
on the basis of the planetesimal accretion mechanism but appears hard to explain using the
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gas disk instability picture. Specifically, the mean molecular weight of the gas disk is always
dominated by H and He even for relatively metal-rich systems like the solar system, so the disk
scale height (and thus the local gas density) should a priori be about the same in all cases.
Whether differences in the radiative opacity of the gas due to the different abundances of heavy
elements could lead to significant differences in the density structure remains unknown.
Planet Migration
Finally, let us return to the time scale argument concerning planetesimal accretion. As we shall
see later on, it is no longer believed that the giant planets were born at the locations where
their orbits are now situated. There are several reasons why planets may have “migrated” by
substantial amounts, when the Solar System was very young. In particular, there is good reason
to think that both Uranus and Neptune have migrated outwards. They may hence have been
formed much closer to the Sun, and this allows to avoid the time scale problem.
Another point concerns the above-mentioned extrasolar planets. These are mostly so-called
hot Jupiters, i.e., giant planets orbiting very close to their parent stars – quite unlike our Solar
System. The explanation is mostly believed to be a special kind of planetary migration that is
relevant for giant planets situated in accretion disks. In case this migration is facilitated by a
high metallicity of the disk, this might explain why hot Jupiters are preferentially found around
metal-rich stars – independent of the way these planets formed.
We have to emphasize that the issue of giant planet formation is still open, though in view
of the general preference for the planetesimal accretion scenario we will focus the following
discussion of physical mechanisms mostly on this picture. However, we will begin with a
discussion of recent results dealing with the gas disk instability mechanism.
Gas disk instability
This picture illustrates variations of the contribution by solids to the total density in a
turbulent gas disk with a mass of about 0.01 M , close to the Minimum Mass Solar Nebula.
These, however, contribute a negligible amount to the total density, which is a smooth function
of distance from the center and the midplane.
From a theoretical stability analysis one finds that a rotating gas disk may become unstable
with respect to wave-like density perturbations, amplified by their own gravity, if its density is
large and its thickness and velocity dispersion are small. This is verified by numerical model
calculations like those represented in the figures below, taken from a 2008 paper by Alan P.
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Boss. Iso-density curves are plotted for different levels, and spiral density waves are easily seen
already after a few hundred years. This time corresponds to ∼ 10 orbits in the region shown,
which corresponds roughly to the giant planet orbits of the Solar System. In the left-hand
panel, the star has one solar mass, and the disk mass is 0.09 M . In the right-hand panel the
star is only half as massive, and the disk mass is 0.065 M .
The clumps seen in these spiral waves correspond to masses similar to that of Jupiter,
and when time proceeds a bit further, they indeed develop into giant planets. However, it is
important to note that the disk masses are ∼ 10% of the stellar mass in both cases, and this is
definitely more than what is indicated by observations of protostellar disks. If masses close to
that of the Minimum Mass Solar Nebula are used, no gravitational instabilities are seen. Hence
we have to take these simulation results with a grain of salt.
It has recently been realized that protoplanetary gas disks may lead a dangerous life, if they
are situated in environments like that of the Orion nebula (where many “proplyds” have been
observed) due to a phenomenon called photo-evaporation. This is due to the presence of nearby
OB stars with a large energy output in the form of far-ultraviolet (FUV) or extreme-ultraviolet
(EUV) radiation. When the disk is irradiated by such energetic photons, the gas may be heated
to temperatures of 103 − 104 K. The thermal velocity of hydrogen molecules may then approach
10 km/s, and this translates into a similar bulk velocity of the gas. At distances ∼ 20 AU from
the Sun, this would exceed the speed of escape, so the entire outer parts of disks might be
“evaporated” and thus lost. Observational evidence for the operation of this process has been
found.
There is reason to believe that the Solar System could have been formed in a similar environment, although many stars do form under much more benign conditions. The argument
is the finding of excess 26 Al in the material that formed chondritic meteorites – a very shortlived isotope that would likely come from massive stars exploding as supernovae, thus revealing
the presence of hot OB stars as well. The solar nebula could possibly have been truncated
bu photo-evaporation, and if this concerned the regions where Uranus and Neptune had been
formed by gas disk instability, the atmospheres of these planets would have been irradiated
too. It has been proposed that this might have stripped away most of their initial inventories
of H and He, leaving the planets with their current composition as “ice giants”. However, in
this case one has to note that the escape velocities from the gas giants (Jupiter and Saturn)
are larger than 30 km/s, so it is doubtful if photo-evaporation could work.
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Grain accretion
By the time the presolar cloud collapsed and the solar nebula was formed, the predominant
type of solid objects was submicron sized grains with refractory cores and icy mantles. In
the warmer parts of the nebula the ices were lost by evaporation. For an order-of-magnitude
estimate of the mass of such a grain we can take Mg ∼ ρg d3 /2, where ρg ∼ 103 kg/m3 is the
density and d ∼ 3 · 10−7 m is the diameter. We thus get Mg ∼ 10−17 kg, which is more than 40
orders of magnitude smaller than the mass of the Earth.
It is thus obvious that in order to explain the formation of the Earth or any other planet
starting from presolar grains, we need to bridge an enormous gap. We likely need to invoke
quite different physical mechanisms for different stages of accretion. A fundamental difference
is that between sizeable objects, whose accretion is assisted by their own gravity and smaller
grains or boulders, whose gravity is negligible. We call objects of the former type planetesimals
or – when they are very large – planetary embryos, and we first need to study how these may
be formed out of the initial, microscopic grains.
Grain-gas coupling
The grains are embedded in a gas that is not extremely rarefied and thus moves according to the
laws of hydrodynamics with a well-defined velocity field. A fundamental question is whether the
grains everywhere follow the motion of the gas or if they may deviate systematically from the
local gas flow. In order to answer this we need to introduce the concept of grain/gas coupling
time scale.
Consider a grain with mass M and geometrical cross-section A moving with velocity V
through a gas with number density n and mean molecular mass µ. We assume that the motion
is subsonic, i.e., V is smaller than the sound speed in the gas, which is of the order of the
mean thermal speed of the molecules. We also assume that the grain is small compared to the
collisional mean free path of the molecules. In this regime we can estimate the gas drag force
and write the equation of motion of the grain:
dV
= −nAV × µuth × CD
(1)
dt
where the first factor of the right-hand member gives the excess number of molecules hitting
the front side of the grain per unit time, the second factor is the average absolute value of the
momentum of each molecule along the direction of the grain motion (uth is the respective thermal velocity component), and the third factor is the drag coefficient expressing the momentum
transfer efficiency per collision – a number of order unity.
The solution of Eq. (1) is:
M
n
V = Vo exp −CD
o
n to
A
nµuth t = Vo exp −
M
τc
6
(2)
with
τc =
M
ACD ρgas uth
(3)
The quantity τc is the exponential time scale of damping of the relative velocity V , or
alternatively, the grain/gas coupling time scale. In Eq. (3) we have introduced the gas density
ρgas = nµ. Treating the grain as a sphere of radius Rgr and density ρgr , we get:
τc =
ρgr Rgr
4ρgr Rgr
∼
3CD ρgas uth
ρgas uth
(4)
where the last expression is an order-of-magnitude approximation.
We see that τc increases linearly with the radius and density of the grain and is otherwise
determined by the density and temperature of the gas. Let us take a numerical example for
grains near Jupiter’s orbit. The minimum mass solar nebula should have had a surface density
Σ ∼ 103 g/cm2 at this distance according to the diagram on p. 17 of the compendium about
the origin of the Solar System. The thermal speed of hydrogen molecules at the temperature
in question (∼ 100 − 200 K) is uth ∼ 103 m/s. In order to derive the gas density, we need to
divide the surface density by the thickness of the disk, and we may estimate this thickness as
follows. A molecule moving with velocity uth in the vertical direction in the midplane will reach
a maximum distance from this plane of H ∼ uth /Ω, where Ω is the frequency of orbital motion,
the so-called Kepler frequency. At Jupiter’s orbit the period is about 12 years, i.e., 3.6 · 10 8 s, so
we get roughly: Ω ∼ 2 · 10−8 s−1 , which translates into H ∼ 5 · 1010 m (approximately, 0.3 AU).
The total thickness of the disk should thus be 2H ∼ 1013 cm, and we get ρgas ∼ 10−10 g/cm3
or ρgas ∼ 10−7 kg/m3 .
With ρgr ∼ 103 kg/m3 , we get τc ∼ 10 s for Rgr ∼ 1 µm and τc ∼ 105 s (about 1 day) for
Rgr ∼ 1 cm. Obviously the grain/gas coupling is extremely quick for tiny grains and starts to
get slow only for meter-sized boulders.
If there were no additional force that accelerated the grain relative to the gas, a few times
τc would be time enough to adapt the grain to the motion of the gas. With a laminar gas flow,
in which the local velocity changes only slowly with position, it would then be relatively easy
for grains of very different sizes (and thus different values of τc ) to be swept up by the gas and
start moving at very low speed (practically zero) relative to each other. On the other hand, if
turbulence develops, grains moving through the gas will experience a rapidly fluctuating flow
velocity, and only the smaller ones may be able to adapt fully to the gas, while larger grains
will decouple from the smallest and fastest eddies. Hence grains of different sizes are set in
relative motion with velocities characteristic of the gas turbulence.
Grain growth
Grain growth by accretion means that the grains need to come together and collide so that they
can stick. Thus the central problem is to explain the emergence of relative velocities between
grains situated in the same small gas element. This process must start from the submicron
grains of the presolar cloud. For such tiny grains a relevant growth mechanism is offered by
the thermal (Brownian) motions caused by collisions with gas molecules. In this regard a
submicron grain of ∼ 10−17 kg can be viewed as a giant gas particle sharing kinetic energy with
the impinging molecules. The difference of a factor ∼ 1010 in mass then leads to thermal grain
velocities ∼ 105 times smaller than those of the hydrogen molecules, i.e., Vgr ∼ 1 cm/s.
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If the collision cross-section of such a grain is Agr and the number density of grains is ngr ,
one can estimate the collision frequency νc as
νc ∼ ngr Agr Vgr .
(5)
The cross-section can be estimated as the square of the grain radius, i.e., Agr ∼ 10−13 m2 .
The number density is given by the fact that the density of solids in the solar nebula must be
∼ 1% of the gas density. If we take the estimate at Jupiter’s orbit that we just performed, we
get ρsolid ∼ 10−9 kg/m3 , and if this is to be represented by single grains of mass 10−17 kg, the
number density comes out as ngr ∼ 108 m−3 . We thus get: νc ∼ 10−7 s−1 , i.e., on the average
a few collisions per year.
It is therefore possible to start the grain growth by this mechanism, but we cannot reach
very far, since the collision frequency decreases very rapidly, if we consider the solids to be made
up of larger and larger grains. Already at the size of a few mm, if all the solids are bound up
in such large grains, they are so few and move so slowly that there would only be one collision
per grain during the age of the Universe! However, it may be enough, if aggregates with a
size range from about 1 µm to more than 10 µm are formed, because these will then react
differently to turbulent gas motions, as mentioned above, and thus be set in relative motion.
As they collide and stick, the larger ones will grow further, while smaller ones get consumed.
Now consider a grain moving with the gas at a certain distance r from the Sun and a certain
distance z from the gas disk midplane. This grain feels the force of gravity from the Sun and
from the gas disk. Let us concentrate on the dominating term, the solar gravity. If the grain
mass is mgr , we can write this force as
FG = mgr Ω2 r
(6)
where Ω is given by Kepler’s third law as
Ω2 r3 = GM
(7)
As seen in the picture above, the force vector has two components – one vertical of magnitude
Fz , and the other horizontal. The latter is nearly offset by the centrifugal force of the rotating
gas disk, but the former is not offset at all for a grain that strictly follows the gas. Its size is
Fz = mgr Ω2 z
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(8)
and it accelerates the grain toward the midplane of the disk. As a result the grain may acquire
a vertical velocity u, and the gas drag force FD can then be obtained from Eq. (1) as
2
FD = nπRgr
u × CD µuth ,
(9)
where Rgr is the radius of the grain. If we introduce ρgas = nµ for the density of the gas and
ρgr for the density of the grain, the force balance Fz = FD yields the condition
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ρgas uth ueq CD = Rgr ρgr Ω2 z
3
(10)
for the equilibrium velocity ueq .
We see from Eq. (10) that ueq is proportional to Rgr , so larger grains will sediment toward
the midplane faster than small ones do. The sedimentation time scale τs can be estimated from
τs ∼
3ρgas uth CD
z
,
=
ueq
4Rgr ρgr Ω2
(11)
since putting ueq = −dz/dt in Eq. (10), the equation is easily solved to get: z = zo exp(−t/τs )
with τs given by Eq. (11). In order to estimate τs at Jupiter’s orbit we take the above values
ρgas ∼ 10−7 kg/m3 , uth ∼ 103 m/s, Ω ∼ 2·10−8 s−1 and ρgr ∼ 103 kg/m3 , and we use Rgr ∼ 1 cm
to obtain τs ∼ 103 yr. We see that as soon as turbulent growth builds grain aggregates with
radii of a mm or larger, sedimentation to the disk midplane proceeds very quickly.
Moreover, as illustrated by the picture above, this sedimentation implies further growth,
because the sedimentation speed is proportional to the particle radius. Thus the larger grains
overtake the smaller ones and sweep them up. We can write the rate of mass growth by this
mechanism in two ways:
dM
2 dRgr
= 4πRgr
(12)
dt
dt
and
dM
2
= πRgr
× αCM ρgas × ueq ,
(13)
dt
where the second equation follows from the assumption that all the other grains have negligible
vertical velocities, and using αCM ρgas for the solid mass density (αCM << 1 is the fraction of
condensible material). Thus, if we put ueq = −dz/dt and integrate from zo = H to z∞ = 0,
combining Eqs. (12) and (13), we get
1
ρgas
∆Rgr = αCM
·H .
4
ρgr
9
(14)
At constant vertical velocity ueq , Rgr would grow exponentially with time since ueq is proportional to Rgr , but taking into account the exponential decrease of ueq (t) obtained from Eq. (11),
we get the limiting ∆Rgr from the integrated amount of available material, as given by Eq. (14).
Using the above estimates for the vicinity of Jupiter’s orbit (ρgas /ρgr ∼ 10−10 ; H ∼ 5·1010 m)
along with αCM ∼ 0.02 as expected from the condensation of refractories and ices, we get
∆Rgr ∼ 2.5 cm. Thus, from a combination of thermal, turbulent and sedimentary growth we
can expect a thin layer of cm-sized grains to develop in the midplane of the solar nebula within
a short time.
But the sedimentation also leads to a large concentration of solid material near the midplane.
Therefore, while the overall fraction of condensible material near Jupiter’s orbit was only ∼ 0.02,
after sedimentation and accumulation of all the solids into a layer of thickness ∼ 0.01H, the
local solid/gas ratio near the midplane was in fact ∼ 2. Under these circumstances, the gas
can no longer move on its own, following its sub-Keplerian speed, but gets accelerated toward
Keplerian speed by the grains. This leads to a vertical shear in the gas disk, since the layers
somewhat away from the midplane are much less charged with solids and therefore keep the
sub-Keplerian motion.
Such a shear leads to turbulence via so-called Kelvin-Helmholtz instability, and this turbulence enforces further growth of the solid grains. We can easily imagine that the chunks of
solid material that rain down toward the midplane have somewhat different sizes or – more
importantly – somewhat different grain/gas coupling time scales. Thus we have to consider the
relative motion of gas and different-sized solid chunks near the solar nebula midplane, keeping
in mind that the gas motion involves turbulent eddies but is essentially a circular flow around
the center.
We then have to deal with the following phenomenon, which is illustrated in the figure
above. The gas moves under the combined influence of the force of gravity, which is directed
inward, and the outward force of the pressure gradient (the density and temperature of the gas
decrease outward). But the grains are not pressure supported unless they are perfectly coupled
to the gas. Therefore, since cm-sized grains tend to be swept along by the gas motion, which
is sub-Keplerian due to the pressure support, they must move too slowly to balance the force of
gravity and hence are driven to spiral inwards. If, on the other hand, a grain grows to meter
size, it decouples from the gas motion on the time scale of its orbital motion. Thus, it does not
have to move too slowly, but when moving at the Keplerian velocity, it feels a gas drag due to
the headwind of the slower moving gas. This will again cause a loss of angular momentum and
an inward spiraling motion.
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Let us now summarize what we have found about grain growth. The growth is due to small
particles sticking by van der Waals forces, as they come together at low relative speeds. This
presents no theoretical problems, and it has been verified by numerous laboratory experiments,
an example of which is seen in the picture above. We observe a highly porous, fractal-looking
structure of the simple aggregate, and we can expect that further growth must lead to some
compaction. But the densities remain relatively low, and a substantial porosity is kept even for
very large aggregates.
In the last stages, we deal with chunks of material approaching 1 meter in size. The increased
decoupling from the gas motions then lead to so large relative speeds that further collisions
may imply destruction rather than accretion. But this is just a small problem compared to the
one which we shall now describe.
Consider a boulder with a radius of 1 m, moving in the solar nebula at Jupiter’s distance
from the Sun. It is essentially decoupled from the gas motion, and thus it moves at close to
Keplerian speed, while the gas is moving slower, causing a drag effect on the boulder. The
orbital angular momentum of the boulder is
q
(15)
τ = rFD = rρgas Au2 CD
(16)
L = M GM r
where r is the distance from the Sun, and we consider the boulder to move in a circular orbit.
The torque caused by the drag force is
considering the relative velocity u between the boulder and the gas to be also the velocity
causing the drag. Inserting usual expressions for M and A (the mass and area of the boulder)
in terms of the radius of a sphere (Rgr ), we get the time scale of orbit decay (loss of all angular
momentum) as
s
ρgr −2 GM
L
Td = ' Rgr
u
(17)
τ
ρgas
r
where the last factor in the right-hand member is the circular Keplerian velocity vc (r).
We may approximate u by taking 1% of vc (assuming that the gas moves on average at 99%
of the circular velocity), and thus we get
ρgr
· 104 vc−1
Td ' Rgr
(18)
ρgas
With Rgr = 1 m, a density ratio of 1010 (see above), and vc ∼ 1 · 104 m/s, we get Td ∼ 1 · 1010 s,
i.e., a few hundred years. This means that we have a formidable problem explaining how the
growth of solid bodies can proceed any further, since in practically no time they all spiral into
the Sun.
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Planetesimal formation
The hardest problem to solve for the planetesimal accretion theory is in fact this one: how to
grow planetesimals (i.e., km-sized bodies) out of meter-sized boulders in practically no time?
And this problem still remains open.
One mechanism that has attracted much attention and was once regarded as the solution
of the problem is gravitational instability arising in the very thin disk of solid chunks at the
midplane pf the solar nebula. These chunks would also have a very small relative velocity
with respect to each other, i.e., a very small local velocity dispersion. Thus the disk would be
unstable to growing density wave modes that would concentrate the solids into large structures
– i.e., planetesimals. However, a problem that has been realized in later times is that the
settling into an extremely thin disk with an extremely small velocity dispersion is prevented by
the turbulence of the gas disk that is caused by the very process of grain settling, as described
above.
Another idea is that the rapid inward drift itself could induce a rapid accretional growth and
thus a slowing down of the drift, since Eq. (18) shows that Td increases with Rgr , for the reason
that larger boulders drift more slowly than smaller ones. Thus their drift could essentially stop
before reaching the Sun by eating their smaller colleagues. This hope only seems vague at best,
as long as we consider the inner Solar System, but for the regions far outside Jupiter’s orbit,
where the time scales are much longer, it may work.
The best ideas for the inner Solar System are recent ones invoking either the MRI-driven
turbulence of the gas (MRI = magneto-rotational instability) or a gas-kinetic streaming instability. Especially in connection with accretional flow around “dead zones”, vortices will develop
with density enhancements and rapidly trap solid boulders in large quantities. An example is
seen in the picture on p. 4. Thus these may easily come together and accrete into large bodies
almost instantaneously.
To be continued
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