The dynamics of dustLecture 9

Lecture Universität Heidelberg WS 11/12Dr. C. Mordasini

Based partially on script of Prof. W. Benz Mentor Prof. T. Henning

Fouchet et al. 2005

Lecture 9 overview1. Settling of dust grains

1.1 Drag laws

1.2 Settling

1.3 Settling with growth

2. Radial drift

2.1 Destruction at the rockline

3. Planetesimal formation by self-gravity

3.1 Gravoturbulent planetesimal formation

4. Collisional growth of planetesimals

4.1 Numerical simulations of coagulative growth

Early phasesSedimentation

Dust grains coagulate and gradually decouple from the gas

Sediment to form a thin mid-plane layer in the disc

Planetesimals form by continued coagulation or self-gravity (orcombination) in dense mid-plane layer

HOWEVER:MRI-driven turbulence very efficient at diffusing dust

Anders Johansen (Lund) Dust growth 21 / 36

The basic picture of the early stage of planet formation (growth from dust to km sized planetesimals) is the following:

A. Johansen

•Dust grains condense, coagulate and gradually decouple from the gas. •The dust grains settle into a thin mid-plane layer in the disk.•Planetesimals form by continued coagulation (two body collisions) or a self-gravitational instability of the dust (or a combination of the two) in the dense mid-plane layer.

1.1 Drag laws

Drag forceOnce the nebula has formed and begins to cool, dust grains will condense out preferentially at high elevation since these are the coldest regions. After formation, these dust grains will settle towards the mid-plane of the nebula as it is the potential minimum. The speed at which the grains settle towards the mid-plane depends upon the interactions between dust and gas, i.e. drag. Settling is necessary to collect the dust in order to build-up the planets.

Drag laws (Weidenschilling 1977, MNRAS, 180, 57)

In order to describe the dust dynamics, we need drag laws. Consider a solid particle that we assume to be spherical with radius s and density ρs moving through a gas of density ρ at velocity v. The mean free path of the gas molecules is defined by λ and the gas is characterized by a thermal velocity vtherm. They are given as

The molecular cross section is where dm is the effective molecular diameter.

The table gives the effective molecular diameter for some gases at standard p-T conditions. As expected, d is of order a few Å.

Drag force IIDepending upon the size of the particle relative to λ, the drag force can be written as:

1) fluid regime (s >> λ) : FD =12CDπs2ρv2

The latter expression is simply (volume swept per unit time ) x (momentum per unit volume). For typical nebula λ~1cm, Epstein drag is the dominant regime for dust.

In the fluid regime, the drag coefficient CD is dimensionless and is a function of the Reynolds number Re which can be written:

Re =2sρv

η; η 1

3ρvthermλ

where η is the (molecular) viscosity of the gas.

The drag coefficient for a sphere can be expressed as (for mach numbers Ma<0.3):

CD 24R−1e for Re < 1 (Stokes drag)

CD 0.44 for Re > 800CD 24R−0.6

e for 1 ≤ Re ≤ 800

2) particle regime (s << λ) : FD 4π

3ρs2vvtherm (Epstein drag)

Inserting CD for the stokes drag gives back the well known formula for small Re:

Friction time and Stokes number The definition of the stopping or friction time comes from the fact that the force is the temporal derivative of the impulse:

For small spherical grains in the Epstein regime, we can write for the stopping (δ is the material density of the dust grains)

For properties typical which are for the solar nebula (δ=3 g/cm3, ρ=5×10−10 g/cm3, vtherm = 2.4×105 cm/s) this corresponds to

Small particles are therefore very well coupled to the gas. Often the nondimensional Stokes number is used as a proxy of the particles size:

1.2 Settling

Dust settlingDue to the small stopping time, dust particles reach very quickly the terminal velocity which can be estimated by setting the z-component of the stellar gravity (cf. the lecture on the vertical structure of the gas disk) equal the drag force:

mΩ2z 43πs2ρvthermvz → vz =

3mΩ2z

4πs2ρvtherm

Thus, the settling timescale can be computed as

τ z

vz=

4πs2ρvtherm

3mΩ2=

4πs2ρvthermvz

3mvz

1Ω2

=1ts

1Ω2

with ts =mvz

FDthe stopping time

Physical insights is gained by writing the timescale:

τ =

1Ω

1

tsΩ

which is roughly: (orbital time)x(orbital time)/(stopping time) as Porb= 2 π /Ω.

The latter ratio can be made more explicit:

Ωts = Ωmvz

43πs2ρvthermvz

= Ω43πs3δ

43πs2ρvtherm

=δs

ρvthermΩ

Dust settling II For the thin disks, we have seen that and therefore

Σ =2ρc

Ω→ ρ vtherm

2ΩΣ3

ThusΩts

32

δ

Σs

Plugging this back into the expression for the settling time, we find the approximation

τ 34π

Σδs

Porb

(c2 =γπ

8v2

therm → c 0.75vtherm for γ =75)

Here we have used that

Note:•For micron sized particles δ s <<1 and therefore the settling time is always much longer than the rotation period.•The settling time is inversely proportional to size of the dust grains, thus smaller grains will fall slower towards the mid-plane than larger ones.•Using the MMSN at 1 AU we have Σ=1500 g/cm2 and we obtain typical settling time for micron-sized particles of density δ=3 g/cm3 : τ≈ 1.2 x 106 years. This time is comparable to the lifetime of the disk itself. Hence settling appears to be too slow a process... •This is especially true as we have neglected turbulence which tends to slow down the settling. Obviously, this appears to be a worrying result.

1.3 Settling with growth

Dust settling with growthFortunately, Safronov has pointed out that collisions between settling dust grains will not only make them grow but also fall more rapidly. These larger dust grains will sweep up the smaller ones and grow more rapidly falling even faster! Let us estimate the size of these bodies by the time they reach the mid-plane using mass conservation. A body of radius s reaching the mid-plane will have swept a column of dust having a mass:

Mcolumn psζΣπs

2

2with ps : sticking prob., ζ : dust gas ratio,

s

2: av. radius

Mass conservation requires that:

43πs3δ = psζΣπ

s2

4→ s 3psζΣ

16δ

As expected, the size becomes large for a high sticking probability, dust to gas ratio and gas surface density.

Using typical numbers for 1 AU: ζ=1/240 (i.e. inside the ice line), Σ=1500 g/cm2, δ=3 g/cm3, we obtain:

s=0.4 ps cm

This is obviously a much larger size (for high sticking probability) than the micro meter size we assumed previously.

Dust settling with growth IIFor this size, the settling time is given by

which for 1 AU gives a typical settling time:

τ 320ps

years Clearly a much shorter timescale!

τ =34π

Σδ

16δ

3psζΣPorb =

4π

1

ζps

Porbτ 3

4π

Σδs

Porb

Of course, assuming s/2 as an average radius for the solid bodies results certainly in a underestimation of the settling time. This is because the settling time is inversely proportional to the size of the bodies. In order to obtain better estimates, we need to compute the growth and settling of the dust grains in a coupled way, which leads to a set of coupled differential equations for dm/dt and dz/dt.

25

FIG. 17 The settling and growth of a single particle in a lami-nar (non-turbulent) protoplanetary disk. The model assumesthat a single particle (with initial size a = 1 µm (solid line),0.1 µm (dashed line), or 0.01 µm (long dashed line) accretesall smaller particles it encounters as it settles toward the diskmidplane. The smaller particles are assumed to be at rest.The upper panel shows the height above the midplane as afunction of time, the lower panel the particle radius a. Forthis example the disk parameters adopted are: orbital radiusr = 1 AU, scale height h = 3 ! 1011 cm, surface density! = 103 g cm!2, dust to gas ratio f = 10!2, and mean ther-mal speed v = 105 cm s!1. The dust particle is taken to havea material density !d = 3 g cm!3 and to start settling froma height z0 = 5h.

where f is the dust to gas ratio in the disk. Substitutingfor the settling velocity one finds,

dm

dt=

3

4

!2f

vzm. (115)

Since z = z(t) this Equation cannot generally be inte-grated immediately11, but rather must be solved in con-cert with the equation for the height of the particle abovethe midplane,

dz

dt= !

!d!

a

v!2z. (116)

Solutions to these equations provide a very simple modelfor particle growth and sedimentation in a non-turbulentdisk.

11 Note however that if the particle grows rapidly (i.e. more rapidlythan it sediments) then the form of the equation implies expo-nential growth of m with time.

Figure 17 shows solutions to equations (115) and (116)for initial particle sizes of 0.01 µm, 0.1 µm and 1 µm. Theparticles settle from an initial height z0 = 5h through adisk whose parameters are chosen to be roughly appro-priate to a (laminar) Solar Nebula model at 1 AU fromthe Sun. Both particle and growth and vertical settlingare extremely rapid. With the inclusion of coagulation,particles settle to the disk midplane on a time scale ofthe order of 103 yr – more than two orders of magnitudefaster than the equivalent time scale in the absence ofparticle growth. By the time that the particles reach themidplane they have grown to a final size of a few mm,irrespective of their initial radius.The single particle model described above is very sim-

ple, both in its neglect of turbulence and because it as-sumes that the only reason that particle-particle colli-sions occur is because the particles have di"erent ver-tical settling velocities. Other drivers of collisions in-clude Brownian motion, turbulence, and di"erential ra-dial velocities. The basic result, however, is confirmedby more sophisticated models (Dullemond & Dominik,2005), which show that if collisions lead to particle adhe-sion growth from sub-micron scales up to small macro-scopic scales (of the order of a mm) occurs rapidly. Thereare no time scale problems involved with the very earliestphases of particle growth. Indeed, what is more problem-atic is to understand how the population of small grains– which are unquestionably present given the IR excessescharacteristic of Classical T Tauri star – survive to latetimes. The likely solution to this quandary involves theinclusion of particle fragmentation in su#ciently ener-getic collisions, which allows a broad distribution of par-ticle sizes to survive out to late times. Fragmentationis not likely given collisions at relative velocities of theorder of a cm s!1 – values typical of settling for micron-sized particles – but becomes more probable for collisionsat velocities of a m s!1 or higher.

3. Radial drift of particles

Previously we showed (equation 48) that the azimuthalvelocity of gas within a geometrically thin disk is closeto the Keplerian velocity. That it is not identical, how-ever, turns out to have important consequences for theevolution of small solid bodies within the disk (Weiden-schilling, 1977b). We can distinguish two regimes,

• Small particles (a < cm) are well-coupled to thegas. To a first approximation we can imagine thatthey orbit with the gas velocity. Since they don’texperience the same radial pressure gradient as thegas, however, this means that they feel a net in-ward force and drift inward at their radial terminalvelocity.

• Rocks (a > m) are less strongly coupled to the gas.To a first approximation we can imagine that theyorbit with the Keplerian velocity. This is faster

Armitage 2009

Results of such a calculation from Armitage 2009 are shown on the left. The figure shows the settling and growth of a single particle in a laminar (non-turbulent) protoplanetary disk. The calculations assumes that a single particle (with initial sizes a = 1 μm (solid), 0.1 μm (dashed), or 0.01 μm (long dashed line) accretes all smaller particles it encounters as it settles toward the disk midplane. The upper panel shows the height above the midplane as a function of time, the lower panel the particle radius s. The disk parameters are: orbital radius r = 1 AU, scale height H/r = 0.02, gas surface density Σ = 1000 g/cm2, dust to gas ratio 1/100, and mean thermal speed 105 cm/s. The dust particle is taken to have a material density δ= 3 g/cm3 and to start settling from a height 5 H.

We note that the results are similar to our simple estimation.

Dust settling with growth IIIUsing such a realistic nebula model and computing the growth of dust particles as they fall towards the mid-plane taking into account experimentally determined sticking probability (which is itself an extremely complex undertaking), we obtain a behavior shown in the figure. Note that the settling time of the dust is a function of distance to the sun. At small distances, it is faster, as we expect from our simple calculations.

These results indicate that the outcome of dust settling is a layer of ~mm sized particles forming within ~104 to 105 years.

Benz et al.Effect of pure growth on SED of disk

The simple particle model neglects the effects especially of turbulence, fragmentation and of radial drift (see below). The basic result of a rapid formation of mm sized particles is however also confirmed in more complex models. This high efficiency of growth at the smallest scales opens the question why IR excess (which must be due to micrometer sized particles) is seen for several million years for T Tauri star. The likely reason for this is that in collision, small dust is constantly regenerated from the destruction of bigger ones.

Dullemond

Effect of pure grain grain growth on SED

Thickness of the dust layerThe final outcome of the settling process in the laminar disk as assumed previously would be a infinitesimally thin layer of the dust in the midplane. From our study of the evolution of the gas disk, we however know that disk are turbulent. The turbulence stirs up the dust.

The thickness of the layer is given by the equilibrium of turbulence which stirs the dust back up and settling. Using the α prescription for turbulence, we can estimate the thickness of the dust layer.

For the simplest possible assumption, namely kd=1, we find for α =0.01, s=1 micron, Σ = 1000 g/cm2, δ= 3 g/cm3 , z/H~223, which would mean that turbulence completely offsets settling. For a lower viscosity, and bigger grains, α =0.001, s=1 cm, z/H~0.7, so the dust layer thickness would be similar as the gas scale height.

Assuming that the effective diffusion coefficient for the dust which are passive tracer particles injected into the turbulent flow is kd times the same as the one of the gas itself (a non-trivial assumption), the timescale for stirring dust to a height z is

Equating this with the settling timescale (without growth) and using

which we can also write as

Thickness of the dust layer IITurbulence stirs dust back up

Dullemond

The usefulness of this estimate is limited due to the assumption that the diffusion of dust is basically the same as the one of the gas. A result of a simulation treating this in a more complex way this is shown in the figure on the left. As we expect from our simple estimate, it is also found that the bigger the particles, the thinner the layer. The figure shows how deep grains of a certain size sediment into the disk after equilibrium between sedimentation and turbulent mixing has set in for a turbulent α-parameter of α = 10-4.

Note that the study of the behavior of dust in a turbulence gas is an active area of research, as it is important for the possible formation of planetesimals by gravitational instability, which we will address further below.

Settling toward equilibrium state

Looking at the temporal evolution of the settling it is found that after a settling phase of ~105 years, a steady state is reached.

2 Radial drift

dust layer

gas

Following Weidenschilling 1977, we investigate now the dynamics of dust (of variable grain size) embedded in a differentially rotating gaseous disk. We begin with the equation for centrifugal equilibrium that we have already seen in a previous lecture:

Once the dust has settled, we can imagine a structured nebula as illustrated in the figure. Of course, this is a schematic figure and one should understand more as an increase of the density towards the mid-plane rather than a separate dust layer...

v2k

r=

GM

r2- centrifugal equilibrium for solids:

v2g

r=

GM

r2+

1ρ

dp

dr<

v2k

r- centrifugal equilibrium for the gas:

Gas rotational velocity

Because of pressure support the gas is orbiting the star at speeds smaller than the Keplerian velocity. In order to compute this velocity difference we approximate the solar nebula by power laws (this is certainly locally valid):

ρ(r) = ρ0

r

r0

−α

; T (r) = T0

r

r0

−β

Gas rotational velocityAssuming an ideal gas, we can write the pressure in terms of a power law as well:

p =k

µmH

ρT =k

µmH

ρ0T0

r

r0

−(α+β)

⇒ dp

dr= − (α + β)k

µmH

ρTr−1

Inserting the pressure gradient in the gas centrifugal equilibrium yields:

v2g

r=

v2k

r− kT

µmH

(α + β)r−1 → v2g

= v2k− kT

µmH

(α + β)

Recalling that the thermal velocity is given by: v2therm

=8kT

πµmH

we can write the rotational velocity of the gas as:

v2g = v2

k (1− 2η(r)) ; η(r) =π

16(α + β)

vtherm

vk

2

Typical numbers for the solar nebula at 1 AU are: ρ0= 10-9 g/cm3, T0=500 K, μ=1.3, α=3/2, β=1/2 which yields a thermal velocity of vtherm= 2.8 km/s. Taking vk=30 km/s yields

η = 3.52× 10−3 ; → ∆v = vk − vg = vk(1− (1− 2η)1/2) ηvk 106 m/s

While this is small relative to vk, this is a non-negligible in absolute terms.

Radial drift IWe now compute the motion of solid bodies moving through the solar nebula. In good approximation, we assume that the drift velocity of solids with respect to the gas velocity is small. In other words, we assume that the trajectories remain always nearly circular.

Let us define the radial and tangential components of the drift velocities (i. e. the relative velocity gas-particle):

→ total drift: v=(u2+w2)1/2 with the assumption that v<<vkep

- radial drift (inward motion towards the star): u

- tangential drift (velocity relative to the gas along a nearly circular orbit of radius r): w

Defining h, the particle's specific angular momentum: h=r(vg+w).

we can write the equations governing the motions in the radial and tangential directions (neglecting all terms quadratic in the small quantities):

1) radial direction: aDrag+acent-aGrav=0

FD

m

u

v+

(vg + w)2

r− v2

k

r= 0

→ −FD

m

u

v− 2ηv2

k

r+

2vkw

r= 0 (1)

The first term is the radial component of the drag.

Radial drift II

dh

dt= −rFD

m

w

v

→ FD

m

w

v+

uvk

2r= 0 (2)

2) tangential direction. Temporal derivative of the angular momentum is a torque, i.e. F x r.

FD w/v is the tangential component of the drag.

These two coupled equations can be solved to give the drift velocity of a solid body as a function of its radius and position in the solar nebula.

Kn>> 1 Kn<<1

In the free molecular regime, the gas molecule directly impinge on the “dust” grain while in the hydrodynamic regime we observe the formation of a boundary layer.

The friction force is just as for settling a function of flow regime in which the body is moving. Since density and temperature depend upon the distance to the star, the flow regime will also be a function of radial position in the nebula. We can estimate if we are in the free molecular regime (Kn >> 1) or in the hydrodynamic regime (Kn << 1) by computing the dimensionless Knudsen number:

Radial drift III

The value of the Knudsen number as a function of size of the “dust” grain is given in the figure to the right for various distances to the sun. To compute the number we assumed: p(d)=2 Pa (d/1AU)-3, T(d)=600K (d/1AU)-1, ρ(d)=10-6 kg/m3 (d/1AU)-2

We see that small particles at large distances are in the free molecular flow regime, as expected.

Using the two equations for the drift velocities and the appropriate drag law, we can compute the drift velocities in the radial and tangential direction as a function of body size. The plot shows drift velocities as a function of body size at 1 AU. For these calculations we set α=2 and β=1.

Benz

Radial drift IIINote:- small bodies move with the gas and have small radial drift velocities. They are well-coupled to the gas. To a first approximation they orbit with the gas velocity (vanishing tangential velocity). But without pressure support as the gas, they undergo slow inward drift at their radial terminal velocity.

- large bodies are less strongly coupled. They move with near Keplerian speed and experience the full headwind which is, as calculated, about 100 m/s.

Bodies of roughly meter-size, which have St~1, experience the largest drift velocity. For these bodies, the inward velocity is of order 100 m/s. Already 10 m sized bodies drift however 100 times slower.

An inward velocity is of order 100 m/s which corresponds to roughly 2 AU in 100 years! In other words, the inward drift of meter sized objects is extremely high.

Benz

For the radial drift velocity of particles in the Epstein regime, one finds:

The solution of the so called “Meter barrier” problem (the fast drift is only one component of it) is one important research subject of planet formation theory today.

Radial drift IVra

dial

drif

t tim

e (y

ears

)

It is interesting to compute the drift timescale which is the time required to reach the sun from the initial position d. Since objects roughly one meter in size move the fastest, they will also have the shortest drift timescale.

Note:-changes of slopes correspond to a change in the drag law.

-one meter objects are removed extremely quickly from the inner parts of the solar nebula. For a 1 meter sized body, it only takes ~50 years!

-bodies growing in the inners regions must grow very rapidly in the meter rage if they want escape the fate of being destroyed near the sun. Or skip this size range...

Benz

2.1 Destruction at the rockline

In a more complete model, we can link the direct integration of the equation of motion of the solid particles with a model describing their destruction at high temperatures. The solid particles get heated as they are immersed in the ambient radiation field of the protoplanetary disk. This radiation is absorbed at the surface of the planetesimals. A part of it is re-radiated, while another part is used to heat up the planetesimals. For sufficiently high ambient temperatures, the surface temperature becomes so hot that vaporization or melting starts (see in one of the next lectures).

Time until destruction

Time until destruction for spherical planetesimals of different initial sizes starting at 1 AU. Two mass ablation mechanisms (melting and vaporization) are considered.

Static powerlaw gas disk models. Model 1: Adachi 1976: α=3, β=1, T0=230 K, ρ0 = 10−8 g/cm3 Model 2: Benz 2005: α=1.5, β=0.5, T0=500 K, ρ0 = 10−9 g/cm3

Mordasini 2007

Note-the times until destruction are similar as in the simple model of Weidenschilling 1977.-the shortest time are again less than 100 years for ~1 m sized bodes.-vaporization is the more likely destruction mechanism due to the high viscosity of rock melt, and the relatively low gas densities. In any case, the difference is small.-For times longer ~105, the evolution of the gaseous disk itself should be considered.

-For melting, no mass loss occurs for Twall < Tmelt . Heat conduction to the interior is neglected, therefore Twall is independent of planetesimal size.

-For vaporization, ddest increases for small bodies. Here, material loss is regulated by the rock vapor pressure. There is no cut-off planetesimal wall temperature Twall below which no mass loss occurs. But with the exponential behavior of the vapor pressure law, dM/dt becomes very small at low wall temperatures. Still, this is enough that small bodies dissolve.

-In absolute terms, ddest is found between 0.08 and 0.3 AU. This will however depend on the specific temperature structure (certainly more complex than power laws) and the material properties.

Orbital distance of destructionOrbital distance ddest at which thermal ablation has completely destroyed the solids, again for both disk models and vaporization and melting.

Mordasini 2007

Note-ddest is largely independent of planetesimal size (for melting for all radii, for vaporization for sizes > ~1 m).

-Not even very massive bodies can pass the rockline which is thus well defined. The reason is that the higher total energy input needed to destroy more massive bodies is offset by their slower inward drift: ablation has more time to act.

Nebula temperature at destruction

Disk gas temperature Tdest at the distance where the solids dissolve, on the left for vaporization, on the right for melting.

-For the more likely ablation mechanism, vaporization, and for large planetesimals, Tdest lies between ~1300 and ~2000 K. This difference is due to the different disk structures. For smaller bodies, Tdest decreases to 900 K.

-For melting, the nebula temperature which leads to the destruction is very similar to the assumed melting temperature of the solids, which is set to 1800 K. Only for massive bodies and disk model 1, Tdest rises by about 100 K.

Some of the vapor gets turbulently mixed back outward and recondenses in the form of dust.

Mordasini 2007

Note: -measured inner dust disk radii: between 0.1 and 0.3 AU. (T=1000 to 1800 K)

-bigger than the size of the magnetospheric cavity (estimated using the measured accretion rates and assuming B=2 kG).

-innermost part: dust free gas disk, between solid destruction radius and magnetospheric cavity radius.

-planet formation is unlikely interior to ~0.1 AU. No in-situ formation of hot Jupiters at ~0.03 AU.

Observed inner disk radiiThe combination of interferometric and SED measurement allows to observationally determine the inner disk edge of the dust disk.

solid: measured inner dust disk radiusdotted: magnetospheric truncation radiusdashed: corotation radii

The table gives measured inner disk radii for accreting T Tauri stars. The theoretical radii are calculated taking into account for the disk temperature structure either stellar irradiation alone or viscous heating and stellar irradiation. Two different disk geometries are considered.

Eisner et al. 2005

3. Planetesimal formation by gravitational instability

The Goldreich & Ward mechanismWe have seen in the previous sections that bodies of order 1 meter are extremely rapidly removed from the solar nebula. An attractive solution to this problem is to simply bypass the critical size range. A mechanism for this has been put forward by Goldreich & Ward 1973. It is based on the idea that the self-gravity of the dust could lead to planetesimal formation. In a modified form, planetesimal formation by self-gravity has seen a revival in the last few years.

Armitage 2009

28

FIG. 20 Illustration of the Goldreich-Ward mechanism forplanetesimal formation. A combination of vertical settlingand (perhaps) radial drift results in a dust sub-disk whosedensity exceeds the local gas density. This sub-disk becomesthin enough to be gravitationally unstable, leading to frag-mentation into planetesimals.

cept. The reader should be aware that planetesimal for-mation is one of the more uncertain aspects of planetformation, and ongoing research continues to study bothpossibilities.

Figure 20 illustrates the basic idea underlying theGoldreich-Ward (1973) mechanism for planetesimal for-mation. A combination of vertical settling and radialdrift of small solid particles results in the formation ofa dense sub-disk within which the solid density exceedsthe local gas density (this obviously requires a very thinsub-disk if the local ratio of gas to dust surface densityis comparable to the fiducial global value of 100). Thesolid sub-disk then becomes gravitationally unstable, andfragments into bound clumps of solid particles that sub-sequently dissipate energy via physical collisions and col-lapse to form planetesimals.

_z

!

x

y

FIG. 21 Geometry for the calculation of the stability of auniformly rotating sheet.

Gravitational instability requires that the disk be mas-sive (high surface density) and / or dynamically cold(low velocity dispersion). The classic analysis of theconditions for gravitational instability is that of Toomre(1964). Here, we consider the stability of a rotating fluidsheet – this is somewhat easier than the collisionless cal-culation, gives the same answer to a small numerical fac-tor when the gas sound speed is identified with the parti-cle velocity dispersion, and carries over to the instabilityof a gas disk that we will discuss later. The simplestsystem to analyze is that of a uniformly rotating sheet– in what follows I follow the notation and approach ofBinney & Tremaine (1987).The setup for the calculation is as shown in Figure 21.

We consider a sheet of negligible thickness in the z =0 plane, with constant surface density !0 and angularvelocity ! = "z. Our aim is to calculate the stability ofthe sheet to in-plane perturbations. Working in a framethat corotates with the (unperturbed) angular velocity", the fluid equations are,

!!

!t+! · (!v) = 0 (129)

!v

!t+ (v ·!)v = "

!p

!"!#" 2!# v

+"2 (xx+ yy) (130)

where the momentum equation picks up terms for theCoriolis and centrifugal forces in the rotating frame.These equations apply in the z = 0 plane only. Thegravitational potential # is given by the Poisson equa-tion,

!2# = 4"G!#(z) (131)

which describes # in all space. In these equations, v =vxx + vyy is the velocity in the rotating frame, ! is thesurface density, and p = p(!) is the vertically integratedpressure. The sound speed is defined via,

c2s $dp

d!

!

!

!

!

!=!0

. (132)

In the unperturbed state, ! = !0, # = #0, v = 0 andp = p0 = p(!0). Substituting these values into the mo-mentum equation yields !#0 = "2(xx+ yy).

The basic idea is that dust settles into a sub-disk thin enough to be gravitationally unstable to itself, leading to fragmentation into planetesimals.

To derive a mathematical criterium, we can use or results on the stability of a rotating (gas) disk. We however now associate the velocity c no more with the sound velocity of the gas, but with the random velocities of the dust grains.

ω2 = 4Ω2 − 2πGΣ0|k| + k2c2 ≥ 0

In lecture five, we found the dispersion relation for the uniformly rotating sheet as:

ω2

unstable region

most unstable wave length

Q =2cΩ

πGΣ0> 1

Stability at the most unstable k gives the Toomre stability criterion k

The Goldreich & Ward mechanism IIWith that, we can estimate the maximum velocity dispersion / turbulent speed of dust grains that would still allow such a local collapse:

cΩGΣ0

≤ π

2→ c ≤ πGΣ0

Ω 7πG

2× 10−7 7 cm/s at 1 AU for MMSN

In other words, in a minimum mass solar nebula the dust must be extremely quiet if gravitational collapse is to take place in the dust disk. It has been argued that such low velocities are unlikely in presence of turbulence and additionally of the strong shear of over 100 m/s present between solids and gas. This shear is likely to stir up the dust and maintain it relatively turbulent (even in absence of background turbulence, as we will see below) and so prevent these instabilities to occur.

The speed of sound of the gas in the disk is of order 105 cm/s. The vertical scale height of the gas is proportional to the speed of sound, while the vertical scale height of the dust disk is proportional to the velocity dispersion. This implies for the unstable dust disk that

We see that an extremely thin disk is required before instability will set in! At r=1 AU, and assuming Hgas~0.05 r, this would require a dust scale height of only about 0.1 Earth radii.

The Goldreich & Ward mechanism III

Mmin = πλ2minΣ0 → R =

Mmin

4π3 δ

1/3

=

3λ2minΣ0

4δ

1/3

4.3 km at 1 AU

We can find the equivalent size of a solid body corresponding to the mass included in a patch with this most unstable wavelength radius:

dω2

dk= o with ω2 = 4Ω2 − 2πGΣ0|k| + k2c2

Solving for the wave number or wavelength:

kmax =πGΣ0

c2→ λmin =

2π

kmax 2.1× 108 cm at 1 AU

If the issue with the low necessary velocity dispersion was not to be the case, instability will proceed on patches of typical size (most unstable / critical wavelenght):

Size of the fragments

In other words, if the dust sheet were to gravitationally collapse, planetesimals with a typical size of approximately 4 km in size (assuming a solid density δ of 3 g/cm3) would be created. These bodies are clearly large enough that radial drift is no more an issue. The collapse itself happens on a very short timescale of about

which is of order less than one year.

It was pointed out (Weidenschilling 1977) that even in absence of any intrinsic turbulence of the gas disk, it is unlikely that the dust could settle into the necessary extremely thin layer. This is due to a classical hydrodynamical instability, the Kelvin-Helmholtz instability. KH instabilities develop at the interface of two shearing fluids of different densities.

Kelvin-Helmholtz instabilities

Such a situation is encountered here:

-the thin midplane dust layer rotates Keplerian (dragging along the gas)-the gas above the midplane layer rotates (as seen before) with sub-Kepler rotation

A classical stability analysis lead to a criterion in terms of the Richardson number:

Kelvin-Helmholtz instability

Vertical

stratification

Shear between dust layer and gas above it:

Two ‘forces’:

1. Shear tries to induce turbulence

2. Vertical stratification tries to stabilize things

Richardson number:

!

Ri = "g

#

$#

$z

$v

$z

%

& '

(

) *

2

Ri>0.25 = Stable

Ri<0.25 = Kelvin-Helmholtz instability: turbulenceFor Ri>0.25, the flow is stable, while for Ri<0.25, the instability develops. Hydrodynamical simulations in a protoplanetary disk indicate rather a critical value of order unity.

Kelvin-Helmholtz instability

Vertical

stratification

Shear between dust layer and gas above it:

Two ‘forces’:

1. Shear tries to induce turbulence

2. Vertical stratification tries to stabilize things

Richardson number:

!

Ri = "g

#

$#

$z

$v

$z

%

& '

(

) *

2

Ri>0.25 = Stable

Ri<0.25 = Kelvin-Helmholtz instability: turbulence

Vertical stratificationdust

gas

Two competing mechanism are at work:

1.Shear tries to drive turbulence 2.Vertical stratification tries to stabilize

Dullemond

Kelvin-Helmholtz instabilities IIKelvin-Helmholtz instability

Ri = 0.07, Re = 300

www.riam.ky ushu-u.ac.jp/ship/STAFF/hu/f low.html

Kelvin-Helmholtz instability

The left plot shows a hydrodynamical solution of the KH instability, while the right shows it occurring above a chain of mountains.

Kelvin-Helmholtz instabilities III

Fig. 11. The onset of the Kelvin-Helmholtz instability for cm-sized pebbles with!0"f = 0.02. The initial Gaussian particle distribution falls towards the mid-plane ofthe disk on the characteristic time-scale of tgrav = 1/(!2

0"f) ! 50!!10 . The increased

vertical shear in the gas rotation velocity eventually makes the disk unstable to theKHI, forming waves that finally break as the turbulence goes into its non-linearstate.

The disk temperature and density decrease with the distance from the centralobject and thus there is a global pressure gradient in the disk pointing radiallyoutward. This radial pressure gradient supports the disk against gravity fromthe central object and the centrifugal force is reduced to a sub-Keplerianrotation rate by a few permille.

Even this small sub-Keplerian rotation can lead to a dramatic ”rain out” ofa certain size particles radially into the central star because they will moveup the pressure gradient. The critical size is reached when the particle growthtime via sweeping up smaller dust grains becomes larger than the radial drifttime (See Fig. 12).

In the case of no external forces the e!ect of turbulent concentration wouldbe simple: Centrifugal forces would expel particles from turbulent eddies andconcentrate them in convergence zones between the vortices (See Fig. 13). In

13

Conclusion: Turbulence (intrinsic or self induced) prevents self gravitational planetesimal formation formation in a global instability as the Goldreich-Ward mechanism.

Klahr 2008

Simulation of the onset of the Kelvin–Helmholtz instability for centimeter-sized pebbles with Ω ts = St = 0.02. The initial Gaussian particle distribution falls towards the mid-plane of the disk on the characteristic time-scale of

~50 Ω-1. The increased vertical shear in the gas rotation velocity eventually makes the disk unstable to the KHI,forming waves that finally break as the turbulence goes into its non-linear state.

τ =

1Ω

1

tsΩ

3.1 Gravoturbulent planetesimal formation

Gravoturbulent planetesimal formationIn the past few years, the interest in gravitational instability as planetesimal formation mechanism was renewed due to the insight that dust may get concentrated locally in a turbulent disk. If its density gets high enough, it collapses due to its gravity, forming gravitationally bound bodies.

The gas does not just passively advect particles, but also suffers a back-reaction which becomes significant at particle concentrations that are much smaller than those needed for the onset of gravitational col- lapse.

!0.0010 !0.0005 0.0000 0.0005 0.0010!/(cs

2"0!2)

0.98

0.99

1.00

1.01

1.02

# d/#0

"0$f=2%10!7

!0.0005 0.0000 0.0005 0.00100.0000!/(cs

2"0!2)

"0$f=2%10!2

0.6

0.8

1.0

1.2

1.4

# d/#0

Fig. 18. Plot of dust-to-gas ratio in bins of the local vortex parameter! ! ["(u · #)u] · f(u) for the turbulent flow. Anticyclonic vortices have a nega-tive value of ! , whereas for cyclonic vortices ! is positive. For the intermediatefriction time run (see right panel:1cm sized particles with !"f $ 10!2), there isa clear anticorrelation between vortex parameter and dust-to-gas ratio. This is anindication that dust is being trapped in anticyclonic vortices which are features ofthe turbulent accretion disk flow. Particles in the left panel are too small to getconcentrated (size is 0.1 µm). Taken from Johansen & Klahr 2005.

0 20 40 60 80 100

t/(2&"0!1)

0

100

200

300

400

500

600

700

max(n)

0

20

40

60

80

max(n)/

n0

50.0 50.5 51.0

200

400

600

Fig. 19. The maximum number of particles in a single grid cell as a function oftime for a run with meter-sized boulders, e.g. a Stokes number !"f of about unity.The maximum density is generally around 20 times the average, but peaks at above80 times the average particle density. The insert shows a magnification of the timebetween 50 and 51 orbits. Taken from Johansen, Klahr & Henning 2006.

and it is di!cult to produce a population of boulders above this meter-sizeregime 3 . So the first defining property of planetesimals is that they move

3 It is convenient to use the term meter-barrier, because for a distance of 5AU fromthe central object in a typical solar nebula the fastest drifting objects are of thisdimension. Yet as the drift velocity depends on the gas density, temperature andlocal orbital period the fastest drifting objects will be smaller at greater distance

20

The plot shows the maximum number of particles in a computational grid cell as a function of time for meter-sized boulders, e.g. a Stokes number of about unity. The maximum density is generally around 20 times the average, but peaks at above 80 times the average particle density.

Dust concentration occurs with two mechanism:

1) Passive concentration of particles by turbulence.

Johansen et al 2006

Instabilities can arise if we consider that the dust acts back on the gas dynamics. This happens already at dust concentrations << necessary for self-gravity.A stability analysis of the two-fluid system shows that it is susceptible to the linear streaming instability.The plot shows numerical simulations of this. Locally, very high dust densities are reached.

2) Two-fluid instabilities.

Youdin, Johansen 2007

Gravoturbulent planetesimal formation IINumerical simulations by Johansen et al. 2007 and by Johansen, Youdin & Mac Low 2009 indicate that the non-linear evolution of the streaming instability can lead to the gravitational collapse, at least if prior to this, collisional growth forms bodies of cm-scale. Cuzzi et al. 2008 propose an alternative model where concentration at the smallest turbulent scales lead to the formation of 10-100 km “sandpile” planetesimals out of chondrule size dust.

The figure shows particle column density as a function of time after self-gravity is turned on at t = 35 Torb. Each gravitationally bound clump is labelled by the mass contained in the its Hill sphere in units of Ceres masses. The insert shows an enlargement of the region around the most massive bound clump. Two clumps condense out within each other’s Hill spheres and quickly merge. At the end of the simulation bound clumps have masses between 0.5 and 7.5 MCeres. Ceres is a 970 km radius dwarf planet (MCeres = 9.5 × 1023 g ≈ 1.5 × 10-4 Mearth).

Joha

nsen

et a

l 201

1

Gravoturbulent planetesimal formation IIIKlahr & Johansen 2008

!10.0 !5.0 0.0 5.0 10.0t/Torb

100

101

102

103

104

max

(!p/!

g)

MHill/MCeres !!!> t

"K#f = 0.25

0.50

0.75

1.00

0.0 1.6 4.0 3.5 6.1 8.4

31.5

40.3

60.767.5

74.9

643, ~G=0.4

!10.0 !5.0 0.0 5.0 10.0t/Torb

100

101

102

103

104

max

(!p/!

g)

MHill/MCeres !!!> t

"K#f = 0.25

0.50

0.75

1.00

2.1 1.8

3.4

4.7

6.7

8.910.0

12.9

17.118.3

!10.0 !5.0 0.0 5.0t/Torb

100

101

102

103

104

max

(!p/!

g)

Self!gravitySedimentation

MHill/MCeres !!!> t

"K#f = 0.25

0.50

0.75

1.00

0.8 0.8 0.9

1.9 2.1

3.0

3.5

2563

Fig. 25. Mass accretion onto a gravitationally bound cluster at three di!erent nu-merical resolutions. The plots show the maximum bulk density of solids as a functionof time, normalized by the average gas density. Drag force and vertical gravity areturned on at t = !10, while self-gravity and collisional cooling are turned on att = 0. The density increases monotonically after the onset of self-gravity becausegravitationally bound clusters of boulders form in the mid-plane. After only sevenorbits peak densities in these clusters approach 104!g or a million times the averageboulder density in the disc. The colored bars show the mass contained within themost massive Hill sphere in the box, in units of the mass of the 970 km radiusdwarf planet Ceres (MCeres = 9.5" 1023 g). The most massive cluster in the highestresolution case accretes about 0.5MCeres per orbit (the entire box contains a totalboulder mass of 50MCeres). The cluster consists of approximately equal fractionsof the three larger boulder sizes. The smallest size, with "K#f = 0.25, is initiallyunderrepresented with a fraction of only 15% because of the stronger aerodynamiccoupling of those particles to the gas, but the fraction of small particles increaseswith time as the cluster grows massive enough to attract smaller particles as well.The mean free path inside the bound clusters is shorter than the size of the clus-ter, so any fragments formed in catastrophic collisions between the boulders will beswept up by the remaining boulders before being able to escape the cluster.

be an understanding of how dust grows to meter-sized boulders in a turbulentdisk, using 3D-simulations that do not oversimplify the relevant physics. Onlyif we have the proper initial conditions of the boulder size distribution and inaddition know the proper turbulent state of protoplanetary disks, will it bepossible to put a solution to the planetesimal formation problem.

8 The core accretion - gas capture model

Once there is a su!cient population of planetesimals, e.g. the majority ofdust was converted into more than kilometer sized objects, a period of oli-garchic growth starts (Thommes et al., 2006). In this period the motion of theplanetesimals is decoupled from the gas and strictly determined by the gravita-tional N-body interactions between the planetesimals. As a result of the grav-itational interaction during close approaches the planetesimals increase theire"ective cross section for collisions (gravitational focusing: Safronov (1969)).

26

The plot on the left shows the maximum bulk density of solids as a function of time, normalized by the average gas density. Drag force and vertical gravity are turned on at t = −10, while self-gravity and collisional cooling are turned on at t = 0. The density increases monotonically after the onset of self-gravity because gravitationally bound clusters of boulders form in the mid-plane. The colored bars show the mass contained within the most massive Hill sphere in the computational box, in units of the mass of Ceres.

Turbulence aided self-gravitational growth thus might proceed directly from pebbles or chondrules to intermediate-sized (100-1000 km) objects.The rapid formation of such large planetesimals are the interesting prediction of the current models. Though it is probably still too early to regard this as a definitive conclusion, it could indicate a paradigm shift from a purely coagulative growth to a growth mode where self-gravity is important.

Note that additional particle concentration mechanisms exist. Examples are long-lived vortices or spiral waves in the gas disk.

Gravoturbulent planetesimal formation IV

cumulative SFD scaled up by a factor 1000 and N0!> Di" is a cumu-lative SFD generated in the Monte-Carlo code. We have found that95.8% of the Monte-Carlo-generated SFDs have D < 6:14. By repeat-ing the Monte-Carlo experiment with different (large) decimationfactors p we have also checked that the value of D is basically inde-pendent on p, while all the curves in Fig. 2 shift along the y-axis pro-portionally to p (so that the solid line coincides with the currentSFD, scaled up by a factor 1=p). These results will be used when test-ing some of our model results in Sections 4 and 5.

The embryos-in-the-Asteroid-belt-model in Wetherill (1992)does not only explain the depletion of the Asteroid belt but alsothe final orbital excitation of eccentricities and inclinations of thesurviving asteroids and the radial mixing of bodies of different tax-onomic types (Petit et al., 2001). In addition, it provides a formida-ble mechanism to explain the delivery of water to the Earth(Morbidelli et al., 2000; Raymond et al., 2004, 2007).

In summary, the ‘‘indigenous embryos” model does a good jobat explaining the orbital and physical properties of the Asteroidbelt within the larger framework of terrestrial planet accretion.To date, it is the only model capable of doing so. Thus, if we trustthis model, embryos of at least one lunar mass had to exist inthe primordial Asteroid belt. This characterizes property (iv) of thepost-accretion main belt population. A successful accretion simula-tion should not only be able to form Asteroid-size bodies in themain belt, but also a significant number of these embryos.

3. The classical scenario: accretion from km-size planetesimals

We start our investigation by simulating the classical version ofStep 2 of the accretion process. In other words, we assume thatkm-size planetesimals managed to form in Step 1, despite the me-ter-size barrier; the accretion of larger bodies occurs in Step 2, bypair-wise collisional coagulation. We simulate this second stepusing our code Boulder. The simulations account for eccentricitye and inclination i excitation due to mutual planetesimal perturba-tions as well as !e; i" damping due to dynamical friction, gas dragand mutual collisions. Collisions are either accretional or disrup-tive depending on the sizes of projectiles/targets and their collisionvelocities. The disruption scaling law used in our simulations, de-fined by the specific dispersion energy function Q #

D, is the one pro-vided by the numerical hydro-code simulations of Benz andAsphaug (1999) for undamaged spherical basaltic targets at impactspeeds of 5 km/s. See the Electronic supplement for the details ofthe algorithm. However, in Section 3.4, we will examine what hap-pens if we use a Q #

D function that allows D < 100 km disruptionevents to occur much more easily than suggested by Benz andAsphaug (1999), as argued in Leinhardt and Stewart (2009) andStewart and Leinhardt (2009).

Here, and in all other simulations (unless otherwise specified),we start with a total of 1.6M$ in planetesimals within an annulusbetween 2 and 3 AU. By assuming a nominal gas/solid mass ratio of200, this corresponds to the Minimum Mass Solar Nebula as de-fined in Hayashi (1981). The bulk mass density of the planetesi-mals is set to 2 g=cm3, the average value between thosemeasured for S-type and C-type asteroids (Britt et al., 2002). Thesimulations cover a time-span of 3 My, consistent with the meanlifetime of proto-planetary disks (Haisch et al., 2001) and hencethe probable formation timescale of Jupiter. The initial velocity dis-persion of the planetesimals is assumed to be equal to their Hillspeed (i.e. vorb%Mobj=!3M&"'1=3, where vorb is the orbital speed ofthe object, Mobj is its mass, and M& is the solar mass). The lowersize limit of planetesimals tracked in our simulation is diameterD ( 0:1 km. Objects smaller than this size are removed from thesimulation. We record the total amount of mass removed in thismanner and, for brevity, refer to it as dust.

The initial size of the planetesimals is assumed to be D ( 2 km.In this simulation, the total mass lost into dust by collisional grind-ing is 7:45) 1027 g, i.e. more than one Earth mass but only 76% ofthe original mass. This is consistent with our claim in Section 2that, even starting with km-size planetesimals, low-velocity colli-sions cannot deplete more than 90% of the initial mass.

The final SFD of the objects produced in the simulation is illus-trated by the black curve in Fig. 3. We find this SFD does not repro-duce the turnover to a shallower slope that the post-accretionAsteroid belt had to have at D * 100 km (i.e. property (ii) of thereconstructed belt). Moreover, in the final SFD shown in Fig. 3,there are about 1.25 million bodies with D > 35 km. Even if wewere to magically reduce this population instantaneously by a fac-tor 200, in order to reduce the number of D > 100 km bodies to thecurrent number, we would still have *6200D > 35 km objectsremaining in the system. Recall (Section 2) that D * 35 km projec-tiles can form mega-basins on Vesta and that the formation of asingle basin is consistent with the existence of 1000 of these ob-jects in the main belt over 4 Gy. Thus, 6200 objects would statisti-cally produce six basins; the probability that only one mega-basinwould form, according to formula (1), is only 1.2%.

For all these reasons, we think that this simulation produces aresult that is inconsistent with the properties of the Asteroid belt.To test whether these results are robust, we performed additionalsimulations as detailed below.

3.1. Extending the simulation timescale

One poorly understood issue is how long the accretion phaseshould last, i.e. the required length of our simulations. Thus, wecontinued the simulation presented above up to 10 My. The resultis illustrated in Fig. 4a. We find that the total amount of mass lostinto dust increases only moderately, reaching at 10 My 78% of themass at t ( 0. Also, SFD does not significantly change between 3and 10 My. The size of the largest embryos does grow from slightlyless than 6000 km to about 7000 km, mostly by agglomerating ob-jects smaller than a few tens of kilometers. Accretion and colli-sional erosion reduce the cumulative number of D > 1 km objects

Fig. 3. The gray lines show the reconstructed (i.e. post-accretion) main belt SFD.The solid gray curve shows the observed main belt SFD for 100 < D < 1000 kmasteroids scaled up 200 times, so that the number of bodies with D > 100 kmmatches that produced in the simulation. The dashed lines show the upper andlower bound of the main belt power law slope in the 20–100 km range (Bottke et al.,2005a). The upper bound corresponds to the current SFD slope. The vertical dottedlines show the sizes of lunar/martian-sized objects for bulk density 2 g cm+3. Thesesize embryos are assumed to have formed across the inner Solar System (Wetherill,1992; O’Brien et al., 2007). The black curve shows the final SFD, starting from1:2) 1012 planetesimals with D ( 2 km, at the end of the 3 My coagulation/grinding process.

A. Morbidelli et al. / Icarus 204 (2009) 558–573 563

is still poorly understood. In the traditional view, planetesimalsgrow progressively from coagulations of dust and pebbles to km-sized objects. Consequently, the simulations of the second step ofthe accretion process, that in which collisional coagulation amongthe planetesimals leads to the formation of planetary embryos andgiant planet cores, usually starts from a population of km-sizedplanetesimals (e.g. Weidenschilling et al., 1997; Kenyon and Brom-ley, 2006).

However, recent paradigm-breaking work (Johansen et al.,2007; Cuzzi et al., 2008) showed that planetesimals might formbig (100 km or larger) thanks to the self-gravity of small bodieshighly concentrated in the turbulent structures of the proto-plan-etary disk. If this is true, then there are no km-sized initial plane-tesimals and the second step of accretion has to be somehowaffected by this change in ‘initial conditions’.

In this work, we have assumed the SFD of the ‘initial planetes-imals’ is unknown and we have attempted to constrain it bymatching the final SFD produced by the second step of the accre-tion process with the SFD of a reconstructed Asteroid belt. Morespecifically, the ‘target SFD’ that we try to reproduce with colli-sional coagulation simulations is the one that the Asteroid beltshould have had just prior to it being dynamical excited and de-pleted of material. The large body of work on the past history ofthe Asteroid belt, which was reviewed in Section 2, allows us to de-fine the shape and size of this target SFD.

While it is impossible to prove a negative result, we believe wehave run enough simulations to understand the response of thecoagulation process to various initial and environmental condi-tions. Based on these results, we find it likely that the SFD of theAsteroid belt cannot be reproduced from an initial population ofkm-sized planetesimals. Instead, we find that the reproduction ofthe Asteroid belt constraints requires that the initial planetesimalshad to span the size range from !100 to several 100 km, probablyup to 1000 km, and that their initial SFD had a slope similar to thatof the current SFD of asteroids in the same size range. Curiously,this result is reminiscent of the original intuition by Kuiper(1958) that the original Asteroid size distribution had to have aGaussian shape centered around 100 km.

Our result provides support for the idea that planetesimalsformed big (Johansen et al., 2007; Cuzzi et al., 2008). The preciseprocess that formed these big planetesimals is still an open issue.Our findings (size range and SFD slope of the initial planetesimals)should help constrain the planetesimal formation models.

We have also shown that, if the initial planetesimals can be asbig as 1000 km, the subsequent collisional coagulation processleading to the formation of planetary embryos is not seriously af-fected by the excitation of eccentricities and inclinations due tothe turbulence in the disk. This may provide a possible solutionfor the problem of planet formation in turbulent disks (Nelson,2005; Ida et al., 2008).

Fig. 8. Like Fig. 7, but starting from initial planetesimals in the 100–1000 km range and an SFD slightly shallower than that of the main belt in this size interval, as shown bythe bullets in panel (a). The solid gray line reproduces the current SFD of the main belt, rescaled so that the total number of D > 100 km objects matches that obtained in eachsimulation. (400 in panel (b), 1000 in all other panels.)

A. Morbidelli et al. / Icarus 204 (2009) 558–573 571

Morbidelli et al. 2009

The gray lines show the reconstructed (post-accretion) SFD. The vertical lines correspond to lunar resp. martian sized bodies, the size of embryos expected to form in the inner Solar System.

On the left, the black curve shows the simulated SFD starting from 1.2 x 1012 planetesimals of 2 km in size, at the end of the 3 Myr coagulation and grinding process. On the right, the initial sizes of the planetesimals are between 100 and 1000 km, with the distribution indicated by the dots.

Clearly, the observed SFD cannot be reproduced with planetesimals that grow (and fragment) starting at a small size but with large initial sizes.

Support for a scenario in which planetesimals are born big comes from the observation of the size frequency distribution SFD in the asteroid belt.

4. Collisional growth

Collisional growthEven if gravitational instabilities might be important at some size scales, collisional growth (also called coagulation for small particles) is still important, as first gravoturbulent growth needs mm to dm sized pebbles to start, and second, growth at large size scales will again occur through two body collision.

This scenario doesn't go without problem either:

Collisions can occur between bodies of different sizes and composition. At small scales, particle stick to each other relatively easily, at large scale gravity will provide the force keeping the bodies together. The problem lies at intermediate scales for which no good sticking mechanism has yet been found: The fact that these scales are not readily accessible to laboratory experiments adds significantly to the difficulties!

Coagulation Collisions can occur between bodies of different sizes and composition as well as at different relative velocities. While the existence of planets seem to indicate that on average collisions lead to growth, it is quite possible that a collision leads to the destruction of the objects involved. In this respect, the extreme events are what matters. The survival of a growing object rests on the probability that this object will not be subjected to a collisional energetic enough to destroy it in its entire lifetime.

To describe the collisional growth at small (dust) sizes, coagulation equations are used (Smoluchowski, 1916).

Here, Kij is the interaction kernel between bodies of mass mi and mj. The first term on the right represents the growth in the number of bodies of mass mk due to collisions of all possible pairs of particles with masses mi and mj which sum up to mk. The second term represents the decrease due to bodies of mass mk being used to form even larger bodies. Fragmentation is neglected.

Coagulation equation

mass1 2 3 4 5 6 7 8 9 10 11 12

Hit and stick between aggregates:

Size distribution function (discrete version):

!

Ni

= Number/cm3 of aggregates with i monomersThe simplest form of the coagulation equation can be written assuming that the masses of bodies are integral multiples of some small mass m1. At a given time t there are nk bodies of mass mk = km1. The coagulation equation in discrete form is then:

Dullemond

n

Coagulation II

Due to the enormous number of mass bins necessary to treat the coagulation equation in its discrete variant, one rather describes the population of dust grains with a continuous distribution function n(m,t) which gives the number density of particles of mass m at time t. The collisional time evolution of this function is described by the integral form of the coagulation equation:

Clearly, the first term on the RHS is the source term for the formation of mass m particles from coagulation of (m-m’) + m’ particles. The second term is the sink term due to the destruction/incorporation of m by collisions with m’.

While this equation doesn’t represent particular mathematical difficulties, the problem is to describe the interaction kernel K(m,m’,t) which describes the collisional outcome, and contains all the physics. This description must clearly include the composition, structure (porosity), collision velocity, collision geometry, etc...

Various lab experiments of dust aggregation have been conducted in order to derive the properties of the kernel. It turns out that various regimes exist.

Lab experiments on stickingLaboratory experiments probe when sticking, bouncing or shattering of particles occurs.

One can then try to compile the different experimental outcomes into a catalogue of regimes, as a function of particle sizes, porosities, and collision velocities.

Collision regimes

Guttler et al. (2010) compiled experimental results for collisionoutcomes with different particle sizes, porosities and speeds

Anders Johansen (Lund) Dust growth 25 / 36

Güttler et al. 2010

Laboratory experimentsLaboratory experiments used to probe sticking, bouncing andshattering of particles (labs e.g. in Braunschweig and Munster)Collisions between equal-sized macroscopic particles lead mostly tobouncing:

From Blum & Wurm (2008)

Anders Johansen (Lund) Dust growth 24 / 36

The image illustrates such an experiment. For collisions between equal-sized macroscopic bodies, mostly to bouncing occurs.

Blum & Wurm 2008

Lab experiments on sticking IIMain result of the laboratory experiments are:

1) There seems to exist a critical sticking velocity vstick. Collisions occurring at velocities smaller than this velocity appear to stick almost always while collision occurring at a velocity above this appear to never stick. Instead, bouncing or shattering occurs. While the exact value of this vstick depends upon the exact nature of the grains, it seems to be of order vstick=1 - 3 m/s. At least for ~ 1 meter bodies, this is a very small velocity compared to their drift velocity (~100 m/s).

2) Sticking may also occur at higher speed if a small impactor hits a large target.

3) Growing dust aggregates have a tentacular shape best described by a fractal dimension:

Here, m is the particle mass, sg is the radius of gyration and Df is the fractal dimension, typically Df≈1.8 - 2.1. This objects are thus very fluffy.LABORATORY EXPERIMENTS ON PREPLANETARY DUST AGGREGATION 275

Figure 6. Examples of fractal agglomerates consisting of monodisperse, spherical SiO2 particlesof 0.95µm radius resulting from sedimentation-induced aggregation in the levitation drum experi-ment. Dust aggregates were collected on a substrate and thereafter imaged in a Scanning ElectronMicroscope. The scale bars denote 10µm.

With this structural information, we modeled the aggregate growth in the TMPusing Smoluchowski’s equation (see Wurm and Blum, 1998, for details). In thereaction kernel K(mi,mj) = !(mi,mj) ! "c(mi,mj) ! vc(mi,mj), the stickingprobabilities were assumed to be !(mi,mj ) = 1, which is well-justified by the lowfractal dimension of the resulting aggregates (no restructuring, see Sect. 2.3). Forthe collision cross section, we took the expression derived by Ossenkopf (1993),and the collision velocity was approximated by

vc(mi,mj) = # |$f1 " $f2|, (7)

equivalent to the expressions for the relative velocities of dust grains in the so-lar nebula (see Weidenschilling and Cuzzi (1993)). This expression is valid forcoagulation driven by either turbulence or settling. Both turbulence and settlingshould produce similar size distributions in the perfect sticking regime. The onlyremaining variable in these equations is the proportionality constant # in Eq. 7which could be determined by fitting the calculated mass distribution curves to theirmeasured counterparts (Fig. 5). Thus, we derived that the mean collision velocityof the grains and aggregates in the TMP experiment was vc # 0.1m/s. From theexperiments presented in Sect. 2.3, we see that these velocities are well within thenon-restructuring aggregation regime.The second type of experiments that was performed in our laboratory simulates

the stage of preplanetary dust aggregation in which relative drift motion determinesthe collision rates among the dust grains. Micron-sized dust grains obtain sedimen-tation velocities in a gas of 10"3 bar pressure and under terrestrial conditions (1 gand 300K) of typically a few cm/s. Thus, the dust grains are lost to the containerwalls (at which they inevitably stick) within less than 1min. In a rotating experi-ment chamber, however, a cloud of dust particles can be aerodynamically levitated

Examples of fractal agglomerates consisting of monodisperse, spherical SiO2 particles (monomers) of 0.95 μm radius resulting from sedimentation-induced coagulation. The scale bars denote 10 μm.

LABORATORY EXPERIMENTS ON PREPLANETARY DUST AGGREGATION 275

Figure 6. Examples of fractal agglomerates consisting of monodisperse, spherical SiO2 particlesof 0.95µm radius resulting from sedimentation-induced aggregation in the levitation drum experi-ment. Dust aggregates were collected on a substrate and thereafter imaged in a Scanning ElectronMicroscope. The scale bars denote 10µm.

With this structural information, we modeled the aggregate growth in the TMPusing Smoluchowski’s equation (see Wurm and Blum, 1998, for details). In thereaction kernel K(mi,mj) = !(mi,mj) ! "c(mi,mj) ! vc(mi,mj), the stickingprobabilities were assumed to be !(mi,mj ) = 1, which is well-justified by the lowfractal dimension of the resulting aggregates (no restructuring, see Sect. 2.3). Forthe collision cross section, we took the expression derived by Ossenkopf (1993),and the collision velocity was approximated by

vc(mi,mj) = # |$f1 " $f2|, (7)

equivalent to the expressions for the relative velocities of dust grains in the so-lar nebula (see Weidenschilling and Cuzzi (1993)). This expression is valid forcoagulation driven by either turbulence or settling. Both turbulence and settlingshould produce similar size distributions in the perfect sticking regime. The onlyremaining variable in these equations is the proportionality constant # in Eq. 7which could be determined by fitting the calculated mass distribution curves to theirmeasured counterparts (Fig. 5). Thus, we derived that the mean collision velocityof the grains and aggregates in the TMP experiment was vc # 0.1m/s. From theexperiments presented in Sect. 2.3, we see that these velocities are well within thenon-restructuring aggregation regime.The second type of experiments that was performed in our laboratory simulates

the stage of preplanetary dust aggregation in which relative drift motion determinesthe collision rates among the dust grains. Micron-sized dust grains obtain sedimen-tation velocities in a gas of 10"3 bar pressure and under terrestrial conditions (1 gand 300K) of typically a few cm/s. Thus, the dust grains are lost to the containerwalls (at which they inevitably stick) within less than 1min. In a rotating experi-ment chamber, however, a cloud of dust particles can be aerodynamically levitated

Blum 2000

Small particles follow the same turbulent eddies and collide at low speeds. Brownian motion only is important. Very large bodies also have low mutual velocities as they decouple from turbulence.

864 F. Brauer et al.: Coagulation and fragmentation of grains

First, let us focus on Brownian motion. Two particles ofmass m1 and m2 in a region of the disk with temperature T havean average statistical relative velocity due to Brownian motiongiven by

!vB =

!8kT (m1 + m2)!m1m2

· (28)

This expression shows that relative thermal velocities are higherfor smaller dust particles than for larger dust particles. Hence,the growth process due to Brownian motion is more e"ective forsmall particles than for large particles. For example, if we as-sume a temperature of 200 K, a solid particle density of 1 g/cm3

and micrometer-sized particles then the relative particle veloc-ity due to Brownian motion is 0.2 cm/s. Particles of centimeterin size lead to a relative velocity of 10!7 cm/s. This particularexample shows that there is practically no coagulation due toBrownian motion for particles much larger than micrometer size.In general, growth by Brownian motion leads to fractal structuresand “flu"y” aggregates (cluster-cluster aggregates) (Ossenkopf1993; Kempf et al. 1999). However, we will ignore these intrin-sic properties of the dust particles in the course of this paper andassume a constant solid material density. See, however, Schmittet al. (1997) or Ormel et al. (2007) for dust particle coagulationmodels including porosity at a fixed radius in the disk.

Di"erential settling is the second process that leads to rela-tive velocities. If we assume that the solid particles are smallerthan the mean free path of the gas then the equilibrium settlingvelocity is given by zSt#k (Dullemond & Dominik 2004). Inthis expression St is the Stokes number introduced in Sect. 2.4.However, for Stokes numbers larger than unity, the equilib-rium settling velocity model loses validity. Very large bodies(St " #) above or below the midplane follow an orbit that istilted with respect to the midplane. The settling velocity towardsthe midplane can not exceed the vertically projected Kepler ve-locity zVk/r corresponding to this inclined orbit. For this reasonwe restrict the settling velocity to be the projected Kepler ve-locity at most. Considering this, we adopt the following settlingvelocity in our model,

vS =zSt#k

1 + St· (29)

The relative settling velocity between two particles of mass miand m j at a height z above the midplane then reads

!vS = z#k

""""""Sti

1 + Sti! St j

1 + St j

"""""" · (30)

The third source for relative velocities of particles in the disk isthe radial drift which was discussed in detail in Sect. 2.5. Therelative velocity in this case is simply the di"erence in the driftvelocities

!vD = |vtotdust,i ! vtot

dust,j|. (31)

The fourth relative velocity between the particles is due to tur-bulence in the disk. Relative particle velocities produced by tur-bulence were calculated numerically by Völk et al. (1980) andMizuno et al. (1988). Weidenschilling (1984) fitted these re-sults with analytical formulas. Current work by Ormel & Cuzzi(2007) shows that these expressions underestimate the turbulentrelative velocities for particles with large Stokes numbers. In thispaper, we will use the expressions calculated by Ormel & Cuzzi(2007).

Fig. 3. Relative velocities of dust particles at 1 AU in the disk as dis-cussed in Sect. 2.6.1. This calculation includes Brownian motion, dif-ferential settling and relative turbulent velocities. In this calculation weadopted a turbulent " value of 10!3.

Fig. 4. As Fig. 3, but now at 10 AU in the disk.

To give an impression of relative dust particle velocities ina protostellar disk, Fig. 3 shows a contour plot of this quantityat 1 AU including Brownian motion, di"erential settling, rela-tive turbulent velocities and relative particle drift velocities. Thesame calculation at 10 AU is shown in Fig. 4.

2.7. Particle fragmentation

Collisions between particle aggregates do not necessarily lead toparticle growth. For su$ciently high relative collision velocities,the aggregates may fragment into smaller bodies. The criticalthreshold velocity for this destructive process generally dependson the mass of the colliding particles. More precisely, fragmen-tation tends to play a non-negligible role if the kinetic collisionalenergy of the particles is of the order of the internal binding en-ergy of the particles (Borkowski & Dwek 1995). Fragmentationvelocities of aggregates are usually of the order of a few cm/s upto several 10 m/s. While smaller particles tend to stick to eachother up to high relative particle velocities (Dominik & Tielens1997) larger bodies show the tendency to fragment even forsmall relative velocities (Benz 2000). For simplicity, we will as-sume a fixed threshold velocity for particle destruction vf whichdoes not depend on the mass of the particles. However, we willinvestigate how the results of the simulations change if vf is var-ied over a wide parameter range. The dependency of vf on the

The plot shows relative velocities of dust particles at 1 AU in a disk with an α value of 10−3. This calculation includes Brownian motion, differential settling and relative turbulent velocities.

Meter barrier

-Roughly meter sized particles do not stick. Instead, for the typically expected high velocities, they shatter each other in collisions.-As seen earlier, roughly meter sized bodies drift extremely rapidly into the star.

The critical sticking velocity must be compared with the expected relative velocities between particles in the nebula.

In the intermediate size regime (0.1-1 meter), relative velocities can however become very high, mostly due to turbulence. These velocities are much larger than vstick. This is the second component of the so called meter barrier:

Brauer et al 2008

4.1 Numerical simulations of coagulative growth

Numerical models of coagulationBrauer et al. 2008 solved the full coagulation equation starting from μm-sized particles, using the results from lab experiments.

866 F. Brauer et al.: Coagulation and fragmentation of grains

Fig. 6. These plots show the particle size distribution at di!erent radii inthe disk after 1 Myr of disk evolution as discussed in Sect. 3.1. The leftand the right plot always belong together. From top to bottom more andmore growth mechanisms are included in the simulations. The upperpanel shows coagulation only due to Brownian motion (BM). The sec-ond panel shows BM and di!erential settling (DS). Finally, BM+DSand turbulent coagulation (TC) are shown in the lowest panel. The leftplots show the surface density of the dust at 1, 10 and 100 AU as a func-tion of particle radius. On the right side the corresponding contour plotsof the dust surface density are shown as a function of the radial locationin the disk and the particle radius. In these simulations the radial driftas well as the fragmentation of the dust particles were neglected.

turbulent ! parameter is 10!3 and the turbulent q parameter is setto be 1/2. At the beginning of each simulation the dust is equallydistributed between a dust particle size of 0.5 µm and 0.8 µm.

Let us first focus on the particle growth due to Brownianmotion at di!erent radii in the disk. The result of this simula-tion, i.e. the particle size distribution after 1 Myr, is shown inthe upper panel of Fig. 6. According to these results dust parti-cles grow from sub-micrometer to "30 µm in radius in 1 Myrat 1 AU in the disk. At 10 AU the particle distribution hasa maximum for a " 4 µm. At 100 AU most of the dust isroughly a micrometer in size. We conclude that particle growthdue to Brownian motion is not very e!ective, which is a wellknown result (Ossenkopf 1993; Schmitt et al. 1997; Dullemond& Dominik 2005). However, Brownian motion is an importante!ect for the following reason. We calculate the relative veloc-ities due to Browian motion, di!erential settling and turbulencefor a = 0.6 µm equal-sized particles at 1 AU in the disk. Whilethe relative particle velocity due to Brownian motion is 0.4 cm/sthe relative turbulent velocity is in the order of 10!8 cm/s. Therelative velocity due to di!erential settling is practically zero.Dust particle growth due to di!erential settling or turbulencegets of importance only for larger particles. Therefore, Brownianmotion is a trigger mechanism for the entire coagulation processwhich was noted before by Weidenschilling (1984).

Now, we will additionally include coagulation due to di!er-ential settling into our model. The result of this simulation isshown in the second panel of Fig. 6. This plot shows that par-ticles have grown to more than 104 cm in radius at 1 AU in

the disk after 1 Myr. This particle size is more than 6 ordersof magnitude larger than the grain size after 1 Myr caused byBrownian motion. Most particles at 10 AU and 100 AU havegrown to sizes of about 1 m and 100 µm, respectively. We con-clude that di!erential settling is an e!ective growth mechanismwhich can create large boulders in the inner parts of the disk.Note that in our model the vertical mixing continuously allowsthe grains to go back up again and grow again by di!erential set-tling. Therefore, the maximal size formula of Safronov (1969)does not apply here.

Apart from the fact that particles grow to much larger sizesif di!erential settling is included, Fig. 6 also shows that there isalways a certain amount of small particles that remains in thedisk and that does not coagulate for at least 1 Myr. After thistime, roughly 6% of the dust between 1 and 75 AU is still presentin grains <1 mm. The reason for this is the following. Not all ofthe dust particles coagulate at the same time. While a certainfraction of the dust has already grown to larger sizes and formeda thinner dust layer according to Eq. (15), a certain fraction ofsmall dust remains in the higher regions above the midplane.These small dust particles high above the midplane are subjectto a rather slow coagulation process. The dust densities abovethe midplane are low after most of the dust already settled closerto the midplane. This leads to long growth time scales accordingto Eq. (24). Larger particles close to the midplane can not sweepup the smaller particles above the midplane since turbulence isnot able to stir them up so far. For this reason small particlesremain in the disk for a long time.

We will now also include relative velocities of the particlescaused by random turbulent motions. The result of this simula-tion is shown in Fig. 6 in the lower panel. This plot indicates thatthe dominant grain size at 1 AU, i.e. the grain size correspond-ing to the surface density maximum, changes by a factor of "10if turbulent coagulation is included in the simulation. The dom-inant particle radius at 1 AU is "105 cm. At 100 AU, randomturbulent motions also speed up the coagulation process whichleads to particles of a few centimeters in radius after 1 Myr ofdisk evolution. Without relative turbulent velocities included inthe simulation, the particle radius was two orders of magnitudesmaller.

3.2. Step 2 – Coagulation and radial motion

We will now include radial motion, both as transport and as extrasource of relative velocities for coagulation. This significantlychanges the results of the last section. We find that the radial driftof solid particles is so high that the dust drift into the evaporationzone long before larger particles in the disk can possibly form.This happens even though an additional source for coagulationis introduced which decreases the coagulation time scales. Wewill investigate if particles can in some way “break through” theradial drift barrier.

3.2.1. Time evolution of the disk

Figure 7 shows the time evolution of the model. This plot in-dicates that cm-dm-sized particles form in the inner regions ofthe disk (<2 AU) within the first 103 yrs. Compared to the outerparts of the disk the formation of these particles appears ratherquickly due to comparatively high gas and dust densities andhigh temperatures. With increasing distance from the central starthe formation of larger particles gets more and more di"cult. At10 AU in the disk, it is still possible to form mm-sized particles

866 F. Brauer et al.: Coagulation and fragmentation of grains

Fig. 6. These plots show the particle size distribution at di!erent radii inthe disk after 1 Myr of disk evolution as discussed in Sect. 3.1. The leftand the right plot always belong together. From top to bottom more andmore growth mechanisms are included in the simulations. The upperpanel shows coagulation only due to Brownian motion (BM). The sec-ond panel shows BM and di!erential settling (DS). Finally, BM+DSand turbulent coagulation (TC) are shown in the lowest panel. The leftplots show the surface density of the dust at 1, 10 and 100 AU as a func-tion of particle radius. On the right side the corresponding contour plotsof the dust surface density are shown as a function of the radial locationin the disk and the particle radius. In these simulations the radial driftas well as the fragmentation of the dust particles were neglected.

turbulent ! parameter is 10!3 and the turbulent q parameter is setto be 1/2. At the beginning of each simulation the dust is equallydistributed between a dust particle size of 0.5 µm and 0.8 µm.

Let us first focus on the particle growth due to Brownianmotion at di!erent radii in the disk. The result of this simula-tion, i.e. the particle size distribution after 1 Myr, is shown inthe upper panel of Fig. 6. According to these results dust parti-cles grow from sub-micrometer to "30 µm in radius in 1 Myrat 1 AU in the disk. At 10 AU the particle distribution hasa maximum for a " 4 µm. At 100 AU most of the dust isroughly a micrometer in size. We conclude that particle growthdue to Brownian motion is not very e!ective, which is a wellknown result (Ossenkopf 1993; Schmitt et al. 1997; Dullemond& Dominik 2005). However, Brownian motion is an importante!ect for the following reason. We calculate the relative veloc-ities due to Browian motion, di!erential settling and turbulencefor a = 0.6 µm equal-sized particles at 1 AU in the disk. Whilethe relative particle velocity due to Brownian motion is 0.4 cm/sthe relative turbulent velocity is in the order of 10!8 cm/s. Therelative velocity due to di!erential settling is practically zero.Dust particle growth due to di!erential settling or turbulencegets of importance only for larger particles. Therefore, Brownianmotion is a trigger mechanism for the entire coagulation processwhich was noted before by Weidenschilling (1984).

Now, we will additionally include coagulation due to di!er-ential settling into our model. The result of this simulation isshown in the second panel of Fig. 6. This plot shows that par-ticles have grown to more than 104 cm in radius at 1 AU in

the disk after 1 Myr. This particle size is more than 6 ordersof magnitude larger than the grain size after 1 Myr caused byBrownian motion. Most particles at 10 AU and 100 AU havegrown to sizes of about 1 m and 100 µm, respectively. We con-clude that di!erential settling is an e!ective growth mechanismwhich can create large boulders in the inner parts of the disk.Note that in our model the vertical mixing continuously allowsthe grains to go back up again and grow again by di!erential set-tling. Therefore, the maximal size formula of Safronov (1969)does not apply here.

Apart from the fact that particles grow to much larger sizesif di!erential settling is included, Fig. 6 also shows that there isalways a certain amount of small particles that remains in thedisk and that does not coagulate for at least 1 Myr. After thistime, roughly 6% of the dust between 1 and 75 AU is still presentin grains <1 mm. The reason for this is the following. Not all ofthe dust particles coagulate at the same time. While a certainfraction of the dust has already grown to larger sizes and formeda thinner dust layer according to Eq. (15), a certain fraction ofsmall dust remains in the higher regions above the midplane.These small dust particles high above the midplane are subjectto a rather slow coagulation process. The dust densities abovethe midplane are low after most of the dust already settled closerto the midplane. This leads to long growth time scales accordingto Eq. (24). Larger particles close to the midplane can not sweepup the smaller particles above the midplane since turbulence isnot able to stir them up so far. For this reason small particlesremain in the disk for a long time.

We will now also include relative velocities of the particlescaused by random turbulent motions. The result of this simula-tion is shown in Fig. 6 in the lower panel. This plot indicates thatthe dominant grain size at 1 AU, i.e. the grain size correspond-ing to the surface density maximum, changes by a factor of "10if turbulent coagulation is included in the simulation. The dom-inant particle radius at 1 AU is "105 cm. At 100 AU, randomturbulent motions also speed up the coagulation process whichleads to particles of a few centimeters in radius after 1 Myr ofdisk evolution. Without relative turbulent velocities included inthe simulation, the particle radius was two orders of magnitudesmaller.

3.2. Step 2 – Coagulation and radial motion

We will now include radial motion, both as transport and as extrasource of relative velocities for coagulation. This significantlychanges the results of the last section. We find that the radial driftof solid particles is so high that the dust drift into the evaporationzone long before larger particles in the disk can possibly form.This happens even though an additional source for coagulationis introduced which decreases the coagulation time scales. Wewill investigate if particles can in some way “break through” theradial drift barrier.

3.2.1. Time evolution of the disk

Figure 7 shows the time evolution of the model. This plot in-dicates that cm-dm-sized particles form in the inner regions ofthe disk (<2 AU) within the first 103 yrs. Compared to the outerparts of the disk the formation of these particles appears ratherquickly due to comparatively high gas and dust densities andhigh temperatures. With increasing distance from the central starthe formation of larger particles gets more and more di"cult. At10 AU in the disk, it is still possible to form mm-sized particles

The plot shows the surface density of bodies as a function of semimajor axis and particle size at a disk age of 1 Myr. Kilometer sized planetesimals have grown.

The dust has grown under the influence of turbulence, brownian motion and differential settling.

1) no drift, no fragmentation

Numerical models of coagulation II2) with drift, no fragmentation

Tim

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Brauer, Dullemond and Henning: Coagulation and fragmentation of grains 11

Fig. 7. The particle size distribution at di!erent radii inthe disk at di!erent times of disk evolution as discussedin Sec. 3.2.1. In this simulation all particle gowth mecha-nisms are included as well as the radial motion of the dust.The fragmentation of particles is neglected. The left andthe right plots always belong together. The left columnshows the surface density as a function of particle radiusat 1, 10 and 100 AU. The right column shows the corre-sponding contour plots of the surface density as a functionof disk radius and particle radius. The white lines in thecontour plots denotes the particle radius for which theStokes number is unity (i.e. largest radial drift and largestradial velocities).

!-parameters instead of di!erent disk masses. The initialdust-to-gas ratio in this simulation is 10!2, the disk massis 10!2 M! and the result is shown in Fig. 9.

One would intuitively think that in a certain time par-ticles can grow to larger sizes in highly turbulent disksthan in low-turbulent disks. Fig. 9 shows, however, thatthe dominant particle size after 104 yrs is only weakly de-pendent on the turbulence parameter !. If ! changes bytwo orders of magnitude then the dominant particle sizeonly changes by a factor of two. This can be understoodby the following consideration.

A high amount of turbulence in the disk leads to highrelative turbulent particle velocities (Volk et al. 1980;Weidenschilling 1984; Cuzzi et al. 2001). These high rela-

Fig. 8. The e!ect of disk mass on the particle growth asdiscussed in Sec. 3.2.2. Shown is the dominant dust par-ticles radius after 104 yrs of disk evolution for di!erentdisk masses between 0.2 and 10 AU. The turbulent ! pa-rameter is 10!3 and the initial dust-to-gas ratio is 10!2.

Fig. 9. Same plot as Fig. 8 but now showing the ef-fect of turbulence on the particle growth as discussed inSec. 3.2.3. Shown is the dominant particles size after 104

yrs of disk evolution for di!erent turbulent ! parametersbetween 0.2 and 10 AU. The disk mass is 10!2 M! andthe initial dust-to-gas ratio is 10!2.

tive velocities cause high collision rates, cf. Eq. (24), whichfavour the process of coagulation. For this reason parti-cles should have grown to larger sizes in highly turbulentdisks. On the other hand, a large amount of turbulenceleads to thick particle layers. The dust is stirred up in thehigher regions of the disk which causes the average dustdensities to decrease. The collision rates in Eq. (24) areproportional to the particle number densities. Lower dustparticle densities lead to longer coagulation time scales.

fdg=1%

Brauer, Dullemond and Henning: Coagulation and fragmentation of grains 13

Fig. 11. This plot shows the results of a simulation inwhich the particles can break through the radial drift bar-rier as discussed in Sec. 3.2.4. Particle fragmentation isneglected in this simulation. Shown is the surface densitydistribution for the first 104 yrs of disk evolution for aninitial dust-to-gas ratio of 0.03 as a function of disk radiusand particle radius. The disk mass is 10!2 M! and theturbulent ! parameter is 10!3. The right side is a contourplot of the surface density. The left side shows the abso-lute values of the surface density for 3 di!erent disk radii(solid - 0.3 AU, dotted - 1 AU, dashed - 3 AU).

also indicates that the disk temperature and intrinsic par-ticle properties like solid density are rather unimportant aslong as the Stokes number of the particles is smaller thanunity and turbulence is the leading process that triggerscoagulation.

However, Ormel et al. (2007) have shown that theporosity of dust particles actually matters in the earlyphases of disk evolution. This discrepancy is due tothe fact that the Eq. (39) only holds if St > ! whileOrmel et al. (2007) considered particles with St < !.Moreover, Brownian motion is the main source for rela-tive particle velocities for small dust grains in the earlydisk evolution while the derivation of Eq. (39) assumesthat turbulence is the major source for relative particlevelocities.

3.2.5. The radial drift barrier

Now we will estimate in which regions of the disk andunder which conditions the solid particles can theoreticallyovercome the radial drift barrier.

In section 3.2.4 we have seen that particle coagulationdue to turbulence in the disk can be described by a =a"k"0. We define a particle growth time scale #g by

#g = $a

a=

$

"k"0. (40)

The parameter $ measures how much the solid particlehas to grow to cross the particle size region of fast radialdrift, i.e. to overcome the radial drift barrier. We assumethis parameter to have a certain value determined by thedisk model and to be a constant throughout the disk. Thelargest radial drift velocity in the disk is approximatelygiven by c2

s/Vk. We define a radial drift time scale #d by

#d =r

c2s/Vk

. (41)

The ratio between these two time scales is given by

#g

#d=

$

"0

!

H

r

"2

. (42)

In the last step we made use of Eq. (15). Now, the particlesmay overcome the radial drift barrier if the ratio #g/#d issmaller than unity, i.e. if the growth time scales are smallerthan the radial drift time scales. The parameter $ is stillindefinite.

To specify this parameter we consider Fig. 10 inSection 3.2.4. These simulation results show for which ini-tial dust-to-gas ratio "0 the particles break through themeter size barrier at a certain radius in the disk. We chosethe parameter $ in a way that the condition #d > #g is inagreement with the results shown in this figure. This leadsto $ ! 12. With this value, the particles should overcomethe radial drift barrier if the inequality

"0 ! 12

!

H

r

"2

(43)

holds.The particles, which break through the radial drift bar-

rier in Fig. 10, have already drifted inwards. For this rea-son, the critical value given by Eq. 43 indicates the initialdust-to-gas ratio for which the particles most likely breakthrough the radial drift barrier. The su#cient "0-value toovercome the radial drift barrier is presumably even lowerthan this value.

3.2.6. Dust mass loss in the disk

When particles drift into the evaporation zone, they arelost for the process of planetesimal formation. Hence, thequestion of how much solid material is actually lost due toits drift into the inner regions is of essential importance.We calculate the mass which is present in small (St < 1)

fdg=3%

In this case it is found that the drift barrier removes all particles for the nominal dust to gas ratio of 1%. Particles do not grow bigger than 0.1 to 1 meter. For a three times higher ratio, it is however possible to break through the drift barrier.

Numerical models of coagulation III3) with drift, with fragmentation

Tim

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Brauer, Dullemond and Henning: Coagulation and fragmentation of grains 15

Fig. 13. As Fig. 7, but now also the fragmentation ofparticles is included in the simulations as discusssed inSec. 3.3.1. The left column shows the surface density as afunction of particle radius at 1, 10 and 100 AU. The rightcolumn shows the corresponding contour plots of the sur-face density as a function of disk radius and particle size.

Fig. 13 indicates that the maximum dominant particlesize amax and the Stokes number St have the same radialbehaviour. This is due to the fact that relative particle ve-locities in our model (except Brownian motion) scale withthis dimensionless number. For this reason, the dominantparticle size follows amax ! r!0.8 which we obtain directlyfrom the definition given by Eq. (13).

Due to destructive collisions a large amount of dust ispresent in small grains as can be clearly seen in Fig. 13.We calculate the amount of dust which is present in grainslarger (smaller) than 10!2 cm after 105 yrs of disk evo-lution. While 18% of the dust mass is present in grainslarger than 10!2 cm, yet 82% of the mass is present insmaller grains. This large population of sub-mm grainsshould have a strong e!ect on the spectrum of the proto-stellar disk. However, we will not investigate the influenceof the fragmentation parameters, i.e. vf and !, on the diskspectrum which goes beyond the scope of this paper. Thiswill be investigated in the near future.

Fig. 14. The influence of the turbulence parameter " onthe dominant particle size after 104 yrs of disk evolutionfor di!erent disk radii between 1 and 20 AU as discussed inSec. 3.3.2. The disk mass is 10!2 M!, the fragmentationvelocity is 103 cm/s and the initial dust-to-gas ratio is10!2. This graph also shows the particle size for whichthe Stokes number ist unity. The # parameter is set to 0.5and $ = 1.

3.3.2. E!ect of turbulence

Di!erent turbulent "-values should lead to di!erent max-imum particle sizes due to destructive collisions. To in-vestigate the influence of turbulence on the fragmentationbarrier, we calculate the dominant particle size for di!er-ent "-values after 104 yrs of disk evolution. In this simula-tion the disk mass is 10!2 M!, the fragmentation velocityis 103 cm/s, the initial dust-to-gas ratio is 10!2 and theresults of the calculation are shown in Fig. 14.

According to this plot, the dominant particle size isfairly dependent on " in moderately turbulent disks. If "is changed from 10!3 to 10!4 then the dominant particlesize adom changes by a factor of " 5. We find that lessturbulence shifts the fragmentation barrier towards largerparticle sizes. Hence, in less turbulent disks particles cangrow to larger sizes than in highly turbulent disks.

However, this statement does not hold for extremelylow turbulent disks. In these disks, turbulence is not themain source for relative velocities and, hence, the frag-mentation barrier should not be dependent on ". If " issmaller than " (cs/2Vk)2 (cf. Eqs. 18 and 35) which is" 10!4 at 1 AU then relative particle velocities due to ra-dial motion exceed relative dust particle motions inducedby turbulence. To illustrate this independency we calcu-late the dominant particle size after 104 yrs for a diskwith a very low "-value of 10!10. The result of this cal-culation is also shown in Fig. 14. In this nearly laminardisk, destructive collsions due to relative drift velocities upto 50 m/s prevent particle growth to sizes of more than" 2 mm at 1 AU.

fdg=1%, vfrag=10m/s18 Brauer, Dullemond and Henning: Coagulation and fragmentation of grains

Fig. 18. These plots show how the particles break throughthe radial drift barrier and the fragmentation barrier asdiscussed in Sec. 3.3.3. Shown are contour plots of thesurface density as a function of disk radius and particleradius at 4 di!erent times of disk evolution. The fragmen-tation velocity is chosen to have the relatively high valueof 30 m/s. In this simulation ! = 2 and " = 0. The initialdust-to-gas ratio is 0.03.

sweep up smaller dust particles drifting inward from theouter regions. For this reason, most of the solid materialafter 1 Myrs has drifted into the evaporation zone and islost for the process of planetesimal formation.

3.3.5. E!ect of disk model

In the introduction we mentioned that the disk modeladopted in this paper di!ers significantly from the MMSNmodel. This leads to the question of how the results of thispaper change if di!erent disk models are considered. Inthis section, we repeat simulations of Sec. 3.3 with otherdisk model parameters, attempting to unveil the basicchanges in the dust particle distribution. Table 1 showsthe disk parameters for the simulations in this section.Model A and B are the MMSN model and the disk modelin this paper, respectively. Model C is our model, but nowwith 10% disk mass instead of 1% compared to M!. Thisleads to gas densities which are comparable to those of

Fig. 19. The mass of the dust disk between 1 AU and150 AU as a function of time for 3 di!erent initial dust-to-gas ratios as discussed in Sec. 3.3.4. In this simulation,particle growth particle fragmentation and radial motionare included. The initial disk mass of gas and solid mate-rial is 10!2 M!, # = 10!4, " = 0.5, ! = 2 and vf = 10 m/s.

Model Surface density Disk Temperaturepower law index ! mass power law index "

A 1.5 0.01 0.50B 0.8 0.01 0.50C 0.8 0.10 0.50D 1.5 0.01 0.62E 0.8 0.01 0.62F 0.8 0.10 0.62

Table 1. Disk parameters for the simulations performedin Sec. 3.3.5. The quantity $ denotes the temperaturepower law index T ! r!" . The Models A and B corre-spond to the MMSN model and the model adopted in thispaper, respectively. Model C is as the model in this pa-per but now with 10% disk mass. The Model D to F areas A to C but with a slighly steeper radial temperaturedependency.

the MMSN model. The mass distribution, however, hasa much flatter radial dependency. The models D to F arethe same as A to C, but with a steeper radial temperaturedependency. Andrews & Williams (2007) observationallyfind radial temperature profiles with a median power lawindex of 0.62. This is slighly higher than the passively ir-radiated disk profile of 0.5 adopted in our model.

Before we come to the results of the simulations,we will qualitatively discuss the di!erence between theMMSN model and the model in the paper at hand. Thegas mass densities of our model are generally smaller thanthose of the MMSN model. This has the following mainimplications. First, solid particles are less coupled to themotions of the gas. The coupling between the gas and the

fdg=3%, vf=30m/sFor the nominal parameters, fragmentation leads to a barrier for growth at very small radii in the mm to cm domain. Only with a implausibly high velocity leading to fragmentation coupled with a higher dust to gas ratio, it is possible to break through the barrier.

We conclude:-Coagulation works well to form cm-sized bodes.-Radial drift, shattering, and bouncing prevent further growth-Either there is something not understood about coagulation (e.g. sticky organic compounds) or a fundamentally different process in necessary.

Questions?