Planet Formation
Topic:
Gravoturbulent planetesimal formation:
Particle clustering,
Streaming instability,
Particle trapping
Lecture by: C.P. Dullemond
Main idea:
• If we can find a way to cluster the dust particles into
small regions („dust clouds“) where the density of dust
would be very large, then perhaps gravity could then
take over and cause the dust cloud to gravitationally
collapse (contract) to form a planetesimal.
• Many ideas have been posed in the literature:
–
–
–
–
The classic Goldreich & Ward model
The streaming instability
Turbulent clustering through intermittency
Dust trapping in local pressure maxima (rings, vortices)
Disk gravitational instability
and the
Goldreich & Ward idea
of planetesimal formation
Gravitational instability of a disk
• When a protoplanetary disk becomes too massive
compared to the central star, it can become
gravitationally unstable.
• The critical quantity is the „Toomre number“:
csWK
Q=
p GS
• If Q>2 then the disk is stable
• If 1<Q<2 then the disk develops huge spiral waves
• If Q<1 then the disk fragments
Gravitational instability of a disk
A perhaps more insight-giving way to write Q is:
æ H p öæ M * ö
csWK
cs GM * 1
Qº
=
@ ç ÷ç
÷
2
p GS WK r r p GS è r øè M disk ø
Where we estimated:
M disk @ p r S
2
Since for protoplanetary disks we typically roughly have H p » 0.05r
we can say that a disk that is more than 2.5% of the stellar mass
is gravitationally unstable, and if more than 5% then it is so unstable
that it will fragment.
This is a possible model of Gas Giant Planet formation!
(but more on that later)
Example of GI in disk
Gravitational instabilities
are global processes. Their
modeling requires global
hydrodynamic disk models.
Here is an example of how
the disk behaves: First
strong global spiral waves
are formed. Then (if Q<1)
gravitationally bound clumps
form.
Image: Quinn et al.
From: http://www.psc.edu/science/quinn.html
The old Goldreich & Ward idea
• In principle the same holds true for the dust subdisk:
æ H dust öæ M * ö
Q@ç
÷
֍
è r øè M dust ø
• In previous chapter we derived that big dust grains will
gather into a thin dust layer near the equator: the dust
subdisk.
• Less turbulence leads to thinner subdisk, i.e. smaller
ratio H/r, i.e. smaller Q, i.e. eventually Q<1, i.e.
formation of clumps = planetesimals.
The old Goldreich & Ward idea
Goldreich & Ward 1973
The old Goldreich & Ward idea
Goldreich & Ward 1973
The old Goldreich & Ward idea
Goldreich & Ward 1973
The old Goldreich & Ward idea
Goldreich & Ward 1973
But there is a problem...
Cuzzi 1993; Johansen, Henning & Klahr 2006; Chiang 2008
But there is a problem...
Cuzzi 1993; Johansen, Henning & Klahr 2006; Chiang 2008
z
Settling leads to ρd>ρg
ϕ
But there is a problem...
Cuzzi 1993; Johansen, Henning & Klahr 2006; Chiang 2008
vgas<vK
z
vlayer≈vK
ϕ
Gas in the disk is subkepler. But in the dust layer, where ρd>ρg,
the pressure-per-mass is lower. This layer will move almost Kepler.
This leads to shear between dust layer and gas above: KH instability!
But there is a problem...
Cuzzi 1993; Johansen, Henning & Klahr 2006; Chiang 2008
Settling leads to ρd>ρg
vlayer=vKepler>vg leads to Kelvin-Helmholtz instability
Induces turbulence, which prevents H from dropping too low,
and thus stabilizes against gravitational instability.
The Streaming Instability of
Youdin & Johansen
(Clustering at large turbulent eddy scales)
Streaming instability
Youdin & Goodman 2005; Johansen & Youdin 2007
Dust particles
with St<1 will
want to move
with the gas,
thus moving
sub-Kepler.
z
vdust≈vgas<vK
vdust≈vgas<vK
vdust≈vgas<vK
ϕ
Streaming instability
Youdin & Goodman 2005; Johansen & Youdin 2007
Dust particles
with St<1 will
want to move
with the gas,
thus moving
sub-Kepler.
z
vcluster≈vK
However, if
they form a
ϕ
cluster, then
this cluster acts as a kind of „large particle“ with St>1, thus wanting
to move Kepler.
Note: Cluster ≠ Aggregate! They don‘t stick, they just move together.
Streaming instability
Youdin & Goodman 2005; Johansen & Youdin 2007
If you have
both clusters
and individual
particles, then
the cluster
will sweep up
the individual
particles.
z
vdust≈vgas<vK
vdust≈vgas<vK
vcluster≈vK
vdust≈vgas<vK
ϕ
Streaming instability
Youdin & Goodman 2005; Johansen & Youdin 2007
If you have
both clusters
and individual
particles,
then the cluster
will sweep up
the individual
particles.
ϕ
This leads to
ever larger and compact clusters. The clusters will also ablate.
Finding out the real „clustering factor“ requires massive numerical
simulations.
z
Streaming instability
Youdin & Goodman 2005; Johansen & Youdin 2007
From: Johansen &
Youdin 2007
Turbulent Particle Clustering
by turbulent intermittency:
Cuzzi & Hogan
(Trapping at small turbulent eddy scales)
Squires & Eaton 1991; Hogan & Cuzzi 2001; Pan et al. 2011
Particle clustering in turbulence
Particles react most strongly to eddies with turn-over time equal
to the stopping time. For St<<1 particles these are the sub-eddies,
i.e. Eddies with leddy << Leddy = a H p
For such small eddies, Coriolis
forces are unimportant (exercise:
why?). And therefore these eddies
are never in geostrophic balance.
Centrifugal forces thus always
win and expel particles from the
eddies, and collect them between
the eddies.
Particle clustering in turbulence
Mathematical derivation (following Pan et al. 2011):
Start from the known equation for the acceleration of a dust particle:
dv particle (t)
dt
=-
1
t stop
(v
particle
(t) - vgas (x(t), t))
Assume τstop<<τeddy, so that
vparticle » vgas + small number
dv particle (t)
We can then replace
with
dt
and obtain (after rewriting slightly):
dv gas (x(t), t)
v particle (t) » vgas (x(t), t) - t stop
dt
in the above equation
dvgas (x(t), t)
dt
Particle clustering in turbulence
v particle (t) » vgas (x(t), t) - t stop
dvgas (x(t), t)
dt
Here the time derivative of the gas velocity is the comoving one:
dvgas
dt
º
¶vgas
¶t
+ (vgas × Ñ)vgas
At scales much below the largest eddy scale, the gas behaves
nearly incompressibly, so let us assume:
Ñ× vgas = 0
If we now take the divergene of the particle velocity (eq above):
Ñ× v particle » -t stop Ñ×[(vgas × Ñ)vgas ]
Using Einstein‘s summation convection we can write:
Ñ× v particle » -t stop (Ñi vgas,k )(Ñk vgas,i )
Particle clustering in turbulence
Ñ× v particle » -t stop (Ñi vgas,j )(Ñ j vgas,i )
One can show that this can be rewritten as:
Ñ × v particle » t stop ( w - sij sij )
1
2
2
Where ω is the vorticity, ω2 is called the enstrophy:
w = Ñ ´ vgas
2
2
and sij is the strain tensor:
w 2 º eijk (Ñ j vgas,k )eilm (Ñl vgas,m )
= (d jldkm - d jmdkl ) (Ñ j vgas,k )(Ñl vgas,m )
sij = 12 ( Ñi vgas, j + Ñ j vgas,i )
Conclusion: particles diverge from regions of high enstrophy and
converge into regions of high strain.
Particle clustering in turbulence
Particles diverge from regions of high enstrophy and converge into
regions of high strain. This happens faster for larger particles (as
long as they have stopping time smaller than the eddy turn-over
time.
high
enstrophy
high
enstrophy
high
strain
high
enstrophy
high
enstrophy
Particle clustering in turbulence
DEMO
Very simple model (yes, you may do this at home! ;-)):
Periodic 2D box with exactly
4 „eddies“ given by the formula:
1
vgas,x = -cos(2p x + f )sin(2p y + y )
vgas,y = sin(2p x + f )cos(2p y + y )
y
The angles φ and ψ are random angles
that are randomly changed between 0 and 0
0
2π every eddy turnover time.
x
Now insert particles with some stopping time and see what happens
1
Particle clustering in turbulence
DEMO
Multiplicator model of Hogan & Cuzzi
Idea:
As eddy splits,
dust is unevenly
distributed. By
repeating this
again and again
(multiplicator)
you can get
(rare) excessive
clusters.
These may
exceed Roche
density and become
a planetesimal.
Note: Still controversial
Figure: Chambers, Icarus 208 (2010) 505–517
Which amount of clustering can
trigger gravitational contraction?
The Roche density
Roche density
• A cluster of matter can only remain gravitationally
bound if it lies within its own Hill radius. If not, the tidal
forces by the Sun will shear the cluster apart.
æ M ö
RHill = ç
÷ a
è 3M * ø
1/3
• Now replace:
M = 43p r R3 and M* = 43p r* R*3
RHill æ r ö a
=ç
÷
R è 3r* ø R*
1/3
Roche density
RHill æ r ö a
=ç
÷
R è 3r* ø R*
1/3
• When R<RHill the cluster is gravitationally sufficiently
bound to withstand the shear (tides of the Sun):
æ R* ö
r > rRoche º 3ç ÷ r*
èaø
3
• For Sun: ρ*=1.4 g cm-3 and R*=6.96x1010 cm, we get
-3
æ a ö
-3
rRoche º 4.3´10 ç
g
cm
÷
è AU ø
-7
Roche density
-3
æ a ö
-3
rRoche º 4.3´10 ç
g
cm
÷
è AU ø
-7
• What is the closest distance a comet (ρ≈1g/cm3) can
get to the Sun without being sheared apart? Assume
that a comet is only bound by its own gravity.
• Answer: About 1.6 R*.
• Example: Comet Lovejoy
Relation Roche density & Toomre Q
• We now have 2 criteria for gravitational collapse:
– Toomre: Q<1
– Roche: ρ>ρRoche
• Which is which?
• First work this out for a gas disk:
3æ
3
ö
4
p
æ
ö
csWK
M*
r
R
4
R
r
* *
*
*
3
ç
÷÷
Q=
= Hp 3 =
=
ç
÷
ç
3
p GS
p a S p 2p a rmidplane 3 2p è a ø è rmidplane ø
• Q<1 leads to:
4 æ R* ö
rmidplane >
ç ÷ r*
3 2p è a ø
• ...which is very close to the Roche condition (only a
factor of ~6 lower critical density).
3
Difficulty of gravoturbulent PF
• The gas must have density much lower than the
Roche density, otherwise the gas disk itself would
become unstable.
• The dust must cluster to a density well above the
Roche density to start the contraction that produces a
planetesimal.
• Therefore: the dust must cluster to densities well
above the gas density!
• Normally the dust-gas feedback (friction) will work
against this.
• Unless the streaming instability is triggered (for that
we need a long-lived gas pressure gradient).
Particle trapping in
global pressure bumps
Particles move toward pressure peak
v drift
æ d lg Pgas ö cs2
1
=
÷
-1 ç
St + St è d lgr ø v K
P(r)
r
Whipple 1972; Barge & Sommeria 1995; Klahr & Henning 1997; Kretke & Lin 2008;
Dzyurkevich et al. 2010; Kato et al. 2010; Johansen et al. 2009; Garaud 2007
New Paradigm: Particle Trapping
Barge & Sommeria; Klahr & Henning; Johansen et al.; Kretke & Lin; Kato et al.; Dzyurkevich et al.
New Paradigm: Particle Trapping
Barge & Sommeria; Klahr & Henning; Johansen et al.; Kretke & Lin; Kato et al.; Dzyurkevich et al.
New Paradigm: Particle Trapping
Barge & Sommeria; Klahr & Henning; Johansen et al.; Kretke & Lin; Kato et al.; Dzyurkevich et al.
Presumably we already observe them...
LkHa 330
SR 21
Brown et al. 2009
HD 135344B
Particle trapping in
local pressure bumps
(anticyclonic vortices)
Vortices as pressure traps
Simple model of a vortex
Remember the local equations of motion of a parcel of gas in
the corotating frame:
GM
x - 2Wy = 3 3 x + f x
r0
y + 2Wx = f y
Let us, for the sake of simplicity, ignore the 3(GM/r^3)x term,
because it makes the solution much harder, and is not critical
for the basic mechanism. The forces are the pressure gradient:
1 ¶P
x - 2Wy = r ¶x
1 ¶P
y + 2Wx = r ¶y
Vortices as pressure traps
Simple model of a vortex
Let us assume that the gas flows in concentric circles with
angular frequency ω:
x(t) = Acos(w t)
y(t) = Asin(w t)
Let us also assume as Ansatz that the pressure goes as:
P(x, y) = P0 + P0 ( x + y
2
2
)
Inserting this into the above equations, and replacing for simplicity
ρ with the average ρ0, yields two identical equations, namely:
-w - 2Ww = 2
2P0
r0
Vortices as pressure traps
Simple model of a vortex
P0 = 12 w (w + 2W) r0
One can see that for
-2W < w < 0
one gets
P0 < 0
meaning that the pressure profile in the vortex has a
local maximum:
P(x, y) = P0 + P0 ( x 2 + y 2 )
Anticyclonic vortices have a pressure maximum, while cyclonic
vortices have a pressure minimum. Anicyclonic vortices can only
exist because of the Coriolis forces (Ω>0).
Vortices as pressure traps
Anti-cyclonic vortex:
Dust gets trapped
Cyclonic vortex:
Dust gets expelled
Star
Barge & Sommeria (1995), Klahr & Henning (1997)
Presumably we also already see these!
Thanks to ALMA!
HD142527
Casassus et al. 2013
IRS 48
Van der Marel et al. 2013