Planetesimal formation in turbulent protoplanetary discs Anders Johansen Planet formation Dust in MHD turbulence Planetesimal formation Zonal flows Analytical model Global models Streaming and self-gravity Dead zones Conclusions Planetesimal formation in turbulent protoplanetary discs Anders Johansen Leiden Observatory, Leiden University “Workshop on the Magnetorotational Instability in Protoplanetary Disks” (Kobe University, June 2009) Collaborators: Andrew Youdin, Hubert Klahr, Wladimir Lyra, Mordecai-Mark Mac Low, Thomas Henning
50
Embed
Planetesimal formation in turbulent protoplanetary discsiccg/pub/associated/fy2009/...Planetesimal formation in turbulent protoplanetary discs Anders Johansen Planet formation Dust
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
cm → km: Sticking or gravitationalinstability(Safronov 1969, Goldreich & Ward 1973, Weidenschilling &
Cuzzi 1993)
Dynamics of turbulent gas importantfor modelling dust grains and boulders
William K. Hartmann
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Overview of planets
Protoplanetary discs
Dust grains
Pebbles
Gas giants andice giants
Terrestrial planets
Dwarf planets+ Countless asteroids and Kuiper belt objects+ Moons of giant planets+ More than 300 exoplanets
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Particle dynamics
Gas accelerates solid particles through drag force:
∂w∂t = . . .− 1
τf(w − u)
@@
@I
Particle velocity @@I
Gas velocity
In the Epstein drag force regime, when the particle is muchsmaller than the mean free path of the gas molecules, thefriction time is (Weidenschilling 1977)
τf =a•ρ•csρg
a•: Particle radius
ρ•: Material density
cs: Sound speed
ρg : Gas density
Important nondimensional parameter in protoplanetary discs:
ΩKτf (Stokes number)
At r = 5 AU we can approximately write a•/m ∼ 0.3ΩKτf .
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Diffusion-sedimentation equilibrium
Diffusion-sedimentationequilibrium:
Hdust
Hgas=
√δt
ΩKτf
Hdust = scale height of dust-to-gasratio
Hgas = scale height of gas
δt = turbulent diffusion coefficient,like α-value
ΩKτf = Stokes number, proportional
to radius of solid particles(Johansen & Klahr 2005)
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Diffusion coefficient
Definition of Schmidtnumber:Sc = νt/Dt = αt/δt
From thescale-height of thedust one cancalculate thediffusion coefficient:δt = δt(Hdust)
Density bumps and zonal flows correlated on tens of orbits
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Analytical model
BxBy(x,t)/<BxBy(t)>
−1.0 −0.5 0.0 0.5 1.0x/H
0
10
20
30
40
50t/T
orb
0.7
1.3
−1.0 −0.5 0.0 0.5 1.0x/H
0
10
20
30
40
50
t/Tor
b
ρ/<ρ> 0.95 1.05
−1.0 −0.5 0.0 0.5 1.0x/H
0
10
20
30
40
50
t/Tor
b
uy/cs −0.05 +0.05
Analytical model of zonal flow excitation and saturation
Need to connect a known (measured) stress and stress variationto amplitude of density bumps and zonal flows
Forcing of the zonal flow by stress variation
Geostrophic balance between pressure bump and zonalflow envelope
Damped random walk model
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Variation in stress
Linearised, axisymmetric evolution equation for uy :
∂u′y∂t
= −1
2Ωu′x + T ′
The tension term T ′ describes momentum transport byMaxwell stress:
T ′ =1
ρ0
1
µ0
∂〈BxBy 〉∂x
M = −µ−10 〈BxBy 〉
In shearing sheet the tension is simply the derivative of theMaxwell stress variation:
T ′ = − 1
ρ0
∂M ′
∂x
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Zonal flow dynamical equations
Linearised equation system for zonal flow excitation(hats denote wave amplitudes):
0 = 2Ωuy −c2s
ρ0ik0ρ
duy
dt= −1
2Ωux + T
dρdt
= −ρ0ik0ux −1
τmixρ
Assumed geostrophic balance between zonal flow andpressure bump
Density evolution includes turbulent diffusion term actingon time-scale τmix
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Solutions
Combine the three equations to get
Master equation
dρdt
=1
1 + k20 H2
(F − ρ(t)
τmix
)
F = −2ik0ρ0Ω−1T
Straight forward solution:
ρeq = τmixF
Only valid if correlation time of stress variation larger thanmixing time-scale. Need to model as damped random walk.Exciting at time-scale τfor and damping on time-scale τmix.
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Damped random walk
Re( )ρ
ρIm( )
Turbulent
diffusion
Stress
Correlation time equal to turbulent diffusion time-scale
What is the ampitude?
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Random walk solution
Solution involves product of forcing and mixing time-scales:
Random walk solution
ρeq
ρ0= 2√
ckτforτmixHk0T
cs
ck =1
1 + k20 H2
ρeq ∝ k−10 for k0H 1
ρeq ∝ const for k0H 1
How to find amplitude of zonal flow:
Take ρ0, H, Ω from disc modelRead off T , τmix and τfor from simulationSolution gives ρeq at a given scale k0
Formula predicts pressure bump amplitude of ρeq ≈ 0.08
In fairly good agreement with the measured ρeq ≈ 0.05
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Global models
Lyra, Johansen, Klahr, & Piskunov (2008):
Global disc with boulders on Cartesian grid (disk-in-a-box)
Gas density (320× 320× 32) Particle density (106 particles)
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Space-time plots
Gas density structure from Lyra et al. (2008):
Gas Density
0
20
40
60
80100
Model A
0.5 1.0 1.5 2.0s
020
40
60
80100
Model C
t/(
2πΩ 0-1
)
0.5
1.0
1.5
Gas Densities - Comparison
0.5 1.0 1.5 2.0 2.5s
0.40.6
0.8
1.0
1.2
1.4
1.6ρ
Model A (cs0=0.05)Model C (cs0=0.20)
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Stress variation
At any given time there are approximately 10% variationsin the α-value
This is enough to launch zonal flows
Similar variations reported in Fromang & Nelson (2006)
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Inverse cascade
Plots show power contribution ofdifferent terms in the inductionequation:
Magnetic energy cascades tolargest scales in the box
Happens through theadvection term
Excites large scale variation inMaxwell stress
Very little large scale activityin the vertical fieldcomponent
−4·10−5
−2·10−5
0
2·10−5
4·10−5
dBx2 /d
t^
AdvAdv (K)Com
StrRes
−4·10−4
−2·10−4
0
2·10−4
4·10−4
dBy2 /d
t^
AdvAdv (K)Com
StrStr (K)Res
1 10k
−2·10−5
−1·10−5
0
1·10−5
2·10−5
dBz2 /d
t^
AdvAdv (K)Com
StrRes
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Streaming instability
Gas rotates slightly slower than KeplerianParticles lose angular momentum due to headwindParticle clumps locally reduce headwind and are fed byisolated particles
Youdin & Goodman (2005) :“Streaming Instabilities in Protoplanetary Disks”
Gas rotates slower than Keplerian because of radialpressure gradientGas and solid components “stream” relative to each otherRadial drift flow of solids is linearly unstableGrowth on dynamical time-scale for marginally coupledsolids (rocks/boulders)
NSH86 equilibrium
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Clumping
Linear and non-linear evolution of radial drift flow ofmeter-sized boulders (ΩKτf = 1):
t=40.0 Ω−1
−20.0 −10.0 +0.0 +10.0 +20.0x/(ηr)
−20.0
−10.0
+0.0
+10.0
+20.0
z/(
ηr)
t=80.0 Ω−1
−20.0 −10.0 +0.0 +10.0 +20.0x/(ηr)
−20.0
−10.0
+0.0
+10.0
+20.0
z/(
ηr)
t=120.0 Ω−1
−20.0 −10.0 +0.0 +10.0 +20.0x/(ηr)
−20.0
−10.0
+0.0
+10.0
+20.0
z/(
ηr)
t=160.0 Ω−1
−20.0 −10.0 +0.0 +10.0 +20.0x/(ηr)
−20.0
−10.0
+0.0
+10.0
+20.0
z/(
ηr)
Strong clumping in non-linear state of the streaming instability(Youdin & Johansen 2007, Johansen & Youdin 2007)
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Clumping in 3-D
3-D evolution of the streaming instability:
DiscSimulation box
Particle clumps have up to 100 times the gas density
Clumps dense enough to be gravitationally unstable
But still too simplified: no vertical gravity
Particle size:
30 cm @ 5 AU or 1 cm @ 40 AU
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Pebbles
t = 0.1Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
0.0 2.0
ρp
Pebbles
t = 10.0Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
t = 20.0Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
t = 30.0Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
Some overdense regions occur, but weak, and couplingwith gas too strong for self-gravity to be important
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Pebbles
t = 0.1Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
0.0 2.0
ρp
Pebbles
t = 10.0Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
t = 20.0Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
t = 30.0Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
Baroclinic instability of uy (z) shear?(Ishitsu & Sekiya 2002; Ishitsu et al. 2009)
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Baroclinic instability?
Particles (Ωτf=0.02)
−0.10 −0.05 0.00 0.05 0.10x/H
−0.04
−0.02
0.00
0.02
0.04
z/H
Single fluid (Ωτf=0)
−0.10 −0.05 0.00 0.05 0.10x/H
−0.04
−0.02
0.00
0.02
0.04
z/H
Ishitsu & Sekiya (2002), Ishitsu et al. (2009)
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Rocks
t = 0.1Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
0.0 2.0
ρp
Rocks
t = 10.0Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
t = 20.0Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
t = 30.0Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
Higher overdensities, due to the streaming instability, butstill with short correlation times
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Boulders
t = 0.1Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
0.0 5.0
ρp
Boulders
t = 10.0Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
t = 40.0Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
t = 100.0Ω−1
−0.10 −0.05 0.00 0.05 0.10x/H
−0.10
−0.05
0.00
0.05
0.10
z/H
Almost no overdensities. Violent turbulent motion puffsup and dilutes mid-plane layer.
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Clumping depends strongly on metallicity
Increase Σpar/Σgas from 0.01 to 0.03
All particles between 1.5 and 15 centimetres
ε=0.01
−0.1 0.0 0.1x/H
0
10
20
30
40
50
t/Tor
b
0.0
0.1
Σp(x,t)/Σg
ε=0.02
−0.1 0.0 0.1x/H
ε=0.03
−0.1 0.0 0.1x/H
0
10
20
30
40
50
t/Tor
b
0 10 20 30 40 50t/Torb
100
101
102
103
104
max
(ρp)
0 10 20 30 40 50t/Torb
0 10 20 30 40 50t/Torb
10−3
10−2
10−1
Hp/
Hg
Johansen, Youdin, & Mac Low (in preparation)
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
The exoplanet zoo
First planet around solar-type star discovered in 1995(Mayor & Queloz)Since then 340 planets discoveredExoplanet probability rises steeply with heavy elementabundance of host star:
(Santos et al. 2004)
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Overdense seeds
Dust column density as a function of radial coordinate x andtime t measured in orbits:
Different-sized particlesconcentrateat the samelocations
t=0.0 Torb t=1.0 Torb t=2.0 Torb t=3.0 Torb
−0.66 0.00 0.66x/H
−0.66
0.00
0.66
y/H
t=4.0 Torb
t=5.0 Torbt=6.0 Torbt=6.5 Torbt=7.0 Torb
ΩKτ f = 0.25 ΩKτ f = 0.50
ΩKτ f = 0.75 ΩKτ f = 1.000.0
20.0
Σ p(i) /<
Σ p(i) >
0.0 20.0Σp/<Σp>
0.0 3.0log10(Σp/<Σp>)
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Forming planet embryos
Time is in Keplerian orbits (1 orbit ≈ 10 years)
6
Keplerian flow
?
Keplerian flow
Johansen et al. 2007 (Nature, 448, 1022)
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Forming planet embryos
Time is in Keplerian orbits (1 orbit ≈ 10 years)
6
Keplerian flow
?
Keplerian flow
Johansen et al. 2007 (Nature, 448, 1022)
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Dead zones
Transition from active accretion to dead zones triggersRossby wave instability in pile up of gas(Varniere & Tagger 2006; Inaba & Barge 2006)Rossby vortices trap particlesFormation of Mars or Earth size planets by self-gravityLyra et al. (2008, 2009)
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Mass spectrum
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Conclusions
MRI can play a crucial role in the formation of planets
Zonal flows are excited by ≈10% radial variation in the Maxwellstress of magnetorotational turbulence
MRI and streaming instability can interact constructively
Convergence zones concentrate solids and allow the formationof 1000 km sized planet embryos by gravity
MRI good for planet formation even in its absence – Rossbyvortices excited at transition from dead to active regions
Planetesimalformation in
turbulentprotoplanetary
discs
AndersJohansen
Planetformation
Dust in MHDturbulence
Planetesimalformation
Zonal flows
Analyticalmodel
Global models
Streaming andself-gravity
Dead zones
Conclusions
Open questions
What sets the scale of zonal flows?
Do collision speeds of MRI turbulence lead to growth or todestruction of dust agglomorates?
Can we even assume MRI to be operative in planetforming regions?
Would turbulent simulations of dead zones lead to Rossbywave instability and vortices?
How do you grow enough pebbles to launch the streaminginstability?
How does coagulation and fragmentation proceed in agravitationally contracting clump?
What is the relative importance of streaming,Kelvin-Helmholtz and baroclinic instabilities in themid-plane layer?