type i migration with stochastic torques fred c. adams & anthony m. bloch university of michigan...
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Type I Migration Type I Migration withwith
Stochastic Stochastic TorquesTorques
Type I Migration Type I Migration withwith
Stochastic Stochastic TorquesTorquesFred C. Adams & Anthony M. Fred C. Adams & Anthony M.
BlochBloch
University of MichiganUniversity of Michigan
Fred C. Adams & Anthony M. Fred C. Adams & Anthony M. BlochBloch
University of MichiganUniversity of Michigan Dynamics of Discs and Dynamics of Discs and PlanetsPlanets Cambridge, England, Cambridge, England, 20092009
OUTLINEOUTLINEOUTLINEOUTLINE The Type I Migration Problem The Type I Migration Problem Solution via Turbulent Torques Solution via Turbulent Torques Fokker-Planck FormulationFokker-Planck Formulation Effects of Outer Disk EdgeEffects of Outer Disk Edge Effects of Initial Planetary LocationsEffects of Initial Planetary Locations Long Term Evolution Long Term Evolution
(eigenfunctions)(eigenfunctions) Time Dependent TorquesTime Dependent Torques
The Type I Migration Problem The Type I Migration Problem Solution via Turbulent Torques Solution via Turbulent Torques Fokker-Planck FormulationFokker-Planck Formulation Effects of Outer Disk EdgeEffects of Outer Disk Edge Effects of Initial Planetary LocationsEffects of Initial Planetary Locations Long Term Evolution Long Term Evolution
(eigenfunctions)(eigenfunctions) Time Dependent TorquesTime Dependent Torques
Previous WorkPrevious WorkPrevious WorkPrevious Work Nelson & Papaloizou 2004: numericalNelson & Papaloizou 2004: numerical Laughlin, Adams, Steinacker 2004: Laughlin, Adams, Steinacker 2004:
basic numerical + back-of-envelopebasic numerical + back-of-envelope Nelson 2005+: longer term numericalNelson 2005+: longer term numerical Johnson, Goodman, Menou 2006: Johnson, Goodman, Menou 2006:
Fokker-Planck treatment Fokker-Planck treatment
Nelson & Papaloizou 2004: numericalNelson & Papaloizou 2004: numerical Laughlin, Adams, Steinacker 2004: Laughlin, Adams, Steinacker 2004:
basic numerical + back-of-envelopebasic numerical + back-of-envelope Nelson 2005+: longer term numericalNelson 2005+: longer term numerical Johnson, Goodman, Menou 2006: Johnson, Goodman, Menou 2006:
Fokker-Planck treatment Fokker-Planck treatment
This work: Effects of outer disk edge, This work: Effects of outer disk edge, long time evolution, time dependentlong time evolution, time dependentforcing terms, predict survival rates…forcing terms, predict survival rates…
Type I Planetary Type I Planetary MigrationMigrationPlanet embedded in Planet embedded in
gaseous disk creates gaseous disk creates spiral wakes. Leading spiral wakes. Leading wake pushes the planet wake pushes the planet outwards to larger semi-outwards to larger semi-major axis, while trailing major axis, while trailing wake pulls back on the wake pulls back on the planet and makes the planet and makes the orbit decay. The planet orbit decay. The planet migrates inward or migrates inward or outward depending on outward depending on distribution of mass distribution of mass within the disk.within the disk.
(Ward 1997)
Net Type I Migration Net Type I Migration TorqueTorque
(Ward 1997; 3D by Tanaka et al. 2002)(Ward 1997; 3D by Tanaka et al. 2002)
€
T1 = f1
mP
M∗
⎛
⎝ ⎜
⎞
⎠ ⎟
2
πΣr2 rΩ( )2 r
H
⎛
⎝ ⎜
⎞
⎠ ⎟2
Type I Migration Type I Migration ProblemProblem
Type I Migration Type I Migration ProblemProblem
€
t1 ≈ J0 /3T1(J 0) where J0 = mP GM∗r0
and T1 = f1
mP
M∗
⎛
⎝ ⎜
⎞
⎠ ⎟
2
πΣr2 rΩ( )2 r
H
⎛
⎝ ⎜
⎞
⎠ ⎟2
For typical parameters, the Type I migrationFor typical parameters, the Type I migrationtime scale is about 0.03 Myr (0.75 Myr) for time scale is about 0.03 Myr (0.75 Myr) for planetary cores starting at radius 1 AU (5 AU). planetary cores starting at radius 1 AU (5 AU). We need some mechanism to save the cores… We need some mechanism to save the cores…
MRI-induced turbulence enforces order-unity surface density fluctuations in the disk. These surface density perturbations provide continuous source of stochastic gravitational torques.
Turbulence -> stochastic torques -> random walk-> outward movement -> some cores saved
We can use results We can use results of MHD simulation of MHD simulation to set amplitude to set amplitude for fluctuations of for fluctuations of angular momentum angular momentum acting on planetsacting on planets(LSA04, NP04, & (LSA04, NP04, & Nelson 2005)Nelson 2005)
Turbulent Torques Turbulent Torques
Working Analytic Model for Working Analytic Model for Characterizing MHD Characterizing MHD
TurbulenceTurbulence
Working Analytic Model for Working Analytic Model for Characterizing MHD Characterizing MHD
TurbulenceTurbulenceMHD instabilities lead to surface density variations in the disk. The gravitational forces from these surface density perturbations produce torques on any nearby planets. To study how this process works, we can characterize the MHD turbulent fluctuations using the following basic of heuristic potential functions:
€
Φk =Aξe−(r−rc )2 /σ 2
r1/ 2cos[mθ −ϕ − Ωct ]sin[π
t
Δt]
(LSA2004)(LSA2004)
Estimate for AmplitudeEstimate for Amplitudedue to Turbulent due to Turbulent
FluctuationsFluctuations
Estimate for AmplitudeEstimate for Amplitudedue to Turbulent due to Turbulent
FluctuationsFluctuations
€
Td = 2π G Σ r mP , (ΔJ) = 4Porb fTTd
€
fT ≈ 0.05 = fraction of physical scale
€
∴ΔJ
J
⎛
⎝ ⎜
⎞
⎠ ⎟k
= fT
16π 2Σr2
M∗
≈10−3 Σ
1000gcm−2
⎛
⎝ ⎜
⎞
⎠ ⎟
see also Laughlin et al. (2004), Nelson (2005), Johnson et al. (2006)see also Laughlin et al. (2004), Nelson (2005), Johnson et al. (2006)
Power-Law DisksPower-Law DisksPower-Law DisksPower-Law Disks
€
Surface Density Σ(r)∝ r− p
Temperature T(r)∝ r−q
Keplerian Rotation Ω(r)∝ r−3 / 2
Scale Height H(r)∝ r(1−q ) / 2
FOKKER-PLANCK FOKKER-PLANCK EQUATIONEQUATION
FOKKER-PLANCK FOKKER-PLANCK EQUATIONEQUATION
€
∂P
∂t= γ
∂
∂x
1
x aP
⎛
⎝ ⎜
⎞
⎠ ⎟+ β
∂ 2
∂x 2x bP( )
€
γ≡ πf1
mP
M∗
⎛
⎝ ⎜
⎞
⎠ ⎟
r
H
⎛
⎝ ⎜
⎞
⎠ ⎟2
GΣr
GM∗r
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
1AU
≈ 10 Myr−1
β ≡ fα fT2 2π( )
3 Σr2
M∗
⎛
⎝ ⎜
⎞
⎠ ⎟ Ω
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
1AU
≈ 1 Myr−1
a = 2( p − q) and b = 7 − 4 p
FOKKER-PLANCK FOKKER-PLANCK EQUATIONEQUATION
FOKKER-PLANCK FOKKER-PLANCK EQUATIONEQUATION
€
∂P
∂t= γ
∂
∂x
1
x 2P
⎛
⎝ ⎜
⎞
⎠ ⎟+ β
∂ 2
∂x 2x P( )
€
γ≡ πf1
mP
M∗
⎛
⎝ ⎜
⎞
⎠ ⎟
r
H
⎛
⎝ ⎜
⎞
⎠ ⎟2
GΣr
GM∗r
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
1AU
≈ 10 Myr−1
β ≡ fα fT2 2π( )
3 Σr2
M∗
⎛
⎝ ⎜
⎞
⎠ ⎟ Ω
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
1AU
≈ 1 Myr−1
p = 3/2, q =1/2, a = 2, and b =1
DIMENSIONLESS DIMENSIONLESS PARAMETERPARAMETER
DIMENSIONLESS DIMENSIONLESS PARAMETERPARAMETER
€
Qmig =8πfT
2ΣH 2
mP
fα
f1
∝ r3− p−q
(depends on radius, time)(depends on radius, time)
Distributions vs TimeDistributions vs TimeDistributions vs TimeDistributions vs Time
time = 0 - 5 Myrtime = 0 - 5 Myr
Survival Probability vs Survival Probability vs TimeTime
(fixed diffusion constant)(fixed diffusion constant)
Survival Probability vs Survival Probability vs TimeTime
(fixed diffusion constant)(fixed diffusion constant)
€
γ =0, 1, 3, 5, 10, 20
Survival Probability vs Survival Probability vs TimeTime
(fixed Type I migration)(fixed Type I migration)
Survival Probability vs Survival Probability vs TimeTime
(fixed Type I migration)(fixed Type I migration)
€
β =0.1, 0.3, 0.5, 1, 3, 10
DIFFUSION DIFFUSION COMPROMISECOMPROMISE
DIFFUSION DIFFUSION COMPROMISECOMPROMISE
If diffusion constant is too small, then planetary cores are accreted, and the Type I migration problem is not solved
If diffusion constant is too large, then the random walk leads to large radial excursions, and cores are also accreted
Solution required an intermediate value of the diffusion constant
If diffusion constant is too small, then planetary cores are accreted, and the Type I migration problem is not solved
If diffusion constant is too large, then the random walk leads to large radial excursions, and cores are also accreted
Solution required an intermediate value of the diffusion constant
Optimization of Diffusion Optimization of Diffusion ConstantConstant
Optimization of Diffusion Optimization of Diffusion ConstantConstant
time=1,3,5,10,20 Myrtime=1,3,5,10,20 Myr
Survival Probability vsSurvival Probability vsStarting Radial LocationStarting Radial LocationSurvival Probability vsSurvival Probability vs
Starting Radial LocationStarting Radial Location
time=1,3,5,10 Myrtime=1,3,5,10 Myr
Distributions in Long Time Distributions in Long Time LimitLimit
Distributions in Long Time Distributions in Long Time LimitLimit
Only the lowest order eigenfunctionOnly the lowest order eigenfunctionsurvives in the asymptotic (large t) limitsurvives in the asymptotic (large t) limit
time=10 - 50 Myrtime=10 - 50 Myr
Lowest Order Lowest Order EigenfunctionsEigenfunctionsLowest Order Lowest Order
EigenfunctionsEigenfunctions
Surviving planets live in the outer disk…Surviving planets live in the outer disk…
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β /γ = 0.01, 0.1, 1
Time Varying TorquesTime Varying TorquesTime Varying TorquesTime Varying Torques
€
Σ(r, t) = Σ0(r) s(t)
s(t) = exp[−t / t0]
€
∂P
∂t= γ s(t)
∂
∂x
1
x aP
⎛
⎝ ⎜
⎞
⎠ ⎟+ β s2(t)
∂ 2
∂x 2x bP( )
Survival Probability withSurvival Probability withTime Varying Surface Time Varying Surface
DensityDensity
Survival Probability withSurvival Probability withTime Varying Surface Time Varying Surface
DensityDensity
€
t0 / Myr =1, 3, 10, 30, ∞
Survival Probability withSurvival Probability withTime Varying Mass and Time Varying Mass and
TorquesTorques
Survival Probability withSurvival Probability withTime Varying Mass and Time Varying Mass and
TorquesTorques
€
γ∝mP exp[−mP /mC ] and mP = m1(t / Myr)3
SUMMARYSUMMARYSUMMARYSUMMARY Stochastic migration saves planetary coresStochastic migration saves planetary cores Survival probability ‘predicted’ 10 percentSurvival probability ‘predicted’ 10 percent Outer boundary condition important -- disk Outer boundary condition important -- disk
edge acts to reduce survival fractionedge acts to reduce survival fraction Starting condition important -- balance Starting condition important -- balance
between diffusion and Type I torquesbetween diffusion and Type I torques Optimization of diffusion constantOptimization of diffusion constant Long time limit -- lowest eigenfunctionLong time limit -- lowest eigenfunction Time dependence of torques and masses Time dependence of torques and masses
Stochastic migration saves planetary coresStochastic migration saves planetary cores Survival probability ‘predicted’ 10 percentSurvival probability ‘predicted’ 10 percent Outer boundary condition important -- disk Outer boundary condition important -- disk
edge acts to reduce survival fractionedge acts to reduce survival fraction Starting condition important -- balance Starting condition important -- balance
between diffusion and Type I torquesbetween diffusion and Type I torques Optimization of diffusion constantOptimization of diffusion constant Long time limit -- lowest eigenfunctionLong time limit -- lowest eigenfunction Time dependence of torques and masses Time dependence of torques and masses
UNRESOLVED ISSUESUNRESOLVED ISSUESUNRESOLVED ISSUESUNRESOLVED ISSUES
Dead zones (turn off MRI, turbulence)Dead zones (turn off MRI, turbulence) Disk structure (planet traps) Disk structure (planet traps) Outer boundary condition Outer boundary condition Inner boundary condition (X-point) Inner boundary condition (X-point) Fluctuation distrib. (tails & black Fluctuation distrib. (tails & black
swans)swans) Competition with other mechanisms Competition with other mechanisms
(see previous talks…) (see previous talks…)
Dead zones (turn off MRI, turbulence)Dead zones (turn off MRI, turbulence) Disk structure (planet traps) Disk structure (planet traps) Outer boundary condition Outer boundary condition Inner boundary condition (X-point) Inner boundary condition (X-point) Fluctuation distrib. (tails & black Fluctuation distrib. (tails & black
swans)swans) Competition with other mechanisms Competition with other mechanisms
(see previous talks…) (see previous talks…)
ReferenceReferenceReferenceReference
F. C. Adams and A. M. Bloch (2008): F. C. Adams and A. M. Bloch (2008): General Analysis of Type I Planetary General Analysis of Type I Planetary Migration with Stochastic Migration with Stochastic Perturbations, ApJ, 701, 1381 Perturbations, ApJ, 701, 1381
[email protected]@umich.edu
