Influence of the disc structure on planet migration · Influence of the disc structure on planet...
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Influence of the disc structureon planet migration
Bertram Bitsch
Collaborators: A. Crida, A. Morbidelli, W. Kley & I. Dobbs-Dixon
Laboratoire Lagrange
15.06.2012
Bertram Bitsch Influence of the disc structure on planet migration
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Outline
IntroductionModel & constant opacity testsNon-constant opacity and implications toplanet migrationSummary & Conclusions
Bertram Bitsch Influence of the disc structure on planet migration
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Orion Nebula
Bertram Bitsch Influence of the disc structure on planet migration
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Hydrodynamical Simulations
Planets in discs:isothermal discs: inward migrationfully radiative discs:Equilibrium structure of disc determined by viscous heatingand cooling (e.g. Paardekooper, S.-J. & Mellema, G. 2006,Kley et al. 2009). Planets can migrate outwards.Zero-torque radius in disc useful for merging protoplanets.Stellar irradiated discs:The energy of the star heats the upper layers of the discand influences the disc structure. The disc is flared withH/r ∝ r2/7 (e.g. Chiang & Goldreich, 1997)⇒ Influence on migration of embedded objects in the disc?
Bertram Bitsch Influence of the disc structure on planet migration
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Model description& constant opacity simulations
Bertram Bitsch Influence of the disc structure on planet migration
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Energy equation for stellar irradiated disc
Coupled energy equation:
∂ER
∂t+∇ · F = ρκP(T ,P)[B(T )− cER][
∂ε
∂t+ (u · ∇)ε
]= −P∇ · u− ρκP(T ,P)[B(T )− cER] + S + Φ
ER radiation energy densityε = cvρT internal energy densityΦ viscous heating, radiative diffusion:
∇F = − λcρκR∇ER
Stellar heating from the star (V Volume of grid cell):
S =R2?
VσT 4
? e−τ (1− e−ρiκOP,i∆r )
Bertram Bitsch Influence of the disc structure on planet migration
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Numerical setup
3D spherical hydrodynamics in r ,θ, φ, but only r -θ planewith 384× 32 active cells used (axisymetric)Star as a point: stellar irradiation propagates on rays alongconstant θ-linesStellar heating absorbed in first two ghost cells, as disccontinues inwards from inner boundaryParameters of the initial setup:
constant viscosity ν = 1015cm2/sM? = 1M Star, R? = 3.0R, T? = 4000Kinitially flared disc profile with H/r ∝ r2/7
constant opacityκ = 1cm2/gvarying opacity by Bell &Lin, 1994
0.5 1 1.5 2 2.5
r [aJup]
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
z in
[a
Ju
p]
1e-11
2e-11
3e-11
4e-11
5e-11
6e-11
7e-11
ρ in
g/c
m3
Bertram Bitsch Influence of the disc structure on planet migration
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Inner disc with constant opacity
Inner disc: possible shielding of outer regions?
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
H/r
r [aJup]
initialnon viscous
viscous
Bertram Bitsch Influence of the disc structure on planet migration
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Outer disc with constant opacity
0.04
0.06
0.08
0.1
0.12
0.14
0.16
1 2 3 4 5 6 7 8 9
H/r
r [aJup]
initialnon viscous, 20 deg
viscous, 20 deg2/7 fit
Bertram Bitsch Influence of the disc structure on planet migration
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Summary so far...
Inner disc: disc structure dominated by viscous heating⇒ Only viscous discs for non-constant opacity sims
Outer disc: follows predicted 2/7th profile (Chiang &Goldreich, 1997)Part of stellar irradiation absorbed by inner disc nearmidplane regions
Bertram Bitsch Influence of the disc structure on planet migration
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Non constant opacity discsand implications to planet
migration
Bertram Bitsch Influence of the disc structure on planet migration
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Opacity by Bell & Lin, 1994
-6
-5
-4
-3
-2
-1
0
1
10 100 1000
log
κ
T in K
ρ = 10-12
g/cm3
ρ = 10-11
g/cm3
ρ = 10-10
g/cm3
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Inner disc with varying opacity
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
0.065
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
H/r
r [aJup]
initialfinal state
Bertram Bitsch Influence of the disc structure on planet migration
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Outer disc with varying opacity
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
1 2 3 4 5 6 7 8 9
H/r
r [aJup]
1000 g/cm2 at 0.2 aJup
3000 g/cm2 at 0.2 aJup
2/7 fit
Bertram Bitsch Influence of the disc structure on planet migration
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Implications for planet migration
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
1 2 3 4 5 6 7 8 9
H/r
r [aJup]
with stellar irradiationwithout stellar irradiation
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Density distribution
1 2 3 4 5 6 7 8 9
r [aJup]
0
0.5
1
1.5
2
2.5
3
z in [a
Ju
p]
-13
-12.5
-12
-11.5
-11
-10.5
-10
-9.5
-9
ρ in g
/cm
3
1 2 3 4 5 6 7 8 9
r [aJup]
0
0.5
1
1.5
2
2.5
3
z in [a
Ju
p]
-13
-12.5
-12
-11.5
-11
-10.5
-10
-9.5
-9
ρ in g
/cm
3
Bertram Bitsch Influence of the disc structure on planet migration
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Migration of the disc via Torque formula
Torque formula by Paardekooper et al. (2011):
Γtot = ΓL + Γc
with ΓL Lindblad torque, Γc corotation torque.Lindblad torque:
γΓL/Γ0 = −2.5− 1.7β + 0.1α , Γ0 =(q
h
)2ΣPr4
p Ω2P ,
where α denotes the negative slope of the surface densityprofile Σ ∝ r−α, β refers to the slope of the temperatureprofile T ∝ r−β, and γ is the adiabatic index of the gas.Corotation part much more complicated!⇒ Gradients in disc determine migration!
Bertram Bitsch Influence of the disc structure on planet migration
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Torque acting on planets
0.5 1 1.5 2 2.5 3 3.5
r [aJup]
0
10
20
30
40
50
60
70
Pla
net m
ass in M
Eart
h
0
2e-05
4e-05
6e-05
8e-05
0.0001
Specific
Torq
ue [a
Jup2 Ω
2]
0.5 1 1.5 2 2.5 3 3.5
r [aJup]
0
10
20
30
40
50
60
70
Pla
net m
ass in M
Eart
h
0
2e-05
4e-05
6e-05
8e-05
0.0001
Specific
Torq
ue [a
Jup2 Ω
2]
Bertram Bitsch Influence of the disc structure on planet migration
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Summary and Conclusions
Viscosity dominates the disc structure in the inner discStellar irradiation determines the disc structure in the outerpartsOuter disc: H/r ∝ r2/7
Shadowing effect of outer disc due to opacity bumpsMigration:
2 different located zero-torque radiiReduced region of outward migration for stellar irradiateddiscsSmaller region where cores can merge: better chance ofgrowing?
Bertram Bitsch Influence of the disc structure on planet migration