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
Outline
IntroductionModel & constant opacity testsNon-constant opacity and implications toplanet migrationSummary & Conclusions
Bertram Bitsch Influence of the disc structure on planet migration
Orion Nebula
Bertram Bitsch Influence of the disc structure on planet migration
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
Model description& constant opacity simulations
Bertram Bitsch Influence of the disc structure on planet migration
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
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
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
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
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
Non constant opacity discsand implications to planet
migration
Bertram Bitsch Influence of the disc structure on planet migration
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
Bertram Bitsch Influence of the disc structure on planet migration
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
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
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
Bertram Bitsch Influence of the disc structure on planet migration
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
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
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
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
