PLANETARY MIGRATION IN WEAKLY MAGNETIZED TURBULENT DISCS Cl´ement Baruteau, DAMTP, University of Cambridge → Baruteau, C., Fromang, S., Nelson, R. P. & Masset, F. 2011, A&A, 583, 84-99
Abstract: In viscous disc models, the migration of planets embedded in their nascent protoplanetary discs may be directed inwards or outwards, depending on the magnitude of the corotation torque. The long-term evolution of the corotation torque is intimately related to viscous and thermal diffusion processes in the planet’s horseshoe region. Through 3D MHD simulations, we show the existence of horseshoe dynamics and an unsaturated corotation torque in weakly magnetized discs with fully developed MHD turbulence.
1. Planetary migration in a nutshell
2. Physical model and numerical setup Aim: investigate the impact of MHD turbulence on the tidal torque (focus on the corotation torque).
In a 2D non-magnetized disc, the tidal torque exerted on the planet encompasses:
How? 3D ideal MHD simulations with the Nirvana and Ramses codes.
1. the differential Lindblad torque (angular momentum carried away by the spiral density waves) → drives migration inwards.
A simplified disc model: no vertical stratification, locally isothermal equation of state, planet on a fixed circular orbit. To get a steady-state density profile, the initial density profile is reinforced on 20 planet orbits.
2. the corotation torque (exchange of angular momentum with the planet’s horseshoe region, powered by advection-diffusion of gas vortensity and entropy) → drives migration inwards or outwards.
3. Disc model with an inner cavity (planet trap) Method: [1] First set up a steady-state density profile with an inner cavity through a preliminary viscous 1D run. [2] Restart as a 3D cylindrical disc model with a Bϕ field. [3] After MHD turbulence is fully developed throughout the disc, restart with a planet.
4. Disc model with a power-law density profile
Torques versus time
-5
3•10
2•10-5
Torque
1•10
0
-1•10-5
200 300 Time (orbits)
400
5. Horseshoe dynamics with weak MHD turbulence Passive contaminant advected with the flow: horseshoe U-turns vs. turbulent diffusion 0.00000
MODEL 1
0.03 0.02 0.01
1.40895
2
3
500
4 5 Radius
6
7
1•10-4 Nirvana Ramses MODEL 1
8•10-5
6•10-5 ρ0
4•10-5
8
1
2
3
4 5 Radius
6
7
8
Radial profiles of the total alpha parameter (left) and midplane density (right), time-averaged over 100 orbits. Bottom: Running time-averaged torque. The stationary torques of 2D non-magnetized laminar disc models (inviscid and viscous) are overplotted, in which the profiles of ρ0 and of α correspond to their time-averaged MHD counterpart. Running time-averaged torque
100
Results: With fully developed MHD turbulence (hαi ≈ 0.02 near the planet), the running-time averaged torque remains positive, suggesting the corotation torque is sustained at a value close to its maximum, unsaturated value. A planet trap maintained in a turbulent disc could prevent the large scale migration of protoplanets.
3.6
0.04
0.00 1
-2•10-5 0
Time-averaged midplane density
-5
Nirvana Ramses
planet
Time-averaged total alpha
0.05
Model 2: ρ0 ∝ r−3/2, T 0 uniform
planet
Model 1: ρ0 ∝ r−1/2, T 0 ∝ r−1
-2.0•10-6
MODEL 1 (Nirvana)
-4.0•10-6
Results: The running-time averaged turbulent torque reaches a stationary value after ∼ 200 orbits, and is in decent agreement with the torque of a laminar, nonmagnetized disc model with similar viscous alpha parameter near the planet.
-6.0•10-6 Turbulent disc -6
-8.0•10
Laminar disc with α = 0.03
-5
-1.0•10
-1.2•10-5 Laminar disc with α = 0
-1.4•10-5 -1.6•10-5 0
1.80713
100
200 300 400 Time (orbits after restart)
500
3.2 Radius
No planet
3.4
6. An additional corotation torque uncovered
3.0 2.8
0.00000
1.40895
1.80713
3.4 3.2 Radius
Mp = 3 × 10−4 M⋆
3.6
3.0 2.8 2.6 2.4 3.6
0.00000
1.40895
0.5
0.0
-0.5 MODEL 1, NIRVANA & FARGO
-1.0 2.0
2.5
3.0 Radius
3.5
4.0
Turbulent disc Laminar disc with α = 0.03 Laminar disc with α = 0
0.5
0.0
-0.5 MODEL 2, NIRVANA & FARGO
-1.0 2.0
2.5
3.0 Radius
3.5
4.0
Torque density distribution time-averaged over 100 orbits. The (stationary) torque distributions of the inviscid and viscous disc models are overplotted. The dashed lines show the approximate location of the separatrices of the planet’s horseshoe region.
3.2 Radius
Turbulent disc Laminar disc with α = 0.03 Laminar disc with α = 0
1.80713
3.4
Mp = 10−3 M⋆
Torque density (arbitrary units)
2.4
Torque density (arbitrary units)
2.6
3.0
Results:
2.8 2.6 2.4 0.0
0.5
1.0
1.5 2.0 Azimuth
2.5
3.0 0.0
0.5
1.0
1.5 2.0 Azimuth
2.5
3.0 0.0
0.5
1.0
1.5 2.0 Azimuth
2.5
3.0
Concentration of the midplane contaminant with superimposed streamlines averaged over 5 orbits.
- Outside the horseshoe region, the very good agreement between the turbulent and laminar torque distributions show that, on time average, the differential Lindblad torque is essentially unchanged by the full development of MHD turbulence. - Inside the horseshoe region, the different torque distributions highlight the existence of an additional corotation torque in weakly magnetized discs, whose properties are under investigation at DAMTP.
