1. 04/11/2014FMasset / MAD20141 Planetary migration in gaseous disks Frédéric Masset ICF-UNAM, Mexico

2. 04/11/2014FMasset / MAD20142 What this talk is not about We exclusively consider the interaction of planets with the gaseous disk. Subsequent migration by planetesimal scattering is not considered here. The former effect is much stronger, but it leaves much less imprint in the planetary system. We will focus on circular, non inclined planets around isolated stars.

3. 04/11/2014FMasset / MAD20143 Disk response to a point-like mass on a circular orbit  The planet tidally excites a one-armed spiral wake that propagates both inwards and outwards. Disk response to an embedded planet

4. 04/11/2014FMasset / MAD20144 What causes planetary migration ? The protoplanet tidally excites a spiral wake. This wake exerts a gravitational force on the planet The semi-major axis, the eccentricity, the inclination vary with time. The most spectacular effect is the variation of the semi-major axis: it is called planetary migration.

5. 04/11/2014FMasset / MAD20145 The inner wake leads the planet. It therefore exerts a positive torque on the latter Inner disk’s torque +

6. 04/11/2014FMasset / MAD20146 The wake in the outer disk lags the planet. It therefore exerts a negative torque on the planet. Outer disk’s torque -

7. 04/11/2014FMasset / MAD20147 Type I migration is inwards  Linear regime (low mass planet - M p < 10 M  )  dubbed « Type I migration »  A (slight) imbalance between inner & outer torque yields a migration of the planet.  The net torque is called the differential Lindblad torque.  In the linear regime (low-mass planet), the differential Lindblad is a negative quantity (Ward 1986).  Type I migration is inwards.

8. 04/11/2014FMasset / MAD20148 Type I migration is fast Timescale conflict problem 10 5 years Central star 1 M  planet Migration timescale Migration 1 M  planet15 M  planet 10 6-7 years Core build-up timescale Build up A one-Earth mass planet, dropped in a MMSN (Minimum Mass Solar Nebula) at 1 AU, should be flushed onto the central object in ~ 10 5 yrs.

9. 04/11/2014FMasset / MAD20149 Type I migration is really inwards The drift rate is quite insensitive to the disk profiles (surface density and temperature). radius Angular velocity Assume that we take a centrally peaked surface density profile. The wake is shifted inwards  this frustrates the effect of the surface density variation. Pressure buffer effect (Ward, 1997)

10. 04/11/2014FMasset / MAD201410 Coorbital dynamics Horseshoe trajectories in the Earth’s corotating frame.

11. 04/11/2014FMasset / MAD201411 On this side a negative torque is exerted onto the planet. On this side a positive torque is exerted onto the planet Horseshoe drag and corotation torque

12. 04/11/2014FMasset / MAD201412 Gap clearance at large planetary mass When the planetary mass becomes sufficiently large, the wake degenerates into a shock in the vicinity of the planet. Horseshoe U-turns then become asymmetric. The gas leaves the horseshoe region in a few libration times and a gap is excavated around the orbit.

13. 04/11/2014FMasset / MAD201413 A Jovian planet clears a gap in approximately 100 orbits. Migration with gap = Type II migration Gap clearance at large planetary mass

14. 04/11/2014FMasset / MAD201414 Type II migration of a giant planet Migration with gap = Type II migration

15. Type II migration paradigm 04/11/2014FMasset / MAD201415 Theoretician’s paradigm: Type II migration is locked in the disk viscous evolution Numericists’ well known secret: type II migration is not locked in the disk viscous evolution Depends on planet vs disk mass, viscosity, aspect ratio. One can have substantial mass flows across the gap.

16. 04/11/2014FMasset / MAD201416 Criteria for gap clearance The amount of residual material in the gap results from a balance between gravitationnal effects, which tend to open the gap, and viscous diffusion, which tends to close it.  The more viscous the disk, the larger the critical mass for gap clearance.  Therefore there exists a viscous criterium: q > 40/R Furthermore, the wake must be a shock in order to clear the gap. This corresponds to the thermal criterium: R H > H These criteria have been revisited by Crida et al (2006) and unified into a unique criterium which reads:

17. 04/11/2014FMasset / MAD201417 The horseshoe drag scales with the vortensity gradient (ratio of vorticity and surface density). It is too weak to counteract the differential Lindblad torque in the linear regime. Horseshoe drag and corotation torque

18. 04/11/2014FMasset / MAD201418 Some words about vortensity Why does the horseshoe drag depend on the vortensity gradient ? The vortensity is conserved in a 2D, inviscid barotropic flow We firstly focus on barotropic flows barotropic = pressure function of density only. E.g. : globally isothermal disks

19. 04/11/2014FMasset / MAD201419 Saturation of corotation torque time Torque Each streamline gives a periodic contribution to the CR torque Each streamline has a different libration time Azimuth Radius  CR torque tends to 0 as a result of phase mixing Viscosity, or turbulence, required to avoid saturation

20. 04/11/2014FMasset / MAD201420 The corotation torque scales with the vortensity gradient, hence is sensitive to a strong surface density gradient. Radius Surface density Orbit Horseshoe separatrices The CR torque can be arbitrarily large and positive at a cavity edge. Planetary trap at a cavity edge

21. 04/11/2014FMasset / MAD201421 Power law disk Differential Lindblad torque Corotation torque TOTAL +0.5 < 0 Cavity edge -4 +5 > 0 x 2 - 4 x 10 - 20 Torques on the cavity edge

22. 04/11/2014FMasset / MAD201422 Surface density profile which includes a surface density jump. * Realized with a jump of kinematic viscosity. * Relaxed on 10 5 orbits Numerical simulations with a cavity

23. 04/11/2014FMasset / MAD201423 Vortensity gradient as a function of radius. Total torque measured in 75 different calculations with 75 different semi-major axis. Fixed point Numerical simulations with a cavity

24. 04/11/2014FMasset / MAD201424 Inner edge of the dead zone. Transition between a Standard Accretion Disk (SAD) and a jet emitting disk (SED). Disc truncated by tidal effects (e.g. at the outer edge of a gap, or disk truncated by a companion star) Note that some degree of turbulence is requested in order to avoid the CR torque’s saturation. Dead zone Example of cavities

25. 04/11/2014FMasset / MAD201425 Mass-period distribution of close-in exoplanets small mass planets get trapped at the cavity edge giant planets destroy the cavity and proceed inward (Benitez-Llambay et al. 2011) Can explain the discontinuity found at 1 M J in mass-period diagrams of exoplanets. Best explained by cavity effects + subsequent drift due to star-planet tide (Q  ~10 7 ).

26. 04/11/2014FMasset / MAD201426 Corotation torque on a migrating planet Planet Fluid element Dotted line : horseshoe separatrices The planet migrates inwards Fluid elements of the inner disk exert a negative torque Positive feedback

27. 04/11/2014FMasset / MAD201427 The planet migrates inwards The horseshoe region exerts a positive torque on the planet Negative feedback Planet Fluid element Dotted line : horseshoe separatrices Corotation torque on a migrating planet

28. 04/11/2014FMasset / MAD201428 Both feedbacks cancel out if the disk profile is unperturbed. If the horseshoe region is depleted, the net feed back is positive. This can therefore happen for planets which deplete their coorbital region (giant planets). Corotation torque on a migrating planet

29. 04/11/2014FMasset / MAD201429 Positive feedback loop Positive feedback  runaway? Yes, if the disk is massive enough.

30. 04/11/2014FMasset / MAD201430 Type III (runaway) migration domain Planetary mass Disk mass

31. 04/11/2014FMasset / MAD201431 Transition to type III migration Line of slope M p /  M Drift rate Normalized torque Normalized Surface Density Normalized drift rate

32. 04/11/2014FMasset / MAD201432 Transition to type III migration Type I Type III bifurcation runaway outwards inwards Type III migration is inwards or outwards: potentially a rich variety of outcomes.

33. 04/11/2014FMasset / MAD201433 Type III migrating Saturn

34. 04/11/2014FMasset / MAD201434 Type III migrating Saturn + - What migrates has an « effective mass » M p - CMD (Coorbital Mass Deficit) Migration is type III when CMD > M p

35. 04/11/2014FMasset / MAD201435 Corotation torque in non-barotropic disks Paardekooper & Mellema (2006) find with nested grid 3D calculations and radiative transfer that a low-mass planet can actually migrate outwards, if the disk is sufficiently opaque, under the action of the CR torque ! In order to understand these results  adiabatic limit. In an adiabatic flow, the entropy is conserved along a fluid element path (provided no irreversibility is introduced). This is true for the coorbital flow of a low mass planet. Entropy gradient must play a key role.

36. 04/11/2014FMasset / MAD201436 The torque excess therefore scales with the entropy gradient. The torque excess is measured by comparing the results to a calculation with an isothermal disk. CR torque in a radiatively inefficient disk Torque excess Entropy gradient This excess can be sufficient to revert migration ! Surface density slope Temperature slope

37. 04/11/2014FMasset / MAD201437 Vortensity field In practice, it is conserved almost everywhere, except at the outgoing separatrices, where there is a strong density gradient. Vortensity stripes at the downstream separatrices. This vortensity field yields the entropy related corotation torque. Exact amount of vortensity created can be calculated by the use of a new invariant. Vortensity not conserved in an adiabatic 2D flow

38. 04/11/2014FMasset / MAD201438 Corotation torque in real disks Disks are neither barotropic (isothermal) nor adiabatic. isothermal limit adiabatic limit thermal diffusion Ultimate torque expression depends on entropy gradient, vortensity gradient, viscosity and thermal diffusion (Paardekooper et al. 2011)

39. Torque formulae and population synthesis 04/11/2014FMasset / MAD201439 Analytic torque formulae (+ eccentricity, inclination damping), N-body integrator. e.g. Cossou et al. (2014) Zone of inward migration - danger Most models have too low yields of giant planets, and too many super Earths (failed giant planet cores).

40. Importance of entropy in the coorbital region 04/11/2014FMasset / MAD201440 Key role of entropy in the coorbital region long recognized (since 2008) Accreting cores release substantial amount of energy in the surrounding nebula (eg. Pollack et al. 1996) Yet for 20 years studies of migration have exclusively considered « passive » point-like masses

41. 04/11/2014FMasset / MAD201441 Conclusions Theories of planetary migration have considerably evolved since the standard type I / type II paradigm and the unavoidable inwards migration. The corotation torque has recently received more attention, and has been shown to play a crucial role in several regimes of migration (planetary trap, type III migration, radiatively inefficient disks, etc.) Is migration overrated ? Torque formulae feature very large torques of opposite signs  demand accurate derivations, and a fine grain knowledge of the disk properties. Migration theories are not overrated, but they are slowly maturing.

42. Thanks 04/11/2014FMasset / MAD201442 Pablo Benítez-Llambay (Univ of Córdoba, Argentina) Clément Baruteau (IRAP, France) Aurélien Crida (OCA, France) Gennaro D’Angelo (SETI Institute) Gloria Koenigsberger (UNAM, Mexico) Alessandro Morbidelli (OCA, France) Judit Szulágyi (OCA, France)