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Planetary migration in gaseous disks

Frédéric Masset

ICF-UNAM, Mexico

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What this talk is not about

We exclusively consider the interaction of planets with thegaseous disk. Subsequent migration by planetesimal scattering is not considered here. The former effect ismuch stronger, but it leaves much less imprint in theplanetary system.

We will focus on circular, non inclined planets aroundisolated stars.

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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

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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.

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The inner wakeleads the planet.

It therefore exertsa positive torqueon the latter

Inner disk’s torque

+

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The wake in theouter disk lags theplanet.

It therefore exertsa negative torqueon the planet.

Outer disk’s torque

-

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Type I migration is inwards

Linear regime (low mass planet - Mp < 10 M) dubbed « Type I migration »

A (slight) imbalance between inner & outer torqueyields a migration of the planet.

The net torque is called the differential Lindblad torque.

In the linear regime (low-mass planet), the differentialLindblad is a negative quantity (Ward 1986). Type I migration is inwards.

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Type I migration is fast

Timescaleconflictproblem

105 years

Central star1 M planet

Migrationtimescale

Migration

1 M planet 15 M planet 106-7 years

Core build-uptimescale

Build up

A one-Earth mass planet, dropped in a MMSN(Minimum Mass Solar Nebula) at 1 AU, shouldbe flushed onto the central object in ~ 105 yrs.

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Type I migration is really inwardsThe drift rate is quite insensitive to the disk profiles(surface density and temperature).

radius

Ang

ular

vel

ocity

Assume that we take acentrally peaked surface density profile.

The wake is shifted inwards this frustrates the effect ofthe surface density variation.

Pressure buffer effect(Ward, 1997)

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Coorbital dynamics

Horseshoetrajectoriesin the Earth’scorotatingframe.

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On thisside anegativetorqueis exertedonto theplanet.

On thisside apositivetorqueisexertedonto theplanet

Horseshoe drag and corotation torque

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Gap clearance at large planetary mass

When the planetary mass becomes sufficiently large,the wake degenerates into a shock in the vicinity ofthe planet. Horseshoe U-turns then become asymmetric.

The gas leavesthe horseshoeregion in a fewlibration times anda gap is excavatedaround the orbit.

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A Jovian planet clears a gapin approximately 100 orbits.

Migration with gap = Type II migration

Gap clearance at large planetary mass

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Type II migration of a giant planet

Migration with gap = Type II migration

Type II migration paradigm

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Theoretician’s paradigm:

Type II migration is lockedin the disk viscous evolution

Numericists’ wellknown secret:type II migration isnot locked in thedisk viscous evolution

Depends on planet vs disk mass, viscosity, aspectratio. One can have substantial mass flows acrossthe gap.

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Criteria for gap clearance

The amount of residual material in the gap results from a balancebetween gravitationnal effects, which tend to open the gap, andviscous 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: RH > H

These criteria have been revisited by Crida et al (2006)and unified into a unique criterium which reads:

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The horseshoe drag scales with thevortensity gradient (ratio of vorticityand surface density).

It is too weak to counteract thedifferential Lindblad torque in thelinear regime.

Horseshoe drag and corotation torque

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Some words about vortensity

Why does the horseshoe drag depend on the vortensity gradient ?

The vortensity is conservedin a 2D, inviscidbarotropic flow

We firstly focus onbarotropic flows

barotropic = pressurefunction of density only.E.g. : globally isothermal disks

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Saturation of corotation torque

time

Tor

que

Each streamline gives a periodic contribution to the CR torque

Each streamline has a different libration time

Azimuth

Rad

ius

CR torque tends to 0 as a result of phase mixing

Viscosity, or turbulence, required to avoid saturation

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The corotation torque scales with the vortensity gradient, henceis sensitive to a strong surface density gradient.

Radius

Surfacedensity

Orbit

Horseshoe separatrices

The CR torque can be arbitrarily large and positive at a cavityedge.

Planetary trap at a cavity edge

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Power law disk

Differential Lindblad torque

Corotation torque

TOTAL

-1

+0.5

< 0

Cavity edge

-4

+5

> 0

x 2 - 4

x 10 - 20

Torques on the cavity edge

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Surface density profile which includes a surfacedensity jump.* Realized with a jump of kinematic viscosity.* Relaxed on 105 orbits

Numerical simulations with a cavity

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Vortensity gradient as a functionof radius.

Total torque measured in 75different calculations with 75different semi-major axis.

Fixed point

Numerical simulations with a cavity

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• 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 requestedin order to avoid the CR torque’s saturation.

Dead zone

Example of cavities

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Mass-period distribution of close-in exoplanets

small mass planets gettrapped at the cavity edge

giant planets destroy thecavity and proceed inward

(Benitez-Llambay et al. 2011)

Can explain the discontinuity found at 1 MJ in mass-perioddiagrams of exoplanets. Bestexplained by cavity effects +subsequent drift due to star-planettide (Q~107).

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Corotation torque on a migrating planet

Planet

Fluid element

Dotted line : horseshoe separatrices

The planetmigrates inwards

Fluid elements of the innerdisk exert a negative torque

Positive feedback

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The planet migrates inwards

The horseshoe region exertsa positive torque on the planet

Negative feedback

Planet

Fluid element

Dotted line : horseshoe separatrices

Corotation torque on a migrating planet

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Both feedbacks cancel out if the disk profileis unperturbed.

If the horseshoe region is depleted, the netfeed back is positive.

This can therefore happen for planets whichdeplete their coorbital region (giant planets).

Corotation torque on a migrating planet

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Positive feedback loop

Positive feedback runaway?

Yes, if the disk is massive enough.

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Type III (runaway) migration domainPlanetary mass

Dis

k m

ass

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Transition to type III migrationLine of slope Mp/M

Drift rate

Nor

mal

ized

tor

que

Normalized Surface Density

Nor

mal

ized

drif

t ra

te

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Transition to type III migration

Type I

Type III

bifu

rcat

ion

runaway

outwards

inwards

Type III migration is inwards or outwards: potentially arich variety of outcomes.

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Type III migrating Saturn

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Type III migrating Saturn

+

-

What migrateshas an « effectivemass » Mp - CMD(Coorbital Mass Deficit)

Migration is type IIIwhen CMD > Mp

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Corotation torque in non-barotropic disks

Paardekooper & Mellema (2006) find with nested grid3D calculations and radiative transfer that a low-massplanet can actually migrate outwards, if the disk issufficiently opaque, under the action of the CR torque !

In order to understand these results adiabatic limit.

In an adiabatic flow, the entropy is conserved alonga fluid element path (provided no irreversibility isintroduced). This is true for the coorbital flow of alow mass planet. Entropy gradient must play a key role.

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The torque excess therefore scales withthe entropy gradient.

The torque excessis measured bycomparing theresults to acalculation with anisothermal disk.

CR torque in a radiatively inefficient disk

Tor

que

exce

ss

Entropy gradient

This excess can be sufficient to revert migration !

Surface density slope

Tem

per

atur

e sl

ope

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Vortensity field

In practice, it is conserved almost everywhere, except atthe outgoing separatrices, where there is a strong densitygradient.

Vortensity stripesat the downstreamseparatrices.

This vortensity field yields theentropy related corotation torque.

Exact amount of vortensity created can be calculated bythe use of a new invariant.

Vortensity not conserved in an adiabatic 2D flow

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Corotation torque in real disks

Disks are neither barotropic (isothermal) nor adiabatic.is

othe

rmal

lim

it

adia

batic

lim

it

thermal diffusion

Ultimate torque expression depends onentropy gradient, vortensity gradient, viscosityand thermal diffusion (Paardekooper et al. 2011)

Torque formulae and population synthesis

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Analytic torqueformulae (+ eccentricity,inclination damping),N-body integrator.

e.g. Cossou et al. (2014)Zone of inwardmigration - danger

Most models have too low yields of giant planets, andtoo many super Earths (failed giant planet cores).

Importance of entropy in the coorbital region

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Key role of entropy in thecoorbital region long recognized(since 2008)

Accreting cores release substantialamount of energyin the surroundingnebula (eg. Pollacket al. 1996)

Yet for 20 years studies of migration haveexclusively considered « passive » point-like masses

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Conclusions

Theories of planetary migration have considerably evolvedsince the standard type I / type II paradigm and the unavoidableinwards migration.

The corotation torque has recently received more attention,and has been shown to play a crucial role in several regimesof migration (planetary trap, type III migration, radiativelyinefficient disks, etc.)

Is migration overrated ? Torque formulae feature very largetorques of opposite signs demand accurate derivations, anda fine grain knowledge of the disk properties. Migration theoriesare not overrated, but they are slowly maturing.

Thanks

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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)