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04/11/2014 FMasset / MAD2014 1
Planetary migration in gaseous disks
Frédéric Masset
ICF-UNAM, Mexico
04/11/2014 FMasset / MAD2014 2
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.
04/11/2014 FMasset / MAD2014 3
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
04/11/2014 FMasset / MAD2014 4
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.
04/11/2014 FMasset / MAD2014 5
The inner wakeleads the planet.
It therefore exertsa positive torqueon the latter
Inner disk’s torque
+
04/11/2014 FMasset / MAD2014 6
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.
04/11/2014 FMasset / MAD2014 8
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.
04/11/2014 FMasset / MAD2014 9
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)
04/11/2014 FMasset / MAD2014 10
Coorbital dynamics
Horseshoetrajectoriesin the Earth’scorotatingframe.
04/11/2014 FMasset / MAD2014 11
On thisside anegativetorqueis exertedonto theplanet.
On thisside apositivetorqueisexertedonto theplanet
Horseshoe drag and corotation torque
04/11/2014 FMasset / MAD2014 12
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.
04/11/2014 FMasset / MAD2014 13
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
04/11/2014 FMasset / MAD2014 15
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.
04/11/2014 FMasset / MAD2014 16
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:
04/11/2014 FMasset / MAD2014 17
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
04/11/2014 FMasset / MAD2014 18
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
04/11/2014 FMasset / MAD2014 19
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
04/11/2014 FMasset / MAD2014 20
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
04/11/2014 FMasset / MAD2014 21
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
04/11/2014 FMasset / MAD2014 22
Surface density profile which includes a surfacedensity jump.* Realized with a jump of kinematic viscosity.* Relaxed on 105 orbits
Numerical simulations with a cavity
04/11/2014 FMasset / MAD2014 23
Vortensity gradient as a functionof radius.
Total torque measured in 75different calculations with 75different semi-major axis.
Fixed point
Numerical simulations with a cavity
04/11/2014 FMasset / MAD2014 24
• 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
04/11/2014 FMasset / MAD2014 25
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).
04/11/2014 FMasset / MAD2014 26
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
04/11/2014 FMasset / MAD2014 27
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
04/11/2014 FMasset / MAD2014 28
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
04/11/2014 FMasset / MAD2014 29
Positive feedback loop
Positive feedback runaway?
Yes, if the disk is massive enough.
04/11/2014 FMasset / MAD2014 31
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
04/11/2014 FMasset / MAD2014 32
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.
04/11/2014 FMasset / MAD2014 34
Type III migrating Saturn
+
-
What migrateshas an « effectivemass » Mp - CMD(Coorbital Mass Deficit)
Migration is type IIIwhen CMD > Mp
04/11/2014 FMasset / MAD2014 35
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.
04/11/2014 FMasset / MAD2014 36
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
04/11/2014 FMasset / MAD2014 37
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
04/11/2014 FMasset / MAD2014 38
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
04/11/2014 FMasset / MAD2014 39
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
04/11/2014 FMasset / MAD2014 40
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
04/11/2014 FMasset / MAD2014 41
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.
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