F. Marzari, Dept. Physics, Padova Univ.

The role of migration and planet-planet The role of migration and planet-planet scattering in shaping planetary systemsscattering in shaping planetary systems

Small semimajor axes Large eccentricities Spin-orbit misalignment

Standard model of planet formation based on Solar system exploration: low eccentricities & inclinations, large semimajor axes for giant planets.....

The standard modelThe standard model

PluginsPlugins

Protostar +Disk

Planetesimal formation by dust coagulation with contribution from turbulence, instability ….

Formation of Terrestrial planets and core of giant planets (subsequent gas infall) by planetesimal accumulation

Gas dissipation – final planetary system

P-P scattering Residual planetesimal scattering Tidal interaction with the star

Planet migration P-P scattering

Planetary migration Planetary migration by interaction with by interaction with the disk: a very the disk: a very complex problemcomplex problem

Type I migration: Small planets (1- 50 M

E)

HS drag

Isothermal, adiabatic, or fully radiative energy equation

Turbulence -> stochastic migration

Saturn-Jupiter and more massive planets: Type II, III migration

Masset & Papaloizou (2003)

Kley & Crida (2008)

The inner wake exerts a positive torque on the planet accelerating it and causing an outward migration

The outer wake exerts a negative torque slowing down the planet and leading to inward migration

The sum of the two torques, the differential Lindblad torque, is negative and causes inward migration.

TYPE I MIGRATION

Wakes (2 arms) are given by superposition of sound waves, excited at Lindblad resonances, in a differentially rotating disk.

(m+ k ) n p−(m±1) n F− k ~̇ω p∓~̇ω F =0

The inner and outer discare responsible for theLindblad torque.

In the horseshoe region,gas particles make U-turns → exchange angular momentum withthe planet→ Corotation torque.

Horseshoe torque

Paardekooper et al. (2010, 2011)

Because of horshoe drag Type I migration can be reversed for radiative

disks (Kley & Crida, 2008) due to the horseshoe

torque.

Up to 40 Earth masses the torque is positive. This is important for Jupiter size

planets where the core is about 10-30 M

E Before gas infall they

migrate outwards and after the gas infall (very rapid, 1 kyr)

they undergo type II migration potentially skipping the critical fast inward migration phase.

Coleman & Nelson (2014) Low mass planets: the horseshoe period long compared to viscous/radiative diffusion timescale: weak corotation torque. Inward migration Intermediate masses: two

timescales comparable, strong corotation torque, outward migration Large masses: saturation of

corotation torque. The viscous/radiative diffusion timescales long compared to horseshoe period, phase mixing, no torque.

Nelson (2005). Large scale MHD-driven turbulence can cause a stochastic migration of planets overcoming the

Lindblad torques. Dead zones?

10 ME

1 ME

Type II migration: Jupiter size planets

ΤOS≈a4Ω2 Σ ( M p

M z)

2

( aΔ )3

Τ ν=−2 πr 3 νΣ (∂ Ω∂ r )

Gap opening criterium: TOS

> T

Δ=max ( H , R Hill )

The gas is pushed away by the resonance perturbations which overcome viscosity.

d ad t

∼32 a

If π a2 Σ⩾M p

e p<h

Crida et al. (2006)

Alibert et al. (2005): migration is included during growth

Movshovitz et al. (2010): detailed grain physics, no migration.

Orbital migration is important not only for changing the architecture of planetary systems but it also influences the growth of a planet in

particular during the gas infall from the disk: shorter formation timescales are obtained.

Even if.....

What about Jupiter and Saturn? Why What about Jupiter and Saturn? Why didn't they migrate very close to the didn't they migrate very close to the

sun? Coupled migration while sun? Coupled migration while trapped in resonance!trapped in resonance!

Masset & Snellgrove (2001): Jupiter and Saturn trapped in a 3:2 resonance migrates outwards.

Jupiter excites inner Lindblad resonances, Saturn the outer ones.

ΤOS≈a J4 Ω2 Σ (

M J

M star

)2

(a J

Δ )3

M S

M J

<( 23)(1 /3)

A positive torque is obtained when T

J > T

S

.

The grand tack scenarioThe grand tack scenario. Recent model by Walsh et al. Nature 2011, assumes that Jupiter migrated to 1.5 AU before reversing the drift direction. This would explains the low mass of Mars and the compositional mixing in the main asteroid belt.

∂∂ t

Σ+1r

∂∂ r

(r Σu r)=−Σ̇ pe

Q + Q irr−Q cool=0

Σ(t , r) , h(t , r)

A model of coupled giant planet migration A model of coupled giant planet migration mustmust account for the account for the disk evolution driven by viscous torques and wind dispersal (EUV disk evolution driven by viscous torques and wind dispersal (EUV photoevaporation, photoevaporation, Dullemond et al. 2007, Clarke 2011...).)

Continuity equation +

thermal balance

The disk evolves with time due to viscosity (→ mass accretion on the star) and photoevaporation. The local gas density decreases with time and when the planets (Jupiter and Saturn for example) begin to migrate outwards they may not go far out. In the GT model Jupiter and Saturn may not have enough time to return to their present position.

Ad example, 1D models predict a superficial density around 100 g/cm2 or lower at 1 AU by the time Jupiter migrates to 1.5 AU.

Only in a minority of disk models (less than 2%) the planets are able to move back to 5 AU or beyond, possible site of their formation

IN ADDITION: In evolved disks with low density, the 2:1 occurs first and the planets move inwards.

D'Angelo & Marzari (2012)

Mass growth of the two planets may lead to violation of the conditions for outward migration

Gas is accreted within 0.1 RH

(under the hypothesis of disk-limited accretion rate).

Time (104 yr)

Σ ~ 1600 gr/cm2 Σ ~ 5000 gr/cm2

Σ ~ 800 gr/cm2Σ ~ 50 gr/cm2 (at 1 AU)

.

Conditions for extended migration of planets in resonance

The interior planet's mass must exceed that of the exterior planet during all migration. If the mass growth changes the ratio, the outward migration is interrupted and possibly reversed.

The gas density has not to be too low at the time of trapping otherwise the planets are captured in the 2:1 resonance. This sets a lower limit on the gas density at the capture.

The outward migration must be fast before the dispersal of the disk.

The planets must not have crossing orbits

Planet growth: Gas from the disk falls on the CPD while crossing the gap

The planet accrete gas from the CPD

For the second step the presence of viscosity in the CPD is relevant for the mass accretion rate dM

p/dt

Potential sources of viscosity:

MRI (inner regions)

Gravitational instability (outer regions)

Spiral waves induced by the sun tidal force (outer regions).

Different scenarios for the CPD:

Inviscid CPD: according to Szulagyi et al. (2014) the disk is MRI inactive. Inferred accretion rate from other mechanisms of the order of 2.5 x 10-6 M

J/yr for the planet. It does not explain why there are planets

with 2 and more Jupiter masses among extrasolar planets.

Viscous CPD: Gressel et al. (2013) performed resistive MHD simulations with adiabatic equation of state. They find that the CPD should be MRI active at least in the inner region with ionization due to XRs and CRs. Accretion rate of the order of 2.5 x 10-5 M

J/yr

Keith & Wardle (2014): self-consistent model of CPD where MRI mostly caused by thermal ionization can drive accretion out to ∼ 200 R

J , beyond which

gravitoturbulence dominates. Large values of are observerd within 30 R

J.

Turner et al. (2014) suggests that CPD have magnetically-active surface layers leading to accretion onto the planet and decoupled interior dead zones.

Images of the planet+CPD surroundings for increasing resolution. The vector indicates the direction of the gas flux. The gas falls onto the planet from high latitudes while the gas in the CPD is not moving radially due to the low viscosity.

1 MJ is not a

threshold value for the mass of giant planets

The growth process must allow for the formation of more massive planets

The Jumping Jupiter model

How do planets achieve large e and small q?

1) Planet-Planet scattering: at the end of the chaotic phase a planet 1) Planet-Planet scattering: at the end of the chaotic phase a planet is ejected, one is injected on a highly eccentric orbit that will be is ejected, one is injected on a highly eccentric orbit that will be tidally circularized to the pericenter, one is sent on an outer orbittidally circularized to the pericenter, one is sent on an outer orbit

2) 3)

Weidenschilling & Marzari (1996), Rasio and Ford (1996),Ida & Lin (1997).......

L=¿

1)

Stability limit for 2 planets

Stability limit for 3 planets

Δ c∼ 2 √3 R Ha i+ 1=a i+K RH

M P=M Neptune

M P=M Jupiter

M P=M Saturn

RH =( m1+m2

3 M s)(1/ 3)

( a1+a2

2 )

Marzari (2014)

Pure N-body P-P Pure N-body P-P scatteringscattering

Tidal interaction with the central

star (Nagasawa et al 2008)

Interaction with the gas of the

circumstellar disk (Marzari et al. 2010)

Interaction with a leftover planetesimal disk (Raymond et al. 2009)

Tidal migration of eccentric orbits Maximum e declines with distance from the star: tidal circularization. Energy is dissipated but the angular momentum J is preserved.

J=m pms

m p+ms√G (M s+M p)√a (1−e p

2 )

a f = a (1− e 2)=q (1+ e p )≈ 2 q

Inclined hot Jupiters due to instability+tide+Kozai with outer planet(s) from Nagasawa et al. (2008).

Example of 'Jumping Jupiters' in presence of the protostellar disk. The density of the disk is MMSN/2. Code used is FARGO (RK5 modified to have variable stepsize). One planet (1 M

J)

merges with another one (0.7 M

J) after a sequence of close

encounters.

Eccentricity evolution after P-P scattering: damping or excitation because of corotation resonance saturation?

Marzari et al. (2010), Lega et al. (2013)

Planetesimal disks Planetesimal disks and P-P scatteringand P-P scattering:

Lower eccentricities and inclinations for outer low-mass planets after P-P scattering (Raymond 2009, 2010) Possible formation of mini Oort

clouds by scattered planetesimals (Raymond & Armitage 2013) Lower fraction of debris disks

co-existing with the final planet system (Marzari 2014)

Green dots are systems which might retain a debris disk.

Synthetic population models: combine all processes to explain the present population of exoplanets.

Benz et al. 2014

Ida et al. 2013

Comparison with observed population (with and without observative bias)

THE FINAL MODEL

There are many weird planets out there, and theory must explain them all!

E=−G M s

2 [ m1

a1

+m2

a2

+m3

a3 ]

a i≈G M smi

2 E

Energy conservation

Eccentricity and inclination excitation. Outcome of many simulations with 3 initial planets within the instability limit by

Chatterjee et al. (2008).

CPD: CircumPlanetary Disk

Size: Defined either by truncation or by the rotation profile. (sw=specific angular momentum)

Mass: Minimum Mass, gas starved, numerical models …..

Gas trajectories in the proximity of the planet orbit. Close to the planet there is a vertical inflow on the CPD