1

Chapter 13

Giant Planet Formation andMigration

Formation of Jovian planets

• Existence of Uranus and Neptune prove that solidcores can form even in the outer reaches of the solarsystem– or they must form elsewhere and be moved out

• Some theoreticians say they formed between Jupiter and Saturn!

• These might accrete gas from the disk to formJupiter/Saturn kind of planets.

• Bottle necks:– Must be able to form a core quickly enough– Must accrete gas fast, before disk disperses

Stages of Core-Accretion

• Core Formation• Hydrostatic Growth• Runaway Growth• Termination of Accretion

2

Characteristic Masses

!

RS =34"

Mp

#m

$

% &

'

( )

!

vesc =2GMp

RS

!

cs =hr"

# $ %

& ' vK

!

Mp >332"#

$ %

&

' ( 1/2 h

r#

$ % &

' ( 3 M*

3/2

)m1/2r 3/2

~ 5*10+4M,Vesc > cs:

So we get some atmosphere already at low masses

Envelope mass

!

dPdR

= "GMp

R2#

Assume isothermal atmosphere, match disk density rho0

!

ln"(R) =GMp

cs21R

+ const

!

Rout =2GMp

cs2

!

"(R) = "0 expGMp

cs21R#12

$

% &

'

( )

!

"0 =12#

$h

Disk midplane density

!

Menv "4#3Rs3$ RS( )Most mass close to surface

!

Mp "3

4#$m

%

& '

(

) *

1/2cs2

G%

& '

(

) *

3/2

ln +$m

$0

%

& '

(

) *

,

- .

/

0 1

3/2

2 0.2M3Want Menv > εMp:

3

Core accretion model

Pollack et al, 1996

Embryoformation(runaway)

Embryoisolation

Rapid gasaccretion

Truncatedby gapformation

Also called ‘nucleated instability model’

Core accretion model

dM/dt

time

10-2M⊕/yr

Rapid gas accretionDeclining accretion as nebulagap develops; onset of satelliteformation

~106 yr

From: Dave Stevensen (2004)

Formation of Jupiter: effect of migration

Alibert, Mousis,Mordasini, Benz(2005)

Model with: - Evolving disk - Migration

Leads to: * Faster growth * Explain Ju +Sa

4

Formation of Jupiter: effect of migration

Withoutmigration: too

slow (diskalready gone)

Withmigration:

fast enough!

Alibert, Mordasini,Benz, Winisdoerfer(2005)

Alternative model: gravitational instab.

Alternative model: gravitational instab.

• ‘Alan Boss model’• Nice:

– Quite natural to form gravitationally unstable disks if there isno MRI-viscosity in the disk

– Avoid problem of dust agglomeration & meter-size barrier– No time scale problem

• Problem:– Can disk get so very unstable? Gravitational spiral waves

quickly lower surface density to marginal stability– Why do we have earth-like planets?

5

Gravitational (in)stabilityIf disk surface density exceeds a certain limit, then disk becomesgravitationally unstable.

Toomre Q-parameter:

!

Q =h"K

2

#G$

!

"hr

M*

Mdisk

For Q>2 the disk is stableFor Q<2 the disk is gravitationally unstable

Instability versus fragmentation

Viscous time scale

!

t" #1

$%K

hr&

' ( )

* + ,2

Thermal timescale

!

tth "1

#$K

In PP disks: tth << tν, so cooling is important

!

" =4

9#(# $1)%Ktcool

!

tcool,crit "3#K

!

tcool =U

dU /dt

α is measure of “self-gravitating turbulence”.Numerical simulations show that acrit marks boundary betweenfragmentation and stable transport of angular momentum

For γ=2

6

Gravitational (in)stability: Viscosity

Spiral waves actas `viscosity’

Rice & Armitage

Gravitational (in)stability: Fragmentatio

Disk cooling through radiative cooling

!

Tc4

Tsurf4 " # ross

For a typical disk at the edge of instability on finds

!

tcool"K # 10$r

5AU%

& '

(

) * +1/2

7

Hot Jupiters: Migration or In Situ?

Migration

rp

Resonances

Interaction happens mostly at resonances:

For example: 1:2 is at 2-2/3rp=0.63 rp

How the disk pushes the planet

log R

log ∑

Viscosity

Gravity

Viscosity tries to close gapGravitation widens gapPlanet acts as bridgePlanet moves with disk gasMigration time ~105 years from 5AU

8

Three types of migration

• Type I: low mass planets

• Type II: high mass planets

• Type III: rare type II variant

Type I migration

• Planet’s gravity launches spiral waves in disk• These spiral waves exert torque on planet:

– Inner spiral wave pushes planet outward– Outer spiral wave pushes planet inward

• Outer spiral wave wins: inward migration

Type I migration

by Frederic Massetwww-star.qmul.ac.uk/~masset/

9

Type I migration

Time scale of inward type I migration (1 solar mass star):

!

tType I =104...105 M10M"

#

$ %

&

' (

)1*gas

100g/cm2

#

$ %

&

' (

)1r

AU#

$ %

&

' ( )1/ 2 h /r

0.07#

$ %

&

' (

2

years

Review Thommes & Duncan in “TheFormation of Planets” 2005

3-D estimates: 105...106

(Tanaka et al. 2002)

Gap opening

Hill sphere: sphere of gravitational influence of planet:

!

rHill =M3M*

"

# $

%

& '

1/ 3

r

If Hill radius larger than h of disk: disk can be regarded as thincompared to potential. This happens for massive enoughplanets.

Planet will affect structure of the disk.

P. Ciecielag

Example: M2/M1=0.1

Effective potential, Lagrange points

r1 r2

!

r1r2

=M2

M1

!

"eff = #GM1r r # r r 1

#GM2r r # r r 2

#12$K2 r r 2

Effective potential in the co-rotating frame:

centrifugalkineticenergy

L1 L2L3

L4

L5

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Effective potential, Lagrange points

r1 r2

!

r1r2

=M2

M1

!

"eff = #GM1r r # r r 1

#GM2r r # r r 2

#12$K2 r r 2

Effective potential in the co-rotating frame:

centrifugalkineticenergy Example: M2/M1=0.01

L1 L2L3

L4

L5

Trojans of Jupiter

Motion of gas / particles in horseshoe

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Gap opening

by Frederic Massetwww-star.qmul.ac.uk/~masset/

Type II migration

• Massive planet opens a gap• Accretion in the disk is stopped by the gap

– If the disk is massive enough: accretion continues, simply bypushing the planet inward. Planet is locked to the diskaccretion. Type II migration

– If the disk is not massive enough: planet will not migrate.Inner disk will deplete.

• Three-dimensional models: accretion can still proceedsomewhat by flowing in 3-D past the planet.

Transition from I to II and gap opening

by Frederic Massetwww-star.qmul.ac.uk/~masset/

12

Type III migration (run-away migration)Masset & Papaoloizou

Type III migration (run-away migration)Masset & Papaoloizou

Type III migration (run-away migration)

• If planet initially moves inward:– Some inner disk material enters horseshoe, gets flung to

outer orbit of horseshoe by planet. Planet loses angularmomentum.

– Some horseshoe material enters outer disk, does not getflung back to inner orbit of horseshoe.

– Netto: one-sided asymmetric angular momentum transportfrom planet to disk: inward push! Run-away!

• If planet initially moves outward: Same thing, but theother way: planet is pushed outward. Also run-away!

13

Type III migration (run-away migration)

www-star.qmul.ac.uk/~masset/

by Frederic Masset

Note: thismovie hasoppositerotation asdiscussionabove.

Why do planets exist everywhere?

• Migration should have depleted all planets• What about bandwagon approach (form planets all the

time, lose most of them via migration, but when diskdissipates some are left)?– Problem: Need plenty of solid disk material to form a planet– Problem: First make rocky core, then accrete gas. This

process takes longer than migration time scale.

• Problem of migration is one of main openquestions of planet formation!

How to planets get in resonance?

G. Bryden

14

Stopping migration with a resonance

Masset & Snellgrove

Formation of Kuiper belt and Oort cloud

Brett Gladmann Science 2005

Debris disks• After about 10 Myrs most gas-rich protoplanetary disks

fade away. Gas is (apparently) removed from the disk ona time scale that is shorter than normal viscous evolution.– Has been removed by accretion onto protoplanet?– Has been removed by photo-evaporation?

• Dust grains are removed from the system by radiationpressure and drag (Poynting-Robertson)

• Yet, a tiny but measurable amount of dust is detected indisk-like configuration around such stars. Such stars arealso called ‘Vega-like stars’.

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Debris disksBeta-PictorisAge: 100 Myr (some say 20 Myr)

Dust is continuously replenished by disuptive collisions betweenplanetesimals. Disk is very optically thin (and SED has very weakinfrared excess).

Basic processes: radiation pressure

!

" =FradFgrav

#0.4µmD

L* /LoM* /Mo

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Grain size limit due to blow-out

Artymowicz 1988

• Radiative blowoutprovides a lower limit tograin sizes

• Smaller grains are onlypresent for a Kepler timeafter production

• Smallest grains are notblown out!

Basic processes: PR drag

• Absorption and reemission of movingbody leads to a force slowing down theparticle

• Associated time scale

!

" PR =400

M* /Mo

(r /AU )2

#years

Radial distribution of dust from a belt

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Basic processes: Collisions• Planetesimal distribution needs to be stirred to get to destructive collisions

(e,i>10-2...-3)• Collisions between large bodies produce a distribution of fragments,

distribution depends on material properties and collision velocity• However, a collisionally dominated cascade develops toward f(a) ~ a-3.5

• Can be used for estimatesof mass ratio dust/comets

Collisions: The devil is in the details

Are there planets in known debris disks?

Wilner, Holman, Kuchner & Ho (2002)

1.3 mm mapSimulation of disk with 3 Mjup planet inhighly eccentric orbit, trapping dust in meanmotion resonances.

Map of the dust around Vega:

18

Vega

JCMT imageHolland et al 1998

Gravitational Perturbations of UnseenPlanet

It is the effect of a planet's gravity on the orbits of planetesimalsand dust in a debris disk which causes structure in it.

The effect of a planet’s gravity can be divided into two groups(e.g., Murray & Dermott 1999)

• Secular Perturbations• Resonant Perturbations

Both are the consequence of Newton’s F=GMdustMpl/r2 law ofgravitation

Secular Perturbations

Cause the disk to be:

• Offsetif the planet has

an eccentric orbit

• Warpedif the planet hasan inclined orbit

Are the long term effect of the planet’s gravity and act on all disk materialover >0.1 Myr timescales

e.g., lobe brightnessasymmetry in HR4796disk (Wyatt et al. 1999;Telesco et al. 2000)

e.g., warp in βPictoris disk (Heap et al.2000; Augereau et al. 2001)

19

Mechanisms for Filling Resonances

While some resonances are very stable, they occupy a small region ofparameter space.

Resonances are filled for two reasons:

• Inward migration of dustDust spirals in toward the stardue to P-R drag and resonancestemporarily halt inward migration

• Outward migration of planet Planet migrates out and planetesimals are swept into the planet’s resonances

Resonant filling causes a ring to form along the planet’s orbit

Pl

Resonance

Star

Pl

Resonance

Star

20

Clumpy Debris Disks

Diameter ofSolar System

Observations show that many debris disks are characterized byclumpy rings

Vega Fomalhaut ε Eridani

Holland et al. (1998) Holland et al. (2003) Greaves et al. (1998)

The only viable explanations for this clumpiness involve planetaryresonances

Vega: Evidence of Planet Migration

Orbit Distribution Spatial Distribution Emission Distribution

• Wyatt (2003) explained Vega’s twoasymmetric clumps by the migration of a17Mearth planet from 40-65AU in 56 Myr

• Most planetesimals end up in the planet’s2:1(u) and 3:2 resonances

Observed

Model

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Plutinos: Migration causes orbital excitation

Neptune P

1:1 3:2 2:1

t=0

today

Pluto’s resonant capture

Malhotra 1993