Gas Disk Migration

Phil Armitage (Colorado)

• Regimes of gas disk migration (review)

• 2 emerging areas where migration may require major

changes to understanding of planet formation

- new work on “Type I” migration [core formation]

- stochastic migration [planetesimal formation]

Recent references:

- Paardekooper et al. (2010, MNRAS in press)

- Ida, Guillot & Morbidelli (2008)

What is gas disk migration?

Planet within gas disk exchanges angular momentum with

the gas due to gravitational torques

Origin of the torque can be understood semi-quantitatively

using impulse approximation (Lin & Papaloizou 1979):

r

b

gas

relative

velocity v

v

2GM

bv

v

1

2v

2GM

bv

2

Gas exterior to the planet has gained specific angular

momentum:

j 2G

2M

2a

b2v

3

Integrate over the disk outside the planet from (a+bmin)

to large distance, get total (exterior) torque on planet:

dJ

dt

8

27

G2M

2a

2b

min

3

…scaling with the square of the planet mass

Numerically, for an Earth mass at 5 AU with = 102 g cm-2,

bmin = h, migration rate due to one sided torque:

J

dJ / dt~ 1 Myr

Elementary argument: unless torques unexpectedly

cancel to high precision, angular momentum transfer

to the gas will lead to significant orbital migration

More rigorous analysis compute the torque as the sum of

partial torques exerted at resonant locations in the disk:

• interior and exterior Lindblad resonances, where

waves are excited

• corotation resonances (co-orbital for circular planet)

Goldreich & Tremaine (1979)… through to Tanaka, Takeuchi

& Ward (2002)… obtain qualitatively same result

Regimes of gas disk migration

For a low mass planet, torques do not affect disk structure

significantly: planet remains embedded within gas

For a high mass planet, torques repel gas from near the

planet, forming a gap in the disk surface density

QuickTime™ and a

YUV420 codec decompressor

are needed to see this picture.

Dividing line depends upon the

angular momentum transport

efficiency and vertical thickness

of the disk: normally a Saturn

mass planet suffice to open a

partial gap…

Type 1 migration

Mp = 1 - 10 Mearth, planet remains fully

embedded in gas

Said to undergo “Type 1” migration

Interaction is weak, standard estimate is linear calculation of

torque at Lindblad + co-orbital corotation resonances

Tanaka, Takeuchi & Ward (2002) find that migration in this

regime is always inward, irrespective of surface density

gradients in the disk.

Numerical simulations of the

same model (isothermal disk

with given structure) agree

with analytic prediction

(Bate et al. 2003)

For any reasonable disk

model, migration of giant

planet cores is very rapid,

faster than their formation

time scale!!!

Critical problem for giant planet formation

models: standard “solution” is to either ignore

migration or to scale the rate down by a large

(~102) and arbitrary factor…

Type 2 migration

Once gap open, planet migrates at rate

that gas tries to flow back into the gap

Orbital evolution occurs on the viscous

time scale of the disk

Alexander & Armitage (2009)

This is “good” - standard

way to end up with hot

Jupiters (Lin et al. 1996)

starting from formation

sites at several AU

Still no direct evidence

for this process

New ideas on Type 1 migration

Main uncertainty has always been computation of the torque

due to co-orbital material

Ward (1991) proposed a

different way to compute

this torque not based on

linear response of the gas

at corotation

Idea is to consider torque

from gas in closed region

where there are horseshoe

orbits

As gas executes horseshoe turns,

changes in density will affect the

torque on the planet

Magnitude of the instantaneous

torque will vary depending upon the

disk’s thermal and structural

properties, e.g.:

Barotropic: p = p()

D

Dt

v

0

depends on gradient

of vortensity in the disk

Adiabatic:

D

Dt

p

0

entropy is conserved along

streamlines, but may vary if

the gas can cool on time scale

of horseshoe turns

On longer time scale viscosity also matters: horseshoe

region is closed so unless viscosity couples to rest of disk

torque must vanish at sufficiently late times

Paardekooper et al. (2010)

If the torque is unsaturated:

2.5 1.7 0.1

1.13

2

7.9

A strong entropy gradient

(declining temperature or

positive ) means the

horseshoe drag beats

Lindblad torque and leads

to outward Type 1 migration

All still quite preliminary - still need to understand better 3D

effects, whether torque remains unsaturated in turbulent disk

BUT does not support the idea that Type 1 migration rate

for ~10 MEarth giant planet cores is slow… adding two large

contributions of opposite sign does not generally give zero!

• cores may migrate outward (rapidly) in optically thick,

almost adiabatic inner disk

• cores migrate inward (fast) in almost isothermal outer

disk, as in “classical” Type 1 migration

Intermediate zone where migration is slow, whose radius

evolves slowly as disk is depleted - perhaps cores accumulate

and stall there to form multiple giant planets?

Stochastic migration

Recall: dJ / dt ~ M2, so Type 1 ~ 1 / M

Negligible angular momentum transfer between very low

mass bodies (“planetesimals”) and a laminar gas disk

QuickTime™ and a

YUV420 codec decompressor

are needed to see this picture.

In a turbulent disk, stochastic

forces from transient surface

density perturbations can:

• excite random motions

of all bodies, even mass

less tracers

• result in random-walk

migration

Nelson & Papaloizou (2004);

Laughlin et al. (2004)

Why does this matter for planetesimal formation?

Conventionally, e / i of planetesimals is set by balance

between:

• excitation by planetesimal / planetesimal scattering

• damping by aerodynamic gas drag

Leads to low equilibrium

eccentricities

Collisions have specific energy

too low to fragment targets, even

km scale planetesimals would

experience runaway growth

Much stronger excitation by

gravitational coupling to

turbulence can change this

picture

Ida et al. (2008)

Using best estimates of the

properties of disks in which

magnetorotational instability

generates turbulence…

Find accretion occurs only

for ~100 km scales and

above, collisions in the

outer disk can be highly

erosive

Preliminary: suggests connections between astrophysics,

planetesimal formation, Solar System planetary science

Astrophysics: can we calculate the strength of disk

turbulence from first principles? Need simulations with

non-ideal MHD (Ohmic + Hall terms)…

Planetesimal formation: if 1 km planetesimals are

destroyed in collisions, must form planetesimals directly

at larger sizes - gravitational collapse. Streaming

instabilities, collection of particles in turbulence etc…

Solar System constraints: is there evidence for large

primordial planetesimals in the size distribution of

asteroids (Morbidelli et al. 2009)? Stochastic migration

would also erase compositional gradients - is that

consistent with observations?