Planet Formation

Topic:

Planet migration

Lecture by: C.P. Dullemond

Planet migration: different kinds

• Type I migration (small mass planets)

• Type II migration (high mass planets)

• Type III migration (rare type II variant)

Two main ways to calculate torque:1. Follow gas packets in time, and see how they

exchange angular momentum with the planet.– Impulse approximation

2. Analyse how azimuthal asymmetries in the steady-state gas distribution in the disk Σ(r,ϕ) gravitationally pull on the planet.

Note: With 2-D/3-D time-dependent hydrodynamic simulations you essentially do both, because you simulate the entire thing in full glory.

Planet-inducedspiral waves

in the protoplanetary disk

Spiral wave: Pitch angle

Δv(a)β

Δvperp(a)

To ensure that the spiral wave isstationary in the reference framecorotating with the planet, the componentof the orbital velocity Δv(a) perpendicularto the spiral wave (i.e. Δvperp(a))must be precisely equal to the sound speed (assuming the wave is not a shock).

spiralwave

gas orbitalvelocity

vector

toward sun

Spiral wave: Launching point

Δv(a)β

Δvperp(a)This angle becomes ≈1 (i.e. very large)when

spiralwave

gas orbitalvelocity

vector

With we can write

So we have: So with the inner/outer wave is launched at:

launchingpoint

Spiral wave: 2-D hydrodynamic models

Frederic Massethttp://www.maths.qmul.ac.uk/~masset/moviesmpegs.html

Spiral wave: 2-D hydrodynamic models

D‘Angelo, Henning & Kley (2002)

Type I migration

Spiral wave: Gravitational „drag“

D‘Angelo, Henning & Kley (2002)

The gravitational force acting on the planetby the material in thespiral arms adds and subtracts angular momentum to/from theplanet.

In general the inwardforce is a tiny bit stronger, and so the planet migrates inward.

Spiral wave: Gravitational „drag“

D‘Angelo, Henning & Kley (2002)

The other way of looking at this is that gas parcels are slungby the planet and spendsome time „behind“ theplanet.

The torque acting on theplanet by these wavesis called theLindblad torque

Time scale of type I migration

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

Review Thommes & Duncan in “The Formation of Planets” 2005

3-D estimates: 105...106 (Tanaka et al. 2002)

In „horseshoe orbits“ the gas parcels „librate“ backand forth.

At turning point A the gasparcels give angularmomentum to the planet(pushing the planet outward).

At turning point B the gasparcels retrieve angularmomentum from the planet(pushing the planetinward).

Normally both forces cancel because each parcel passes as many times point A as point B.

A

B

Horseshoe drag

Horseshoe drag: close-up

Close-up view of theplanet fly-bys thatadd and remove angular momentumto/from the planetpushing the planetoutward / inward.

Image: D‘Angelo, Henning & Kley (2002)

„Unsaturated“ horseshoe drag

r

s(entropy)

Suppose, as is to be expected, that the specific entropy s of the gasin the disk increases with radial distance from the star. At the turningpoints (the fly-by points) gas parcels change radius, but (if they do notcool/heat quickly) retain their entropy.

horseshoeregion

The inward moving gas parcel finds itself with „too much“ entropywhile the outward moving gas parcel has „too little“ entropy comparedto the local „standard“.

„Unsaturated“ horseshoe drag

Image: D‘Angelo, Henning & Kley (2002)

The fact that the fly-bygas parcels keep theirentropy, but have toadjust their density tokeep in local pressurebalance, they willcreate an imbalancein the two torques.

NOTE: After many libration periods thiswould „saturate“.

IF gas radiatively cools/heats during libration,it can remain unsaturated.

Radiation-hydro problem!

Excess entropy:under-density.

Deficitentropy:over-density.

Gap openingand

Type II migration

Hill sphere: sphere of gravitational influence of planet:

If Hill radius larger than h of disk: disk can be regarded as thin compared to potential. This happens for massive enough planets.

Planet may then open a gap. But this depends also on other things, e.g.viscosity.

P. Ciecielag

Gap opening in a disk

Gap opening in a disk

by Frederic Massethttp://www.maths.qmul.ac.uk/~masset/moviesmpegs.html

Role of disk viscosity:

Planet pushes gasaway, out of the co-orbital region.

Viscosity tries to move gas back in tothe co-orbital region.

Low viscosity largergap, extending beyond the co-orbitalregion less gas near the planet less torque.

Behavior of Type II migration• Case of Mplanet<<Mdisk:

– Planet will automatically get pushed to the center of the gap. If, for example, it is too close to the outer gap edge, the outer torque (pushing the planet inward) is stronger than the inner torque, so the planet is pushed inward. Planet is „locked to the disk“.

– The viscous evolution of the disk will dictate the planet‘s migration. Planet migration goes on viscous time scale (much slower than type I migration)

• Case of Mplanet>>Mdisk:– Disk cannot push planet. Planet migration is very slow.– Gap can be very deep, completely halting inward gas

flow through the gap: inner disk „choked“ and vanishes on the viscous time scale. Large inner hole forms.

Type III migrationMasset & Papaoloizou

Type III migration takesplace when the planetmigration time acrossthe co-orbital regionis shorter than thelibration time.

Type III migrationMasset & Papaoloizou

Type III migration takesplace when the planetmigration time acrossthe co-orbital regionis shorter than thelibration time.

By the time a parcel haslibrated to the otherfly-by point, it might finditself no longer inside the co-orbital region.

A strong asymmetric horseshoe drag follows.

Type III migration

by Frederic Masset

Note: this movie has opposite rotation as discussion above.

http://www.maths.qmul.ac.uk/~masset/moviesmpegs.html