Planet Formation - PowerPoint Presentation

Slide1

Planet Formation

Topic:

Planet migration

Lecture by: C.P. DullemondSlide2

Planet migration: different kindsType I migration (small mass planets)Type II migration (high mass planets)

Type III migration (rare type II variant)Slide3

Two main ways to calculate torque:Follow gas packets in time, and see how they exchange angular momentum with the planet.

Impulse approximation

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.Slide4

Planet-inducedspiral wavesin the protoplanetary diskSlide5

Spiral wave: Pitch angle

Δv(a)

β

Δv

perp

(a)

To ensure that the spiral wave is

stationary in the reference frame

corotating with the planet, the component

of the orbital velocity Δv(a) perpendicular

to the spiral wave (i.e. Δv

perp

(a))

must be precisely equal to the sound

speed (assuming the wave is not a shock).

spiral

wave

gas orbital

velocity

vector

toward sunSlide6

Spiral wave: Launching point

Δv(a)

β

Δv

perp

(a)

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

when

spiral

wave

gas orbital

velocity

vector

With we can write

So we have:

So with the inner/outer wave is launched at:

launching

pointSlide7

Spiral wave: 2-D hydrodynamic models

Frederic Masset

http://www.maths.qmul.ac.uk/~masset/moviesmpegs.htmlSlide8

Spiral wave: 2-D hydrodynamic models

D‘Angelo, Henning & Kley (2002) Slide9

Type I migrationSlide10

Spiral wave: Gravitational „drag“

D‘Angelo, Henning & Kley (2002)

The gravitational force

acting on the planet

by the material in the

spiral arms

adds

and

subtracts

angular

momentum to/from the

planet. In general the

inwardforce is a tiny bit stronger, and so the

planet migrates inward.Slide11

Spiral wave: Gravitational „drag“

D‘Angelo, Henning & Kley (2002)

The other way of

looking at this is that

gas parcels are slung

by the planet and spend

some time „behind“ the

planet.

The torque acting on theplanet by these wavesis called theLindblad torqueSlide12

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: 10

5

...10

6

(Tanaka et al. 2002)Slide13

In „horseshoe orbits“ the

gas parcels „librate“ back

and forth.

At turning point

A

the gas

parcels give angular

momentum to the planet

(pushing the planet

outward).

At turning point

B

the gas

parcels retrieve angular

momentum from the planet

(pushing the planet

inward).

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

B.

ABHorseshoe dragSlide14

Horseshoe drag: close-up

Close-up view of the

planet fly-bys that

add

and

remove

angular momentum

to/from the planet

pushing the planetoutward / inward.

Image: D‘Angelo, Henning & Kley (2002) Slide15

„Unsaturated“ horseshoe drag

r

s

(entropy)

Suppose, as is to be expected, that the specific entropy s of the gas

in the disk increases with radial distance from the star. At the turning

points (the fly-by points) gas parcels change radius, but (if they do not

cool/heat quickly) retain their entropy.

horseshoe

region

The

inward

moving gas parcel finds itself with „too much“ entropy

while the

outward

moving gas parcel has „too little“ entropy compared

to the local „standard“. Slide16

„Unsaturated“ horseshoe drag

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

The fact that the fly-by

gas parcels keep their

entropy, but have to

adjust their density to

keep in local pressure

balance, they will

create 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.

Deficit

entropy:

over-

density.Slide17

Gap openingandType II migrationSlide18

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 diskSlide19

Gap opening in a disk

by Frederic Masset

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

Role of disk viscosity:

Planet pushes gas

away, out of the

co-orbital region.

Viscosity tries to move gas back in to

the co-orbital region.

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

less torque. Slide20

Behavior of Type II migrationCase of Mplanet<<M

disk

:

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 M

planet

>>M

disk: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.Slide21

Type III migration

Masset & Papaoloizou

Type III migration takes

place when the planet

migration time across

the co-orbital region

is shorter than the

libration time.Slide22

Type III migration

Masset & Papaoloizou

Type III migration takes

place when the planet

migration time across

the co-orbital region

is shorter than the

libration time.

By the time a parcel has

librated to the other

fly-by point, it might find

itself no longer inside

the co-orbital region.

A strong asymmetric

horseshoe drag follows.Slide23

Type III migration

by Frederic Masset

Note: this movie has opposite rotation as discussion above.

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