Planet Formation - PowerPoint Presentation

Slide1

Planet Formation

Topic:

Formation of

rocky planets from

planetesimals

Lecture by: C.P. DullemondSlide2

Standard model of rocky planet formationStart with a sea of planetesimals (~1...100 km)

Mutual gravitational stirring, increasing „dynamic temperature“ of the planetesimal swarm.

Collisions, growth or fragmentation, dependent on the impact velocity, which depends on dynamic temperature.

If velocities low enough: Gravitational focusing: Runaway growth: „the winner takes it all“

Biggest body will stir up planetesimals: gravitational focusing will decline, runaway growth stalls.

Other „local winners“ will form: oligarchic growth

Oligarchs merge in complex N-body „dance“Slide3

Gravitational stirring of planetesimalsby each other and by a planetSlide4

Describing deviations from Kepler motionWe can describe an inclined elliptic orbit as an in-plane circular orbit with a „perturbation“ on top:

For the z-component we have:

So the mean square is:

For bodies at the midplane (maximum velocity):Slide5

Describing deviations from Kepler motionWe can describe an inclined elliptic orbit as an in-plane circular orbit with a „perturbation“ on top:

guiding

center

epicycle

For the x,y-components we have epicyclic

motion.

But notice that compared to the local (shifted) Kepler velocity

(green dashed circle in diagram), the y-velocity is lower:Slide6

„Dynamic temperature“ of planetesimalsMost massive bodies have smallest Δv. Thermalization is fast.So if we have a planet in a sea of planetesimals, we can assume

that the planet has e=i=0 while the planetesimals have e>0, i>0.

If there are sufficient gravitational interactions between the bodies

they „thermalize“. We can then compute a dynamic „temperature“:

Example: 1 km planetesimals at <i>=0.1, <e>=0.2, have a

dynamic temperature around 10

44

Kelvin!

Now that is high-energy physics! ;-)Slide7

Gravitational stirringWhen the test body comes very close to the bigger one, thebig one can strongly „kick“ the test body onto another orbit.

This leads to a jump in

a, e

and

i

. But there are relations

between the „before“ and „after“ orbits:

From the constancy ofthe Jacobi integralone can derive the Tisserand relation,

where ap is the a of the big planet:

Conclusion: Short-range „kicks“ can change e, i and a

before

afterSlide8

Gravitational stirringOrbit crossings: Close encounters can only happen if the orbitsof the planet and the planetesimal cross.

Given a semi-major axis a and eccentricity e, what are the smallest

and largest radial distances to the sun?Slide9

Gravitational stirring

Figure: courtesy of Sean Raymond

Can have close encounter

No close

encounter

possible

No close

encounter

possibleSlide10

Gravitational stirring

Ida & Makino 1993

Lines of constant

Tisserand numberSlide11

Gravitational stirring

Ida & Makino 1993

Lines of constant

Tisserand numberSlide12

Gravitational stirring

Ida & Makino 1993Slide13

Gravitational stirring: Chaotic behaviorSlide14

Gravitational stirring: resonancesWe will discuss resonances later, but like in ordinary dynamics,there can also be resonances in orbital dynamics. They make

stirring particularly efficient.

Movie: courtesy of Sean RaymondSlide15

Limits on stirring: The escape speedA planet can kick out a small body from the solar system by a single „kick“ if (and only if):

Jupiter can kick out a small body from the solar system,

but the Earth can not.Slide16

Collisions and growthSlide17

Feeding the planet

Feeding dynamically

„cool“ planetesimals.

The „shear-dominated regime“Slide18

Close encounters and collisions

Greenzweig & Lissauer 1990

Hill SphereSlide19

Feeding the planet

Feeding dynamically

„warm“ planetesimals.

The „dispersion-dominated regime“

with gravitational focussing (see

next slide).

Note: if we would be in the ballistic dispersion

dominated regime: no gravitational focussing („hot“ planetesimals).Slide20

Gravitational focussing

Due to the gravitational pull by the (big) planet, the smaller

body has a larger chance of colliding. The effective cross

section becomes:

M

m

Where the escape velocity is:

Slow bodies are easier captured! So: „keep them cool“!Slide21

Collision

Collision velocity of two bodies:

Rebound velocity:

v

c

with 1: coefficient of restitution.

v

c

 v

e

Two bodies remain gravitationally bound: accretion

v

c

 v

e

Disruption / fragmentation

Slow collisions are most likely to lead to merging.

Again: „Keep them cool!“Slide22

Example of low-velocity mergingFormation of Haumea (a Kuiper belt object)

Leinhardt, Marcus & Stewart (2010) ApJ 714, 1789Slide23

Example of low-velocity mergingFormation of Haumea (a Kuiper belt object)

Leinhardt, Marcus & Stewart (2010) ApJ 714, 1789Slide24

Growth of a planet



sw

= mass density of swarm of planetesimals

M = mass of growing protoplanet

v = relative velocity planetesimals

r = radius protoplanet

 = Safronov number



p

= density of interior of planet

Increase of planet mass per unit time:

Gravitational focussingSlide25

Growth of a planet

Estimate properties of planetesimal swarm:

Assuming that all planetesimals in feeding zone finally end up in planet

R = radius of orbit of planet

R = width of the feeding zone

z = height of the planetesimal swarm

Estimate height of swarm:Slide26

Growth of a planet

Remember:

Note: independent of



v!!

For M<<M

p

one has linear growth of rSlide27

Growth of a planet

Case of Earth:

v

k

= 30 km/s,

=6, M

p

= 6x10

27

gr, R = 1 AU, R = 0.5 AU, 

p

= 5.5 gr/cm

3

Earth takes 40 million years to form (more detailed models: 80 million years).

Much longer than observed disk clearing time scales. But debris disks can live longer than that.Slide28

Runaway growth

So for Δv<<v

esc

we see that we get:

The largest and second largest bodies separate in mass:

So: „The winner takes it all“!Slide29

End of runaway growth: oligarchic growthOnce the largest body becomes planet-size, it starts to stir upthe planetesimals. Therefore the gravitational focussing

reduces eventually to zero, so the original geometric cross

section is left:

Now we get that the largest and second largest planets

approach each other in mass again:

Will get locally-dominant „oligarchs“ that have similar masses,

each stirring its own „soup“. Slide30

Gas damping of velocitiesGas can dampen random motions of planetesimals if they are < 100 m - 1 km radius (at 1AU).

If they are damped strongly, then:

Shear-dominated regime (

v < 

r

Hill

)

Flat disk of planetesimals (h << r

Hill)One obtains a 2-D problem (instead of 3-D) and higher capture chances.

Can increase formation speed by a factor of 10 or more. This can even work for pebbles (cm-size bodies): “pebble accretion” is a recent development.Slide31

Isolation mass

Once the planet has eaten up all of the mass within its reach, the growth stops.

Some planetesimals may still be scattered into feeding zone, continuing growth, but this depends on presence of scatterer (a Jupiter-like planet?)

with

b = spacing between protoplanets in units of their Hill radii. b

 5...10.