Planet Formation

Topic:

Formation ofrocky planets from

planetesimals

Lecture by: C.P. Dullemond

Standard model of rocky planet formation

1. Start with a sea of planetesimals (~1...100 km)

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

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

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

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

6. Other „local winners“ will form: oligarchic growth

7. Oligarchs merge in complex N-body „dance“

Gravitational stirring of planetesimalsby each other and by a planet

Describing deviations from Kepler motion

We 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):

Describing deviations from Kepler motion

We can describe an inclined elliptic orbit as an in-plane circular orbit with a „perturbation“ on top:

guidingcenter

epicycle

For the x,y-components we have epicyclicmotion.

But notice that compared to the local (shifted) Kepler velocity (green dashed circle in diagram), the y-velocity is lower:

„Dynamic temperature“ of planetesimals

Most massive bodies have smallest Δv. Thermalization is fast.So if we have a planet in a sea of planetesimals, we can assumethat the planet has e=i=0 while the planetesimals have e>0, i>0.

If there are sufficient gravitational interactions between the bodiesthey „thermalize“. We can then compute a dynamic „temperature“:

Example: 1 km planetesimals at <i>=0.1, <e>=0.2, have adynamic temperature around 1044 Kelvin!

Now that is high-energy physics! ;-)

Gravitational stirring

When 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 relationsbetween 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

after

Gravitational stirring

Orbit 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 smallestand largest radial distances to the sun?

Gravitational stirring

Figure: courtesy of Sean Raymond

Can have close encounter

No closeencounterpossible

No closeencounterpossible

Gravitational stirring

Ida & Makino 1993

Lines of constant Tisserand number

Gravitational stirring

Ida & Makino 1993

Lines of constant Tisserand number

Gravitational stirring

Ida & Makino 1993

Gravitational stirring: Chaotic behavior

Gravitational stirring: resonances

We will discuss resonances later, but like in ordinary dynamics,there can also be resonances in orbital dynamics. They makestirring particularly efficient.

Movie: courtesy of Sean Raymond

Limits on stirring: The escape speed

A 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.

Collisions and growth

Feeding the planet

Feeding dynamically„cool“ planetesimals.

The „shear-dominated regime“

Close encounters and collisions

Greenzweig & Lissauer 1990

Hill Sphere

Feeding the planet

Feeding dynamically„warm“ planetesimals.

The „dispersion-dominated regime“with gravitational focussing (seenext slide).

Note: if we would be in the ballistic dispersiondominated regime: no gravitational focussing („hot“ planetesimals).

Gravitational focussing

Due to the gravitational pull by the (big) planet, the smallerbody has a larger chance of colliding. The effective crosssection becomes:

Mm

Where the escape velocity is:

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

Collision

Collision velocity of two bodies:

Rebound velocity: vc with 1: coefficient of restitution.

vc veTwo bodies remain gravitationally bound: accretion

vc veDisruption / fragmentation

Slow collisions are most likely to lead to merging.Again: „Keep them cool!“

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

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

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

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

Growth of a planet

sw = mass density of swarm of planetesimalsM = mass of growing protoplanetv = relative velocity planetesimalsr = radius protoplanet = Safronov number

p = density of interior of planet

Increase of planet mass per unit time: Gravitational focussing

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 planetR = width of the feeding zonez = height of the planetesimal swarm

Estimate height of swarm:

Growth of a planet

Remember:

Note: independent of v!!

For M<<Mp one has linear growth of r

Growth of a planet

Case of Earth:

vk = 30 km/s, =6, Mp = 6x1027 gr, R = 1 AU, R = 0.5 AU, p = 5.5 gr/cm3

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.

Runaway growth

So for Δv<<vesc we see that we get:

The largest and second largest bodies separate in mass:

So: „The winner takes it all“!

End of runaway growth: oligarchic growth

Once the largest body becomes planet-size, it starts to stir upthe planetesimals. Therefore the gravitational focussing reduces eventually to zero, so the original geometric crosssection 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“.

Gas damping of velocities• Gas 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 < rHill)– Flat disk of planetesimals (h << rHill)

• 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.

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.