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Planet Formation. Topic: Formation of rocky planets from planetesimals Lecture by: C.P. Dullemond. Standard model of rocky planet formation. Start with a sea of planetesimals (~1...100 km) Mutual gravitational stirring, increasing „dynamic temperature“ of the planetesimal swarm.

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Planet Formation

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Planet Formation

Topic:

Formation of

rocky planets from

planetesimals

Lecture by: C.P. Dullemond


Standard model of rocky planet formation

  • Start 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“


Gravitational stirring of planetesimals

by 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:

guiding

center

For the x,y-components we have epicyclic

motion.

epicycle

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

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


„Dynamic temperature“ of planetesimals

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 1044 Kelvin!

Now that is high-energy physics! ;-)

Most 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.


Gravitational stirring

When the test body comes very close to the bigger one, the

big 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 of

the Jacobi integral

one 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 orbits

of 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?


Gravitational stirring

Can have close encounter

No close

encounter

possible

No close

encounter

possible

Figure: courtesy of Sean Raymond


Gravitational stirring

Lines of constant

Tisserand number

Ida & Makino 1993


Gravitational stirring

Lines of constant

Tisserand number

Ida & Makino 1993


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 make

stirring 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

Hill Sphere

Greenzweig & Lissauer 1990


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).


Gravitational focussing

m

M

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

body has a larger chance of colliding. The effective cross

section becomes:

Where the escape velocity is:

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


Two bodies remain gravitationally bound: accretion

vc  ve

Disruption / fragmentation

vc  ve

Collision

Collision velocity of two bodies:

Rebound velocity: vc with 1: coefficient of restitution.

Slow collisions are most likely to lead to merging.

Again: „Keep them cool!“


Example of low-velocity merging

Formation of Haumea (a Kuiper belt object)

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


Example of low-velocity merging

Formation of Haumea (a Kuiper belt object)

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


Increase of planet mass per unit time:

Gravitational focussing

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


Estimate height of swarm:

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


Growth of a planet

Remember:

Note: independent of v!!

For M<<Mp one has linear growth of r


Case of Earth:

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

Growth of a planet

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 up

the 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“.


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.


with

Isolation mass

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

b = spacing between protoplanets in units of their Hill radii. b  5...10.

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


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