Planet formation
This presentation is the property of its rightful owner.
Sponsored Links
1 / 31

Planet Formation PowerPoint PPT Presentation


  • 120 Views
  • Uploaded on
  • Presentation posted in: General

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.

Download Presentation

Planet Formation

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Planet formation

Planet Formation

Topic:

Formation of

rocky planets from

planetesimals

Lecture by: C.P. Dullemond


Standard model of rocky planet formation

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“


Planet formation

Gravitational stirring of planetesimals

by each other and by a planet


Describing deviations from kepler motion

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 motion1

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

„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

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 stirring1

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 stirring2

Gravitational stirring

Can have close encounter

No close

encounter

possible

No close

encounter

possible

Figure: courtesy of Sean Raymond


Gravitational stirring3

Gravitational stirring

Lines of constant

Tisserand number

Ida & Makino 1993


Gravitational stirring4

Gravitational stirring

Lines of constant

Tisserand number

Ida & Makino 1993


Gravitational stirring5

Gravitational stirring

Ida & Makino 1993


Gravitational stirring chaotic behavior

Gravitational stirring: Chaotic behavior


Gravitational stirring resonances

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

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.


Planet formation

Collisions and growth


Feeding the planet

Feeding the planet

Feeding dynamically

„cool“ planetesimals.

The „shear-dominated regime“


Close encounters and collisions

Close encounters and collisions

Hill Sphere

Greenzweig & Lissauer 1990


Feeding the planet1

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

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“!


Collision

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

Example of low-velocity merging

Formation of Haumea (a Kuiper belt object)

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


Example of low velocity merging1

Example of low-velocity merging

Formation of Haumea (a Kuiper belt object)

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


Growth of a planet

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


Growth of a planet1

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 planet2

Growth of a planet

Remember:

Note: independent of v!!

For M<<Mp one has linear growth of r


Growth of a planet3

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

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

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 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

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?)


  • Login