Planet Formation

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 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

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?)

Planet Formation