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Planet Formation Topic: Formation of rocky planets from planetesimals Lecture by: C.P. Dullemond

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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“

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Gravitational stirring of planetesimals by each other and by a planet

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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):

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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 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:

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„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 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 =0.1, =0.2, have a dynamic temperature around 10 44 Kelvin! Now that is high-energy physics! ;-)

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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 a p is the a of the big planet: Conclusion: Short-range „kicks“ can change e, i and a before after

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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?

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Gravitational stirring Figure: courtesy of Sean Raymond Can have close encounter No close encounter possible No close encounter possible

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Gravitational stirring Ida & Makino 1993 Lines of constant Tisserand number

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Gravitational stirring Ida & Makino 1993 Lines of constant Tisserand number

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Gravitational stirring Ida & Makino 1993

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Gravitational stirring: Chaotic behavior

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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

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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.

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Collisions and growth

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Feeding the planet Feeding dynamically „cool“ planetesimals. The „shear-dominated regime“

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Close encounters and collisions Greenzweig & Lissauer 1990 Hill Sphere

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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).

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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“!

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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!“

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Example of low-velocity merging Formation of Haumea (a Kuiper belt object) Leinhardt, Marcus & Stewart (2010) ApJ 714, 1789

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Example of low-velocity merging Formation of Haumea (a Kuiper belt object) Leinhardt, Marcus & Stewart (2010) ApJ 714, 1789

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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 focussing

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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:

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Growth of a planet Remember: Note: independent of v!! For M<<M p one has linear growth of r

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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.

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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“!

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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“.

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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 < 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.

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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.

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