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Planet Formation Topic: Formation of rocky 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 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: v body = v K + Dv body For the z-component we have: Dv (z) body (t) = sini v K sin(Wt) So the mean square is: Dv (z) body = 12 sini v K » 12 v K i For bodies at the midplane (maximum velocity): Dv (z) body midplane = sini v K » v K i Describing deviations from Kepler motion We can describe an inclined elliptic orbit as an in-plane circular orbit with a „perturbation“ on top: v body = v K + Dv body For the x,y-components we have epicyclic motion. Dv Dv ( x) body ( y) body (t) » ev K sin(Wt) (t) » 2ev K cos(Wt) But notice that compared to the local (shifted) Kepler velocity (green dashed circle in diagram), the y-velocity is lower: Dv ( y) body,local (t) » 12 ev K cos(Wt) guiding center epicycle „Dynamic temperature“ of planetesimals If there are sufficient gravitational interactions between the bodies they „thermalize“. We can then compute a dynamic „temperature“: Dv 2 body = kBTdyn mbody 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: before ap ap a a' 2 Tº + (1- e ) cosi » + (1- e'2 ) cosi' 2a ap 2a' ap Conclusion: Short-range „kicks“ can change e, i and a 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? a(1- e2 ) a(1- e)(1+ e) r(v) = = 1+ ecos(v) 1+ ecos(v) rmin = a(1- e) rmax = a(1+ e) Gravitational stirring Can have close encounter No close encounter possible Figure: courtesy of Sean Raymond No close encounter possible Gravitational stirring Lines of constant Tisserand number e2 + i 2 Ida & Makino 1993 a Gravitational stirring Lines of constant Tisserand number e2 + i 2 Ida & Makino 1993 a 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): v esc,planet º 2GM planet Rplanet 2GM * > v esc,system º aplanet 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. Dv £ v Hill GM p v Hill = RHill The „shear-dominated regime“ Close encounters and collisions Hill Sphere Greenzweig & Lissauer 1990 Feeding the planet Feeding dynamically „warm“ planetesimals. vHill £ Dv £ vesc GM p v Hill = RHill v esc = 2GM p Rp The „dispersion-dominated regime“ with gravitational focussing (see next slide). Note: if Dv>vescwe 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: s eff 2ù é æ v esc ö 2 = p ( R + r ) ê1+ ç ÷ú êë è Dv ø úû Where the escape velocity is: v esc = 2G ( M + m) (R + r) Slow bodies are easier captured! So: „keep them cool“! Collision Collision velocity of two bodies: 2 vc = Dv 2 + vesc Rebound velocity: vc with 1: coefficient of restitution. vc ve Two bodies remain gravitationally bound: accretion vc ve Disruption / fragmentation 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 Growth of a planet Increase of planet mass per unit time: sw M v r Gravitational focussing = mass density of swarm of planetesimals = mass of growing protoplanet = relative velocity planetesimals = radius protoplanet = Safronov number dr rsw Dv = (1+ q ) dt 4rp p = density of interior of planet dM = 4p r 2 rp dr 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: Growth of a planet Remember: dr rsw Dv = (1+ q ) dt 4rp dr v K (1+ q )(M p - M ) = dt 16p R 2 DR rp Note: independent of v!! For M<<Mp one has linear growth of r æ dM M ö 2/3 ÷÷ µ M (1+ q ) çç1dt è Mp ø Growth of a planet dr v K (1+ q )(M p - M ) = dt 16p R 2 DR rp Case of Earth: vk = 30 km/s, =6, 1 AU, R = 0.5 AU, dr = 15 cm/year dt Mp = 6x1027 gr, R = p = 5.5 gr/cm3 t growth = 40 Myr 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 2ù é æ ö dM v µ M 2/3 (1+ q )= M 2/3 ê1+ ç esc ÷ ú dt êë è Dv ø úû 2GM v = µ M 2/3 R 2 esc So for Δv<<vesc we see that we get: dM µ M 4/3 dt The largest and second largest bodies separate in mass: d æ M1 ö 1/3 1/3 lg ç ÷ µ M1 - M 2 > 0 dt è M 2 ø 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: dM µ M 2/3 (1+ q )® M 2/3 dt Now we get that the largest and second largest planets approach each other in mass again: d æ M1 ö -1/3 -1/3 lg ç µ M M <0 ÷ 1 2 dt è M 2 ø 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. M iso æ Sm (t = 0) ö1/ 3 =ç ÷ è ø B with 31/ 3 M*1/ 3 B= 2p bR 2 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?)