The Formation of The Solar System Re’em Sari (Caltech) Yoram Lithwick (UCB) Peter Goldreich (Caltech & IAS) Ben Collins (Caltech)

The Final Chapter of Planet Formation Re’em Sari (Caltech) Yoram Lithwick (UCB) Peter Goldreich (Caltech & IAS)

Minimum Mass Solar Nebula Surface density (g cm -2 )

When did oligarchy end? What set the number of solar system planets? How long did it take them to form? Why are their orbits circular and coplanar? Why exoplanets occupy eccentric orbits? Why Jupiter does not!? What happened after planet formation? When and how did the Oort cloud form? How long do debris disks live? Questions

If collision cross section is geometric Scale height h~v/  –n  /h In terms of surface density: –Independent of velocity For MMSN: –Earth (6,400km) 10 8 yr –Jupiter’s core 10 9 yr –Neptune (25,000km) yr Geometric Accretion

Need Gravitational Focusing Larger collision cross section Growth rate depends on Safronov number (v esc /v) 2 Safronov: velocity, v, is not a free parameter. –Controlled by interactions between bodies. From MMSN: geometricrequired required time (yr)time (yr)eccentricity –Earth (6,400km) 10 8 yrt<10 8 e<1 –Jupiter’s core 10 9 yrt<10 7 e<0.1 –Neptune (25,000km)10 12 yr10 6 <t< <e<0.4

Physical Processes Setup: many small bodies, few big bodies , u , v viscous stirring dynamical friction accretion viscous stirring collisions = =

Hill sphere –Tidal effects from the Sun –Sets a minimum drift velocity –Sets the maximum binary separation Viscous stirring –Radial and tangential velocity are coupled - eccentricity –Even elastic deflections increase velocity dispersion –Results in much faster heating: temperature doubles in one deflection timescale Disk Effects VrVr VtVt Elastic scat. V r ->V t Initial velocity dispersion Elastic scat. V t ->V r

Solar Angle -   - Angle subtended by Solar radius. –  = at 1AU. –  ~10 -4 at Kuiper belt. Significance: spacing of scales: –Hill radius: R H   -1 R –Hill velocity v H   1/2 v esc Kuiper Belt binaries: –Possible separations 2R<a<R H –From Earth, R H for a 100km body subtends 20’’ R  100km R H   -1 R  10 6 km   AU= cm v H  1 m/s

Rates Frequent small kicks dominate stirring.

Physical Processes: velocities Setup: many small bodies, few big bodies , u , v viscous stirring dynamical friction accretion viscous stirring collisions = ~ =  solve for v

Physical Processes: accretion Setup: many small bodies, few big bodies , u , v viscous stirring accretion collisions = ~ Very small bodies (s«R)  very fast accretion rate ~

Gravitational Instability of Particle Disk From Q~1: Required size for setup of such velocities:

Gravitational Disk Instabilities In very cold disk: Largest unstable wavelength Quick collapse No angular momentum lost Energy dissipates by collisions

Interaction with a cold surrounding

Early Times: Runaway Accretion Without gravitational focusing –Bodies tend to become of equal size With focusing –Few large bodies become larger than their peers. Most of the Bodies & mass Exponential cutoff Runaway tail m m N(m)

Late Times: Oligarchic Growth Big bodies heat their own food  larger u around bigger bodies  bigger bodies grow more slowly  big bodies are: equal mass, uniformly spaced Number of big bodies decreases as they grow Super Hill - win by competition Sub Hill - large-large accretion Thin disk - No sustained Oligarchy ( Kokubo and Ida 1998) “Classical” oligarchy: More refined oligarchy - battles:

Formation time of Uranus & Neptune fast enough (10 Myr) if small bodies are very small (< 1 m) and very cold (e<0.05) In cold (sub Hill) accretion, might expect: Observe: Uranus & Neptune = End of Oligarchy? 30, 000 km 20 AU 30 AU few Hill radii 20 AU 30 AU 50, 000 km 15 Hill radii

Formation time of earth is fast (100 Myr) even with no focusing. In cold accretion, might expect: Observed: Earth = End of Oligarchy? 4, 000 km 0.72 AU 1 AU ~3 Hill radii=0.01AU 0.72 AU 1 AU 13, 000 km ~40 Hill radii ~30 sub-mars objects

Isolation when  ~  we assume u~v H  Uranus & Neptune: Beyond Isolation 50, 000 km 20 AU 30 AU few Hill radii

After 10 million years,  Heating > Cooling  runaway heating Planets are ejected in time Uranus & Neptune: Ejection 50, 000 km 20 AU 30 AU few Hill radii

Complete N-body simulations –e.g. Kokubo & Ida (1998) –Can not realistically treat enough small bodies. Statistical treatments –Interactions between mass bins Hybrid Codes –e.g. Kenyon –Still difficult to have very small bodies –Gravitational instability N-body + simple accretion and dynamical friction –Preliminary simulations (Collins and Sari) Numerical Approaches

New n-body code Accretion Dynamical friction Numerical n-body Simulation

Non thermal distribution –Contrary to current statistical models Extended tail up to V H Most of energy around V H Most bodies at v<<V H Velocity Distribution Encounters within R H

Simulation - Oligarchy Begins

Beginning of Oligarchy Radius of circle = Hill sphere ~ 3000R

Oligarchic Growth

End of Oligarchy Radius of circle = Hill sphere ~ 3000R At end of oligarchy  ~ , v~V H

Simulation - Oligarchy Ends

Simple Collisionless Solutions Consistency: –We assume u>v H & v<v H which requires –We neglected collisions between small bodies Generalization yields the complete velocity spectrum. Mass spectrum more subtle Agrees with detailed simulations (Kenyon) (Kenyon 2002)

L 2 s Channel Initial separation 1.0 R H –Bottleneck: 2 bodies together + enough dynamical friction. –Formation rate per large body: (  /  R) 2 (  /  )  -2  = 3 / My

L 2 s Channel Initial separation 0.9 R H –Bottleneck: 2 bodies together + enough dynamical friction. –Formation rate per large body: (  /  R) 2 (  /  )  -2  = 3 / My

Binary Evolution = Separation Distribution 3’’ L 2 s or L 3 Create binaries here L 2 s every 3·10 5 yr 0.2’’ Probability a p(a) separation 5% Orbit decays by dynamical friction. With u=3m/s, timescale is 10 3 years Orbit decays by dynamical friction. Larger u therefore longer timescale Contact achieved over 1Myr = growth timescale 300% 20’’

Only a few remaining bodies (Uranus & Neptune) No Chaos = no heating Cooled by remaining small bodies (explains small eccentricity) Uranus & Neptune: regularization 50, 000 km 20 AU 30 AU

V esc <V orbit ejection unlikely Collisions Formation on Geometric timescale (100 Myr) Giant impacts! Venus & Earth: Beyond isolation 0.72 AU 1 AU 13, 000 km ~40 Hill radii

When did oligarchy end? –When !! GENERAL RESULT What set the number of solar system planets? –Stability: No wide range chaos. How long did it take them to form? –About 100 Myr at a~1AU. –Faster at a~30AU ! –More time until ejection at a~30AU. Why are their orbits circular and coplanar? –Velocity damping by small bodies. Answers

Preliminary Simulation 5 planets, about Neptune masses,  =0.25 t/P

Collide or Eject? Collide Eject

Small bodies: –Inner planetary region: Most will be accreted. MMSN is sufficient! –Outer solar system: ~5 MMSN to form Neptune. Large ejected bodies: –~3 bodies >M earth could be in Oort cloud. –Origin in outer planetary system. –Survival depends on their long term stability and solar environment. Giant impacts: –Expected internal to 3AU. –Our moon More Implications

Eccentricity decays due to leftover small bodies. –Initial timescale = ejection (outer) or collision (inner) timescale Gas effects? –Could have helped in cooling the small bodies during oligarchy –Unlikely to be present at the end stages 100Myr for inner solar system 10Myr-1Gyr after ejection in outer solar system –Must rely on small bodies. Residual mass (of small bodies) during regularization? –Of order the initial mass in outer solar system Uncertainties: separation and instability onset –Perhaps somewhat smaller in inner solar system delicate balance between accretion and shattering Orbital Regularization

Once e decays (e<  a/a) planet disk torque open gaps. Size of gap: Clean-sharp or dirty-smooth gaps? Accretion through the gaps? For Exoplanets, depends on viscosity: Gap Formation

Smooth Gaps? 50, 000 km 20 AU30 AU Competition between diffusion and repulsion from planets  =0.1 

Sharp Gaps? If small bodies are mm or smaller Optical depth is high. Negative “viscosity” Angular momentum diffuses inward. NeptuneUranusSaturn a high  low 

Eccentricity evolution in clean gaps Similar question to exoplanets. Coorbital Lindblad resonances are absent. BUT DIFFERENT ANSWER! Two issues: –Are corotation resonances saturated? –Will the apsidal wave be effective?

e=0.2 expansion in m & e 1 st order in e

Resonances Out of resonance: disk particles are drifting relative to the potential torque averages to 0! Corotation resonance: particles move together with the potential feel a constant force. Lindblad resonance: each disk particle experiences both signs of force but at different phases of its epicycle each epicyclic period it drifts one potential peak. Inner Lindblad resonance (ILR) Outer Lindblad resonance (OLR)

Planet disk interactions in gaps Slow component  pattern <  planet r planet Fast component  pattern >  planet

Planet disk interactions in gaps a Little effect on e Slow component Fast component

Planet disk interaction in gaps saturation saturation

Criterion for Saturation Torque acts on libration timescale Viscosity timescale over corotation width or

Planet - Disk Symbiosis

Apsidal Wave With Self Gravity

Eccentricity evolution in clean gaps Similar question to exoplanets. BUT DIFFFERENT ANSWER! Two issues: –Corotation resonance saturates in both. –Relative importance of apsidal waves:

Numerical simulations!? All simulations show eccentricity damping! –Except for very wide gaps beyond the 3:1 resonance. Width of corotation resonances To appropriately resolve saturation requires Long integration time to set up apsidal wave loop in equilibrium.

Clean Up? Tight limits on residual mass in our solar system. –For a>30 AU, M disk <10M earth (a/100) 3 Halley’s If all bodies are very small (as we suggest): –Gaps are clean - no accretion. –No ejection. –Cannot form the Oort cloud. What size is large enough to allow ejection? –S cross : no collision in planet crossing time. –S eject : no collision before direct ejection.

Ejection & Oort Cloud a/a p =1 Time Low efficiency High efficiency Jupiter 10 5 years Neptune 3x10 8 years

Sizes of comets Meech, Hainaut and Marsden 2001

Summary Oligarchy ends when With small bodies, last stage of isolation is fast: Could be 0.1Myr for Uranus and Neptune Even faster for Earth. Ejection (outer) or collisions (inner). Orbital regularization by small bodies Regularization works also in clean gaps - unlike exoplanets Difficult (but possible) to get rid of small bodies Gravitational instability - key player: Second generation of 1km size bodies. Sets minimum to velocity dispersion. May set minimum to size of small bodies. Oort cloud: most likely ejection by Jupiter (no direct ejection by U-N). Total mass too high - collision will happen first. 1km size can be ejected out of Jupiter-Saturn without collisions. Collisional limited ejection: MMSN 5 MMSN Dones Chambers

Summary - II Oort cloud: most likely ejection by Jupiter. Cannot form Oort cloud directly out of U-N region Total mass too high - collision will happen first. 1km size can be ejected out of Jupiter-Saturn without collisions. Unlikely to have sub-km bodies in Oort cloud Eccentricity of exoplanet can be excited in clean gaps. Because apsidal wave is inefficient + corotation resonances saturate. Signature of collision limited ejection may be found in the Oort cloud