Published on October 16, 2007
Accretion of Planetesimals in the Inner and Outer Solar System: Accretion of Planetesimals in the Inner and Outer Solar System S. J. Weidenschilling Planetary Science Institute Nobel Symposium 135 June 18-22, 2007 Current Paradigm for Accretion of Terrestrial Planets: Current Paradigm for Accretion of Terrestrial Planets Initial population of ~ km-sized planetesimals Runaway growth produces multiple “oligarchic” sub-planetary embryos Mutual perturbations lead to giant impacts until a few planets are left in stable orbits Is the process different in the outer solar system? Collisional Coagulation of Planetesimals: Collisional Coagulation of Planetesimals Rate of mass gain Consider two bodies with m1 >m2 I f V < Ve (runaway growth) If V > Ve (orderly growth) Runaway Growth: Runaway Growth If m1/m2 increases with time, the largest body becomes much larger than the second largest The embryo dominates some region around its orbit Runaway depends on velocities remaining low; V < Ve Embryo’s growth slows when its own perturbations stir the background population to V ~ Ve Slide5: Hill radius RH = (M/3MSun)1/3 R Embryo can accrete particles out to distance from its orbit (restricted 3-body problem) Limiting “isolation mass” Miso = 2.1x10-3 R3 3/2 MEarth If varies as R-, Miso increases with R if < 2 The process repeats a few RH away… N-body simulations (Kokubo & Ida 1998): N-body simulations (Kokubo & Ida 1998) “Oligarchy” of embryos of similar size, M ~ Miso Orbital spacing ~ 10 RH Growth slows due to increasing e, i of small population ~ half of total mass in embryos, the rest in small bodies Planetesimal Accretion Simulations: N-body Integrations: Planetesimal Accretion Simulations: N-body Integrations Accurate treatment of gravitational interactions and collision rates Heavy computational demands limit range of heliocentric distance and/or model times Number of bodies limits dynamical range of sizes and fragmentation Particle-in-a-box Simulations: Particle-in-a-box Simulations Statistical calculation of collision rates and gravitational stirring based on spatial density of bodies Size distribution represented by “bins” or “batches” Large size range allows fragmentation No spatial resolution; hard to account for separation of orbits Hybrid Multi-zone Accretion Code: Hybrid Multi-zone Accretion Code Radial zones of semimajor axis In each zone, size distribution of small bodies modeled with logarithmic mass bins Each bin has mean value of e, i and range about mean Impact rates and velocities selected from appropriate distributions Gravitational interactions: Viscous stirring and dynamical friction Discrete Bodies: Discrete Bodies Bodies larger than threshold mass (~1024 g) are treated as discrete Individual values of M, a, e, i Interact with continuum by viscous stirring, dynamical friction Angular elements are averaged Mutual collisions and gravitational scattering treated as stochastic events, allows impulsive changes in orbital elements Slide12: Radial zoning allows spatial resolution Collisions and stirring occur when orbits overlap, even for different zones (not a series of separate particle-in-box simulations) Explicit transport of mass between zones due to collisions, gas drag, tidal interactions with nebula, and “shepherding” by massive bodies Long-range perturbations by discrete bodies stir each other, and the continuum, where orbits don’t overlap Impact Strength Q* = energy/mass yielding 1/2 of target mass (monolithic or rubble pile): Impact Strength Q* = energy/mass yielding 1/2 of target mass (monolithic or rubble pile) D < 1 km: material strength dominates, decreases with size due to presence of flaws D > 1 km: gravitational binding energy increases with size Minimum strength at D ~ 1 km Solar Nebula Model: Solar Nebula Model Nebular surface density varies as 1/R At 1 AU, surface density of gas = 2500 g/cm2 , solids 8.4 g/cm2 ~ 7 MEarth of solids between 0.5 and 4 AU Gas deviates from VK by 1.8x10-3 Simulation 1: Inner solar system, 0.5 - 4 AU: Simulation 1: Inner solar system, 0.5 - 4 AU Initial planetesimal diameter 1 km Fragments smaller than 1/8 km are lost Aerodynamic drag acts on small bodies Large bodies interact tidally with gas, causing decay of semimajor axes (Type 1 migration), and damping of eccentricities and inclinations Gas decays exponentially on timescale 2 My Slide39: Cumulative mass to distance R: About half of the original mass is lost by fragmentation Fragments smaller than 1/8 km are lost into the Sun in ~ 104 y Simulation 2: Inner solar system, 0.5 - 4 AU: Simulation 2: Inner solar system, 0.5 - 4 AU Same as Case 1, but initial planetesimal diameter 10 km Results in Inner Solar System: Results in Inner Solar System Runaway growth proceeds as a “wave” moving outward Growth rate and speed of wave front are inversely dependent on initial planetesimal size Oligarchic embryos are produced, with masses a few times Miso If Type 1 migration is effective, gas must dissipate within a few My, or embryos are lost by migration into Sun Planetesimal Size Matters: Planetesimal Size Matters For initial size 1 km, about half of starting mass is ground down to small fragments during the first My Lifetimes of sub-km fragments due to gas drag are short; probably lost from the system For initial size 10 km, runaway growth is slower. The small bodies are stronger due to gravitational binding energy, and little mass is lost by grinding More mass survives, yielding larger embryos; their higher mass leads to more migration Terrestrial Region:: Terrestrial Region: Runaway growth is a local phenomenon, leading to oligarchy of planetary embryos Wave of growth propagates outward Mass loss by collisional grinding may be significant, depending on initial planetesimal size Type 1 migration can cause loss of embryos into the Sun, unless gas dissipates on timescale ~ 1 My Outer Solar System: Outer Solar System Inaba et al. (2003): particle-in-a-box code in 6 zones 5.2-29 AU Inward drift of fragments by gas drag; no other interaction Found runaway growth qualitatively similar; timescales a problem Hybrid Multi-zone Simulations of Entire Disk: Hybrid Multi-zone Simulations of Entire Disk Radial range 0.5 - 30 AU “Snow Line” at 4.5 AU; surface density of solids increased ~ 4 x beyond this distance Initial planetesimal size 1 km (no change in density or strength at snow line) Planetesimal Scattering in the Outer Disk: Planetesimal Scattering in the Outer Disk For most nebular models, Miso increases with heliocentric distance; is much greater beyond the snow line Gravitational scattering in close encounters with larger embryos causes larger V Lower VK means larger changes in a, e; bodies are more mobile in outer disk Outward scattering is easier than inward Seeded Runaway Growth: Seeded Runaway Growth Bodies can be scattered from region near the snow line into the dynamically cold outer disk Scattered body (~ 1024 g) is much more massive (~103 - 106 x ) than the local population Dynamical friction circularizes its orbit Isolated body is less effective at stirring the disk (Jacobi constant is conserved) Rapid growth in shear-dominated regime to ~ Miso in a few x 103 orbital periods Slide90: Seeded runaway occurs only in a dynamically cold region of the swarm, where Keplerian shear dominates over random velocities (eccentricities) Perturbations by a massive embryo increase e’s and prevent other scattered bodies from initiating runaway growth Only a few planets can be seeded by scattered bodies Inhibition of Runaway Growth in the Outer Disk: Inhibition of Runaway Growth in the Outer Disk Rates of gravitational stirring and energy exchange by dynamical friction between bodies in crossing orbits are proportional to the local space density of bodies Stirring by synodic encounters with distant bodies in non-crossing orbits is proportional to the surface density of those bodies; falls off more slowly with R Long-range perturbations become relatively more important in the outer disk Slide95: Long-range perturbations increase e, but have little effect on i Eccentricities excited by long-range perturbations do not depend on the masses of the perturbed bodies Planetesimal velocities are ~ independent of mass; V > Ve Higher V shuts off runaway growth beyond some distance (~ 10 - 15 AU) Conclusions: Outer Region: Conclusions: Outer Region Beyond the snow line, accretion is not a local phenomenon Bodies are more mobile, subject to scattering Some scattered bodies may nucleate extreme runaway growth while swarm is dynamically cold (only a few planetary cores) Long-range perturbations can excite e’s and inhibit runaway growth over large distances Few giant planets inside ~ 15 AU and swarm of small bodies beyond: Nice model?