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Supplementary Information doi: 10.1038/nature08094 SUPPLEMENTARY INFORMATION Contents: S1) The Orbital Dynamics Calculations. S2) Collisional and Dynamical Depletion Evolution Model (CoDDEM) Calculations. S3) The Micrometeorite Origin Problem. S4) The Collisional Evolution of the Primordial Cometary Disk. S1. The Orbital Dynamics Calculations The goal of these simulations is to follow the long-term orbital evolution of a large number of massless test particles (which represent primitive objects originally from the distant proto-planetary disk) under the gravitational influence of the giant planets. Initially the orbits of the planets change (sometimes violently), but they eventually settle into their current configuration. The first step models the evolution of the system during the time when the gas giant planets migrate and primitive bodies are injected into the Jupiter-Saturn region by Uranus and Neptune as they penetrate the primitive planetesimal disk. Because Uranus and Neptune are far from the Sun (beyond ∼15 AU) and we are interested in the dynamical evolution of objects in the ∼2 to ∼5 AU zone, we only include Jupiter and Saturn in the simulations. As the Nice model specifies, these planets start slightly interior to the 1:2 mean motion resonance (MMR) with one another. They were forced to migrate by including a suitably chosen acceleration term in the planets equations of motion. In particular, we chose an evolution similar to the ‘fast migration’ run in Ref. [3]. In these simulations, however, we took increased care to reproduce the observed eccentricities of the giant planets and the period of the Great Inequality at the end of the simulation. The temporal evolution of the planetary orbits employed in our simulations is shown in Fig. 1. We followed the system for 10 million years using the SyMBA orbital integrator [31]. We removed a disk particle if it either impacted a planet, or reached a heliocentric distance greater than 15 AU or less than 1.5 AU (i.e. became Mars-crossing). We added the last requirement to account for the terrestrial planets, which were not directly included in the simulations. Every time that a test particle was dynamically eliminated, it was reintroduced on its original trans- Saturnian orbit rescaled to the current position of Saturn. Thus, we supplied a steady flux of planetesimals through the Jupiter-Saturn system during the entire integration. In this way, the number of particles in the simulation at any time was held constant at 834,500. In total ∼ 31 million disk particles were eliminated and reintroduced during the considered time-span. At the www.nature.com/nature 1 doi: 10.1038/nature08094 SUPPLEMENTARY INFORMATION – 2 – 10 Saturn 9 8 2:1 Resonance 7 Semi-Major Axis (AU) 6 Jupiter 5 Time (yr) Fig. 1.— The black curves show the temporal evolution of the perihelion distance, semi-major axes and aphelion distance of Jupiter (from bottom to top, respectively) in the first stage of our dynamical simulations. The red curves are the same for Saturn. The blue curve shows the location of the 1:2 MMR with Jupiter. The system starts with Jupiter and Saturn interior to this resonance, but since the planets diverge, they soon cross it. According to the Nice model this resonance crossing triggers an instability in the ice giant orbits, which, in turn, scatters primitive planetesimals throughout the planetary system. A small fraction of these objects are then captured into the Hilda, Trojan, and OMB regions. end of the 10 Myr simulation, there were 1147 particles captured onto orbits that were isolated from encounters with the planets. Next, we determined the long-term stability of the particles captured during the migration simulations by integrating the system from 107 to 4×109 years using the integrator of Ref. [32] with a timestep of 0.1 yr. We removed any particle that impacted a planet, became Mars-crossing, reached a heliocentric distance of 15 AU, or entered a planet’s Hill’s sphere. Fig. 2 shows the number of test bodies in the long-term simulations as a function of time. Initially, there were 1147 particles — 218 were Trojans, 196 were Hildas, and 395 were in the OMB. At the end of 4 billion years of dynamical evolution there were 32 Trojans, 19 Hildas, and 158 OMB particles remaining. Note that these numbers do not include the effects of collisional grinding. www.nature.com/nature 2 doi: 10.1038/nature08094 SUPPLEMENTARY INFORMATION – 3 – Fig. 2.— The temporal evolution of the number of objects in our long-term dynamical simulations scaled to the number of model Trojans that survived to the current epoch. The black curve shows all the captured objects, while the red, blue, and green curves show the three main captured populations in the Hilda, Trojans, and the main asteroid belt regions, respectively. The tan curve shows the temporal evolution of the indigenous main belt as described in §S2.1. S2. Collisional and Dynamical Depletion Evolution Model (CoDDEM) Calculations The size-frequency distribution (SFD) evolution results described in the main text were com- puted using the 1-dimension collisional and dynamical depletion evolution model called CoDDEM [13, 14]. CoDDEM was modified to use a more modern and complete algorithm for calculating the outcomes of collisions developed by Ref. [15], which, in turn, was based on 160 impact experiments performed using smooth particle hydrodynamic (SPH) codes coupled with N-body codes [33, 34]. In these runs, Ref. [33, 34] tracked D = 10-46 km projectiles slamming into D = 100 km basaltic spheres at a wide range of impact speeds (2.5-7 km/s) and impact angles (15◦-75◦). In addition, CoDDEM now includes the effects of cratering events. Using CoDDEM, we tracked the evolution of 5 independent populations: (1) indigenous inner main belt (IMB), (2) indigenous outer main belt (I-OMB), (3) captured outer main belt (C-OMB), (4) Hildas, and (5) Trojans. Each of the above populations is represented by several parameters in the code: i) the SFD, ii) the collision probabilities, iii) the impact velocities, iv) the bulk densities, and v) the disruption scaling law. The code follows the evolution of the SFD while holding the rest www.nature.com/nature 3 doi: 10.1038/nature08094 SUPPLEMENTARY INFORMATION – 4 – of the parameters constant. In addition, each population was forced to decay over time due to the dynamical effects de- scribed in §S1 and in the main text. Note that we follow the collisional evolution of the system only after the first 10 Myr, i.e. after migration is over. Thus, we are assuming that collisional grinding is not important during the implantation process. We believe that this is reasonable because the dynamical lifetime of particles once they cross the orbit of Saturn is very short, ∼ 105 yrs, and thus, there is not really much time for collisional evolution to occur. Indeed, there is observational evidence to this effect in the fact that the size-distributions of the Trojans and the cold classical Kuiper belt are similar [35]. We now discuss the initial values of the parameters for each of our populations. In particular, we discuss the indigenous and captured populations separately. S2.1 Indigenous Main Belt Populations Here we describe the indigenous main belt populations 1 (IMB) and 2 (I-OMB) used in our simulations. The IMB is defined as the collection of objects with 2.1≤a<2.82 AU, while the OMB is defined as those objects with 2.82≤a<3.27 AU. Collision probabilities and impact velocities. To quantify the collision probabilities and impact velocities of populations in the main belt region striking either each other or the Hilda population (the main belt does not collide with the Trojans) we used the techniques and code described in Ref. [36], which computes the average “intrinsic collision probability” Pi and mean impact velocity Vimp for any two populations based on the semi-major axes, eccentricities and inclinations of their members. We represented the IMB and both the OMB populations using the 270 and 412 known asteroids, respectively, with D > 50 km. For the Hildas, we chose the 110 known asteroids with H < 12, which is likely close to a observationally complete set of objects. The values of Pi and Vimp derived from these calculations are found in Tables 1 and 2. Population: 1 2 3 4 5 IMB I-OMB C-OMB Hildas Trojans 1 4.9 × 10−18 2.3 × 10−18 2.3 × 10−18 2.6 × 10−19 0 2 2.3 × 10−18 2.8 × 10−18 2.8 × 10−18 9.1 × 10−19 0 3 2.3 × 10−18 2.8 × 10−18 2.8 × 10−18 9.1 × 10−19 0 4 2.6 × 10−19 9.1 × 10−19 9.1 × 10−19 2.3 × 10−18 2.4 × 10−19 5 0 0 0 2.4 × 10−19 3.0 × 10−18 Table 1: The intrinsic impact probability (Pi) of the target population being struck by the impactor population. These numbers are in units of km−2 yr−1. Note that this matrix is symmetric, as it should be. www.nature.com/nature 4 doi: 10.1038/nature08094 SUPPLEMENTARY INFORMATION – 5 – Population: 1 2 3 4 5 IMB I-OMB C-OMB Hildas Trojans 1 5.6 5.4 5.4 5.7 — 2 5.4 5.3 5.3 4.7 — 3 5.4 5.3 5.3 4.7 — 4 5.7 4.7 4.7 3.1 4.6 5 — — — 4.6 5.2 Table 2: The impact velocity (Vimp) in units of km/s. Note that this matrix is symmetric, as expected. Bulk densities. As a simplifying approximation, we assumed the indigenous asteroids have a mean bulk density of 2.7 g cm−3 [37].
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