Making the Terrestrial Planets: N-Body Integrations of Planetary Embryos in Three Dimensions

We simulate the late stages of terrestrial-planet formation using N-body integrations, in three dimensions, of disks of up to 56 initially isolated, nearly coplanar planetary embryos, plus Jupiter and Saturn. Gravitational perturbations between embryos increase their eccentricities,e, until their orbits become crossing, allowing collisions to occur. Further interactions produce large-amplitude oscillations ineand the inclination,i, with periods of ∼105years. These oscillations are caused by secular resonances between embryos and prevent objects from becoming re-isolated during the simulations. The largest objects tend to maintain smallereandithan low-mass bodies, suggesting some equipartition of random orbital energy, but accretion proceeds by orderly growth. The simulations typically produce two large planets interior to 2 AU, whose time-averagedeandiare significantly larger than Earth and Venus. The accretion rate falls off rapidly with heliocentric distance, and embryos in the “Mars zone” (1.2 2 AU) is efficiently cleared as objects scatter one another into resonances, where they are lost via encounters with Jupiter or collisions with the Sun, leaving, at most, one surviving object. Accretional evolution is complete after 3 × 108years in all simulations that include Jupiter and Saturn. The number and spacing of the final planets, in our simulations, is determined by the embryos' eccentricities, and the amplitude of secular oscillations ine, prior to the last few collision events.