Onset of oligarchic growth and implication for accretion histories of dwarf planets

We investigate planetary accretion that starts from equal-mass planetesimals using an analytic theory and numerical simulations. We particularly focus on how the planetary mass Moli at the onset of oligarchic growth depends on the initial mass m0 of a planetesimal. Oligarchic growth commences when the velocity dispersion relative to the Hill velocity of the protoplanet takes its minimum. We find that if m0 is small enough, this normalized velocity dispersion becomes as low as unity during the intermediate stage between the runaway and oligarchic growth stages. In this case, Moli is independent of m0. If m0 is large, on the other hand, oligarchic growth commences directly after runaway growth, and Moli∝m03/7. The planetary mass Moli for the solid surface density of the Minimum Mass Solar Nebula is close to the masses of the dwarf planets in a reasonable range of m0. This indicates that they are likely to be the largest remnant planetesimals that failed to become planets. The power-law exponent q of the differential mass distribution of remnant planetesimals is typically −2.0 and −2.7 to −2.5 for small and large m0. The slope, q≃−2.7, and the bump at 1021 g (or 50 km in radius) for the mass distribution of hot Kuiper belt objects are reproduced if m0 is the bump mass. On the other hand, small initial planetesimals with m0 ∼ 1013 g or less are favored to explain the slope of large asteroids, q≃−2.0, while the bump at 1021 g can be reproduced by introducing a small number of asteroid seeds each with mass of 1019 g.