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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 $M_{\rm oli}$ at the onset of oligarchic growth depends on the initial mass $m_0$ 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 $m_0$ 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, $M_{\rm oli}$ is independent of $m_0$. If $m_0$ is large, on the other hand, oligarchic growth commences directly after runaway growth, and $M_{\rm oli} \propto m_0^{3/7}$. The planetary mass $M_{\rm oli}$ 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 $m_0$. 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 $m_0$. The slope, $q \simeq -2.7$, and the bump at $10^{21}$ g (or 50 km in radius) for the mass distribution of hot Kuiper belt objects are reproduced if $m_0$ is the bump mass. On the other hand, small initial planetesimals with $m_0 \sim 10^{13}$ g or less are favored to explain the slope of large asteroids, $q \simeq -2.0$, while the bump at $10^{21}$ g can be reproduced by introducing a small number of asteroid seeds each with mass of $10^{19} $ g.