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After Runaway: The Trans-Hill Stage of Planetesimal Growth

When planetesimals begin to grow by coagulation, they enter an epoch of runaway, during which the biggest bodies grow faster than all the others. The questions of how runaway ends and what comes next have not been answered satisfactorily. Here we show that runaway is followed by a `trans-hill stage' that commences once the bodies become trans-hill, i.e. once the Hill velocity of the bodies that dominate viscous stirring matches the random speed of the small bodies they accrete. Subsequently, the small bodies' speed grows in lockstep with the big bodies' sizes, such that the bodies remain in the trans-hill state. Trans-hill growth is crucial for determining the efficiency of growing big bodies, as well as their growth timescale and size spectrum. We work out the properties of trans-hill growth analytically and confirm these numerically. Trans-hill growth has two sub-stages. In the earlier one, collisional cooling is irrelevant, in which case the efficiency of forming big bodies remains very low (0.1% in the Kuiper belt) and their mass spectrum is flat. This explains results from previous coagulation simulations for both the Kuiper belt and the asteroid belt. The second sub-stage commences when small bodies begin to collide with one another. Collisional cooling then controls the evolution, in which case the efficiency of forming big bodies rises, and their size spectrum becomes more top-heavy. Trans-hill growth terminates in one of two ways, depending on parameters. First, mutual accretion of big bodies can become significant and conglomeration proceeds until half the total mass is converted into big bodies. This mode of growth may explain the size distributions of minor bodies in the Solar System, and is explored in forthcoming work. Second, if big bodies become separated by their Hill radius, oligarchy commences. This mode likely precedes the formation of fully-fledged planets.