When planetesimals begin to grow by coagulation, they first 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. We show that runaway is followed by a new stage—the "trans-Hill stage"—that commences when the bodies that dominate viscous stirring ("big bodies") become trans-Hill, i.e., when their Hill velocity matches the random speed of the small bodies they accrete. Subsequently, the small bodies' random speed grows in lockstep with the big bodies' sizes, such that the system remains 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. Trans-Hill growth has two sub-stages. In the earlier one, which occurs while the stirring bodies remain sufficiently small, the evolution is collisionless, i.e., collisional cooling among all bodies is irrelevant. The efficiency of forming big bodies in this collisionless sub-stage is very low, ~10α 1, where α ~ 0.005(a/AU)–1 is the ratio between the physical size of a body and its Hill radius. Furthermore, the size spectrum is flat (equal mass per size decade, i.e., q = 4). This collisionless trans-Hill solution explains results from previous coagulation simulations for both the Kuiper Belt and the asteroid belt. The second trans-Hill sub-stage commences once the stirring bodies grow big enough (α–1 × the size of the accreted small bodies). After that time, collisional cooling among small bodies controls the evolution. The efficiency of forming big bodies rises and the size spectrum becomes more top heavy. Trans-Hill growth can terminate in one of two ways, depending on the sizes of the small bodies. First, mutual accretion of big bodies can become significant and conglomeration proceeds until half of the total mass is converted into big bodies. This mode of growth may explain the observed size distributions of small bodies in the solar system and is explored in our subsequent work. Second, if the big bodies' orbits become separated by their Hill radius, oligarchy commences. This mode likely precedes the formation of fully fledged planets.
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