We present a systematic examination of the changes in semimajor axis of a protoplanet as it interacts with other protoplanets in the presence of eccentricity dissipation. For parameters relevant to the oligarchic stage of planet formation, dynamical friction keeps the typical eccentricities small and prevents orbit crossing. Interactions at impact parameters greater than several Hill radii cause the protoplanets to repel each other; if the impact parameter is instead much less than the Hill radius, the protoplanets shift slightly in semimajor axis but remain otherwise unperturbed. If the orbits of two or more protoplanets are separated by less than a Hill radius, they are each pushed toward an equilibrium spacing between their neighbors and can exist as a stable co-orbital system. In the shear-dominated oligarchic phase of planet formation, we show that the feeding zones contain several oligarchs instead of only one. Growth of the protoplanets in the oligarchic phase drives the disk to an equilibrium configuration that depends on the mass ratio of protoplanets to planetesimals, Σ/σ. Early in the oligarchic phase, when Σ/σ is low, the spacing between rows of co-orbital oligarchs are about 5 Hill radii wide, rather than the 10 Hill radii cited in the literature. It is likely that at the end of oligarchy, the average number of co-orbital oligarchs is greater than unity. In the outer solar system, this raises the disk mass required to form the ice giants. In the inner solar system, this lowers the mass of the final oligarchs and requires more giant impacts than previously estimated. This result provides additional evidence that Mars is not an untouched leftover from the oligarchic phase, but must be composed of several oligarchs assembled through giant impacts.
- planets and satellites: formation
- solar system: formation