Conglomeration of kilometre-sized planetesimals

We study the efficiency of forming large bodies, starting from a sea of equal-sized planetesimals. This is likely one of the earlier steps of planet formation and relevant for the formation of the asteroid belt, the Kuiper belt and extra-solar debris disks. Here we consider the case that the seed planetesimals do not collide frequently enough for dynamical collisional to be important (the collisionless limit), using a newly constructed conglomeration code, and by carefully comparing numerical results with analytical scalings. In the absence of collisional cooling, as large bodies grow by accreting small bodies, the velocity dispersion of the small bodies ($u$) is increasingly excited. Growth passes from the well-known run-away stage (when $u$ is higher than the big bodies' hill velocity) to the newly discovered trans-hill stage (when $u$ and big bodies both grow, but $u$ remains at the big bodies' hill velocity). We find, concurring with the analytical understandings developed in Lithwick (2014), as well as previous numerical studies, that a size spectrum $dn/dR \propto R^{-4}$ results, and that the formation efficiency, defined as mass fraction in bodies much larger than the initial size, is $\sim {\mathrm{ a \,\, few}}\times R_\odot/a$, or $\sim 10^{-3}$ at the distance of the Kuiper belt. We argue that this extreme inefficiency invalidates the conventional conglomeration model for the formation of both our Kuiper belt and extra-solar debris disks. New theories, possibly involving direct gravitational collapse, or strong collisional cooling of small planetesimals, are required.