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At the final stage of terrestrial planet formation, known as the giant impactstage, a few tens of Mars-sized protoplanets collide with one another to formterrestrial planets. Almost all previous studies on the orbital and accretionalevolution of protoplanets in this stage have been based on the assumption ofperfect accretion, where two colliding protoplanets always merge. However,recent impact simulations have shown that collisions among protoplanets are notalways merging events, that is, two colliding protoplanets sometimes move apartafter the collision (hit-and-run collision). As a first step towards studyingthe effects of such imperfect accretion of protoplanets on terrestrial planetformation, we investigated the merging criteria for collisions of rockyprotoplanets. Using the smoothed particle hydrodynamic (SPH) method, weperformed more than 1000 simulations of giant impacts with various parametersets, such as the mass ratio of protoplanets, $\gamma$, the total mass of twoprotoplanets, $M_{\rm T}$, the impact angle, $\theta$, and the impact velocity,$v_{\rm imp}$. We investigated the critical impact velocity, $v_{\rm cr}$, atthe transition between merging and hit-and-run collisions. We found that thenormalized critical impact velocity, $v_{\rm cr}/v_{\rm esc}$, depends on$\gamma$ and $\theta$, but does not depend on $M_{\rm T}$, where $v_{\rm esc}$is the two-body escape velocity. We derived a simple formula for $v_{\rmcr}/v_{\rm esc}$ as a function of $\gamma$ and $\theta$, and applied it to thegiant impact events obtained by \textit{N}-body calculations in the previousstudies. We found that 40% of these events should not be merging events.

Merging Criteria for Giant Impacts of Protoplanets