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From Planetesimals to Planets in Turbulent Protoplanetary Disks I. Onset of Runaway Growth

When planetesimals grow via collisions in a turbulent disk, stirring through density fluctuation caused by turbulence effectively increases the relative velocities between planetesimals, which suppresses the onset of runaway growth. We investigate the onset of runaway growth in a turbulent disk through simulations that calculate the mass and velocity evolution of planetesimals. When planetesimals are small, the average relative velocity between planetesimals, $v_{\rm r}$, is much greater than their surface escape velocity, $v_{\rm esc}$, so that runaway growth does not occur. As planetesimals become large via collisional growth, $v_{\rm r}$ approaches $v_{\rm esc}$. When $v_{\rm r} \approx 1.5 v_{\rm esc}$, runaway growth of the planetesimals occurs. During the oligarchic growth subsequent to runaway growth, a small number of planetary embryos produced via runaway growth become massive through collisions with planetesimals with radii of that at the onset of runaway growth, $r_{\rm p,run}$. We analytically derive $r_{\rm p,run}$ as a function of the turbulent strength. Growing $\sim 10\,M_\oplus$ embryos that are suitable to become the cores of Jupiter and Saturn requires $r_{\rm p,run} \sim 100$\,km, which is similar to the proposed fossil feature in the size distribution of main belt asteroids. In contrast, the formation of Mars as quickly as suggested from Hf-W isotope studies requires small planetesimals at the onset of runaway growth. Thus, the conditions required to form Mars, Jupiter, and Saturn and the size distribution of the main-belt asteroids indicate that the turbulence increased in amplitude relative to the sound speed with increasing distance from the young Sun.