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When planetesimals grow via collisions in a turbulent disk, stirring throughdensity fluctuation caused by turbulence effectively increases the relativevelocities between planetesimals, which suppresses the onset of runaway growth.We investigate the onset of runaway growth in a turbulent disk throughsimulations that calculate the mass and velocity evolution of planetesimals.When planetesimals are small, the average relative velocity betweenplanetesimals, $v_{\rm r}$, is much greater than their surface escape velocity,$v_{\rm esc}$, so that runaway growth does not occur. As planetesimals becomelarge via collisional growth, $v_{\rm r}$ approaches $v_{\rm esc}$. When$v_{\rm r} \approx 1.5 v_{\rm esc}$, runaway growth of the planetesimalsoccurs. During the oligarchic growth subsequent to runaway growth, a smallnumber of planetary embryos produced via runaway growth become massive throughcollisions with planetesimals with radii of that at the onset of runawaygrowth, $r_{\rm p,run}$. We analytically derive $r_{\rm p,run}$ as a functionof the turbulent strength. Growing $\sim 10\,M_\oplus$ embryos that aresuitable to become the cores of Jupiter and Saturn requires $r_{\rm p,run} \sim100$\,km, which is similar to the proposed fossil feature in the sizedistribution of main belt asteroids. In contrast, the formation of Mars asquickly as suggested from Hf-W isotope studies requires small planetesimals atthe onset of runaway growth. Thus, the conditions required to form Mars,Jupiter, and Saturn and the size distribution of the main-belt asteroidsindicate that the turbulence increased in amplitude relative to the sound speedwith increasing distance from the young Sun.

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