  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.

Earth and Planetary Astrophysics