 Solution

In the strong-focusing, small-projectile limit, write $R_M=(3M/(4\pi {\rho }_s){)}^{1/3}$ with constant solid-material density. ThenIntegrating $dM/M^{4/3}=Adt$ from $M_0$ to $M$ givesfor $M\gg M_0$. The fractional growth rate $\dot{M}/M\propto M^{1/3}$ increases with mass: larger bodies gain an increasing advantage, hence gravitationally focused growth of a planetary core is called runaway accretion in this regime.

The formal finite time to unlimited mass is not physical. Growth depletes the feeding region; embryos also stir planetesimal velocities and reduce focusing, while fragmentation, orbital competition and stellar tides alter the encounter geometry. A growing core can enter an oligarchic rather than runaway regime, and eventually reach a supply-limited isolation mass.

Dynamical friction transfers random kinetic energy from a massive body to a swarm and tends toward $M{v}_M^2\sim m{v}_m^2$. Larger bodies therefore have smaller random eccentricities and inclinations, remain concentrated near the planetesimal layer, and can retain favorable encounter conditions. When their own random motion contributes appreciably to relative speed, damping it also strengthens focusing through $f\propto v_0^{-2}$. If the small-body dispersion already dominates $v_0$, equipartition of the core alone does not further lower that encounter speed: the displayed growth law assumed a fixed $v_0$, not an automatically imposed $v_0\propto M^{-1/2}$.