Gravitationally focused growth of a planetary core

Conservation of orbital energy and angular momentum in a two-body encounter gives $\sigma =\pi R_c^2(1+v_{\mathrm{esc}}^2/v_{\infty }^2)$. At fixed density and encounter speed in the strong-focusing limit, $R_c\propto M_c^{1/3}$ and $\sigma \propto M_c^{4/3}$. This yields runaway fractional growth until stirring or supply changes the assumptions.