Gravitational focusing

Gravitational focusing increases a physical collision cross-section because mutual gravity bends initially non-colliding trajectories toward one another.

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.

For a point projectile approaching an absorbing spherical body of radius $R$ in a prescribed exterior inverse-square force, conservation of energy and angular momentum give $b^2=r_p^2+2GM{r}_p/v_{\infty }^2$. This is increasing in the periapsis radius $r_p$, so $r_p=R$ gives the collision threshold displayed above. The gravitational scattering angle obeys $\tan (\chi /2)=GM/(b{v}_{\infty }^2)$. On the grazing family its deflection approaches zero at high asymptotic speed and $\pi$ as $v_{\infty }\to 0^{+}$. Strict noncollision requires $b>b_{\mathrm{crit}}$; the equality is the tangency limit.