Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 316 / 2 / iii / Solution

In the gravitational focusing regime, take $d\ll D$ and $v_{\mathrm{rel}}\ll v_{\mathrm{esc}}$ at large separation. Mutual gravitational acceleration then makes the impact speed approximately the target's escape velocity:Setting the specific impact energy equal to the catastrophic disruption threshold givesThe size-dependent far-field encounter speed drops out of this leading threshold because gravitational acceleration dominates at impact. Gravitational focusing also increases the collision cross-section by approximately $1+v_{\mathrm{esc}}^2/v_{\mathrm{rel}}^2$, but this concerns the rate rather than the energy threshold. These formulas assume the disruption-energy convention remains per unit target mass and that gravitationally accelerated impact energy couples to disruption as specified by $Q_D^{\ast }$.