Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 1

A body of diameter $D$ has mass $m(D)=\pi \rho D^3/6$. Integrating the power-law size distribution therefore givesHenceWhen $D_{\max }\gg D_{\min }$ and $3<\alpha <4$, the belt mass is dominated by its largest bodies and $K\simeq 6M(4-\alpha )/(\pi \rho D_{\max }^{4-\alpha })$.

The impactors' number density per diameter is $K{D}_{\mathrm{im}}^{-\alpha }/V$. With gravitational focusing neglected, the geometric collision cross-section is $\pi (D+D_{\mathrm{im}}{)}^2/4$. The planetesimal collision rate is consequentlywhere

For equal mass density, the specific impact energy isThus the smallest catastrophic impactor has $D_{\mathrm{im}}=X_{c}D$, whereIntegrating the collision rate givesFor $X_c\ll 1$, the last term is the dominant lower-limit contribution, so the catastrophic planetesimal collision rate is

The definition of $X_c$ givesThe cratering collision prescription therefore removes the target-mass fraction

Cratering impactors satisfy $D_{\mathrm{im}}<X_{c}D$. For $X_c\ll 1$, they also satisfy $D_{\mathrm{im}}\ll D$, so $R_{\mathrm{col}}\simeq A{D}^2{D}_{\mathrm{im}}^{-\alpha }$. The fractional mass-loss rate is thenUsing the catastrophic planetesimal collision rate found above,

For $D_{\mathrm{im}}\ge X_{c}D$, the fraction removed from the largest remnant isbecause $Q/Q_D^{\ast }=(D_{\mathrm{im}}/X_{c}D{)}^3$. Again using $R_{\mathrm{col}}\simeq A{D}^2{D}_{\mathrm{im}}^{-\alpha }$ near the dominant lower limit,Consequently the catastrophic collision mass-loss timescale is

A size-independent $Q_D^{\ast }$ gives the steady Dohnanyi collisional cascade exponent $\alpha =7/2$. The preceding results becomeThus, under these idealized prescriptions, cratering collisions remove target mass about three times faster than catastrophic collisions. The conclusion concerns mass removed from targets; both processes contribute fragments to the collisional cascade.