Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 316 / 2

Write the distributed fragment power-law size distribution as $n(D_f)=K_f{D}_f^{-{\alpha }_f}$. A spherical fragment has mass $\pi \rho D_f^3/6$. Since this population contains one half of the target mass,and henceThe distributed fragments have total geometric cross-sectionThe single fragment containing the other half of the mass has diameter $2^{-1/3}{D}_t$ and cross-section $\pi D_t^2/(42^{2/3})$. Therefore the exact result within the model isFor $D_{\max ,f}\gg D_{\min ,f}$ and $3<{\alpha }_f<4$, area is dominated by the smallest distributed fragments while mass is dominated by the largest, so normally the first term is negligible and

The half-target mass assigned to equal fragments containsfragments. Their total geometric cross-section is consequentlyIgnoring the negligible single largest fragment in ${\sigma }_{\mathrm{tot},1}$,For example, if $D_f=D_{\min ,f}$, this ratio is $({\alpha }_f-3)(D_{\max ,f}/D_{\min ,f}{)}^{4-{\alpha }_f}/(4-{\alpha }_f)$ and is large for a broad size range. This reflects the inverse-size cross-section per unit mass of equal-density spherical fragments: concentrating the mass at small $D_f$ creates more area.

Let the belt contain $n(D)dD=C{D}^{-\alpha }dD$ bodies. Its total mass fixesFor equal mass density, an impactor of diameter $D$ supplies specific impact energyThus catastrophic disruption requires $D\ge D_{\mathrm{cc}}=X_c{D}_t$, whereThe impactor number density per diameter is $C{D}^{-\alpha }/V$. Neglecting gravitational focusing, the geometric collision cross-section is $\pi (D_t+D{)}^2/4$, so the exact catastrophic planetesimal collision rate in this model isEquivalently, with $A=\pi C{v}_{\mathrm{rel}}/(4V)$,If $D_{\min }\ll X_c{D}_t\ll D_t\ll D_{\max }$, the steep distribution makes impactors just above $X_c{D}_t$ dominate and their collision cross-section is approximately $\pi D_t^2/4$. HenceThis approximation also assumes a common $v_{\mathrm{rel}}$ and size-independent catastrophic disruption threshold $Q_D^{\ast }$.

A collision producing cross-section $\sigma$ has target diameterby part (ii). The production rate of clumps above ${\sigma }_{\mathrm{tot}}$ is the number of targets in each size interval times their per-target catastrophic rate. Therefore, with $D_0=\max (D_{\min },D_{{\sigma }_{\mathrm{tot}}})$,Using the small-impactor approximation from part (iii) givesWhen $D_{\min }\ll D_{{\sigma }_{\mathrm{tot}}}\ll D_{\max }$, this reduces toAssigning each event to its much larger target avoids double-counting collisions in this $X_c\ll 1$ approximation.

For the Dohnanyi collisional cascade exponent $\alpha =7/2$ and $D_{\max }\gg D_{\min }$,Substitution in part (iv) yieldsThusThe $M^2/V$ dependence is the pair-collision scaling. Longer-lived clumps are more numerous, stronger bodies disrupt less often, and smaller fragments put more geometric cross-section into each event. At fixed total mass, increasing $D_{\max }$ lowers the normalization of the power-law size distribution and therefore lowers the event rate.

The impact gives fragments a spread of orbital energy and specific angular momentum. Their resulting spread of mean motion lets Keplerian shear stretch a compact dust clump into an arc and then a ring, while the spread of orbital frequencies causes phase mixing. Size-dependent radiation-pressure coefficients immediately place small grains on different eccentric or even radiation-pressure blowout orbits; Poynting–Robertson drag and stellar-wind drag then alter their orbits on longer timescales. Further collisions grind or disperse the clump, and planetary perturbations can accelerate mixing.

These processes depend strongly on $D_f$. Small grains have larger radiation-force-to-gravity ratios and generally shorter collisional or drag lifetimes, while larger fragments remain closer to the parent orbit but can preserve a velocity-dispersion-driven clump for longer. The lifetime also depends on collision location, ejection velocities, optical depth, and orbital radius. A universal fixed $\Delta t$ is therefore a useful population-model approximation, not a literal property of every collision; a size- and event-dependent lifetime distribution is more realistic.