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

Assume spherical bodies of common mass density $\rho$, a well-mixed belt, and that $n(D)dD$ counts bodies in the entire belt rather than per unit volume. A body of diameter $D$ has mass $\pi \rho D^3/6$. The mass normalization of a power-law size distribution isConsequentlyThe thin-torus volume is circumference times radial width times full vertical height,The local number-density distribution needed for a planetesimal collision rate is $n(D)/V$. Thus belt geometry does not enter this total-number $K$ but does enter collision rates. This approximation treats the stated height as uniform, independent of size.

Let $d$ be an impactor diameter and assume it is much smaller than target diameter $D$, with equal material mass density. Neglect gravitational focusing and take the encounter speed to be the stated target-size law $v_{\mathrm{rel}}=v_p{D}^p$. The specific impact energy isA catastrophic planetesimal collision requires $Q\ge Q_D^{\ast }=Q_b{D}^b$. Therefore the catastrophic projectile diameter isUse $\max (D_{\mathrm{cc}},D_{\min })$ as the actual lower size cutoff when integrating a rate. If this exceeds $D_{\max }$, the specified population contains no destructive impactors. The small-impactor approximation itself requires $D_{\mathrm{cc}}\ll D$.

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 $Q_D^{\ast }$.

Neglect gravitational focusing and assume a fixed well-mixed volume $V$, $\alpha >1$, and $D_{\min }\ll D_{\mathrm{cc}}\ll D\ll D_{\max }$. For the small impactors that dominate the rate, the geometric collision cross-section is approximately $\pi D^2/4$. ThusEvaluating and neglecting the upper-end term gives the catastrophic planetesimal collision rateUsing part (ii), its size dependence isIf the destructive cutoff is near an endpoint or comparable to $D$, retain the full integral and the cross-section $\pi (D+d{)}^2/4$. The simple power law then need not apply.

For equal logarithmic width $d\log D$, the number of bodies is $n(D)Dd\log D$. Their mass is thereforeThe mass removed by catastrophic planetesimal collisions per logarithmic bin per unit time scales asAssume a steady collisional cascade with supply only at the large end, loss only at the small end, negligible reassembly or cratering contributions, and self-similar fragmentation that transfers mass through comparable logarithmic intervals. Mass conservation requires the downward mass flux across every intermediate size boundary to be the same; otherwise mass would accumulate or deplete there. Under the scale-invariant bin-transfer assumption, this is proportional by a size-independent factor to ${\dot{M}}_{\mathrm{loss}}$, soThis is constant mass flux in a collisional cascade. Steady state alone fixes the net flux, while the identification with the destruction rate of each bin additionally uses the stated fragmentation assumptions.

Combining constant mass flux in a collisional cascade with the rate in part (iv) givesSet the exponent to zero and multiply by three:or $21+b+p-\alpha (6+b-2p)=0$. Therefore the velocity-dependent collisional cascade hasFor $b=p=0$, this recovers the Dohnanyi collisional cascade slope $\alpha =7/2$. The denominator must be nonzero, and the resulting slope and destructive cutoff must satisfy the assumptions used for the collision-rate integral. A size-dependent vertical belt thickness would supply an additional volume factor and change this result.