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

Table of contents
- Solution iv

Solution

A collision producing cross-section

σ

has target diameter

D_{σ} = (\frac{8 D _{f} σ}{π})^{1/3}

by part (ii). The production rate of clumps above

σ_{tot}

is the number of targets in each size interval times their per-target catastrophic rate. Therefore, with

D_{0} = max (D_{m i n}, D_{σ_{tot}})

,

N (> σ_{tot}) = Δ t \int_{D_{0}}^{D_{m a x}} C D_{t}^{- α} R_{cc} (D_{t}) d D_{t} .

Using the small-impactor approximation from part (iii) gives

N (> σ_{tot}) = \frac{π C ^{2} v _{rel} Δ t}{4 V ( α - 1 ) ( 2 α - 4 )} X_{c}^{1 - α} (D_{0}^{4 - 2 α} - D_{m a x}^{4 - 2 α}) .

When

D_{m i n} ≪ D_{σ_{tot}} ≪ D_{m a x}

, this reduces to

N (> σ_{tot}) ≃ \frac{π C ^{2} v _{rel} Δ t}{4 V ( α - 1 ) ( 2 α - 4 )} X_{c}^{1 - α} (\frac{8 D _{f} σ _{tot}}{π})^{(4 - 2 α) /3} .

Assigning each event to its much larger target avoids double-counting collisions in this

X_{c} ≪ 1

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