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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-316/1/iv/solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 316 / 1 / iv / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

With negligible comet radius and mass, the planet's radius and escape velocity are

R_{p} = (\frac{3 M _{p}}{4 π ρ _{p}})^{1/3}, v_{esc, p}^{2} = \frac{2 G M _{p}}{R _{p}} .

Gravitational focusing changes the limiting collision impact parameter from

R_{p}

to

R_{p} 1 + v_{esc, p}^{2} / v_{rel}^{2}

. Thus it is negligible when

v_{esc, p}^{2} ≪ v_{rel}^{2}

.

Put

s = q / a_{p}

and

γ = 3 - 2 2 s

in the

Q ≫ a_{p}

prograde approximation. Substituting

v_{rel} = γ G M_{⋆} / a_{p}

gives

M_{p} ≪ (\frac{γ ^{2} M _{⋆}}{2 a _{p}})^{3/2} (\frac{3}{4 π ρ _{p}})^{1/2} .

Equivalently

2 (M_{p} / M_{⋆}) (a_{p} / R_{p}) ≪ γ^{2}

. For

q ≪ a_{p}

, use

γ^{2} ≃ 3

. If negligible gravitational focusing is required for every coplanar near-parabolic orbit, including nearly tangential prograde encounters, use the smaller value

γ^{2} = (2 - 1)^{2}

.

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