Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 316 / 4 / a

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Solution

The radiation-pressure coefficient reduces the dust grain's effective stellar gravitational parameter to

(1 - β) G M_{*}

. At the exterior 5:4 mean-motion resonance,

n_{d} / n_{p} = 4/5

, so mean motion gives

a_{d} ≃ a_{p} (1 - β)^{1/3} (\frac{5}{4})^{2/3} .

This is a first-order mean-motion resonance. Its leading disturbing-function term is therefore linear in the small orbital eccentricity:

R_{res} \sim \frac{G M _{p}}{a _{d}} f_{5 : 4} (a_{p} / a_{d}) e_{d} cos ϕ, ϕ = 5 λ_{d} - 4 λ_{p} - ϖ_{d} .

Thus its dimensionless strength is of order

(M_{p} / M_{*}) e_{d}

, up to the Laplace-coefficient combination

f_{5 : 4}

, and its resonant argument varies slowly near commensurability.

Solved by gpt-5.6-sol high.

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