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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-316/3/vi/solution

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Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 316 / 3 / vi / Solution by

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

Put

ϵ = m_{2} / m_{1} ≪ 1

and

t_{s 2} / t_{s 1} = ϵ^{k}

, with

0 < k < 1

. The gas-driven contraction of a planetesimal binary and gas-driven migration of a binary centre of mass times obey

\frac{τ}{τ _{b}} = η \frac{m _{1} / t _{s 1} + m _{2} / t _{s 2}}{m _{1} / t _{s 2} + m _{2} / t _{s 1}} = η \frac{1 + ϵ ^{1 - k}}{ϵ ^{- k} + ϵ} .

(1)

Therefore

\frac{τ}{τ _{b}} ≃ η ϵ^{k} ≪ 1.

(2)

The low-mass secondary has the shorter aerodynamic stopping time and chiefly controls internal damping, whereas the primary chiefly controls centre of mass drag. The pair can contract substantially, potentially merging, before its stellar orbital radius changes appreciably. This conclusion is restricted to a pair that remains bound and within the adopted linear-drag regime.

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