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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ii/paper-1/9d/d/solution

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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 9D / d / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

Estimate

T \sim M ⟨ v^{2} ⟩ /2

and

V \sim - G M^{2} / ⟨ R ⟩

. The virial theorem gives

M \sim \frac{⟨ R ⟩ ⟨ v ^{2} ⟩}{G}

(1)

up to a profile-dependent factor. If this exceeds the stellar/gas luminous mass, the difference is attributed primarily to dark matter (with modelling uncertainties and non-equilibrium effects checked).

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