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

Estimate $T\sim M⟨v^2⟩/2$ and $V\sim -G{M}^2/⟨R⟩$. The virial theorem givesup 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).