Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 1 / e / Solution

For $r\ll GM/B$, the Mach number tends to a constant ${\mathcal{M}}_1$. Since $A{\dot{M}}_{\mathrm{crit}}^{1/2}=GM/4$, its algebraic equation isor equivalentlyThe central Bernoulli balance givessoFinally $u={\mathcal{M}}_1{v}_s$. Using the algebraic relation to rewrite its coefficient gives