Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 347 / 2 / c

Let $\epsilon =H/R\ll 1$. Vertical hydrostatic balance gives $c_s\sim H{\Omega }_K$, whereas leading radial force balance gives $u_{\varphi }\sim R{\Omega }_K$. HenceUsing part (a) and $\nu =\alpha c_{s}H$ far from the inner edge,Thus subsonic radial drift in a thin disk coexists with highly supersonic orbital motion. Gas follows nearly circular, pressure-coherent orbits and loses angular momentum only slowly, completing of order $(\alpha {\epsilon }^2{)}^{-1}$ revolutions during its viscous inflow.