Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 347 / 2 / c / Solution

Let $l(R)=R^2\Omega =\sqrt{GMR}$ be the Keplerian specific angular momentum. In a source-free steady interval, the sum of advected and viscously transported angular momentum is constant:Inside the injection radius, $\dot{m}={\dot{m}}_0$. The zero-torque inner boundary condition at $R_{\mathrm{ISCO}}$ sets $C={\dot{m}}_0{l}_{\mathrm{ISCO}}$, and therefore

Outside $R_0$ there is no net mass flow in the stated steady distribution, but it must carry outward the angular momentum deposited by matter moving from $R_0$ to the ISCO. Its constant viscous torque in an accretion disk is thereforeThusThe two expressions agree at $R_0$; the jump in mass flux there is exactly the injected rate.