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

At fixed radiative efficiency, the Eddington ratio implies $\dot{m}\propto f_{\mathrm{Edd}}M$. Far outside the innermost stable circular orbit, part (a) gives $\Sigma \propto \dot{m}/\nu$. The supplied viscosity law therefore gives, in physical radius,The disk mass follows by radial integration:Since the Schwarzschild radius satisfies $R_S\propto M$, replacing $R$ by $(R/R_S)R_S$ contributes another factor $M^{5/4}$. Thusso