Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 321 / 1 / b / Solution

Take $v>0$ to mean the magnitude of the inward radial velocity. In the Gaussian units used for the magnetic pressure, approximate the surface density of a disk by $\Sigma \simeq 2H\rho$. The mass accretion rate and free-fall speed then giveAt the magnetospheric truncation radius, balance the magnetic pressure of the magnetic dipole field with the specified ram pressure:ThusDiscarding the convention-dependent numerical factor givesThe magnetic pressure grows inward as $r^{-6}$, faster than the ram pressure $\propto r^{-5/2}$, so the field disrupts the flow inside this radius. The numerical coefficient depends on the vertical averaging and on treating the inflow as radial free fall, as stipulated here; a detailed thin disk boundary model need not have the same coefficient.