Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 314 / 4 / b / Solution

Let $M$ be the central stellar mass and take positive Keplerian rotation, $\Omega (R)=\sqrt{GM/R^3}$. The initial poloidal magnetic field stays fixed by part (a), while $R{\Omega }^{\prime }(R)=-(3/2)\Omega (R)$. Integrating the toroidal induction equation with $B_{\varphi }(R,0)=0$ givesThe initial field is a solenoidal vector field on the disk region $R>0$, since ${\partial }_R(R{B}_R)=0$. Assuming $B_0\ne 0$, the amplitude ratio isIt reaches one atThus the equality time is the same fraction $1/(3\pi )\approx 0.106$ of the orbital period at every radius. If “orbital time” instead means ${\Omega }^{-1}$, the corresponding fraction is $2/3$. These are kinematic solutions with prescribed rotation and the neglected magnetic back-reaction.