Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 1 / c / ii / Solution

Order-of-magnitude vertical balance givesMagnetic support also gives ${\Omega }^2{H}^2\sim r\Omega T^{1/2}$, so $T\sim {\Omega }^2{H}^4/r^2$. Eliminating $P,F,\rho ,T$ givesFor Keplerian $\Omega \propto r^{-3/2}$,The alpha disk stress implies $\bar{\nu }\sim \alpha P/(\rho \Omega )\sim \alpha \Omega H^2$, and therefore