Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 321 / 1 / c

For $\beta =1/2$, the exact profiles areIntegration through both disk faces givesSince $\mu =\alpha P/\Omega$ and $\bar{\nu }\Sigma =\int \mu dz$,

For $\beta =1/2$, part (b) givesParts (a) and (c)(i) imply $P_{0}H/\Omega \propto \dot{M}f$. With constant $\dot{M}$,For Keplerian rotation, $\Omega \propto r^{-3/2}$, and consequentlyIt follows thatand