For $\beta=1/2$, part (b) gives
$$
H\propto T_0\Omega^{-3/2},
\qquad
P_0\propto T_0^4,
\qquad
\rho_0\propto\Omega T_0^2.
$$
Parts (a) and (c)(i) imply $P_0H/\Omega\propto\dot M f$. With constant $\dot M$,
$$
T_0^5\Omega^{-5/2}\propto f.
$$
For <Keplerian rotation>, $\Omega\propto r^{-3/2}$, and consequently
$$
\boxed{T_0\propto f^{1/5}r^{-3/4}}.
$$
It follows that
$$
\boxed{H\propto T_0\Omega^{-3/2}
\propto f^{1/5}r^{3/2}},
$$
and
$$
\boxed{\Sigma\propto\rho_0H
\propto\Omega^{-1/2}T_0^3
\propto f^{3/5}r^{-3/2}}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-321/1/c/ii/solution