Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 1 / b / ii

For constant

v_{m}

M_{r}

\partial_{t} M_{r} + v_{m} \partial_{r} M_{r} = 0.

M_{r} (r, t) = M_{r} (r - v_{m} t, 0) .

The localized profile translates inward without changing shape: material initially concentrated near

r_{0}

reaches the central star after

t_{acc} ≃ \frac{r _{0}}{∣ v _{m} ∣} .

Each annulus loses specific angular momentum as its characteristic moves to smaller

r

. The negative magnetic torque transmits that angular momentum along the large-scale field to the anchored interstellar medium. Thus the disk's angular momentum decreases even before mass crosses

r = 0

; angular momentum is conserved only after the field and its anchoring medium are included.

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