Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 2 / b

Table of contents
- Solution b

Solution

If no mass is accreted,

M

is constant and

Σ \sim M / R^{2}

. The viscous-time estimate becomes

t \propto R^{2 - m + 2 n}

, while

J \sim M (G M_{⋆} R)^{1/2}

. Thus

R \propto t^{1/ (2 - m + 2 n)}, J \propto t^{1/ [2 (2 - m + 2 n)]} .

The central torque supplies the angular momentum that allows the fixed disk mass to spread.

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