Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 321 / 2 / d

Solution

The longest unstable disturbance has mixing length

ℓ \sim k_{1}^{- 1}

and grows on the orbital timescale

Ω^{- 1}

. Its characteristic turbulent velocity is therefore

u_{turb} \sim Ω ℓ

\overset{ν}{ˉ} \sim u_{turb} ℓ \sim k_{1}^{- 2} Ω.

For

Q ≪ 1

k_{1} \sim \frac{Q}{H} = \frac{Ω ^{2}}{π G Σ},

up to an unimportant numerical factor. Hence

\overset{ν}{ˉ} \sim \frac{G ^{2} Σ ^{2}}{Ω ^{3}} \propto r^{9/2} Σ^{2}

For a disk of characteristic radius

R

M_{D}

Σ \sim M_{D} / R^{2}

t_{ν} \sim \frac{R ^{2}}{ν ˉ} \sim \frac{R ^{6} Ω ^{3}}{G ^{2} M _{D}^{2}} .

Using

Ω^{2} = G M_{*} / R^{3}

gives

t_{ν} \sim (\frac{M _{*}}{M _{D}})^{2} Ω^{- 1} .

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