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 Incoming links: Keplerian disk

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 321 / 2 / a / Solution 2026-09-28

The three terms represent competing physical effects:

$Ω^{2}$ is the stabilizing epicyclic motion of a radially displaced fluid element in a Keplerian disk.
$- 2 π G Σ∣ k ∣$ is the destabilizing self-gravity of a surface-density perturbation, obtained from the razor-thin disk Poisson kernel.
$c_{s}^{2} k^{2}$ is the stabilizing pressure response, strongest at short wavelength.

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 321 / 2 / d / Solution 2026-09-28

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

, so the gravitoturbulent viscosity estimate is

\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}

in a Keplerian disk.

For a disk of characteristic radius

R

and mass

M_{D}

,

Σ \sim M_{D} / R^{2}

. The viscous timescale is

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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