Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 321 / 1 / b / i

Table of contents
- Solution i

Solution

For a perfect gas,

c_{s}^{2} \sim k T / (μ_{m} m_{p})

. At the margin of the Toomre stability criterion,

c_{s} \sim \frac{G Σ}{Ω}, T \sim \frac{μ _{m} m _{p}}{k} \frac{G ^{2} Σ ^{2}}{Ω ^{2}} .

Integrating viscous heating through the disk and applying radiative diffusion gives

F \sim α P Ω H \sim α Σ c_{s}^{2} Ω, F \sim \frac{σ T ^{4}}{κ ρ H} \sim \frac{σ T ^{4}}{κ Σ} .

Equating these fluxes yields

α \sim \frac{σ}{κ} (\frac{μ _{m} m _{p} Σ}{k})^{4} G^{6} Ω^{- 7} .

The kinematic viscosity is

\overset{ν}{ˉ} \sim α c_{s}^{2} /Ω

, so

\overset{ν}{ˉ} \propto Σ^{4} Ω^{- 7}, S i g m a^{2} Ω^{- 3} = Σ^{6} Ω^{- 10} .

Because a Keplerian accretion disk has

Ω \propto r^{- 3/2}

,

\overset{ν}{ˉ} \propto r^{15} Σ^{6} .

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