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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-347/3/c/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 347 / 3 / c / Solution by

Codex  0 2026-09-29

Far from the inner edge, use

u_{R} ≃ - 3 ν / (2 R)

and the alpha-viscosity prescription

ν = α c_{s} H = α \frac{c _{s}^{2}}{Ω}, H = \frac{c _{s}}{Ω} .

Mass conservation then gives

Σ = \frac{m ˙ Ω}{3 π α c _{s}^{2}}, ρ ≃ \frac{Σ}{H} = \frac{m ˙ Ω ^{2}}{3 π α c _{s}^{3}} .

At the onset of disk self-gravity,

ρ = M_{BH} / R^{3}

. Since

Ω^{2} = G M_{BH} / R^{3}

, this condition reduces to

c_{s}^{3} = \frac{G m ˙}{3 π α} .

The blackbody approximation and local radiative balance on one face give

σ_{SB} T_{eff}^{4} = \frac{3 G M _{BH} m ˙}{8 π R ^{3}},

while the stated ideal-gas estimate is

c_{s}^{2} = γ k_{B} T_{eff} / μ

. Inserting these expressions into the self-gravity condition gives

(\frac{γ k _{B}}{μ})^{3/2} (\frac{3 G M _{BH} m ˙}{8 π σ _{SB} R ^{3}})^{3/8} = \frac{G m ˙}{3 π α} .

Solving for

R

produces

R_{crit} = (3 π α)^{8/9} (\frac{γ k _{B}}{μ})^{4/3} (\frac{3 M _{BH}}{8 π σ _{SB}})^{1/3} (G \overset{m}{˙})^{- 5/9} .

Beyond this radius, the disk's own gravity rivals the central vertical gravity; fragmentation and star formation can then invalidate the smooth steady thin-disk model.

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