Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 321 / 1 / b / ii

Table of contents
- Solution ii

Solution

Write

K = 11 a /12

, so

P = K T^{4}

. With

ρ \sim Σ/ H

, vertical balance gives

\frac{P}{H} \sim ρ Ω^{2} H, H \sim \frac{K T ^{4}}{Σ Ω ^{2}} .

At fixed

Σ

and radius, the volumetric viscous heating is

C_{+} = \frac{9}{4} Ω^{2} μ = \frac{9}{4} α Ω P \propto T^{4} .

The neutrino cooling is

C_{-} = A ρ T^{1/ β} \sim A \frac{Σ}{H} T^{1/ β} \propto T^{1/ β - 4} .

Thermal stability of an accretion disk requires the cooling rate to have the larger logarithmic temperature slope:

\frac{1}{β} - 4 > 4.

Since

β > 0

, the stable range is

0 < β < \frac{1}{8} .

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